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\title{Reductive group schemes}
\date{\today}
\author{Brian Conrad}

\address{Math Dept., Stanford University\\
Stanford, CA 94305, USA}
\email{conrad@math.stanford.edu}
\urladdr{http://smf.emath.fr/}
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%
%\title{Reductive group schemes (SGA3 summer school, 2011)}
%\author{Brian Conrad}
%\address{Department of Mathematics\\
%Stanford University\\
%Stanford, CA 48109, USA}
%\email{conrad@math.stanford.edu}
%\author{Benedict Gross}
%\address{Department of Mathematics\\
%Harvard University\\
%Cambridge, MA 02138, USA}
%\email{gross@math.harvard.edu}
%\author{Jiu-Kang Yu}
%\address{Department of Mathematics\\
%Purdue University\\
%West Lafayette, IN 47907, USA}
%\email{jyu@math.purdue.edu}
%
%\date{March 5, 2014} 
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\begin{document}
\def\smfbyname{}

\begin{abstract} We develop the relative theory of reductive group schemes, using
dynamic techniques and algebraic spaces to streamline the original development
in SGA3.
\end{abstract}

\begin{altabstract}
Nous d\'eveloppons la th\'eorie relative des sch\'emas en groupes r\'eductifs, \`a l'aide da 
techniques dynamiques et des espaces alg\'ebriques afin de simplifier le d\'eveloppement original 
dans SGA3.
\end{altabstract}


\maketitle


\begin{center}

{\em{To the memory of Robert Steinberg}}

\end{center}

\section*{Introduction}
%\vskip1mm
%\centerline{\sc{Introduction}}
%\vskip1mm


These notes present the theory of reductive group schemes,
simplifying the original proofs
via tools developed after 1963 (see ``What's new?'' at the end of
this Introduction). 
We assume familiarity with the 
structure theory over an algebraically closed field $k$
(as developed in \cite{borelag}, \cite{humphreys}, \cite{springer}), but
a review is given in \S\ref{review} to fix 
terminology, set everything in the framework of $k$-schemes
(rather than classical varieties), and provide a convenient reference
for the scheme-theoretic developments.   

We give complete proofs of the main results in the theory (conjugacy theorems, scheme of maximal tori, 
construction of root groups and root datum, structure of open cell, parameterization of
parabolics, schemes of Borel and parabolic subgroups,
 Existence and Isomorphism theorems, existence of automorphism scheme),
apart from some calculations with low-rank root systems.
We do {\em not} assume the Existence and Isomorphism Theorems over a general algebraically closed field $k$
because the scheme-theoretic approach proves these over any non-empty base
scheme granting only the Existence Theorem over $k = \C$
(which we prove in Appendix \ref{csec}).

Although the structure theory of connected reductive groups is developed 
over a general field in \cite{borelag} and \cite{springer}, we 
take on faith only the case of an algebraically closed ground field (as in \cite{sga3} via its reliance on \cite{bible}).
Some results are proved here in less generality than 
in \cite{sga3}, since our aim is
to reach the structure theory of  reductive group schemes as quickly as possible. 
In a small number of places we refer to \cite{sga3} for omitted proofs, 
and the interested reader can readily check
that this does not create circular arguments in our development of the general theory.

\subsection*{Background}
We assume familiarity with smooth and \'etale morphisms
(e.g., functorial criteria for each; cf.\ \cite[2.1--2.2]{neron}), faithfully flat descent 
(see \cite[6.1--6.2]{neron}), and the functorial
approach to  group schemes (e.g., scheme-theoretic kernels,
intersections of closed subgroup schemes, and quotient morphisms via sheaves for a 
Grothendieck topology). We also use Cartier duality for
finite flat commutative group schemes (of finite presentation).

In the arguments after \S\ref{review} we use multiplicative
type group schemes \cite[IX, X]{sga3}. 
\index{multiplicative type}
This material
is covered in Appendix \ref{multsec}, building on Oesterl\'e's lectures \cite{oesterle}. 
We use a more restricted notion of ``multiplicative type''
than in \cite{sga3} and \cite{oesterle}. 
In these notes, by definition {\em multiplicative type groups are required to be of finite type}
over the base, and are required
to split {\em fppf-locally} on the base: 
they are fppf-local forms of diagonalizable
groups of finite type.  The fpqc topology is used
instead of the fppf topology in the definition in \cite[IX, 1.1]{sga3}
and \cite{oesterle} (allowing fpqc groups that are not of finite type).
Our restriction to the fppf topology is harmless, as we explain in Appendix \ref{multsec}
(due to the ``finite type'' requirement that we impose).
We often use that multiplicative type groups are necessarily split 
\'etale-locally on the base.  It is inconvenient to include  this in the {\em definition} of ``multiplicative type'', so
a proof of \'etale-local splitting is given in Proposition \ref{fppfet}
(as a mild variant of the proof of an analogous result in \cite[IX, 4.5]{sga3}).
The equivalence with fpqc-local triviality (under a finite-type hypothesis on the group)
is proved in Corollary \ref{fpqcwin}(1) but is never used.

We require the notion of Lie algebra \index{Lie algebra}
for a group scheme (not necessarily smooth). 
A reference that covers what we need (and beyond) is \cite[A.7]{pred}.
One of the properties we use for the Lie functor
is that it commutes with fiber products of groups (and so is left exact), as is verified
by considering points valued in the dual numbers over the base.


For a reader unfamiliar with algebraic spaces, in a few places
it will be necessary to accept that algebraic spaces \index{algebraic spaces} are 
a useful mechanism to equip certain set-valued functors with enough ``geometric structure''
that it makes sense to carry over concepts for schemes
(e.g., properness, flatness, quasi-finiteness, etc.) to such functors;
an excellent reference for this is \cite{knutson}.  All algebraic spaces we use will almost
immediately be proved to be schemes, so our use of algebraic spaces
will be similar to the use of distributions to provide function solutions to elliptic
partial differential equations (i.e., they appear in the middle of a construction,
the end result of which is an object of a more familiar type that we prefer to use).


\subsection*
{Theory over a field}
In Example \ref{chevgp} we relate the approach in \cite{sga3} to other constructions
in the literature (e.g., \cite{steinyale}) for the group of field-valued points of a split semisimple group. 
 Many arithmetic applications
(as well as applications over $\mathbf{R}$) require a structure theory 
for connected reductive groups over
general fields $k$ without assuming the existence of a split
(geometrically) maximal $k$-torus.  Such a structure theory is 
due to Borel and Tits (see \cite{boreltits} and \cite[\S20--24]{borelag}),
and is not discussed here. 
In \cite[XXVI, 6.16--6.18, \S7]{sga3} and \cite[\S4--\S5]{gillesga3} the Borel--Tits theory is partially generalized to 
reductive groups over connected semi-local non-empty schemes;
see \cite{petrov} for recent work in this direction. 

There is a result at the foundation of
the Borel--Tits structure theory for which we do provide a proof:
Grothendieck's theorem  \cite[XIV, 1.1]{sga3} (later proved in more elementary 
terms by Borel and Springer) that any smooth 
affine group $G$ over a field $k$ admits a $k$-torus $T \subset G$
such that $T_{\overline{k}}$ is maximal in $G_{\overline{k}}$. 
This existence result implies
that for {\em every} $k$-torus $T'$ 
in $G$ not contained in a strictly larger $k$-torus
and for every extension field $K/k$, 
$T'_K$ is not contained in a strictly larger $K$-torus of $G_K$
(see Remark \ref{remtorus} for a proof).  In particular, the concept
of ``maximal $k$-torus'' in a smooth affine group over $k$ is insensitive to ground field extension.
(We only use this in the trivial case $k = \overline{k}$.)
\index{maximal tori}

Since Grothendieck's existence theorem for maximal tori over arbitrary fields
is not needed in the development of the theory of reductive
groups over general schemes, we have relegated our discussion
of his theorem to Appendix \ref{torusapp} (which provides a scheme-theoretic version
of the Borel--Springer proof of Grothendieck's theorem). 
What matters for our purposes is the existence of a ``geometrically maximal''
torus defined over a finite {\em separable} extension of the ground field.  We
present Grothendieck's  construction of such a torus, using the existence
and smoothness 
properties of a ``scheme of maximal tori''. 
The proof of Grothendieck's finer result that such a torus exists 
over the ground field uses a detailed study of
Lie algebras. It can also be deduced from the deeper
result that the scheme of maximal tori is a rational variety \cite[XIV, 6.1]{sga3}
(coupled with special arguments for finite ground fields).


\subsection*
{What's new?}
We take advantage of three post-1963 developments to streamline or simplify some of the original
proofs:


\medskip

(i) (Artin approximation)\, In the study of lifting problems, infinitesimal methods
allow one to build liftings
\index{Artin approximation}
over the {\em completion} of a local noetherian ring (when starting with an algebro-geometric
object over the residue field).  The {\em Artin approximation theorem} (\cite{Artapprox}, \cite[3.6/16]{neron}), 
whose statement we recall in Theorem \ref{approx},
provides a method to use such lifts over a ``formal'' neighborhood of a point to construct lifts over
an \'etale neighborhood of a point.  This solves global problems
over an \'etale cover, a dramatic improvement on having solutions only in formal neighborhoods of points.
We will use this reasoning to study liftings of tori in smooth affine groups (e.g., see the proof of Theorem \ref{schemetori}).

\medskip

(ii) (algebraic spaces)\, A representability result of Murre  \cite[\S3, Thm.\,2, Cor.\,2]{murre}
 is used in \cite{sga3} to build certain quotients by 
flat equivalence relations (see \cite[XVI, \S2]{sga3}).  
\index{algebraic spaces}
This underlies Grothendieck's quotient constructions
in \cite[XVI, 2.4]{sga3}.  
Murre's result is a precursor to Artin's criteria for a functor on schemes to be 
an algebraic space.  The work of
Artin (\cite{alg}, \cite{versal}) and Knutson \cite{knutson} on algebraic spaces 
provides an ideal framework for  a geometric  theory of  
quotients by flat equivalence relations in algebraic geometry (and includes a sufficient 
criterion for an algebraic space to be a scheme), so we use algebraic spaces
in place of Murre's result; e.g., see the proof of Theorem \ref{xi5.3bis}.

\medskip

(iii) (dynamic method)\, There is a ``dynamic'' approach to describing
parabolic subgroups as well as their unipotent radicals and Levi factors
in connected reductive groups over an algebraically closed field.  
\index{dynamic method}
This method 
involves the limiting behavior along orbits under the conjugation
action of a 1-parameter subgroup; it is a standard tool in the classical setting
(see \cite[13.8(1),(2)]{borelag}, \cite[8.4.5]{springer}) and also arises in Mumford's GIT.

The relative version of the dynamic viewpoint
for group schemes over any ring was introduced
and developed in \cite[2.1]{pred}, where it was used
to study pseudo-reductive groups over imperfect fields.
In the present paper, this leads to simplifications in several places.  For instance, 
arguments in \cite[XX]{sga3} for constructing closed root groups 
in split reductive group schemes and 
classifying the split semisimple-rank 1 case over any scheme rest on 
elaborate computations. 
The relative dynamic method, which rests on elementary
arguments, eliminates the need for most of the 
computations in \cite[XX]{sga3} (see \S\ref{corootsec}). 
We review what we need from the dynamic method in \S\ref{dynsec}, referring to
\cite[2.1]{pred} for proofs. 

\medskip

{\bf Acknowledgements}. I am grateful to the referees and 
A.\:Auel, O.\:Gabber, S.\:Garibaldi, B.\,Gross, L.\:Illusie, J.\:Parson, P.\:Polo, G.\:Prasad, 
J-P.\:Serre, A.\:Venkatesh, and J-K.\:Yu for many helpful suggestions. 
K\k{e}stutis \v{C}esnavi\v{c}ius and Jessica Fintzen provided a huge amount of expository and mathematical
feedback on an earlier version, for which I am especially appreciative.  This work was supported by NSF grants DMS-0917686 and DMS-1100784.


\newpage

\tableofcontents

\newpage

\section{Review of the classical theory}\label{review}

We fix an algebraically closed field $k$ and 
consider smooth affine group schemes over $k$ (not necessarily connected).  Such groups are called
{\em linear algebraic groups} over $k$ because
\index{linear algebraic group}
they are precisely the smooth $k$-groups $G$ of finite type
for which there exists a $k$-homomorphism 
$$j:G \hookrightarrow \GL(V) \simeq \GL_n$$
that is a closed immersion \cite[1.10]{borelag}.  
Although
we will be working in the classical setting, we use
the viewpoint of schemes rather than classical varieties.
For instance, we do not use any ``universal domain'' as in \cite{borelag}, and a linear
algebraic group over $k$ is a scheme rather than
a set of $k$-points equipped with extra structure.
Likewise, the {\em kernel} of a $k$-homomorphism
\index{kernel}
always means the scheme-theoretic kernel 
(e.g., $\ker({\rm{SL}}_n \rightarrow {\rm{PGL}}_n)$ is identified
with the $k$-group scheme $\mu_n = \ker(t^n:\Gm \rightarrow \Gm)$
\index{ aaamun@$\mu_n$}
even if ${\rm{char}}(k)|n$), and 
an intersection of closed linear algebraic $k$-subgroups
of a linear algebraic $k$-group always 
means scheme-theoretic intersection (which may be non-smooth when ${\rm{char}}(k) > 0$;
e.g., the central $\Gm$ in $\GL_n$ meets ${\rm{SL}}_n$ in precisely $\mu_n$, even if
${\rm{char}}(k)|n$).

We do not require linear algebraic groups to be connected
because there are interesting disconnected examples (such as orthogonal groups)
and group-theoretic operations (centralizers, intersection of subgroups, kernels, etc.)
can lead to disconnected (possibly non-smooth) groups.  The identity
component $G^0$ 
\index{ aaagzero@$G^0$}
of any linear algebraic $k$-group is irreducible
(as for any smooth connected non-empty $k$-scheme). 

The basic structure theory of linear algebraic 
groups is developed in Borel's book \cite{borelag}
and Springer's book \cite{springer}, as well
as Chevalley's book \cite{bible}.  In this section we summarize
some of the highlights of this theory, as a review of classical
notions and results that are needed in the developments over a general base scheme in \cite{sga3},
and it is assumed that the reader has prior experience with the classical case.
We also discuss some aspects of representation theory over an algebraically closed field.

Before we get started, it will be helpful to check the consistency between the notions of 
$G/H$ in the classical and scheme-theoretic theories, with $H$ a linear algebraic subgroup of 
a linear algebraic group $G$.   This goes as follows. Classically, a smooth quasi-projective quotient 
$G/H$ is built so that $q:G \rightarrow G/H$ identifies
$(G/H)(k)$ with $G(k)/H(k)$ and the map 
${\rm{Tan}}_e(q)$ is the natural surjection
$\mathfrak{g} \twoheadrightarrow \mathfrak{g}/\mathfrak{h}$
\cite[6.7, 6.8]{borelag}. In the scheme-theoretic approach the
``quotient'' morphism $G \rightarrow Q$ by the right $H$-action on $G$ is uniquely characterized
(if it exists!) by the condition that it that represents the fppf-sheafification of 
the functor $S \rightsquigarrow G(S)/H(S)$ on the category of $k$-schemes.
We claim that the classical quotient construction satisfies this property,
so it is the desired quotient in the sense of \cite{sga3}.

To compare these concepts, first observe (via $G(k)$-translations) that 
$q$ is surjective on tangent spaces at all $g \in G(k)$, so $q$ 
is a smooth morphism (``submersion theorem'' \cite[2.2/8(c)]{neron}). 
The agreement with the scheme-theoretic
notion of quotient amounts to saying 
that $q$ is the quotient of $G$ modulo the flat equivalence relation
$R := G \times H \hookrightarrow  G \times G$  defined by
$(g,h) \mapsto (g,gh)$.  The map $q$ is fppf, so by descent
theory $q$ is the quotient of $G$ modulo the flat equivalence relation
$R' = G \times_{G/H} G \rightarrow G \times G$.  The relation $R'$ is
a closed subscheme that is smooth (since ${\rm{pr}}_1:R' \rightarrow G$ is a base change
the smooth $q$, and $G$ is smooth), 
and the same clearly holds for $R$.  We just have to show that
$R = R'$ as subschemes of $G \times G$, so by smoothness
it is equivalent to show $R(k) = R'(k)$ inside $G(k) \times G(k)$.
Since $G(k)/H(k) \rightarrow (G/H)(k)$ is a bijection, 
the desired equality is clear.

\subsection{Solvable groups and reductive groups}

By the closed orbit lemma \cite[1.8]{borelag}, if $f:G' \rightarrow G$ is
a $k$-homomorphism between linear algebraic $k$-groups then 
the image $f(G')$ is a smooth closed $k$-subgroup of $G$.
In particular, since $G' \rightarrow f(G')$ is a surjective
map between smooth $k$-schemes, by homogeneity 
using translation by $k$-points over $k  = \overline{k}$
it follows that generic flatness propagates everywhere,
so $G' \rightarrow f(G')$ is faithfully flat.
Thus, descent theory gives that $G'/(\ker f) \simeq f(G')$;
i.e., $f(G')$ represents the quotient sheaf $G'/(\ker f)$ for the fppf
topology, where $\ker f$ is the scheme-theoretic kernel. 
If $\mathbf{P}$ is a property of scheme morphisms that is fppf local
on the base (e.g., being finite, smooth, or an isomorphism) then
the fppf map $G' \rightarrow G'/(\ker f)$ satisfies $\mathbf{P}$
if and only if $\ker f \rightarrow \Spec k$ does, since $G' \times_{G'/(\ker f)} G' \simeq
\ker f \times G'$ via $(g'_1, g'_2) \mapsto (g'_1 {g'}_2^{-1}, g'_2)$. In particular, 
since $\ker f$ is finite as a $k$-scheme if and only if
$(\ker f)(k)$ is finite (as $k = \overline{k}$), we get:

\begin{proposition}\label{propker} If $(\ker f)(k)$ is finite then $f$ is finite flat onto its smooth closed image,
and if $\ker f = 1$ then $f$ is a closed immersion.
If $f$ is surjective and $\ker f = 1$ then $f$ is an isomorphism.
\end{proposition}

\begin{example} The natural $k$-homomorphism 
$\pi:{\rm{SL}}_n \rightarrow {\rm{PGL}}_n$ between finite type $k$-schemes is surjective on $k$-points, 
hence surjective as a scheme map, and 
the scheme-theoretic kernel $\ker \pi = \mu_n$ is  a finite $k$-scheme of degree $n$, so 
$\pi$ is finite flat of degree $n$.  If $n = p = {\rm{char}}(k) > 0$ then
$\ker \pi$ is infinitesimal, $\pi$ is purely inseparable, and $\pi$ is bijective on $k$-points but is not 
an isomorphism.
\end{example}

It follows from Proposition \ref{propker} that if a linear algebraic group $G'$
is a $k$-subgroup of another such group $G$ in the sense
that there is given a monic homomorphism $G' \rightarrow G$
then $G'$ is a closed $k$-subgroup of $G$.  In particular, if a linear representation 
$G \rightarrow \GL(V)$ is {\em faithful} in the sense
that it has trivial (scheme-theoretic) kernel then it is necessarily
a closed immersion. 

\begin{example} The action of ${\rm{SL}}_n$ on the vector space $V = {\rm{Mat}}_n$ 
of $n \times n$ matrices via conjugation defines a linear representation
$\rho:{\rm{SL}}_n \rightarrow \GL(V)$ whose kernel is $\mu_n$.
In particular, if $n = {\rm{char}}(k) = p > 0$ then 
this representation is injective on $k$-points but it is not faithful
in the scheme-theoretic sense (i.e., $\ker \rho \ne 1$).
\end{example}

\begin{remark}\label{classicalmonic} It is important that Proposition \ref{propker} has a variant 
without smoothness hypotheses:  if $f:G' \rightarrow G$ is a $k$-homomorphism between affine
$k$-group schemes of finite type and $\ker f = 1$ then $f$ is a closed immersion.
We do not use this in \S\ref{review}, but it is used in the development of
the relative theory over rings.  See \cite[VI$_{\rm{B}}$, 1.4.2]{sga3} for a proof.
\end{remark}

The closed immersion property
for monic homomorphisms 
\index{linear algebraic group!monic homomorphism}
\index{monic homomorphism}
between linear algebraic
$k$-groups is a wonderful feature of
the theory over fields; it is not true for smooth affine 
groups over rings (see Example \ref{weird})
and criteria to ensure it over rings can lie quite deep
(see Theorem \ref{adimm}). 

We now recall notions related to Jordan decomposition
\index{linear algebraic group!Jordan decomposition}
\index{Jordan decomposition}
for a linear algebraic group $G$ (see \cite[4.4]{borelag} for proofs).
Fix a faithful linear representation $j:G \hookrightarrow \GL(V)$.  
For any $g \in G(k)$, 
we say that $g$ is {\em semisimple}  if the linear endomorphism $j(g)$
of $V$ is semisimple in the sense of linear algebra; i.e., $j(g)$ is diagonalizable.
Likewise, we say that $g$ is {\em unipotent}  if
$j(g)$ is unipotent as a linear endomorphism of
$V$. 
These properties are independent of the choice of $j$
and are preserved under any $k$-homomorphism
$f:G \rightarrow H$ to another linear algebraic $k$-group;
that is, if $g \in G(k)$ is semisimple (resp.\,unipotent) 
then so is $f(g) \in H(k)$. 

In general for any $g \in G(k)$ there are unique commuting
elements $g_{\rm{ss}}, g_{\rm{u}} \in G(k)$
such that $g_{\rm{ss}}$ is semisimple, $g_{\rm{u}}$
\index{ aaagss@$g_{\rm{ss}}$, $g_{\rm{u}}$}
is unipotent, and $g = g_{\rm{ss}} g_{\rm{u}} = g_{\rm{u}} g_{\rm{ss}}$.
We call these the {\em semisimple part} and {\em unipotent part}
of $g$ respectively, and refer to these product expressions
for $g$ as its {\em Jordan decomposition}.  The existence of this 
Jordan decomposition demonstrates an advantage of 
linear algebraic groups over Lie algebras: the former is defined
and studied in a characteristic-free way (over algebraically closed fields), whereas the development of the latter 
is entirely different in characteristic 0 and in positive characteristic
(see \cite[V.7.2]{seligman} and \cite{humams}).

The formation of the
Jordan decomposition is functorial
in the sense that if $f:G \rightarrow H$ is a $k$-homomorphism
to another linear algebraic $k$-group then
$$f(g_{\rm{ss}}) = f(g)_{\rm{ss}},\,\,\,f(g_{\rm{u}}) = f(g)_{\rm{u}}.$$
In particular, if $H = \GL(W)$ for a finite-dimensional vector space
$W$ then $f(g)$ and $f(g_{\rm{ss}})$ have
the same characteristic polynomial.   This is a very useful fact. 

\begin{definition}
A linear algebraic group $G$ is {\em solvable} if $G(k)$
is solvable, and is {\em unipotent} if $g = g_{\rm{u}}$
for all $g \in G(k)$.
\end{definition}
\index{linear algebraic group!solvable}
\index{linear algebraic group!unipotent}

\begin{remark}
A more elegant definition of unipotence, avoiding the crutch of $\GL_n$-embeddings,
is given in \cite[XVII, 1.3]{sga3}; its equivalence with
the above definition is given in \cite[XVII, 2.1]{sga3}.
\end{remark}

By the Lie--Kolchin theorem \cite[10.5]{borelag}, if $G$ is solvable and connected
then every linear representation
$G \rightarrow \GL_N$
 can be conjugated to have image inside the upper triangular subgroup $B_N$.  
Also, {\em without} a connectedness hypothesis, 
$G$ is unipotent precisely when it occurs as a closed subgroup of
the strictly upper-triangular subgroup $U_n \subset \GL_n$ for some $n$
\cite[4.8]{borelag}. In particular, every unipotent linear algebraic group is solvable.

\begin{example} The group $\Add$ is unipotent, due to the faithful representation
$x \mapsto (\begin{smallmatrix} 1 & x \\ 0 & 1 \end{smallmatrix})$.
\end{example}

The strictly upper triangular
subgroup $U_n$ of $\GL_n$ admits a composition series whose
successive quotients are $\Add$, so every
unipotent linear algebraic group $U$ admits a composition
series whose successive quotients are $\Add$ or a finite \'etale subgroup of
$\Add$ (as $k = \overline{k}$, so we can use the smooth
underlying reduced schemes of intersections of $U$ with the composition
series for some $U_n$).  When $U$ is connected,
we may arrange the composition series so that the successive quotients are all
equal to $\Add$ \cite[10.6(2), 10.9]{borelag}. 

Functoriality of the Jordan decomposition implies that every homomorphic image or smooth closed
subgroup of a unipotent linear algebraic group is unipotent,
and likewise that if a linear algebraic group $G$ contains a unipotent normal linear algebraic 
subgroup $U$ such that $G/U$ is also unipotent then $G$ is unipotent.
Since $\Add$ has no
nontrivial finite subgroups when ${\rm{char}}(k) = 0$
and its finite \'etale subgroups are $p$-groups when
${\rm{char}}(k) = p > 0$, the unipotence of
the finite \'etale component group $U/U^0$ of a unipotent
linear algebraic group $U$ implies that
unipotent linear algebraic $k$-groups are 
connected when ${\rm{char}}(k) = 0$ and have component group that is a $p$-group
when ${\rm{char}}(k) = p > 0$.

A key result at the start of the theory of linear algebraic groups is
\cite[11.1, 11.2]{borelag}:

\begin{theorem}\label{bpar} Let $G$ be a linear
algebraic group over $k$. The maximal connected solvable linear algebraic $k$-subgroups
of $G$ are all $G(k)$-conjugate to each other, and these
are precisely the connected solvable linear algebraic subgroups
$B$ such that the quasi-projective quotient scheme $G/B$ is projective.
\end{theorem}

A {\em Borel subgroup} of $G$ is a maximal connected solvable linear
algebraic subgroup of $G$.  
A {\em parabolic subgroup} of $G$ is a linear algebraic
subgroup $P \subset G$ such that $G/P$ is projective.
\index{Borel subgroup}
\index{parabolic subgroup}
\index{linear algebraic group!Borel subgroup}
\index{linear algebraic group!parabolic subgroup}
All Borel subgroups are parabolic, by Theorem \ref{bpar}, so any
linear algebraic subgroup containing a Borel subgroup is also parabolic.  In fact, 
the parabolic subgroups are precisely the subgroups that contain Borel subgroups
\cite[11.2, Cor.]{borelag}.  Note that the definition of parabolicity 
does not require connectedness.  The following result is fundamental
(see \cite[11.16]{borelag} for a proof):

\begin{theorem}[Chevalley]\label{parconn} The parabolic subgroups
$P$ of any connected linear algebraic group $G$ are connected,
and $N_{G(k)}(P) = P(k)$.
\end{theorem}

In Corollary \ref{parnorm} this will be improved to a scheme-theoretic equality
$N_G(P) = P$ when $G$ is connected reductive. 

\begin{example}\label{pargln} For $G = {\rm{SL}}_n$, the subgroup $B$ of upper triangular
matrices is clearly connected and solvable, and under the natural transitive $G$-action
on the complete variety $F$ of full flags in $k^n$
the smooth group $B$ is the stabilizer scheme of the standard flag $F_0$.
Thus, the orbit map $G \rightarrow F$
through $F_0$ is faithfully flat (even smooth) and induces
a scheme isomorphism $G/B \simeq F$, so $B$ is also parabolic, hence a Borel subgroup.

Some obvious parabolic subgroups of $G$ containing $B$ are labelled  by ordered partitions
$\vec{a} = (a_1,\dots,a_r)$ of $n$ into non-empty parts (i.e., all $a_i > 0$, $\sum a_i = n$): 
we associate to $\vec{a}$ the subgroup
$P_{\vec{a}}$ consisting of points of ${\rm{SL}}_n$
that preserve each of the subspaces $V_j$ spanned
by the first $b_j := a_1 + \dots + a_j$ standard
basis vectors.  (Note that $\vec{a} \mapsto \{b_1,\dots,b_{r-1}\}$
is a bijection from the set of such $\vec{a}$ onto 
the set of subsets of $\{1,\dots,n-1\}$.) These $2^{n-1}$ parabolic subgroups are
the only parabolic subgroups containing $B$, so they represent (without repetition;
see Corollary \ref{147}) 
the conjugacy classes of parabolic subgroups of $G$.
(In Example \ref{stdpar} this example is addressed more fully.)
\end{example}

Let $G$ be a linear algebraic group over $k$. 
If $U, U' \subset G$ are connected {\em normal} unipotent linear
algebraic subgroups of $G$ then the normal closed subgroup $U \cdot U'$ that they generate
is unipotent (as it is a quotient of $U \ltimes U'$). 
Hence, by dimension considerations there exists a unique maximal  connected unipotent
normal linear algebraic subgroup $\mathscr{R}_u(G) \subset G$, called the
{\em unipotent radical} of $G$.  In a similar manner, there
is a maximal connected solvable normal linear algebraic subgroup $\mathscr{R}(G)$,
called the {\em radical} of $G$.
\index{linear algebraic group!radical, unipotent radical}
\index{radical!linear algebraic group}
\index{unipotent radical!linear algebraic group}
\index{ aaarug@$\mathscr{R}_u(G)$}
\index{ aaarg@$\mathscr{R}(G)$}

If $H$ is a normal linear algebraic subgroup of $G$ then
$\mathscr{R}_u(H)$ is normal in $G$, so 
$\mathscr{R}_u(H) = (H \bigcap \mathscr{R}_u(G))^0_{\rm{red}}$; the same holds
for radicals.  (Note that the formation of the underlying
reduced scheme is a local operation for the Zariski topology, so
the formation of identity component and underlying reduced scheme
of a finite type $k$-group scheme commute.  In particular, there is no ambiguity in notation such
as $\mathscr{G}^0_{\rm{red}}$ for such a group scheme
$\mathscr{G}$.) These notions also 
behave well with respect to quotients: if $\pi:G \twoheadrightarrow G'$
is a surjective homomorphism between linear algebraic groups
then $\mathscr{R}_u(G') = \pi(\mathscr{R}_u(G))$ and likewise
for radicals.  However, the proof of this result for quotient maps
is nontrivial when ${\rm{char}}(k) > 0$ (if $\ker \pi$ is not smooth); see \cite[14.11]{borelag}
for an argument that works regardless of the smoothness properties of $\pi$.

\begin{definition} A {\em reductive} $k$-group is a linear algebraic $k$-group $G$
such that $\mathscr{R}_u(G) = 1$ (i.e., $G$ contains no nontrivial
unipotent normal  connected linear algebraic $k$-subgroup).
A {\em semisimple} $k$-group is a linear algebraic $k$-group $G$
such that $\mathscr{R}(G) = 1$.
\end{definition}
\index{linear algebraic group!reductive, semisimple}
\index{reductive group}
\index{semisimple group}

It is immediate from the good behavior of radicals and unipotent radicals with
respect to normal subgroups and quotients that semisimplicity and reductivity
are inherited by normal linear algebraic subgroups and images of homomorphisms.

\begin{example}
A basic example of a connected reductive group is the open unit group
$\GL(V)$ in the affine space of linear
endomorphisms of $V$.  This follows from the Lie--Kolchin theorem; it
recovers $\Gm$ when $\dim V = 1$.  The radical
$\mathscr{R}(\GL(V))$ coincides with the ``scalar'' $\Gm$ (see Example \ref{cisog}). 

Examples of
connected semisimple groups
are: the smooth irreducible hypersurface ${\rm{SL}}(V)$ 
in $\GL(V)$ (which recovers ${\rm{SL}}_2$ when $\dim V = 2$),
its quotient ${\rm{PGL}}(V)$, 
and symplectic groups ${\rm{Sp}}(V,\psi)$
for non-degenerate alternating forms $\psi$ on a nonzero finite-dimensional vector space $V$.
(The connectedness of symplectic groups
can be proved via induction on $\dim V$ via a fibration argument, and the smoothness
can be proved by the infinitesimal criterion.)

Special orthogonal groups ${\rm{SO}}(V,q)$
associated to non-degenerate quadratic spaces $(V,q)$
of dimension $\ge 3$ are also connected semisimple, but special care
is needed to give a characteristic-free development of such groups that works well in
characteristic 2 without parity restrictions
on $\dim V$; see Exercise \ref{1.4} and Definition \ref{soqdef}. 
Smoothness and connectedness of
special orthogonal groups are proved in Appendix \ref{sosec}
(see \S\ref{cliffso}--\S\ref{connZ}).

For proofs of reductivity of these groups, see Exercises \ref{1.6}(i) and \ref{redexer}.
\end{example}


\begin{remark} There is no universal convention as to whether or not
reductive groups should be required to be connected. If $G$ is connected and reductive then
the centralizer in $G$ of any torus is connected and reductive (see Theorem \ref{torithm}(3)),
so for arguments with torus centralizers there is no harm in requiring 
connectedness.  
A more subtle case is the centralizer
$Z_G(g)$ for connected reductive $G$ and semisimple $g \in G(k)$.
(The $k$-group $Z_G(g)$ is smooth even when
defined to represent a ``centralizer functor'', due to \cite[9.2, Cor.]{borelag}
and Exercise \ref{gross1}(i); a vast generalization is provided by
Lemma \ref{centralizer}, recovering $Z_G(g)$ by taking $Y$ there to be the Zariski closure
of $g^{\Z}$.) 
Such centralizers appear in orbital integrals in the trace formula for automorphic forms.  
In general $Z_G(g)^0$ is reductive 
(see Theorem \ref{torithm}(3))  but $Z_G(g)$ may be disconnected (e.g., $G = {\rm{PGL}}_2$
and $g = (\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix})$
with ${\rm{char}}(k) \ne 2$; for ${\rm{SL}}_2$
the analogous centralizer is connected).

 In \cite{borelag}
and \cite{springer}, as well as in these notes, reductive groups 
over a field are not assumed to be connected.  This is a contrast with \cite[XIX, 2.7]{sga3},
and it may seem to present a slightly inconsistency with the general definition
over schemes, but no real confusion should arise. 
For reductive groups over schemes, the usefulness of a fibral connectedness condition 
is more compelling than in the theory over a field (see Example \ref{weird}).
\end{remark}

\begin{remark} For linear algebraic groups
over an algebraically closed field of characteristic 0, the property
of reductivity for the identity component is equivalent to the semisimplicity
of all linear representations. This fails in positive characteristic; see Exercise \ref{1.6}. 
\end{remark}

In the context of connected solvable linear algebraic groups, the opposite
extreme from  the unipotent groups is the following class of groups:

\begin{definition} A {\em torus} over the algebraically closed
field $k$ is a $k$-group $T$ that is isomorphic
to a power $(\Gm)^r$ for some $r \ge 0$.
\end{definition}

In view of the general structure of connected solvable linear algebraic groups
(as $T \ltimes U$ for a torus $T$ and unipotent radical $U$ \cite[10.6(4)]{borelag}),
we see that the solvable connected reductive groups are precisely the tori.

\begin{example}\label{cisog}
Let $G$ be connected reductive.  The solvable connected $\mathscr{R}(G)$ is a torus
(as it must be reductive).  Since the automorphism scheme ${\rm{Aut}}_{T/k}$ of a 
$k$-torus $T$ is a constant group (see the proof of Theorem \ref{torithm}),
a normal torus $T'$ in any {\em connected} linear algebraic group $G'$ must be central
(as the $k$-homomorphism $G' \rightarrow {\rm{Aut}}_{T'/k}$ giving the conjugation
action has to be trivial).  Hence, 
$Z := \mathscr{R}(G)$ is a central torus such that $G/Z$ is semisimple.
In other words, every connected reductive group is a central extension of
a connected semisimple group by a torus.  For example, if 
$G = \GL(V)$ then $G/Z = {\rm{PGL}}(V)$.

Deeper structure theory of reductive groups \cite[14.2]{borelag} ensures that the derived group
$\mathscr{D}(G)$ is semisimple 
and that the commutative reductive quotient $G/\mathscr{D}(G)$
(which must be a torus) is an isogenous quotient of $Z$.  In other words,
$G$ is also canonically an extension of a torus by a connected semisimple group.
For $G = \GL(V)$ this is the exact sequence
$$1 \rightarrow {\rm{SL}}(V) \rightarrow \GL(V) \stackrel{\Det}{\rightarrow} \Gm \rightarrow 1,$$
and the isogeny of tori $Z \rightarrow G/\mathscr{D}(G)$ in this case is identified with the endomorphism 
$t \mapsto t^n$ of $\Gm$ for $n = \dim V$.

The natural homomorphism $Z \times \mathscr{D}(G) \rightarrow G$ 
is an isogeny for any connected reductive $k$-group $G$.
Because of this, for many (but not all!) problems
in the classical theory of reductive groups one can reduce to a separate treatment of
tori and semisimple groups. 
Note also that the semisimple $\mathscr{D}(G)$ must in fact be equal to its
own derived group.  Indeed, the quotient
$\mathscr{D}(G)/\mathscr{D}^2(G)$ is connected, semisimple (inherited from $\mathscr{D}(G)$), and commutative,
hence trivial. 
\end{example}

A fundamental result (whose
proof rests on the theory of Borel
subgroups) is that a connected linear algebraic group $G$ over $k = \overline{k}$ is a torus 
when all $k$-points are semisimple \cite[11.5(1)]{borelag}.
If instead $G$ contains no nontrivial tori
then it must be unipotent \cite[11.5(2)]{borelag}.  This is very useful: it implies that 
a general connected linear algebraic $k$-group $G$ either
admits a strictly upper triangular faithful representation
or it contains a nontrivial $k$-torus.  (This dichotomy is also true without
assuming $k = \overline{k}$, but requires Grothendieck's deep result
on the existence of geometrically maximal $k$-tori, proved in Appendix \ref{torusapp}.) Thus, 
if $\dim G > 1$ then $G$ contains either $\Add$ or $\Gm$ as a proper $k$-subgroup
(since $k = \overline{k}$).

A very effective way to work with tori is by means of some associated lattices.
To be precise, since ${\rm{End}}(\Gm) = \Z$ via the endomorphisms $t \mapsto t^n$,
for any torus $T$ the commutative groups
\index{ aaaxt@${\rm{X}}(T)$, ${\rm{X}}_{\ast}(T)$}
$${\rm{X}}(T) = \Hom_{k\mbox{-}{\rm{gp}}}(T,\Gm),\,\,\,
{\rm{X}}_{\ast}(T) = \Hom_{k\mbox{-}{\rm{gp}}}(\Gm,T)$$
are finite free $\Z$-modules of rank $\dim T$ that are
respectively contravariant and covariant in $T$.  
(Elements of ${\rm{X}}_{\ast}(T)$ are called {\em cocharacters} of $T$.) 
For historical reasons via the theory of compact Lie groups, 
it is a standard convention to use additive notation when discussing
elements of the character and cocharacter lattices of $T$.
For example, if $a, b:T \rightrightarrows \Gm$ are two characters
then $a+b$ denotes $t \mapsto a(t)b(t)$ and $-a$ denotes
$t \mapsto 1/a(t)$ (and $0$ denotes the trivial character).
For this reason, $a(t)$ is often denoted as $t^a$.

Evaluation defines a perfect duality of lattices
$$\langle \cdot, \cdot \rangle:{\rm{X}}(T) \times {\rm{X}}_{\ast}(T) \rightarrow {\rm{End}}(\Gm) = \Z$$
via $\langle a, \lambda \rangle = a \circ \lambda$.
(Here, by ``perfect'' we mean that the $\Z$-bilinear form identifies each lattice with
the $\Z$-dual of the other. In terms of matrices relative to a $\Z$-basis, it means that 
the matrix of the bilinear form has determinant in $\Z^{\times} = \{\pm 1\}$.) 
In terms of the cocharacter group ${\rm{X}}_{\ast}(T)$ we have
$${\rm{X}}_{\ast}(T) \otimes_{\Z} k^{\times} \simeq T(k)$$
via $\lambda \otimes c \mapsto \lambda(c)$.
In fact the algebraic group $T$ (and not just its group of $k$-points) 
can be reconstructed from its cocharacter group, by considering the functor
${\rm{X}}_{\ast}(T) \otimes_{\Z} \Gm$ that assigns to any $k$-algebra $A$
the group ${\rm{X}}_{\ast}(T) \otimes_{\Z} A^{\times} = \Hom({\rm{X}}(T), A^{\times})$.  More specifically, 
the contravariant functors 
$$T \rightsquigarrow {\rm{X}}(T),\,\,\,
M \rightsquigarrow M^{\vee} \otimes_{\Z} \Gm$$
(where $M^{\vee} := \Hom(M, \Z)$) 
are inverse anti-equivalences between the categories
of tori and finite free $\Z$-modules.

A fundamental fact about tori is that their linear representations
are completely reducible in {\em any} characteristic.  This can be expressed in the following
canonical form.

\begin{proposition}
Let $T$ be a torus, $M = {\rm{X}}(T)$ its character group,
and $V$ a finite-dimensional linear representation of $T$ over $k$.
For each $a \in M$, let $V_{a}$ be the space of
$v \in V$ such that $t.v = a(t)v$ for all $t \in T(k)$.
The natural $T$-equivariant map $\bigoplus_{a \in M} V_{a} \rightarrow V$
is an isomorphism.

In this way, the category of linear representations of $T$
is equivalent to the category of $M$-graded $k$-vector spaces.
\end{proposition}

We call $V_{a}$ the {\em $a$-weight space} for $T$ acting on $V$.
\index{weight space}
Note that these vanish for all but finitely many $a$; the {\em weights}
for the $T$-action on $V$ are the $a$ such that $V_{a} \ne 0$.
For the dual representation space $V^{\ast}$, we have
$(V^{\ast})_{a} = (V_{-a})^{\ast}$ for all $a \in M$. 
Hence, if $V$ is a self-dual representation of $T$ then its set of
weights is stable under negation.

\begin{example} For $T = \Gm$ we have ${\rm{X}}(T) = {\rm{End}}(\Gm) = \Z$, so
a $k$-homomorphism $\Gm \rightarrow \GL(V)$ is the ``same'' as
a $k$-linear $\Z$-grading $V = \bigoplus_{n \in \Z} V(n)$, with $t \in \Gm = T$ acting on $V(n)$
via $t^n$-scaling. (There is an analogous result for linear $\Gm^r$-actions using $\Z^r$-gradings,
as well as for linear $\mu_n$-actions using $\Z/n\Z$-gradings. A common generalization
for linear representations
of any diagonalizable $k$-group $D_k(M)$ is expressed
in terms of $M$-gradings, but we will not need it here.  See \cite[A.8.1--A.8.9]{pred}.) 

Choose $v \in V$, and write $v = \sum_{n \in \Z} v_n$ with $v_n \in V(n)$, so 
$t.v = \sum_{n \in \Z} t^n v_n$.  Viewing $V$ as an affine space
over $k$, clearly the orbit map $\Gm \rightarrow V$ defined by
$t \mapsto t.v$ extends to a $k$-scheme morphism $\mathbf{A}^1 \rightarrow V$
if and only if $v \in V_{\ge 0} := \bigoplus_{n \ge 0} V(n)$.
For any such $v$, we define $\lim_{t \rightarrow 0} t.v$ to be the image of
$0 \in \mathbf{A}^1(k) = k$ under the extension $\mathbf{A}^1 \rightarrow V$
of the orbit map.  This limiting value is $v_0$, so the space of $v \in V$
for which $\lim_{t \rightarrow 0} t.v$ exists and vanishes is $V_{>0} := \bigoplus_{n > 0} V(n)$.
\end{example}

By dimension considerations, any torus in a linear algebraic
$k$-group is contained in a maximal such torus.   
Here are some important properties of tori in linear algebraic groups: 

\begin{theorem}\label{torithm} Let $G$ be a linear algebraic group over $k$. 
\begin{enumerate}
\item For any torus $T'$ in $G$, the Zariski-closed centralizer $Z_G(T')$
has finite index in the Zariski-closed normalizer $N_G(T')$,
and if $G$ is connected then $Z_G(T')$ is connected.
\item All maximal tori $T$ in $G$ are $G(k)$-conjugate.
\item Assume $G$ is connected reductive.  The centralizer $Z_G(T')$
is connected reductive for any torus $T'$ in $G$, and if $T$ is a maximal
torus in $G$ then $Z_G(T) = T$.  If $g \in G(k)$ is semisimple then
$Z_G(g)^0$ is reductive.
\end{enumerate}
\end{theorem}

In this theorem we take $N_G(T')$ and $Z_G(T')$ to have the classical
meaning, as smooth closed subgroup schemes corresponding to
$N_{G(k)}(T')$ and $Z_{G(k)}(T')$ respectively. 
(This coincides
with the functorial viewpoint on normalizers and centralizers that is 
addressed in Proposition \ref{xi2.4bis}, Definition \ref{zgy}, Lemma \ref{centralizer}, and Exercise \ref{2.7}.)

\begin{proof}
Since the endomorphism functor $R \rightsquigarrow {\rm{End}}_{R\mbox{-}{\rm{gp}}}(\Gm)$
on $k$-algebras 
is represented by the constant $k$-group $\Z$ \cite[I, \S5.2]{oesterle}, the endomorphism functor
of $\Gm^r$ is represented by the constant $k$-group ${\rm{Mat}}_r(\Z)$.
Thus, the automorphism functor of $T' \simeq \Gm^r$ 
is represented by the locally finite type constant $k$-group
$\GL_r(\Z)$.  In particular, all quasi-compact
$k$-subschemes of this automorphism scheme are finite and closed.  Since $N_G(T')$ is finite type, its image 
in the automorphism scheme must therefore be finite.  But the kernel of the action of $N_G(T')$ on
$T'$ has underlying space $Z_G(T')$ (in fact this kernel is smooth, but we do not need it here),
so the finite-index claim in (1) is proved.   The connectedness in (1) is \cite[11.12]{borelag}. 
The conjugacy of maximal tori is \cite[11.3(1)]{borelag}.  The assertions in (3) concerning
torus centralizers are
part of \cite[13.17, Cor.\,2]{borelag}, and the reductivity of $Z_G(g)^0$ for
semisimple $g \in G(k)$ is \cite[13.19]{borelag} 
\end{proof}

\begin{remark}\label{borelcover} For $G = \GL_n$, every element can be conjugated into an upper triangular
form and every semisimple element can be diagonalized.  In other words,
$G(k)$ is covered by the subgroups $B(k)$ as $B$ varies through the Borel
subgroups, and the subset of semisimple elements in $G(k)$ is covered by the subgroups
$T(k)$ as $T$ varies through the maximal tori.  These properties
are valid for every connected linear algebraic group $G$.
Indeed, for solvable $G$ the result can be deduced
from the structure of connected solvable groups (see \cite[10.6(5)]{borelag}),
and so by the general conjugacy of Borel
subgroups and maximal tori it suffices to show that all elements of
$G(k)$ lie in a Borel subgroup.  See \cite[11.10]{borelag} for this result.
\end{remark}

In our later work with reductive groups over schemes we will define scheme-theoretic notions
of centralizer and normalizer by closed subgroup schemes of a linear
algebraic group, and in the case of a torus $T$ in a linear algebraic group
$G$ over $k$ we will show that the scheme-theoretic notions of
$Z_G(T)$ and $N_G(T)$ are smooth (and so coincide with their classical counterparts);
see Proposition \ref{xi2.4bis} and Lemma \ref{centralizer}.
The quotient $N_G(T)/Z_G(T)$ will then be a finite \'etale $k$-group,
denoted as $W_G(T)$, and it is the constant
\index{ aaawgt@$W_G(T)$}
group associated to its group of $k$-points,
which is $N_{G(k)}(T)/Z_{G(k)}(T)$.  This latter group is
the {\em Weyl group} associated to $(G,T)$ in the classical theory,
\index{Weyl group}
and it agrees with the group of $k$-points of the \'etale quotient construction that will
be used in the relative scheme-theoretic theory.
The Weyl group $W_G(T)$ is especially important when $T$ is {\em maximal}
in a connected reductive $k$-group $G$,
in which case it is $N_G(T)/T$. 

For a linear algebraic group $G$, the centralizers $Z_G(T)$ of maximal tori $T$ 
are called the {\em Cartan subgroups}. 
\index{linear algebraic group!Cartan subgroup}
\index{Cartan subgroup}
These are visibly $G(k)$-conjugate to each other; 
their common dimension is called the {\em nilpotent rank} of $G$,
\index{nilpotent rank}
and the common dimension of the maximal tori is called the {\em reductive rank} of $G$.
\index{reductive rank}
In the special case of connected reductive $G$ the Cartan subgroups are precisely the maximal
tori, and their common dimension is then simply called the {\em rank} of $G$.
\index{reductive group!rank}

\subsection{Roots and coroots}

Let $G$ be a connected reductive $k$-group, $T$ a maximal torus in $G$,
and $\mathfrak{g}$ and $\mathfrak{t}$ the respective Lie algebras.
In contrast with characteristic zero, the adjoint action of $G$ on $\mathfrak{g}$
can sometimes fail to be semisimple when ${\rm{char}}(k) > 0$.  For instance,
if ${\rm{char}}(k) = p > 0$
and $G = {\rm{SL}}_p$
then $\mathfrak{g} = \mathfrak{sl}_p = \mathfrak{gl}_p^{{\rm{Tr}}=0}$
(the kernel of the trace map) and this 
contains the diagonal scalar subspace of
$\mathfrak{gl}_p$ on which the
adjoint action of $G$ is trivial.  This line admits no $G$-equivariant complement
(see Exercise \ref{slp}).  

However,
the $T$-action on $\mathfrak{g}$ is completely
reducible, as for any linear representation of a torus.
When $\mathfrak{g}$ is equipped with this extra structure then 
it becomes a useful invariant of $G$ in any characteristic.
To see this, consider the weight space decomposition
\begin{equation}\label{wtg}
\mathfrak{g} = \mathfrak{g}_0 \oplus (\oplus_{a \in \Phi} \mathfrak{g}_{a})
\end{equation}
for a finite subset $\Phi = \Phi(G,T) \subset {\rm{X}}(T) - \{0\}$.
\index{ aaafigt@$\Phi(G,T)$}
The following properties hold: the subspace $\mathfrak{g}_0 = \mathfrak{g}^T$ coincides
with $\mathfrak{t}$, for each
$a \in \Phi$ the weight space $\mathfrak{g}_{a}$ is $1$-dimensional,
\index{ aaaga@$\mathfrak{g}_a$}
and $\Phi$ is stable under negation in ${\rm{X}}(T)$
\cite[13.18(1),(4a),(4b)]{borelag}.  Beware
that even though the set of $T$-weights on $\mathfrak{g}$
is stable under negation, just like self-dual representations of $G$, 
when ${\rm{char}}(k) > 0$ the $G$-representation space
$\mathfrak{g}$ can fail to be self-dual; see Exercise \ref{slp}. 

Letting $r = \dim T$ denote the rank of $G$, the 1-dimensionality of
the weight spaces $\mathfrak{g}_{a}$ for
the nontrivial $T$-weights on $\mathfrak{g}$ implies that the characteristic polynomial of
the $T$-action on $\mathfrak{g}$ is
$$\Det(xI - {\rm{Ad}}_G(t)|\mathfrak{g}) = (x-1)^r \prod_{a \in \Phi}(x - a(t))$$
for $t \in T(k)$.  For this reason, the elements
of $\Phi$ are called the {\em roots} of $(G,T)$; the corresponding
\index{reductive group!root}
\index{root space}
\index{root}
1-dimensional weight spaces $\mathfrak{g}_a$ are called {\em root spaces}
in $\mathfrak{g}$.

\begin{example} Take $G = \GL(V) = \GL_n$ with $n > 0$ and let $T = \Gm^n$ be the
torus consisting of diagonal matrices
$$t = {\rm{diag}}(c_1,\dots, c_n).$$
It is easy to check that $T(k) = Z_{G(k)}(T)$, so $T$ is a maximal torus. 
We have $${\rm{X}}(T) = \bigoplus \Z e_i,\,\,\,
{\rm{X}}_{\ast}(T) = \bigoplus \Z e_i^{\vee}$$
where $e_i({\rm{diag}}(c_1,\dots,c_n)) = c_i$
and $e_i^{\vee}(c) = {\rm{diag}}(1,\dots,c,\dots,1)$ (with
$c$ as the $i$th diagonal entry, all others being 1).
The normalizer $N_{G(k)}(T)$ is the group of invertible monomial matrices
(i.e., one nonzero entry in each row and in each column) and 
$N_{G(k)}(T)/T(k) = \mathfrak{S}_n$
\index{ aaasn@$\mathfrak{S}_n$} is represented by the group of $n \times n$ permutation matrices
(relative to the standard basis).

The Lie algebra $\mathfrak{g}$ is ${\rm{End}}(V) = {\rm{Mat}}_n(k)$, and the 
Lie subalgebra $\mathfrak{t}$ is the subspace of diagonal matrices.  The 
roots are the characters
$$a_{ij}({\rm{diag}}(c_1,\dots,c_n)) = c_i/c_j$$
for $1 \le i \ne j \le n$;
in other words, $a_{ij} = e_i - e_j$. The corresponding root
space $\mathfrak{g}_{a_{ij}} \subset {\rm{Mat}}_n(k)$
consists of matrices with vanishing entries away from the $ij$-entry.
\end{example}

\begin{example} For $G = {\rm{SL}}_2$, the diagonal torus
$T = \{{\rm{diag}}(c, 1/c)\}$ is maximal
and is usually identified with $\Gm$ via $\lambda:c \mapsto
{\rm{diag}}(c, 1/c)$.  The Lie algebra
$\mathfrak{g}$ is the space 
$\mathfrak{sl}_2 = \mathfrak{gl}_2^{{\rm{Tr}}=0}$ of traceless $2 \times 2$ matrices over $k$,
in which a basis is given by
$$H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},\,\,\,
E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\,\,\,
F = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}.$$
The line $k H$ coincides with $\mathfrak{t}$, and the lines
$k E$ and $k F$ are the root spaces.  Explicitly, 
$${\rm{Ad}}_G(\lambda(c))(E) = \lambda(c)E \lambda(c)^{-1} = c^2 E,\,\,\,
{\rm{Ad}}_G(\lambda(c))(F) = \lambda(c)F \lambda(c)^{-1} = c^{-2}F,$$
so the roots in ${\rm{X}}(T) = {\rm{X}}(\Gm) = \Z$ are $\pm 2$
with corresponding root spaces $\mathfrak{g}_2 = k E$
and $\mathfrak{g}_{-2} = kF$.
Note that $H$ is central in $\mathfrak{sl}_2$ when ${\rm{char}}(k) = 2$
(and the central $\mu_2$ in ${\rm{SL}}_2$ has
Lie algebra ${\rm{Lie}}(\mu_2) = \mathfrak{t} = k H$ when ${\rm{char}}(k) = 2$). 
\end{example}

Although our survey of the classical theory in \S\ref{review}
largely uses the classical convention to treat normalizers and centralizers
as reduced schemes, we now briefly digress to discuss a special case in the classical theory
for which schematic centralizers are smooth.  (In \S\ref{normsec} we will take up the relative
scheme-theoretic notions of centralizer and normalizer.)

For a smooth affine $k$-group $G$ and $g \in G(k)$, define
the {\em schematic centralizer} $Z_G(g)$ to be the scheme-theoretic
\index{centralizer!scheme-theoretic}
fiber over $e$ for the morphism $G \rightarrow G$ defined by $x \mapsto gxg^{-1}x^{-1}$.  (For any $k$-algebra $R$, 
$Z_G(g)(R)$ consists of those $x \in G(R)$ that commute with $g_R \in G(R)$.)  For any smooth closed $k$-subgroup
$H$, consider the closed subgroup scheme
\index{ aaazgh@$Z_G(H)$}
$$Z_G(H) := \bigcap_{h \in H(k)} Z_G(h)$$
in $G$, where $Z_G(h)$ is taken in the scheme-theoretic sense as just defined.

\begin{proposition}\label{zrep}
The $k$-subgroup scheme $Z_G(H)$ represents the functorial centralizer of $H$ in $G$:
\index{centralizer!functorial}
for any $k$-algebra $R$, $Z_G(H)(R)$ coincides with the set of $g \in G(R)$ such that $g$-conjugation on
$G_R$ restricts to the identity on $H_R$.
Moreover, ${\rm{Lie}}(Z_G(H)) = \mathfrak{g}^H$ inside $\mathfrak{g}$.
\end{proposition}

\begin{proof}
We first prove that $Z_G(H)$ represents the functorial centralizer.  We have to show for any $k$-algebra $R$ and $g \in G(R)$,
the $R$-morphism $H_R \rightarrow G_R$ defined by 
$x \mapsto xgx^{-1}g^{-1}$ is the constant map $x \mapsto 1$
if and only if $g \in Z_G(H)(R) = \bigcap_{h \in H(k)} Z_G(h)(R)$, which is to say
if and only if $hgh^{-1}g^{-1} = 1$ in $G(R)$ for all $h \in H(k)$.

More generally, for any reduced affine $k$-scheme $X$ of finite type and (possibly non-reduced)
$k$-algebra $R$, we claim that an
$R$-morphism $f:X_R \rightarrow Y$ to an affine $R$-scheme $Y$ is uniquely determined by
the collection of values $f(x) \in Y(R)$ for all $x \in X(k)$.  It suffices to show that 
an $R$-morphism $X_R \rightarrow Y \times_{\Spec R} Y$ factors through the diagonal
if it does so at all points in $X(R)$ arising from $X(k)$.  Since the diagonal is cut out by an ideal
in the coordinate ring of $Y \times Y$, it suffices to show that if an element $a \in R \otimes_k k[X]$
vanishes in $R$ after specialization at all $k$-points of the coordinate ring $k[X]$ then $a = 0$.  In other words, it suffices
to show that the map of $R$-algebras
$$h_R:R \otimes_k k[X] \rightarrow \prod_{x \in X(k)} R$$
defined by $a \mapsto (a(x))$ is injective for any $k$-algebra $R$.
(This says that $H(k)$ is ``relatively schematically dense'' in $H$ over $k$ in the sense of
\cite[IV$_3$, 11.10.8]{ega}; see \cite[IV$_3$, 11.9.13]{ega}.)

Since $k[X]$ is a reduced
$k$-algebra of finite type, by the Nullstellensatz 
the map $h_k$ is injective.  The map $h_R$ is the composition of $R \otimes_k h_k$
and the natural map
$$R \otimes_k \prod_{x \in X(k)} k \rightarrow \prod_{x \in X(k)} R$$
defined by $r \otimes (a_x) \mapsto (r a_x)$.  Hence, it suffices
to prove a general fact in linear algebra:  if $W$ is a (possibly infinite-dimensional) vector space over a field $k$
and if $\{V_i\}$ is a collection of $k$-vector spaces then the natural map
$W \otimes_k \prod_i V_i \rightarrow \prod_i (W \otimes_k V_i)$ is injective.
Any element in the kernel is a finite sum of elementary tensors, so we easily
reduce to the case when $W$ is finite-dimensional.  The case $W = 0$
is obvious, and otherwise by choosing a $k$-basis of $W$ we reduce to the 
trivial case $W = k$.  This completes the proof that $Z_G(H)$ represents the functorial
centralizer. 
%
%By definition, for $g \in G(k)$ the adjoint action ${\rm{Ad}}_G(g)$ on $\mathfrak{g}$
%is the restriction to $\mathfrak{g}$ of $g$-conjugation on $G(k[\epsilon])$ (as for computing
%the induced map on tangent spaces at $k$-points for any map between $k$-schemes of finite type).

To prove ${\rm{Lie}}(Z_G(H)) = \mathfrak{g}^H$, we give an argument using just the functor of points of $Z_G(H)$
and not the smoothness of $H$. 
Since $Z_G(H)$ represents
the functorial centralizer, ${\rm{Lie}}(Z_G(H))$ is the subset of elements in $\mathfrak{g} \subset G(k[\epsilon])$
on which $H_{k[\epsilon]}$-conjugation is trivial. 
Thus, for $v \in \mathfrak{g}$
we have to show that ${\rm{Ad}}_G(h)(v) = v$ in $\mathfrak{g}_R \subset G(R[\epsilon])$
for all $k$-algebras $R$ and $h \in H(R)$ if and only if
$H_{k[\epsilon]}$-conjugation on $G_{k[\epsilon]}$ leaves $v$ fixed.
Using the universal point of $H_{k[\epsilon]}$ (namely, its identity automorphism),
for the latter condition it suffices to check triviality on $v_R$ under conjugation against
$H(R[\epsilon])$ for all $k$-algebras $R$ (such as $R = k[H]$).
For any $h \in H(R[\epsilon])$, the specialization $h_0 \in H(R)$ at $\epsilon=0$ can be promoted to
an $R[\epsilon]$-point  (still denoted $h_0$) via $R \rightarrow R[\epsilon]$, so $h = h_0 h'$
for $h' \in \ker(H(R[\epsilon]) \rightarrow H(R)) = {\rm{Lie}}(H_R)$.  But the commutative addition on ${\rm{Lie}}(G_R) = \mathfrak{g}_R$ 
is induced by the group law on $G(R[\epsilon])$, so $v_Rhv_R^{-1}h^{-1} = v_Rh_0 v_R^{-1}h_0^{-1} = v - {\rm{Ad}}_G(h_0)(v)$.
Since every point in $H(R)$ arises in the form $h_0$, the desired equivalence is proved.
\end{proof}


\begin{corollary}\label{zgood}
Let $G$ be a smooth affine $k$-group and 
$T$ a $k$-torus in $G$. The schematic centralizer $Z_G(T)$ is $k$-smooth
with Lie algebra $\mathfrak{g}^T$, and if $G$ is connected
reductive and $T$ is maximal in $G$ then $Z_G(T) = T$
$($i.e., for any $k$-algebra $R$, if $g \in G(R)$ centralizes $T_R$ then $g \in T(R)$$)$.
In particular, for connected reductive $G$ and maximal tori $T$ in $G$, every central closed subgroup scheme of $G$ lies in $T$.
\end{corollary}

The smoothness is proved in another way
in Lemma \ref{centralizer} (via the infinitesimal criterion), avoiding recourse to the classical theory. 

\begin{proof}
%By direct limit considerations we can assume $R$ is finite type over $k$.
%A standard application of the Krull Intersection Theorem, as we now explain, permits
%further reduction to the case that $R$ is a finite local
%$k$-algebra.  For $R$ of finite type over $k$, suppose
%$g \in G(R)$ centralizes $T_R$.  We want to show that
%$g \in T(R)$.   It suffices to check the localization
%$g_{\mathfrak{m}} \in G(R_{\mathfrak{m}})$ lies in $T(R_{\mathfrak{m}})$ for each maximal ideal 
%$\mathfrak{m}$ of $R$.  This asserts the vanishing of the pullback of the ideal of $T$
%in $G$ under the map $g_{\mathfrak{m}}:{\rm{Spec}}(R_{\mathfrak{m}}) \rightarrow G$.
%By the Krull Intersection Theorem, it suffices to check this vanishing modulo
%each power of the maximal ideal of $R_{\mathfrak{m}}$, and that in turn amounts
%to solving our problem for the reduction of $g$ in 
%$G(R/\mathfrak{m}^n)$  for all $\mathfrak{m}$ and all $n \ge 1$.
%In other words, we may replace $R$ with the local finite $k$-algebras $R/\mathfrak{m}^n$.
%
%Now assume $R$ is $k$-finite and local. Since $T(k) = Z_{G(k)}(T)$,
%a point $g \in G(R)$ that centralizes $T_R$ is physically supported on $T$, so
%$g$ factors through the open cell $\Omega = U' \times T \times U$ relative to a choice
%of Borel subgroup $B$ containing $T$ (so $B = T \ltimes U$). 
%Hence, $g = u'tu$ for $u' \in U'(R)$, $t \in T(R)$, and $u \in U(R)$.
%We aim to prove that $u' = u = 1$.
%
%Since $T_R$ centralizes $g$ and normalizes $U_R$ and $U'_R$,
%it follows that $u'$ and $u$ are each centralized by $T_R$.
%It therefore suffices to show that the schematic centralizer
%$Z_G(T)$ has trivial schematic intersection with $U$ and $U'$.
%We argue for $U$, and $U'$ is handled in the same way.
%
First we consider the case of connected reductive case with maximal $T$.
In such cases, ${\rm{Lie}}(Z_G(T)) =
\mathfrak{g}^T = \mathfrak{t}$.  Thus, the inclusion of group schemes
$T \hookrightarrow Z_G(T)$ that is an equality on $k$-points (by the classical theory) 
is also an equality on Lie algebras (again, by the classical theory), so $\dim Z_G(T) = \dim T = \dim \mathfrak{t} = 
\dim {\rm{Lie}}(Z_G(T))$. Hence, $Z_G(T)$ is $k$-smooth by the tangential criterion.   The equality 
$T(k) = Z_{G(k)}(T) = Z_G(T)(k)$ of $k$-points therefore implies an equality 
$T = Z_G(T)$ as $k$-schemes.

In general, by the same argument, the smoothness of $Z_G(T)$ amounts to 
showing that ${\rm{Lie}}(Z_G(T)_{\rm{red}}) = \mathfrak{g}^T$. This is a special case
of \cite[Cor.\,9.2]{borelag} (setting $H, L$ there equal  to $G, T$ respectively, and working throughout
with reduced $k$-schemes).
%
%Choose $\lambda \in {\rm{X}}_{\ast}(T)$ such that $\langle a, \lambda \rangle \ne 0$
%for all $a \in \Phi = \Phi(G,T)$, and fix an enumeration of $\Phi^+ := \Phi(B,T)$
%such that $\lambda(a) > \lambda(b)$ if $a$ occurs after $b$ in the enumeration. 
%For $n > 0$ let $U_{\ge n} = \prod_{\lambda(a) \ge n} U_a$ be the subvariety of $U$
%directly spanned by 
%
%Using the direct spanning of $U$ and $U'$ by  the root groups (which are normalized by $T$), 
%it suffices to check that if $a \in \Phi(G,T)$ and
%$u \in U_a(R)$ is centralized by the $T_R$-action on $(U_a)_R$ then
%$u = 1$.  The character $a:T \rightarrow \Gm$ is an fppf covering,
%so the map $a_R:T_R \rightarrow \Gm$ of $R$-groups is an fppf covering.  Hence, 
%every point of $\Gm$ valued in an $R$-algebra arises from a point of $T_R$ valued in
%an fppf cover.  It follows via a choice of isomorphism $U_a \simeq \Ga$ that 
%the triviality on $u \in U_a(R)$ by $T_R$-conjugation implies
%the corresponding point $u \in \Ga(R)$ is invariant
%under the $\Gm$-action on $\Ga$ (over the category of $R$-algebras).  We want to show that a point
%of the latter sort must vanish. Since $R$ has an algebraically closed residue field,
%it contains a unit $c$ such that $c - 1 \in R^{\times}$, so the equality $cu = u$ forces $u = 0$.
\end{proof}


For $a \in \Phi$, there is a unique subgroup
$U_a \subset G$ normalized by $T$
\index{root group}
\index{ aaaua@$U_a$}
such that $U_a \simeq \Add$ and ${\rm{Lie}}(U_a) = \mathfrak{g}_a$
\cite[13.18(4d)]{borelag}.  This is the
{\em root group} associated to $a$.
Explicitly, by Exercise \ref{gmga}(iv) and the $T$-equivariant identification
${\rm{Lie}}(U_a) \simeq \mathfrak{g}_a$, the $T$-action on $U_a \simeq \Add$
is $t.x = a(t)x$ (so $T \cap U_a \subset U_a^T = 1$ as $k$-schemes). 

\begin{example}\label{rootsl2} For $G = {\rm{SL}}_2$ and $T = D$ the diagonal torus,
the root groups are the strictly upper and lower triangular unipotent subgroups $U^{\pm}$.
The same holds for ${\rm{PGL}}_2$ and its diagonal torus $\overline{D}$,
using the strictly upper and lower triangular unipotent subgroups $\overline{U}^{\pm}$.
\end{example}

The following lemma will turn out to be a generalization of the classical fact
that ${\rm{SL}}_2$ is generated by the root groups $U^{\pm}$.

\begin{lemma}\label{dgt}
Let $T_a = (\ker a)^0_{\rm{red}}$
be the unique codimension-$1$ torus in $T$ killed by $a \in \Phi$.
The root groups $U_a$ and $U_{-a}$ generate $G_a := \mathscr{D}(Z_G(T_a))$,
and $G_a$ is a closed subgroup admitting ${\rm{PGL}}_2$ as an isogenous quotient.
\end{lemma}

\begin{proof}
Fix an isomorphism $u_{\pm a}:\Add \simeq U_{\pm a}$, so 
$t u_{\pm a}(x)t^{-1} = u_{\pm a}(a(t)^{\pm 1}x)$ for $t \in T$,
by consideration of the Lie algebra and Exercise \ref{gmga}(iv). 
Hence, $T_a$ must centralize $U_{\pm a}$, so 
these root groups lie in $Z_G(T_a)$.
The group $Z_G(T_a)$ is a connected reductive subgroup of $G$ in which
the maximal torus $T$ contains the central subtorus $T_a$ of codimension 1.
The Lie algebra ${\rm{Lie}}(Z_G(T_a))$ is equal to $\mathfrak{g}^{T_a}$
\cite[9.4]{borelag}, and this in turn 
is equal to $\mathfrak{t} \oplus \mathfrak{g}_a \oplus \mathfrak{g}_{-a}$
\cite[13.18(4a)]{borelag}.  In particular, $T$ is noncentral in $Z_G(T_a)$
since its adjoint action on ${\rm{Lie}}(Z_G(T_a))$ is nontrivial, 
so $T_a$ is the maximal central torus in $Z_G(T_a)$ (and hence
by Example \ref{cisog} it coincides
with $\mathscr{R}(Z_G(T_a))$).  We conclude that the quotient $Z_G(T_a)/T_a$ is semisimple
with the 1-dimensional $T/T_a$ as a maximal torus.  Equivalently,
the isogenous $G_a$ is semisimple
with a 1-dimensional maximal torus $T'_a = (T \bigcap G_a)^0_{\rm{red}}$ 
(an isogeny-complement to $T_a$ in $T$);
since $T$ is maximal in $Z_G(T_a)$, the maximality of
$T'_a$ in $G_a$ is a special case of Exercise \ref{normaltorus}.   

Since the only nontrivial $T$-weights on ${\rm{Lie}}(Z_G(T_a))$ 
are $\pm a$, the semisimple
$G_a$ must have Lie algebra $\mathfrak{t}'_a \oplus \mathfrak{g}_a \oplus \mathfrak{g}_{-a}$,
with $\mathfrak{t}'_a := {\rm{Lie}}(T'_a)$. 
Thus, $G_a$ is 3-dimensional.  By \cite[13.13(5)]{borelag},
there exists an isogeny $G_a \rightarrow {\rm{PGL}}_2$,
and by conjugacy of maximal tori it can be arranged to carry
$T'_a$ onto the diagonal torus $\overline{D}$.
Thus, this isogeny carries $U_{\pm a}$ onto the root
groups $\overline{U}^{\pm}$ for $\overline{D}$.
But the pair of subgroups $\overline{U}^{\pm}$ visibly generates ${\rm{PGL}}_2$
(since the subgroups $U^{\pm}$ generate ${\rm{SL}}_2$), so we conclude
that indeed $U_{\pm a}$ generate $G_a$.
\end{proof}

We wish to introduce coroots:  to each root $a \in \Phi$  we will attach a canonical 
nontrivial cocharacter
\index{coroot}
$a^{\vee}:\Gm \rightarrow T$ that generalizes the
cocharacter $c \mapsto {\rm{diag}}(c,1/c)$ in
${\rm{SL}}_2$ attached to the root ${\rm{diag}}(c,1/c) \mapsto c^2$
for $D$. The definition of coroots rests 
on the classification of semisimple $k$-groups of rank 1.
This classification
is the assertion that any such group $G$ is isomorphic to
either ${\rm{SL}}_2$ or ${\rm{PGL}}_2$.
In other words, there exists an isogeny 
${\rm{SL}}_2 \rightarrow G$ whose kernel is contained in the central $\mu_2$.
When combined with Lemma \ref{dgt}, this provides interesting homomorphisms from
${\rm{SL}}_2$ into nontrivial connected semisimple groups. 
Here is the statement in the form that we will need. 

\begin{theorem}\label{rtgp} For each $a \in \Phi(G,T)$,
there exists a homomorphism
$\varphi_a:{\rm{SL}}_2 \rightarrow G$
carrying the diagonal torus $D$ into $T$ and 
the strictly upper triangular and strictly lower triangular unipotent
subgroups $U^{\pm}$ isomorphically onto the respective root groups $U_{\pm a}$.

Such a homomorphism $\varphi_a$ 
is an isogeny onto $G_a$ with $\ker \varphi_a \subset \mu_2$,
it is unique up to $T(k)$-conjugation on $G$,
and it carries the standard Weyl element
$w = (\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix})$
to an element $n_a \in N_{G(k)}(T) - T(k)$. 
\end{theorem}

\begin{proof}
Let $T'_a = (T \bigcap G_a)^0_{\rm{red}}$, 
a maximal torus of $G_a$ that is an isogeny complement to $T_a$ in $T$.
Assume there is a homomorphism $\varphi_a$
with the desired properties on $D$ and $U^{\pm}$. 
By Lemma \ref{dgt} we know $U_{\pm a} \subset G_a$,
so the classical fact that $U^{\pm}(k)$ generate
${\rm{SL}}_2(k)$ implies that $\varphi_a$ must land
inside $G_a$. Likewise,
$\varphi_a(D)$ must be contained in
$(T \bigcap G_a)_{\rm{red}}^0 = T'_a$.
Thus, we can replace $(G,T)$ with $(G_a,T'_a)$.

Now $G$ is semisimple with a 1-dimensional maximal torus $T$
and $\Phi(G,T) = \{\pm a\}$.  In particular,
$\mathfrak{g} = \mathfrak{t} \oplus \mathfrak{g}_a \oplus \mathfrak{g}_{-a}$
is 3-dimensional, so $\dim G = 3$.
The key point is to show that $G$ is isomorphic to
either ${\rm{SL}}_2$ or ${\rm{PGL}}_2$.  This 
is not proved in \cite{borelag} (and correspondingly, coroots are not discussed
in \cite{borelag}), so we provide a proof below. 

{\bf Step 1}.  We first show that $G$ admits an isogeny onto
${\rm{PGL}}_2$ with scheme-theoretic kernel $Z$ that is isomorphic
to 1 or $\mu_2$ and is scheme-theoretically central in $G$
(i.e., $Z(R)$ is central in $G(R)$ for every $k$-algebra $R$).
Note that the case ${\rm{char}}(k) = 2$ is  ``non-classical'' since $\mu_2$ is non-reduced
for such $k$, but our arguments
will be characteristic-free.

Since $T \cap U_a = 1$ as $k$-schemes inside $G$, the map $B := T \ltimes U_a \rightarrow G$
is a closed $k$-subgroup (see Proposition \ref{propker}). 
By dimension considerations, $B$ is 
a Borel subgroup of $G$ containing $T$, 
and another is $B' := T \ltimes U_{-a}$.
The multiplication map
\begin{equation}\label{uab}
\mu:U_{-a} \times B = U_{-a} \times T \times U_a \rightarrow G
\end{equation}
is \'etale at the identity due to the tangential criterion, so (by left $U_{-a}(k)$-translation
and right $B(k)$-translation) it is \'etale everywhere.
Since the closed $k$-subgroup  scheme
$U_{-a} \bigcap B$ is \'etale (due to transversality: $\mathfrak{u}_{-a} \bigcap \mathfrak{b} = 0$) 
and normalized
by $T$ (see the $T$-equivariant description of root groups at the start of the proof
of Lemma \ref{dgt}), yet  the $k$-group $U_{-a} \simeq \Add$ clearly contains
no nontrivial finite \'etale subgroups normalized by $T$, it follows
that the \'etale map $\mu$ is injective on $k$-points.
Thus, $\mu$ is \'etale and radiciel, hence (by \cite[IV$_4$, 17.9.1]{ega}) an isomorphism onto its open image $\Omega$; i.e., $\mu$ 
is an open immersion.  We conclude  that $B \bigcap B' = T$ scheme-theoretically.

Since $G$ is semisimple of rank 1, $G/B \simeq \mathbf{P}^1$ \cite[13.13(4)]{borelag}.
The left translation action of $G$ on $G/B$ then defines
a $k$-homomorphism to the automorphism scheme 
$$f:G \rightarrow {{\rm{Aut}}}_{\mathbf{P}^1_k/k} = {\rm{PGL}}_2$$
(see Exercise \ref{1.1}(iv),(v)) whose scheme-theoretic
kernel $K$ is a normal subgroup scheme that is contained in $B$.
We shall now prove that $K = \ker a$ and that this is a finite central subgroup scheme of $G$
(so $f$ is an isogeny with central kernel).
By normality $K$ is contained in the $G(k)$-conjugate $B'$ of $B$, so $K \subset B \bigcap B' = T$
as closed subschemes of $G$.
The left translation action by $T$ on $G$ preserves
the open subscheme $\Omega = U_{-a} \times T \times U_a$
via the formula
$$t.(u_{-} t' u_{+}) = (tu_{-}t^{-1})(tt')u_{+},$$
so the left $T$-action on $G/B$ preserves the open subscheme
$\Omega/B = U_{-a} \simeq \Add$ on which it acts via
scaling through $-a:T \rightarrow \Gm$.  Hence,
$K$ must be contained in $\ker a$.
But the group scheme $\ker a$ 
visibly centralizes the dense open subscheme $\Omega$ in 
the smooth group $G$, so it centralizes $G$.  Since $\ker a \subset T \subset B$,
we obtain the reverse inclusion $\ker a \subset K$.

To summarize, we have built a short exact sequence of group schemes
\begin{equation}\label{zext}
1 \rightarrow Z \rightarrow G \stackrel{f}{\rightarrow} {\rm{PGL}}_2 \rightarrow 1
\end{equation}
with $Z = \ker a$ a finite subgroup scheme of $T = \Gm$ that is central in $G$.
In particular, $Z \simeq \mu_n$ for some $n \ge 1$.
The Weyl group $W_G(T) := N_{G(k)}(T)/T(k)$ has order 2 \cite[13.13(2)]{borelag}, so
conjugation by a representative $n \in N_{G(k)}(T)$ of the nontrivial element
in $W_G(T)$ acts on $T = \Gm$ by its only nontrivial automorphism, namely
inversion.  But such conjugation must be trivial on
the {\em central} subgroup scheme $Z$ in $G$, so inversion on $\mu_n$ is trivial.
This forces $n|2$.   If $n = 1$ then $f$ is an isomorphism,
and if $n = 2$ then the conjugation action by
$T = \Gm$ on $U_{\pm a} = \Add$ must be scaling
by the only two characters of $\Gm$ with kernel $\mu_2$, namely
$t \mapsto t^{\pm 2}$.

{\bf Step 2}.  Now
we relate $G$ to ${\rm{SL}}_2$
in case $Z = \mu_2$, and in general we adjust
$f$ so that it relates
$T$ and $U_{\pm a}$
to $\overline{D}$ and $\overline{U}^{\pm}$ respectively.
The isogeny $f$ must carry $T$ onto a maximal torus of ${\rm{PGL}}_2$,
so by composing $f$ with a suitable conjugation we can arrange that $f(T)$ is the
diagonal torus $\overline{D}$.  It then follows that $T = f^{-1}(\overline{D})$
scheme-theoretically because $Z \subset T$. 

Since the $k$-subgroup $Z = \ker a$ has trivial scheme-theoretic intersection with 
the root groups $U_{\pm a}$ (as even $U_{\pm a} \bigcap T = 1$), these root groups 
are carried {\em isomorphically} by $f$ onto their images in 
${\rm{PGL}}_2$.  By the unique characterization of root groups
\cite[13.18(4d)]{borelag}, 
it follows that (as unordered pairs) $\{f(U_a), f(U_{-a})\}$
is the set of root groups $\{\overline{U}^{+}, \overline{U}^{-}\}$ for $\overline{D}$, so
by composing $f$ with conjugation by the standard Weyl element of
$({\rm{PGL}}_2, \overline{D})$ if necessary
we can arrange that $f(U_a) = \overline{U}^+$ and
$f(U_{-a}) = \overline{U}^{-}$.

Suppose $Z = \mu_2$. There is a unique isomorphism
$T \simeq \Gm$ carrying the degree-2 isogeny $a:T \rightarrow \Gm$
over to the map $t^2:\Gm \rightarrow \Gm$.  Combining this with 
the isomorphisms
$U_{\pm a} \simeq \overline{U}^{\pm} = \Add$ arising from $f$ 
and the natural isomorphisms $U^{\pm} \simeq \overline{U}^{\pm}$, 
we identify the open subscheme $\Omega = U_{-a} \times T \times U_a$
in $G$ with the standard open subscheme
\begin{equation}\label{explicitcell}
\Add \times \Gm \times \Add \simeq U^{-} \times D \times U^{+} \subset {\rm{SL}}_2
\end{equation}
where the isomorphism (\ref{explicitcell}) is defined by
$$(x',c,x) \mapsto \begin{pmatrix} 1 & 0 \\ x' & 1 \end{pmatrix}
\begin{pmatrix} c & 0 \\ 0 & 1/c \end{pmatrix} \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}.$$

Continuing to assume $Z = \mu_2$, let $V \subset \Omega \times \Omega$ be the dense open locus of points
$(\omega, \omega')$ such that $\omega \omega' \in \Omega$ inside $G$. 
We claim that the 
open immersion $j:\Omega \simeq U^{-} \times D \times U^{+} \hookrightarrow {\rm{SL}}_2$
is a ``birational homomorphism'' in the sense that
$j(\omega \omega') = j(\omega)j(\omega')$
for $(\omega, \omega') \in V$. 
Since the composition of $j$ with the canonical isogeny $q:{\rm{SL}}_2 \rightarrow {\rm{PGL}}_2$
is a homomorphism (namely, $f$), 
the map $V \rightarrow {\rm{SL}}_2$
defined by $j(\omega\omega')j(\omega')^{-1}j(\omega)^{-1}$
factors through $\ker q = \mu_2$ and so 
is identically 1.  Hence, indeed $j$ is a birational homomorphism, so it extends uniquely to an isomorphism of $k$-groups
(Exercise \ref{trans})!

Allowing either possibility for $Z$, $f:G \rightarrow {\rm{PGL}}_2$ is either
an isomorphism carrying $T$ onto $\overline{D}$ and carrying $U_{\pm a}$ onto
$\overline{U}^{\pm}$ or else it factors through $q:{\rm{SL}}_2 \rightarrow {\rm{PGL}}_2$
via an isomorphism $G \simeq {\rm{SL}}_2$ carrying
$T$ onto $D$ and carrying $U_{\pm a}$ onto $U^{\pm}$.
Either way, there is a unique homomorphism $\varphi:{\rm{SL}}_2 \rightarrow G$
factoring $q$ through $f$, and $\varphi$ satisfies the desired
properties to be $\varphi_a$
except possibly uniqueness up to $T(k)$-conjugation (e.g., $\varphi(D) = T$
since $T = f^{-1}(\overline{D})$). 

{\bf Step 3}.
Finally, we prove the uniqueness of $\varphi_a$
up to $T(k)$-conjugation.  First suppose $f$ is an isomorphism.
By using composition with $f$,
it suffices to show that the only homomorphisms
$\pi:{\rm{SL}}_2 \rightarrow {\rm{PGL}}_2$
carrying $D$ into $\overline{D}$ and
$U^{\pm}$ isomorphically onto
$\overline{U}^{\pm}$ respectively are
$\overline{D}(k)$-conjugates of $q$.
Since the roots for ${\rm{PGL}}_2$ have trivial kernel
whereas the roots for ${\rm{SL}}_2$ have kernel
equal to the central $D[2] = \mu_2$, it follows
from the isomorphism condition on root groups 
that any such $\pi$ must kill $D[2]$ and so factors through
$q$.  In other words, to prove the uniqueness
of $\pi$ up to $\overline{D}(k)$-conjugation it
suffices to treat the analogous
assertion for endomorphisms $\overline{\pi}:
{\rm{PGL}}_2 \rightarrow {\rm{PGL}}_2$
that satisfy $\overline{\pi}(\overline{D}) = \overline{D}$
and $\overline{\pi}:\overline{U}^{\pm} \simeq \overline{U}^{\pm}$.

Since ${\rm{Aut}}(\Add) = k^{\times}$ and $\overline{\pi}$ carries
$\overline{U}^{+} = \Add$ isomorphically
onto $\overline{U}^{+}$, by composing
with a $\overline{D}(k)$-conjugation (which makes
${\rm{diag}}(t,1) \in \overline{D}(k)$ act on
$\overline{U}^+ = \mathbf{G}_{\rm{a}}$ via $t.x = tx$) we may arrange
that $\overline{\pi}$ is the {\em identity} map on $\overline{U}^+$.
By hypothesis $\overline{\pi}$ carries $\overline{D}$ into $\overline{D}$,
so the faithfulness of the $\overline{D}$-action on
$\overline{U}^+ = \Add$ implies that $\overline{\pi}$ restricts
to the identity on $\overline{D}$.  We will prove that
$\overline{\pi}$ restricts to the identity map on
the dense open $\overline{\Omega} :=
\overline{U}^{-} \times \overline{D} \times \overline{U}^{+}$,
thereby forcing $\overline{\pi}$ to be the identity map.

The restriction of $\overline{\pi}$ to the dense open direct product subscheme $\overline{\Omega} \subset
{\rm{PGL}}_2$
is visibly an automorphism of $\overline{\Omega}$,
so $\ker \overline{\pi}$ is \'etale.  But an \'etale closed
normal subgroup of a connected linear algebraic group is central,
yet ${\rm{PGL}}_2$ has no nontrivial central finite \'etale
subgroup (as $\overline{D}$ is its own centralizer
on $k$-points and acts faithfully under conjugation on
$\overline{U}^{+}$), so $\ker \overline{\pi} = 1$.  Thus, 
$\overline{\pi}$ is an isomorphism.
In particular, $\overline{\pi}^{-1}(\overline{\Omega}) = \overline{\Omega}$.

Let $\overline{u}_{\pm}:\Add \simeq \overline{U}^{\pm}$ be the 
parameterizations $x \mapsto (\begin{smallmatrix} 1 & x \\ 0 & 1 \end{smallmatrix})$
and $x \mapsto (\begin{smallmatrix} 1 & 0 \\ x & 1 \end{smallmatrix})$, so the calculations
\begin{eqnarray*}
\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}\begin{pmatrix}1 & 0 \\ y & 1 \end{pmatrix} &=&
\begin{pmatrix} 1+xy & x \\ y & 1 \end{pmatrix},\\
\begin{pmatrix} 1 & 0 \\ x' & 1 \end{pmatrix}\begin{pmatrix} a & 0 \\ 0 & a^{-1} \end{pmatrix} \begin{pmatrix} 1 & y' \\ 0 & 1 \end{pmatrix}
&=& \begin{pmatrix} a & ay'\\ ax' & ax'y' + a^{-1} \end{pmatrix}
\end{eqnarray*}
imply that the product $\overline{u}_{+}(x)\overline{u}_{-}(y)$ lies in $\overline{\Omega} = 
\overline{U}^{-} \times \overline{D} \times \overline{U}^{+}$
if and only if $1 + xy \in \Gm$.
The restriction $\overline{\pi}:\overline{U}^{-} \simeq \overline{U}^{-}$
corresponds to an automorphism of $\Add$, which is
to say $\overline{\pi}(\overline{u}_{-}(y)) = \overline{u}_{-}(cy)$
for some $c \in k^{\times}$.  Thus,
$\overline{\pi}(\overline{u}_{+}(x)\overline{u}_{-}(y)) = 
\overline{u}_{+}(x)\overline{u}_{-}(cy)$, 
so the equality $\overline{\pi}^{-1}(\overline{\Omega}) = \overline{\Omega}$
implies that $1 + xy \in \Gm$ if and only if $1 + x\cdot cy \in \Gm$.
It follows that $c = 1$, so $\overline{\pi}$ is the identity on
$\overline{U}^{-}$ and we are done when $f$ is an isomorphism.

Suppose instead that $f$ is not an isomorphism, so (as we have seen above)
$(G, T) \simeq ({\rm{SL}}_2, D)$ carrying $U_{\pm a}$ to $U^{\pm}$ respectively. 
It therefore suffices to show that the only endomorphisms $\varphi$ of ${\rm{SL}}_2$ carrying each of
$D, U^{+}, U^{-}$ into themselves are conjugation by elements of $D(k)$. 
By the uniqueness established above, $q \circ \varphi:{\rm{SL}}_2 \rightarrow {\rm{PGL}}_2$
is the composition of $q$ with conjugation against some $\overline{d} \in \overline{D}(k)$. Hence, if
$d \in D(k)$ lifts $\overline{d}$ then $g \mapsto \varphi(g)(dgd^{-1})^{-1}$
is a scheme morphism from the smooth connected ${\rm{SL}}_2$ into $\mu_2$ and thus
is the trivial map $g \mapsto 1$.  That is, necessarily $\varphi(g) = dgd^{-1}$ for all $g$, as desired. 
\end{proof}

Using any $\varphi_a$ as in Theorem \ref{rtgp}, the cocharacter 
$$a^{\vee}:\Gm \rightarrow D \rightarrow T$$
defined by $a^{\vee}(c) = \varphi_a({\rm{diag}}(c,1/c))$ is unaffected
by $T(k)$-conjugation on $G$, so it is intrinsic.
\index{ aaaav@$a^{\vee}$}
\index{coroot}
\index{ aaafigtv@$\Phi(G,T)^{\vee}$}

\begin{definition} The {\em coroot}
associated to $(G,T,a)$ is
the cocharacter $a^{\vee} \in {\rm{X}}_{\ast}(T) - \{0\}$.
The finite subset of coroots in ${\rm{X}}_{\ast}(T)$ is denoted
$\Phi^{\vee}$.
\end{definition}

Concretely, 
$a^{\vee}$ is a parameterization (with kernel 1 or $\mu_2$)
of the 1-dimensional torus 
$(T \bigcap \mathscr{D}(Z_G(T_a)))_{\rm{red}}^0$
that is an isogeny complement to $T_a$ in $T$. 
The composition of any $\varphi_a$
with transpose-inverse on ${\rm{SL}}_2$ satisfies
the requirements to be $\varphi_{-a}$.
Since transpose-inverse acts by inversion of the diagonal torus
of ${\rm{SL}}_2$, we conclude that $(-a)^{\vee} = -a^{\vee}$.

\begin{example}\label{sl2coroot}
Suppose $G = {\rm{SL}}_2$ and $T$ is the diagonal torus $D$.
The roots for $(G,T)$ are ${\rm{diag}}(c,1/c) \mapsto c^{\pm 2}$.
Let $a$ be the root ${\rm{diag}}(c,1/c) \mapsto c^2$, so
$\mathfrak{g}_a$ is the subspace of upper triangular
nilpotent matrices in $\mathfrak{sl}_2$
and $\mathfrak{g}_{-a}$ is the subspace of lower triangular
nilpotent matrices in $\mathfrak{sl}_2$.
By Example \ref{rootsl2}, the corresponding
root groups are $U_a = U^{+}$ and $U_{-a} = U^{-}$, 
so $\varphi_a$ can be taken to be the identity map.
In particular, $a^{\vee}(c) = {\rm{diag}}(c,1/c)$.
Note that $\langle a, a^{\vee} \rangle = 2$ in
${\rm{End}}(\Gm) = \Z$ (i.e., $a(a^{\vee}(c)) = c^2$): this follows from the calculation
$$\begin{pmatrix} c & 0 \\ 0 & 1/c \end{pmatrix}
\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} c & 0 \\ 0 & 1/c \end{pmatrix}^{-1} = 
\begin{pmatrix} 1 & c^2 x \\ 0 & 1 \end{pmatrix},$$
which implies that the adjoint action of $a^{\vee}(c)$ on
$\mathfrak{g}_a = {\rm{Lie}}(U_a)$ is scaling by $c^2$.
Observe also that in $\mathfrak{t}$ we have 
$${\rm{Lie}}(a^{\vee})(t\partial_t) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = 
H_a = [E_a, F_a]$$
(where $t$ is the standard coordinate on $\Gm \simeq D$ via the 
isomorphism $c \mapsto {\rm{diag}}(c,1/c)$). 

For $G = {\rm{PGL}}_2$ and $T = \overline{D}$,
we have an isomorphism $\Gm \simeq T$ via $c \mapsto {\rm{diag}}(c,1) \bmod \Gm$.
The inverse isomorphism $\overline{a}:T \simeq \Gm$ is a root whose
root space consists of the upper triangular nilpotent
matrices in $\mathfrak{pgl}_2 = \mathfrak{gl}_2/\mathfrak{gl}_1$, 
and similarly for the root $-\overline{a}$ using lower triangular nilpotent matrices.  Thus, 
$\varphi_{\overline{a}}:{\rm{SL}}_2 \rightarrow G$ can be taken to be the canonical
projection.  In particular, $\overline{a}^{\vee}:\Gm \rightarrow T$ is
$c \mapsto {\rm{diag}}(c,1/c) \bmod \Gm = {\rm{diag}}(c^2,1) \bmod \Gm$,
so $\langle \overline{a}, \overline{a}^{\vee} \rangle = 2$.
\end{example}

\begin{remark}
It follows from Example \ref{sl2coroot} (for both
${\rm{SL}}_2$ and ${\rm{PGL}}_2$) that in general $\langle a, a^{\vee} \rangle = 2$
for any connected reductive $k$-group $G$ and maximal torus $T \subset G$. 
\end{remark}

\begin{example}\label{glncoroot}
Let $G = \GL(V) = \GL_n$,
and take $T$ to be the diagonal torus $D_n$.
For the root $a = e_i - e_j$ ($1 \le i \ne j \le n$),
the associated codimension-1 subtorus 
$T_a$ is defined by the equality $\{c_i = c_j\}$
between $i$th and $j$th diagonal entries.
The centralizer $Z_G(T_a)$ consists
of elements of $\GL_n$ whose off-diagonal entries vanish away from
the $ij$ and $ji$ positions,
and the subgroup $G_a := \mathscr{D}(Z_G(T_a))$ is
${\rm{SL}}(ke_i \oplus k e_j) = {\rm{SL}}_2$.  
In particular, $a^{\vee} = e_i^{\vee} - e_j^{\vee}$.
This makes explicit that $\langle a, a^{\vee}\rangle = 2$ for all $a \in \Phi(\GL_n,D_n)$.
\end{example}

\subsection{Root datum, root system, and classification theorem}

Let $G$ be a connected reductive $k$-group, and let
$T \subset G$ be a maximal torus.  In the dual lattices
$X = {\rm{X}}(T)$ and $X^{\vee} = {\rm{X}}_{\ast}(T)$
we have defined finite subsets
$\Phi = \Phi(G,T) \subset X - \{0\}$
and $\Phi^{\vee} = \Phi(G,T)^{\vee} \subset X^{\vee} - \{0\}$
\index{ aaaxxfifi@$({\rm{X}}, \Phi, {\rm{X}}^{\vee}, \Phi^{\vee})$}
and a bijection
$a \mapsto a^{\vee}$ between $\Phi$ and $\Phi^{\vee}$
such that (i) $\langle a, a^{\vee} \rangle = 2$ for all
$a$, (ii) for each $a \in \Phi$, $\Phi \bigcap \Q a = \{\pm a\}$ inside 
$X_{\Q}$.  

There are additional properties
satisfied by this combinatorial data.  To formulate
them, we introduce some reflections.  
(If $V$ is a nonzero finite-dimensional vector space
over a field of characteristic 0, a {\em reflection} $r:V \rightarrow V$
is an automorphism with order 2 such that $-1$ occurs
as an eigenvalue with multiplicity one, or equivalently
such that $V^{r=1}$ is a hyperplane.) 
\index{reflection}
For each 
$a \in \Phi$, define the linear endomorphisms
$s_a:X \rightarrow X$ and
$s_{a^{\vee}}:X^{\vee} \rightarrow X^{\vee}$
by \index{ aaasasav@$s_a$, $s_{a^{\vee}}$}
\begin{equation}\label{refl}
s_a(x) = x - \langle x, a^{\vee}\rangle a,\,\,\,
s_{a^{\vee}}(\lambda) = \lambda - \langle a, \lambda \rangle a^{\vee}.
\end{equation}
Since $\langle a, a^{\vee} \rangle = 2$,
it is easy to check that 
$s_a(a) = -a$ and $s_{a^{\vee}}(a^{\vee}) = -a^{\vee}$.
Clearly $s_a$ fixes pointwise the hyperplane $\ker a^{\vee} \subset X_{\Q}$ complementary to $\Q a$,
and similarly for $s_{a^{\vee}}$ and $\ker a \subset X_{\Q}^{\vee}$.  
Moreover, $s_a^2 = 1$ since 
$$s_a^2(x) = x - \langle x, a^{\vee} \rangle a - \langle x - \langle x, a^{\vee} \rangle a, a^{\vee} \rangle a = 
x - 2\langle x, a^{\vee} \rangle a + \langle a, a^{\vee} \rangle \langle x, a^{\vee} \rangle a = x,$$
and similarly $s_{a^{\vee}}^2 = 1$, 
so on $X_{\Q}$ and $X^{\vee}_{\Q}$
the automorphisms $s_a$ and $s_{a^{\vee}}$ are reflections
in the lines spanned by $a$ and $a^{\vee}$ respectively.
It is also easy to check that $s_{a^{\vee}}$ is dual to $s_a$;
this amounts to the identity
$$\langle x - \langle x, a^{\vee} \rangle a, \lambda \rangle = \langle x, \lambda - \langle a, \lambda \rangle a^{\vee} \rangle.$$ 
(Some introductory accounts of the theory of root systems impose
a Euclidean structure on $X_{\Q}$ or $X_{\RRR}$ at the outset, such as in \cite[III]{humlie},
but this is not necessary.  To that end, note that we have not imposed any positive-definite
quadratic form on $X_{\Q}$ or $X_{\RRR}$.)


\begin{example}
Let $G = \GL_n$
and let $T$ be the diagonal torus.
For $1 \le i \ne j \le n$ and $1 \le h \le n$, we have 
$s_{e_i - e_j}(e_h) = e_h$ when $h \ne i, j$
and the reflection $s_{e_i - e_j}$ swaps $e_i$ and $e_j$.
In particular, $s_a(\Phi) = \Phi$ for all $a \in \Phi$.

For $n=2$ this amounts to the fact that the standard Weyl element
$w = (\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix})$
acts on the diagonal torus of ${\rm{SL}}_2$ via inversion.
\end{example}

The importance of the reflections $s_a$ and $s_{a^{\vee}}$ in general is:

\begin{proposition}\label{saprop} For all $a \in \Phi$, 
$s_a(\Phi) = \Phi$ and $s_{a^{\vee}}(\Phi^{\vee}) = \Phi^{\vee}$.
\end{proposition}

\begin{proof}
Visibly $T_a \cap T'_a$ is finite (as $T'_a$ is 1-dimensional and $a|_{T'_a} \ne 1$), so 
$T'_a \times T_a \rightarrow T$ is an isogeny.  This 
identifies ${\rm{X}}(T)_{\Q}$ with ${\rm{X}}(T'_a)_{\Q} \oplus
{\rm{X}}(T_a)_{\Q}$, carrying ${\rm{X}}(T_a)_{\Q}$ onto the hyperplane
spanned over $\Q$ by the characters
that kill $T'_a = a^{\vee}(\Gm)$ and carrying ${\rm{X}}(T'_a)_{\Q}$ onto the line
spanned over $\Q$ by the characters that kill $T_a$ (i.e.,
it is the $\Q$-span of $a$).

Let $n_a \in G_a = \mathscr{D}(Z_G(T_a))$
be a representative in $N_{G_a(k)}(T'_a)$ for the nontrivial element
in the group $W_{G_a}(T'_a)$ of order 2.  The conjugation action by
$n_a$ on the almost direct product $T = T'_a \cdot T_a$
is trivial on $T_a$ and inversion on $T'_a = a^{\vee}(\Gm)$,
so $n_a$ acts on ${\rm{X}}(T)$ as an involution
whose effect on ${\rm{X}}(T)_{\Q}$ negates the line
spanned by $a$ and fixes pointwise
the hyperplane ${\rm{X}}(T_a)_{\Q}$.  

We conclude that 
$n_a$ acts on ${\rm{X}}(T)$ as a reflection $r_a$.
This reflection 
visibly preserves $\Phi$ (as does the effect of any element of $N_{G(k)}(T)$),
and we claim that $r_a = s_a$ (so $s_a(\Phi) = \Phi$). 
The reflections $r_a$ and $s_a$ negate the same line $\Q a$, so it suffices
to show that they restrict to the identity on a common
hyperplane, namely ${\rm{X}}(T_a)_{\Q}$.
But this hyperplane is the annihilator of 
$\Q a^{\vee} \subset {\rm{X}}_{\ast}(T)_{\Q}$,
so the definition of $s_a$ makes it clear
that $s_a$ is the identity on ${\rm{X}}(T_a)_{\Q}$.

The equality $r_a = s_a$ implies that the dual reflection
$s_a^{\vee} = s_{a^{\vee}}$ on ${\rm{X}}_{\ast}(T)$ is also induced by 
$n_a$-conjugation on $T$.  Thus,
$s_{a^{\vee}}$ preserves $\Phi^{\vee}$.
\end{proof}

We have shown that $R(G,T) := ({\rm{X}}(T), \Phi(G,T), {\rm{X}}_{\ast}(T), \Phi(G,T)^{\vee})$
\index{ aaargt@$R(G,T)$}
satisfies the requirements in the following definition
introduced in \cite[XXI]{sga3}:

\begin{definition}[Demazure]
A {\em root datum} is a 4-tuple
\index{root datum}
$(X,\Phi, X^{\vee}, \Phi^{\vee})$ consisting
of a pair of finite free $\Z$-modules $X$ and $X^{\vee}$
equipped with 
\begin{itemize}
\item a perfect duality
$\langle \cdot, \cdot \rangle:X \times X^{\vee} \rightarrow \Z$,
\item finite subsets $\Phi \subset X - \{0\}$
and $\Phi^{\vee} \subset X^{\vee} - \{0\}$
stable under negation 
\end{itemize}
for which there exists a bijection $a \mapsto a^{\vee}$
from $\Phi$ to $\Phi^{\vee}$
such that $\langle a, a^{\vee} \rangle = 2$
and the resulting reflections
$s_a:X \simeq X$ and $s_{a^{\vee}}:X^{\vee} \simeq X^{\vee}$
as in (\ref{refl}) 
satisfy $s_a(\Phi) = \Phi$ and $s_{a^{\vee}}(\Phi^{\vee}) = \Phi^{\vee}$.
If moreover $\Q a \bigcap \Phi = \{\pm a\}$ inside $X_{\Q}$ for
all $a \in \Phi$
then the root datum is {\em reduced}.
\index{root datum!reduced}
\end{definition}

\begin{remark}\label{rem324}
The bijection $a \mapsto a^{\vee}$ in the definition 
of a root datum is uniquely determined.
For a proof, see \cite[Lemma 3.2.4]{pred}.
\end{remark}

\begin{remark}\label{remdual} It is immediate from the axioms
that if $(X,\Phi,X^{\vee},\Phi^{\vee})$ is a root datum
then so is $(X^{\vee},\Phi^{\vee},X,\Phi)$. This is
called the {\em dual} root datum.
\index{root datum!dual}
\end{remark}

In the study of connected semisimple $k$-groups ``up
to central isogeny'' (see Exercise \ref{scclassical}), it is convenient to work with a coarser
notion than a root datum, in which we relax the
$\Z$-structure to a $\Q$-structure and remove the explicit mention of
the coroots.  This leads to the notion of a {\em root system}
\index{root system}
(which historically arose much earlier than the notion of a root datum,
in the classification of semisimple Lie algebras over $\C$,
and is extensively studied in \cite[Ch.\,VI]{lie4}): this is 
a pair $(V,\Phi)$ consisting of a finite-dimensional
$\Q$-vector space $V$ and a finite spanning
set $\Phi \subset V - \{0\}$ such that for each
$a \in \Phi$ there exists a reflection $s_a:v \mapsto v - \lambda(v)a$
with $\lambda \in V^{\ast}$ such that $s_a(\Phi) = \Phi$, 
$s_a(a) = -a$ (equivalently, $\lambda(a)=2$), 
and $\lambda(\Phi) \subset \Z$.  Such a reflection is {\em unique} (even
without the integrality condition on $\lambda$) because if $s'$ is another then by inspecting the effects
on $\Q a$ and $V/\Q a$ we see that 
$s' \circ s^{-1}$ is a unipotent automorphism of $V$ yet it preserves the
{\em finite} spanning set $\Phi$ and hence has finite order, forcing $s' \circ s^{-1} = 1$
(as ${\rm{char}}(\Q)=0$).


\begin{example} If $(X,\Phi, X^{\vee}, \Phi^{\vee})$ is a root datum
then the $\Q$-span $V$ of $\Phi$ in $X_{\Q}$
equipped with the subset $\Phi$ is a root system.
\end{example}

The difference between
root systems and root data is analogous
to the difference between connected semisimple $k$-groups
considered up to central isogeny and connected reductive $k$-groups
considered up to isomorphism:  the possible failure
of $\Phi$ to span $X_{\Q}$ is analogous to the possibility
that a connected reductive group may have a nontrivial central torus
(i.e., fail to be semisimple), 
and the use of $\Q$-structures rather than $\Z$-structures amounts to considering
groups up to  central isogeny.  

More explicitly, if $(G,T)$ is a connected reductive
$k$-group equipped with a maximal torus $T$, then
the saturation in ${\rm{X}}(T)$ of the $\Z$-span $\Z \Phi$ of $\Phi$ 
is ${\rm{X}}(T/Z)$ where $Z$ is the maximal central torus of $G$
(since this saturation is
${\rm{X}}(T/\mathscr{T})$ for the largest torus
$\mathscr{T}$ killed by $\Phi$, and a torus $T'$ in $G$ is central if and only if
$a(T') = 1$ for all $a \in \Phi$ \cite[14.2(1)]{borelag}). 
Thus, $\Phi$ spans ${\rm{X}}(T)_{\Q}$ if and only if
$G$ has no nontrivial central torus, which is to say that $G$ is semisimple.

\begin{remark} If a root datum is not
reduced then $\Q a \bigcap \Phi$ equals $\{\pm a\}$, 
$\{\pm a, \pm 2a\}$ (the {\em multipliable} case), 
or $\{\pm a, \pm a/2\}$ (the {\em divisible} case).
Indeed, this is a property of the underlying root system,
so it holds by 
\cite[VI, \S1.3, Prop.\,8(i)]{lie4}. 
The study of connected reductive groups over
fields $k \ne k_s$ gives rise to non-reduced root data; e.g., 
this occurs for $k = \RRR$ in the study of non-compact connected semisimple Lie
groups, as well as in the study of special unitary groups over general $k$
admitting a separable quadratic extension. 
\end{remark}


In general, the quotient ${\rm{X}}(T)/\Z \Phi$
is the Cartier dual $\Hom(Z_G, \Gm)$ of the scheme-theoretic
\index{ aaazg@$Z_G$}
\index{center}
center $Z_G$ of $G$. This asserts that the inclusion
$Z_G \subset \bigcap_{a \in \Phi} \ker a$ is an equality, and holds because
$G$ is generated by $T$ and the root groups $U_a$ (as even holds for Lie algebras).
As a special case,  $\Z \Phi$ has finite index in ${\rm{X}}(T)$ 
if and only if $Z_G$ is finite; i.e., 
there is no nontrivial central  torus (equivalently, $G$ is semisimple).
Here are some illustrations of the relations between
the root datum and the (scheme-theoretic) center. 

\begin{example} The set $\Phi$ spans ${\rm{X}}(T)$ over $\Z$ if and only if
the scheme-theoretic center is trivial (the ``adjoint semisimple'' case).
For example, $({\rm{PGL}}_2,\overline{D})$ has roots  $\pm \overline{a}$
that are isomorphisms $\overline{D} \simeq \Gm$,
so these each span ${\rm{X}}(\overline{D}) = \Z$, 
whereas $({\rm{SL}}_2,D)$ has roots $\pm a:D \rightarrow \Gm$
with kernel $D[2] = \mu_2$, so these each span
the unique index-2 subgroup of ${\rm{X}}(D) = \Z$.
This encodes the fact that ${\rm{SL}}_2$ has scheme-theoretic
center $\mu_2$ whereas ${\rm{PGL}}_2$ has trivial scheme-theoretic center.
\end{example}

\begin{example} The set $\Phi(G,T)$ is empty if and only if $G$ is a torus (i.e., $G = T$),
or equivalently $G$ is solvable.
This is immediate from the weight space decomposition (\ref{wtg})
since $\mathfrak{g}_0 = \mathfrak{t}$.
\end{example}

\begin{example}\label{exsc}
Suppose $G$ is semisimple.  In this case we have the containment of lattices
\begin{equation}\label{xphi}
Q := \Z \Phi \subset {\rm{X}}(T) \subset (\Z \Phi^{\vee})^{\ast} =: P,
\end{equation}
where the dual lattice $(\Z \Phi^{\vee})^{\ast}$ in ${\rm{X}}(T)_{\Q}$ is
$\Z$-dual to the lattice $\Z \Phi^{\vee} \subset {\rm{X}}(T)^{\vee} = 
{\rm{X}}(T)^{\ast}$.
Thus, the two ``extreme'' cases are
${\rm{X}}(T) = \Z \Phi$ and ${\rm{X}}(T) = (\Z \Phi^{\vee})^{\ast}$
(i.e., $\Z \Phi^{\vee} = {\rm{X}}_{\ast}(T)$).  In the language
of root systems, these cases respectively
correspond to the cases when (i) the base
for a positive system of roots $\Phi(G,T)$ (see Definition \ref{posdef})
is a basis of the character group of $T$,
and (ii) the base for a positive system of coroots
is a basis of the cocharacter group of $T$.
The first of these two extremes is the case of adjoint $G$ (such as
${\rm{PGL}}_n$) and the second is the case of
simply connected $G$ (such as ${\rm{SL}}_n$); see
Exercise \ref{scclassical}(ii). 
\end{example}

The above examples (and the theory over $\C$; see Proposition \ref{scgt})
inspire: 

\begin{definition}\label{sssc} A reduced root datum $R = (X,\Phi,X^{\vee},\Phi^{\vee})$
is {\em semisimple} if $\Phi$ spans $X_{\Q}$ over $\Q$.  In the semisimple case, it 
is {\em adjoint} if $\Z\Phi = X$ and is {\em simply connected} if
$\Z \Phi^{\vee} = X^{\vee}$.
\end{definition}
\index{root datum!adjoint, simply connected}
\index{root datum!semisimple}
\index{simply connected!root datum}
\index{adjoint!root datum}


%By the general theory of root systems, for any reduced root datum
%$R = (X,\Phi,X^{\vee},\Phi^{\vee})$ and $a, b \in \Phi$ we have 
%$|\langle a, b^{\vee} \rangle| \le 3$ 
%(see the start of \cite[VI, \S1.3]{lie4}).

If $(\Q \Phi^{\vee})^{\perp}$ denotes the annihilator 
in $X_{\Q}$ of the subspace $\Q \Phi^{\vee} \subset X_{\Q}^{\vee}$
then the natural map $(\Q \Phi) \oplus (\Q \Phi^{\vee})^{\perp} \rightarrow
X_{\Q}$ is an isomorphism \cite[Exer.\,7.4.2]{springer}
(mirroring  the isogeny decomposition of a connected reductive
group into the almost direct product of a torus and a connected
semisimple group as in Example \ref{cisog}).
Thus, the subgroup
\index{ aaawr@$W(R)$}
\index{root datum!Weyl group}
\index{Weyl group}
$$W(R) = \langle s_a\,|\,a \in \Phi \rangle \subset {\rm{Aut}}(X)$$
is trivial on $(\Q\Phi^{\vee})^{\perp}$
and acts on $\Q \Phi$ through permutations of
a finite spanning set.  It follows that  $W(R)$ is finite; it is
called the {\em Weyl group} of the root datum, 
and is naturally isomorphic to the Weyl group of the associated
root system: 

\begin{example}
If $(G,T)$ is a connected reductive $k$-group equipped with a maximal torus $T$
then since $Z_{G(k)}(T) = T(k)$, the action of $N_{G(k)}(T)$ on $T$ identifies $W_G(T) = N_{G(k)}(T)/T(k)$
with a finite subgroup of ${\rm{Aut}}({\rm{X}}(T))$.  This subgroup
is the Weyl group of the root datum associated to $(G,T)$
\cite[14.8]{borelag}.
Here is the idea of the proof.
In the proof of Proposition \ref{saprop}
we showed each reflection $s_a$ in the Weyl group
of the root datum $R(G,T)$ lies in $W_G(T)$, so 
$W(R(G,T)) \subset W_G(T)$.  The reverse containment
is proved in \S\ref{possec} by  relating Borel subgroups to positive systems of roots
in the associated root system (see Definition \ref{posdef} and Proposition \ref{borpos}) and using 
that the Weyl group of a root system acts simply transitively on
the set of positive systems of roots (\cite[VI, \S1.5, Thm.\,2(i); \S1.6, Thm.\,3]{lie4}).
\end{example}

\begin{definition}
A root system $(V,\Phi)$ is {\em non-empty} if $\Phi \ne \emptyset$ (equivalently, $V \ne 0$).
The {\em direct product} of root systems $(V_1,\Phi_1)$ and $(V_2, \Phi_2)$ is 
$(V_1, \Phi_1) \times (V_2, \Phi_2) = (V_1 \oplus V_2, \Phi_1 \coprod \Phi_2)$.
A root system $(V,\Phi)$ is {\em irreducible}
if it is non-empty and not a direct
product of two non-empty root systems. 
\end{definition}

\begin{remark}\label{remprod}
 Every non-empty root system
is uniquely a direct product of irreducible root systems \cite[VI, \S1.2, Prop.\,6]{lie4}
and there is a classification of irreducible root systems 
(\cite[VI, \S4.2, Prop.\,1, Thm.\,3]{lie4} in the reduced
case, and \cite[VI, \S1.4, Prop.\,13]{lie4} 
and \cite[VI, \S4.14]{lie4} in the non-reduced case).
This is very useful in the study of root data.

For each irreducible root system $\Phi$, there is a positive-definite $\mathbf{Q}$-valued 
quadratic form $Q$ invariant
under the action of the Weyl group  \cite[VI, \S1.1, Prop.\,3]{lie4}.
This quadratic form is unique up to scaling whether
we take $V$ to be finite-dimensional over $\Q$ (as we have done above)
or over $\RRR$ (as in most literature) \cite[VI, \S1.2, Prop.\,7]{lie4}.
In particular, it is intrinsic to compare ratios $Q(a)/Q(b)$ for $a, b \in \Phi$
when $(V, \Phi)$ is irreducible. 
After extending scalars to $\RRR$,
this equips irreducible root systems with a canonical inner product
(up to scaling), and thereby puts the study of root systems
into the framework of Euclidean geometry.

For example, suppose $R = (X,\Phi,X^{\vee},\Phi^{\vee})$
is a root datum such that $\Phi$ spans $V := X_{\Q}$
and $(V, \Phi)$ is an irreducible root system.
Choose a $W(R)$-invariant inner product
$(\cdot|\cdot)$
on $V_{\RRR}$.  We use this inner product
to identify $X_{\RRR}$ with its linear dual $X^{\vee}_{\RRR}$.
In this way the coroot $a^{\vee}$ is identified with the element
$2a/(a|a) \in X_{\RRR}$ \cite[VI, \S1.1, Lemma 2]{lie4}. This is the most convenient way for drawing
pictures of low-rank irreducible root data. 
Also, it is intrinsic to compare {\em ratios} of root lengths,
and in cases with distinct root lengths there
are exactly two, with $(a|a)/(b|b) \in \{2, 3\}$ when $(a|a) > (b|b)$
\cite[VI, \S1.4, Prop.\,12]{lie4}. 
\end{remark}

There is an evident notion of {\em isomorphism}
between root data (required to respect the given perfect duality between
$X$ and $X^{\vee}$).  If
$f:(G,T) \simeq (G',T')$ is an isomorphism, then
it is clear that the induced isomorphisms
${\rm{X}}(T') \simeq {\rm{X}}(T)$ and
${\rm{X}}_{\ast}(T) \simeq {\rm{X}}_{\ast}(T')$
respect the dualities and the subsets of roots and coroots,
so we get an isomorphism of root data $R(G',T') \simeq R(G,T)$.
If we change $f$ via composition with a $T'(k)$-conjugation or
$T(k)$-conjugation then the isomorphism between the root data is unchanged.
Now we can finally state a fundamental result
in the classical theory over $k = \overline{k}$:

\index{Existence Theorem}
\index{Isomorphism Theorem}
\begin{theorem}[Existence and Isomorphism Theorems] The reduced root datum $R(G,T)$ associated
to a connected reductive $k$-group $G$ and maximal torus
$T \subset G$ determines
$(G,T)$ uniquely up to isomorphism.
More precisely, for any two
pairs $(G,T)$ and $(G',T')$, every isomorphism $R(G',T') \simeq R(G,T)$
arises from an isomorphism $(G,T) \simeq (G',T')$ that is
unique up to the conjugation actions of $T'(k)$ and $T(k)$,
and every reduced root datum is isomorphic to $R(G,T)$ for some
pair $(G,T)$ over $k$.
\end{theorem}

A remarkable aspect of this theorem is that the root datum
has nothing to do with $k$ or ${\rm{char}}(k)$.
There is a finer version of the theorem that also classifies 
isogenies in terms of a notion of ``isogeny'' between root data;
this encodes characteristic-dependent concepts (such as the Frobenius
isogeny in characteristic $p > 0$), but we postpone the statement and proof of this Isogeny Theorem
until we discuss the Existence and Isomorphism Theorems
over a general non-empty base scheme in \S\ref{existsec}.
The proof of these theorems over a general base scheme
will {\em not} require the classical version over a general algebraically closed
field.  It only requires the Existence Theorem for connected semisimple
groups over a single algebraically
closed field of characteristic 0; see Appendix \ref{csec}.

\begin{example}
For any $(G,T)$, consider the root datum $R'$ dual to $R(G,T)$
in the sense of Remark \ref{remdual}.
By the Existence Theorem, there exists another pair
$(G',T')$ over $k$ unique up to isomorphism
for which $R(G',T') \simeq R'$.  Since
the isomorphism in $(G',T')$ is only ambiguous up to a $T'(k)$-conjugation,
and such conjugation has {\em no} effect on $T'$, 
it is reasonable to consider the representation theory of
$(G',T')$ (incorporating $T'$-weight space information) as a structure that is intrinsically associated to $(G,T)$.
This is a version of {\em Langlands duality}.

A basic example is $G = \GL_n$, in which case $G' = \GL_n$.
Slightly more interesting is the case $G = {\rm{SL}}_n$, for which $G' = {\rm{PGL}}_n$.
In general,  Langlands duality in the semisimple case swaps
adjoint and simply connected groups (see Example \ref{exsc}).
\end{example}

\begin{remark}
Operations with root systems have analogues for connected semisimple groups.
For example, the decomposition of a non-empty root system
into its irreducible components corresponds to the fact that
every nontrivial connected semisimple $k$-group $G$ 
has only finitely many minimal (necessarily semisimple) nontrivial normal smooth connected subgroups $G_i$,
each $G_i$ is simple (i.e., has no nontrivial proper
connected normal linear algebraic subgroup), and these pairwise commute
and define a central isogeny $\prod G_i \rightarrow G$
via multiplication.  See \cite[14.10]{borelag} for proofs based
on the structure of automorphisms of semisimple groups, 
and see Proposition \ref{simpleprod}ff.\,for a simple proof based on the structure of the ``open cell''
(using (\ref{27eq})). 
\end{remark}



\subsection{Positive systems of roots and parabolic subgroups}\label{possec}

Let $G$ be a connected reductive $k$-group and $T$ a maximal torus.
There are only finitely many parabolic subgroups $P$ of $G$ containing
$T$, and in particular only finitely many Borel subgroups $B$ of $G$ containing $T$.
These can be described in terms of the following combinatorial notion
applied to the root system $\Phi(G,T)$:

\begin{definition}\label{posdef}
Let $(V,\Phi)$ be a non-empty root system.
A {\em positive system of roots} is a subset $\Phi^+ \subset \Phi$
\index{positive system of roots}
\index{root system!positive system of roots}
\index{ aaafigtplus@$\Phi^{+}$, $\Phi_{\lambda > 0}$}
such that $\Phi^+ = \Phi_{\lambda > 0} := \{a \in \Phi\,|\, \lambda(a) > 0\}$
for some $\lambda \in V^{\ast}$ that is non-vanishing on $\Phi$;
i.e., $\Phi^+$ is the part of $\Phi$ lying in an open half-space of $V_{\RRR}$
whose boundary hyperplane is disjoint from $\Phi$.
\end{definition}

For any positive system of roots $\Phi^+$ in $\Phi$, 
the subset $\Delta \subset \Phi^+$ of elements of
$\Phi^+$ that cannot be expressed as a sum of
two elements of $\Phi^+$ turns out to be a basis of $V$
and every element of $\Phi$ has the form $\sum_{a \in \Delta} m_a a$
with integers $m_a$ that are either all $\ge 0$ or all $\le 0$ 
\cite[VI, \S1.6, Thm.\,3]{lie4}. 
The elements of $\Delta$ are called the {\em simple positive roots}
relative to $\Phi^+$, and any such $\Delta$ is called a {\em base} for
the root system $\Phi$. 
\index{root system!base}
\index{base of root system}
\index{ aaadelta@$\Delta$}

Note that if we fix an enumeration $\{a_i\}$ of such a $\Delta$
and equip $V = \bigoplus \Q a_i$ with the lexicographical ordering
then $\Phi^+$ consists of the elements of $\Phi$ that are
positive relative to this ordering.  Conversely, 
for any ordered vector space structure on $V$
the set of positive elements of $\Phi$
turns out to be a positive system of roots \cite[VI, \S1.7, Cor.\,2]{lie4} 
(this is useful in the study of ``highest weight vectors''
in representation theory; see \S\ref{repsec}).

\begin{example} For $G = \GL_n$ ($n \ge 2$) and $T = \Gm^n$ the diagonal torus,
identify ${\rm{X}}(T)_{\Q} = {\rm{X}}(\Gm)_{\Q}^n$ with $\Q^n$ via the canonical
identification ${\rm{X}}(\Gm) = \Z$.  In other words, to each
$\chi:\Gm^n \rightarrow \Gm$ associate the $n$-tuple $(c_1,\dots,c_n) \in \Z^n$
for which $\chi(t_1,\dots,t_n) = \prod t_i^{c_i}$.
In this way, $\Phi(G,T)$ is a root system in the hyperplane $V = \{\vec{c} \in \Q^n\,|\,\sum c_j = 0\}$.
Equip $\Q^n$ with the lexicographical ordering, and $V$ with the induced ordering.
The root system $\Phi(G,T)$ consists of the differences
$e_i - e_j$ for $1 \le i \ne j \le n$, and
the positive ones for this ordering are 
$e_i - e_j$ for $i < j$.  
The corresponding base $\Delta$ consists of the roots
$e_i - e_{i+1}$ for $1 \le i \le n-1$.

This positive system of roots is exactly
the set of roots that occur as nontrivial $T$-weights on the Lie algebra
of the Borel subgroup $B$ of upper triangular matrices in $G$.
See Proposition \ref{borpos} for the general result of which
this is a special case.
\end{example}

Every element of $\Phi$ is a simple positive root
for some choice of $\Phi^+$ (equivalently, for any $\Delta$
the $W(\Phi)$-orbits of the elements of $\Delta$ cover $\Phi$) \cite[VI, \S1.5, Prop.\,15]{lie4}, 
and if we fix a choice of $\Phi^+$
and let $\Delta$ be the corresponding base
then $W(\Phi)$ is generated by the reflections $s_a$ for
$a \in \Delta$.  In fact, $W(\Phi)$ has a presentation as a reflection group
generated by the reflections in the simple positive roots
\cite[VI, \S1.5, Thm.\,2(vii), Rem.\,3, (11)]{lie4}:
\begin{equation}\label{wpres}
W(\Phi) = \langle \{s_a\}_{a \in \Delta}\,|\, (s_a s_b)^{m_{ab}} = 1 \mbox{ for } a, b \in \Delta \rangle
\end{equation}
where $m_{aa} = 1$ for all $a \in \Delta$, $m_{ab} = 2$ (equivalently, 
$s_a s_b = s_b s_a$) when $a$ and $b$ are in distinct irreducible components 
of $\Phi$ or are orthogonal in
the same component, and otherwise $m_{ab} = 3$ for non-orthogonal
$a, b \in \Delta$ with the same length and 
$m_{ab} = 2 \langle a, b^{\vee}\rangle \langle b, a^{\vee} \rangle \in \{4, 6\}$
for non-orthogonal $a, b \in \Delta$ with distinct lengths.

\begin{remark}\label{diagram}
An important invariant of a reduced root system $(V,\Phi)$ is its {\em Dynkin diagram}
${\rm{Dyn}}(\Phi)$, a graph with extra structure on certain edges.
\index{Dynkin diagram}
\index{ aaadynfi@${\rm{Dyn}}(\Phi)$}
This intervenes in the classification of root systems,
which is an ingredient in the proof of the Existence
and Isomorphism Theorems (at least in low rank),
and it can be defined in terms of Euclidean geometry or combinatorics
with a root system (see \cite[VI, \S4.2]{lie4}).  We will give both definitions. 
In all cases,  
the vertices of the graph are the elements of a base $\Delta$
for a positive system of roots (and the simply transitive action of $W(\Phi)$ on the set of all $\Delta$'s
identifies vertices with certain $W(\Phi)$-orbits in $\Phi$, eliminating the dependence on the choice of $\Delta$).

For the Euclidean definition we shall define ${\rm{Dyn}}(\Phi) = \coprod {\rm{Dyn}}(\Phi_i)$
for the irreducible components $(V_i,\Phi_i)$ of
$(V,\Phi)$, so suppose $\Phi$ is irreducible. 
Recall from Remark \ref{remprod} that 
there exists a $W(\Phi)$-invariant inner product
$(\cdot|\cdot)$ on $V_{\RRR}$ (even a $\Q$-valued $W(\Phi)$-invariant positive-definite
quadratic form on $V$), it is unique up to scaling,
and as we vary through $a, b \in \Phi$
with $(a|a) \ge (b|b)$, the ratio $(a|a)/(b|b)$ is $\Z$-valued and takes on at most two possible
values.   The diagram ${\rm{Dyn}}(\Phi)$ has as its vertices the elements
of $\Delta$, and there exists an edge linking
vertices $a$ and $b$ precisely when 
$(a|b) \ne 0$.  If  $(a|a) > (b|b)$ then this edge is assigned multiplicity $(a|a)/(b|b)$
and a direction pointing from $a$ to $b$.  The 
graph is always connected (for irreducible $\Phi$). 

In the combinatorial definition we do not need to
pass to the irreducible components: ${\rm{Dyn}}(\Phi)$ is a graph whose
vertices are the elements of $\Delta$, and an edge
joins $a$ and $b$ precisely when $a+b \in \Phi$. 
(Note that $a - b \not\in \Phi$ since $a, b \in \Delta$.)
For such $a$ and $b$, the multiplicity and direction of the edge joining $a$ and $b$ are defined
in terms of the sets 
$$I_{a,b} = \{j \in \Z\,|\,a + jb \in \Phi\},\,\,\,
I_{b,a} = \{j \in \Z\,|\,b+ja \in \Phi\}$$
as follows.  Since $a, b \in \Delta$, we have
$I_{a,b} = \Z \bigcap [0, \dots, -\langle a, b^{\vee} \rangle]$
and $I_{b,a} = \Z \bigcap [0,\dots, -\langle b, a^{\vee} \rangle]$
\cite[VI, \S1.3, Prop.\,9]{lie4}. 
An inspection of cases \cite[VI, \S1.3, Rem.]{lie4}
shows that either both sets coincide with $\{0,1\}$, in which case
$a$ and $b$ are joined by a single undirected edge,
or one of them is $\{0,1\}$ and the other is either $\{0,1,2\}$ or $\{0,1,2,3\}$.
In these latter cases if we arrange the labels
so that $I_{a,b}$ contains 2 or 3 then $a$ and $b$ are joined by an edge
with multiplicity $-\langle a, b^{\vee} \rangle$ pointing
from $a$ to $b$.
\end{remark}

Returning to the setting of a connected reductive $k$-group $G$
and a maximal torus $T$, if $B$ is a Borel subgroup containing $T$ then the 
subalgebra $\mathfrak{b} = {\rm{Lie}}(B) \subset \mathfrak{g}$
contains $\mathfrak{t}$ and so has a weight space
decomposition $\mathfrak{b} = \mathfrak{t} \bigoplus (\bigoplus_{a \in \Phi(B,T)} \mathfrak{g}_a)$
for a subset $\Phi(B,T) \subset \Phi(G,T)$.
Such subsets are positive systems of roots 
in $\Phi(G,T)$ \cite[14.1]{borelag}.
If $U = \mathscr{R}_u(B)$ then for
any enumeration $\{a_i\}$ of $\Phi(B,T)$ the multiplication map
\begin{equation}\label{27eq}
\prod U_{a_i} \rightarrow U
\end{equation}
is an isomorphism of schemes (see \cite[14.4]{borelag}
or \cite[3.3.6, 3.3.7, 3.3.11]{pred});
we say $U$ is ``directly spanned in any order'' by the
root groups contained in $U$.
\index{directly spanned}

Since
$W_G(T)$ acts simply transitively on the set
of all $B \supset T$ \cite[13.10(2)]{borelag}, and
$W(\Phi)$ acts simply transitively on the set of positive systems
of roots in $\Phi$ \cite[VI, \S1.5, Thm.\,2(i)]{lie4}, 
the inclusion $W(\Phi) \subset W_G(T)$ is forced
to be an equality and we obtain:

\begin{proposition}\label{borpos}
The map $B \mapsto \Phi(B,T)$ is a bijection
from the set of $B \supset T$ onto the set of positive systems of roots in $\Phi(G,T)$.
\end{proposition}

The generalization of Proposition \ref{borpos} to the case of parabolic subgroups
containing $T$ requires another class of distinguished subsets of a root system:

\begin{definition}\label{parsubdef} Let $(V,\Phi)$ be a root system.
A {\em parabolic subset} of $\Phi$ is a subset $\Psi \subset \Phi$ of
the form $\Psi = \Phi_{\lambda \ge 0} := \{a \in \Phi\,|\,\lambda(a) \ge 0\}$
for some $\lambda \in V^{\ast}$; i.e., $\Psi$ is the part of $\Phi$ lying
in a closed half-space of $V$ (or $V_{\RRR}$).
\end{definition}
\index{parabolic set of roots}
\index{root system!parabolic subset}
\index{ aaafigtge@$\Phi_{\lambda \ge 0}$}

The reason for this terminology is that it will turn out that for any
parabolic subgroup $P \subset G$ containing $T$, the set
$\Phi(P,T)$ of nontrivial $T$-weights occurring in $\mathfrak{p} = {\rm{Lie}}(P)$
is a parabolic subset of $\Phi(G,T)$.   Before we can state the precise
bijective correspondence in this direction, it will be convenient to discuss
some alternative formulations of the definition of parabolicity for a subset of a root system $(V,\Phi)$.

It is clear from the definition that any parabolic subset $\Psi$ of
$\Phi$ satisfies the following two properties: (i) $\Phi = \Psi \bigcup -\Psi$,
and (ii) $\Psi$ is a {\em closed} set in $\Phi$ (i.e., if $a, b \in \Psi$ and $a +b \in \Phi$ then
\index{closed set of roots}
\index{root system!closed subset}
$a+b \in \Psi$).  Conversely, any subset of $\Psi$ satisfying (i) and (ii) 
is parabolic.  The equivalence is proved in \cite[2.2.8]{pred} (where
parabolicity is defined using (i) and (ii), as in \cite[Ch.\,IV]{lie4}), and 
a more explicit description of parabolic sets is provided there:
they are precisely the subsets $\Phi^+ \bigcup [I]$,
where $\Phi^+$ is a positive system of roots,
$I$ is a subset of the corresponding base $\Delta$,
and $[I]$ denotes the set of roots that are 
\index{ aaaII@$[I]$}
$\Z$-linear combinations of elements of $I$.
In particular, every parabolic set contains a positive system of roots.

\begin{remark}\label{closedspan}
Let $\Psi \subset \Phi := \Phi(G,T)$ be a closed set of roots that is
contained in a positive system of roots $\Phi^+$ for $\Phi$.
For the unique Borel subgroup $B \subset G$ containing $T$ that satisfies
$\Phi(B,T) = \Phi^+$, the smooth connected subgroup $U_{\Psi} \subset G$
generated by the root groups $\{U_a\}_{a \in \Psi}$ is contained
in $\mathscr{R}_u(B)$ and hence is unipotent.    But much more is true:
the group $U_{\Psi}$ is {\em directly spanned in any order}
\index{directly spanned} by the groups $\{U_a\}_{a \in \Psi}$
in the same sense as for the case $\Psi = \Phi^+$ considered in (\ref{27eq}).
That is, for {\em any} enumeration $\{a_1,\dots,a_m\}$ of $\Psi$, the multiplication map
$\prod U_{a_i} \rightarrow U_{\Psi}$ between pointed schemes is an isomorphism.
A proof using the structure theory of reductive groups is given
in \cite[14.5, Prop.\,(2)]{borelag}, and a proof via general dynamical
principles is given in \cite[3.3.11, 3.3.13(1)]{pred}.
\end{remark}

A given parabolic set $\Psi \subset \Phi$ can contain more than one positive system of roots,
just as a parabolic subgroup $P$ can contain more than one Borel subgroup
containing a fixed maximal torus $T$ in $P$. Nonetheless, for {\em any} positive system of roots $\Phi^+$
(with corresponding base $\Delta$) 
contained in $\Psi$, we have $\Psi = \Phi^+ \bigcup [I]$ for a unique
$I \subset \Delta$  \cite[VI, \S1.7, Lemma 3]{lie4}.

\begin{proposition}\label{parpos}
The map $P \mapsto \Phi(P,T)$ is a bijective correspondence
between the set of parabolic subgroups of $G$ containing $T$ and the set of parabolic
subsets of $\Phi = \Phi(G,T)$, and the following are equivalent: $P \subset P'$,
$\Phi(P,T) \subset \Phi(P',T)$, and ${\rm{Lie}}(P) \subset {\rm{Lie}}(P')$
inside $\mathfrak{g}$.
\end{proposition}

%This proposition is a (mild)
%refinement of the classification of parabolic subgroups $P$ of $G$ containing $T$ in terms of 
%``standard parabolic subgroups''
%in \cite[14.18ff.]{borelag} because that classification
%only considers $P$ that contain a fixed Borel subgroup $B$ of $G$ containing $T$.
%In contrast, Proposition \ref{parpos} does not presuppose the existence
%of a common Borel subgroup (and in positive characteristic, containments between connected
%linear algebraic subgroups cannot generally be checked at the level of Lie algebras).

\begin{proof} 
Since ${\rm{Lie}}(P)$ is spanned by ${\rm{Lie}}(T)$
and the weight spaces $\mathfrak{g}_c = {\rm{Lie}}(U_c)$
for $c \in \Phi(P,T)$, to prove the equivalence of containment assertions
it suffices to prove that if $c \in \Phi(P,T)$  then
$U_c \subset P$ (as then $P$ is generated by $T$
and such $U_c$ by Lie algebra considerations
since $P$ is connected). We reduce to the rank-1 case as follows.

Any $P$ contains a Borel subgroup
$B$ of $G$ that contains $T$ (due to the conjugacy of
maximal tori in $P$).  For any subtorus $S \subset T$, the group 
$(P \bigcap Z_G(S))_{\rm{red}} = Z_P(S)$ contains the subgroup
$(B \bigcap Z_G(S))_{\rm{red}} = Z_B(S)$ that is a Borel subgroup of $Z_G(S)$
\cite[11.15]{borelag}, so $(P \bigcap Z_G(S))_{\rm{red}}$ is a parabolic subgroup
of $Z_G(S)$ containing $T$.  The bijective
correspondence between the sets of parabolic subgroups
of a connected reductive group $H$ and of
its derived group $\mathscr{D}(H)$ is defined
by $P' \mapsto (P' \bigcap \mathscr{D}(H))_{\rm{red}}$, so $(P \bigcap \mathscr{D}(Z_G(S)))_{\rm{red}}$
is a parabolic subgroup of $\mathscr{D}(Z_G(S))$
containing the maximal torus $(T \bigcap \mathscr{D}(Z_G(S)))_{\rm{red}}$.

Taking $S$ to be the codimension-1 subtorus $T_c := (\ker c)^0_{\rm{red}} \subset T$,
$P_c := (P \bigcap \mathscr{D}(Z_G(T_c)))_{\rm{red}}$ is a parabolic subgroup
of the 3-dimensional connected semisimple group $\mathscr{D}(Z_G(T_c)) = \langle U_c, U_{-c} \rangle$
containing the maximal torus $T \bigcap \mathscr{D}(Z_G(T_c)) = c^{\vee}(\mathbf{G}_{\rm{m}})$. 
Since the root groups of $\mathscr{D}(Z_G(T_c))$ relative to $c^{\vee}(\mathbf{G}_{\rm{m}})$
are $U_{\pm c}$, we can replace $(G,T,P)$ with $(\mathscr{D}(Z_G(T_c)), c^{\vee}(\mathbf{G}_{\rm{m}}), P_c)$
to reduce to the case when $G$ is semisimple of rank 1.  We can then
choose an isomorphism from $G$ onto either ${\rm{SL}}_2$ or ${\rm{PGL}}_2$
such that $T$ is carried to the diagonal torus and $U_c$ is carried to the upper triangular
unipotent subgroup.  There are three parabolic subgroups
containing the diagonal torus: the entire group and the upper and lower triangular
Borel subgroups.  The condition $c \in \Phi(P,T)$ rules out the lower
triangular Borel subgroup, and inspection of the two remaining possibilities shows that $U_c \subset P$.

It remains to show each parabolic subset $\Psi$ of $\Phi$ is $\Phi(P,T)$ for some
parabolic subgroup $P$ containing $T$.  Since $W_G(T)$ acts transitively
on the set of positive systems of roots in $\Phi$, we may restrict
attention to $\Psi$ that contain $\Phi^+ := \Phi(B,T)$ for a fixed
Borel subgroup $B$ of $G$ containing $T$.  Thus, $\Psi = \Phi^+ \cup [I]$
for a unique subset $I$ of the base $\Delta$ of $\Phi^+$.
Since $T$-conjugation on $U_c \simeq \Add$ is scaling through $c$, 
for the subtorus $T_I = (\cap_{a \in I} \ker a)^0_{\rm{red}}$
we see that $Z_G(T_I)$ contains as its $T$-root groups exactly
$U_c$ for $c \in \Phi$ that kill $T_I$.
Let $U_I$ be the smooth connected subgroup
of $\mathscr{R}_u(B)$ directly spanned in any order
by the root groups $U_b$ for $b$ in the closed subset
$(\Phi - [I]) \cap \Phi^+ \subset \Phi^+$ (Remark \ref{closedspan}).

The connected reductive group $Z_G(T_I)$
normalizes $U_I$.  Indeed, its root system with respect
to $T$ is $[I]$ (as $\Delta$ is a basis of the character group of the adjoint torus
$T/Z_G$), so it is generated by $T$ and the root
groups $U_c$ for $c \in [I]$.  Clearly $T$ normalizes $U_I$,
and if $c \in -\Phi^+ \bigcap [I]$  then $U_c$ even centralizes $U_I$ since
$c+b$ cannot be a root for any $b \in \Phi^+ \cap (\Phi - [I])$.
If instead $c \in [I] \cap \Phi^+$ then for any $b \in (\Phi - [I]) \cap \Phi^+$
the roots of the form $ib + jc$ with $i, j \ge 1$ obviously lie 
in $(\Phi - [I]) \cap \Phi^+$, so points of $U_c$ conjugate $U_b$ into $U_I$
for all such $b$.  That is, $U_c$ normalizes $U_I$ for all such $c$,
so $Z_G(T_I)$ normalizes $U_I$ as claimed.
Moreover, the schematic intersection $Z_G(T_I) \cap U_I$ is trivial.
Indeed, $T$-weight space considerations show that the Lie algebra
of this intersection vanishes, so the intersection is \'etale, yet it is
also connected since $Z_{T_I \ltimes U_I}(T_I) = T_I \times (Z_G(T_I) \cap U_I)$
and torus centralizers in smooth connected affine groups are connected.
Thus, $P_I := Z_G(T_I) \ltimes U_I$ makes
sense as a subgroup of $G$. 

The smooth subgroup $P_I$ clearly contains $B$
(so it is a parabolic subgroup containing $T$) 
and satisfies $\Phi(P_I, T) = \Phi^+ \cup [I] = \Psi$.
(This recovers the parameterization of ``standard'' parabolic
subgroups of $G$ in \cite[14.18]{borelag} via a different proof.
Note that $\mathscr{R}_u(P_I) = U_I$.)
\end{proof}


\begin{example}\label{stdpar}
Let $G = {\rm{SL}}_n$ and consider the upper triangular $B$
and diagonal $T$ with character
group $\Z^n/{\rm{diag}}(\Z)$.  Then $\Delta = \{e_{i+1}- e_i\}_{1 \le i \le n-1} = \{1,\dots,n-1\}$, 
and for $I  \subset \{1,\dots,n-1\}$ the parabolic $P_I$ corresponds via Example \ref{pargln}
to the ordered partition $\vec{a} = (a_1,\dots,a_r)$ of $n$ into non-empty parts 
for which the associated subset $\{b_j = a_1 + \dots + a_j\}_{1 \le j < r}$ in
$\{1,\dots,n-1\}$ is the {\em complement} of $I$; e.g., $P_{\emptyset} = B$.
\end{example}

%
%
%
%We conclude that $\Phi(P,T)$ is
%a parabolic subset of $\Phi$ in general, and that every parabolic
%subset of $\Phi$ arises in this way.  This shows that the correspondence $P \mapsto \Phi(P,T)$
%between parabolic subgroups and parabolic subsets is well-defined and surjective.
%
%It is clear via weight space decompositions relative to the $T$-action that
%$\Phi(P,T) \subset \Phi(P',T)$ if and only if $\mathfrak{p} \subset \mathfrak{p}'$.
%Thus, to prove the bijectivity of the correspondence $P \mapsto \Phi(P,T)$, as well as the asserted equivalence
%of containment conditions, it remains to show that
%if $\Phi(P,T) \subset \Phi(P',T)$ then $P \subset P'$.
%The subtlety here is that we are not given that $P$ and $P'$
%contain a common Borel subgroup of $G$ containing $T$.
%To circumvent this problem, we will prove a result about individual
%parabolic subgroups. 
%More specifically, it suffices to prove that any parabolic subgroup $P$ of $G$ containing $T$
%is generated by $T$ and the root groups $U_a$ for $a \in \Phi(P,T)$.
%The Lie algebras $\mathfrak{t}$ and $\mathfrak{u}_a$ span $\mathfrak{p}$,
%so we just have to check that $U_a \subset P$ for each $a \in \Phi(P,T)$.
%
%Choose a Borel subgroup $B$ of $G$ containing $T$ such that $B \subset P$.
%Thus, $P = P_I$ for a subset $I$ of the base $\Delta$ of $\Phi^+ := \Phi(B,T)$.
%Explicitly, $\Phi(P,T) = \Phi^+ \bigcup [I]$ and $P = Z_G(T_I) \ltimes U_I$.
%For $a \in [I]$ not in $\Phi^+$ we have $U_a \subset U_I$ by the definition of $U_I$.
%Likewise, since $\mathscr{R}_u(B)$ is directly spanned by the root groups $U_a$
%for $a \in \Phi(B,T) = \Phi^+$, we have $U_a \subset B$ for 
%all $a \in \Phi^+$.  Since $B$ and $U_I$ are contained in $P$, we are done.
%\end{proof}

\begin{corollary}\label{147} The number of $G(k)$-conjugacy classes of parabolic subgroups
of $G$ is $2^{r_{\rm{ss}}}$, where $r_{\rm{ss}} = {\rm{rank}}(\mathscr{D}(G))$
is the dimension of the maximal tori of $\mathscr{D}(G)$.
$($Equivalently, $r_{\rm{ss}} = \#\Delta$ for a base $\Delta$ of a positive system
of roots in $\Phi(G,T)$ for maximal tori $T$ of $G$.$)$
\end{corollary}

An explicit description of conjugacy class representatives is in Exercise \ref{gross6}.

\begin{proof}
We just have to show that for a fixed Borel subgroup $B \subset G$
and a maximal torus $T \subset B$, parabolic subgroups
$P_I$ and $P_J$ associated to subsets $I$ and $J$ of
the base $\Delta$ of $\Phi^+ = \Phi(B,T)$ are $G(k)$-conjugate
if and only if $I = J$.  More conceptually,
if $P$ and $P'$ are parabolic subgroups of $G$ containing
$B$ and if $P' = gPg^{-1}$ for some $g \in G(k)$ then we claim
that $P' = P$.  The Borel subgroups $B$ and $gBg^{-1}$
in $P'$ are $P'(k)$-conjugate, so by replacing $g$ with a suitable left $P'(k)$-translate
we can arrange that $gBg^{-1} = B$.  But then $g \in N_{G(k)}(B) = B(k) \subset P(k)$,
so $P' = P$.
\end{proof}

\begin{remark}\label{parrem}
Note the remarkable consequence of Proposition \ref{parpos}
that in arbitrary characteristic, a parabolic subgroup $P$ is uniquely determined by its
Lie algebra $\mathfrak{p}$ as a subalgebra of $\mathfrak{g}$ {\em given}
that $P$ is a parabolic subgroup containing a specific maximal torus $T$. 
In fact, $P$ is determined by $\mathfrak{p}$ without reference
to $T$:  under the adjoint representation of $G$ on
$\mathfrak{g}$, $P(k)$ is the stabilizer in $G(k)$ of the subspace $\mathfrak{p}$.
This goes beyond the classical result $P(k) = N_{G(k)}(P)$
in Theorem \ref{parconn}, but there are some
subtleties because the scheme $N_G(\mathfrak{p})$ can be non-reduced
in characteristic 2; see Exercise \ref{parlie} for further discussion.
%
%Beware that there exist connected
%semisimple $G$ and a Borel subgroup $B \subset G$
%and maximal torus $T \subset G$ such that $\mathfrak{t} \subset \mathfrak{b}$
%inside $\mathfrak{g}$ but $T \not\subset B$ inside $G$.
%For example, consider $G = {\rm{SL}})_n$ over
%an algebraically closed field $k$ of characteristic $p|n$.
%Let $T$ be the diagonal torus, and $S$ the codimension-1 subtorus
%of $T$ consisting of diagonal matrices whose lower-right entry is 1.
%The centralized $H = Z_G(S)$ is a commutative reductive subgroup
%that is not a torus
%the diagonal torus $T = \Gm$ has the same Lie algebra
%as the central subgroup scheme $\mu_2$, so by the conjugacy
%of maximal tori we see that all maximal tori of $G$ have the same Lie algebra.
%Thus, every Borel subgroup of $G = {\rm{SL}}_2$ contains the common Lie algebra of every maximal
%torus, but only two Borel subgroups contain any given maximal torus of $G$.
\end{remark}

We call $\Phi(B,T)$ the positive system of roots {\em associated to $B$} (and $T$).
For each $B$, the unique Borel subgroup $B' \supset T$ for which $\Phi(B',T) = -\Phi(B,T)$
is the {\em opposite Borel subgroup} (relative to $T$),
\index{opposite Borel subgroup}
\index{Borel subgroup!opposite}
and its unipotent radical $U'$ is directly spanned in any order by
the negative root groups $U_{-a}$ for $a$ in the positive system of
roots $\Phi(B,T)$.  (Explicitly, $B' = U' \rtimes T$.) 
For example, if $G = \GL_n$, $T$ is the diagonal torus, and $B$ is the Borel subgroup of
upper triangular matrices then $B'$ is the Borel subgroup of lower triangular matrices. 

\begin{proposition}
The natural multiplication map
\begin{equation}\label{opencell}
U' \times T \times U = U' \times B \rightarrow G
\end{equation}
is an open immersion.
\end{proposition}

\begin{proof} This map 
is visibly \'etale at the origin, corresponding to the decomposition
$\mathfrak{g} = \mathfrak{g}_{<0} \oplus \mathfrak{t} \oplus \mathfrak{g}_{>0}$
relative to an ordering on $\Q \Phi$ that makes $\Phi(B,T)$ positive.
Thus, it is everywhere \'etale, due to $U'(k)$-translation on the left and $B(k)$-translation
on the right.  Now we can argue exactly as in the analysis of (\ref{uab}) provided
that $U'(k)$ has no nontrivial finite subgroup $\Gamma$ normalized by $T(k)$.  Since 
$T$ is connected, such a subgroup must be centralized by $T$, yet $Z_{G(k)}(T) = T(k)$
and $T(k) \cap U'(k) = 1$.
\end{proof}

The image $\Omega$ of (\ref{opencell}) is the {\em open cell}
\index{open cell}
\index{reductive group!open cell}
for $(G,T,B)$, though this terminology is also often used
for the ``inverse'' $U \times T \times U' \subset G$ involving multiplication
in the opposite order.  It is also commonly
called the {\em big cell}. 
%Here is a useful application of the open cell:

The open cell naturally arises
in the description of double cosets for $B$ in $G$.  To 
make this precise, first note that by the orbit lemma (applied
to $B \times B$ acting on $G$ through the commuting operations
of left and right multiplication by $B$), 
each double coset $B g B$ in $G$ (for $g \in G(k)$) is locally closed.
Recall that $W(\Phi)$ acts simply transitively on the set of positive systems of roots.
In particular, there is a unique $w_0 \in W(\Phi)$ such that $w_0(\Phi^+) = -\Phi^+$.
\index{Weyl group!long element}
\index{long Weyl element}
This $w_0$ is the unique longest element
relative to the reflections in the simple positive roots for
$\Phi^+ = \Phi(B,T)$, by \cite[VI, \S1.6, Cor.\,3 to Prop.\,17]{lie4}.  For any representative $n_0 \in N_{G(k)}(T)$ of $w_0$ we have
$U' = n_0^{-1} U n_0$, so 
$B n_0 B = n_0 \Omega$.  This is the only open double coset since distinct double cosets
are disjoint. 

\index{Bruhat decomposition}
\begin{theorem}[Bruhat decomposition]\label{bruhat}  Let $G$ be a connected reductive group,
$T$ a maximal torus, and $B$ a Borel subgroup.  For each
$w \in W := W_G(T)$, let $n_w \in N_G(T)(k)$ be a representative.
Each locally closed double coset $Bn_w B \subset G$ only depends on $w$, and set-theoretically 
$$G = \coprod_{w \in W} B n_w B.$$
If $\Phi^+ := \Phi(B,T)$ then $B n_w B = U_w \times n_w B$ as $k$-schemes 
via multiplication, 
where $U_w$ is a unipotent smooth connected 
subgroup directly spanned in any order
by the positive root groups $U_a$ for $a$ in the closed set $\Phi^+_w$ of $a \in \Phi^+$ such that
$w^{-1}(a) \in -\Phi^+$.
\end{theorem}

\begin{proof} Since $T \subset B$ and $n_w$ is well-defined modulo
$T(k)$, it is clear that $B n_w B$ only depends on $w$.
The asserted decomposition of the underlying set of the scheme $G$ as a disjoint union of locally closed sets amounts
to an equality on $k$-points: $G(k) = \coprod_{w \in W} B(k) n_w B(k)$.
This equality is proved in \cite[\S14.12]{borelag} (and it 
is a special case of a general decomposition for any abstract
group equipped with a ``$BN$-pair'' \cite[IV, \S2.1, Def.\,1]{lie4}, which
in turn was inspired by axiomatizing arguments for reductive groups;
see \cite[14.15]{borelag}).
To establish the desired description of each $B n_w B$ as a product scheme, 
since $B = U \rtimes T$ and the $k$-group $U$ is directly spanned in any order by
the $U_a$'s for $a \in \Phi^+$, we see via the equalities $n_w^{-1}T n_w = T$ and 
$n_w^{-1}U_a n_w = U_{w^{-1}(a)}$ that $B n_w B = U_w n_w B$
where $U_w$ is directly spanned (in any order) by 
the root groups $U_a$ for $a \in \Phi_w^+$ (Remark \ref{closedspan} applied
to $\Psi = \Phi^+_w$). 

The multiplication map $U_w \times n_w B \rightarrow G$ is a locally closed
immersion because applying left translation by $n_w^{-1}$ yields the map
$(n_w^{-1} U_w n_w) \times B \rightarrow G$ (via multiplication) that is the restriction to a closed
subscheme of the open immersion
$U^{-} \times B \rightarrow G$. 
%It remains to show that $U_w$ is a $k$-subgroup of $U$
%and that it is independent of the enumeration of $\Phi^+_w$. 
%The subset $\Phi^+_w$ of $\Phi^+$ is closed
%(i.e., if $a, b \in \Phi^+_w$ and $a + b \in \Phi$ then $a+b \in \Phi^+_w$),
%so the desired property of $U_w$ is a special case
%of a general result on the direct spanning by root groups associated to any closed
%set of positive roots (see \cite[14.5, Prop.\,(2)]{borelag} or \cite[3.3.11, 3.3.13(1)]{pred}). 
\end{proof}

\begin{corollary}\label{borpoor} For a Borel subgroup $B$ of a connected
reductive $k$-group $G$, the stabilizer of $\mathfrak{b}$ under the adjoint
action of $G(k)$ on $\mathfrak{g}$ is $B(k)$.
\end{corollary}

\begin{proof}
Choose a maximal torus $T$ in $B$ and let 
$\Phi^+ = \Phi(B,T)$ be the positive system of roots associated to $B$.
Choose $g \in G(k)$ whose adjoint action preserves $\mathfrak{b}$.
By the Bruhat decompostion, $g = b n b'$ for some $b, b' \in B(k)$
and $n \in N_{G(k)}(T)$.  It is harmless to replace $g$ with $b^{-1} g {b'}^{-1}$,
so $g$ normalizes $T$.  Then $g$ represents some $w \in W_G(T)$,
and the adjoint action of $g$ carries $\mathfrak{b} = \mathfrak{t} \oplus
(\oplus_{a \in \Phi^+} \mathfrak{g}_a)$ to 
$\mathfrak{t} \oplus (\oplus_{a' \in w(\Phi^+)} \mathfrak{g}_{a'})$.
Hence, the preservation of $\mathfrak{b}$ forces
$w(\Phi^+) = \Phi^+$, so $w = 1$ and hence $g \in T(k) \subset B(k)$.
\end{proof}

In translation arguments with split reductive group schemes over
a general scheme $S$, the following covering result on geometric
fibers will be very useful.

\begin{corollary}\label{bruhattranslate} For any $(G,T,B)$, with associated open
cell $\Omega = U' \times B$ and Weyl group
$W = W_G(T)$, $G = \bigcup_{w \in W} n_w \Omega$ for any
set of representatives $\{n_w\}_{w \in W} \subset N_{G(k)}(T)$
of $W$.
\end{corollary}

\begin{proof}
For each $w \in W$, $n_w^{-1}U_w n_w B \subset U' \times B = \Omega$.
Thus, $B n_w B = U_w n_w B \subset n_w \Omega$.
\end{proof}


\subsection{Based root datum, pinnings, and representation theory}\label{repsec}

In this section, we will explore consequences of
the Existence and Isomorphism Theorems over $k$.
Some 
concepts introduced (such as a ``based root datum'' and a ``pinning'' of
a triple $(G,T,B)$) will be used in the proof
of the Existence and Isomorphism Theorems over general base schemes
in \S\ref{existsec}.

One often considers triples $(G,T,B)$ rather than pairs $(G,T)$
(for a connected reductive $k$-group $G$), and likewise 
for root data $R = (X, \Phi, X^{\vee}, \Phi^{\vee})$
it is often convenient to choose a positive system of roots
$\Phi^+ \subset \Phi$.  By Proposition \ref{borpos}, the choice
of $\Phi^+$ corresponds to a choice of $B \supset T$ when
$R$ arises from a connected reductive $k$-group $G$ equipped
with a maximal $k$-torus $T$.  Another way to express
the choice of $\Phi^+$ is to specify its base $\Delta$.
The set $\Delta^{\vee}$ of coroots
$a^{\vee}$ for $a \in \Delta$ is a base for $\Phi^{\vee}$
(Exercise \ref{dualexer}), and we call the 
6-tuple $(X, \Phi, \Delta, X^{\vee}, \Phi^{\vee}, \Delta^{\vee})$ 
(or the triple $(R, \Delta, \Delta^{\vee})$, or simply $(R,\Delta)$) a
{\em based root datum}.  
\index{based root datum}

There are evident notions
of {\em isomorphism} and {\em duality} for based root data.
For a based root datum $(R, \Delta, \Delta^{\vee})$ with associated root system $\Phi$, 
$W(\Phi)$ is normal in ${\rm{Aut}}(R)$ because
$\theta s_a \theta^{-1} = s_{\theta(a)}$ for each
$\theta \in {\rm{Aut}}(R)$, as is seen from (\ref{refl}). Thus, if
$\Theta$ is 
the automorphism group of the based root datum then 
\begin{equation}\label{autr}
{\rm{Aut}}(R) = W(\Phi) \rtimes \Theta
\end{equation}
because the action of
$W(\Phi)$ on $X$ is simply transitive on the set of choices
of $(\Delta, \Delta^{\vee})$.
If $\Phi$ spans $X_{\Q}$ (or equivalently, $\Delta$ is a $\Q$-basis of
$X_{\Q}$) and the root system
$(X_{\Q},\Phi)$ is reduced then $\Theta$ is naturally identified with a subgroup of
the automorphism group of the Dynkin diagram ${\rm{Dyn}}(\Phi)$ of $\Phi$.
The interest in $\Theta$ is due to the fact (discussed below; see (\ref{autg}))
that it computes the outer automorphism group of a connected reductive $k$-group,
so for the connected semisimple case (corresponding to the condition that $\Phi$ spans $X_{\Q}$)
it is useful to know sufficient criteria
(in terms of terminology in Definition \ref{sssc}) for $\Theta$ to exhaust the automorphism group of the Dynkin diagram:

\begin{proposition}\label{autdyn}
Assume $\Phi$ spans $X_{\Q}$ and that $(X_{\Q},\Phi)$ is reduced. 
The inclusion $\Theta \subset {\rm{Aut}}({\rm{Dyn}}(\Phi))$ is an equality
if the root datum is adjoint or simply connected, 
or if $(X_{\Q}, \Phi)$ is irreducible
and $(\Z \Phi^{\vee})^{\ast}/\Z \Phi$ is cyclic.
\end{proposition}

\begin{proof}
In the adjoint cases of Example \ref{exsc}, the root datum
is a direct product of root data whose associated root system
is irreducible and adjoint; we have likewise in the simply connected case.  Thus,
we can assume $(X_{\Q}, \Phi)$ is irreducible. 
Hence, there exists a $W(\Phi)$-invariant inner product on
$X_{\RRR}$ that is unique up to scaling, and it is uniquely
determined by imposing the condition that the shortest root length is 1.
The resulting inner products among the roots are encoded
in the Dynkin diagram, as is the resulting
identification of the coroots with elements of $X_{\RRR}$ (see Remark \ref{remprod}
and Remark \ref{diagram}).

Identify the set of vertices of the diagram with $\Delta$. The diagram encodes
the Cartan matrix of integers $\langle a, b^{\vee}\rangle$
for $a, b \in \Delta$, so any automorphism of the diagram
uniquely extends to an automorphism of $X_{\RRR}$ preserving $\Phi$ and hence 
$\Phi^{\vee}$ \cite[VI, \S1.5, Cor.]{lie4}.
This automorphism of $X_{\RRR}$ carries
$X$ onto itself (or equivalently (by duality) carries $X^{\vee}$ onto itself).
Indeed, the adjoint and simply connected cases are obvious,
and if $\Pi := (\Z\Phi^{\vee})^{\ast}/\Z \Phi$ is cyclic 
then every automorphism of $\Pi$
carries each subgroup of $\Pi$ onto itself (with
$X$ uniquely determined by the corresponding subgroup
in $\Pi$).
\end{proof}

Here is a visualization of Proposition \ref{autdyn}
when $R = R(G,T)$
for a connected semisimple group $G$.
In Exercise \ref{scclassical}(ii) we define the
{\em simply connected central cover} $\widetilde{G} \rightarrow G$;
\index{simply connected!central cover}
\index{reductive group!simply connected cover}
this is a central isogeny onto $G$ from a connected semisimple group
that is initial among such central isogenies onto $G$.  
(If ${\rm{char}}(k)=0$ then $\widetilde{G}$ is simply connected as a variety,
in the sense that it has trivial \'etale fundamental group; this ultimately
rests on comparison with the theory of compact Lie groups for $k = \mathbf{C}$.
In contrast, if ${\rm{char}}(k) > 0$ then $\widetilde{G}$ never has trivial \'etale fundamental group if $G \ne 1$.) 
Every automorphism of a connected
semisimple group $G$ lifts (uniquely) to $\widetilde{G}$, but 
an automorphism $\widetilde{f}$ of 
$\widetilde{G}$ descends to $G$ if and only if its restriction to $Z_{\widetilde{G}}$
preserves the central subgroup $\mu = \ker(\widetilde{G} \rightarrow G)$.
If $\widetilde{f}$ is inner or 
$Z_{\widetilde{G}}$ is cyclic then preservation of $\mu$ is automatic
(because any automorphism of a finite cyclic group preserves every subgroup). 
In general, $\Theta$ is the group of outer automorphisms of $\widetilde{G}$ that arise from lifting
automorphisms of $G$ (see (\ref{autg}) below).

\begin{example}\label{d2n}
The only cases
with {\em irreducible} $\Phi$ when non-cyclicity occurs in Proposition \ref{autdyn}
are type ${\rm{D}}_{2n}$ ($n \ge 2$), with
$(\Z\Phi^{\vee})^{\ast}/(\Z\Phi) = (\Z/2\Z)^2$. 
Let $\widetilde{G} = {\rm{Spin}}_{4n}$ (type ${\rm{D}}_{2n}$).
Choose a maximal torus $T \subset \widetilde{G}$ and label ${\rm{Dyn}}(\Phi(\widetilde{G},T))$ as follows:
%Label the nodes in order along the ``long arm'' of ${\rm{Dyn}}(\Phi)$ as 
%$\{a_1,\dots,a_{2n-2}\}$ (with $a_{2n-2}$ the ``central node'' in ${\rm{Dyn}}(\Phi)$) 
%and label the other two nodes as $a_{2n-1}$ and $a_{2n}$.\edit{Insert a picture of ${\rm{D}}_{2n}$ with
% nodes labelled as just described; then the preceding sentence can be
% deleted.}
\begin{center}
\[
\xymatrix@M=2pt@R=-2pt{
&&&&&&&&\mathop{\circ}\limits_{\hbox to 0pt{~\(\scriptstyle
    a_{2n-1}\)\hss}}^t\\
%(t,t') \in \mu_2 \times\mu_2 \simeq Z_{\widetilde{G}}:\quad&
\mathop{\circ}\limits_{a_1}^{tt'} \ar@{-}[r]&\mathop{\circ}\limits_{a_2}^{1}\ar@{-}[r]
&\mathop{\circ}\limits_{a_3}^{tt'}\ar@{-}[r]&\mathop{\circ}\limits_{a_4}^{1}\ar@{-}[r]&\cdots\ar@{-}[r]
&\mathop{\circ}\limits_{\hbox to  0pt{\(\scriptstyle a_{2n-3}\quad\)\hss}}^{tt'}
&\mathop{\circ}\limits_{\hbox to  0pt{\(\scriptstyle a_{2n-2}\quad\)\hss}}^{1}\ar@{-}[rru]\ar@{-}[rrd]\ar@{-}[l]\\
&&&&&&&&\mathop{\circ}\limits_{\hbox to 0pt{~\(\scriptstyle a_{2n}\)\hss}}^{t'}
}\]
\end{center}
where $(t,t') \in \mu_2 \times\mu_2 \simeq Z_{\widetilde{G}}$.
We have $T = \prod_j a_j^{\vee}(\Gm)$, and 
the center $Z_{\widetilde{G}}$ is $\mu_2 \times \mu_2$ 
embedded in $T[2]$ by diagonally mapping 
each $\mu_2$ into the 2-torsion of the indicated coroot groups.

There are three intermediate groups $G$ strictly between $\widetilde{G}$
and $\widetilde{G}/Z_{\widetilde{G}}$ (one for each copy $\mu$
of $\mu_2$ in $\mu_2 \times \mu_2 \subset Z_{\widetilde{G}}$),
and ${\rm{Aut}}({\rm{Dyn}}(\Phi))$ has order 2 when
$n \ge 3$ and is $\mathfrak{S}_3 = {\rm{GL}}_2(\F_2)$ when $n = 2$.
Thus, if $n = 2$ then each $\mu \simeq \mu_2 \subset Z_{\widetilde{G}}$ has
${\rm{Aut}}({\rm{Dyn}}(\Phi))$-stabilizer of order 2, 
and if $n \ge 3$ then only one such $\mu$ is 
preserved under the order-2 group ${\rm{Aut}}({\rm{Dyn}}(\Phi))$; we claim that this corresponds to
the non-adjoint quotient $G = {\rm{SO}}_{4n}$.

More generally, noting that ${\rm{Spin}}_{2n}$ has center $\mu_4$ for odd $n \ge 3$ (e.g., ${\rm{Spin}}_6 = {\rm{SL}}_4$),
it suffices to show that for all $n \ge 2$, the group ${\rm{SO}}_{2n}$ has
$\Theta$ of order 2, rather than of order 1.  (In contrast, ${\rm{SO}}_{2n+1}$ for $n \ge 1$ has $\Theta = 1$
since ${\rm{Aut}}({\rm{Dyn}}({\rm{B}}_n)) = 1$ for $n \ge 1$; see Example \ref{so3} for $n = 1$.) 
In view of (\ref{autg}) below, this says exactly that the outer
automorphism of ${\rm{SO}}_{2n}$ has order 2 (rather than order 1).
Where does a non-inner automorphism come from?
Conjugation by ${\rm{O}}_{2n}$!  Indeed,
if $g \in {\rm{O}}_{2n}(k) - {\rm{SO}}_{2n}(k)$ 
and $g$-conjugation on ${\rm{SO}}_{2n}$ is inner then
by replacing $g$ with a suitable left ${\rm{SO}}_{2n}(k)$-translate
we would get that $g$ centralizes ${\rm{SO}}_{2n}$.  
But the ``diagonal'' maximal torus $T$ in ${\rm{SO}}_{2n}$ has ${\rm{O}}_{2n}$-centralizer
equal to $T$ (by explicit computation), so this is impossible.
\end{example}

The Isomorphism Theorem and (\ref{autr}) lead to the determination of the automorphism group of
a connected reductive $k$-group $G$, as follows.  
Inside ${\rm{Aut}}(G)$ there is the normal subgroup $G(k)/Z_{G(k)}$ of inner automorphisms;
for $g \in G(k)$ let $c_g$ denote the inner automorphism $x \mapsto gxg^{-1}$.
To describe the quotient group ${\rm{Out}}(G)$ of outer automorphisms,
fix a choice of $(B,T)$ as usual and consider an automorphism $\varphi$ of $G$.
By composing $\varphi$ with a suitable inner automorphism, we can
arrange that $\varphi(B) = B$ and $\varphi(T) = T$.  The only $g \in G(k)$ such that $\varphi \circ c_g$
preserves $B$ and $T$ are $g \in N_{G(k)}(B) \bigcap N_{G(k)}(T) = B(k) \bigcap N_{G(k)}(T) = T(k)$,
and this is precisely the ambiguity that arises by passing from $\varphi$ to the
induced automorphism of the root datum $R(G,T)$. 
The automorphism $\varphi$ induces an automorphism $\overline{\varphi}$ of the based root datum
associated to $(G,T,B)$, and by (\ref{autr}) the element 
$\overline{\varphi} \in \Theta$ determines the outer automorphism class of $\varphi$.
Moreover, every element of $\Theta$ arises in this way, due to the Isomorphism Theorem.
We thereby obtain a short exact sequence of abstract groups 
\begin{equation}\label{autg}
1 \rightarrow G(k)/Z_{G(k)} \rightarrow {\rm{Aut}}(G) \rightarrow \Theta \rightarrow 1.
\end{equation}

\begin{corollary}\label{autaut} The natural map ${\rm{Aut}}(G,T) \rightarrow {\rm{Aut}}(R(G,T))$
is surjective with kernel $T(k)/Z_{G(k)}$.
\end{corollary}

\begin{proof}
Since ${\rm{Aut}}(G,T) \cap (G(k)/Z_{G(k)}) = N_{G(k)}(T)/Z_{G(k)}$ inside ${\rm{Aut}}(G)$,
the quotient group ${\rm{Aut}}(G,T)/(T(k)/Z_{G(k)})$ contains
$W_G(T)$ as a subgroup that is carried isomorphically
onto the subgroup $W(\Phi) \subset {\rm{Aut}}(R(G,T))$.  The quotient
of ${\rm{Aut}}(G,T)/(T(k)/Z_{G(k)})$ modulo its normal subgroup $W_G(T)$
is clearly ${\rm{Out}}(G)$, and likewise
${\rm{Aut}}(R(G,T))/W(\Phi) = \Theta$.
This is compatible with the isomorphism
$\Theta \simeq {\rm{Out}}(G)$ defined via (\ref{autg}). 
\end{proof}

Remarkably, the short exact sequence (\ref{autg}) splits as a semi-direct product.
To formulate this, we need to more
structure beyond the triple $(G,T,B)$:

\begin{definition} A {\em pinning} of $(G,T,B)$
\index{pinning}
\index{reductive group!pinning}
is the specification of an isomorphism $p_a:\Add \simeq U_{a}$
for each $a \in \Delta$; equivalently, it is the choice of 
a nonzero  $X_a \in \mathfrak{g}_{a}$ for each $a \in \Delta$
(via ${\rm{Lie}}(p_a)(\partial_x) = X_a$, with $x$ the standard coordinate on $\Add$).
The data $(G,T,B,\{X_a\}_{a \in \Delta})$ is a {\em pinned} connected reductive group.
\end{definition}

(In \cite[IX, \S4.10, Def.\,3]{lie7} the analogous notion for a connected
compact Lie group is called a {\em framing}. Kottwitz, Langlands,
and Shelstad use the terminology {\em splitting}.)
There is an evident notion of {\em isomorphism} for pinned connected
reductive groups. 
Pinnings remove $T(k)$-conjugation ambiguity in the Isomorphism Theorem:

\begin{proposition}\label{pinsd} For $(G,T,B,\{X_a\}_{a \in \Delta})$ and $(G',T',B',\{X'_{a'}\}_{a' \in \Delta'})$, 
the natural map
\begin{equation*}
\xymatrix{
{\rm{Isom}}((G,T,B,\{X_a\}_{a \in \Delta}), (G',T',B',\{X'_{a'}\}_{a' \in \Delta'})) \ar[d] \\
{\rm{Isom}}((R(G,T),\Delta,\Delta^{\vee}), (R(G',T'), \Delta', {\Delta'}^{\vee}))}
\end{equation*}
is bijective.  In particular, if $f$ is an automorphism of $(G,T,B)$
that is the identity on $T$ and on the simple positive root groups
then $f$ is the identity on $G$, and a choice of pinning
on $(G,T,B)$ defines a homomorphic section to the quotient map ${\rm{Aut}}(G) \rightarrow 
{\rm{Out}}(G) \simeq \Theta$.
\end{proposition}

\begin{proof}
Since the elements of $\Delta$ are linearly independent in ${\rm{X}}(T)$, 
the map $T \rightarrow \Gm^{\Delta}$ defined by $t \mapsto (a(t))_{a \in \Delta}$
is surjective.  
(Indeed, otherwise the cokernel would map onto $\Gm$ via a quotient map
$\Gm^{\Delta} \rightarrow \Gm$ corresponding to a nonzero map $\Delta \rightarrow \Z$
and thereby give a nontrivial dependence relation on $\Delta \subset {\rm{X}}(T)$.) Thus, 
the group of inner automorphisms by $T(k)$ acts transitively on the set of
possible pinnings.  Hence, the Isomorphism Theorem guarantees that 
if the based root data are isomorphic then there exists an isomorphism between
the pinned data.  We may therefore pass to the case of automorphisms.

Since the $T(k)$-action is trivial on the root datum,
it follows from the Isomorphism Theorem that
any automorphism of the based root datum
arises from an automorphism of $(G,T,B,\{X_a\}_{a \in \Delta})$;
i.e., surjectivity is proved.
It remains to prove injectivity, so it suffices to show that if $f$ is an 
automorphism of $(G,T,B)$ that is the identity on $T$ (encoding
being the identity on ${\rm{X}}(T)$) 
and on the simple positive root groups (encoding preservation of the pinning) then $f$ is the identity on $G$.

Corollary \ref{autaut} implies that $f$ must be conjugation against
some $t \in T(k)$, but the condition on the simple positive root groups
implies that $a(t) = 1$ for all $a \in \Delta$, so $a(t) = 1$
for all $a \in \Phi$.  It follows that $t$ centralizes
every root group, and hence an open cell, so
$t$ is central in $G$.  That is, the conjugation action $f$ by $t$ is the identity map.
\end{proof}

We now discuss representation theory, which will only play a role in 
our proof of the Existence Theorem over $\C$ via analytic methods in Appendix \ref{csec}
(and in a few exercises in \S\ref{sec1Exer} that are not used anywhere else).  
The proof of the Existence Theorem over $\Z$ in \S\ref{existsec} rests on
the Existence Theorem over $\C$.

Fix a triple $(G,T,B)$, with $G$ a connected reductive
group over an algebraically closed field $k$ of any characteristic.  Let
$\Delta$ be the base of simple positive roots of 
$\Phi^+ = \Phi(B,T)$ in $\Phi = \Phi(G,T)$, and $W = W_G(T) = W(\Phi)$.
By  \cite[VI, \S1.5, Thm.\,2(ii),(vi)]{lie4}, ${\rm{X}}(T)$ is covered by
$W$-translates of the closed Weyl chamber 
\begin{equation}\label{weylc}
C = \{\lambda \in {\rm{X}}(T)\,|\,\langle \lambda, a^{\vee} \rangle \ge 0 \mbox{ for all }
a \in \Delta\}.
\end{equation}

For an irreducible representation $G \rightarrow \GL(V)$,
consider the $W$-stable finite set $\Omega_V \subset {\rm{X}}(T)$ of $T$-weights on $V$.

\index{highest weight theory|(}
\begin{theorem}[Theorem of the highest weight: group version]\label{highwt}
 There is a unique weight $\lambda_V \in \Omega_V \bigcap C$
that is highest in the sense that all weights in $\Omega_V$ have the form
$$\lambda_V - \sum_{a \in \Delta} n_a a$$
with integers $n_a \ge 0$, and the $\lambda_V$-weight space is $1$-dimensional
and $B$-stable.
For each $\lambda \in {\rm{X}}(T)$ there exists a unique irreducible 
representation $V_{\lambda}$ of $G$ with highest weight $\lambda$.
\end{theorem}

\begin{proof} 
See \cite[31.2--31.4]{humphreys}.
The relationship between constructions in characteristic 0
and positive characteristic is in \cite[II, 2.2, 2.4, 2.6]{jantzen}.
\end{proof}

Theorem \ref{highwt} is not used later, but to prove
the Existence Theorem over $\C$ by analytic methods we
need a link between the notions of 
``simply connected'' in the sense of topology and in the sense of
root data.  This rests on a variant of Theorem \ref{highwt}
for Lie algebras that is used in the proof of Proposition \ref{scgt}.
To state this variant, let $\mathfrak{g}$ be a semisimple Lie algebra
over an algebraically closed field $k$ of characteristic $0$,
and let $\mathfrak{t}$ be a Cartan subalgebra.
Fix a positive system of roots $\Phi^+$ in $\Phi(\mathfrak{g}, \mathfrak{t}) \subset \mathfrak{t}^{\ast} - \{0\}$,
and for each $a \in \Phi$ define an ``infinitesimal coroot''
$h_a \in \mathfrak{t}$ using $\mathfrak{sl}_2$ similarly to how
coroots are defined in reductive $k$-groups using ${\rm{SL}}_2$.
(See \cite[VI, 3.1(iii)]{serreclie} for another formulation.)
A linear form $\lambda \in \mathfrak{t}^{\ast}$ is
{\em integral} if $\lambda(h_a) \in \Z$ for all $a \in \Phi$,
and {\em dominant integral} if $\lambda(h_a) \in \Z_{\ge 0}$ for
all $a \in \Phi^+$ (equivalently, for all simple positive $a$).

\begin{theorem}[Theorem of the highest weight: Lie algebra version]\label{highwtlie}
For each dominant integral $\lambda \in \mathfrak{t}^{\ast}$,
there exists an irreducible finite-dimensional $\mathfrak{g}$-representation $V_{\lambda}$ 
whose highest weight for $\mathfrak{t}$ is $\lambda$
in the sense that its $\mathfrak{t}$-weights 
have the form $\lambda - \sum_{a \in \Delta} n_a a$ with
integers $n_a \ge 0$.  It is unique up to isomorphism and has a $1$-dimensional $\lambda$-weight space. 
\end{theorem}

\begin{proof}
See \cite[20.2, 21.1--21.2]{humlie} (and see \cite{humams} for a version in positive characteristic).
\end{proof}
\index{highest weight theory|)}

Existence  in Theorem \ref{highwtlie} is the key construction
in the analytic proof of the Existence Theorem for connected reductive groups over $\C$.
The character of  $V_{\lambda}$ 
 in Theorem \ref{highwt} 
is determined by its restriction to the regular semisimple locus of $G(k)$, a dense open locus
introduced in Exercise \ref{gross1}. Since the character is conjugation-invariant,
it is even determined by its restriction to the dense open set  of elements of $T(k)$ that are 
{\em regular} in $G(k)$ (see Exercise \ref{gross1}(ii)). 
This restriction is given by: 

\index{Weyl character formula}
\begin{theorem}[Weyl character formula]
Let $G$ be a nontrivial connected semisimple group over an algebraically closed
field $k$ of characteristic $0$, $T$ a maximal torus ,and $B$ a Borel subgroup containing $T$.
Assume that $G$ is simply connected.
Let $W = W_G(T)$, and let $\varepsilon = \varepsilon_B:W \rightarrow \{1, -1\}$ be the unique 
quadratic character carrying each simple reflection to $-1$ $($see $(\ref{wpres})$$)$.

For each $\lambda$ in the Weyl chamber of ${\rm{X}}(T)$ corresponding to $B$ 
$($see $(\ref{weylc})$$)$  
and for each regular semisimple $t \in T(k)$, the character of $t$ on 
$V_{\lambda}$ is given by
$${\rm{Tr}}(t|V_{\lambda}) = 
\frac{\sum_{w \in W} \varepsilon(w) t^{w(\lambda + \rho)}}{\sum_{w \in W} \varepsilon(w) t^{w\rho}},$$
where $t^a := a(t)$ for $a \in {\rm{X}}(T)$ and
$\rho := (1/2)\sum_{a \in \Phi^+} a \in {\rm{X}}(T)$.
\end{theorem}

\begin{proof}  
Using the weight space decomposition $V_{\lambda} = \bigoplus_{\mu \in {\rm{X}}(T)}
V_{\lambda}(\mu)$, the character $\chi_{\lambda}(t) = {\rm{Tr}}(t|V_{\lambda})$
is equal to $\sum_{\mu} \dim V_{\lambda}(\mu) t^{\mu}$.  Hence, the assertion of the formula is the identity
$$\left( \sum_{\mu \in {\rm{X}}(T)} \dim V_{\lambda}(\mu) t^{\mu}\right)
\left(\sum_{w \in W} \varepsilon(w) t^{w\rho}\right) = 
\sum_{w \in W} \varepsilon(w) t^{w(\lambda + \rho)}$$
in the group ring $\Z[{\rm{X}}(T)]$.
The $T$-weight spaces for a finite-dimensional $G$-representation $V$ are
the same as the $\mathfrak{t}$-weight spaces for $V$ viewed as a $\mathfrak{g}$-module
since $G$ is connected and ${\rm{char}}(k) = 0$.  Thus, by embedding 
${\rm{X}}(T) = \Hom(T,\Gm)$ into $\mathfrak{t}^{\ast}$ in the natural way, it is equivalent
to prove the analogue of the Weyl character formula
for $V_{\lambda}$ viewed as a $\mathfrak{g}$-module
(by identifying $\Phi(G,T)$ with $\Phi(\mathfrak{g}, \mathfrak{t})$).
For a proof of the Lie algebra version, see \cite[VIII, 9.1, Thm.\,1]{lie7}.
(The group version can be reduced to the case $k = \C$, in which case
it can be proved using integration on a compact form of the group; see
\cite[26.2]{fultonharris}.)
\end{proof}

\begin{remark} The regularity condition on $t$ is needed in the Weyl character formula 
to ensure that the denominator is nonzero; 
see Example \ref{weylsl2}.
Also, the integrality of $\rho$ can be made explicit: it is 
the sum of the elements in ${\rm{X}}(T)$ dual to the $\Z$-basis $\Delta^{\vee}$ of 
${\rm{X}}_{\ast}(T)$ (a $\Z$-basis
due to the simply connected condition on $G$; see Exercise \ref{scclassical}(ii)).  This is the ``dual'' of
the situation in Exercise \ref{gross6}(iii) (which treats
adjoint semisimple groups).
\end{remark}

\begin{example}\label{weylsl2}
Consider $G = {\rm{SL}}_2$ over an algebraically closed
field $k$ of characteristic 0, and let $T$ be the diagonal torus. Identify $\Gm$ with $T$
via $c \mapsto {\rm{diag}}(c,1/c)$, so ${\rm{X}}(T) = \Z$.  
Take $B$ to be the upper triangular Borel subgroup,
so $\Phi(B,T) = \{2\}$.  The Weyl chamber $C$ is $\Z_{\ge 0}$ 
and $\rho = 2/2 = 1$.

The irreducible representation of $G$ with highest weight $n \ge 0$ is the symmetric power
$V_n = {\rm{Sym}}^n(k^2)$. 
A regular semisimple element $t \in T(k)$ is any $t \in k^{\times}$ such that $t^2 \ne 1$.
Clearly 
$${\rm{Tr}}(t|V_n) = t^n + t^{n-2} + \dots + t^{-n} = 
\frac{t^{n+1} - t^{-(n+1)}}{t - t^{-1}},$$
and the right side is exactly the Weyl character formula in this case (likewise
illustrating the need for the regularity condition on $t$ in order that the denominator be nonzero).

If ${\rm{char}}(k) = p > 0$ then
these symmetric powers can be non-semisimple;
e.g., ${\rm{Sym}}^p(k^2)$ is not semisimple (Exercise \ref{1.6}(iii)) but
it has a unique irreducible quotient with
highest weight $p$.  In general,  the dimensions
of ``highest weight'' representations are not known when ${\rm{char}}(k) > 0$,
nor is there a formula for the character at regular semisimple points of $T(k)$.
\end{example}

\begin{example}  
\label{sp4picture} Consider \(G=\Sp_4\) over an algebraically closed
  field of characteristic \(0\).  The Dynkin diagram is
  \(\xymatrix{\mathop{\circ}\limits_\alpha\ar@{<=}[r]&\mathop{\circ}\limits_\beta}\).
  The weight lattice \({\rm{X}}(T)\) is represented
  by dots in the diagram below, and the arrows represent positive
  roots.  Note that \({\rm{X}}(T) = \mathbf{Z}\alpha + \mathbf{Z}(\beta/2)\) contains \(\mathbf{Z}\alpha+\mathbf{Z}\beta\) as a
  subgroup of index \(2\) (so the half-sum
  $\rho = 2\alpha + (3/2)\beta$ of the positive roots 
  lies in ${\rm{X}}(T)$ as it should).  
  The weights in the Weyl chamber (\ref{weylc}) are denoted by black dots
  and labelled with the dimension of the corresponding irreducible
  representation.
\begin{center}
\[
\xymatrix@M=1pt{
\circ&&\hbox to 0pt{\hss14~}\bullet&&\hbox to 0pt{\hss35~}\bullet&&\hbox to 0pt{\hss35~}\bullet\\
&\circ&&\hbox to 0pt{\hss16~}\bullet\hbox to
0pt{~\(\scriptstyle\rho\)}&&\hbox to 0pt{\hss 20~}\bullet&\\
\circ\hbox to 0pt{~\(\scriptstyle \beta\)\hss}&&
\hbox to 0pt{\hss5~}\bullet\hbox to 0pt{~\(\scriptstyle \beta+\alpha\)\hss}&&
\hbox to 0pt{\hss10~}\bullet\hbox to 0pt{~\(\scriptstyle \beta+2\alpha\)\hss}&&\circ\\
&\circ&&\hbox to 0pt{\hss 4~}\bullet&&\circ&\\
\circ&&\hbox to 0pt{\hss 1~}\bullet\ar@{->}[rr]\ar@{->}[rruu]\ar@{->}[uu]\ar@{->}[lluu]
%\ar@{->}[l]\ar@{->}[ld]\ar@{->}[d]\ar@{->}[rd]
&&\circ\hbox to 0pt{~\(\scriptstyle\alpha\)\hss}&&\circ\\
&\circ&&\circ&&\circ&
}\]
\end{center}
We can describe some irreducible representations of small dimension 
in more familiar terms:
\begin{itemize}
\item The \(4\)-dimensional representation \(V_4\) is the standard
  representation of \(G=\Sp_4\).
\item The \(5\)-dimensional representation \(V_5\) is \(\wedge^2V_4/L\),
  where \(L\) is the line in \(\wedge^2V_4\) fixed by \(G\) (i.e., the line
  spanned by the symplectic form). This can
  be regarded as the standard representation of \(\mathop{\rm SO}_5 \simeq \Sp_4/\mu_2\) (see
  Example \ref{so5}).
\item The \(10\)-dimensional representation is
  \(\mathop{\rm Sym}^2(V_4)\).
\item The \(14\)-dimensional representation is
  \(\mathop{\rm Sym}^2(V_5)/L'\), where \(L'\) is the line 
  \(\mathop{\rm Sym}^2(V_5)^{{\rm SO}_5}\) corresponding to
  the quadratic form. 
\item The \(20\)-dimensional representation is \(\mathop{\rm Sym}^3(V_4)\).
\end{itemize}
%\edit{We should include
%the picture of highest weights for various irreducible representations of ${\rm{Sp}}_4$
%in characteristic 0.}
\end{example}

\newpage

\subsection{Exercises}\label{sec1Exer} The first seven exercises
below require no specific background
in the classical theory beyond basic definitions that we assume are familiar
to the reader. However, some of those exercises are referenced in \S\ref{review}
to clarify features of the classical theory. 


\begin{exer}\label{gross3}
Let $G \subset \GL_3$ be the non-reductive connected solvable subgroup 
$$\begin{pmatrix} t & x & z \\ 0 & 1 & y \\ 0 & 0 & t^{-1} \end{pmatrix} \subset \GL_3$$
in which a maximal torus is given by $T = \Gm$ via $t \mapsto {\rm{diag}}(t, 1, 1/t)$.
Consider the weight space decomposition for $\mathfrak{g}$ under
the adjoint action of $T$. 

Show that the nontrivial character $\chi(t) = t$ occurs
with multiplicity 2, and that its inverse $\chi(t)^{-1} = 1/t$ occurs with
multiplicity 0.  (This contrasts with two important features of the reductive case,
namely that root spaces are 1-dimensional and that the set of roots
is stable under negation in the character lattice.)
\end{exer}

\begin{exer}\label{gmga}
(i) Prove that the only automorphisms of $\Add$ over a field $F$
are the usual $F^{\times}$-scalings, and conclude
that over a reduced ring $R$ the only automorphisms of
$\Add$ over $R$ are the usual $R^{\times}$-scalings.
(Hint: reduce to the case of noetherian $R$.)

(ii) Prove $\Gm$ represents the automorphism functor 
of $\Add$ on the category of $\Q$-algebras. (Hint: reduce to the noetherian
case, then induct on the nilpotence order of the nilradical.) 

(iii) Let $k$ be a field with ${\rm{char}}(k) = p > 0$. 
Prove that $\Gm$ does not represent the automorphism functor
of $\Add$ on $k$-algebras by giving 
an example of an automorphism of $\Add$ over the dual numbers
$k[\epsilon]$ that does {\em not} arise from the usual $\Gm$-action on
$\Add$.  

(iv) Despite (iii), prove any action by a torus $T$ on $\Add$ over
a field $k$ of any characteristic is given by $t.x = \chi(t)x$ for a homomorphism
of $k$-groups $\chi:T \rightarrow \Gm$.
(Hint: use (i) to get a homomorphism $\chi:T(k_s) \rightarrow k_s^{\times}$, 
and work over the function field $K = k(T)$ of $T$ to  prove $\chi$ is ``algebraic''
over $k_s$ and defined over $k$. Algebraicity is the main point.)
\end{exer}

\begin{exer}\label{1.1}
Define $\Z[x_{ij}]_{(\Det)}$ to be the degree-0 part of $\Z[x_{ij}][1/\Det]$
(i.e., the ring of fractions $f/\Det^e$ with $f$ homogenous and $\deg f = e\deg(\Det) = e n$).
Define ${\rm{PGL}}_n = \Spec(\Z[x_{ij}]_{(\Det)}) = 
\{\Det \ne 0\} \subset \mathbf{P}^{n^2-1}_{\Z}$. 
\index{ aaapgln@${\rm{PGL}}_n$}

(i) Construct
an injective map 
${\rm{GL}}_n(R)/R^{\times} \rightarrow
{\rm{PGL}}_n(R)$ 
natural in rings $R$.

(ii) Prove ${\rm{PGL}}_n$ is the Zariski-sheafification of 
$S \rightsquigarrow {\rm{GL}}_n(S)/\Gm(S)$ on the category of schemes, and that 
it has a unique $\Z$-group structure making 
${\rm{GL}}_n \rightarrow {\rm{PGL}}_n$ a homomorphism. 

(iii) Prove ${\rm{GL}}_n(R)/R^{\times} = 
{\rm{PGL}}_n(R)$ for local $R$, and construct a {\em counterexample} with $n = 2$
for any Dedekind domain $R$ with ${\rm{Pic}}(R)[2] \ne 1$. 

(iv) For local $R$ show ${\rm{Pic}}(\mathbf{P}^N_R) = \Z$
(generated by $\mathscr{O}(1)$) by using deformation from the
residue field and the theorem on formal functions
(after reducing to the case of noetherian $R$). 
Deduce that the evident action of ${\rm{PGL}}_n$ on
$\mathbf{P}^{n-1}_{\Z}$ via (ii) makes ${\rm{PGL}}_n$ represent the automorphism
functor $S \rightsquigarrow {\rm{Aut}}_S(\mathbf{P}^{n-1}_S)$ on the category of schemes.
(For example, the $\Z$-group ${\rm{PGL}}_2$ represents the automorphism functor of $\mathbf{P}^1_{\Z}$.)  By using intersection theory on $\mathbf{P}^1 \times \mathbf{P}^1$ over a field, adapt
the deformation arguments to prove that 
$({\rm{PGL}}_2 \times {\rm{PGL}}_2) \rtimes (\Z/2\Z)$ represents
the automorphism functor of $\mathbf{P}^1 \times \mathbf{P}^1$ in the evident manner.
(Hint: prove that ${\rm{Pic}}_{({\mathbf{P}}^1 \times \mathbf{P}^1)/\Z}$
is the constant $\Z$-group associated to $\Z \oplus \Z$.)

(v) Give a pre-Grothendieck proof (i.e., no functors, 
non-reduced schemes, or cohomology)
that if a linear algebraic group $G$ over a field $k$ acts
on $\mathbf{P}^{n-1}_k$ then the resulting homomorphism $G(k_s) \rightarrow 
{\rm{Aut}}_{k_s}(\mathbf{P}^{n-1}_{k_s}) = {\rm{PGL}}_n(k_s)$
arises from a $k$-homomorphism $G \rightarrow {\rm{PGL}}_n$.
\end{exer}

\begin{exer}\label{slp}
Construct natural Lie algebra
isomorphisms over $\Z$ between
$\mathfrak{pgl}_n := {\rm{Lie}}({\rm{PGL}}_n)$ and $\mathfrak{gl}_n/\mathfrak{gl}_1$, as well as between
$\mathfrak{sl}_n := {\rm{Lie}}({\rm{SL}}_n)$ with $\mathfrak{gl}_n^{{\rm{Tr}}=0}$ 
(kernel of the trace). 

(i) Construct a $\GL_n$-equivariant
duality between $\mathfrak{sl}_n$ and $\mathfrak{pgl}_n$ over $\Z$.

(ii) Over any field $k$ of characteristic $p > 0$, prove that
$\mathfrak{sl}_p$ and $\mathfrak{pgl}_p$ are not isomorphic
as representation spaces for the diagonal torus of ${\rm{SL}}_p$.
In particular, neither is self-dual as an ${\rm{SL}}_p$-representation space.

(iii) In the setup of (ii), prove that the central line ${\rm{Lie}}(\mu_p) \subset
\mathfrak{sl}_p$ consisting of scalar diagonal matrices
does not admit an ${\rm{SL}}_p$-equivariant complement.

(iv) Note that the conclusion of (ii) holds with 
$\mathfrak{sl}_p$ and $\mathfrak{pgl}_p$
respectively replaced by $\mathfrak{sl}_{p^e}$ and $\mathfrak{pgl}_{p^e}$.  
Verify that for $0 < f < e$, the central line in ${\rm{Lie}}({\rm{SL}}_{p^e}/\mu_{p^f})$
does have an ${\rm{SL}}_{p^e}$-equivariant complement, namely
the image of ${\mathfrak{sl}}_{p^e}$.
\end{exer}

\begin{exer}\label{geomconn}
Let $X$ be a connected scheme of finite type over a field $k$, and assume that $X(k) \ne \emptyset$.
Prove that $X$ is geometrically connected over $k$, which is to say that $X_K$ is connected for any field extension $K/k$,
or equivalently that $X_{\overline{k}}$ is connected. (Hint: prove connectedness of $X_{k_s}$ by considering the 
fiber over $x_0 \in X(k)$ for the open and closed projection map $X_K \rightarrow X$ with finite Galois extensions $K/k$.)
Deduce that a connected $k$-group scheme of finite type is geometrically connected over $k$;
this fact is often used without comment when working with fibers of finitely presented group schemes
in relative situations.
\end{exer}

\begin{exer}\label{trans}
Let $G$ and $G'$ be smooth connected groups over a field $k$, and 
$f:\Omega \rightarrow G'$ a $k$-morphism defined
on a dense open subset $\Omega \subset G$.  Assume
\index{rational homomorphism}
$f$ is a ``rational homomorphism'' in the sense that for a dense open subset $V \subset \Omega \times \Omega$ for which
$\omega\omega' \in \Omega$ for all $(\omega, \omega') \in V$, the morphism $V \rightarrow G$ given by
$(\omega, \omega') \mapsto f(\omega\omega')f(\omega')^{-1}f(\omega)^{-1}$ is identically 1.
Use pre-Grothendieck arguments (i.e., no
descent theory or scheme-theoretic methods) to prove
that $f$ uniquely extends to a $k$-homomorphism $\widetilde{f}:G \rightarrow G'$ and that if
$f$ is birational then $\widetilde{f}$ is an isomorphism.
\end{exer}


\begin{exer}\label{2.2}
Let a smooth finite type $k$-group $G$ act
linearly on a finite-dimensional $k$-vector space $V$.  Let $\underline{V}$ be the affine space
over $k$ whose $A$-points are $V_A := V \otimes_k A$ for any $k$-algebra $A$.  Define
$\underline{V}^G(A)$ to be the set of
$v \in V_A$ on which the $A$-group $G_A$ acts trivially
(i.e., $g.v = v$ in $V_R$ for all $A$-algebras $R$ and $g \in G(R)$).

(i) Prove $\underline{V}^G$ is represented by the closed subscheme 
associated to a $k$-subspace of $V$ (denoted $V^G$).  Hint:  use Galois descent to reduce
to the case $k = k_s$ and then prove $V^{G(k)}$ works.

(ii) For an extension field $K/k$, prove $(V_K)^{G_K} = (V^G)_K$ inside  $V_K$. 
\end{exer}

\begin{exer}\label{frobex} Let $G$ be a connected reductive group over an algebraically closed
field $k$ of characteristic $p > 0$, and let $T \subset G$ be a maximal $k$-torus. 
For any affine $k$-scheme $X$ of finite type, let 
\index{ aaaxpn@$X^{(p^n)}$}
$X^{(p^n)}$ denote the scalar extension by the $p^n$-power endomorphism of $k$,
and define $F_{X/k,n}:X \rightarrow X^{(p^n)}$ to be the natural $k$-morphism
\index{relative Frobenius morphism}
\index{ aaafxkn@$F_{X/k,n}, F_{X/k}, F_{G/k}$}
induced by the absolute $p^n$-Frobenius $F_X^n:X \rightarrow X$ over
the $p^n$-Frobenius $F_k^n:\Spec k \rightarrow \Spec k$.
This is the {\em $n$-fold relative Frobenius morphism for $X$ over $k$},
also denoted $F_{X/k}$ when $n=1$.

(i) Using the isomorphism $(\mathbf{A}_k^d)^{(p^n)} \simeq \mathbf{A}_k^d$
via the natural $\F_p$-descent of $\mathbf{A}^d_k$, compute $F_{\mathbf{A}^d_k,n}$
as an explicit $k$-endomorphism of $\mathbf{A}^d_k$.  Do the same for $\mathbf{P}^d_k$.

(ii) Prove that $F_{X/k,n}$ is functorial in $X$, compatible with direct products in $X$ (over $k$),
and compatible with extension of the ground field.   Deduce that if $X$ is
a $k$-group and $X^{(p^n)}$ is made into a $k$-group via scalar extension then
$F_{X/k,n}$ is a $k$-homomorphism. 

(iii) If $X$ is smooth of pure dimension $d > 0$ then prove that $F_{X/k,n}$ is finite flat
of degree $p^{dn}$.  In particular, $F_{G/k,n}:G \rightarrow G^{(p^n)}$ is
an isogeny of degree $p^{dn}$ carrying $T$ onto $T^{(p^n)}$.

(iv) Compute $F_{G/k,n}$ for ${\rm{GL}}(V)$ and ${\rm{SO}}(q)$, and prove
${\rm{Lie}}(F_{G/k}) = 0$ for any $G$.  In general, compute the effect on root data arising from
$F_{G/k,n}:(G,T) \rightarrow (G^{(p^n)},T^{(p^n)})$.
\end{exer}




\begin{exer}\label{gross1}
Let $G$ be a connected linear algebraic group over an algebraically closed field $k$.
By Remark \ref{borelcover}, the set of semisimple elements of
$G(k)$ is the union $\bigcup_T T(k)$ as $T$ varies through the maximal tori of $G$.

(i) Prove that for semisimple $g \in G(k)$, ${\rm{Lie}}(Z_G(g)) = \mathfrak{g}^{g=1}$ and 
$Z_G(g)^0$ contains some Cartan subgroup $Z_G(T)$ of $G$
(these are the maximal tori when $G$ is reductive). Deduce that
$\dim Z_G(g)$ coincides with the common dimension of the Cartan subgroups if and only if 
$Z_G(g)^0$ is a Cartan subgroup.

(ii) Prove that there exist semisimple $g \in G(k)$ such that $Z_G(g)^0$ is a Cartan subgroup.
(Hint: For a maximal torus $T$, consider the {\em finitely many} nontrivial
$T$-weights that occur on $\mathfrak{g}$.)
An element $g \in G(k)$ is {\em regular} when $Z_G(g_{\rm{ss}})^0$ is a Cartan subgroup.
Using that $g \in Z_G(g_{\rm{ss}})^0$ \cite[11.12]{borelag}, deduce that
$g$ is regular if and only if it belongs to a unique Cartan subgroup.
(For $G = \GL(V)$, these are precisely the $g \in G(k)$ whose characteristic
polynomial has non-zero discriminant since $g$ and $g_{\rm{ss}}$ have
the same characteristic polynomial, so this is a Zariski-dense open locus in $\GL(V)$.)

(iii) By considering the multiplicity of $x-1$ as a factor of the characteristic polynomial for the adjoint
action of $G$ on $\mathfrak{g}$, prove that the regular 
locus of $G(k)$ is  a (non-empty) Zariski-open
subset.  (Hint: for any $g \in G(k)$, ${\rm{Ad}}_G(g)$ and ${\rm{Ad}}_G(g_{\rm{ss}})$ have the same
characteristic polynomial on $\mathfrak{g}$.)

(iv) Steinberg proved that all regular elements in $G(k)$ are semisimple when $G$ is reductive.
\index{Steinberg!regular elements in reductive groups}
Without using this fact, prove that for reductive $G$ the non-empty locus of regular semisimple elements in $G(k)$ is
Zariski-open.  (This is false if $G$ is nontrivial and unipotent!)  In other words, within
the dense Zariski-open locus of regular elements, prove that the non-empty semisimple locus is open. 
(Hint: consider $\dim Z_G(g)$ rather than $\dim Z_G(g_{\rm{ss}})$, and apply
semicontinuity of fiber dimension to a ``universal centralizer scheme'' over $G$.) 
\end{exer}

\begin{exer}\label{1.4}
Let $k$ be a field.  To define special orthogonal groups over rings in a uniform way, 
we need a characteristic-free definition of
non-degeneracy for a quadratic form
$q:V \rightarrow k$ on a finite-dimensional $k$-vector space
$V$ of dimension $d \ge 2$.  We say $q$ is {\em non-degenerate} when
$q \ne 0$ and $(q=0)$ is smooth in $\mathbf{P}(V^{\ast}) := {\rm{Proj}}({\rm{Sym}}(V^{\ast})) 
\simeq \mathbf{P}^{d-1}_k$.  
\index{quadratic form!non-degenerate}
\index{non-degenerate quadratic form}

(i) Let $B_q:V \times V \rightarrow k$ be the symmetric bilinear form
$(v,v') \mapsto q(v+v') - q(v) - q(v')$, and define 
$V^{\perp} = \{v \in V\,|\,B_q(v,\cdot)= 0\}$; we call $\delta_q := \dim V^{\perp}$ the {\em defect}
of $q$.  Prove that $B_q$ uniquely factors through a non-degenerate symmetric bilinear form
on $V/V^{\perp}$, and that $B_q$ is non-degenerate precisely when the defect is 0. 
Also show that if ${\rm{char}}(k) = 2$ then $B_q$ is alternating, and deduce
that $\delta_q \equiv \dim V \bmod 2$ for such $k$ (so $\delta_q \ge 1$ if $\dim V$ is odd). 

(ii) Prove that if $\delta_q = 0$ then $q_{\overline{k}}$ 
admits one of the following ``standard forms'':
$\sum_{i=1}^n x_{2i-1} x_{2i}$ if $\dim V = 2n$ ($n \ge 1$), and 
$x_0^2 + \sum_{i=1}^n x_{2i-1} x_{2i}$ if $\dim V = 2n+1$ ($n \ge 1$). 
Do the same if ${\rm{char}}(k) = 2$ and $\delta_q = 1$. (Distinguish whether or
not $q|_{V^{\perp}} \ne 0$.) 
How about the converse?  

(iii) If ${\rm{char}}(k) \ne 2$, prove $q$ is non-degenerate
if and only if $\delta_q = 0$. For ${\rm{char}}(k) = 2$, prove $q$ is non-degenerate
if and only if $\delta_q \le 1$ with $q|_{V^{\perp}} \ne 0$
when $\delta_q = 1$. Hint: use (ii) to simplify calculations.
(In \cite[\S7]{ekm} $(V,q)$ is called {\em regular} if $q$ has no nontrivial zeros on $V^{\perp}$.  This is equivalent to
non-degeneracy if ${\rm{char}}(k) \ne 2$ or if
${\rm{char}}(k)=2$ with $\dim V^{\perp} \le 1$. In general regularity is preserved by separable extension
on $k$ and is equivalent to 
the zero scheme $(q=0) \subset \mathbf{P}(V^{\ast})$ being regular at $k$-points.) 
\end{exer}


\begin{exer}\label{1.6}
Let $G$ be a smooth affine group over a
field $k = \overline{k}$ (allow $G \ne G^0$). 

(i) If all finite-dimensional linear representations of $G$ are completely reducible, or
if there is even a single faithful semi-simple representation of $G$, then prove
that $G^0$ is reductive.  (Hint: use Lie--Kolchin and the behavior of semisimplicity under restriction
to a normal subgroup.)  Deduce that ${\rm{GL}}(V)$ is reductive
and ${\rm{SL}}(V)$ is semisimple. 

(ii) Conversely, assume $G^0$ is reductive and ${\rm{char}}(k) = 0$. 
Prove that finite-dimensional linear representations
of $G$ are completely reducible. (Hint: prove
${\rm{Lie}}(\mathscr{D}(G))$ is semisimple.)

(iii) Let $V$ be the standard 2-dimensional representation of
${\rm{SL}}_2$ over $k$, and assume ${\rm{char}}(k) = p > 0$.
In ${\rm{Sym}}^p(V)$, prove that the line of $p$th powers has
no ${\rm{SL}}_2$-equivariant complement (so ${\rm{Sym}}^p(V)$
is not semisimple as a representation, and hence $V^{\otimes p}$ is not semisimple).
\end{exer}

\begin{exer}\label{normaltorus} Let $G$ be a linear algebraic group over an algebraically
closed field $k$, $N$ a normal linear algebraic
subgroup (e.g., $N = \mathscr{D}(G)$), and $T$ a maximal torus in $G$.
Prove that $(T \bigcap N)^0_{\rm{red}}$ is a maximal torus in $N$.
Hint: argue in reverse, starting with a maximal torus in $N$
and extending it to a maximal torus in $G$, which in turn is conjugate to $T$. 
(See Example \ref{cent} and Exercise \ref{5.3} for
smoothness and connectedness properties
of $T \bigcap N$.) 
\end{exer}

\begin{exer}\label{scclassical}
Let $G$ be a connected semisimple group over an algebraically closed field $k$. 
The {\em central isogeny class} of $G$ is 
\index{central isogeny}
the equivalence class of $G$ generated by the relation on connected semisimple $k$-groups
$G'$ and $G''$ that there exists a central isogeny $G' \rightarrow G''$ or
$G'' \rightarrow G'$ (i.e., an isogeny whose scheme-theoretic kernel is
central).  Composites of central isogenies between
connected reductive groups are central (due to Corollary \ref{qtcenter} applied
over $k$, where the proof simplifies); this is false
for general smooth connected affine groups in positive characteristic (Exercise \ref{cenexer}(ii)).

(i) Prove that any central isogeny $f:G' \rightarrow G''$ can be arranged via composition with a suitable conjugation 
to satisfy $f(T') = T''$ and $f(B') = B''$ for any choices of Borel subgroups and maximal
tori that they contain, and use the open cells to show that such an $f$ must induce isomorphisms
between root groups (false for Frobenius isogenies!).  
Deduce via Corollary \ref{zgood}, the Isomorphism Theorem, 
and (\ref{xphi}) that if $G' \simeq G''$ abstractly then $f$ must be an isomorphism
and that if $G' \not\simeq G''$ then up to conjugacy $f$ is the only central isogeny between
$G'$ and $G''$ (in either direction!).  In particular, a central
isogeny class is {\em partially ordered} (when members
are considered up to abstract $k$-isomorphism). 

(ii) Using the Existence and Isomorphism Theorems over $k$, prove
the equivalence of the following conditions on $G$:
the only central isogenies $G' \rightarrow G$ from  connected semisimple groups are isomorphisms,
$G$ dominates all other members of its central isogeny class, and
the simple positive coroots of $(G,T)$ (relative
to a choice of $\Phi^+$) are a $\Z$-basis of the cocharacter group of $T$.
Under these conditions, we say $G$ is {\em simply connected}
(e.g., ${\rm{Sp}}_{2n}$ and ${\rm{SL}}_n$).
\index{reductive group!simply connected}

Likewise prove the equivalence of:
${\rm{Ad}}_G$ is a closed immersion, $G$ is dominated by all members of its central isogeny class,
and the simple positive roots of $(G,T)$ (relative to a choice
of $\Phi^+$) are a $\Z$-basis of the character group of $T$.
Under these conditions we say $G$ is {\em adjoint} (e.g., ${\rm{PGL}}_n$
and ${\rm{SO}}_{2n+1}$). 
\index{reductive group!adjoint}

(iii) Let $T \subset G$ be a maximal torus.  For each $a \in \Phi(G,T)$, let
$T_a = (\ker a)^0_{\rm{red}}$ be the unique codimension-1 torus in $T$ killed by $a$.
By the structure theory underlying
the {\em definition} of coroots, $G_a := \mathscr{D}(Z_G(T_a))$ is either ${\rm{SL}}_2$ or
${\rm{PGL}}_2$ with maximal torus $a^{\vee}(\Gm)$
having root groups $U_a$ and $U_{-a}$.

If $G$ is simply connected, prove that $G_a = {\rm{SL}}_2$ for all $a \in \Phi(G,T)$.
Show that the converse is false by proving that 
$G_a = {\rm{SL}}_2$ for all $a \in \Phi(G,T)$ when $G = {\rm{PGL}}_n$
with $n \ge 3$.
\end{exer}

\def\SO{\mathop{\rm SO}\nolimits}
\def\Sp{\mathop{\rm Sp}\nolimits}
\def\SL{\mathop{\rm SL}\nolimits}
\def\injto{\hookrightarrow}
\begin{exer} 
Let \(U\) be a nonzero finite-dimensional vector space over
a field  \(k\), and \(U^*\) its dual.  Define \(W=U\bigoplus U^*\). 

(i)  Define $\psi_0:W \times W \rightarrow k$ by \(\psi_0(
  (v,f),(v',f')) =f'(v)-f(v')\).  Show that \((W,\psi_0)\) is a non-degenerate symplectic space.  Let \(\GL(U)\)
  act on \(W\) by \(g.(v,f)=(g.v,f\circ g^{-1})\).  Show that this
  defines a monic homomorphism (hence closed immersion) of \(k\)-groups \(\GL(U) \to
  \GL(W)\) and that the image lies in \(\Sp(W,\psi_0)\).

(ii) Define a quadratic form \(q_0\)
  on \(W\) by \(q_0(v,f)=f(v)\).  Show that \(q_0\) is non-degenerate and that
 the image of \(\GL(U) \to \GL(W)\) also lies in \(\SO(W,q_0)\).
 \end{exer}

\begin{exer}\label{vuex} Let \(U=k^2\) for an algebraically closed field \(k\) and define \(W,\psi_0,q_0\) as in
  the previous exercise.  Thus, we have inclusions \(\SL_2 \subset
  \GL_2 \subset \Sp(W,\psi_0)\) and \(\SL_2 \subset
  \SO(W,q_0)\).  In this exercise, you can take for granted that the
  groups \(\SO(V,q)\) and \(\Sp(V,\psi)\) are connected reductive groups.
For such groups, we shall interpret the maps $\varphi_a$ from Theorem \ref{rtgp}.


(i) Let \((V,q)\) be a non-degenerate quadratic space of dimension
\(2n\) over \(k\) with \(n \ge 2\).  Consider the diagonal maximal  torus
$T$ in $G = {\rm{SO}}(V,q)$ as in Proposition \ref{typeBD} relative to a basis of $V$ putting $q$ into the ``standard form''
as in Exercise \ref{1.4}(ii). 
Compute \(\Phi(G,T)\).  For each \(a \in\Phi(G,T)\), show that
\(\varphi_a:\SL_2\to G\) arises from the following construction: take
an embedding \(i: (W,q_0) \injto (V,q)\) of quadratic spaces, show $W \oplus W^{\perp} = V$, and use this to define
an embedding \(\SO(W,q_0)\injto \SO(V,q)\) in an obvious
manner (yielding \(\SL_2 \injto \SO(W,q_0) \injto \SO(V,q)\)).

(ii) Let \((V,q)\) be a non-degenerate quadratic space of dimension
\(2n+1\) over \(k\) with \(n \ge 1\).  
 Consider the diagonal maximal  torus
$T$ in $G = {\rm{SO}}(V,q)$ as in Proposition \ref{typeBD} relative to a basis of $V$ putting $q$ in ``standard form''
as in Exercise \ref{1.4}(ii). 
Compute \(\Phi(G,T)\).  For each \(a \in\Phi(G,T)\), show that
\(\varphi_a:\SL_2\to G\) arises from one of the following two
constructions: (a) take
an embedding \(i: (W,q_0) \injto (V,q)\) of quadratic spaces and form
\(\SL_2 \injto \SO(W,q_0) \injto \SO(V,q)\) as in (i); (b) take a
\(3\)-dimensional subspace \(V_3\) of \(V\) such that \((V_3,q)\) is
non-degenerate, so \(\SO(V_3,q)\simeq\PGL_2\) (Example \ref{so3}), 
and show $V_3 \oplus V_3^{\perp} = V$, yielding a homomorphism \(\SL_2 \to \SO(V_3,q)\injto
\SO(V,q)\). Case (a) is for the long roots, and (b)
is for the short roots. 

(iii) Let \((V,\psi)\) be a non-degenerate symplectic
space of dimension \(2n\) over \(k\).  Construct a maximal torus \(T\)
of \(G=\Sp(V,\psi)\) and compute \(\Phi(G,T)\).  For each
\(a \in \Phi(G,T)\), show that \(\varphi_a:\SL_2\to G\) arises from
one of the following two constructions: (a) take an embedding
\(i:(W,\psi_0) \injto (V,\psi)\) and form
\(\SL_2 \injto \Sp(W,\psi_0)\injto \Sp(V,\psi
)\); (b) take a \(2\)-dimensional subspace \(V_2\) of \(V\) such that
\((V_2,\psi)\) is non-degenerate, so we have 
\(\SL_2=\Sp(V_2,\psi)\injto \Sp(V,\psi)\).
(Determine which of (a) or (b) corresponds to long roots and 
short roots.) 

\end{exer}

\def\grg{\mathfrak{g}}
\begin{exer}\label{redexer}
 (i) Let \(G\) be a connected linear algebraic group over an  algebraically closed
field $k$, and
let  \(T\) be a torus of \(G\).  Let \(\grg=\bigoplus_{a\in
    {\rm{X}}(T)}\grg_a\) be the weight space decomposition of \(\grg=\Lie G\) for
  the action of \(T\).  Let $\Phi = \{a \in {\rm{X}}(T) - \{0\}\,|\,\grg_a \ne 0\}$. Assume:
  \begin{itemize}
  \item \(\grg_0=\Lie T\) (so $T$ is maximal as a torus of $G$) and \(\dim \grg_a\le  1\) for all $a \in \Phi$; 
    \item $\Phi = -\Phi$ and if
    $a, b \in \Phi$ are $\Z$-linearly dependent then $b = \pm a$;
      \item for all $a \in \Phi$ there exists a homomorphism
  \(\varphi_a:\SL_2 \to G\) with finite kernel such that the image
  commutes with the codimension-1 torus $T_a = (\ker a)^0_{\rm{red}} \subset T$. 
  \end{itemize}
Show that \(G\) is a reductive group.  (Hint: prove \(\Lie
(\mathscr{R}_u(G_{\overline{k}}))=0\). Also see Lemma \ref{redcrit}.)

(ii) Prove ${\rm{SO}}_n$ and ${\rm{Sp}}_{2n}$ are reductive
(grant each is connected and smooth).
\end{exer}

\begin{exer}\label{dualexer} Let $\Delta$ be the base of a positive system of roots $\Phi^+$ 
in a reduced root system $(V,\Phi)$.  Consider
the reduced dual root system $(V^{\ast}, \Phi^{\vee})$ arising
from the coroots (where $a^{\vee}$ is the linear form that computes
the unique reflection $s_a:V \simeq V$ preserving $\Phi$
and negating $a$; i.e., $s_a(v) = v - a^{\vee}(v)a$). 

(i) Assume $(V,\Phi)$ is irreducible and fix a $W(\Phi)$-invariant positive-definite quadratic
form $Q:V \rightarrow \Q$ on $V$ (unique up to scaling, by
Remark \ref{remprod}).  Prove that under the resulting isomorphism $V^{\ast} \simeq V$,
$a^{\vee}$ is identified with $2a/Q(a)$.

(ii) Let $\Delta^{\vee}$ be the set of coroots $a^{\vee}$
for $a \in \Delta$. Using (i), prove that $\Delta^{\vee}$ is a basis of
$V^{\ast}$ and that every element of $\Phi^{\vee}$ is
a linear combination of $\Delta^{\vee}$ with all coefficients in $\Q_{\ge 0}$ or in $\Q_{\le 0}$.
Use the equivalent characterizations of bases of root systems
in \cite[XXI, 3.1.5]{sga3} to deduce that $\Delta^{\vee}$ is the base of a positive system
of roots in $\Phi^{\vee}$.  

(iii) Assume $(V,\Phi)$
is irreducible and choose a nonzero $\lambda \in V^{\ast}$ 
such that $\Phi_{\lambda > 0} = \Phi^+$ (so $\langle \lambda, a \rangle \ne 0$ for
all $a \in \Phi$).  For $Q$ as in (i) and $v \in V$
satisfying $B_Q(v,\cdot) = \lambda$, prove that $v$ is not orthogonal to
any coroot and that $\Delta^{\vee}$ is a base
for the positive system of roots $\Phi^{\vee}_{v > 0}$. 

(iv) Using (iii), prove that if $(V,\Phi)$ is irreducible then
the Dynkin diagrams for $\Phi$ and $\Phi^{\vee}$ coincide 
in the simply laced case (i.e., one root length)
and otherwise are related by swapping the direction of the unique multiple edge.
(Hence, by the classification of irreducible reduced 
root systems, the diagrams coincide except that ${\rm{B}}_n$ and ${\rm{C}}_n$ are swapped
for $n \ge 3$.) 
\end{exer}

\newpage

\section{Normalizers, centralizers, and quotients}\label{normsec}

To motivate the need for a theory of reductive groups over rings, 
consider a connected reductive group $G$ over a number field $K$
with ring of integers $R$,
Choose a faithful representation $G \hookrightarrow {\rm{GL}}_{n,K}$ over $K$.
The schematic closure of $G$ in the $R$-group ${\rm{GL}}_{n,R}$
is a flat closed $R$-subgroup $\mathscr{G}$ of ${\rm{GL}}_{n,R}$.
By direct limit considerations (and Exercise \ref{geomconn}), if $a \in R$ is  nonzero and
sufficiently divisible (i.e., $\Spec R[1/a]$ is
a sufficiently small neighborhood of the generic point in $\Spec R$)
then $\mathscr{G}_{R[1/a]}$ is smooth over $\Spec R[1/a]$ with (geometrically) connected
fibers (see \cite[IV$_3$, 9.7.8]{ega}). 
Is $\mathscr{G}_{\overline{s}}$ reductive for all $s \in \Spec R[1/a]$,  perhaps after
making $a$ more divisible in $R$, and if so then is
the isomorphism type (of the associated root datum) independent of $s$?  
In such cases we wish to say that $\mathscr{G}_{R[1/a]}$ is a reductive
group scheme over $R[1/a]$.  

Although $G$ may not have a Borel $K$-subgroup, $G_{K_v}$ has
a Borel $K_v$-subgroup for all but finitely many $v$.  This
assertion only involves the theory of reductive groups over fields (of
characteristic 0), but its proof 
uses the notion of reductive group over localized rings of integers.
The theory of reductive groups
over discrete valuation rings links the theories over
$\F_p$ and $\Q$, as well as relates finite groups of Lie
type to complex semisimple Lie groups.

We also seek a conceptual understanding
of the smooth affine $\Z$-groups constructed by Chevalley
in \cite{chevbour} (with $\Q$-split semisimple generic fiber).
The Bruhat--Tits structure theory for $p$-adic groups, the study of integral models of
Shimura varieties, 
and Galois deformation theory valued in reductive groups provide further
motivation for the notion of reductive group over rings. 

The main result that we are aiming for is this: 
over {\em any} non-empty scheme $S$, there are analogues
of the Existence and Isomorphism Theorems for connected reductive groups
over algebraically closed fields.  More specifically, we will show that
the category of ``split'' reductive $S$-groups (equipped with a suitable notion of
isomorphism as the morphisms) is the same as the category of root data
(equipped with isomorphisms as the morphisms); the latter has nothing to do with $S$!

\medskip

In the classical theory, normalizers and centralizers of tori are ubiquitous tools for creating interesting
subquotients of a non-solvable 
smooth connected affine group.  In the relative theory these constructions
remain essential, but we need to
 use torsion-levels in a  torus because those are {\em finite} flat over the base (unlike the torus).
Such finiteness makes the torsion-levels very useful in proving representability results for functorial
normalizers and centralizers of tori.  The method of passing to torsion-levels in tori 
has a useful variant when considering normalizers and centralizers
of smooth closed subgroups with connected fibers: passing to infinitesimal neighborhoods of
the identity (which are often finite flat closed subschemes, but generally not subgroups).  
The construction of quotients by normalizers of subgroups will also use the consideration of
 ``finite flat approximations'' to subgroups. 

Our use of torsion levels in tori forces us to 
consider general (possibly non-smooth) groups of multiplicative type.
See Appendix \ref{multsec} for the basics of the theory of group schemes of multiplicative type, including definitions
and notation (and precise references within \cite{oesterle}
for proofs of some fundamental results). The reader should
learn the material in Appendix \ref{multsec} before continuing on to the discussion that follows.
(For example, by Lemma \ref{ix2.5}, any monic homomorphism $H \rightarrow G$ from a multiplicative type
$S$-group to an $S$-affine $S$-group of finite presentation is a closed immersion.)

\subsection{Transporter schemes and Hom schemes}


Let $G$ be an $S$-group, and $Y, Y' \rightrightarrows G$ monic $S$-morphisms from $S$-schemes.
The transporter functor on $S$-schemes is defined to be
\index{transporter functor, scheme}
$$\underline{\rm{Transp}}_G(Y, Y'):  S' \rightsquigarrow \{g \in G(S')\,|\,g(Y_{S'})g^{-1} 
\subset Y'_{S'}\}.$$
 In the special case
$Y' = Y$ with $Y$ of finite presentation over $S$, this coincides with the normalizer functor 
$$\underline{N}_G(Y):S' \rightsquigarrow  \{g \in G(S')\,|\, g Y_{S'} g^{-1} = Y_{S'}\}$$
because the monic endomorphism of $Y_{S'}$ given by $g$-conjugation is 
\index{normalizer!functor, scheme}
an automorphism, as for any finitely presented scheme over any base \cite[IV$_4$, 17.9.6]{ega}.
When these functors are representable, we denote
representing objects as ${\rm{Transp}}_G(Y,Y')$ and $N_G(Y)$ respectively.
\index{ aaatransp@$\underline{\rm{Transp}}_G(Y,Y')$, ${\rm{Transp}}_G(Y,Y')$}
\index{ aaangy@$\underline{N}_G(Y)$, $N_G(Y)$}

\begin{definition}\label{normdef} A finitely presented $S$-subgroup $G'$ in $G$ is {\em normal} if
\index{normal subgroup scheme}
$\underline{N}_G(G') = G$, or equivalently for all $S$-schemes $S'$ the 
conjugation on $G_{S'}$ by any $g \in G(S')$ carries
$G'_{S'}$ into (and hence isomorphically onto) itself.
\end{definition}

Another formulation of this definition is that $G'(S')$ is a normal subgroup of
$G(S')$ for every $S$-scheme $S'$.  Normality
of $G'$ in $G$ {\em cannot} be checked on geometric points
even when $G'$ is smooth and $S = \Spec k$
for an algebraically closed field $k$; see Example \ref{centerex}. 



%
%The multiplicative type $S$-subgroups $H$ and $\mathscr{H}'$ in the commutative 
%$\widehat{H}'$
%satisfy the containment relation $H_0 \subset \mathscr{H}'_0$ inside 
%$\widehat{H}'_0$ for their special fibers.
%We claim that $H \subset \mathscr{H}'$.   The product map
%$H \times \mathscr{H}' \rightarrow \widehat{H}'$ makes sense as an 
%$\widehat{S}$-homomorphism since
%$\widehat{H}'$ is commutative, and by \cite[IX, 6.8]{sga3} this map factors
%through an fppf quotient onto a multiplicative type
%$\widehat{S}$-subgroup $K$.  By construction $K$ contains $\mathscr{H}'$, but its special fiber is
%an fppf quotient of $H_0 \times \mathscr{H}'_0$ via addition in $\widehat{H}'_0 = H'_0$,
%so since $H_0 \subset \mathscr{H}'_0$ inside $H'_0$ we conclude
%that the containment $\mathscr{H}' \subset K$ of $\widehat{S}$-subgroups 
%of $\widehat{H}'$ induces an equality between special fibers.  But $K$ and $\mathscr{H}'$ are both
%of multiplicative type
%and $\widehat{A}$ is complete local noetherian, so necessarily $\mathscr{H}' = K$.  
%Thus, $H \subset K = \mathscr{H}'$ as desired.
%
%Consider the map $\mathscr{H}' \rightarrow \overline{H}' := \widehat{H}'/H$ between 
%$\widehat{S}$-groups of
%multiplicative type.  On special fibers, this is the quotient map $\mathscr{H}'_0 = H'_0 \rightarrow
%\overline{H}'_0$, so the induced map between constant character
%groups is injective.  Hence,  the $\widehat{S}$-homomorphism $\mathscr{H}' \rightarrow \overline{H}'$ 
%is dual to an injection between character groups, so it is faithfully flat.
%Thus, $\mathscr{H}' \rightarrow \overline{H}'$ is an fppf quotient homomorphism whose
%kernel is $H$, yet $\mathscr{H}'$ is an $\widehat{S}$-subgroup of 
%the $\widehat{S}$-group 
%$\widehat{H}'$ that is {\em also} an extension of $\overline{H}'$ by $H$ for the fppf topology
%(compatibly with the analogous such extension structure on $\mathscr{H}'$).
%A snake lemma argument with fppf group sheaves then implies that the inclusion
%$\mathscr{H}' \subset \widehat{H}' = H'_{\widehat{S}}$ is an equality, so 
%H'_{\widehat{S}}$ is of multiplicative type 
%and (by construction of $\mathscr{H}'$) it is split by a {\em finite \'etale} cover of $\widehat{S}$. 
%
%Now we descend from $\widehat{A}$ to $A$.  The categories
%of finite \'etale covers of $A$ and $\widehat{A}$ coincide (via comparison
%through their residue fields) since $A$ is henselian local.  Thus,
%the description of $H'_{\widehat{S}} = \mathscr{H}'$ in
%terms of ``finite \'etale twisting data'' on a split multiplicative type group
%over $\widehat{S}$ descends uniquely over $S$.  That is,
%$H'_{\widehat{S}}$ uniquely descends to an $S$-group 
%$T$ of multiplicative type that is split by a finite \'etale cover of $S$.
%It suffices to show that the $\widehat{S}$-isomorphism $T_{\widehat{S}} \simeq H'_{\widehat{S}}$
%uniquely descends to an $S$-isomorphism $T \simeq H'$.
%Uniqueness is clear, so for existence we may replace $S$ with a local finite \'etale
%cover that splits $T$.  In other words, we can assume
%that 
%there is an isomorphism
%$\widehat{f}:H'_{\widehat{S}} \simeq D_{\widehat{S}}(M) = D_S(M)_{\widehat{S}}$ for
%a finitely generated abelian group $M$, and we seek
%to prove that $f$ uniquely descends to an isomorphism $H' \simeq D_S(M)$.
%Since $A$ is the henselization of a local ring on a finite type $\Z$-scheme,
%by the Artin approximation theorem there is
%an $S$-isomorphism $f:H' \simeq D_S(M)$ that agrees
%with $\widehat{f}$ on the special fibers.  Hence,
%$\widehat{f} \circ f_{\widehat{S}}^{-1}$ is an automorphism
%of $D_{\widehat{S}}(M)$ that is the identity on the special fiber,
%so it is the identity map!  That is, $\widehat{f} = f_{\widehat{S}}$, so we
%are done.
%
%
%\end{proof}


\begin{proposition}\label{xi2.4bis}
Let $G$ be a finitely presented $S$-affine group scheme. 
Let $Y, Y' \subset G$ be finitely presented closed subschemes.  Assume
$Y$ is either a multiplicative type subgroup or is finite flat over $S$.
The scheme ${\rm{Transp}}_G(Y,Y')$ exists as a finitely presented closed subscheme of $G$.
In particular, the normalizer $N_G(Y)$ exists as a finitely presented closed subgroup of $G$.

If $G$ is smooth and $Y$ and $Y'$ are both multiplicative type
subgroups of $G$ then these subschemes are smooth.
\end{proposition}
\index{normalizer!smoothness}
\index{normalizer!multiplicative type}
\index{multiplicative type!smooth normalizer}

In \cite[XI, 2.4bis]{sga3}, the smoothness aspect of this proposition is proved
without affineness hypotheses on $G$.  The proof in the affine case
is much simpler, and this case suffices for our needs.
In Proposition \ref{conntrans} we will adapt the method of proof to apply
to smooth closed subschemes $Y, Y' \subset G$ with geometrically connected fibers, but the {\em smoothness}
of  ${\rm{Transp}}_G(Y,Y')$ may then fail, even if $G$ is smooth.
Smoothness of ${\rm{Transp}}_G(H,H')$
for any $H$ and $H'$ of multiplicative type (when $G$ is smooth) is remarkably useful.

\begin{proof}
By ``standard'' direct limit arguments, we may and do assume that $S$ is noetherian. 
To be precise, the problems are Zariski-local on the base, so we can assume $S = {\rm{Spec}}(A)$ is affine.
Writing $A = \varinjlim A_i$ for the directed system of finitely generated $\Z$-subalgebras $A_i \subset A$,
the finite presentation hypotheses imply that for sufficiently large $i_0$
there is a finite type affine $A_{i_0}$-group scheme $G_{i_0}$ descending $G$ 
and finite type closed $A_{i_0}$-subgroup schemes $Y'_{i_0}$ and $Y_{i_0}$
of $G_{i_0}$ that descend $Y', Y \subset G$.  By increasing $i_0$ if necessary,
it can be arranged that the property of being finite flat over the base descends, 
and likewise for smoothness by \cite[IV$_4$, 17.8.7]{ega}.   

To descend the multiplicative type property,
for a closed $A$-subgroup $H \subset G$ of multiplicative type and $i \ge i_0$
such that there is a closed $A_i$-subgroup $H_i \subset
G_i := G_{i_0} \otimes_{A_{i_0}} A_i$ descending $H$, pick an fppf cover $\Spec A' \rightarrow \Spec A$ such that
$H_{A'} \simeq D_{A'}(M)$ for some finitely generated abelian group $M$.
By increasing $i$ we can arrange that $\Spec A' \rightarrow \Spec A$
descends to an fppf cover $\Spec A'_i \rightarrow \Spec A_i$.
Then the $A'_i$-groups $H_i \otimes_{A_i} A'_i$
and $D_{A'_i}(M)$ become isomorphic over $A'$.  However,
$A'$ is the direct limit of the rings $A'_{j} := A'_{i} \otimes_{A_{i}} A_{j}$
over $j \ge i$, so by increasing $i$ some more
we can arrange that $H_i \otimes_{A_i} A'_i \simeq D_{A'_i}(M)$.
Since $\Spec A'_i$ is an fppf cover of $\Spec A_i$, it follows
that $H_i$ is multiplicative type. 
Now it suffices to treat the case of noetherian $S$, so all closed subschemes of $G$ are finitely presented over $S$
and hence we do not need to keep
track of the ``finitely presented'' property.

Granting the
representability by a closed subscheme, let's address the $S$-smoothness when $G$ is smooth
and $Y$ and $Y'$ respectively coincide with multiplicative type subgroups $H$ and $H'$ of $G$.
We shall verify the functorial criterion for smoothness.
The condition is that for an affine scheme $S'$ over $S$ (which we may and do take to be noetherian,
or even artinian)
and a closed subscheme $S'_0$ defined
by a square-zero quasi-coherent ideal on $S'$, 
any $g_0 \in G(S'_0)$ conjugating $H_{S'_0}$ into $H'_{S'_0}$
admits a lift  $g \in G(S')$ conjugating $H_{S'}$ into $H'_{S'}$.  We may rename 
$S'$ as $S$, and define $X_0 := X_{S_0}$ for $S$-schemes $X$.
By $S$-smoothness of $G$ we can lift
$g_0$ to some $g \in G(S)$ but perhaps $gHg^{-1}$ is not contained in $H'$.   
Nonetheless, these two subgroups of multiplicative type in $G$ satisfy
$(gHg^{-1})_0 = g_0 H_0 g_0^{-1} \subset H'_0$, so 
by renaming $gHg^{-1}$ as $H$ we may assume that $H_0 \subset H'_0$ and $g_0 = 1$.  

Since $H'$ is of multiplicative type, the multiplicative type $S_0$-subgroup
$H_0$ in $H'_0$ uniquely lifts to a multiplicative type $S$-subgroup
$\widetilde{H}$ of $H'$ (as the uniqueness allows us to work \'etale-locally on
$S$, so it suffices to consider the easy case that $H'$ is $S$-split and $H_0$ is
$S_0$-split).  The multiplicative type $S$-subgroups
$H$ and $\widetilde{H}$ in $G$ have the same reduction in $G_0$, so by 
\cite[IV, \S1]{oesterle} these subgroups are abstractly isomorphic  in a manner
lifting their residual identification inside $G_0$.  Hence, if we choose such
an isomorphism $H \simeq \widetilde{H}$ then by 
Corollary \ref{h2obs} (applied to $H \hookrightarrow G$ and
$H \simeq \widetilde{H} \hookrightarrow G$) and the vanishing of higher
Hochschild cohomology in \cite[III, 3.3]{oesterle} there exists $g' \in G(S)$ satisfying
$g'_0 = 1$ and $g'H{g'}^{-1} = \widetilde{H} \subset H'$.  This completes
the proof of smoothness of the transporter scheme (granting its existence).

It remains to prove that the transporter functor is represented by a closed
subscheme of $G$ (necessarily of finite presentation
since $S$ is noetherian) when $Y$ is either finite flat
or a closed subgroup of multiplicative type.  
We first reduce the second case to the first,
so suppose $Y = H$ is a closed subgroup of
multiplicative type.  By the relative schematic density of $\{H[n]\}_{n \ge 1}$ in $H$
\index{relative schematic density}
(in the sense of \cite[IV$_3$, 11.10.8--11.10.10]{ega}), for any $S$-scheme $S'$
the only closed subscheme $Z \subset H_{S'}$ containing every $H[n]_{S'}$ is $Z = H_{S'}$.  Thus, 
$$\underline{\rm{Transp}}_G(H,Y') = \bigcap_{n > 0} \underline{\rm{Transp}}_G(H[n], Y')$$
as subfunctors of $G$, 
so it suffices to treat each of the pairs $(H[n], Y')$ separately.  Since
an arbitrary intersection of closed subschemes is a closed subscheme, now we may and do
assume that $Y$ is finite and flat over $S$ (and $S$ is noetherian). 

To build ${\rm{Transp}}_G(Y,Y')$ for finite flat $Y$, we will use Hom-schemes.  Hence, we now 
prove the representability of Hom-functors
in a special case.

\begin{lemma}\label{finitehom}
Let $S$ be a scheme, $X \rightarrow S$ a finite flat and finitely presented map,
and $Y \rightarrow S$ an affine morphism of finite presentation.
The functor on $S$-schemes defined by
$$S' \rightsquigarrow \Hom_{S'}(X_{S'}, Y_{S'})$$
is represented by an $S$-affine $S$-scheme of finite presentation.

Likewise, if $G$ and $G'$ are finitely presented $S$-groups with
$G$ finite flat and $G'$ affine over $S$ then the functor 
$S' \rightsquigarrow \Hom_{S'\mbox{-}{\rm{gp}}}(G_{S'}, G'_{S'})$
classifying group scheme homomorphisms is
represented by an $S$-affine $S$-scheme of finite presentation.
\end{lemma}

The representing schemes are  denoted
$\underline{\Hom}(X,Y)$ and $\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(G,G')$
respectively (since the notation without underlining has a categorical meaning).
\index{ aaahomxy@$\underline{\rm{Hom}}(X,Y)$,
$\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(G,G')$}

\begin{proof}
We may and do assume that $S$ is affine, and then noetherian, say
$S = {\rm{Spec}}(A)$, with $X = {\rm{Spec}}(B)$ for $B$ that is finite
free as an $A$-module (admitting an $A$-basis containing 1) and $Y = {\rm{Spec}}(C)$
for $C$ a finitely generated $A$-algebra.
Once the case of the functor of scheme
morphisms is settled, the refinement for group schemes
amounts to the formation of several fiber products
(see Exercise \ref{fiberex}). 
Thus, we just focus on the assertion for scheme
morphisms with $X$ and $Y$, avoiding any involvement of group schemes.

The basic idea of the construction of $\underline{\Hom}(X,Y)$ is 
similar to the construction of Weil restriction of scalars ${\rm{R}}_{S'/S}(X')$ 
for a finite flat map $S' \rightarrow S$ and affine finite type $S'$-scheme $X'$
for affine noetherian $S$ (see \cite[7.6]{neron}).
The reason for the similarity is that $\underline{\Hom}(X,Y) = {\rm{R}}_{X/S}(X \times_S Y)$, since
for any $S$-scheme $S'$ the set 
$\Hom_S(S', \underline{\Hom}(X,Y))$ is identified with
$$\Hom_S(S' \times_S X, Y) = 
\Hom_X(S' \times_S X, X \times_S Y) = \Hom_S(S', {\rm{R}}_{X/S}(X \times_S Y)).$$

Let $\{e_1,\dots,e_r\}$ be an $A$-basis of $B$ with $e_1 = 1$, and let
$$A[t_1,\dots,t_n]/(f_1,\dots,f_m) \simeq C$$
be a presentation of $C$.  
For any $A$-algebra $A'$ and the associated
scalar extensions $B' = A' \otimes_A B$ and $C' = A' \otimes_A C$ over $A'$, we identify
$\Hom_{A'\mbox{-}{\rm{alg}}}(C',B')$ with the set of ordered $n$-tuples
$b' = (b'_1,\dots,b'_n) \in {B'}^n$ such that
$f_j(b') = 0$ in $B'$  for all $j$.
By expressing the $A$-algebra structure on the $A$-module
$B = \bigoplus A e_{\alpha}$ in terms of ``structure constants'' in $A$, 
the specification of $b'$ amounts to the specification of
an ordered $nr$-tuple in $A'$, and the relations $f_j(b') = 0$
amount to a ``universal'' system of polynomial conditions over $A$ 
on this ordered $nr$-tuple.   These polynomial conditions define
the desired representing object as a closed subscheme of an affine space over $A$.
\end{proof}

\begin{remark}\label{remdeep}
In \cite[XI, 4.1, 4.2]{sga3}, a fundamental result is proved:
for any smooth $S$-affine $S$-group $G$ and multiplicative type group $H$,
the functor $\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$ classifying 
$S'$-group homomorphisms $H_{S'} \rightarrow G_{S'}$ over $S$-schemes $S'$ is represented by
a smooth separated $S$-scheme
and the functor $\underline{\rm{Mult}}_{G/S}$ classifying subgroups of $G$ of multiplicative
type is likewise represented by a smooth separated $S$-scheme.
(Beware that each of these representing schemes is generally
just {\em locally} of finite presentation over $S$; they are typically
{\em not} quasi-compact over $S$.)  

The construction of schemes representing $\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$
and $\underline{\rm{Mult}}_{G/S}$ rests on 
deep representability criteria for functors in \cite[XI, \S3]{sga3}
that are specially designed for such applications. These moduli schemes in turn 
underlie Grothendieck's construction of
quasi-affine quotients $G/N_G(H)$ and $G/Z_G(H)$
for multiplicative type subgroups $H$ in $G$ (see Remark \ref{qaffdef} for a review
of the notion of quasi-affine morphism). 
The quasi-affineness property of these quotients \cite[XI, 5.11]{sga3} is crucial 
for applications with descent theory (as it ensures effectivity of descent).

Later we will use a variant on Grothendieck's method, 
requiring only the more elementary case of Hom-schemes as in Lemma \ref{finitehom}
and establishing the existence and quasi-affineness of the quotient schemes
$G/N_G(H)$ and $G/Z_G(H)$ for smooth $G$ via an alternative procedure.  
The reason that we can succeed in this way is that we will appeal
to general theorems from the theory of algebraic spaces to understand
representability and geometric properties of quotient sheaves.  
\end{remark}

Returning to the proof of Proposition \ref{xi2.4bis}, by Lemma \ref{finitehom} 
the Hom-functors $\underline{\Hom}(Y,G)$ 
and $\underline{\Hom}(Y,Y')$ classifying scheme homomorphisms 
(over varying $S$-schemes) 
are represented by $S$-affine $S$-schemes of finite type.  There
is a natural $G$-action on $\underline{\Hom}(Y,G)$ via 
$G$-conjugation on $G$, so 
the $S$-point
$j \in \underline{\Hom}(Y,G)(S) = \Hom(Y,G)$
corresponding to the given inclusion yields a $G$-orbit map
\begin{equation}\label{gorbit}
G \rightarrow \underline{\Hom}(Y,G)
\end{equation}
over $S$ defined by $g \mapsto (y \mapsto g j(y)g^{-1})$.  Consider the pullback of 
the natural monomorphism
\begin{equation}\label{monichom}
\underline{\Hom}(Y,Y') \rightarrow \underline{\Hom}(Y,G)
\end{equation}
(defined by composition with the inclusion $Y' \hookrightarrow G$)
under the map (\ref{gorbit}).  This pullback recovers the subfunctor
$\underline{\rm{Transp}}_G(Y,Y')$; i.e., we have cartesian diagram of functors 
$$\xymatrix{
{\underline{\rm{Transp}}_G(Y,Y')} \ar[r] \ar[d] & {\underline{\Hom}_S(Y,Y')} \ar[d] \\
G \ar[r] & {\underline{\Hom}_S(Y,G)}}$$
This establishes the representability of the transporter functor, and to prove
that it is a closed subscheme of $G$ it suffices to prove that 
(\ref{monichom}) is a closed immersion.
Exactly as in the construction of these Hom-schemes in the proof of Lemma \ref{finitehom}, the condition
on an $S'$-scheme morphism $Y_{S'} \rightarrow G_{S'}$ (for an $S$-scheme $S'$)
that it factors through the closed subscheme $Y'_{S'}$ is represented by
an additional system of universal Zariski-closed conditions arising from
generators of the ideal of $Y'$ in $G$ (Zariski-locally over $S$).   This establishes
the required closed immersion property. 
\end{proof}

%It is a consequence of Zariski's Main Theorem
%that a finitely presented monomorphism is a closed immersion if (and only if) 
%it is proper \cite[IV$_3$, 8.11.5]{ega}
%(or see \cite[IV$_4$, 18.12.6]{ega} without finite presentation hypotheses), 
%so we just need to check that (\ref{monichom}) satisfies the valuative criterion for properness.
%This amounts to the assertion that if $S = \Spec R$ for a discrete valuation ring $R$
%then a map $f:Y \rightarrow G$ as $R$-schemes 
%factors through the closed subscheme $Y'$ over $R$ if and only if it does so on generic fibers.
%In other words, we claim that the closed subscheme $Z := f^{-1}(Y')$ in $Y$ 
%coincides with $Y$ if it does so on generic fibers over $\Spec R$.
%This in  turn follows from the evident fact that if a quasi-coherent
%ideal on a flat $R$-scheme (such as $Y$) vanishes on
%the generic fiber over $\Spec R$ then it is 0.
%\end{proof}

See Exercise \ref{2.7} for further discussion of constructions
of transporters and normalizers.  
As an application of Proposition \ref{xi2.4bis}, we have a mild refinement
of \cite[XI, 5.4bis]{sga3}:

\begin{corollary}\label{etconj} Let $G$ be a smooth $S$-affine $S$-group, and $H$ and $H'$ a pair
of subgroups of multiplicative type.  For any $s \in S$, if there exists
an extension field $K/k(s)$ such that $(H_s)_K$ is $G_s(K)$-conjugate
to $(H'_s)_K$ then there exists an \'etale neighborhood
$U \rightarrow S$ of $s$ such that $H_U$ is $G(U)$-conjugate to $H'_U$.
\end{corollary}
\index{multiplicative type!conjugacy}
\index{conjugacy! multiplicative type subgroups}

\begin{proof}
The hypothesis is that the smooth map
${\rm{Transp}}_G(H,H') \rightarrow S$ hits a $K$-point over $s$,
so its open image $V$ contains $s$.  Any surjective smooth map
of schemes admits sections \'etale-locally on the base.
Applying this to ${\rm{Transp}}_G(H,H') \twoheadrightarrow V$
provides an \'etale neighborhood $(U,u)$ of $(S,s)$ such that
${\rm{Transp}}_G(H,H')(U)$ is non-empty.  This $U$ does the job.
Indeed, for $g \in G(U)$ that conjugates $H_U$ into $H'_U$, the inclusion
$gH_Ug^{-1} \subset H'_U$ is an equality on $u$-fibers since
$H_s$ and $H'_s$ are abstractly isomorphic, and any containment between multiplicative type groups that is an equality on
geometric fibers at one point is an equality over an open neighborhood (as
we see by passing to an fppf or \'etale covering that splits both groups). 
\end{proof}

In our later study of parabolic subgroups $P$ of reductive group
schemes $G$, it will be important to establish that $P$ is self-normalizing,
which is to say that $P$ represents $\underline{N}_G(P)$
(a scheme-theoretic improvement on the result $N_{G(k)}(P) = P(k)$
in the classical theory over an algebraically closed field $k$).
The proof of this property will rest on knowing a priori that $\underline{N}_G(P)$
is represented by {\em some} finitely presented closed subscheme of $G$, so 
we need a variant on Proposition \ref{xi2.4bis} that is applicable
to smooth closed subschemes $Y$ of $G$ such that all fibers $Y_s$ are geometrically
connected (e.g., $Y = P$):

\begin{proposition}\label{conntrans}
Let $G$ be a smooth $S$-affine $S$-group and
$Y, Y' \subset G$ finitely presented closed subschemes 
such that $Y$ is smooth with non-empty and geometrically connected fibers over $S$. 
The transporter functor $\underline{\rm{Transp}}_G(Y,Y')$
is represented by a finitely presented closed subscheme of $G$.
In particular, the normalizer functor
$$\underline{N}_G(Y): S' \rightsquigarrow \{g \in G(S')\,|\,g Y_{S'} g^{-1} = Y_{S'}\}$$
is represented by a finitely presented closed subgroup of $G$.
\end{proposition}
\index{normalizer!smooth subscheme}

The main idea in the proof of this result
is to reduce to the finite flat case treated in Proposition \ref{xi2.4bis}.
The role of relative schematic density of the torsion-level subgroups
in the proof of Proposition \ref{xi2.4bis} will be replaced by 
an alternative notion of ``relative density'' applicable to the collection of infinitesimal neighborhoods of
a section $y:S \rightarrow Y$ (which exists \'etale-locally on the base).  This is easiest to understand
in the classical setting.  Before explaining this case, we introduce some convenient terminology:

\begin{definition}\label{weakdense}
For a scheme $S$, a collection $\{S_{\alpha}\}$ of closed subschemes of $S$ is
{\em weakly schematically dense} if the only closed subscheme $Z \subset S$ containing
every $S_{\alpha}$ is $Z = S$.  
\end{definition}

This notion is {\em not} Zariski-local on $S$, as the following example illustrates.

\begin{example}\label{ptex}
Let $Y$ be a smooth connected non-empty scheme over
an algebraically closed field $k$ (so $Y$ is irreducible and reduced).  Pick $y \in Y(k)$,
and let $Y_n$ denote the $n$th infinitesimal neighborhood of $y$;
i.e., $Y_n$ is the infinitesimal closed subscheme defined by the vanishing of
$\mathscr{I}_y^{n+1}$, where $\mathscr{I}_y$ is the ideal of $y$ in $\mathscr{O}_Y$.
We claim that the collection $\{Y_n\}$ is weakly schematically dense in $Y$. 
(This collection is supported at a single point, so it is rarely schematically dense
in the sense of \cite[IV$_3$, 11.10.2]{ega}.) Let $Z$ be a closed subscheme of $Y$ containing every $Y_n$. 
If $\mathscr{J}$ is the ideal of $Z$ in $Y$
then the stalk $\mathscr{J}_y$ in the local ring $\mathscr{O}_{Y,y}$
vanishes in the completion $\mathscr{O}_{Y,y}^{\wedge}$,
so $\mathscr{J}_y = 0$.  But $Y$ is integral, so $\mathscr{J} = 0$ as desired.
\end{example}

\begin{example}\label{weakdenseex}
If $H \rightarrow S$ be a group scheme of multiplicative type
then $\{H[n]\}_{n \ge 1}$ is weakly schematically dense in $H$.  Indeed, we may
pass to an \'etale cover of $S$ so that $S = \Spec(A)$ is affine and $H = \Gm^N \times \prod \mu_{d_i}$
for some $\{d_1, \dots, d_r\}$.  Thus, it suffices to observe the elementary fact that if an element 
$$b \in A[T_1^{\pm 1}, \dots, T_N^{\pm 1}, X_1,\dots,X_r]/(X_1^{d_1} - 1, \dots, X_r^{d_r}-1)$$
vanishes modulo $(T_j^n - 1)_j$ for all $n$ divisible by the $d_i$'s then $b = 0$.
\end{example}

Now we turn to the proof of Proposition \ref{conntrans}.

\begin{proof}
Since \'etale descent is effective for closed subschemes, and the functors
in question are sheaves for the \'etale topology, the 
problem is \'etale-local on $S$.  Thus, we may assume
there exists $y \in Y(S)$.
For each $n \ge 0$, let $Y_n$ denote the $n$th infinitesimal neighborhood
of $y$ in $Y$; this is the closed subscheme defined by the $(n+1)$th-power of the ideal of
the closed immersion $y:S \rightarrow Y$. 
The proof of Proposition \ref{xi2.4bis} 
will be adapted by using these infinitesimal neighborhoods
in the role of the $n$-torsion subgroups in that earlier proof.

By direct limit arguments we may assume that $S$ is noetherian
(see \cite[IV$_3$, 8.3.3, 9.7.9; IV$_4$, 17.8.7]{ega}). 
We claim that each $Y_n$ is finite flat over $S$, so its formation commutes with any base change,
and that the collection $\{Y_n\}$ is 
weakly schematically dense in $Y$ (in the sense of Definition \ref{weakdense}) and remains so after any base change.  Once
this is established, the resulting equality
$$\underline{\rm{Transp}}_G(Y,Y') = \bigcap_{n \ge 0} \underline{\rm{Transp}}_G(Y_n,Y')$$
as subfunctors of $G$ and the representability of
$\underline{\rm{Transp}}_G(Y_n,Y')$ will complete the proof.

To prove that $Y_n$ is flat over $S$, we can reduce to the case $S = \Spec A$ for an artin local ring $A$
(by the local flatness criterion; see \cite[22.3(1),(5)]{crt}).  Now $y$ has a single physical point $y_0$, and 
by $A$-smoothness of $Y$ near $y$ the completion 
of $\mathscr{O}_{Y,y_0}$ along $y$ is identified with
$A[\![t_1,\dots,t_d]\!]$ carrying the ideal of $y$ over to the ideal $(t_1,\dots,t_d)$.
(This is seen by using an \'etale map $f:(Y,y) \rightarrow (\mathbf{A}^d_S,0)$.)
Thus, $Y_n \simeq A[\![t_1,\dots,t_d]\!]/(t_1,\dots,t_d)^n$, which is visibly
$S$-flat.  

Returning to a general noetherian $S$, $Y_n$ is $S$-finite
since $(Y_n)_{\rm{red}} = S_{\rm{red}}$ and $Y_n$ is finite type
over $S$. 
It remains to show $\{Y_n\}$ is weakly schematically dense in $Y$
and remains so after any base change.  By localizing we may assume
$S$ is local with closed point $s_0$.  Let $U$ be an affine open neighborhood
of $y(s_0)$ in $Y$, so $y \in U(S)$ inside $Y(S)$.  Since
$Y \rightarrow S$ is smooth surjective with geometrically connected fibers, 
$U \rightarrow S$ is fiberwise dense and hence $U \rightarrow Y$ is relatively schematically dense
over $S$ \cite[IV$_3$, 11.10.10]{ega}.  Thus, a closed subscheme of $Y$ containing $U$
coincides with $Y$, and likewise after base change on $S$, so we may
replace $Y$ with $U$ to reduce to the case that $Y$ is affine.

It now suffices to show that if $A$ is a noetherian local ring and $B$ is a smooth $A$-algebra
such that $\Spec(B) \rightarrow \Spec(A)$ is surjective with geometrically connected fibers
and there is an $A$-algebra section $s:B \rightarrow A$ 
then for any $\{t_1,\dots,t_n\}$ in $J = \ker(s)$ lifting an $A$-basis of $J/J^2$
and any local homomorphism $A \rightarrow A'$, 
the natural map $h:B \otimes_A A' \rightarrow A'[\![T_1,\dots,T_n]\!]$ to the $J \otimes_A A'$-adic
completion is injective.   Writing $A'$ as a direct limit of noetherian local $A$-subalgebras $A'_i$
with local inclusion $A'_i \hookrightarrow A'$, we reduce to the case that $A'$ is noetherian
and so we may assume $A' = A$.    We may also assume $A$ is complete, so
$h$ is identified with the natural map from $B$ to its completion
at the closed point of the section.   Hence, any $b \in \ker(h)$ vanishes
in the local ring of $B$ at that closed point, so $b$ vanishes on an open neighborhood
of the section in $\Spec(B)$.  The schematic density argument as above then implies that $b = 0$.
\end{proof}
%
%It suffices to test this using base change to 
%noetherian $S$-schemes. This amounts
%to the claim that for any noetherian $S$-scheme $S'$ and $g \in G(S')$, if
%$g(Y_n)_{S'} g^{-1} \subset (Y'_n)_{S'}$ for all $n$ then $g Y_{S'} g^{-1} \subset Y'_{S'}$.
%By renaming $S'$ as $S$ and renaming $gYg^{-1}$ as $Y$, this becomes the general claim that
%if $Y_n \subset Y'$ for all $n$ then $Y \subset Y'$.
%The hypothesis says that the ideal $J$ of $Y' \bigcap Y$ in $Y$ vanishes in every $Y_n$,
%and we wish to prove that $J = 0$.  This follows from Lemma \ref{ideal0} below.
%\end{proof}

%\begin{lemma}\label{ideal0}
%Let $S$ be a noetherian scheme, $Y \rightarrow S$ a smooth map
%with connected fibers and a section $y \in Y(S)$, and $J$ a coherent ideal on $Y$ that vanishes in every 
%infinitesimal neighborhood $Y_n$ of $y$.  Then $J = 0$.
%\end{lemma}

%\begin{proof}
%By the Krull intersection theorem we may and do assume
%that $S$ is artin local, so $Y$ has the same topology as its special fiber $\overline{Y}$
%(a smooth connected scheme over the residue field).  The 
%vanishing hypothesis on the $Y_n$'s implies that the stalk of $J$ at the closed point $\overline{y}$ of $y$
%vanishes in the completion of $\mathscr{O}_{Y,\overline{y}}$, 
%so this stalk vanishes.  That is, the zero scheme $Z$ of $J$ in $Y$ is a closed
%subscheme that contains an open subscheme $U$ around the section $y$.

%The connectedness and smoothness of $\overline{Y}$ 
%implies that $\overline{Y}$ is irreducible,  so the special fiber $\overline{U} = U_{\rm{red}}$ 
%of $U$ is topologically dense in $\overline{Y}$.
%If we were working over a field then we would be done, since
%restriction of functions on $\overline{Y}$ to functions on $\overline{U}$ is faithful.  
%The smoothness of $Y$ allows us to carry over the same reasoning
%with an artin local base:  smoothness implies (by \cite[IV$_3$, 11.10.10]{ega}) 
%that the open immersion $i:U \hookrightarrow Y$ is schematically dominant
%in the sense that $\mathscr{O}_Y \rightarrow i_{\ast}(\mathscr{O}_U)$
%has vanishing kernel.  Since the ideal $J$ lies in the kernel, we conclude that $J = 0$.
%\end{proof}

\begin{example}\label{219}
In contrast with the smoothness of normalizers in Proposition \ref{xi2.4bis}
(and centralizers in \S\ref{centralscheme}) for multiplicative type subgroups
in smooth affine groups,
the normalizers in Proposition \ref{conntrans} can fail to be flat 
when $G$ is smooth and $Y$ is a smooth subgroup with connected
fibers.  In particular, normality on a fiber does not
imply normality on nearby fibers, even when 
working with smooth groups.  More specifically, a family of non-normal smooth closed subgroups
can degenerate to a normal subgroup. 
\index{normalizer!non-smooth example}

We give an example over $S = \mathbf{A}^1$ (with coordinate $t$) 
using the $S$-group $G = \Add \times {\rm{SL}}_2$. 
Consider the closed $S$-subgroup $G' = \Add$ defined by the closed immersion
$$j:(u,t) \mapsto (u, (\begin{smallmatrix} 1 & tu \\ 0 & 1 \end{smallmatrix}), t)$$
over the $t$-line $S$.  The fiber map over $t = 0$ is the inclusion of $\Add$ into the first factor
of $\Add \times {\rm{SL}}_2$,
but for $t \ne 0$ the fiber map $j_t$ has non-normal image. Hence,
the normalizer $N_G(G')$ has fiber $G_0$ over $t = 0$ but its fiber
over any $t \ne 0$ has strictly smaller dimension.  It follows that $N_G(G')$ is not $S$-flat.
\end{example}

\subsection{Centralizer schemes}\label{centralscheme}

In the classical theory of linear algebraic groups over an algebraically closed
field $k$, the centralizer of a smooth closed subscheme $Y$ of a smooth
affine group $G$ is defined by brute force: it is the reduced (hence smooth)
Zariski-closed subgroup structure on $\bigcap_{y \in Y(k)} Z_G(y)$.
This definition makes it unclear what the Lie algebra is when $Y$ is a subgroup of $G$.
The scheme-theoretic approach defines the centralizer in a more functorial
way that makes it easy to identify the Lie algebra, but shifts
the burden of work to proving that (in favorable circumstances) this
subgroup is actually {\em smooth}. 

\begin{definition}\label{zgy}
Let $G \rightarrow S$ be a group scheme and $Y$ a closed subscheme.
The {\em functorial centralizer} $\underline{Z}_G(Y)$ on $S$-schemes
assigns to any $S$-scheme $S'$ the set of $g \in G(S')$ such that $g$ centralizes
$Y_{S'}$ inside $G_{S'}$.

A closed subgroup of $G$ representing $\underline{Z}_G(Y)$, if one exists, is
denoted $Z_G(Y)$ and is called the {\em  centralizer} of $Y$ in $G$.
In the special case $Y = G$, such a subgroup scheme is called the {\em 
center} of $G$ (if it exists!) and is denoted $Z_G$.
\end{definition}
\index{centralizer!functor, scheme}
\index{center}
\index{ aaazgy@$\underline{Z}_G(Y)$, $Z_G(Y)$}

The general existence of  centralizers is delicate,
but in special cases there are affirmative results.  One favorable situation is
when $S = \Spec(k)$ for a field $k$, in which case
$Z_G(Y)$ exists for any $Y$; see Exercise \ref{2.7}.  The 
case of smooth $Y$ with connected fibers is
\cite[XI, 6.11]{sga3} (subject to some mild hypotheses on $Y$ and $G$), but for later purposes 
over a general base $S$ we must allow any $Y$ of multiplicative type
(so $Y$ may not be $S$-smooth and may have disconnected fibers).
Such cases are handled in Lemma \ref{centralizer} below, where 
we also show that the centralizer of $Y$ is
smooth when $G$ is smooth (even if $Y$ is not smooth!).  

\begin{definition}
A subgroup scheme $G'$ in $G$ is 
{\em central}
when  $G$-conjugation on $G$
is trivial on $G'$ (equivalently, $\underline{Z}_G(G') = G$). 
\end{definition}
\index{central subgroup scheme}

Another way to express the centrality condition is that for
every $S$-scheme $S'$ and $g' \in G'(S')$, the
$g'$-conjugation action on $G_{S'}$ is trivial.

\begin{example}\label{centerex}
If $G$ is a {\em smooth} finite type group over an algebraically closed field $k$,
a closed subgroup scheme $G'$ is central
if $G'$ is centralized by all elements of
$G(k)$; see Exercise \ref{2.7}(iv).   This is false when $G$ is not assumed to be smooth, even 
when $G'$ is smooth.  For example, in characteristic $p > 0$
the usual semi-direct product $G = \Gm \ltimes \alpha_p$
and its smooth closed 
subgroup $G' = \Gm$ have the same geometric points, but $G'$ is not normal in $G$.

As another example of a central subgroup scheme, 
for any scheme $S$  the diagonal $\mu_n$ in the $S$-group ${\rm{SL}}_n$ is central for
any $n > 1$.  See Example \ref{slncenter} for a stronger centrality property of $\mu_n$ in ${\rm{SL}}_n$.
\end{example}

\begin{lemma}\label{centralizer}
Let $G \rightarrow S$ be a finitely presented $S$-affine $S$-group, and $Y$ a finitely
presented closed subscheme of $G$.  Assume that $Y$ is a subgroup of multiplicative
type or that $Y \rightarrow S$ either is smooth with each fiber non-empty and connected 
$($hence geometrically connected$)$ or is finite flat. 
Then $Z_G(Y)$ exists as a finitely presented closed subgroup; it is smooth when $G$ is smooth
and $Y$ is a subgroup of multiplicative type, in which case ${\rm{Lie}}(Z_G(Y)) = {\rm{Lie}}(G)^Y$
and this Lie algebra 
represents the functor of $Y$-invariants under ${\rm{Ad}}_G$.
\end{lemma}
\index{centralizer!existence, smoothness}
\index{multiplicative type!centralizer}

For $G$ smooth and $Y$ a subgroup of multiplicative type, this lemma is part of \cite[XI, 5.3]{sga3} (aside from the description of
${\rm{Lie}}(Z_G(Y))$ in such cases).

\begin{proof}
We may assume $S$ is noetherian, and for the representability
assertions it suffices to restrict to functors on the category of
noetherian $S$-schemes (since for affine $S = \Spec(A)$, the functor 
$\underline{Z}_G(Y)$ on $A$-algebras commutes with the formation of direct limits).

First we treat the existence when $Y$ is finite flat
over $S$.  In this case Lemma \ref{finitehom} provides
an $S$-scheme $\underline{\Hom}(Y,G)$ that is affine of finite type over 
$S$.   The given
inclusion of $Y$ into $G$ corresponds to an $S$-point $j$ of 
$\underline{\Hom}(Y,G)$, and $j:S \rightarrow \underline{\Hom}(Y,G)$ is a closed
immersion since it is a section to a separated map.
The pullback of the morphism $j$ under the
orbit map $G \rightarrow \underline{\Hom}(Y,G)$ through $j$ (via the $G$-action on $\underline{\Hom}(Y,G)$
through conjugation) is a closed subscheme of
$G$ representing $\underline{Z}_G(Y)$.

Suppose $Y$ is a subgroup $H$ of multiplicative type. Since 
$\{H[n]\}$ is weakly schematically dense in $H$ in the sense of Definition \ref{weakdense} and remains
so after any base change on $S$ (Example \ref{weakdenseex}), $\underline{Z}_G(H) = \bigcap_{n > 0}
\underline{Z}_G(H[n])$ as subfunctors of $G$ since the condition of equality for two maps
$H \rightrightarrows G$ can be expressed using the closed relative diagonal of $G$ over $S$.  Each $H[n]$ is finite flat, 
and $\bigcap_{n > 0} Z_G(H[n])$ represents $\underline{Z}_G(H)$.
Likewise, if $Y$ is a smooth closed subscheme with geometrically connected non-empty fibers
then by working \'etale-locally on $S$ we can assume that $Y \rightarrow S$ admits a section $y$.
Then we can argue exactly as in the multiplicative type case
by using the finite flat infinitesimal neighborhoods $Y_n$
of $y$ in the role of the torsion-levels $H[n]$
(due to the weak schematic density of
$\{Y_n\}$ in $Y$ that persists after any base change, as established in the proof
of Proposition \ref{conntrans}).  

Finally, for smooth $G$ and general $H$ of multiplicative type
we prove that $Z_G(H)$ is smooth and compute its Lie algebra inside $\mathfrak{g} = {\rm{Lie}}(G)$. 
By the functorial smoothness criterion and the generality of the base scheme, to prove smoothness it suffices 
to show that if $S$ is affine and $S_0$ is a closed subscheme of $S$ defined by a square-zero quasi-coherent
ideal then any $g_0 \in G_0(S_0)$ centralizing $H_0$ 
lifts to some $g \in G(S)$ that centralizes $H$.
By smoothness of $G$ we can pick some $g \in G(S)$ lifting $g_0$, so
$gHg^{-1}$ and $H$ are multiplicative type subgroups of $G$ with the same
reduction in $G_0$.    As in the proof of Proposition \ref{xi2.4bis}, by Corollary \ref{h2obs} and \cite[III, 3.3]{oesterle}  
there exists $g' \in G(S)$ lifting $1 \in G_0(S_0)$
such that $g'$ conjugates $gHg^{-1}$ to $H$, so $g'g$ normalizes $H$.
But over $S_0$ this conjugation endomorphism of $H_0$
is conjugation by $(g'g)_0 = g_0$, which is the trivial action on $H_0$.
Since $H$ is multiplicative type and $S_0$ is defined by a nilpotent ideal
on $S$, it follows (by Corollary \ref{rigidhom}) 
that $g'g$ must centralize $H$,
so $g'g \in Z_G(H)(S)$ and this lifts $g_0 \in Z_G(H)(S_0)$.

Since $Z_G(H)$ is $S$-smooth,
${\rm{Lie}}(Z_G(H))$ is a subbundle of $\mathfrak{g}$ whose formation commutes
with base change.   Likewise, by fppf descent from the case of split $H = D_S(M) := \Spec(\calO_S[M])$
\index{ aaadsm@$D_S(M)$}
(for which a linear representation of $H$ on a vector bundle corresponds to an $M$-grading, by
\cite[III, 1.5]{oesterle} or \cite[Lemma A.8.8]{pred}),
there is a subbundle $\mathfrak{g}^H$ of $\mathfrak{g}$ representing the functor of $H$-invariants under
${\rm{Ad}}_G$ and its formation also commutes with any base change on $S$.  There is an evident
inclusion ${\rm{Lie}}(Z_G(H)) \subset \mathfrak{g}^H$ as subbundles of $\mathfrak{g}$, so to prove it is an equality
it suffices to check on geometric fibers over $S$.  Thus, we may assume $S = \Spec k$ for an algebraically closed field $k$,
in which case the equality ${\rm{Lie}}(Z_G(H)) = \mathfrak{g}^H$ is shown in the proof of Proposition \ref{zrep}
(where that part of the proof was specifically written to only use the existence of $Z_G(H)$ as a closed subgroup of $G$
and not any smoothness hypothesis on $H$).
\end{proof}


\begin{remark}
An important special case of Lemma \ref{centralizer} is $Y = G$, for which this lemma
asserts the existence of the scheme-theoretic center $Z_G$ of a smooth
$S$-affine $S$-group $G$ such that all $G_{\overline{s}}$ are connected.
(See \cite[XI, 6.11]{sga3} for the removal of the fibral connectedness
condition, taking $G = H$ there.) 
In general $Z_G$ can fail to be flat (see the end of \cite[XVI, \S3]{sga3}
for an example), in which case it is not very useful.

See Exercise \ref{2.1}(ii) for examples of Lemma \ref{centralizer} with $Z_G$
of multiplicative type,  and Exercises \ref{2.7} and  \ref{2.6} for generalizations
of Lemma \ref{centralizer} over fields.  In Theorem \ref{redcenter} we will show $Z_G$ is of multiplicative type (hence flat) for any
reductive group scheme $G \rightarrow S$. 
\end{remark}


\begin{example}\label{cent}
Here are two applications of the smoothness aspect of
Lemma \ref{centralizer} in the context of smooth affine groups over a field $k$.
Let $G$ be such a group, and $T$  a torus in $G$ (not necessarily
maximal).  The lemma ensures that the  centralizer
$Z_G(T)$ is always smooth (and it is connected when $G$ is connected, by
the classical theory over $\overline{k}$).  This is the scheme-theoretic proof
of a fact in the classical theory \cite[9.2, Cor.]{borelag}:  
the reduced structure on the closed subset of $G_{\overline{k}}$ 
corresponding to the centralizer of $T_{\overline{k}}$ in $G(\overline{k})$ descends
to a smooth closed $k$-subgroup of $G$ (namely $Z_G(T)$).  
As a special case, if $G$ is connected reductive and $T$ is a (geometrically) maximal torus
then $Z_G(T) = T$ because such an equality between smooth closed
subgroups can be checked on $\overline{k}$-points, where it follows from the classical theory.

For another application, if $G'$ is a smooth closed $k$-subgroup of $G$ that is normalized by $T$
(e.g., a normal $k$-subgroup of $G$) 
then the scheme-theoretic intersection $Z_G(T) \bigcap G'$ represents the functorial centralizer
for the $T$-action on $G'$.  Although $T$ may not be a $k$-subgroup of $G'$ inside $G$,
so Lemma \ref{centralizer} does not literally apply to $T$ acting on $G'$, there
is a standard trick with semi-direct products that enables us to apply the lemma anyway
to prove $Z_G(T) \bigcap G'$ is smooth:
form the semi-direct product $G' \rtimes T$ in which $T$ embeds
along the second factor.  Lemma \ref{centralizer} can be applied
to this semi-direct product.  Thus, $Z_{G' \rtimes T}(T)$ is smooth, and we know it is connected when $G'$ is
connected.  But clearly 
$$Z_{G' \rtimes T}(T) = (Z_G(T) \bigcap G') \rtimes T$$
as $k$-schemes, and the left side is smooth (and connected when $G'$ is connected).
Thus, the direct factor scheme $Z_G(T) \bigcap G'$ is smooth
(and connected when $G'$ is).  For example, if $N \subset G$ is any smooth
closed normal subgroup then $N \bigcap Z_G(T)$ is smooth (and connected when $N$ is connected).
\end{example}

\begin{remark} Our construction of centralizers and normalizers
relied on passage to $Y$ that are finite flat over the base.
For an alternative approach in the presence of enough ``module-freeness'' for
some coordinate rings (as algebras over a base ring), 
see \cite[VIII, \S6]{sga3}, 
\cite[A.8.10(1)]{pred}, and Exercise \ref{2.7}.
\end{remark}

\subsection{Some quotient constructions}\label{qtsec}

As was noted in Remark \ref{remdeep}, for the construction
of quotients in the relative setting we will bypass some of Grothendieck's techniques
in \cite{sga3} in favor of the theory of algebraic spaces.  This is illustrated in the following
results, the first of which is a variant on \cite[XI, 5.3bis]{sga3}.

\begin{theorem}\label{xi5.3bis} Let $G \rightarrow S$ be a smooth $S$-affine group scheme,
and $H$ a subgroup of multiplicative type.  The quotients $G/Z_G(H)$
and $G/N_G(H)$ exist as smooth quasi-affine $S$-schemes.
Moreover, the quotient $W_G(H) := N_G(H)/Z_G(H)$ exists as a separated
and \'etale $S$-group of finite presentation.
\index{quasi-affine morphism}

In particular, if $H$ is normal in $G$ $($i.e., $N_G(H) = G$$)$ and the fibers $G_s$ are connected
$($so $W_G(H)_s = W_{G_s}(H_s) = 1$ for all $s \in S$ and hence $W_G(H) = 1$$)$ 
then $H$ is central in $G$.
\end{theorem}

The final centrality assertion admits a more elementary proof; see the self-contained
Lemma \ref{multsub}(1).  
Before we prove Theorem \ref{xi5.3bis}, we briefly digress to make some remarks.

\begin{remark}\label{qaffdef} As in \cite[II, 5.1.1]{ega}, a map of schemes $f:X \rightarrow Y$ is {\em quasi-affine}
if, over the constituents of some affine open cover of $Y$, it factors as a quasi-compact open immersion into
an affine scheme (so $f$ is quasi-compact and separated).  
In \cite[II, 5.1.2, 5.1.6]{ega} there are several equivalent versions of
this definition.  A more ``practical'' description is provided in \cite[II, 5.1.9]{ega}
when $f:X \rightarrow Y$ is finite type and $Y$ is noetherian
(or more generally, when $Y$ is quasi-compact and quasi-separated):  such an 
$f$ is quasi-affine if and only if $f$ factors as a quasi-compact open immersion followed by 
an affine map $Y' \rightarrow Y$ {\em of finite type}.  

Since 
fppf descent is always effective for schemes that are 
quasi-affine over the base, the quasi-affineness
in Theorem \ref{xi5.3bis} is useful; e.g., it will underlie our later construction of
the ``scheme of maximal tori'' in a reductive group scheme (even when there is
no torus over the given base scheme that is maximal on all geometric fibers).
\end{remark}

\begin{remark}
We will build the quotients in Theorem \ref{xi5.3bis}
via an orbit argument, as is also done when
$S = \Spec(k)$ for a field $k$.  However, unlike the case over a field, in the relative
setting we do {\em not} have a plentiful supply of linear representations. In fact,
we do not even know if every smooth affine group over the dual numbers $k[\epsilon]$
is a closed subgroup scheme of some ${\rm{GL}}_n$!  
(See \cite[VI$_{\rm{B}}$, 13.2, 13.5]{sga3}
and \cite[XI, 4.3]{sga3} for further discussion in this direction.) 
Thus, the construction
of quotients $G/H$ modulo flat and finitely presented
closed subgroups $H$ is rather subtle when the base is not a field.

Generally such quotients $G/H$ are algebraic spaces. 
In some cases we will prove that $G/H$ is a scheme by
using the following modification of the classical orbit argument.
We will identify the quotient sheaf $G/H$ as a $G$-equivariant subfunctor 
of a scheme $X$ on which $G$ acts, but  we have no general analogue in the relative setting of the result in the classical case that
$G$-orbits are always smooth and locally closed when $G$ is smooth. 
To show that the algebraic space $G/H$ is a scheme, we will use a general result of
Knutson which only requires that the subfunctor inclusion $j:G/H \hookrightarrow X$ into a scheme is 
quasi-finite and separated (with $X$ noetherian).  In the situations that arise
in the proof of Theorem \ref{xi5.3bis} one can prove (see Remark 
\ref{immqt}) that the morphism $j$ is \'etale (hence an open immersion, so
{\em a fortiori} $G/H$ is locally closed in $X$), but this fact is not used in our proof
of Theorem \ref{xi5.3bis}. 
%In some cases we will prove that $G/H$ is a scheme by
%using an orbit argument with a slightly different
%flavor from the scheme case:  we will {\em not} directly build the quotient scheme as
%a subscheme of a scheme $X$ on which $G$ acts (or rather, we will identify the quotient sheaf $G/H$
%as a subfunctor of such an $X$ and we do not try to represent the subfunctor
%inclusion $G/H \hookrightarrow X$ by a subscheme of $X$)
%because instead we will appeal to general criteria for an algebraic space to be a scheme.
%Our  deviation from the  method of construction of $G/H$ in 
%the classical case is forced by the fact 
%that there is no general analogue in the relative setting of the result in the classical case that
%$G$-orbits are always smooth and locally closed when $G$ is smooth. In Remark \ref{immqt} we will 
%prove that the orbit maps arising in the cases of interest to us
%in Theorem \ref{xi5.3bis} are open immersions, but this is never used in what follows.
\end{remark}

Now we turn to the proof of Theorem \ref{xi5.3bis}.

\begin{proof}
We may and do assume that $S$ is noetherian.
The relative schematic density of $\{H[n]\}$ in $H$ implies
that $Z_G(H) = \bigcap_{n > 0} Z_G(H[n])$
and $N_G(H) = \bigcap_{n > 0} N_G(H[n])$
as closed subschemes of $G$. 
But the noetherian condition on $G$ implies
that any descending chain of closed subschemes of $G$ stabilizes,
so $Z_G(H) = Z_G(H[n])$ and $N_G(H) = N_G(H[n])$ for
sufficiently large $n$.  Hence, it suffices to treat
each $H[n]$ in place of $H$, so we may assume that $H$ is $S$-finite.
Now Lemma \ref{finitehom} provides
the scheme $\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$
that is $S$-affine of finite type.  
By Proposition \ref{xi2.4bis} and Lemma \ref{centralizer}, $N_G(H)$ and $Z_G(H)$ are $S$-smooth.

Consider the natural $G$-action on this Hom-scheme via
composition with the conjugation action of $G$ on itself,
and the $S$-point corresponding to the given inclusion $j:H \rightarrow G$.
The orbit map $G \rightarrow \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$
through $j$ is right-invariant by the stabilizer scheme 
$Z_G(H)$ of the $S$-point $j$, so
the quotient sheaf $G/Z_G(H)$ for the fppf (or \'etale) topology is naturally a subfunctor
of the Hom-scheme.   By a general theorem of Artin \cite[Cor.\,6.3]{versal},
for any  finite type $S$-scheme $X$ and  equivalence
relation $R$ on $X$ that is represented by a closed subscheme in $X \times_S X$ and for which 
both projections $R \rightrightarrows X$ are flat,
the fppf quotient sheaf $X/R$ is a separated algebraic space
of finite type over $S$.  Thus,
$G/Z_G(H)$ is such an algebraic space.
The monomorphism $G/Z_G(H) \rightarrow \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$ over $S$
must be separated and finite type with 
finite fibers, so $G/Z_G(H)$ is separated and quasi-finite over a scheme.

By a result of Knutson \cite[II, 6.15]{knutson}, 
an algebraic space that is quasi-finite and separated over
a noetherian scheme is a scheme. (See \cite[Thm.\,A.2]{lmb} for
a generalization without noetherian hypotheses.)
Hence, $G/Z_G(H)$ is a scheme that is moreover
quasi-finite and separated over the scheme $\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$. 
By Zariski's Main Theorem \cite[IV$_3$, 8.12.6]{ega}, 
any quasi-finite and separated map between noetherian
schemes is quasi-affine, so $G/Z_G(H)$ is quasi-affine
over the $S$-affine $\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$
 and hence it is quasi-affine over $S$. 
The quotient $G/Z_G(H)$ is $S$-smooth, since $G \rightarrow G/Z_G(H)$ is an fppf
cover by the smooth $G$.
(In Remark \ref{immqt} we show the orbit map $G/Z_G(H) \rightarrow
\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$ is an open immersion.)

By the same reasoning, the quotient sheaves $G/N_G(H)$ and $W_G(H) = N_G(H)/Z_G(H)$ are smooth and separated 
algebraic spaces of finite type over $S$.  
The map $W_G(H) \rightarrow G/Z_G(H)$ is a closed immersion, via fppf descent of
the closed immersion property for its pullback $N_G(H) \rightarrow G$ along the
smooth covering $G \rightarrow G/Z_G(H)$, so $W_G(H)$ is a scheme as well.
To prove it is $S$-\'etale we may pass to geometric fibers over
$S$.  With $S = \Spec(k)$ for an algebraically closed field $k$, 
the automorphism functor of $H$ is represented by a disjoint union 
$\Aut_{H/k}$ of
rational points (opposite to the automorphism functor of
the constant dual of $H$) and $N_G(H)/Z_G(H)$ is a finite type $k$-group
equipped with a monic homomorphism to the \'etale $k$-group $\Aut_{H/k}$.
This forces the smooth $N_G(H)/Z_G(H)$ to be {\em finite}, hence \'etale.

Exhibiting $G/N_G(H)$ 
as quasi-finite and separated over a quasi-affine $S$-scheme
requires a new idea. We shall exhibit it as a subfunctor 
of a scheme that is finite type and quasi-affine over $S$.  For this, 
we replace the scheme $\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$ 
classifying {\em homomorphisms} of $H$ into $G$ with the scheme classifying
{\em closed subgroup schemes} of $G$ that are ``twists'' of $H$:  

\begin{lemma}\label{hilb} Let  
$G \rightarrow S$ be a smooth $S$-affine group scheme, and
$H \subset G$ a subgroup of multiplicative type with
finite fibers.  There is a quasi-affine $S$-scheme ${\rm{Twist}}_{H/G}$ of finite presentation
that represents the functor $\underline{\rm{Twist}}_{H/G}$ assigning to any $S$-scheme $S'$ the 
set of multiplicative type subgroups $H' \subset G_{S'}$ such that $H'_{\overline{s}'} \simeq
H_{\overline{s}'}$ for all geometric points $\overline{s}'$ of $S'$.
\end{lemma}
\index{ aaatwist@$\underline{\rm{Twist}}_{H/G}$, ${\rm{Twist}}_{H/G}$}

This lemma is a special case of the deeper result \cite[XI, 4.1]{sga3}
that the functor classifying {\em all} multiplicative type subgroups of $G$ is represented
by a smooth and separated $S$-scheme (without requiring that
$G$ contains any such nontrivial subgroups over $S$).  Whereas
Lemma \ref{hilb} will be deduced from Proposition \ref{finitehom},
which is a special case of \cite[XI, 4.2]{sga3},
in \cite{sga3} the logic goes the other way:  
\cite[XI, 4.2]{sga3} is deduced from \cite[XI, 4.1]{sga3}.

\begin{proof}  We may and do assume $S$ is noetherian. Since \'etale descent
is effective for schemes that are quasi-affine over the base,
to construct the quasi-affine ${\rm{Twist}}_{H/G} \rightarrow S$
representing $\underline{\rm{Twist}}_{H/G}$ 
we may work \'etale-locally on $S$ so that $H$ has {\em constant} Cartier dual $M$.
Thus, $\underline{\rm{Aut}}_{S\mbox{-}{\rm{gp}}}(H) = \Gamma_S$ 
for the ordinary finite group $\Gamma = {\rm{Aut}}(M^{\vee})$, where
$M^{\vee} := \Hom(M,\Q/\Z)$. 

Inside $\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$, the monicity condition on $S'$-homomorphisms
$H_{S'} \rightarrow G_{S'}$ is represented by an open subscheme $V$.
Indeed, if $B$ is noetherian and $f:K\rightarrow B$ is a  finite group scheme with $K_b = 1$ for some $b \in B$
then $K|_U = 1$ for some open $U \subset B$ around $b$ (by applying
Nakayama's Lemma to the stalks of the ideal sheaf $\ker(e^{\ast}:f_{\ast}(\calO_K) \rightarrow \calO_B)$
of the identity section over $B$).  Applying this to the kernel of the universal
homomorphism over $B := \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$ gives the open $V$.
The natural right action of $\Gamma_S$ on the finite type $S$-affine scheme
$\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$ via
$\gamma.f = f \circ \gamma$ leaves $V$ stable and is free on $V$, so by \cite[V, Thm.\,4.1(iv)]{sga3} there exists a finite
\'etale quotient map $$V \rightarrow 
Q := V/\Gamma_S.$$ 
This quotient is constructed via $\Gamma$-invariants over open affines in $S$;
it is quasi-affine over $S$ by 
\cite[II, 6.6.1, 5.1.6(c$'$)]{ega} (applied to the finite \'etale cover $V$ that is quasi-affine over $S$). 

It remains to show $V/\Gamma_S$ represents
$\underline{\rm{Twist}}_{H/G}$.
For the evident $\Gamma_S$-invariant map $V \rightarrow \underline{\rm{Twist}}_{H/G}$,
the induced map
$V/\Gamma_S \rightarrow \underline{\rm{Twist}}_{H/G}$ is a monomorphism because if 
$j, j':H_{S'} \rightrightarrows G_{S'}$
are $S'$-subgroup inclusions whose images agree as closed subschemes
then $j' = j \circ \gamma$ for some $\gamma \in {\rm{Aut}}_{S'}(H_{S'}) = \Gamma_S(S')$.

It remains to prove that the map $V \rightarrow \underline{\rm{Twist}}_{H/G}$ between sheaves for the
\'etale topology is a surjection (thereby forcing the inclusion $V/\Gamma_S 
\hookrightarrow \underline{\rm{Twist}}_{H/G}$ to
be an equality).  Pick an $S$-scheme $S'$ and an $S'$-subgroup $H'$ of $G_{S'}$ such that
$H'_{\overline{s}'} \simeq H_{\overline{s}'}$ for all geometric points $\overline{s}'$ of $S'$.
In particular, $H'$ has finite fibers.
It suffices to find an \'etale cover $S''$ of $S'$ over which the pullbacks of $H'$ and $H$ become
isomorphic as group schemes. 
Passing to an \'etale cover brings us to the trivial case that $H'$ and $H$ each have constant
Cartier dual.
\end{proof}

To finish the proof of Theorem \ref{xi5.3bis}, note that 
the conjugation action of $G$ on itself defines a left action of $G$ 
on ${\rm{Twist}}_{H/G}$, and $N_G(H)$ is the stabilizer of the $S$-point 
of ${\rm{Twist}}_{H/G}$ corresponding
to the given copy of $H$ in $G$.  Thus, 
the separated and finite type algebraic space $G/N_G(H)$
over $S$ is a subfunctor of 
${\rm{Twist}}_{H/G}$, so applying Knutson's schematic criterion \cite[II, 6.15]{knutson}
proves that $G/N_G(H)$ is a scheme 
that is separated and quasi-finite over the $S$-scheme
${\rm{Twist}}_{H/G}$ that we know is quasi-affine
over $S$.  Applying Zariski's Main Theorem, the monomorphism 
$G/N_G(H) \hookrightarrow {\rm{Twist}}_{H/G}$ must be quasi-affine
(in Remark \ref{immqt} we show it is an open immersion).
Hence, $G/N_G(H)$ is quasi-affine over $S$ since ${\rm{Twist}}_{H/G}$ is.
\end{proof}

\begin{remark}\label{immqt}
A common difficulty with algebraic spaces is the intervention
of monomorphisms $j:X \rightarrow Y$
such that it is not obvious if $j$ is a (locally closed) immersion.  
In the proof of Theorem \ref{xi5.3bis} we encountered
two such maps, namely the orbit maps $j:G/Z_G(H) \rightarrow \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G)$
and $j':G/N_G(H) \rightarrow {\rm{Twist}}_{H/G}$ for $H$ a {\em finite} $S$-group
of multiplicative type (with noetherian $S$).  Ignorance of the immersion
property for these maps is irrelevant for our purposes. 
We shall now prove that these monomorphisms are open immersions.

Let $f:X \rightarrow Y$ be a map of finite type between noetherian schemes.
It is an open immersion if and only if it is an \'etale monomorphism \cite[IV$_4$, 17.9.1]{ega},
or equivalently a smooth monomorphism. 
We apply this criterion for open immersions to the orbit maps
$$j:G/Z_G(H) \rightarrow \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,G),\,\,\,
j':G/N_G(H) \rightarrow {\rm{Twist}}_{H/G}$$
with $G$ and $H$ as in Theorem \ref{xi5.3bis} and $H$ finite over $S$ (a noetherian scheme). 
These maps are monomorphisms of finite type, 
and (by \cite[IV$_4$, 17.14.2, 17.7.1(ii)]{ega}) to verify the functorial 
criterion for \'etaleness we may assume $S = \Spec R$ for an artin local 
ring $(R,\mathfrak{m})$ with algebraically closed residue field $k$.  Let
$R_0 = R/J$ for an ideal $J \subset R$ satisfying $J^2 = 0$. 
Consider an $R$-homomorphism $f:H \rightarrow G$ 
(resp.\:an $R$-subgroup $H' \subset G$  as in Lemma \ref{hilb} with $S'=S$).  For any $R$-algebra $A$ and $g \in G(A)$,
let $c_g$ denote conjugation on $G_A$ by $g$.
Assuming that $f_0$ arises from $(G/Z_G(H))(R_0)$ (resp.\:$H'$ arises from
$(G/N_G(H))(R_0)$), we seek to prove that $f$ arises from
$(G/Z_G(H))(R)$ (resp.\:$H'$ arises from $(G/N_G(H))(R)$).

The quotient maps $G \rightarrow G/Z_G(H)$ and
$G \rightarrow G/N_G(H)$ are smooth since
$Z_G(H)$ and $N_G(H)$ are smooth. 
Any $R_0$-point lifts through a smooth surjection since the residue field
is algebraically closed, so $(G/Z_G(H))(R_0) = G(R_0)/Z_G(H)(R_0)$
and similarly for $G/N_G(H)$.

Consider the case of $G/Z_G(H)$, so $f:H \rightarrow G$ is an $R$-homomorphism
that lifts $c_{g_0}|_{H_0}$ for some $g_0 \in G(R_0)$.  By $R$-smoothness of $G$, $g_0$ lifts to some $g \in G(R)$,
so $c_g|_H:H \rightarrow G$ and $f$ are $R$-homomorphisms
with the same reduction.  By Corollary \ref{biglift}, 
we can change the choice of $g$ lifting $g_0$ if necessary so that $f  = c_g|_H$.  

The case of $G/N_G(H)$ goes similarly.
Indeed, by hypothesis $H'_0 = c_{g'_0}(H_0)$
for some $g'_0 \in G(R_0)$, and we choose $g' \in G(R)$ lifting $g'_0$, so 
$c_{g'}(H)$ and $H'$ are multiplicative type subgroups of $G$
that lift $H'_0$.   By the deformation theory of
multiplicative type subgroups of smooth affine groups (Corollary \ref{h2obs} and
\cite[III, 3.3]{oesterle}) 
we can change the choice of $g'$ lifting $g'_0$ if necessary
so that $c_{g'}(H) = H'$. 
\end{remark}

Whereas Theorem \ref{xi5.3bis} concerns quasi-affine quotients, 
we know from the classical theory that it is also necessary to consider
quotients $G/H$ that turn out to be projective. The criterion we will use
to make such quotients as schemes, and not merely as algebraic spaces,
is a self-normalizer hypothesis (see Corollary \ref{parnorm} for an important class of examples): 

\begin{theorem}\label{normqt} 
Let $G \rightarrow S$ be a smooth $S$-affine group scheme with connected
fibers, and $H$ a smooth closed 
subgroup with connected fibers such that $H = N_G(H)$.
\begin{enumerate}
\item The quotient sheaf $G/H$ is represented by a smooth $S$-scheme
that is quasi-projective Zariski-locally over $S$, and it
coincides with the functor $\underline{\rm{Twist}}_{H/G}$ of closed 
subgroups of $G$ that are conjugate to $H$ \'etale-locally on the base.
\item
Assume that the geometric fibers $(G/H)_{\overline{s}} = G_{\overline{s}}/H_{\overline{s}}$
are projective.  The morphism $G/H \rightarrow S$
is proper and admits as a canonical $S$-ample line bundle 
$\Det({\rm{Lie}}(\mathscr{H}))^{\ast}$
where the $G/H$-subgroup $\mathscr{H} \hookrightarrow G \times (G/H)$ is the universal
\'etale-local conjugate of $H$ in $G$.  In particular, $G/H \rightarrow S$ is
projective Zariski-locally on the base.
\end{enumerate}
\end{theorem}

The following preliminary remarks should clarify aspects of Theorem \ref{normqt} before
we undertake the proof.  By Proposition \ref{conntrans} (and Exercise \ref{geomconn}) the given smoothness and connectedness
hypotheses on $H$ imply that the normalizer $N_G(H)$ does exist a priori. 
If we do not assume that $N_G(H) = H$  then typically
$N_G(H)$ may not be $S$-flat (so there would not be a useful 
notion of quotient $G/N_G(H)$); see Example \ref{219}. 
By Corollary \ref{etconj}, if $H$ is of multiplicative type then the criterion defining
the functor $\underline{\rm{Twist}}_{H/G}$ in part (1)
can be expressed on geometric fibers as in Lemma \ref{hilb}.  

The \index{ampleness|(}
notion of ``$S$-ample'' that we use in part (2) means ``ample on fibers''; by 
\cite[IV$_3$, 9.6.4]{ega}, for proper and finitely presented $S$-schemes this implies the usual notion of
ampleness over affine opens in $S$.   
Part (2) of Theorem \ref{normqt} is \cite[XXII, 5.8.2]{sga3}, apart from the explicit
$S$-ample line bundle $G/H$ (which is borrowed from the proof
of \cite[XVI, 2.4]{sga3}).  

The existence of a canonical $S$-ample
line bundle on $G/H$ in part (2) will be crucial in our later
construction of the ``scheme of Borel subgroups'' of
a reductive group scheme $G$.  The reason is that in general 
$G$ does not admit a Borel subgroup over $S$, so 
it is necessary to pass to an \'etale cover $S' \rightarrow S$
in order that there exists a Borel subgroup $B' \subset G_{S'}$.
We will then be faced with a descent problem for $G_{S'}/B'$ relative
to $S' \rightarrow S$.  The {\em canonical} $S'$-ample line bundle
on $G_{S'}/B'$ (arising from the Lie algebra of the universal Borel subgroup
in the $G_{S'}/B'$-group $G_{S'} \times(G_{S'}/B')$) will ensure
the effectivity of the descent.
The fibral projectivity hypothesis in part (2) is a familiar condition in the classical theory.

\begin{remark}\label{antican} 
In the classical theory over an algebraically closed field $k$, it is well-known that 
the quotient $G/P$ modulo a parabolic subgroup has a canonical ample
line bundle: the anti-canonical bundle $\Det(\Omega^1_{(G/P)/k})^{\ast}$.
This is the line bundle in Theorem \ref{normqt}(2) when $S = \Spec k$, up to a twist 
against the $k$-line $\Det({\rm{Lie}}(G))^{\ast}$.  To explain this link in the relative setting, 
view $f:X = G/H \rightarrow S$ as the moduli scheme classifying
closed subgroups of $G$ that are conjugate to $H$ \'etale-locally on the base.
We claim that the $S$-ample line bundle $\Det({\rm{Lie}}(\mathscr{H}))^{\ast}$ 
arising from the universal closed $X$-subgroup $j:\mathscr{H} \hookrightarrow G \times_S X$
is canonically isomorphic to $(\Det_X(\Omega^1_{X/S}))^{\ast}
\otimes f^{\ast}(\Det_S {\rm{Lie}}(G))^{\ast}$.  

Let $\mathscr{I}_j$ be the ideal defining $j$,
so we obtain (see \cite[IV$_4$, 17.2.5]{ega}) an exact sequence of vector bundles on $\mathscr{H}$:
$$0 \rightarrow \mathscr{I}_j/\mathscr{I}_j^2 \rightarrow j^{\ast}(\Omega^1_{(G \times_S X)/X})
\rightarrow \Omega^1_{\mathscr{H}/X} \rightarrow 0.$$
 Pulling back along the identity section $e:X \rightarrow \mathscr{H}$
yields an exact sequence
$$0 \rightarrow e^{\ast}(\mathscr{I}_j/\mathscr{I}_j^2) \rightarrow f^{\ast}({\rm{Lie}}(G)^{\ast})
\rightarrow {\rm{Lie}}(\mathscr{H})^{\ast} \rightarrow 0$$
of vector bundles on $X$.  But the left term is identified with $\Omega^1_{X/S}$ due to 
the cartesian square
$$\xymatrix{\mathscr{H} \ar[r]^-{j} \ar[d]_-{\pi} & {G \times_S X} \ar[d]^-{a} \\
X \ar[r]_{\Delta_{X/S}} & {X \times_S X}}$$
in which $a(g, x) := (gx, x)$ and $\pi$ satisfies $\pi \circ e = {\rm{id}}_X$.  
(Indeed, $\mathscr{I}_j/\mathscr{I}_j^2 \simeq \pi^{\ast}(\mathscr{I}_{\Delta}/\mathscr{I}_{\Delta}^2) =
\pi^{\ast}(\Omega^1_{X/S})$, so applying $e^{\ast}$ gives the identification.)  Thus, 
$$\Det \Omega^1_{X/S} \otimes \Det({\rm{Lie}}(\mathscr{H}))^{\ast} \simeq
f^{\ast}({\Det}_S {\rm{Lie}}(G))^{\ast},$$ yielding the asserted description of $\Det({\rm{Lie}}(\mathscr{H}))^{\ast}$.
\index{ampleness|)}
\end{remark}

\begin{proof}[Proof of Theorem $\ref{normqt}$]
Since $N_G(H) = H$ and $H$ is smooth (so $G \rightarrow G/H$ admits
sections \'etale-locally on $G/H$), it follows by effective descent for
closed subschemes that the quotient sheaf
$G/H$ coincides with the functor of 
smooth closed subgroups of $G$ that are conjugate to $H$
\'etale-locally on the base.
To prove the representability of this functor and its
properties as asserted in (1) and (2),
we may and do assume that $S$ is noetherian.

For $n \ge 0$, let $H_n$ denote the $n$th infinitesimal neighborhood of the identity in $H$
(i.e., the closed subscheme defined by the $(n+1)$th power of the ideal of the identity section).
The construction of $N_G(H)$ in the proof of Proposition \ref{conntrans} gives
a description of $N_G(H)$ as an infinite descending intersection,
namely the intersection of the normalizers of the closed subschemes $\{H_n\}_{n \ge 1}$ that are finite flat
over $S$ (where we define the ``normalizer'' of $H_n$ in the evident manner; this
makes sense even though $H_n$ is usually not a subgroup scheme of $H$,
and it exists by Proposition \ref{xi2.4bis}).
The noetherian property of $G$ implies that the intersection stabilizes
for large $n$.  In other words, for sufficiently large $n$ we have an equality
$$N_G(H) = N_G(H_n)$$
inside $G$.  Fix such an $n > 0$.  
By hypothesis $H = N_G(H)$, so   $N_G(H_n) = H$.

Since $G$ and $H$ are smooth, with $H$ closed in $G$, 
we can write the finite flat $S$-schemes $G_n$ and $H_n$ in the form
$G_n = \Spec_S(A_n)$ and $H_n = \Spec_S(B_n)$ for coherent 
$\calO_S$-algebras $A_n$ and $B_n$ that are locally free over $\calO_S$, with $B_n$ a quotient of $A_n$.
The degrees of $H_n$ and $G_n$ over $S$ are given 
by some universal formulas in terms of $n$ and the relative dimensions of $H$ and $G$;
let $N$ denote the degree of $H_n$ over $S$.
Consider the Grassmannian ${\rm{Gr}}_N(G_n)$ that classifies
quotient vector bundles of $A_n$ with rank $N$. 
The conjugation action of $G$ on itself induces an action of $G$ on $G_n$, and hence
an action of $G$ on ${\rm{Gr}}_N(G_n)$.  Under this action, the
$S$-point $\xi$ of ${\rm{Gr}}_N(G_n)$ corresponding to $H_n \subset G_n$
has functorial stabilizer $N_G(H_n) = N_G(H) = H$.
We conclude that the orbit map
$G \rightarrow {\rm{Gr}}_N(G_n)$ through $\xi$
identifies the quotient sheaf $G/H$ for the \'etale (or equivalently, fppf) topology with a subfunctor of 
the projective $S$-scheme ${\rm{Gr}}_N(G_n)$.  
Now
we can run through the same argument with algebraic spaces
(and Zariski's Main Theorem) as in the proof of Theorem \ref{xi5.3bis}
to conclude that $G/H$ is a smooth $S$-scheme that is quasi-affine over
${\rm{Gr}}_N(G_n)$.  This completes the proof of (1).

Finally, assume every geometric fiber $G_{\overline{s}}/H_{\overline{s}} = (G/H)_{\overline{s}}$
is projective.  The properness of $G/H \rightarrow S$
 in such cases is a consequence of the following general fact: 
 if $f:X \rightarrow S$ is a separated flat surjective map of finite
type (with $S$ noetherian) and if the fibers $X_s$ are proper and geometrically connected
then $f$ is proper. To prove this fact (a special
case of \cite[IV$_3$, 15.7.10]{ega}), by direct limit considerations 
we may pass to local rings on $S$; i.e., we can assume $S$ is local.  Then by fpqc descent
for the properness property of morphisms (which we only need in the quasi-projective case,
for which it reduces to the topological property of closedness for a locally closed immersion), 
we may assume $S = \Spec(A)$ for a {\em complete} local noetherian ring $A$.

By a deep result of Grothendieck on algebraization for formal $A$-schemes (see
\cite[III$_1$, 5.5.1]{ega}), properness of the special fiber $X_0$ provides
an $S$-proper open and closed subscheme $Z \subset X$ with $Z_0 = X_0$.
(The existence of $Z$ is a simple consequence of the theorem 
on formal functions if we assume
$X$ is open in a proper $A$-scheme $\overline{X}$, as
is automatic when $X$ is quasi-projective.  To see this, note that 
the open subscheme $X_0 \subset \overline{X}_0$ is closed
by properness of $X_0$, so it is the zero scheme of an idempotent $e_0$
on $\overline{X}_0$. Idempotents uniquely lift
through infinitesimal thickenings, so by the theorem on formal functions
we can lift $e_0$ uniquely to an idempotent $e$ on $\overline{X}$.
Then the $S$-proper open and closed subscheme $\overline{Z}$ of $\overline{X}$
defined by the vanishing of $e$ meets $X$ in an open subscheme $Z$ of $\overline{Z}$
that is open and closed in $X$ and has special fiber $Z_0 = \overline{Z}_0 \cap X_0 = \overline{Z}_0$,
so the closed complement of $Z$ in the $S$-proper $\overline{Z}$ is empty because
its special fiber is empty.  Such an $A$-proper $\overline{X}$ actually exists
even when $X$ is not quasi-projective, due to the Nagata compactification theorem,
but that lies much deeper than Grothendieck's construction of $Z$
in general.  Anyway, we only need the case of quasi-projective $X$, namely $G/H$ above.)
Since $f$ is an open and closed map (as it is flat and proper) and $Z_0 = X_0 \ne \emptyset$, so
$f(Z) = S$ because the local $S$ is connected, we conclude that $Z_s$ is non-empty for all $s \in S$.
But each $X_s$ is connected by hypothesis, and $Z_s$ is open and closed in $X_s$,
so $Z_s = X_s$ for all $s$.  Hence, $Z = X$, so $X$ is $S$-proper.
 
 We conclude from the properness of $G/H$ that the monomorphism 
$i:G/H \rightarrow {\rm{Gr}}_N(G_n)$ constructed above over $S$ is proper,
and hence by
\cite[IV$_3$, 8.11.5]{ega}
$i$ is a {\em closed immersion} (compare with Remark \ref{immqt}).  Let $\mathscr{O}_{G_n}$ denote
the structure sheaf on $G_n$, viewed as a vector bundle on $S$.
The canonical $S$-ample line bundle $\mathscr{N}$ on 
the Grassmannian equips $G/H$ with a line bundle $\mathscr{L} = i^{\ast}(\mathscr{N})$ that is
$S$-ample (since $i$ is a closed immersion).  Although $\mathscr{L}$
depends on the  choice of Grassmannian ${\rm{Gr}}_N(G_n)$
(i.e., depends on the choice of $n$), so it is not canonically attached
to $(G, H)$, nonetheless $\mathscr{L}$ serves a useful purpose: we will
prove that it is a non-negative power of $\Det({\rm{Lie}}(\mathscr{H}))^{\ast}$,
so this dual determinant bundle is also $S$-ample, as desired.  

By construction of the Pl\"ucker embedding of the Grassmannian, $\mathscr{N}$ is the
determinant of the universal rank-$N$ quotient bundle $\mathscr{Q}$ of 
$\mathscr{O}_{G_n}$ over ${\rm{Gr}}_N(G_n)$. 
The $n$th infinitesimal neighborhood $\mathscr{H}_n$
of $\mathscr{H}$ along its identity section is finite locally free over $G/H$ of rank $N$,
so the structure sheaf $\mathscr{O}_{\mathscr{H}_n}$ may be viewed
as a rank-$N$ vector bundle quotient of $\mathscr{O}_{G_n \times (G/H)}$ over $G/H$.
By the definition of $i$, we have $i^{\ast}(\mathscr{Q}) = \mathscr{O}_{\mathscr{H}_n}$
as quotients of $\mathscr{O}_{G_n \times (G/H)}$, so
$\mathscr{L} = \Det_{G/H}(\mathscr{O}_{\mathscr{H}_n})$.

There is an evident filtration of $\mathscr{O}_{\mathscr{H}_n}$ by powers of the augmentation
ideal of $\mathscr{H}$,
and this filtration has successive quotients ${\rm{Sym}}^i_{G/H}({\rm{Lie}}(\mathscr{H})^{\ast})$ for $0 \le i \le n$.
For any vector bundle $\mathscr{E}$ of rank $r$,
naturally $\Det({\rm{Sym}}^i(\mathscr{E})) \simeq (\Det \mathscr{E})^{m(i,r)}$
for an exponent $m(i,r) \ge 0$ depending only on $i$ and $r$.
Hence, the $S$-ample line bundle $\mathscr{L} = \Det \mathscr{O}_{\mathscr{H}_n}$ is a non-negative power of 
$\Det({\rm{Lie}}(\mathscr{H}))^{\ast}$. 
(The power depends on $n$ and  on the relative dimensions of
$H$ and $G$ over $S$.)
\end{proof}

\newpage

\subsection{Exercises}

\begin{exer}\label{closedmult}
Let $M$ be a finitely generated abelian group, and $k$ a field.
Prove that every closed $k$-subgroup scheme $H$ of $D_k(M)$
has the form $D_k(M/N)$ for a subgroup $N \subset M$.
(Hint: reduce to the case $k = \overline{k}$, so $H_{\rm{red}}$
is a {\em smooth} subgroup.  By considering $H_{\rm{red}}^0$
and a decomposition of $M$ into a product of a finite free $\Z$-module
and a finite abelian group, 
first treat the case when $M$ is free and $H$ is smooth and connected.  Then 
pass to $D_k(M)/H^0_{\rm{red}}$ in general to reduce to the case of finite $H$,
for which Cartier duality can be used.) 
\end{exer}
%
%\begin{exer}\label{drexer} This exercise proves the result of Deligne and Rapoport used in the proof
%of Proposition \ref{conntrans}. Let $f:X \rightarrow Y$ be 
%a quasi-finite flat and separated map between noetherian
%schemes, and assume its fiber rank is constant.  We seek to prove that $f$ is finite.
%
%(i) Using that a proper quasi-finite map is finite, reduce to the case $Y = \Spec R$ for
%a discrete valuation ring $R$ (hint: use the valuative criterion for properness). 
%
%(ii) By Zariski's Main Theorem, the quasi-finite separated $X$ over $Y$ admits an open
%immersion $j:X \hookrightarrow \overline{X}$ into a finite $Y$-scheme $\overline{X}$.
%With $Y = \Spec R$ as in (i), arrange that $\overline{X}$ is also $R$-flat
%and has the same generic fiber as $X$.
%
%(iii) Using constancy of fiber rank, deduce that $j$ is an isomorphism and conclude.
%
%(iv) In case $f$ is \'etale, express the result in the language of specialization for
%constructible \'etale sheaves.
%
%(v) Remove the noetherian hypotheses without requiring $f$ to be of finite presentation.
%
%\end{exer}
%

\begin{exer}\label{ix6.4}
Let $H \rightarrow S$ be a group of multiplicative type,
and let $j:K \hookrightarrow H$  be a finitely presented quasi-finite closed subgroup.  This exercise proves
$K$ is {\em finite} over $S$ (a special case of \cite[IX,  6.4]{sga3}).

(i) Reduce to the case of noetherian $S$. Using that proper monomorphisms
are closed immersions, reduce to $S = \Spec R$ for a discrete valuation ring $R$ (hint: valuative
criterion). Further reduce to the case when $H = D_R(M)$ for a finitely generated
abelian group $M$. 

(ii) With $S = \Spec R$ as in (i), use Exercise \ref{closedmult} to show 
that the schematic closure in $H$ of the generic fiber
of $K$ is $D_R(M/M')$ for $M' \subset M$ of finite index.

(iii) Pass to quotients by $D_R(M/M')$ to reduce to the case $M' = 0$, and conclude via Zariski's Main Theorem.
\end{exer}

\begin{exer}\label{ix6.8}
Let $f:H \rightarrow G$ be a homomorphism from a multiplicative type group $H$ to an $S$-affine 
group $G$ of finite presentation over a scheme $S$.  Prove as follows that
$K := \ker f$ is multiplicative type (so $f$ factors through the multiplicative type fppf quotient $H/K$
that is an $S$-subgroup of $G$); this is part of \cite[IX,  6.8]{sga3}.

(i) Reduce to split $H = D_S(M)$
and $S = \Spec A$ for local noetherian $A$.

(ii) Prove that $K$ is closed in $H$, so by Exercise \ref{closedmult} the special fiber 
$K_0$ equals $D_k(M/M')$ for a subgroup $M' \subset M$ (with $k$ the residue field of $A$).
Prove that the map $D_A(M/M') \rightarrow G$ vanishes
(hint: use Corollary \ref{biglift}), so $D_A(M/M') \subset K$.

(iii) Replace $H$ with $H/D_A(M/M') = D_A(M')$ so that $K_0 = 1$.  By
considering the special fiber of each {\em finite} (perhaps non-flat?) $S$-group $K[n]$ with $n \ge 1$,
prove $K_s = 1$ for all $s \in S$.  
Use Lemma \ref{flatclosed} to show that $e:S \rightarrow K$ is an isomorphism.

(iv) Relax the affineness hypothesis on $G$ to separatedness.
\end{exer}

\begin{exer}\label{2.7}
Let $X$, $Y$, $Z$ be schemes over a ring $k$, and $\alpha:X \times Y \rightarrow Z$
a $k$-morphism.   For a closed subscheme $\iota:Z' 
\hookrightarrow Z$ and $k$-algebra $R$, let $\underline{\rm{Transp}}_X(Y,Z')(R)$ be 
the set of $x \in X(R)$ such that $\alpha(x,\cdot)$ carries $Y_R$ into $Z'_R$. 

(i) If $Y$ is affine and $k[Y]$ is $k$-free, prove 
$\underline{\rm{Transp}}_X(Y,Z')$ is represented by a closed subscheme 
${\rm{Transp}}_X(Y,Z')$ of $X$.
(Hint: Reduce to affine $X$. If $\{e_i\}$ is a $k$-basis of $k[Y]$ and 
$\{f_j\}$ generates the ideal in $k[X \times Y] = k[X] \otimes_k k[Y]$ 
of the pullback of $\iota$, with $f_j = \sum h_{ij} \otimes e_i$, 
consider the zero scheme of the $h_{ij}$ in $X$.)
Remove the affineness hypothesis on $Y$ if $k$ is artinian and $Y$ is $k$-flat
(and see \cite[VIII, \S6]{sga3} for further generalizations).

(ii) Let $Y$ be a closed subscheme of a separated $k$-group $G$. Using (i)
with $\Delta_{G/k}$ as $\iota$, construct 
a closed subscheme $Z_G(Y) \subset G$ if 
$k$ is a field or if $Y$ is affine and $k[Y]$ is $k$-free. 
Discuss the case $Y = G$.
For a closed subscheme $Y' \subset G$, construct
${\rm{Transp}}_G(Y,Y')$. How about $N_G(Y)$?

(iii) Compute equations for $N_G(G') \subset G$ over $k = \Z[t]$ in Example \ref{219}.

(iv) Consider a finite type group $G$ over a field $k = k_s$ and a closed subscheme
$Y \subset G$. If $Y$ is smooth then prove 
$Z_G(Y) = \bigcap_{y \in Y(k)} Z_G(y)$, and if $G$ is smooth then prove 
$Y$ is normalized
(resp.\,centralized) by $G$ if it is normalized (resp.\,centralized) by $G(k)$.
\end{exer}

\begin{exer}\label{fiberex}
Let $\pi:G \rightarrow S$ a finite group scheme, with $\pi_{\ast}(\calO_G)$  locally free over
$\calO_S$.  
Let $G'$ be an $S$-affine $S$-group of finite presentation.
Recall that the functor $\underline{\Hom}(X,Y)$ is represented 
by an $S$-affine $S$-scheme of finite presentation 
under the hypotheses of the first part of 
Lemma \ref{finitehom}. Use several fiber
products to represent
$\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(G,G')$ by 
an $S$-affine $S$-scheme of finite presentation,
thereby proving the second
part of Lemma \ref{finitehom}. 
\end{exer}

\begin{exer}\label{2.1}
(i) Let $G$ be ${\rm{SL}}_n$ or ${\rm{PGL}}_n$ 
 over a ring $k$ and $T$ the diagonal torus,
 or let $G = {\rm{Sp}}_{2n}$ and $T$ the torus
of points $\left(\begin{smallmatrix} t & 0 \\ 0 & t^{-1} \end{smallmatrix}\right)$
for diagonal $t \in {\rm{GL}}_n$.  In all cases prove $Z_G(T) = T$ (so $T$ is a maximal torus
on all geometric fibers)
by using the Lie algebra.

(ii) Using (i), prove $Z_{{\rm{SL}}_n} = \mu_n$, $Z_{{\rm{PGL}}_n} = 1$, and
$Z_{{\rm{Sp}}_{2n}} = \mu_2$ as schemes.   
\end{exer}

\begin{exer}\label{2.3}
Consider a field $k$ and a $k$-group $H$ acting on a separated $k$-scheme $Y$.
For a $k$-scheme $S$, let 
$Y^H(S)$ be the set of $y \in Y(S)$ invariant by the $H_S$-action on $Y_S$.

(i) Adapt Exercise \ref{2.7}(ii) to prove that $Y^H$ is represented by 
a closed subscheme of $H$. In case $H$ is smooth
and $k = k_s$, prove $Y^H = \bigcap_{h \in H(k)} Y^h$
where $Y^h := \alpha_h^{-1}(\Delta_{Y/k})$ for the map $\alpha_h:Y \rightarrow Y \times Y$ defined by 
$y \mapsto (y, h.y)$.
Relate this to Exercise \ref{2.2}.

(ii) For $Y$ of finite type over $k$ and $y \in Y^H(k)$, prove
${\rm{Tan}}_y(Y^H) = {\rm{Tan}}_y(Y)^H$ for a suitable $H$-action on ${\rm{Tan}}_y(Y)$.

(iii) Assume $H$ is a closed subgroup of a $k$-group $G$ of finite type. Let 
$\mathfrak{g} := {\rm{Lie}}(G)$ and $\mathfrak{h} := {\rm{Lie}}(H)$.
Prove ${\rm{Tan}}_e(Z_G(H)) = \mathfrak{g}^H$ (schematic invariants under the adjoint action).  Also prove 
${\rm{Tan}}_e(N_G(H)) = \bigcap_{h \in H(k)} ({\rm{Ad}}_G(h) - 1)^{-1}(\mathfrak{h})$ when $k = k_s$
and $H$ is smooth.
\end{exer}

\begin{exer}\label{parlie} Let $G$ be a smooth connected affine group over a field $k$.
For a smooth connected $k$-subgroup $H \subset G$,
the proof of Theorem \ref{normqt} 
constructs $n > 0$ so that the $n$th-order
infinitesimal neighborhood $H_n$ of 1
in $H$ satisfies $N_G(H_n) = N_G(H)$.  

(i) For any subspace $V \subset \mathfrak{g}$, show that the ${\rm{Ad}}_G$-stabilizer
$N_G(V)$ of $V$ in $G$ has Lie algebra $\mathfrak{n}_{\mathfrak{g}}(V)$ equal to the
normalizer of $V$ in $\mathfrak{g}$.

(ii) Assume $G$ is reductive and let $P \subset G$ be parabolic with Lie algebra $\mathfrak{p}
\subset \mathfrak{g}$.  Show the inclusion $P \subset N_G(\mathfrak{p})$ identifies
$P$ with $N_G(\mathfrak{p})_{\rm{red}}$.  (Hint: Reduce to $k = \overline{k}$
and choose a maximal torus $T \subset P$. Let $Q := N_G(\mathfrak{p})_{\rm{red}}$,
so $Q$ is parabolic since $P \subset Q$.  For a Borel $B \subset P$ containing $T$
and the associated basis $\Delta$ of $\Phi(G,T)$, $P = P_I$ and $Q = P_J$ for subsets
$I \subset J$ of $\Delta$.  If there exists $a \in J - I$
then by identifying 
$G_a = \mathscr{D}(Z_G(T_a))$ with ${\rm{SL}}_2$ or ${\rm{PGL}}_2$
show for $u \in U_{-a}(k) - \{1\}$ that ${\rm{Ad}}_G(u)(\mathfrak{g}_a)$ has nonzero 
component in the $-a$-weight space.)
If ${\rm{char}}(k) \ne 2$
(so $\mathfrak{sl}_2 = \mathfrak{pgl}_2$!) then show
$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{p}) = \mathfrak{p}$ and deduce
$N_G(\mathfrak{p}) = P$ as schemes.  

(iii) Assume ${\rm{char}}(k) = 2$
and $G = {\rm{SL}}_2$.  Let $B$ be the upper triangular Borel subgroup.
Show $N_G(B)$ has Lie algebra $\mathfrak{sl}_2$,
and for the $k$-algebra $R = k[\epsilon]$ of dual numbers
and $g = (\begin{smallmatrix} 1 & 0 \\ \epsilon & 1 \end{smallmatrix}) \in G(R)$
show $gB_Rg^{-1}$ and $B_R$ are distinct Borel $R$-subgroups of
${\rm{SL}}_2$ with the same Lie algebra over $R$. 
Also show that all maximal $k$-tori $T$ in ${\rm{SL}}_2$
have the {\em same} Lie algebra in $\mathfrak{sl}_2$ (hint: ${\rm{Lie}}(Z_{{\rm{SL}}_2}) = {\rm{Lie}}(T)$),
 so membership of $T$ in $B$
{\em cannot} be detected on Lie algebras. Show that if $n=2$ then 
$N_{{\rm{SL}}_2}(D_n) = N_{{\rm{SL}}_2}(D)$ for the diagonal torus $D$.

(iv) Let $G$ be semisimple of {\em adjoint type} over $k = \overline{k}$. Consider 
$(P, T, B, \Delta)$ as in (ii), so $\Delta$ is a $\Z$-basis of
${\rm{X}}(T)$ and hence $\{{\rm{Lie}}(a)\}_{a \in \Delta}$ is a basis
of the dual space $\mathfrak{t}^{\ast}$ via the canonical identification
${\rm{Lie}}(\Gm) = k$.  Using that 
$[x_a, v] = -{\rm{Lie}}(a)(v) x_a$ for $x_a \in \mathfrak{g}_a$
and $v \in \mathfrak{t}$, show the $T$-stable $\mathfrak{g}$-transporter
of $\mathfrak{t}$ into $\mathfrak{p}$ is exactly $\mathfrak{p}$.
Adapt the argument for (ii) to prove $N_G(\mathfrak{p}) = P$
as schemes, without restriction on ${\rm{char}}(k)$.
(See \cite[XXII, 5.1.7(a), 5.3.2]{sga3}.)
\end{exer}

\begin{exer}\label{2.4}
Let $\Gamma = {\rm{Gal}}(k_s/k)$ for a field $k$.
For a $k$-group $M$ of multiplicative type, the {\em character group}
${\rm{X}}(M) = {\rm{Hom}}_{k_s}(M_{k_s}, \Gm)$ is 
a discrete $\Gamma$-module in an evident manner.

(i) If $k'/k$ is a finite subextension of $k_s$, prove the Weil restriction 
${\rm{R}}_{k'/k}(M')$ is of multiplicative type over $k$ when $M'$ is of multiplicative type over $k'$. (For $M' = \Gm$ this is
``${k'}^{\times}$ viewed as a $k$-group''.)  
By functorial considerations,  
prove ${\rm{X}}({\rm{R}}_{k'/k}(M')) = {\rm{Ind}}_{\Gamma'}^{\Gamma}({\rm{X}}(M'))$
with $\Gamma' = {\rm{Gal}}(k_s/k')$. 
For every $k$-torus $T$, construct a surjective $k$-homomorphism
$\prod_i {\rm{R}}_{k'_i/k}({\mathbf{G}}_m) \twoheadrightarrow T$ 
for finite separable extensions $k'_i/k$.
Conclude that $k$-tori are {\em unirational} over $k$. 

(ii) For a local field and $k$-torus $T$, prove $T$ 
is $k$-anisotropic if and only if $T(k)$ is compact.

(iii) For a finite extension field $k'/k$, define a {\em norm} map
${\rm{N}}_{k'/k}:{\rm{R}}_{k'/k}(\Gm) \rightarrow {\mathbf{G}}_m$. Prove
its kernel is a torus when $k'/k$ is separable. 
What if $k'/k$ is not separable?
\end{exer}
%
%\begin{exer}\label{2.5}
%Let $T$ be a $k$-torus in a $k$-group $G$ of finite type.  
%This exercise uses the automorphism scheme ${\rm{Aut}}_{T/k}$ (which is an \'etale $k$-group). 

%(i) Construct a $k$-morphism $N_G(T) \rightarrow {\rm{Aut}}_{T/k}$ with kernel $Z_G(T)$.
%Prove $W(G,T) := 
%N_G(T)(\overline{k})/Z_G(T)(\overline{k})$ is naturally a {\em finite} subgroup of
%${\rm{Aut}}_{\Z}({\rm{X}}(T))$.  If $f:G' \rightarrow G$ is surjective with finite kernel and $T'$ is a $k$-torus in 
%$G'$ containing $\ker f$ with $f(T') = T$ then prove 
%$W(G',T') \rightarrow W(G,T)$ is an isomorphism. 

%(ii) For $G = {\rm{GL}}_n, {\rm{PGL}}_n, {\rm{SL}}_n, {\rm{Sp}}_{2n}$ and $T$ the $k$-split diagonal maximal $k$-torus
%(so $Z_G(T) = T$), respectively identify
%${\rm{X}}(T)$ with $\Z^n$, $\Z^n/{\rm{diag}}$, $\{m \in \Z^n\,|\,\sum m_j = 0\}$,
%and $\Z^n$.  Prove $N_G(T)(k)/Z_G(T)(k) \subset
%{\rm{Aut}}_{\Q}({\rm{X}}(T)_{\Q})$ is $S_n$ for the first three, and
%$S_n \ltimes \langle -1 \rangle^n$ for ${\rm{Sp}}_{2n}$, all with natural action. 
%(Hint: to control $N_G(T)$, via 
%$G \hookrightarrow {\rm{GL}}(V)$ decompose $V$ as a direct sum of
%$T$-stable lines with {\em distinct} eigencharacters.)
%\end{exer}

\begin{exer}\label{2.6}
Let $X$ be a smooth separated scheme locally of finite type over a field $k$, and 
$T$ a $k$-group of multiplicative type with a left action on $X$.  This exercise is devoted
 to proving that $X^T$ (as in Exercise \ref{2.3}) is {\em smooth}, generalizing
 Lemma \ref{centralizer} over fields.

(i) Reduce to the case $k = \overline{k}$.
Fix a finite local $k$-algebra $R$
with residue field $k$, and an ideal $J$ in $R$ with $J \mathfrak{m}_R = 0$.
Choose $\overline{x} \in X^T(R/J)$, and for $R$-algebras $A$ let $E(A)$ be
the fiber of $X(A) \twoheadrightarrow X(A/JA)$ over $\overline{x}_{A/JA}$.
Let $x_0 = \overline{x} \bmod \mathfrak{m}_R \in X^T(k)$ and $A_0 = A/\mathfrak{m}_R A$.
Prove $E(A) \ne \emptyset$ and make it a torsor over the $A_0$-module
$F(A) := JA \otimes_k {\rm{Tan}}_{x_0}(X) = JA \otimes_{A_0} (A_0 \otimes_k {\rm{Tan}}_{x_0}(X))$
naturally in $A$ (action $v.x$ denoted as $v+x$). 

(ii) Define an $A_0$-linear $T(A_0)$-action on $F(A)$ (hence a $T_R$-action on $F$),
and prove that $E(A)$ is $T(A)$-stable in 
$X(A)$ with $t.(v+x) = t_0.v + t.x$ for $x \in E(A)$, $t \in T(A)$, $v \in F(A)$, 
and $t_0 = t \bmod \mathfrak{m}_R A$.

(iii) Choose $\xi \in E(R)$ and define a map of functors
$h:T_R \rightarrow F$ by $t.\xi = h(t) + \xi$ for 
points $t$ of $T_R$; check
it is a 1-cocycle, and is a 1-coboundary if and only if $E^{T_R}(R) \ne \emptyset$.
For $V_0 = J \otimes_k {\rm{Tan}}_{x_0}(X)$
use $h$ to define a 1-cocycle $h_0:T \rightarrow \underline{V}_0$, and prove
$t.(v,c) := (t.v + c h_0(t), c)$ is a $k$-linear representation of $T$ on $V_0 \bigoplus k$.
Use a $T$-equivariant $k$-linear splitting (!) to prove $h_0$ (and then $h$) is a 1-coboundary;
deduce 
$X^T$ is smooth.
\end{exer}

\begin{exer}\label{h1ex}
Let $S$ be a scheme, $\mathscr{G}$ an fppf $S$-affine $S$-group, and 
${\rm{H}}^1(S,\mathscr{G})$ 
\index{ aaah1sg@${\rm{H}}^1(S,G)$}
the set of isomorphism classes of right $\mathscr{G}$-torsors over $S$
for the fppf topology.  For any homomorphism
$\mathscr{G} \rightarrow \mathscr{G}'$ between such groups,
define  ${\rm{H}}^1(S,\mathscr{G}) \rightarrow {\rm{H}}^1(S,\mathscr{G}')$
via pushout of torsors:  $E \mapsto E \times^{\mathscr{G}} \mathscr{G}'$
(the quotient of $E \times \mathscr{G}'$ by the anti-diagonal $\mathscr{G}$-action
$(e,g').g = (e.g, g^{-1}g')$). 

(i) Prove that any right $\mathscr{G}$-torsor $E$ is necessarily $S$-affine (and fppf), 
and that the quotient $E \times^{\mathscr{G}} \mathscr{G}'$
exists as a scheme. Also use the affineness to prove that 
${\rm{H}}^1(S,\mathscr{G})$ can be 
computed (functorially in $\mathscr{G}$!) by a non-commutative version of the usual \v{C}ech-type
procedure generalizing non-abelian degree-1 Galois cohomology.

(ii) For any fppf $S$-affine $S$-group $G$
and fppf closed $S$-subgroup $H$ such that the fppf quotient sheaf
$G/H$ is represented by a scheme (see Theorem \ref{normqt} for a sufficient criterion), identify
$G(S) \backslash (G/H)(S)$ with the kernel of the map
${\rm{H}}^1(S,H) \rightarrow {\rm{H}}^1(S,G)$.
\end{exer}

\begin{exer}\label{oferex}  
For $n \ge 1$, let $X_n = \Spec A_n$ be the $\C$-scheme obtained by gluing $2^n$ affine lines
in a loop, with 0 on the $i$th line glued to 1 on the $(i+1)$th line ($i \in \Z/2^n\Z$). 

(i) Prove that $\pi_1(X_n) = \widehat{\Z}$ and the finite \'etale covers
of $X_n$ split Zariski-locally on $X_n$.

(ii) Define $X_{n+1} \rightarrow X_n$ by collapsing odd-indexed lines to points
and sending the $2j$th line in $X_{n+1}$ to the $j$th line in $X_n$. Prove
$X_{\infty} := \invlim X_n$ is 
reducible and its local rings are discrete valuation rings or fields.
(This is a slight variant of an example in the Stacks Project.)

(iii) Construct a nontrivial $\Z$-torsor $E_1 \rightarrow X_1$ that is split Zariski-locally on $X_1$ but
not by any finite \'etale cover of $X_1$.  Prove $E_{\infty} := E_1 \times_{X_1} X_{\infty} \rightarrow X_{\infty}$ is a
non-split $\Z$-torsor that splits Zariski-locally on $X_{\infty}$, and construct
a rank-2 torus $T \rightarrow X_{\infty}$ that splits Zariski-locally on $X_{\infty}$ but is not isotrivial.
Thus, ``irreducible''  cannot be relaxed to ``connected'' in Corollary \ref{mikecor}.  
(The preceding construction was suggested by Gabber.) 
\end{exer}

\newpage

\section{Basic generalities on reductive group schemes}\label{basicsec}


\subsection{Reductivity and semisimplicity}\label{redss}

In \cite[XIX, 2.7]{sga3} a connectedness condition is
imposed in the relative theory of reductive groups: 

\begin{definition} Let $S$ be a scheme.  An {\em $S$-torus} is an $S$-group $T \rightarrow S$
of multiplicative type with smooth connected fibers.  A {\em reductive $S$-group}
is a smooth $S$-affine group scheme $G \rightarrow S$ such that the geometric
fibers $G_{\overline{s}}$ are connected reductive groups.
A {\em semisimple $S$-group} is a reductive $S$-group
whose geometric fibers are semisimple.
\end{definition}
\index{reductive group}
\index{semisimple group}
\index{torus}

In this definition, it suffices to check reductivity (resp.\:semisimplicity) for a single geometric
point over each $s \in S$ because for any linear algebraic group $H$ over an algebraically closed field $k$
and any algebraically closed extension $K/k$ the inclusions $\mathscr{R}_u(H)_K \subset \mathscr{R}_u(H_K)$
and $\mathscr{R}(H)_K \subset \mathscr{R}(H_K)$ are equalities (see Exercise \ref{geomrad}).
By Proposition \ref{fppfet}, any $S$-torus becomes a power of $\Gm$ \'etale-locally on $S$
(also see Corollary \ref{fpqcwin}(1) and \cite[II, \S1.3]{oesterle}).

In the theory of linear algebraic groups $G$ over an algebraically closed field $k$, 
reductive groups are often permitted to be disconnected.  One reason is that if 
$g \in G(k)$ is semisimple then $Z_G(g)$ may be disconnected (as happens already
for ${\rm{PGL}}_2$) but $Z_G(g)^0$ is always reductive.  Also, 
the Galois cohomological
 classification of  connected semisimple groups $G$ over a field $k$ leads
to the consideration of the automorphism scheme
$\Aut_{G/k}$, and this is 
a smooth affine $k$-group whose identity component is semisimple
but is usually disconnected when the Dynkin diagram has
nontrivial automorphisms (e.g., if $n > 2$
then $\Aut_{{\rm{SL}}_n/k} = {\rm{PGL}}_n \rtimes \Z/2\Z$ with component group generated
by transpose-inverse).  In \S\ref{straut} we will discuss the existence
and structure of automorphism schemes of
reductive group schemes.

In the relative theory over a scheme that is not a single point, 
the disconnectedness of fibers presents new phenomena not seen in the classical case.  For example, 
if $G \rightarrow S$ is a smooth $S$-affine group scheme then the orders of the 
geometric fibral component groups $\pi_0(G_{\overline{s}})$ can
vary with $\overline{s}$, so these component groups can fail to arise as the fibers of
a {\em finite} \'etale $S$-group (see Example \ref{best}).  

Requiring connectedness of fibers
is not unreasonable.  By Exercise \ref{geomconn}, a connected
group scheme of finite type over a field is geometrically connected (as for
any connected finite type scheme $X$ over a field $k$ when $X(k)$ is non-empty),
so for a group scheme of finite type
the property of having connected fibers is preserved by any base change.
Also, the identity component
varies well in smooth families of
groups:  for any smooth group scheme $G \rightarrow S$ of finite presentation 
there exists a unique open subgroup scheme $G^0 \subset G$ such that $(G^0)_s$ is the identity
\index{ aaagzero@$G^0$}
component of $G_s$ for all $s \in S$ \cite[IV$_3$, 15.6.5]{ega}.   The 
formation of $G^0$ commutes with any base change on $S$ since each $G_s^0$
is geometrically connected, so by reduction to the case of noetherian $S$
we see that $G^0$ is finitely presented over $S$.  Imposing 
connectedness of fibers amounts to passing to $G^0$ in place of $G$.
Beware that passage to $G^0$ can exhibit some
peculiar behavior relative to the theory over a field:

\begin{example}\label{weird}
For smooth $S$-affine $S$-groups $G$, the open subgroup $G^0$ may not be closed.
An interesting example is given in \cite[XIX, \S5]{sga3} over
$\Spec k[t]$ for any field $k$ of characteristic 0. To describe this example, 
let $\mathfrak{g}$ be the Lie algebra over $k[t]$ whose
underlying $k[t]$-module is free with basis $\{X,Y,H\}$ satisfying the bracket relations
$$[H,X]=X, \,\,\,[H,Y]=-Y,\,\,\,[X,Y]=2tH.$$
Over $\{t \ne 0\}$ this becomes isomorphic to 
$\mathfrak{sl}_2$ over the degree-2 finite \'etale cover given by $\sqrt{t}$, using
the $\mathfrak{sl}_2$-triple $(X/\sqrt{t}, Y/\sqrt{t},2H)$, but the fiber at $t=0$ is solvable.
Explicit computations (see \cite[XIX, 5.2--5.10]{sga3}) show
that  the group scheme $G$ of automorphisms of $\mathfrak{g}$
that lie in ${\rm{SL}}(\mathfrak{g})$ is smooth, and that $G|_{t\ne 0}$ is
an \'etale form of ${\rm{PGL}}_2$ but  the fiber $G_0$ at $t=0$  is solvable with two geometric components.
Consequently, $G^0$ cannot be closed since it is a dense open subscheme of $G$
that is distinct from $G$.
\index{monic homomorphism}

 In this example the inclusion morphism $G^0 \rightarrow G$ is affine (see \cite[XIX, 5.13]{sga3}), 
 so $G^0$ is $S$-affine. There are pairs $(G,S)$
with smooth $S$-affine $G$ such that $G^0$ is not 
$S$-affine (so it is not closed in $G$).
See \cite[VII, \S3, (iii)]{raynaud} for such an example over $S = \mathbf{A}^2_k$
with $k$ of characteristic 0.
\end{example}

It turns out that with a reductivity hypothesis on fibral identity components,
the problems in Example \ref{weird} do not arise.  This is made precise
by the following result that we will never use and which rests on many later
developments in the theory: 

\begin{proposition}\label{disconn}
Let $G \rightarrow S$ be a smooth separated group scheme of finite presentation
such that $G^0_{\overline{s}}$ is reductive for all $s \in S$.
Then $G^0$ is a reductive $S$-group
that is open and closed in $G$, and $G/G^0$ exists as a separated \'etale $S$-group
of finite presentation.
\end{proposition}

\begin{proof}
The open subgroup $G^0 \rightarrow S$ is smooth
and finitely presented with connected reductive fibers. 
Incredibly, in the definition of a reductive group scheme
we can replace ``affine'' with ``finitely presented'' \cite[XVI, 5.2(i)]{sga3}, so $G^0$ is a reductive $S$-group
(in particular, $G^0$ is $S$-affine).
Moreover, in Theorem \ref{adimm} we will show that any monic homomorphism from
a reductive group scheme to a finitely presented and separated group scheme is always a closed
immersion.  Hence, for such $G$ we see that $G^0$ is both open {\em and} closed in $G$,
so the quotient $E := G/G^0$ that is initially a finitely presented and \'etale algebraic space
over $S$ is also {\em separated} over $S$.  Thus, after a reduction to the case of
noetherian $S$, we may apply Knutson's
criterion (as in the proof of Theorem \ref{xi5.3bis}) to conclude that $E$ is a scheme.  
\end{proof}

\begin{example}\label{best}
The conclusion in Proposition \ref{disconn} 
 is ``best possible'', in the sense that the relative component group $G/G^0$, which
 is always quasi-finite, separated, and \'etale over $S$, may not be 
finite over $S$.  For example, let $S$ be a connected $\Z[1/2]$-scheme and ${\rm{O}}(q)$ 
the orthogonal group of a nondegenerate quadratic space $(V,q)$ over $S$, so 
${\rm{O}}(q)^0 = {\rm{SO}}(q)$ and ${\rm{O}}(q)/{\rm{SO}}(q) = (\Z/2\Z)_S$.  Let 
$U \subset S$ be a finitely presented non-empty open subscheme with $U \ne S$, so $U$ is not closed in $S$.
The open subgroup $E \subset (\Z/2\Z)_S$
obtained by removing the closed non-identity locus over $S - U$ is not $S$-finite,
so its open preimage $G \subset {\rm{O}}(q)$ satisfies the hypotheses in Proposition \ref{disconn} but 
$G/G^0 = E$ is not $S$-finite.
\end{example}

In the classical theory of connected reductive groups, it is a fundamental fact
that torus centralizers are again connected reductive \cite[13.17, Cor.\,2]{borelag}.  In the relative case
this remains valid: 
if $G$ is a reductive $S$-group scheme and $T$ is an $S$-torus in $G$
then the closed subgroup $Z_G(T)$ is $S$-smooth
(see Lemma \ref{centralizer}) and its geometric fibers are connected
reductive by the classical theory, so $Z_G(T)$ is reductive over $S$.
This can be pushed a bit further, as explained in the following Remark that we will never use.

\begin{remark}
Let $H$ be a multiplicative type subgroup of a reductive $S$-group $G$.
Even over an algebraically closed field, $H$ need not lie in a maximal torus of $G$.
Nonetheless, $Z_G(H)$ is smooth (by Lemma \ref{centralizer}) 
and the geometric fibers $Z_G(H)_{\overline{s}}$ have reductive identity component
(by \cite[A.8.12]{pred}, which applies to a wider class of
fibers $H_s$; its proof rests on 
a hard affineness theorem of Borel and Richardson for coset spaces modulo
 connected reductive groups (\cite{borelaff},  \cite{ri})).  
The fibers of $Z_G(H)$ can be disconnected. 
By Proposition \ref{disconn}, the open subgroup
$Z_G(H)^0$ is reductive (hence affine) over $S$ and closed in $Z_G(H)$.
\end{remark}

To affirm that the notion of reductive group scheme is
reasonable, we want to prove that
reductivity of a fiber is inherited by nearby fibers
for any smooth affine group scheme with connected fibers.   This requires an improvement
on the lifting of tori over adic noetherian rings in Corollary \ref{biglift}, replacing completions
with \'etale neighborhoods:

\begin{proposition}\label{lifttorus}  Let $G \rightarrow S$ be a smooth $S$-affine
group scheme, and $H_0$ a multiplicative type subgroup of the fiber $G_s$
over some $s \in S$.  There exists an \'etale neighborhood
$(S',s')$ of $(S,s)$ with
$k(s') = k(s)$ and a multiplicative type subgroup $H' \subset G_{S'}$
such that $H'_{s'} = H_0$ inside $(G_{S'})_{s'} = G_s$.
\end{proposition}
\index{multiplicative type!deformation in smooth group}

This result is \cite[XI, 5.8(a)]{sga3}. Note that by Lemma \ref{ix2.5}, the monomorphism
$H' \rightarrow G_{S'}$ must be a closed immersion.  

\begin{proof}
We may assume $S$ is noetherian and affine, and even finite type over $\Z$
(by expressing a noetherian ring as a direct limit of its $\Z$-subalgebras
of finite type; see %\cite[IV$_3$, 8.3.4(c), 9.7.7(ii)]{ega} and
\cite[IV$_4$, 17.8.7]{ega} for the descent of smoothness through such direct limits).  
Let $A$ denote the completion $\mathscr{O}_{S,s}^{\wedge}$.
By Corollary \ref{biglift}, there exists a multiplicative type $A$-subgroup $\widehat{H}$ in $G_A$
lifting $H_0$ in the special fiber $G_s$.  
Let $\{A_{\alpha}\}$
be the directed system of finite type $\mathscr{O}_{S,s}$-subalgebras of $A$,
so $A = \varinjlim A_{\alpha}$.  By the argument at the start of the proof of Proposition \ref{xi2.4bis},
we can choose $\alpha$ large enough so that $\widehat{H}$ descends to an
$A_{\alpha}$-group $H_{\alpha}$ of multiplicative type.
%For sufficiently large $\alpha$, the closed $A$-subgroup $\widehat{H} \subset G_A$ of multiplicative type 
%descends to a closed $A_{\alpha}$-subgroup $H_{\alpha} \subset
%G_{\alpha} := G_{A_{\alpha}}$.  By increasing $\alpha$ we can arrange that $H_{\alpha}$ is multiplicative type.
%Indeed, there exists an fppf cover $\Spec A' \rightarrow \Spec A$ such that
%$\widehat{H}_{A'} \simeq D_{A'}(M)$ for some finitely generated abelian group $M$,
%and by increasing $\alpha$ we can arrange that $\Spec A' \rightarrow \Spec A$
%descends to an fppf cover $\Spec A'_{\alpha} \rightarrow \Spec A_{\alpha}$.
%Then the $A'_{\alpha}$-groups $H_{\alpha} \otimes_{A_{\alpha}} A'_{\alpha}$
%and $D_{A'_{\alpha}}(M)$ become isomorphic over $A'$.  However,
%$A'$ is the direct limit of the rings $A'_{\beta} := A'_{\alpha} \otimes_{A_{\alpha}} A_{\beta}$
%over $\beta \ge \alpha$, so by increasing $\alpha$ some more
%we can arrange that $H_{\alpha} \otimes_{A_{\alpha}} A'_{\alpha} \simeq D_{A'_{\alpha}}(M)$.
%Since $\Spec A'_{\alpha}$ is an fppf cover of $\Spec A_{\alpha}$, it follows
%that $H_{\alpha}$ is multiplicative type.
%Fix such an $\alpha$. 
Since $\mathscr{O}_{S,s}$ is
essentially of finite type over $\Z$, and $A_{\alpha}$ is 
finite type over $\mathscr{O}_{S,s}$, we can apply the powerful Artin approximation theorem:
\index{Artin approximation}

\begin{theorem}[Artin]\label{approx}  Let
$R$ be a local ring that is essentially of finite type over $\Z$,
and $B$ a finite type $R$-algebra equipped with a map
$f:B \rightarrow \widehat{R}$ over $R$.
Pick $N \ge 0$. 

The map $f$ admits an $N$th-order ``\'etale'' approximation over $R$ 
in the sense that there exists a residually trivial local-\'etale extension
$R \rightarrow R'$ and an $R$-algebra map
$\varphi:B \rightarrow R'$ such that the induced map to the completion
$$\widehat{\varphi}:B \rightarrow \widehat{R'} = \widehat{R}$$
agrees with $f$ modulo $\mathfrak{m}_{\widehat{R}}^{N+1}$.
\end{theorem}

This theorem says that any solution in $\widehat{R}$ to a finite system
of polynomial equations over $R$ is well-approximated by a solution in the henselization of $R$
(equivalently, a solution in some residually trivial local-\'etale extension of $R$).
The Artin approximation theorem actually allows any excellent Dedekind domain in place of $\Z$ in
Theorem \ref{approx}; see \cite[3.6/16]{neron}
for the proof in that generality.

We apply Theorem \ref{approx} to $R = \mathscr{O}_{S,s}$, $B = A_{\alpha}$, $N = 0$, 
and the inclusion $B \rightarrow \widehat{R}$ 
to obtain a residually trivial local-\'etale extension $R \rightarrow R'$
and an $R$-algebra map $A_{\alpha} \rightarrow R'$ that agrees
residually with the reduction of the given inclusion $A_{\alpha} \hookrightarrow \widehat{R}$.
Thus, the $R'$-group $H_{\alpha} \otimes_{A_{\alpha}} R'$ of multiplicative type in $G_{R'}$
has special fiber $H_0$ in $G_s$!  In other words,
we have found a multiplicative type subgroup lifting $H_0$ over a local-\'etale neighborhood of $(S,s)$.
Spreading this out over an \'etale neighborhood of $(S,s)$ then does the job.
\end{proof}

\begin{corollary}\label{redtorus}
Let $G \rightarrow S$ be an fppf $S$-affine group scheme with connected
fibers, and assume that $G_s$ is a torus 
for some $s \in S$.  Then $G_U$ is a torus for some open neighborhood
$U$ of $s$ in $S$.  
\end{corollary}

This result is \cite[X, 4.9]{sga3}.

\begin{proof}
First we assume that $G$ is smooth, and then we reduce the general case to the smooth case.
Since the property of being a torus is \'etale-local on the base, we may work
in an \'etale neighborhood of $(S,s)$.  Hence, by the smoothness of $G$
we can use Proposition \ref{lifttorus}
to arrange that $G$ contains a multiplicative type subgroup $H$
such that $H_s = G_s$, and that $G$ has constant fiber
dimension.  Passing to a further \'etale neighborhood
makes $H$ split, say $H = D_S(M)$ for a finitely generated abelian group $M$.
Since $H_s$ is a torus, $M$ is free.  Hence, $H$ is a torus.  But
the inclusion $H \hookrightarrow G$ between smooth $S$-affine groups 
with connected fibers is
an equality on $s$-fibers, so by smoothness and constancy of fiber dimensions
over $S$ it follows that $H_{s'} = G_{s'}$ for all $s' \in S$.  
That is, $H \rightarrow G$ induces an isomorphism on fibers,
so it is an isomorphism by the fibral isomorphism criterion (Lemma \ref{flatclosed}). 

In general (with $G$ only assumed to be fppf rather than smooth over $S$), we just need to prove
that $G$ is automatically smooth over an open neighborhood of $s$.
We may reduce to the case when $S$ is local noetherian, and the smoothness of
the $s$-fiber and the fppf hypothesis on $G$ implies that $G \rightarrow S$ is smooth
at all points of $G_s$, and so on an open neighborhood of $G_s$ in $G$.
This open neighborhood has open image in $S$, so by shrinking around $s \in S$ 
we can arrange that this open image is equal to $S$.  In particular, each fiber
group scheme $G_{s'}$ has a non-empty smooth locus, so the fibers
are smooth (due to homogeneity considerations on geometric fibers). 
\end{proof}

\begin{proposition}\label{cartan} Let $G \rightarrow S$ be a smooth $S$-affine group scheme 
and suppose $G_{\overline{s}}^0$ is reductive for some $s \in S$.
\begin{enumerate}
\item There is an open $U$ around $s$ in $S$ such that $G_{\overline{u}}^0$ is
reductive for all $u \in U$.  The same holds for semisimplicity. 
\item If $T \subset G$ is a torus such that $T_{\overline{s}}$ is maximal in
$G_{\overline{s}}^0$ for some $s \in S$ then there exists an open $V$ around $s$ in $S$ such that
$Z_G(T)^0_V = T_V$ and $T_{\overline{v}}$ is a maximal torus in $G_{\overline{v}}^0$
for all $v \in V$.
\end{enumerate}
\end{proposition}

This result is essentially \cite[XIX, 2.6]{sga3} (where it is assumed
that each $G_s$ is connected). Note that in part
 (2) we {\em assume} $T$ exists. The existence of 
such a $T$ \'etale-locally around $s$ follows from Proposition \ref{lifttorus} if we admit
Theorem \ref{grthm}, but we will prove such \'etale-local existence for reductive $G$
in Corollary \ref{buildtori} without using Theorem \ref{grthm}; also see Exercise \ref{maxfield}.
(See \cite[XIV, 3.20]{sga3} for a generalization using the Zariski topology,
building on Theorem \ref{grthm}.)

\begin{proof}
First we prove (2).
The centralizer $Z_G(T)$ in $G$ is a smooth closed
$S$-subgroup by Lemma \ref{centralizer}, and obviously
$T \subset Z_G(T)^0$.  By working Zariski-locally around $s$,
we may assume that the smooth $S$-groups
$T \rightarrow S$ and $Z_G(T) \rightarrow S$
have constant fiber dimension.   These fiber dimensions agree
at $s$, so they agree on all fibers.  
For any $\xi \in S$, the closed subgroup $Z_{G_{\xi}}(T_{\xi}) = Z_G(T)_{\xi}$ in
$G_{\xi}$ is smooth and contains 
$T_{\xi}$.  But the dimensions agree, 
so $T_{\xi} = (Z_G(T)_{\xi})^0 = (Z_G(T)^0)_{\xi}$ for all $\xi \in S$. 
We conclude that the $S$-map
$T \hookrightarrow Z_G(T)^0$ between smooth $S$-schemes is an isomorphism on fibers over $S$,
so it is an isomorphism (Lemma \ref{flatclosed}). 

Although $G^0$ might not be $S$-affine, for any torus
$T' \subset G$ it is clear that $G^0 \cap Z_G(T')$ represents
$\underline{Z}_{G^0}(T')$, so we denote it as $Z_{G^0}(T')$.
In the classical theory it is shown that the centralizer of {\em any} torus in a smooth connected
affine group over an algebraically closed field is  connected, so 
$Z_{G^0}(T) = Z_G(T)^0 = T$.
Also, for any geometric point
$\overline{\xi}$ over $\xi \in S$ we have $T_{\overline{\xi}} = Z_{G^0}(T)_{\overline{\xi}} = 
Z_{G^0_{\overline{\xi}}}(T_{\overline{\xi}})$, so $T_{\overline{\xi}}$ must be maximal as a torus
in $G_{\overline{\xi}}^0$.  This proves (2).

We next turn to (1), and we may assume $G_s^0 \ne 1$. 
Fix an algebraic geometric point $\overline{s} = \Spec(\overline{k(s)})$ over $s$.
Any maximal torus in $G_{\overline{s}}^0$ descends to a split
torus in $G_K^0$ for some finite extension $K/k(s)$ contained in $\overline{k(s)}$, and it 
is harmless to work fppf-locally around $s$.  Thus, we can 
pass to an fppf neighborhood of $(S,s)$ to increase $k(s)$ to coincide with
such a $K$.  Now $G_s^0$ contains a split torus $T_s$ such that $T_{\overline{s}}$
is maximal in $G_{\overline{s}}^0$.  By Proposition \ref{lifttorus},
we may make a further \'etale base change on $S$ around $s$
to get to the case that $T_s$ lifts to a torus $T$ in $G$ (hence in $G^0$), and that $T$ is even split.  By (2),
after some further Zariski-localization around $s$ we may assume
$Z_{G^0}(T) = T$ and that 
$$T \simeq D_S(M) := \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(M_S,\Gm)$$ for a finite free
$\Z$-module $M$.  

The $T$-action on the vector bundle $\mathfrak{g} = {\rm{Lie}}(G) = {\rm{Lie}}(G^0)$
over $S$ decomposes
it into a direct sum of quasi-coherent weight spaces indexed by elements of $M$, 
and the formation of these weight spaces commutes with base change on $S$.
These weight spaces are vector bundles (being direct summands of $\mathfrak{g}$),
so the weight space decomposition on the $s$-fiber encodes
the weight spaces on the nearby fibers (by Nakayama's Lemma at $s$).
Thus, by shrinking around $s$ we can arrange that all
weight spaces have constant rank, so the only characters $m \in M$
for which the weight space $\mathfrak{g}_m$ is nonzero
are $m = 0$ and the elements of the root system $\Phi = \Phi(G^0_s,T_s)$ for
the connected reductive geometric fiber $G^0_{\overline{s}} \ne 1$.

For each $\xi \in S$, the weight space $\mathfrak{g}_0$ for the trivial weight
has $\xi$-fiber 
$${\rm{Lie}}(Z_{G^0_{\xi}}(T_{\xi})) = {\rm{Lie}}(Z_{G^0}(T)_{\xi}) = {\rm{Lie}}(T_{\xi}) = {\rm{Lie}}(T)_{\xi},$$
but  the subbundle $\mathfrak{t} := {\rm{Lie}}(T)$ in $\mathfrak{g}$ is clearly contained in
$\mathfrak{g}_0$, so the inclusion $\mathfrak{t} \subset \mathfrak{g}_0$
between subbundles 
must be an equality over $S$ for rank reasons.  In other words, the weight space decomposition over $S$ is
$$\mathfrak{g} = \mathfrak{t} \bigoplus (\bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha})$$
where each $\mathfrak{g}_{\alpha}$ is a line bundle on $S$. 

For each $\alpha \in \Phi$, let $T_{\alpha} \subset T = D_S(M)$ be the unique
relative codimension-1 subtorus contained in $\ker \alpha$;
explicitly, $T_{\alpha} = D_S(M/L)$ where $L \subset M$ is the saturation of
$\Z \alpha$ in $M$. 
Since $\Phi$ is a reduced root system (by the classical theory, applied to 
$(G^0_{\overline{s}}, T_{\overline{s}})$),
elements of $\Phi$ apart from $\pm \alpha$ are linearly independent from $\alpha$ and so 
cannot vanish on any fiber of $T_{\alpha}$.   By the following lemma, 
which is a variant of Exercise \ref{redexer}(ii), 
a geometric fiber $G^0_{\overline{\xi}}$ is reductive provided
that each fiber $Z_{G^0}(T_{\alpha})_{\overline{\xi}} = 
Z_{G^0_{\overline{\xi}}}((T_{\overline{\xi}})_{\alpha})$ is reductive. 

\begin{lemma}\label{redcrit} Let $G$ be a $($not necessarily reductive$)$ smooth connected affine group
over an algebraically closed field $k$, and $T$ a maximal torus in $G$
such that $Z_G(T) = T$.
For each nonzero $T$-weight $\alpha$ on $\mathfrak{g}$, 
let $T_{\alpha}$ be the codimension-$1$ subtorus $(\ker \alpha)_{\rm{red}}^0$.

The group $G$ is reductive if and only if the smooth connected subgroup 
$Z_G(T_{\alpha})$ is reductive for each $\alpha$. 
\end{lemma}

\begin{proof}
The implication ``$\Rightarrow$'' is part of the classical theory of reductive groups
(cf.\:Theorem \ref{torithm}(3)). 
For the converse, let $U = \mathscr{R}_u(G)$, so $U$ is connected by definition. 
Thus, $\mathfrak{u} := {\rm{Lie}}(U)$ is a $T$-stable subspace
of $\mathfrak{g}$, so it has a weight space decomposition.
Each intersection $Z_G(T_{\alpha}) \bigcap U$ is smooth and connected
by Example \ref{cent}, yet is also visibly unipotent
and normal in $Z_G(T_{\alpha})$ since $U$ is unipotent and normal in $G$.
The reductivity of $Z_G(T_{\alpha})$ then forces
$Z_G(T_{\alpha}) \bigcap U = 1$.  The formation of Lie
algebras of closed subgroup schemes is compatible with the formation of
intersections (as one sees by consideration of dual numbers), so
we conclude that $\mathfrak{u}$ has trivial intersection with
${\rm{Lie}}(Z_G(T_{\alpha}))$.  But
the functorial definition of $Z_G(T_{\alpha})$ implies
that ${\rm{Lie}}(Z_G(T_{\alpha})) = \mathfrak{g}^{T_{\alpha}}$ (see Proposition \ref{zrep}), 
so this contains the entire weight space for $\alpha$ on $\mathfrak{g}$.
In particular, $\mathfrak{u}$ has vanishing intersection with each such weight space. 

We conclude that $\mathfrak{u}$ supports no nontrivial $T$-weights,
so $\mathfrak{u} \subset \mathfrak{g}^T = {\rm{Lie}}(Z_G(T))$.
Thus,
$${\rm{Lie}}(U \bigcap Z_G(T)) = \mathfrak{u} \bigcap {\rm{Lie}}(Z_G(T)) = \mathfrak{u}.$$
But $U \bigcap Z_G(T)$ is smooth and connected (by Example \ref{cent}),
yet we have just seen that the containment $U \bigcap Z_G(T) \subset U$ induces
an equality on Lie algebras, so it must be an equality.  In other words,
necessarily $U \subset Z_G(T)$.  We assumed $Z_G(T) = T$, so
the smooth connected unipotent $U$ must be trivial.
\end{proof}

Returning to the relative setting,
by smoothness of $Z_{G^0}(T_{\alpha})$ the inclusion
${\rm{Lie}}(Z_{G^0}(T_{\alpha})) \subset \mathfrak{g}^{T_{\alpha}} = \mathfrak{t} \bigoplus \mathfrak{g}_{\alpha} \bigoplus
\mathfrak{g}_{-\alpha}$ as subbundles of $\mathfrak{g}$
is an equality since this holds on geometric fibers.   To verify that $G^0$ has reductive fibers at all
points of $S$ near $s$, it suffices to treat each $Z_G(T_{\alpha})$ separately
in place of $G$ (by Lemma \ref{redcrit}), so we have reduced to the case 
$\Phi = \{\alpha, -\alpha\}$.

Clearly $G^0 \cap N_G(T)$ represents $\underline{N}_{G^0}(T)$, so
we denote this as $N_{G^0}(T)$.  Likewise, $W_{G^0}(T) := N_{G^0}(T)/T$
is an open and finitely presented
$S$-subgroup of the $S$-group $W_G(T) = N_G(T)/T$ that is separated, \'etale, and quasi-finite
(Theorem \ref{xi5.3bis}), so $W_{G^0}(T)$ inherits these properties.
The fiber $W_{G^0}(T)_{\overline{s}}$ has order 2 and nontrivial element that acts by inversion on
$(T/T_{\alpha})_{\overline{s}}$ (and so swaps $\alpha$ and $-\alpha$).
\'Etaleness of $W_{G^0}(T)$ implies 
that by passing to an \'etale neighborhood of $(S,s)$ we can arrange
that $W_{G^0}(T) \rightarrow S$ admits a section $w$ that is the nontrivial point in the $s$-fiber.
Thus, the $w$-action on the rank-1 torus $T/T_{\alpha}$ is inversion over a neighborhood
of $s$ since it is inversion on the $s$-fiber.   

By localizing
more around $s$ to lift $w$ to a section of $N_{G^0}(T)$, we obtain  that for {\em all} geometric points $\overline{\xi}$ of $S$
the fiber $N_{G^0}(T)_{\overline{\xi}}$ contains an element that does
not centralize $T_{\overline{\xi}}$.  But in any smooth connected {\em solvable}
group over an algebraically closed field, the normalizer of a maximal torus
is equal to its centralizer (as is immediately verified by considering the description of 
any such group as a semi-direct product of a maximal torus against the unipotent radical).
It follows that all fibers $G^0_{\overline{\xi}}$ are {\em non-solvable}. 

We have arranged that each fiber $G^0_{\overline{\xi}}$ contains
the central torus $(T_{\alpha})_{\overline{\xi}}$ with codimension 3
and is not solvable.  Thus, the quotient $(G/T_{\alpha})^0_{\overline{\xi}}$
is a  non-solvable 3-dimensional smooth connected affine group.  
This leaves no room for a nontrivial unipotent radical (as the quotient by such a radical
would be a smooth connected group of dimension at most 2, forcing solvability).
Hence, every $G^0_{\overline{\xi}}$ is reductive.  

It remains to check that when $G^0 \rightarrow S$ has
reductive geometric fibers, semisimplicity of a geometric fiber 
$G^0_{\overline{s}}$ implies semisimplicity of geometric fibers at
points near $s$.  In the presence of reductivity, semisimplicity of a connected 
linear algebraic group $H$ over an algebraically closed field can be read off
from the root system $\Psi$:  it is equivalent to
the condition that the $\dim H = \#\Psi + {\rm{rank}}(\Psi)$.
The preceding arguments (using a weight space decomposition of
$\mathfrak{g}$ fppf-locally near $s$) show that these invariants are inherited
by geometric fibers at points near $s$, so we are done. 
\end{proof}

In later arguments involving reduction to the noetherian case, we need to ensure
that reductivity (and semisimplicity) hypotheses can be descended:

\begin{corollary}\label{descend}  Let
$\{A_i\}$ be a directed system of rings with direct limit $A$,
and $G_{i_0}$ a smooth affine $A_{i_0}$-group for some $i_0$.
For all $i \ge i_0$ define $G_i = G_{i_0} \otimes_{A_{i_0}} A_i$
and $G = G_{i_0} \otimes_{A_{i_0}} A$.  The fibers of
$G^0 \rightarrow \Spec A$ are reductive if and only if the fibers of $G_i^0 \rightarrow
\Spec A_i$ are reductive for all sufficiently large $i \ge i_0$, and 
$G$ is a reductive $A$-group if and only if $G_i$ is a reductive $A_i$-group for 
all sufficiently large $i \ge i_0$.  The same holds for semisimplicity.
\end{corollary}

\begin{proof}
For $i \ge i_0$, let $U_i \subset \Spec A_i$ be the locus of points at
which the geometric fiber of $G_i^0$ is reductive (resp.\,semisimple),
and define $U \subset \Spec A$ similarly for $G^0$.  By Proposition \ref{cartan}(1),
these loci are open subsets.  Our first problem is to prove that 
$U = \Spec A$ if and only if $U_i = \Spec A_i$ for all sufficiently large $i$.  

Under the transition maps $\Spec A_{i'} \rightarrow \Spec A_i$
(resp.\,the maps $\Spec A \rightarrow \Spec A_i$), the preimage of $U_i$
is $U_{i'}$ (resp.\,$U$), 
so the same holds for the respective closed complements $Z_i$
of $U_i$ in $\Spec A_i$ and $Z$ of $U$ in $\Spec A$.  We wish to show
that $Z$ is empty if and only if $Z_i$ is empty for all sufficiently large $i$.  Letting
$J_i \subset A_i$ and $J \subset A$ be the respective radical ideals
of these closed sets, we have $A/J = \varinjlim A_i/J_i$.  Hence,
$A/J = 0$ if and only if $A_i/J_i = 0$ for all sufficiently large $i$
(by considering the equation $1 = 0$).

It remains to show that if $G^0 = G$ then $G_i^0 = G$ for
all sufficiently large $i$.  By \cite[IV$_3$, 9.7.7(ii)]{ega},
the subset $Y_i \subset \Spec A_i$ of points
at which $G_i$ has a geometrically connected fiber
is constructible, and if $i' \ge i$ then the preimage of $Y_i$ in $\Spec A_{i'}$
is $Y_{i'}$.  The common preimage of all $Y_i$ in $\Spec A$ is 
the entire space, so by \cite[IV$_3$, 8.3.4]{ega} we have $Y_i = \Spec A_i$ for 
large $i$.
\end{proof}


In the proof of Proposition \ref{cartan} we showed that when
each $G^0_{\overline{s}}$ is reductive and there
is a split torus $T \subset G$ that is maximal
on geometric fibers then for each $s_0 \in S$
and varying $s \in S$ near $s_0$ the root systems $\Phi(G_{\overline{s}}^0,T_{\overline{s}})$
may be identified with $\Phi(G_{\overline{s}_0}^0, T_{\overline{s}_0})$.
This has the following interesting consequence:

\begin{proposition}\label{connfiber}
Let $G \rightarrow S$ be a smooth $S$-affine group such that each $G_s^0$ is reductive.
The locus of $s \in S$ such that $G_s$ is connected is closed in $S$.
In particular, if $S$ is irreducible and the generic
fiber is connected then all fibers are connected.
\end{proposition}


Proposition \ref{connfiber}
follows immediately from the claim in Proposition \ref{disconn} that $G/G^0$ is a finitely presented
{\em separated} 
\'etale $S$-group (ensuring that the locus $\{s \in S\,|\,\#(G/G^0)_{\overline{s}} \ge n\}$ is open in $S$).
The proof of  such separatedness 
rests on Theorem \ref{adimm} (which is proved much later),
so we avoid Proposition \ref{disconn} in our proof of Proposition \ref{connfiber}.

\begin{proof} We may assume $S$ is affine, and then by Corollary 
\ref{descend} we can assume $S$ is noetherian.  
Since $G_s$ is connected if and only if it is geometrically
connected over $k(s)$, the set $Y$ of points $s \in S$
such that $G_s$ is connected is constructible
\cite[IV$_3$, 9.7.7(ii)]{ega}.  For any map $S' \rightarrow S$,
the preimage of $Y$ in $S'$ is the locus of connected
fibers for $G_{S'} \rightarrow S'$, so by the specialization criterion
for closedness of a constructible set 
we may assume $S = \Spec R$ for a discrete valuation ring
$R$ and that the generic fiber is {\em connected}. 
We seek to prove that the special fiber is connected.

Without loss of generality, $R$ is complete with an algebraically
closed residue field $k$, so 
$G_k^0$ contains a maximal torus $T_0$ and
by Proposition \ref{lifttorus} there is
an $R$-torus $T \subset G$ lifting $T_0 \subset G_0$.
The completeness of $R$ ensures that $T$ is split.
Clearly $T \subset Z_G(T)$, so $T$ is closed
in $Z_G(T)$.  Consider the special fibers $T_0$
and $Z_G(T)_k = Z_{G_k}(T_0)$.  The reductive identity component 
$G_k^0$ meets $Z_{G_k}(T_0)$ in $Z_{G_k^0}(T_0) = T_0$
(equality due to the maximality of $T_0$).  

Let $K$ be the fraction field of $R$. 
The closed subgroups $T$ and $Z_G(T)$ in $G$
are both $R$-smooth, and we have shown that their special fibers
have the same identity component, so their relative dimensions agree.
Hence, on $K$-fibers the inclusion $T_K \subset Z_G(T)_K = Z_{G_K}(T_K)$
must be an equality, as $Z_{G_K}(T_K)$ is smooth and connected (since $G_K$ is connected).
In other words, the complement 
$Z_G(T) - T$ is the union of the non-identity components of the special fiber of $Z_G(T)$, so
it is closed in $Z_G(T)$.  Hence, $T$ is also open in $Z_G(T)$. 
But $Z_G(T)$ is $R$-flat with irreducible generic fiber, so the total space of $Z_G(T)$ is
connected (even irreducible).  Thus, the open and closed subset $T$ in $Z_G(T)$ 
must be the entire space, so
the closed immersion $T \hookrightarrow Z_G(T)$ must be an isomorphism due to $R$-smoothness.

By the observation immediately preceding the present proposition, 
the split reductive pairs $(G_K, T_K)$ and $(G^0_k, T_k)$
have isomorphic root systems.  
Let $\Phi$ denote the common isomorphism class of 
$\Phi(G_K,T_K)$ and $\Phi(G_k^0, T_k)$. 
The normalizer $N_G(T)$ is a smooth closed subgroup of $G$,
so the quotient $$W_G(T) := N_G(T)/Z_G(T) = N_G(T)/T$$ is a separated \'etale $R$-group
of finite type (Theorem \ref{xi5.3bis}).
The generic fiber $W_G(T)_K = W_{G_K}(T_K)$ is the finite
constant group over $K$ associated to the ordinary finite group $W(\Phi)$.
Likewise, $W_G(T)_k = W_{G_k}(T_k)$ is finite \'etale over $k$ and contains
as an open subgroup $W_{G^0_k}(T_k) \simeq W(\Phi)$.
But the fiber degree for a separated \'etale map of finite type can only ``drop''
under specialization (by Zariski's Main Theorem \cite[IV$_3$, 8.12.6]{ega}), so we conclude that $W_G(T)_k = W_{G^0_k}(T_k)$.
In other words, $N_G(T) \subset G^0$.

For any $g \in G_k$, the conjugate
$gT_kg^{-1}$ is a maximal torus of $G^0_k$.  Hence,
there exists $g' \in G^0_k(k)$ such that $gT_k g^{-1} = g'T_k {g'}^{-1}$, 
so $g^{-1}g' \in N_G(T)_k \subset G^0_k$.  This forces $g \in G^0_k$.
\end{proof}

See \cite[Thm.\,1.2]{py} for a generalization of Proposition \ref{connfiber}, and note
that the locus of connected fibers can fail to be open
(Example \ref{best}). 

\subsection{Maximal tori}

There are several ways to define 
the notion of ``maximal torus'' in a smooth affine group scheme. Different approaches
do lead to the same concept for reductive groups, at least Zariski-locally
on the base.  For our purposes, 
the following definition (taken from \cite[XII, 1.3]{sga3}) is suitable for developing the general theory. 

\begin{definition}\label{maxtori} A {\em maximal torus} in a smooth $S$-affine group scheme
$G \rightarrow S$ is a torus $T \subset G$ such that for each
geometric point $\overline{s}$ of $S$ the fiber $T_{\overline{s}}$ is not contained
in any strictly larger torus in $G_{\overline{s}}$. 
\end{definition}
\index{maximal tori}
\index{reductive group!maximal tori}

There are two sources of ambiguity in this definition.  One is rather minor, namely
to check that for any $s \in S$ it suffices to consider a single geometric point over
$s$.  This is a special case of the general principle that properties of finitely presented
structures in algebraic geometry are insensitive to
scalar extension from one algebraically closed ground field to a bigger such field;
this principle pervades the classical approach to linear algebraic groups
(``independence of the universal domain''). In the case of interest, it comes out as follows.

\begin{proposition}\label{maxrem}
If $K/k$ is an extension of algebraically
closed fields and $T \subset G$ is a torus in a smooth affine
$k$-group $G$ then $T$ is not contained in a strictly
larger torus of $G$ if and only if $T_K$ is not contained in a strictly larger torus of $G_K$. 
\end{proposition}

An immediate consequence of Proposition \ref{maxrem} is that if $T \subset G$ is a torus in a smooth
relatively affine group scheme over a scheme $S$ and if $S' \rightarrow S$ is surjective
then $T$ satisfies Definition \ref{maxtori} in $G$ over $S$ if and only if $T_{S'}$ does in $G_{S'}$ over $S'$.

\begin{proof} The implication ``$\Leftarrow$'' is obvious.  To verify the converse, 
we shall use the general technique of ``spreading out and specialization''. 
Express $K$ as the direct limit of its $k$-subalgebras $A_i$ of finite type over $k$,
so by limit considerations, any strict containment $T_K \subsetneq T'$  between $K$-tori in $G_K$
descends to a strict containment $T_{A_i} \subsetneq T'_i$ between $A_i$-tori in
$G_{A_i}$ for some $i$.  (This is the ``spreading out'' step, as
$\Spec(A_i)$ is a finite type $k$-scheme, unlike $\Spec(K)$ in general.
Note that an argument is required to verify that $T'_i$ is a torus for large $i$;
e.g., it follows from the limit argument used at the start of the proof of
Proposition \ref{xi2.4bis}.)  

Viewing $G_{A_i} = G \times \Spec(A_i)$ as a reduced $k$-scheme of finite type in which
$T_{A_i}$ and $T'_i$ are reduced closed subschemes (due to smoothness over the reduced
$\Spec(A_i)$), since $k = \overline{k}$ and
$T_{A_i}$ is a proper closed subset of $T'_i$ it
follows from the Nullstellensatz that some $k$-point of $T'_i$ is not contained in $T_{A_i}$.
This point lies over some $\xi \in \Spec(A_i)(k)$, so we specialize there:  passing to $\xi$-fibers
yields a strict containment $T \subsetneq (T'_i)_{\xi}$ between $k$-tori in $G$, a contradiction.
\end{proof}

A more serious ambiguity in the terminology of Definition \ref{maxtori}
is to determine its relation with maximality relative to inclusions among tori in $G$ over $S$. 
Any $T$ that is maximal in the sense of Definition \ref{maxtori}
is maximal among tori of $G$; i.e., any containment
$T \subset T'$ among tori of $G$ must be an equality.
Indeed, the equality on fibers is clear, so by Lemma \ref{flatclosed} 
equality holds in $G$ (and likewise
$T_U$ is maximal in $G_U$ for all open subschemes $U$ of $S$).

Consider  the converse:  if a torus $T \subset G$ is not contained
in a strictly larger torus over $S$ then is $T$ maximal in
the sense of Definition \ref{maxtori} (i.e., is $T_{\overline{s}}$ maximal
in $G_{\overline{s}}$ for all geometric points $\overline{s}$ of $S$)?
There are Zariski-local obstructions:  $T_U$ may lie in a strictly larger $G_U$-torus
for some open $U \subset S$.  
This is due to the fact that the dimension of the maximal tori in each
geometric fiber $G_{\overline{s}}$ may not be locally constant
in $s$.  For instance, 
in the Example at the end of \cite[XVI, \S3]{sga3} there is given
an explicit smooth affine group over a discrete
valuation ring such that the generic fiber is $\mathbf{G}_{\rm{m}}$
and the special fiber is $\Add$ (so $T = 1$ is maximal relative
to containment over the entire base, but not over the open generic point).  Here is a more natural
example.

\begin{example}\label{rankjump} Let 
$S = \Spec(V)$ for a complete discrete valuation ring $V$
with $K := {\rm{Frac}}(V)$, 
and consider a separable quadratic extension $K'/K$ that is
not unramified. Let $V'$ be the valuation ring of $K'$.
The Weil restriction $G = {\rm{R}}_{V'/V}(\mathbf{G}_{\rm{m}})$
\index{Weil restriction}
(which represents the functor 
$B \rightsquigarrow \mathbf{G}_{\rm{m}}(B \otimes_V V') = (B \otimes_V V')^{\times}$
on $V$-algebras) is a smooth affine $V$-group of relative dimension 2;
it is even an open subscheme of the $V$-scheme ${\rm{R}}_{V'/V}(\mathbf{A}^1_{V'}) = \mathbf{A}^2_V$
(see \cite[Prop.\,A.5.2]{pred}). In particular, the fibers of $G$ are 
connected (as is also a consequence of general connectedness
results for Weil restriction of {\em smooth} schemes \cite[Prop.\,A.5.9]{pred}). 
The fibral connectedness for $G$ 
 can also be seen by inspection: $G_K = {\rm{R}}_{K'/K}(\mathbf{G}_{\rm{m}})$ 
is a torus since $K'/K$ is separable, and since $K'/K$ is not unramified the geometric special fiber is 
${\rm{R}}_{\overline{k}[\epsilon]/\overline{k}}(\mathbf{G}_{\rm{m}}) =
\mathbf{G}_{\rm{m}} \times \Add$ (via $(t,x) \mapsto t(1 + x\epsilon)$).

Thus, the evident $\mathbf{G}_{\rm{m}}$ as a $V$-subgroup of $G$ is
a torus not contained in any strictly larger torus of $G$ (due to consideration of
the special fiber),
but its geometric generic fiber is not maximal in that of $G$.
\end{example}

For our purposes, ``maximal torus'' will always be taken in the sense
of Definition \ref{maxtori}.  See Remark \ref{remtorus}
for the equivalence with another possible definition (using actual fibers rather than geometric fibers).
  The case of most interest to us
is maximal tori in reductive group schemes.  In such groups,
the maximality property is robust with respect to the Zariski topology:

\begin{example} Proposition \ref{cartan}(2) implies that if
$T$ is a torus in a reductive group scheme $G \rightarrow S$ and it is maximal
on the $\overline{s}$-fiber for some $s \in S$
then it is maximal in the sense of Definition \ref{maxtori}
over a Zariski-open neighborhood of $s$. 
\end{example}

To study maximal tori via ``reduction to the noetherian case'', we require:

\begin{lemma}\label{limitmax} Let $\{A_i\}$ be a directed system of rings with
direct limit $A$, and let $G$ be a smooth affine $A$-group 
equipped with an $A$-torus $T \subset G$.
Pick $i_0$ so that $G$ descends to
a smooth affine $A_{i_0}$-group $G_{i_0}$ 
and $T$ descends to an $A_{i_0}$-torus $T_{i_0}$ in $G_{i_0}$. 
For $i \ge i_0$, define the pair $(G_i, T_i)$ over $A_i$
by scalar extension of the pair $(G_{i_0}, T_{i_0})$ over $A_{i_0}$.

If $T$ is maximal in $G$ then $T_i$ is maximal in $G_i$ for sufficiently large $i$.
\end{lemma}

The proof of this lemma uses  standard direct limit
arguments as well as constructions
that are specific to group schemes.  

\begin{proof}
We will use  centralizers for tori (as in Lemma \ref{centralizer})
and elementary affineness results for quotients by central tori (see \cite[VIII, 5.1; IX, 2.3]{sga3}).
It is harmless to replace $G_{i_0}$ with the finitely presented affine 
 centralizer $Z_{G_{i_0}}(T_{i_0})$
(and replace $G_i$ with $Z_{G_{i_0}}(T_{i_0})_{A_i} = Z_{G_i}(T_i)$
for all $i \ge i_0$, and replace $G$ with their common base change $Z_G(T)$ over $A$), so we may assume
that $T_{i_0}$ is central in $G_{i_0}$ over $A_{i_0}$
(and similarly over $A_i$ for $i \ge i_0$, as well as over $A$).
Then by passing to the affine quotient $G_{i_0}/T_{i_0}$ over $A_{i_0}$,
and similarly over every $A_i$ and over $A$, 
we reduce to the case that all $T_i$ and $T$ are trivial.  

Letting $S = \Spec(A)$ and $S_i = \Spec(A_i)$ for all $i$, we are brought to the case
that the smooth affine geometric fibers $G_{\overline{s}}$ all have no nontrivial tori
(equivalently, $G_{\overline{s}}^0$ is unipotent for all $s \in S$).  We
seek to prove the same property holds for all geometric fibers of
$G_i \rightarrow S_i$ when $i$ is sufficiently large. We may and do assume that $S$
is non-empty, and hence likewise all $S_i$ are non-empty.
Since $G_{i_0} \rightarrow S_{i_0}$ is finitely presented,
the number of connected components
for its geometric fibers is bounded \cite[IV$_3$, 9.7.9]{ega}.
The same bound is valid for 
the geometric fibers of every $G_i \rightarrow S_i$
for $i \ge i_0$, as well as for the geometric fibers of $G \rightarrow S$.
Thus, if we choose a prime number $p$ larger than such a bound
then the unipotence of all $G_{\overline{s}}^0$ implies
that for all $s \in S$ there are no nontrivial group homomorphisms $\mu_p \rightarrow G_{\overline{s}}$.
In other words, for the affine and finitely presented
$S$-scheme $Y := \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(\mu_p,G)$ and 
the canonical section $\sigma:S \rightarrow Y$ over $S$ corresponding to 
the {\em trivial} $S$-map $\mu_p \rightarrow G$, the map $\sigma$ is surjective.

Let $Y_i = \underline{\Hom}_{S_i\mbox{-}{\rm{gp}}}(\mu_p,G_i)$, so
$Y_i = Y_{i_0} \otimes_{A_{i_0}} A_i$ compatibly with change in $i \ge i_0$
and the limit of these affine schemes is $Y$.  Since the canonical sections 
$\sigma_i:S_i \rightarrow Y_i$ defined by the trivial $S_i$-maps $\mu_p \rightarrow G_i$
yield the surjective map $\sigma$ in the limit, for sufficiently large $i \ge i_0$
the map $\sigma_i$ is surjective \cite[IV$_3$, 8.10.5(vi)]{ega}.
Fix such a large $i$, so the geometric fibers of the smooth $S_i$-group 
$G_i \rightarrow S_i$ do not contain $\mu_p$ as a subgroup scheme over $S_i$
(since $S_i$ is non-empty). 
It follows that the identity components of these fibers cannot contain a nontrivial torus.
Hence, these identity components are unipotent, as desired.
\end{proof}

The following fundamental result concerning the ``scheme of maximal tori'' is the engine that makes 
the relative theory of reductive groups work (and it
is proved in \cite[XII, 1.10]{sga3} under weaker hypotheses than we impose):

\begin{theorem}\label{schemetori}
Let $G \rightarrow S$ be a smooth $S$-affine group scheme
such that in the identity component $G^0_{\overline{s}}$
of each geometric fiber the maximal tori 
are their own centralizers.  Then the functor on $S$-schemes
$$\underline{\rm{Tor}}_{G/S}:S' \rightsquigarrow \{\mbox{maximal tori in } G_{S'}\}$$
is represented by a smooth quasi-affine $S$-scheme ${\rm{Tor}}_{G/S}$, 
and ${\rm{Tor}}_{G/S} \rightarrow S$ is surjective.

If $T$ is a maximal torus of $G$ then the map $G/N_G(T) \rightarrow {\rm{Tor}}_{G/S}$
defined by $G$-conjugation against $T$ is an isomorphism.  In particular, any two maximal tori of $G$
are conjugate \'etale-locally on $S$.
\end{theorem}
\index{ aaatorgs@$\underline{\rm{Tor}}_{G/S}$, ${\rm{Tor}}_{G/S}$}
\index{reductive group!maximal tori}
\index{conjugacy!maximal tori}
\index{maximal tori!scheme of}

The main case of interest for which we can verify the hypothesis ``Cartan subgroups of 
(identity components of) geometric
fibers are tori'' is the case of reductive group schemes, but another interesting
case that will arise later is parabolic subgroups of reductive group schemes
(to be defined and studied in \S\ref{parsec}). 
In \cite[XII, 5.4]{sga3} it is shown that  ${\rm{Tor}}_{G/S}$ is actually $S$-affine
in Theorem \ref{schemetori}.
The proof of this finer property 
uses a hard representability theorem \cite[XI, 4.1]{sga3} that we are avoiding.
We do not need such affineness, so we say nothing further about it.
See Exercise \ref{maxfield} for a generalization of Theorem \ref{schemetori} over fields.

\begin{proof}
By effectivity of fppf descent for schemes that are quasi-affine over the base, it suffices
to work fppf-locally on $S$ (since the functor in question is clearly an fppf sheaf of sets).
Choose $s \in S$.  As in the proof of Proposition \ref{cartan}(1), we may work fppf-locally
around $s$ to arrange
that there exists a torus $T \subset G$ such that $T_{\overline{s}}$ is maximal in $G^0_{\overline{s}}$.
The inclusion $T \subset Z_G(T)^0$ between smooth $S$-groups is an isomorphism
on geometric fibers at $\overline{s}$, so by connectedness and fiber dimension
considerations it follows that the same holds on geometric fibers over points
near $s$.  
Hence, by passing to a Zariski-open neighborhood
of $s$ in $S$ we may arrange that $T = Z_G(T)^0$, so $T$ is maximal in $G$ (in the sense
of Definition \ref{maxtori}). Now we 
will just use that $G \rightarrow S$ is a smooth $S$-affine group. 

By Theorem \ref{xi5.3bis},
the quotient $G/N_G(T)$ exists as a smooth quasi-affine $S$-scheme,
and clearly $G/N_G(T) \rightarrow S$ is surjective.  There is an evident
$G$-action on $\underline{\rm{Tor}}_{G/S}$ via conjugation, and the point 
$T \in \underline{\rm{Tor}}_{G/S}(S)$
has stabilizer subfunctor in $G$ represented by 
the transporter scheme ${\rm{Transp}}_G(T,T) = N_G(T)$.
Thus, $G/N_G(T)$ is a subfunctor of $\underline{\rm{Tor}}_{G/S}$, and we claim that it equals
 $\underline{\rm{Tor}}_{G/S}$. 
It suffices to prove that the map $G \rightarrow \underline{\rm{Tor}}_{G/S}$ between
 fppf sheaves of sets is surjective,
which is to say that for any $S$-scheme $S'$ and $T' \in \underline{\rm{Tor}}_{G/S}(S')$
there exists an fppf covering $S'' \rightarrow S'$ and a $g \in G(S'')$ such
that $g T'_{S''} g^{-1} = T_{S''}$. Put another way, we are claiming
that the maximal tori $T_{S'}$ and $T'$ in $G_{S'}$ are conjugate fppf-locally on $S'$.
Upon renaming $S'$ as $S$, our task is to prove that 
any two maximal tori $T$ and $T'$ in $G$ are conjugate fppf-locally on $S$.

Direct limit arguments (along with Lemma \ref{limitmax})
allow us to arrange that $S$ is noetherian and affine, and even
finite type over $\Z$ (which will be relevant when we apply Artin approximation later in the argument). 
Pick $s \in S$, so $T_{\overline{s}}$ and $T'_{\overline{s}}$ are $G(\overline{s})$-conjugate
for an algebraic geometric point $\overline{s}$ over $s$.  Express 
$k(\overline{s})$ as a direct limit of subextensions of finite degree over $k(s)$,
so there exists a finite extension $K/k(s)$ such that $T_K$ and $T'_K$ are $G(K)$-conjugate
and split.
Passing to an fppf neighborhood of $(S,s)$ then brings us to the case that
$T_s$ and $T'_s$ are $G(s)$-conjugate and split.  Let $A$ denote the completion $\mathscr{O}_{S,s}^{\wedge}$.

By the same style of Artin approximation argument as used in the proof of Proposition
\ref{lifttorus} (this is where we need that
$S$ is finite type over $\Z$), if $T_A$ is $G(A)$-conjugate to $T'_A$ then we
can build an \'etale neighborhood $U$ of $(S,s)$ such that $T_U$ is $G(U)$-conjugate
to $T'_U$, so we would be done.   Thus, we may and do
replace $S$ with $\Spec(A)$ to arrange that (i) $S$ is the spectrum of a complete
local noetherian ring $(A,\mathfrak{m})$ with residue field denoted $k$,  and 
(ii) the special fibers $T_0$ and $T'_0$
are $G(k)$-conjugate and split.  By lifting
a conjugating element from $G(k)$ into $G(A)$
(as we may do since the local noetherian $A$ is complete and $G$ is smooth and affine),
we can arrange that $T_0 = T'_0$ inside the special fiber $G_k$.  

It follows from the completeness of $A$ that
$T$ and $T'$ are split, hence abstractly isomorphic.
The splitting isomorphisms $T \simeq \mathbf{G}_{\rm{m}}^r$
and $T' \simeq \mathbf{G}_{\rm{m}}^r$ can be chosen to coincide on the special
fibers (i.e., they induced the same splitting of the torus $T_0 = T'_0$  in $G_0$), 
so the inclusions of $T$ and $T'$ into $G$ may be viewed
as a pair of inclusions from a common $A$-torus $\mathbf{G}_{\rm{m}}^r$ into
$G$ that  agree residually.  Thus, by Corollary \ref{biglift}, these inclusions
are $G(A)$-conjugate.
\end{proof}

In earlier arguments we have built maximal tori in reductive group schemes by working fppf-locally
on the base.  Now we show that it always suffices to work \'etale-locally:

\begin{corollary}\label{buildtori} Any reductive group scheme $G \rightarrow S$ admits
a maximal torus \'etale-locally on $S$.  In particular, any connected
reductive group over a field $k$ admits a geometrically maximal torus 
defined over a finite separable extension of $k$.
\end{corollary}
\index{reductive group!existence of maximal tori}

\begin{proof}  The structural morphism 
${\rm{Tor}}_{G/S} \rightarrow S$ from the scheme of maximal tori
is a smooth surjection, so it admits sections \'etale-locally on $S$.  But to give
a section over an $S$-scheme $U$ is to give an element of the set
${\rm{Tor}}_{G/S}(U)$, which is to say a maximal torus of $G_U$. 
\end{proof}

The following result is a special case of part of  \cite[XII, 5.4]{sga3}.

\begin{proposition}\label{finitew} Let $T$ be a maximal torus in a reductive group scheme 
$G \rightarrow S$.
The Weyl group $W_G(T) := N_G(T)/Z_G(T)$ is finite \'etale over $S$.
\end{proposition}
\index{Weyl group!finite \'etale}
\index{reductive group!Weyl group}

Recall from Theorem \ref{xi5.3bis} that a priori (without using reductivity) 
the quotient sheaf $W_G(T)$ is represented by an $S$-scheme that is 
separated, \'etale, and finitely presented. 

\begin{proof}
By Proposition \ref{cartan} (and Corollary \ref{descend}) we use direct limit
arguments to reduce to the case that $S$ is noetherian, so we can apply 
to $W_G(T) \rightarrow S$
a general finiteness criterion for quasi-compact separated \'etale maps: 
it suffices that the number of points in the geometric fibers is
Zariski-locally constant on the base.  This finiteness criterion follows
from \'etale localization on the base and the local structure theorem
for such morphisms over a henselian local base in \cite[IV$_4$, 18.5.11(c)]{ega}.
(See Exercise \ref{drexer} for a generalization.)
In other words, we claim that the order of the fibral Weyl group 
$W_G(T)_{\overline{s}} = W_{G_{\overline{s}}}(T_{\overline{s}})$
is locally constant in $s$. 

Since the Weyl group of a connected
reductive group over an algebraically closed
field coincides with the Weyl group of the associated root system,
it suffices to check that the isomorphism class of the root system 
$\Phi(G_{\overline{s}},
T_{\overline{s}})$ is locally constant in $s$.  By passing to 
an \'etale neighborhood of a chosen point of $S$ we can arrange that 
$$T = D_S(M) = \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(M_S, \mathbf{G}_{\rm{m}})$$
for a finite free $\Z$-module $M$ (so the abelian sheaf 
$\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(T,\mathbf{G}_{\rm{m}})$
is thereby identified with $M_S$).  In the proof of Proposition \ref{cartan}(1)
we saw that the vector bundle $\mathfrak{g}$ with its $T$-action
decomposes into a direct sum of  ``root subbundles'' and that this decomposition identifies the root
system $\Phi(G_{\overline{\xi}},T_{\overline{\xi}}) \subset {\rm{X}}(T_{\overline{\xi}})
= M = {\rm{X}}(T_{\overline{s}})$
with $\Phi(G_{\overline{s}}, T_{\overline{s}})$ for all
$\xi$ near $s$.
\end{proof}

\subsection{Scheme-theoretic and reductive center}

The final topic in this section is the functorial center
of a reductive group scheme.  This material is developed
in \cite[XII, \S4]{sga3}, and we will navigate our way towards
a single result for reductive group schemes (which in turn is
a special case of \cite[XII, 4.11]{sga3}). We begin with two lemmas.

\begin{lemma}\label{multsub} Let $G$ be a smooth $S$-affine $S$-group with connected
fibers, and $H$ a subgroup of multiplicative type.
\begin{enumerate}
\item If $H$ is normal in $G$ then it is central in $G$.
\item Assume $H$ is normal in $G$, and let $\overline{G} = G/H$ be the associated
smooth $S$-affine quotient {\rm{\cite[VIII, 5.1; IX, 2.3]{sga3}}}.  For every central multiplicative type subgroup
$\overline{H}'$ in $\overline{G}$, the preimage $H'$ of $\overline{H}'$ in $G$
is a central multiplicative type subgroup of $G$. 
\end{enumerate}
\end{lemma}

Part (2) of this lemma serves a role in our treatment akin to the role of \cite[XII, 4.7]{sga3}
in the  general development of \cite[XII]{sga3}.

\begin{proof}
The normality of $H$ in $G$ implies that of each $H[n]$ in $G$, 
and the weak schematic density of $\{H[n]\}$ in $H$ after any base change
(Example \ref{weakdenseex}) 
implies that $H$ is central in $G$ if all $H[n]$ are central in $G$.  Hence,
to prove (1) we replace $H$ with $H[n]$ for arbitrary $n > 0$ so
that $H$ is $S$-finite.  Then the automorphism functor
of $H$ is identified (up to inversion) with that of its finite \'etale Cartier dual, so
this functor is represented by a finite \'etale $S$-scheme (via effective descent
for {\em finite} \'etale schemes over the base, applied 
to the automorphism functor of the constant Cartier dual over an \'etale cover of $S$). 

The conjugation action of $G$ on $H$ is classified by an $S$-group map from
$G$ to the finite \'etale automorphism scheme $E$ of $H$.  The identity section
of $E \rightarrow S$ is an open and closed immersion (as for any section to a finite \'etale map),
so the kernel of the action map $G \rightarrow \underline{\rm{Aut}}_{S\mbox{-}{\rm{gp}}}(H) = E$ 
is an $S$-subgroup scheme of
$G$ that is both open and closed.  Thus, (1) is reduced to the obvious fact that 
for any map of schemes $X \rightarrow S$ with connected fibers,
the only open and closed subscheme of $X$ that maps onto $S$ is $X$. 

To prove (2), first note that $H'$ is clearly finitely presented and flat (and affine) over $S$,
and it is normal in $G$ since $\overline{H}'$ is central in $\overline{G}$. 
In particular, by (1) it suffices to prove that $H'$ is multiplicative type.
We shall check directly that $H'$ is central in $G$ and hence is commutative, so then
Corollary \ref{fpqcwin}(2) will ensure that $H'$ is of multiplicative type.   Since $H$ is central in $G$ and
the quotient $H'/H = \overline{H}'$ is central in $G/H = \overline{G}$, the $G$-action on
$H'$ via conjugation is classified by a homomorphism from $G$ to the Hom-functor
$$\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(\overline{H}',H)$$
via $g \mapsto (\overline{h}' \mapsto gh'g^{-1})$ where $h'$ is an fppf-local lift of
$\overline{h}'$ to $H'$, the choice of which does not matter.
 This map to the Hom-functor 
vanishes if and only if $H'$ is central in $G$, so by the weak schematic density of
$\{\overline{H}'[n]\}$ in $\overline{H}'$ after any base change it suffices to
replace $\overline{H}'$ with $\overline{H}'[n]$ (for an arbitrary $n > 0$)
and to replace $H'$ with the preimage $H'_n$ of $\overline{H}'[n]$ in $H'$.
But then we have the equality of functors
$$\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(\overline{H}',H) = 
\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(\overline{H}',H[n]),$$
and via Cartier duality the right side is identified with a Hom-functor between
finite \'etale $S$-groups.  In particular, this is represented
by a finite \'etale $S$-group, so as in the proof of (1)
any $S$-homomorphism from $G$ to this Hom-scheme
vanishes.  Thus, $H'$ is central in $G$.
%
%It now follows that the fibers of $H'$ are of multiplicative type.  Indeed, we may check
%on geometric fibers, and over a 
%field any commutative extension of a multiplicative type group
%by another is again of multiplicative type (Exercise \ref{3.7}(ii)). 
%Our goal is to prove that $H'$ is of multiplicative type.
%
%To bootstrap from the fibers, we use deformation theory,
%as follows. The initial hypotheses
%are compatible with standard limit arguments on the base, so we
%may and do reduce to the case that $S$ is noetherian at the outset,
%and then local, say $S = \Spec(A)$.  
%Let $\widehat{S} = \Spec(\widehat{A})$.
%Suppose we can prove that $H'_{\widehat{S}}$ is of multiplicative type.
%By Proposition \ref{fppfet}, this group
%is split by an \'etale cover of $\widehat{S}$.
%Any such \'etale cover is refined by a local finite \'etale cover
%$\widehat{S}' \rightarrow \widehat{S}$
%(since $\widehat{A}$ is henselian local).  But
%the categories of finite \'etale covers of $A$ and $\widehat{A}$
%are equivalent (since $A$ is henselian local), so
%$\widehat{S}'$ uniquely descends to a local
%finite \'etale cover $S' \rightarrow S$.  We may rename
%$S'$ as $S$ to reduce to the case that  there is
%an isomorphism $f:D_{\widehat{S}}(M) \simeq H'_{\widehat{S}}$
%for a finitely generated abelian group $M$.
%Since $H'$ is commutative, we may argue exactly as in the proof of Proposition \ref{fppfet}
%to descend $f$ to an isomorphism $D_S(M) \simeq H'$.
%
%We may now rename $\widehat{S}$ as $S$ to reduce to the case
%that $A$ is complete.  Thus, by the final assertion in Theorem \ref{torlift}, there
%exists an open and closed 
%$S$-subgroup scheme $\mathscr{H}'$ in $H'$ of multiplicative
%type such that the special fibers of $H'$ and $\mathscr{H}'$ coincide inside the special fiber of 
%$G$.
%We will prove that $\mathscr{H}' = H'$.
%
%The multiplicative type $S$-subgroups $H$ and $\mathscr{H}'$ in the commutative 
%$H'$
%satisfy the containment relation $H_0 \subset \mathscr{H}'_0$ inside 
%${H}'_0$ for their special fibers.
%We claim that $H \subset \mathscr{H}'$.   The product map
%$H \times \mathscr{H}' \rightarrow {H}'$ makes sense as an 
%$S$-homomorphism since
%${H}'$ is commutative, and by Exercise \ref{ix6.8} this map factors
%through an fppf quotient onto a multiplicative type
%$S$-subgroup $K$.  By construction $K$ contains $\mathscr{H}'$, but its special fiber is
%an fppf quotient of $H_0 \times \mathscr{H}'_0$ via addition in $H'_0$,
%so since $H_0 \subset \mathscr{H}'_0$ inside $H'_0$ we conclude
%that the containment $\mathscr{H}' \subset K$ of $S$-subgroups 
%of ${H}'$ induces an equality between special fibers.  But $K$ and $\mathscr{H}'$ are both
%of multiplicative type
%and $A$ is complete local noetherian, so $K$ and $\mathscr{H}'$ split
%over a finite \'etale extension of $S$ (due to Proposition \ref{fppfet}
%and the cofinality of such covers among all \'etale covers of $S$). Hence, it
%is clear that $\mathscr{H}' = K$, so $H \subset K = \mathscr{H}'$ as desired.
%
%Consider the map $\mathscr{H}' \rightarrow \overline{H}' := {H}'/H$ between 
%$S$-groups of
%multiplicative type.  On special fibers, this is the quotient map $\mathscr{H}'_0 = H'_0 \rightarrow
%\overline{H}'_0$, so the induced map between constant character
%groups is injective.  Hence,  the $S$-homomorphism $\mathscr{H}' \rightarrow \overline{H}'$ 
%is dual to an injection between character groups, so it is faithfully flat.
%Thus, $\mathscr{H}' \rightarrow \overline{H}'$ is an fppf quotient homomorphism whose
%kernel is $H$, yet $\mathscr{H}'$ is an $S$-subgroup of 
%${H}'$ that is {\em also} an extension of $\overline{H}'$ by $H$ for the fppf topology
%(compatibly with the analogous such extension structure on $\mathscr{H}'$).
%A snake lemma argument with fppf group sheaves then implies that the inclusion
%$\mathscr{H}' \subset H'$ is an equality, so 
%$H'$ is of multiplicative type.
\end{proof}

The following lemma identifies an interesting property for multiplicative
type subgroups that are central on geometric fibers. 

\begin{lemma}\label{redcent}  Let $G$ be a smooth $S$-affine $S$-group with
connected fibers, and $H \subset G$ an $S$-subgroup of multiplicative type.
For all geometric points $\overline{s}$ of $S$,
assume $H_{\overline{s}}$ is central in $G_{\overline{s}}$
and contains all central multiplicative type subgroups of
$G_{\overline{s}}$.  Then $H$ is central in $G$ and 
for any $S$-scheme $S'$, every central multiplicative type subgroup of
$G_{S'}$ is contained in $H_{S'}$.
\end{lemma}
\index{multiplicative type!and centrality}

%By a  ``spreading out and specialization'' argument (as in the proof
%of Proposition \ref{maxrem}), the fibral hypothesis in Lemma \ref{redcent}
%is open on the base. 

\begin{proof}
By Lemma \ref{centralizer}, the functorial centralizer of $H$ in $G$ is represented
by a smooth closed $S$-subgroup $Z_G(H)$.  
The inclusion $Z_G(H) \hookrightarrow G$ between smooth $S$-affine $S$-groups  is
an isomorphism on $s$-fibers for all $s \in S$ and hence (by Lemma \ref{flatclosed})  is an isomorphism.
Thus, $H$ is central in $G$.  Since each $G_s$ is connected, by
dimension and smoothness considerations the same method shows that 
the centrality of $H_{\overline{s}}$ in $G_{\overline{s}}$ for a single geometric point $\overline{s}$ over $s$
implies that $H_U$ is central in $G_U$ for some open neighborhood $U$ of $s$ in $S$.  
%Any central subgroup scheme of $G$ lies in $Z_G(H)$, so we may replace
%$G$ with $Z_G(H)$ to reduce to the case that $H$ is central in $G$.

The fppf quotient sheaf $G/H$ has a natural $S$-group structure due to centrality of $H$ in $G$, and by  
\cite[VIII, 5.1; IX, 2.3]{sga3} it is represented by an $S$-affine $S$-scheme of finite presentation.
By Exercise \ref{ix6.8} and Lemma \ref{multsub}(2) we may rename $G/H$ as $G$ so that
each $G_{\overline{s}}$ contains no nontrivial central closed subgroup of multiplicative type.
This property persists after any base change $S' \rightarrow S$, so upon renaming $S'$
as $S$ it remains to show that a central multiplicative type subgroup $H' \subset G$ must be trivial.
Every geometric fiber $H'_{\overline{s}}$ is trivial, so consideration of the character group
of the multiplicative type $H'$ forces $H'$ to be the trivial $S$-group. 
\end{proof}

In view of the preceding lemma, the following definition (taken
from \cite[XII, 4.1]{sga3}) is reasonable as well as checkable in practice.

\begin{definition} 
Let $G \rightarrow S$ be a smooth $S$-affine group scheme with connected fibers.
A {\em reductive center} of $G$ 
is a central multiplicative type subgroup $H \subset G$ that satisfies the conditions
in Lemma $\ref{redcent}$.
\end{definition}
\index{reductive center}

It is clear that if a reductive center exists then it is unique; we then call it {\em the} reductive center.
Finally, we arrive at a special case of \cite[XII, 4.11]{sga3}: 

\begin{theorem}\label{redcenter}  Any reductive group scheme $G \rightarrow S$
admits a reductive center $Z$, and $Z$ coincides with the
scheme-theoretic center $Z_G$ of $G$.  In particular,
$Z_G$ is $S$-flat.
Moreover, $Z$ represents the kernel of the action map
$$u:G \rightarrow \underline{\rm{Aut}}_S({\rm{Tor}}_{G/S}).$$
\end{theorem}
\index{center}
\index{reductive group!center}

\begin{proof}
Since the formation of the reductive center commutes with base change if it exists,
by effective fppf descent for schemes affine over the base we may work fppf-locally on $S$.
Hence, we may and do assume that $G$ contains a split maximal torus $T$. 
This identifies $G/N_G(T)$ with ${\rm{Tor}}_{G/S}$ 
via $g \mapsto gTg^{-1}$, so the natural $G$-action on ${\rm{Tor}}_{G/S}$ via conjugation
goes over to the left translation action of $G$ on $G/N_G(T)$.  We will 
first construct a reductive center $Z$, and then prove that it is the
center and represents the functorial kernel $\ker u$.

We may and do assume $S$ is affine and $T = D_S(M) \simeq \mathbf{G}_{\rm{m}}^r$ for 
$M \simeq \Z^r$ with some $r \ge 0$.
Consider the $\calO_S$-linear $M$-graded decomposition 
$\bigoplus_{m \in M} \mathfrak{g}_m$ of $\mathfrak{g} = {\rm{Lie}}(G)$
into weight spaces under the $T$-action, with $t.v = m(t) v$ for $v \in \mathfrak{g}_m$
(see \cite[III, 1.5]{oesterle} or \cite[Lemma A.8.8]{pred}).  These weight spaces are vector bundles
since they are direct summands of $\mathfrak{g}$, and the formation of this
decomposition commutes with base change on $S$. 
By working Zariski-locally on $S$, we may arrange that the weight spaces
$\mathfrak{g}_{m_0}$ that are nonzero
have constant rank (necessarily rank 1 for $m_0 \ne 0$, by the reductivity hypothesis). 
Let $\Phi \subset M$ be the finite set of nontrivial weights that arise.  Define
$$H = \bigcap_{\alpha \in \Phi} \ker(\alpha:T \twoheadrightarrow \mathbf{G}_{\rm{m}}).$$
This intersection of finitely many multiplicative type $S$-subgroups of $T$
is a multiplicative type $S$-subgroup of $T$ (corresponding to the quotient of
$M$ by the $\Z$-submodule spanned by $\Phi$), and by construction
the $H$-action on $G$ via conjugation induces the trivial action on $\mathfrak{g}$.

The centralizer $Z_G(H)$ is a smooth closed $S$-subgroup of $G$ (Lemma \ref{centralizer})
and its Lie algebra is $\mathfrak{g}^H$ (Proposition \ref{zrep}).
By design $\mathfrak{g}^H = \mathfrak{g}$, 
so $Z_G(H) \hookrightarrow G$ is an isomorphism between $s$-fibers for all $s \in S$ and hence 
is an equality by Lemma \ref{flatclosed}.  This shows that
$H$ is a central multiplicative type $S$-subgroup of $G$.  But for any $S$-scheme $S'$
and $g' \in G(S')$ that centralizes $G_{S'}$, clearly 
$g' \in Z_{G(S')}(T_{S'}) = (Z_G(T))(S') = T(S')$ (Proposition \ref{cartan}(2)) 
and the adjoint action of $T_{S'}$ on $\mathfrak{g}_{S'} = {\rm{Lie}}(G_{S'})$
makes the point $g' \in T(S')$
act trivially, so $g'$ is an $S'$-point of $\bigcap_{\alpha \in \Phi} \ker(\alpha_{S'}) = H_{S'}$.
This shows that $H$ is a reductive center of $G$, and that it represents the functorial center of $G$.
The centrality also forces $H \subset \ker u$, so it
remains to prove (after making a base change by any $S' \rightarrow S$ and renaming $S'$ as $S$)
that any $g \in G(S)$ with trivial conjugation action on $\mathscr{T} := {\rm{Tor}}_{G/S}$ necessarily lies in $H(S)$.

By the usual limit arguments (including Corollary \ref{descend})
we may reduce to the case when $S$ is noetherian,
and then artin local (by the Krull intersection theorem relative to the ideal 
of $H$ in $\calO_G$), and finally artin local with
algebraically closed residue field (via faithfully flat base change \cite[0$_{\rm{III}}$, 10.3.1]{ega}). 
Writing $S = \Spec(A)$ for an artin local ring $A$, the
set of $A$-points of the $A$-smooth $\mathscr{T}$ 
is relatively schematically dense \cite[IV$_3$, 11.9.13, 11.10.9]{ega};
i.e., for any $A$-algebra $A'$,
an $A'$-morphism from $\mathscr{T}_{A'}$ to a separated $A'$-scheme
is uniquely determined by its effect on the points of $\mathscr{T}(A)$ (viewed as $A'$-points
of $\mathscr{T}_{A'}$).  In other words, $\ker u$ is {\em represented}
by the  intersection of the closed subschemes
$N_G(T_{\sigma})$ where $T_{\sigma}$ is the maximal $A$-torus in $G$
corresponding to the varying point $\sigma \in \mathscr{T}(A)$.

We have $H \subset \ker u$ as closed subschemes of $G$, and  seek
to prove this is an equality.   By Lemma \ref{flatclosed},
this reduces to the consideration of the special fiber (as the artin local $S$ has only one point).
Now we are considering a connected
reductive group $G$ over an algebraically closed field $k$, so every central closed $k$-subgroup
scheme is of multiplicative type since any maximal torus in $G$ is its own schematic centralizer
(Corollary \ref{zgood}).  The above description of
$\ker u$ becomes $\ker u = \bigcap_{T'} N_G(T')$ with $T'$ varying through the maximal
$k$-tori of $G$. Since $G/Z_G$ is perfect, a normal closed subgroup scheme of $G$ is central
if and only if its identity component is central (see \cite[Lemma 5.3.2]{pred}
for a self-contained proof of this elementary fact).  Thus, since $(\ker u)^0 \subset \bigcap_{T'} T'$, to show 
$\ker u = H$ it suffices to prove that $\bigcap_{T'} T'$ is central in $G$.

Consider the smooth closed subgroup $N$ of $G$ generated by the maximal tori.
This is normal, so $G/N$ is a reductive group that contains no nontrivial tori
(since quotient maps between linear algebraic $k$-groups carry
maximal tori onto maximal tori).  Hence, $G/N$ is unipotent and therefore
trivial.  In other words, $G$ is generated
by its maximal tori,  so there exists a finite set $\{T'_i\}$ of maximal tori of $G$
such that the multiplication map of $k$-schemes $q:\prod T'_i \rightarrow G$
is dominant.  A dominant map between $k$-varieties is generically flat, so
there are dense open $V \subset G$  and $V' \subset q^{-1}(V)$
such that $V' \rightarrow V$
is faithfully flat.  Any (functorial) point of $\bigcap_{T'} T' = \ker u$
centralizes all $T'_i$ and hence centralizes $q(V') = V$, so $\ker u$ centralizes
$G$ (as any $S$-endomorphism of $G_S$ for a $k$-scheme $S$ is determined by its restriction
to $V_S$, by \cite[IV$_3$, 11.9.13, 11.10.1(d)]{ega}).
\end{proof}

\begin{corollary}\label{qtcenter}
Let $G$ be a reductive $S$-group scheme, and $Z$ a multiplicative type subgroup scheme
of the center $Z_G$.  The reductive quotient $G/Z$ has center $Z_G/Z$;
in particular, $G/Z_G$ has trivial center.  

Moreover,
$T \mapsto T/Z$ defines a bijective correspondence
between the set of maximal tori of $G$ and the set of maximal tori of $G/Z$,
with inverse given by scheme-theoretic preimage under the quotient map
$G \rightarrow G/Z$.
\end{corollary}

Any such $T$ contains the central $Z$ in $G$ since $T = Z_G(T)$,
so $T/Z$ makes sense.  This corollary is a special case of \cite[XII, 4.7(b),(c)]{sga3}. 

\begin{proof}
By Lemma \ref{multsub}, the set of central
subgroups of multiplicative type in $G/Z$ is in bijection with the set of central subgroups of multiplicative type in $G$
containing $Z$, which is to say the multiplicative type subgroups of $Z_G$
containing $Z$ (as $Z_G$ is the reductive center of $G$).  Those subgroups
correspond to multiplicative type subgroups of $G/Z$
contained in $Z_G/Z$.  Thus, $Z_G/Z$ is the reductive center of $G/Z$,
so it is the center of $G/Z$.

To establish the bijective correspondence for maximal tori,  we first
note that the classical theory ensures that $T/Z$ is maximal in $G/Z$ when
$T$ is maximal in $G$.  Hence, the proposed correspondence
makes sense. It is also clear that $T$ is the preimage of $T/Z$ under
the quotient map $G \rightarrow G/Z$, so the only problem is to prove
that every maximal torus $\overline{T}$ in $\overline{G} := G/Z$ has the form $T/Z$ for some
(necessarily unique) maximal torus $T$ in $G$.  Consider
the preimage $\widetilde{T}$ of $\overline{T}$ in $G$. 
We seek to prove that $\widetilde{T}$ is a maximal torus in $G$.

Since $\overline{G}$ is the quotient of $G$ modulo a central subgroup scheme,
the conjugation action of $G$ on itself factors through a left action of $\overline{G}$ on $G$.
As such, we get a left action of the torus $\overline{T}$ on $G$, so
the functorial centralizer $G^{\overline{T}}$ for this action
is represented by a smooth closed subgroup of $G$ with connected geometric fibers
(by arguing as in Example \ref{cent}:  we form
the $\overline{T}$-centralizer in the reductive semi-direct product
$G \rtimes \overline{T}$).    Consideration of geometric
fibers shows that $G^{\overline{T}}$ is reductive over $S$.
But $Z \subset G^{\overline{T}}$, so we get
a natural monomorphism 
$$j:G^{\overline{T}}/Z \rightarrow Z_{\overline{G}}(\overline{T}) = \overline{T}$$
between smooth $S$-affine $S$-groups.  To show $j$ is 
an isomorphism, we pass to geometric fibers (by
Lemma \ref{flatclosed}) so the monic $j$ is a closed immersion (Proposition \ref{propker}). 
Then $j$ is surjective since surjections
between smooth connected affine groups over a field
induce surjections between centralizers for a torus action
\cite[11.14, Cor.\,2]{borelag}. Thus, $j$ is an isomorphism. 

The isomorphism $G^{\overline{T}}/Z = \overline{T}$ implies
that $G^{\overline{T}}$ is the preimage $\widetilde{T}$ of $\overline{T}$ in $G$.
Apply Lemma \ref{multsub}(2) to the quotient map
$G^{\overline{T}} \rightarrow G^{\overline{T}}/Z = \overline{T}$ to
conclude that the reductive $G^{\overline{T}}$ is central in itself (i.e., commutative),
so $G^{\overline{T}}$ is a torus by Corollary \ref{buildtori}.  It suffices to prove that it is a maximal torus in $G$
(in the sense of Definition \ref{maxtori}).
Passing to geometric fibers over $S$, now $S = \Spec(k)$ for
an algebraically closed field $k$. 
Consider a torus $T'$ of $G$ containing $G^{\overline{T}}$.  The image $T'/Z$ in $\overline{G}$
is a torus (as it is multiplicative type and smooth), yet this image contains the 
torus $G^{\overline{T}}/Z = \overline{T}$ that is maximal by hypothesis, so 
$T'/Z = G^{\overline{T}}/Z$ and hence $T' = G^{\overline{T}}$.  Thus,
$G^{\overline{T}}$ is a maximal torus in $G$. 
\end{proof}

\begin{corollary}\label{computez}
Let $G \rightarrow S$ be a reductive group scheme, and
$T$ an $S$-torus in $G$ that is maximal
$($in the sense of Definition $\ref{maxtori}$$)$. 
\begin{enumerate}
\item The center $Z_G$ is the  kernel of the adjoint action 
$T \rightarrow {\rm{GL}}(\mathfrak{g})$.
\item If  $S = \Spec k$ for an algebraically closed field $k$ then 
$Z_G$ coincides with the scheme-theoretic
intersection of all maximal tori $T'$ in $G$.
\end{enumerate}
\end{corollary}

This corollary is a special case of \cite[XII, 4.7(d), 4.10]{sga3}. 

\begin{proof}
To prove (1) we may work \'etale-locally on $S$ so that $T$ is split.
In that case the equality in (1) was shown in the course of proving Theorem \ref{redcenter}.
For (2), since $Z_G(T') = T'$ for all $T'$ (as subschemes of $G$), clearly every $T'$ contains 
$Z_G$.  It remains to show that the  intersection of all $T'$
is central in $G$ (as a subgroup scheme), and we showed this in the proof of Theorem \ref{redcenter}.
\end{proof}

\begin{example}\label{slncenter} 
Corollary \ref{computez}(1) provides a way to compute $Z_G$, since 
it is often easy to find a torus $T$
such that $Z_G(T) = T$ (equivalently $\mathfrak{g}^T = \mathfrak{t}$,
due to smoothness and fibral connectedness of torus centralizers in $G$).

We illustrate with $G = {\rm{SL}}_n$ over any scheme $S$.
The diagonal torus $T$ is maximal, since the case of
geometric fibers is well-known.   The action of $T$
on the Lie algebra $\mathfrak{sl}_n$ over any base $S$
is the given by the habitual formulas, from which we
see that the  kernel of the action is $\mu_n \subset T$, so
$\mu_n = Z_{{\rm{SL}}_n}$.
\end{example}


\begin{proposition}\label{kerAd}
Let $G \rightarrow S$ be a reductive group scheme. 
The center $Z_G$ equals the kernel of
the adjoint representation ${\rm{Ad}}_G: G \rightarrow {\rm{GL}}(\mathfrak{g})$.
\end{proposition}
\index{center! and adjoint representation}
\index{ aaaadg@${\rm{Ad}}_G$}

\begin{proof} The schematic center $Z_G$ is fppf over $S$ (Theorem \ref{redcenter})
and $\ker {\rm{Ad}}_G$ is finitely presented over $S$,
so by Lemma \ref{flatclosed} the inclusion
$Z_G \hookrightarrow \ker {\rm{Ad}}_G$ is an isomorphism
if it is so on (geometric) fibers over $S$.  Thus, we may and do
assume $S = \Spec(k)$ for an algebraically closed field $k$.
As we discussed in the proof of Theorem \ref{redcenter}, a normal closed
$k$-subgroup of $G$ is central if its identity component is central.   It therefore suffices 
to show that $(\ker {\rm{Ad}}_G)^0$ is central in $G$.
Thus, by Corollary \ref{computez}(2), 
it suffices to show that $(\ker {\rm{Ad}}_G)^0$ is contained in each
maximal torus $T$ of $G$, or equivalently is contained in the
schematic centralizer $Z_G(T) = T$ for each such $T$.

Normality of $\ker {\rm{Ad}}_G$ in the smooth affine $k$-group $G$ implies
the normality of its identity component in $G$.  Hence, there is 
a $T$-action on this identity component via conjugation, and we just need
to show that this action is trivial.
Since $T$ is of multiplicative type, its action on a connected
$k$-group scheme $H$ of finite type is trivial if and only its induced
action on ${\rm{Lie}}(H)$ is trivial, by \cite[Cor.\:A.8.11]{pred} (whose proof
simplifies significantly for the action by a torus).  Thus, we only need
to verify that the adjoint action of $T$ on ${\rm{Lie}}(\ker {\rm{Ad}}_G)$ is trivial. 
But ${\rm{Lie}}(\ker {\rm{Ad}}_G) = \ker({\rm{ad}}_{\mathfrak{g}})$ \cite[Prop.\:A.7.5]{pred}, so it
\index{adjoint representation!relation with ${\rm{ad}}_{\mathfrak{g}}$}
suffices to prove that $\ker({\rm{ad}}_{\mathfrak{g}}) \subset {\rm{Lie}}(T)$, or
equivalently (via $T$-weight space considerations) that
$\ker({\rm{ad}}_{\mathfrak{g}})$ does not contain any root
line $\mathfrak{g}_a$ for $a \in \Phi(G,T)$.

For any root $a$, consider the rank-1 semisimple subgroup
$G_a = \mathscr{D}(Z_G(T_a))$ with maximal torus $a^{\vee}(\Gm)$
whose root groups are $U_{\pm a}$. 
By functoriality of the adjoint representation (applied to the inclusion
$G_a \hookrightarrow G$), if $\mathfrak{g}_a \subset \ker({\rm{ad}}_{\mathfrak{g}})$
then the analogue holds for $(G_a, a^{\vee}(\Gm), a)$ in place of $(G, T, a)$.
Thus, to get a contradiction we may replace $G$ with $G_a$, so 
it suffices
to treat the groups ${\rm{SL}}_2$ and ${\rm{PGL}}_2$,
taking $T$ to be the diagonal torus and $a$ to correspond 
to the upper triangular unipotent subgroup $U^{+}$.  Choose nonzero 
$v^{\pm} \in \mathfrak{u}^{\pm}$ and a nonzero $t \in \mathfrak{t}$.  In the ${\rm{SL}}_2$-case
$[v^{+},v^{-}] \ne 0$ and in the ${\rm{PGL}}_2$-case $[v^{+},t] \ne 0$.
\end{proof}

Passing to the Lie algebra, we conclude from Proposition \ref{kerAd} that 
$${\rm{Lie}}(Z_G) =
\ker({\rm{Lie}}({\rm{Ad}}_G)) = \ker({\rm{ad}}_{\mathfrak{g}}).$$
We say $G$ is {\em adjoint} if $Z_G = 1$; this can be checked
on geometric fibers since $Z_G$ is multiplicative type, and is equivalent to each $G_s$
being adjoint semisimple. 
\index{reductive group!adjoint}
\index{adjoint!reductive group}


%The remainder of this section, apart from some self-contained definitions, will not be used until \S\ref{clsec} (and it rests
%upon results proved in \S\ref{parsec}).  We include it here because 
%this discussion provides an opportunity to introduce the notion of ``adjoint type''. 
%
%Let $G \rightarrow S$ be a reductive group scheme. By Corollary \ref{computez}(1)
%and the existence of a maximal torus \'etale-locally on the base
%(Corollary \ref{buildtori}), $Z_G$ equals the kernel of
%the adjoint representation ${\rm{Ad}}_G: G \rightarrow {\rm{GL}}(\mathfrak{g})$
%{\em provided} that $\ker {\rm{Ad}}_G$ is central in $G$.  We shall now prove the centrality of this
%kernel in general, using results from \S\ref{parsec}.
%
%By standard arguments with finitely presented
%structures, it suffices to check the centrality 
%in $G$ of any $g \in (\ker {\rm{Ad}}_G)(S)$ with $S = \Spec A$ for an artin local ring $A$
%having algebraically closed residue field $k$.  The key ingredient is: 
%
%\begin{lemma}\label{bortor} For $S = \Spec A$ as above  and any Borel subgroups $B, B' \subset G$
%such that ${\rm{Lie}}(B) = {\rm{Lie}}(B')$, there is a maximal torus of $G$ contained in $B \cap B'$.
%\end{lemma}
%
%In this lemma, the notion of {\em Borel subgroup} in $G$ over $A$ means a smooth closed $A$-subgroup
%whose special fiber is a Borel subgroup of $G_k$.  (See Definition \ref{qsdef} for the analogue over
%a general base scheme.) Later we will prove the stronger property
%that $B = B'$ in such cases.
%
%Granting the lemma, we may conclude as follows.   Since $A$ is artin local with residue field $k = \overline{k}$,
%by Theorem \ref{borel}(3) 
%there is a Borel subgroup $B \subset G$.  The hypothesis ${\rm{Ad}}_G(g) = 1$
%implies that ${\rm{Lie}}(gBg^{-1}) = {\rm{Lie}}(B)$.  By Lemma \ref{bortor},
%there is a maximal torus $T$ of $G$ contained in $B$ and $gBg^{-1}$.   Hence, we may apply
%Corollary \ref{parfiber}(2) to conclude that $B = gBg^{-1}$.  But $B$ represents its own functorial normalizer
%by Corollary \ref{parnorm}, so $g \in B(S)$.  By Proposition \ref{oppborel}, there is a unique
%Borel subgroup $B' \subset G$ such that $B \cap B' = T$.  The same reasoning
%shows that $g \in B'(S)$, so $g \in T(S)$.  Since $\ker({\rm{Ad}}_G|_T) = Z_G$ by Corollary \ref{computez}(1),
%the desired centrality then follows.  Now we prove Lemma \ref{bortor}.
%
%\begin{proof}
%By Theorem \ref{borel}(2), there exists $g \in G(A)$ such that $B' = gBg^{-1}$.
%The classical theory provides a maximal torus $T_0 \subset G_k$
%contained in $B_k \cap B'_k$.   By the Bruhat decomposition for
%$G_k$ relative to $(B_k, T_0)$, we can write $g_0 = b_0 n_0$
%for some $b_0 \in B_k(k)$ and $n_0 \in N_{G_k}(T_0)(k)$.   Hence,
%$B'_k$ contains $b_0 T_0 b_0^{-1}$, and obviously $B_k$ contains
%this maximal $k$-torus of $G_k$ too.   The hypothesis of equality of Lie algebras
%over $A$ implies that the Borel subgroups $B_k, B'_k$ in $G_k$ have the same
%Lie algebra inside ${\rm{Lie}}(G_k)$.  These Borel subgroups contain
%a common maximal torus, so $B'_k = B_k$.  Hence, $g_0$ normalizes $B_k$,
%so $g_0 \in B_k(k)$.  
%
%By smoothness, $g_0$ lifts to $b \in B(A)$.  We may replace $g$ with $gb^{-1}$ to arrange
%that $g_0 = 1$.   Since the Borel subgroups of $G$ constitute a single $G(A)$-conjugacy class
%(by Theorem \ref{borel}(2)), 
%the relative dynamic method as reviewed in \S\ref{dynsec}  (see especially
%Theorem \ref{215}(4)) provides a cocharacter $\lambda:\Gm \rightarrow B$ such that 
%$B = P_G(\lambda)$.  The reductive group scheme $Z_G(\lambda) \subset B$ is a maximal torus of $G$
%since this can be checked over the residue field, so it is a maximal torus of $B$.
%The maximal tori of $B$ constitute a single $B(A)$-conjugacy class (Theorem \ref{schemetori}), 
%so we may apply such a conjugation to arrange that $Z_G(\lambda) = T$.  
%For the long Weyl element $w_0 \in W(G_k,T_k)$ and $n \in N_G(A)$ lifting a representative of $w_0$, 
%$nBn^{-1} = P_G(w_0.\lambda) = T \ltimes U_G(w_0.\lambda)$ (see Theorem \ref{215}).  
%
%In \S\ref{corootsec} and  \S\ref{splitopen}
%we will develop the notions of root group and open cell 
%in the relative split case.  For each $a \in \Phi(G_k,T_k)$
%the pairing $\langle a, w_0.\lambda \rangle$ is nonzero and has the same sign as $\langle a, -\lambda \rangle$,
%so by (the proof of) Theorem \ref{bigcell} we have $U_G(w_0.\lambda) = U_G(-\lambda)$.
%In particular, $nBn^{-1} = P_G(-\lambda)$.
%Hence, by Theorem \ref{215}(4), the multiplication map $\Omega := nBn^{-1} \times U_G(\lambda) \rightarrow G$
%is an open immersion onto a neighborhood of 1.  We have arranged that $g_0 = 1$, so
%$g \in \Omega(A) = n B(A)n^{-1} U_G(\lambda)(A)$ with $U_G(\lambda) \subset P_G(\lambda) = B$.
%Applying a suitable residually trivial right $B(A)$-translation to $g$ then brings us to the case
%that $g = nbn^{-1}$ for some $b \in B(A)$ such that $b_0=1$.   The group $B'$ contains
%$gTg^{-1} = nbTb^{-1}n^{-1}$
%
%The maximal torus $bTb^{-1}$ of $G$ is contained in $B$.
%It is also contained in $B' = gBg^{-1}$ since $bTb^{-1} = bn^{-1}Tnb^{-1} = n^{-1}gTg^{-1}n$
%But $bTb^{-1} = n^{-1}gTg^{-1}n \subset n^{-1}B'n$
%\end{proof}
%
%Passing to the Lie algebra, we conclude that 
%$${\rm{Lie}}(Z_G) =
%\ker({\rm{Lie}}({\rm{Ad}}_G)) = \ker({\rm{ad}}_{\mathfrak{g}})$$
%(see \cite[Prop.\,A.7.5]{pred} for the relation 
%between ${\rm{Lie}}({\rm{Ad}}_G)$ and ${\rm{ad}}_{\mathfrak{g}}$).
%We say $G$ is {\em adjoint} when $Z_G = 1$; this property can be checked
%on geometric fibers since $Z_G$ is multiplicative type, and is equivalent to the geometric
%fibers being semisimple of adjoint type (so $G$ is semisimple). 
%
%For semisimple $G$ of adjoint type, 
%the $S$-homomorphism ${\rm{Ad}}_G:G \rightarrow {\rm{GL}}(\mathfrak{g})$
%has trivial kernel.  In view of Remark \ref{immrem} it is not obvious if ${\rm{Ad}}_G$
%is a closed immersion in the adjoint semisimple case over a general scheme.  In fact, 
%it is always a closed immersion for such $G$, because
%any monomorphism from
%a {\em reductive} group scheme to a separated group of finite presentation
%is a closed immersion (Theorem \ref{adimm}).
%We will not use this fact. 

%The quotient map $G \rightarrow G/Z_G$ will be a focus of interest later, but for now
%we consider quotient maps with finite flat kernel.

\begin{definition}\label{ceniso} A homomorphism $f:G' \rightarrow G$ between smooth $S$-affine
$S$-groups is an {\em isogeny} if it is a finite flat surjection,
and is a {\em central isogeny} if also $\ker f$ is central in $G'$.
\end{definition}
\index{isogeny}
\index{central isogeny}

Over any field of characteristic $> 0$, the Frobenius isogeny
of a nontrivial connected semisimple group is non-central.
In the classical setting there exist examples of isogenies 
between connected semisimple groups such that the kernel is
{\em commutative} and non-central, though these only exist in characteristic 2
(see Remark \ref{remblah}
and \cite[Lemma 2.2]{py}).

%\begin{remark}\label{borelcen}
%The definition of ``central isogeny'' in \cite[22.3]{borelag} looks 
%more complicated than Definition \ref{ceniso}, but it is the same concept.
%The complications are due to encoding the centrality of 
%$\ker f$ in terms of a language that avoids non-smooth group schemes.
%\end{remark}

If $f:G' \rightarrow G$ is a homomorphism
between smooth $S$-affine $S$-groups and if $f_s$ is an isogeny for all $s \in S$ then is
$f$ finite flat?  Such an $f$ is certainly surjective, and also
flat due to the fibral flatness criterion \cite[IV$_3$, 11.3.10]{ega}.
Hence, $\ker f$ is a quasi-finite flat closed normal $S$-subgroup of $G'$,
and $G = G'/\ker f$ in the sense of fppf sheaves. 
By fppf descent, $f$ is finite if and only if $\ker f$ is $S$-finite (cf.\,Exercise \ref{3.6}(iii)). 
But is $\ker f$ actually 
 $S$-finite? And if moreover $\ker f_s$ is central in $G_s$ for all $s \in S$ then is $\ker f$ central in $G'$?
We shall give affirmative answers in the reductive case.
The case of central isogenies will be treated now; the general case
lies a bit deeper (see Proposition  \ref{isogfiber}). 

\begin{proposition}\label{central}
A surjective homomorphism $f:G' \rightarrow G$ between reductive groups over a scheme $S$ 
is a central isogeny if and only if $f_{\overline{s}}$ has finite central kernel for all $s \in S$.
\end{proposition}

\begin{proof}
The implication ``$\Rightarrow$'' is obvious.  For the converse, the flatness
of $f$ follows from the fibral flatness criterion \cite[IV$_3$, 11.3.10]{ega}, so it remains to show that 
$K := \ker f$ is $S$-finite and central in $G'$.  We may assume that $S$ is noetherian,
then local (by direct limit considerations), and finally complete (by faithfully flat descent).

First we show $K$ is central.  This asserts that the $G'$-action on $K$ by conjugation
is trivial, an identity  that is sufficient to check on artin local points over $S$.
Thus, we may assume $S = \Spec A$ for an artin local ring $A$, so 
the quasi-finite flat $K$ is finite flat.  The special fiber $K_0 \subset Z_{G'_0}$
uniquely lifts to a finite multiplicative type subgroup $\widetilde{K} \subset Z_{G'}$ since $A$ is artin local
and $Z_{G'}$ is of multiplicative type, 
and the map $\widetilde{K} \rightarrow G$ induced by $f$ is trivial on the special fiber,
so it is trivial by Corollary \ref{rigidhom}. 
This implies $\widetilde{K} \subset \ker f = K$ inside $G'$.  But this inclusion
between finite flat $A$-groups induces an equality on special fibers (by construction of $\widetilde{K}$),
so it is an equality. Hence, $K$ is central in $G'$.

Over a general $S$, $K$ is closed
in the multiplicative type 
$Z_{G'}$.  By Exercise \ref{ix6.4}, all finitely presented quasi-finite closed subgroups of a multiplicative
type group are finite.  Thus, $K$ is $S$-finite.  
\end{proof}

\newpage

\subsection{Exercises}

\begin{exer}\label{geomrad}
Let $K/k$ be an extension of algebraically closed fields, and $G$ a connected reductive $k$-group.
This exercise proves by contradiction that the smooth connected affine $K$-group $G_K$ is reductive;
the same method also handles semisimplicity.

(i) Assume $G_K$ is not reductive.  Show that $G_K$ contains a nontrivial normal $K$-subgroup $U$
admitting a finite composition series whose successive quotients are isomorphic to $\Add$.

(ii) In the setup of (i), by expressing $K$ as a direct limit of its
finitely generated $k$-subalgebras show that there is a finitely generated $k$-subalgebra $A \subset K$
such that $G_A$ has a smooth affine normal closed $A$-subgroup $\mathscr{U} \subset G_A$
admitting an increasing finite sequence of smooth closed $A$-subgroups 
$1 = \mathscr{U}_0 \subset \dots \subset  \mathscr{U}_n = \mathscr{U}$ such that $n > 0$ and 
$\mathscr{U}_i$ is identified with the kernel of an fppf $A$-homomorphism $\mathscr{U}_{i+1} \twoheadrightarrow
\Add$ for $0 \le i < n$.

(iii) By specializing at a $k$-point of $\Spec(A)$, deduce that $G$ is not reductive.
\end{exer}

\begin{exer}\label{drexer} This exercise proves a very useful
lemma of Deligne and Rapoport \cite[II, 1.19]{dr} that is a generalization of
the finiteness criterion used in the proof
of Proposition \ref{finitew}. Let $f:X \rightarrow Y$ be 
a quasi-finite flat and separated map between noetherian
schemes, and assume its fiber degree is constant.  We seek to prove that $f$ is finite.

(i) Using that a proper quasi-finite map is finite, reduce to the case $Y = \Spec R$ for
a discrete valuation ring $R$ (hint: use the valuative criterion for properness). 

(ii) By Zariski's Main Theorem, the quasi-finite separated $X$ over $Y$ admits an open
immersion $j:X \hookrightarrow \overline{X}$ into a finite $Y$-scheme $\overline{X}$.
With $Y = \Spec R$ as in (i), arrange that $\overline{X}$ is also $R$-flat
and has the same generic fiber as $X$.

(iii) Using constancy of fiber degree, deduce that $j$ is an isomorphism and conclude.

(iv) If $f$ is \'etale, express the result in terms of specialization for
constructible \'etale sheaves.

(v) Remove the noetherian hypotheses without requiring $f$ to be of finite presentation.
\end{exer}

\begin{exer}\label{flatexer} This exercise directly proves the fibral isomorphism criterion
(Lemma \ref{flatclosed}) when $Y$, $Y'$, and $S$ are noetherian.
(The case of general $S$ reduces to this case by standard limit arguments.)

(i) Reduce to the case of separated $h$ by using that $\Delta_h:Y \rightarrow Y \times_{Y'} Y$ satisfies
the given hypotheses and is separated.  

(ii) Now taking $h$ to be separated, use the result from Exercise \ref{drexer} to reduce to proving
$h$ is flat, and then reduce to the case when $S$ is artin local.

(iii) For artin local $S = \Spec A$, prove $h$ is a closed immersion, and use $S$-flatness of $Y$
to deduce that the ideal defining it in $Y'$ vanishes modulo $\mathfrak{m}_A$, so $Y = Y'$.
\end{exer}

\begin{exer}\label{cenexer} (i) Prove that Corollary \ref{qtcenter} is valid when $G$ is replaced with
a parabolic subgroup of a connected reductive group.

(ii) Over any field $k$, show that the Heisenberg group
$U \subset {\rm{GL}}_3$ (the standard upper triangular
unipotent subgroup) is a central extension of
$U'' \simeq \Add \times \Add$ by $U' \simeq \Add$, with $U'$ the scheme-theoretic
center of $U$. Assuming ${\rm{char}}(k) = p > 0$, 
show that the Frobenius kernel $\ker F_{U/k}$ (see Exercise \ref{frobex}) is likewise a central extension of
$\ker F_{U''/k} \simeq \alpha_p \times \alpha_p$ by $\ker F_{U'/k} \simeq \alpha_p$,
and deduce that $U \rightarrow \overline{U} := U/(\ker F_{U'/k})$
and $\overline{U} \rightarrow U/(\ker F_{U/k}) = U^{(p)}$
are central isogenies between
smooth connected affine $k$-groups such that the composite isogeny is not
a central isogeny.
\index{relative Frobenius morphism}

(iii) Using any nontrivial smooth connected unipotent group
over a field, show both parts of Corollary \ref{computez} fail 
when ``reductive'' is relaxed to ``smooth connected affine''.
\end{exer}

\begin{exer}\label{3.1}
Let $A$ be a finite-dimensional associative algebra over a field $k$.
Consider the ring functor 
$\underline{A}:R \rightsquigarrow A \otimes_k R$ and the group functor
$\underline{A}^{\times}: R \rightsquigarrow (A \otimes_k R)^{\times}$
on $k$-algebras.
\index{ aaaaunderline@$\underline{A}$, $\underline{A}^{\times}$}

(i) Prove that $\underline{A}$ is represented by an affine space over $k$.
Using the $k$-scheme map ${\rm{N}}_{A/k}:
\underline{A} \rightarrow \mathbf{A}^1_k$ defined functorially by
$u \mapsto \Det(m_u)$, where $m_u:A \otimes_k R \rightarrow A \otimes_k R$ is left multiplication
by $u \in \underline{A}(R)$, prove that
$\underline{A}^{\times}$ is represented by the open {\em affine} subscheme 
${\rm{N}}_{A/k}^{-1}(\Gm)$.  (This is often called
``$A^{\times}$ viewed as a $k$-group'', a phrase that
is, strictly speaking, meaningless,  since $A^{\times}$ does not encode the $k$-algebra $A$.) 

(ii) For $A = {\rm{Mat}}_n(k)$ prove 
$\underline{A}^{\times} = {\rm{GL}}_n$, and for $k = \Q$ and $A = \Q(\sqrt{d})$ identify it with
an explicit $\Q$-subgroup of ${\rm{GL}}_2$ (depending on $d$).
Prove $\underline{A}^{\times}$ is connected reductive in general.

(iii) For $A = {\rm{Mat}}_n(k)$, show that ${\rm{N}}_{A/k}:\underline{A}^{\times} \rightarrow
\Gm$ is $\Det^n$. 
\end{exer}

\begin{exer}\label{3.6}
A diagram $1 \rightarrow G' \stackrel{j}{\rightarrow} G \stackrel{\pi}{\rightarrow} G'' \rightarrow 1$ of
fppf groups over a ring $k$ is {\em exact} if $\pi$ is faithfully flat and $G' = \ker \pi$.
\index{exact sequence of group schemes}

(i)  For any such diagram, 
prove $G'' = G/G'$ via $\pi$ as fppf sheaves on the category of $k$-schemes.  Prove a diagram of $k$-groups of multiplicative type
$1 \rightarrow H' \rightarrow H \rightarrow H'' \rightarrow$ is exact if and only if
the associated diagram of $\Gm$-dual \'etale sheaves is exact.

(ii) Prove that $G''$ is smooth when $G$ is smooth, even if $G'$ is not smooth.

(iii) If $G'$ is finite then prove that $\pi$ is 
{\em finite flat} with fibral degree locally constant on $\Spec k$, and that $\pi_n:{\rm{SL}}_n \rightarrow
{\rm{PGL}}_n$ has degree $n$.  Compute ${\rm{Lie}}(\pi_n)$; when is it surjective?

(iv) Assume $k$ is a field. 
Prove that the left exact sequence
of Lie algebras arising from 
a short exact sequence of finite type $k$-groups as above is short exact
if $G'$ is smooth.  Give a counterexample to short exactness on Lie algebras
with smooth $G$ and non-smooth $G'$.
\end{exer}




\begin{exer}\label{3.7}
Let $k$ be a ring, and let $1 \rightarrow H' \rightarrow E \rightarrow H'' \rightarrow 1$
be a short exact sequence of fppf group sheaves with  
$H''$ and $H'$ group schemes of multiplicative type.

(i) Using descent theory and the affineness of $H'$, prove that $E$ is affine
and fppf over $k$.

(ii) Assume $E$ is commutative. Prove that if $k$ is a field then $E$ is of multiplicative type.
What if $k$ is an arbitrary ring? (See Corollary \ref{fpqcwin}(2).) 

(iii) By considering the $E$-conjugation action on $H'$
and \'etaleness of the automorphism functor of $H'$, prove
that if $H''$ has connected fibers (e.g., a torus) then 
$H'$ is central in $E$ and in fact $E$ is commutative (hint: once
centrality is proved, show the commutator of $E$ factors through a bi-additive
pairing $H'' \times H'' \rightarrow H'$).  What can we then conclude via (ii)?
\end{exer}

\begin{exer}\label{maxfield}
Let $G$ be a smooth affine group over a field $k$.  Using fppf descent and the existence
of a (geometrically!) maximal torus over some finite extension, 
generalize Theorem \ref{schemetori} to apply to $G$ without restriction on
the Cartan subgroups of $G_{\overline{k}}$.  Deduce that $G$ admits such a torus
over a finite {\em separable} extension $k'/k$. 
\end{exer}

\newpage

\section{Roots, coroots, and semisimple-rank 1}

Let $G \rightarrow S$ be a reductive group scheme.  By Corollary \ref{buildtori}, 
at the cost of passing to an \'etale cover on $S$ we may arrange that
$G$ contains a maximal torus $T$. 
By deeper work with Cartan subalgebras of $\mathfrak{g}$ 
one can even make a maximal torus {\em Zariski-locally} on the original $S$
\cite[XIV, 3.20]{sga3}; we do not use this result (but see Exercise \ref{maxtb}(i)). 

Suppose for a moment that $T$ is split, so there is 
an isomorphism 
\index{ aaadsm@$D_S(M)$}
$$T \simeq D_S(M) := \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(M_S, \mathbf{G}_{\rm{m}})$$
for a finite free $\Z$-module $M$. 
In the evident manner, we get a map of groups 
$M \rightarrow \Hom_{S\mbox{-}{\rm{gp}}}(T,\mathbf{G}_{\rm{m}})$.
The $T$-action on $\mathfrak{g} = {\rm{Lie}}(G)$
then corresponds to an $\calO_S$-linear $M$-grading $\bigoplus_{m \in M} \mathfrak{g}_m$
of the vector bundle $\mathfrak{g}$, where
$t \in T$ acts on the subbundle $\mathfrak{g}_m$ 
via multiplication by the unit $m(t)$ (see \cite[III, 1.5]{oesterle} or \cite[Lemma A.8.8]{pred}).
The formation of  each subbundle $\mathfrak{g}_m$ in $\mathfrak{g}$ commutes with any base change on $S$.

By passing to geometric fibers and using the classical theory of
root spaces  for connected reductive groups, we
see that the locally constant rank of each vector bundle $\mathfrak{g}_m$ takes
values in $\{0, 1\}$ when $m \ne 0$ and that $\mathfrak{g}_0 = {\rm{Lie}}(T)$
(since ${\rm{Lie}}(T) \subset \mathfrak{g}_0$ as subbundles of $\mathfrak{g}$,
with equality on geometric fibers over $S$).
We will use these observations to develop a general theory
of root spaces and root groups, leading (in \S\ref{splitsec}) 
to both a complete classification in the split case when the geometric
fibers have semisimple-rank 1 as well as a deeper
understanding of the Weyl group scheme $W_G(T)$ in
the split case.

\subsection{Roots and the dynamic method}\label{dynsec}

When $T$ is split as above, any character $T \rightarrow \mathbf{G}_{\rm{m}}$ arises from
an element of $M$ Zariski-locally on $S$ since (i) $T \simeq \mathbf{G}_{\rm{m}}^r$ for
some $r \ge 0$, and (ii) any endomorphism
of $\mathbf{G}_{\rm{m}}$ over $S$ is given, Zariski-locally on $S$, by $t \mapsto t^n$ for some $n \in \Z$.
This leads to the construction of some subbundles of $\mathfrak{g}$:

\begin{definition}\label{rootdef} 
Let $S$ be a non-empty scheme. Assume that $G$ admits a split maximal torus $T$ over $S$
and fix an isomorphism $T \simeq D_S(M)$ 
for a finite free $\Z$-module $M$.  A {\em root} for $(G,T)$ is a nonzero element $a \in M$
such that $\mathfrak{g}_a$ is a line bundle.
We call such $\mathfrak{g}_a$ a {\em root space} for $(G,T,M)$.
\end{definition}
\index{root space}
\index{root}
\index{reductive group!root}

We may view roots as fiberwise nontrivial characters $T \rightarrow \Gm$
corresponding to {\em constant} sections of 
the \'etale sheaf $\underline{\Hom}_{S\mbox{-}{\rm{gp}}}(T,\Gm) = M_S$
that are fiberwise nonzero and induce roots in the classical sense
on geometric fibers.  In Exercise \ref{rootex} it is shown that any root
for $(G_{\overline{s}}, T_{\overline{s}})$ arises from
a root for $(G_U, T_U)$ in the above sense for some
Zariski-open neighborhood $U$ of $s$ in $S$.  

We will later 
study root systems arising from reductive group schemes equipped
with a split maximal torus, but for now we study a single
root  $a:T \rightarrow \mathbf{G}_{\rm{m}}$.   Since $a$ is fiberwise nontrivial, 
its  kernel $\ker a$ is $S$-flat by the fibral flatness criterion \cite[IV$_3$, 11.3.10]{ega}
and hence is an $S$-group
of multiplicative type (by Corollary \ref{moreflat}). 

\begin{example}\label{fakens}
In the relative theory, we encounter a new phenomenon that is never seen in the theory
over a field: the root spaces $\mathfrak{g}_a$ are line bundles on $S$ but they may be {\em nontrivial}
as such, even in the presence of a split maximal torus.  
The most concrete version is seen over a Dedekind domain $A$
with nontrivial class group:  if $J$ is a non-principal integral ideal of $A$
then for the rank-2 vector bundle $M = A \oplus J$ the
$A$-group ${\rm{GL}}(M)$ has generic fiber ${\rm{GL}}_2$
and its points valued in any $A$-algebra $R$ consist of matrices
in $(\begin{smallmatrix} R & J^{-1} \otimes_A R\\ J \otimes_A R & R \end{smallmatrix})$
with unit determinant.  This is a form of ${\rm{GL}}_2$ over $A$
whose root spaces relative to the split diagonal torus are $J^{\pm 1}$ inside  the standard
root spaces for $\mathfrak{gl}_2$ over the fraction field of $A$.

More generally, consider a nontrivial
line bundle $\mathscr{L}$ over a scheme $S$ (such as $\mathscr{L} = \mathscr{O}(1)$ on $S = \mathbf{P}^1_k$
for a field $k$).  Let $G = \GL(\mathscr{E})$ for
$\mathscr{E} = \mathscr{O} \bigoplus \mathscr{L}$.
This is a Zariski-twisted form of $\GL_2$ over $S$, and it contains
the split maximal torus $T = \Gm^2 = D_S(\Z^2)$ acting as ordinary unit scaling on both
$\mathscr{O}$ and $\mathscr{L}$.  In this case
the Lie algebra of $G$ is
$$\mathfrak{g} = \underline{\rm{End}}(\mathscr{E}) = 
\mathscr{E} \otimes \mathscr{E}^{\ast} = \mathscr{O}^{\oplus 2} 
\oplus \mathscr{L}^{-1} \oplus \mathscr{L},$$
where $\mathscr{O}^{\oplus 2} = {\rm{Lie}}(T)$ and the root spaces
$\mathfrak{g}_{\pm a}$ are the subbundles $\mathscr{L}^{\mp 1}$
(with roots $\pm a:T \rightarrow \Gm$ corresponding to
$(c_1,c_2) \mapsto (c_1/c_2)^{\pm 1}$). 
Since $\mathscr{L}$ a nontrivial line bundle, 
the root spaces are nontrival line bundles.
\end{example}

As in the classical theory, we describe characters of $T$ using additive notation,
so we write $-a$ rather than $1/a$, and $a+b$ rather than $ab$ (and 0 denotes the trivial element
of $\Hom_{S\mbox{-}{\rm{gp}}}(T,\mathbf{G}_{\rm{m}})$).

\begin{lemma}\label{ss1} Let $T = D_S(M)$ be a split maximal torus of 
a reductive group scheme $G$ over a non-empty scheme $S$. 
A  fiberwise nontrivial $a:T \rightarrow \mathbf{G}_{\rm{m}}$
is a root of $(G,T)$ if and only if $-a$ is,
in which case the common 
kernel $\ker a = \ker(-a)$ contains a unique subtorus $T_a = T_{-a}$
of relative codimension $1$ in $T$.

In such cases, the reductive centralizer $G_a := Z_G(T_a)$
has geometric fibers with
semisimple-rank $1$ and Lie algebra $\mathfrak{t} \bigoplus \mathfrak{g}_a \bigoplus \mathfrak{g}_{-a}$
inside $\mathfrak{g}$.
\end{lemma}

This is essentially \cite[XIX, 3.5]{sga3}.  Beware that our notation now deviates from the classical case,
with $G_a$ denoting $Z_G(T_a)$ whereas in the classical theory it denotes $\mathscr{D}(Z_G(T_a))$.
There is no serious risk of confusion because in the relative setting for smooth affine group
schemes there is no concept of ``derived
group'' in the same generality as over fields.  (This problem is overcome 
for reductive group schemes in Theorem \ref{dergp}ff.)

\begin{proof}
An element $a \in M$ is a root if and only if 
its negative $-a \in M$ is a root, by the theory on geometric fibers.
Suppose these are both roots (over $S$).  To prove the rest, consider 
the relative codimension-1 torus $T_a$ in $T$ corresponding to the maximal torsion-free
quotient of $M/(\Z a)$, so $T_a \subset \ker a$.
Its uniqueness as a relative codimension-1 torus
killed by $a$ is clear on (geometric) fibers by the classical theory, 
and so uniqueness holds over $S$ by the duality between tori and \'etale sheaves
of finite free $\Z$-modules on $S$.

The centralizer $Z_G(T_a)$ has Lie algebra $\mathfrak{g}^{T_a}$
that is a subbundle of $\mathfrak{g}$ whose formation commutes with any base change.
This subbundle contains the subbundle
$\mathfrak{t} \bigoplus \mathfrak{g}_a \bigoplus \mathfrak{g}_{-a}$ of $\mathfrak{g}$,
so to prove that the containment is an equality we may pass
to geometric fibers and use the classical theory. 
The semisimple-rank 1 property of the geometric fibers of $G_a$ over $S$ is likewise classical 
(and is evident
from the description of the Lie algebra).
\end{proof}

Now that we have built
root spaces $\mathfrak{g}_a$,
the next step is to build root groups.
First we review the classical perspective on root groups so we can 
see why it cannot be used when working over a base scheme $S$.  

Over an algebraically closed field $k$, if $a \in \Phi(G,T)$
and $T_a = (\ker a)_{\rm{red}}^0$ is the unique codimension-1
torus in $T$ killed by $a$, 
then the centralizer scheme $Z_G(T_a)$ is a connected reductive subgroup of $G$
with Lie algebra $\mathfrak{g}^{T_a}$ containing the nonzero $a$-weight space $\mathfrak{g}_a$.
Thus, the codimension-1 torus $T_a$ in $T$ is the maximal central torus in $Z_G(T_a)$,
so ${\rm{Lie}}(Z_G(T_a)) = \mathfrak{g}^{T_a}$
has as its $T$-roots precisely the nonzero $\Q$-multiples of $a$
in $\Phi(G,T) \subset {\rm{X}}(T)_{\Q}$ since
$a$ is a nontrivial character of 
$T/T_a \simeq \Gm$.  In particular, 
$\mathscr{D}(Z_G(T_a))$ is a semisimple group
having as a maximal torus the 1-dimensional isogeny complement 
$T' := (T \bigcap \mathscr{D}(Z_G(T_a)))_{\rm{red}}^0$
to $T_a$ in $T$.  By the semisimple rank-1 classification,
the group $\mathscr{D}(Z_G(T_a))$ is isomorphic to either
${\rm{SL}}_2$ or ${\rm{PGL}}_2$,
and the isomorphism can be chosen to carry $T'$ over to the diagonal torus.
In particular, the roots for $(G,T)$ that 
are $\Q$-multiples of $a$ are precisely $\pm a$, and 
(by composing with conjugation on ${\rm{SL}}_2$
or ${\rm{PGL}}_2$ if necessary) we obtain a central isogeny
$q_a:{\rm{SL}}_2 \rightarrow \mathscr{D}(Z_G(T_a))$
carrying the diagonal torus $D$ onto $T'$ and carrying the standard
upper triangular unipotent subgroup $U^{+}$ isomorphically onto
a subgroup $U_a \subset \mathscr{D}(Z_G(T_a))$
that is $k$-isomorphic to $\Add$ and normalized by $T$
with ${\rm{Lie}}(U_a) = \mathfrak{g}_a$. This $q_a$ is unique up to $D(k)$-conjugation.

A direct inspection of ${\rm{SL}}_2$ and ${\rm{PGL}}_2$
then shows that $U_a$ is uniquely determined by
these properties relative to $(G,T,a)$, and 
it is called the {\em root group} for $a$.
\index{root group}
The composition of $q_a$ with the standard parameterization $t \mapsto {\rm{diag}}(t,1/t)$
yields a unique cocharacter $a^{\vee}:\Gm \rightarrow T'$ satisfying $\langle a, a^{\vee} \rangle = 2$,
and this is called the {\em coroot} for $(G,T,a)$.
\index{coroot}
The definitions of root groups and coroots
in the classical theory 
rest on notions of  ``derived group'' and ``unipotent subgroup''
that are not (yet) available over a general base.
Also, in the classical theory the definition of a coroot rests on 
the semisimple-rank 1 classification.

To introduce root groups and coroots attached to roots
of a reductive group scheme over a general scheme $S$,
we require an entirely different construction technique. 
This will {\em not} give a new
approach in the classical case because in our proofs over $S$ 
we will appeal to the known theory of root groups and coroots on geometric fibers. 
The current absence  of a ``derived group'' for reductive $S$-groups 
(which will only become available in Theorem \ref{dergp}ff.)
also makes the statement of the split semisimple-rank 1 classification 
over $S$ more complicated than in the classical case,
as we do not yet have a way to ``split off'' the maximal central torus (up to isogeny) 
in the relative setting. 

In the relative theory, root groups will be instances of a general group construction
that we now explain. 
For any finite-rank vector bundle $\mathscr{E}$ on $S$, let
$\mathbf{W}(\mathscr{E}) \rightarrow S$ denote the associated additive $S$-group
whose set of $S'$-points is the additive group of global sections of
$\mathscr{E}_{S'}$ for any $S$-scheme $S'$.  Explicitly, 
double duality for $\mathscr{E}$ provides a canonical isomorphism 
\index{ aaawe@$\mathbf{W}(\mathscr{E})$}
$$\mathbf{W}(\mathscr{E}) = \Spec_{S}({\rm{Sym}}(\mathscr{E}^{\ast}))$$
as $S$-groups, and ${\rm{Lie}}(\mathbf{W}(\mathscr{E})) \simeq
\mathscr{E}$ as vector bundles on $S$ (with trivial Lie bracket)
respecting base change on $S$.  
The relative approach to root groups is:

\begin{theorem}\label{rootgp} Let $G \rightarrow S$ be a reductive group
scheme over a non-empty scheme $S$, 
$T \simeq D_S(M)$ a split maximal torus, and $a \in M$ a root.  Let $T$ act on
$\mathbf{W}(\mathfrak{g}_a)$ via $t.v = a(t)v$ using the vector bundle structure on
$\mathfrak{g}_a$.  There is a unique $S$-group homomorphism
\index{ aaaexpa@$\exp_a$}
\index{root group!existence}
$$\exp_a:\mathbf{W}(\mathfrak{g}_a) \rightarrow G$$
inducing the canonical 
inclusion $\mathfrak{g}_a \hookrightarrow \mathfrak{g}$ on Lie algebras and intertwining
the $T$-action on $G$ via conjugation and
the $T$-action on $\mathbf{W}(\mathfrak{g}_a)$ via $a$-scaling.

The map $\exp_a$ is also a closed immersion factoring through $Z_G(T_a)$, its formation commutes
with base change on $S$, and the multiplication map
\begin{equation}\label{wtw}
\mathbf{W}(\mathfrak{g}_{-a}) \times T \times \mathbf{W}(\mathfrak{g}_a) \rightarrow Z_G(T_a)
\end{equation}
defined by $(X',t,X) \mapsto \exp_{-a}(X')t\exp_a(X)$
is an isomorphism onto an open subscheme $\Omega_a \subset Z_G(T_a)$. 
Moreover, the semi-direct product of $T$ against each $\mathbf{W}(\mathfrak{g}_{\pm a})$
is a closed $S$-subgroup of $G$.
\end{theorem}

The closed subgroup $\exp_a(\mathbf{W}(\mathfrak{g}_a)) \subset G$ will
be called the {\em $a$-root group} for $(G,T,M)$.
\index{aroot group@$a$-root group}
The proof of Theorem \ref{rootgp} (to be given in \S\ref{corootsec}) uses a  ``dynamic method''
entirely different from the approach given in 
\cite[XX, 1.5--1.14]{sga3} 
(which rests on Hochschild cohomology, deformation theory, and fpqc descent). 
To be precise, the method in \cite[XX, \S1]{sga3} gives slightly less; 
the properties that the maps $\exp_{\pm a}$ are closed immersions
(rather than mere monomorphisms) and 
that the semi-direct products $T \ltimes \mathbf{W}(\mathfrak{g}_{\pm a})$
are closed $S$-subgroups of $G$ are not obtained until \cite[XX, 5.9]{sga3}.
The dynamic approach will yield these properties
immediately and provide a more streamlined route through
the semisimple-rank 1 classification over $S$.  Thus, we now digress
and introduce the dynamic method.

\index{dynamic method|(}
As motivation for what will follow, we first explain how to construct the standard upper triangular Borel
subgroup and its unipotent radical in ${\rm{SL}}_2$ over an algebraically closed
field 
by means of a $\Gm$-action on ${\rm{SL}}_2$ without ever needing
to say ``solvable subgroup'' or ``unipotent subgroup'' as in the classical theory. 
Rather generally, if $\Gm \times G \rightarrow G$ is an action of
$\Gm$ on a separated group scheme $G$ over a base scheme $S$,
for any $g \in G(S)$ we say {\em $\lim_{t \rightarrow 0} t.g$ exists}
if the orbit map $\Gm \rightarrow G$ defined by $t \mapsto t.g$
extends to an $S$-scheme morphism $\mathbf{A}^1_S \rightarrow G$;
such an extension is unique if it exists since $G$ is separated
and $k[x] \subset k[x, 1/x]$ for any ring $k$.
In such cases, the image of $0$ in $G(S)$ is called $\lim_{t \rightarrow 0} t.g$.
This limit concept has the following interesting application for ${\rm{SL}}_2$:

\begin{example}\label{sl2ex} Let $\lambda:\Gm \rightarrow G := {\rm{SL}}_2$ be 
$t \mapsto (\begin{smallmatrix} t & 0 \\ 0 & 1/t \end{smallmatrix})$.
Define a $\Gm$-action on $G$ as follows: 
for any $k$-algebra $k'$, $g = (\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}) \in G(k')$, 
and $t \in \Gm(k') = {k'}^{\times}$, let 
$$t.g = \lambda(t) \begin{pmatrix} a & b \\ c & d \end{pmatrix} \lambda(t)^{-1} = 
\begin{pmatrix} a & t^2 b \\ t^{-2} c & d \end{pmatrix}.$$
Thus, $\lim_{t \rightarrow 0} t.g$ exists if and only if $c = 0$ in $k'$, and
this limit exists and equals 1 if and only if $c = 0$ and $a = d = 1$ in $k'$.

In other words, the upper triangular subgroup $B \subset G$ represents
the functor of points $g$ of $G$ for which $\lim_{t \rightarrow 0} t.g$ exists,
and the strictly upper triangular subgroup $U$ represents the functor
of points $g$ of $G$ for which $\lim_{t \rightarrow 0} t.g$ exists and equals 1.
This gives a mechanism for constructing $B$ and $U$ entirely
in terms of $G$ and the $\Gm$-action on it, without reference
to notions such as solvability or unipotence that are well-suited to working
over a field but are 
not available (at least not in a useful manner) when the base is 
anything more complicated than a field (such as a discrete valuation ring
or non-reduced artin local ring).

Note that if we replace $\lambda$ with its reciprocal 1-parameter
subgroup $-\lambda:t \mapsto \lambda(1/t)$ then the analogous
limiting process above recovers  the  lower triangular subgroup $B'$
and its strictly lower triangular subgroup $U'$.
\end{example}

The subgroup constructions in the preceding example can be carried out more generally.
In \cite[Ex.\,2.1.1]{pred} the case $G  = {\rm{GL}}_n$ is worked out over any ring $k$
when using a diagonal 1-parameter subgroup
$$\lambda(t) = {\rm{diag}}(t^{e_1}, \dots, t^{e_n})$$
for integers $e_1 \ge \dots \ge e_n$.  In such cases
one gets various ``parabolic'' subgroups and their ``unipotent radicals''
(keep in mind that the base ring $k$ may not be a field, hence the quotation marks), all 
depending on the $e_j$'s. 

A general setting for these limit considerations uses
a separated $S$-scheme $X$ equipped with a left action by the $S$-group $\Gm$
(only the case of $S$-affine $X$ will be used below).
For an $S$-scheme $S'$ and $x \in X(S')$, it makes sense to ask if the orbit map
$\Gm \rightarrow X_{S'}$
over $S'$ defined by $t \mapsto t.x$ extends to an $S'$-map
$\mathbf{A}^1_{S'} \rightarrow X_{S'}$.  If such an extension exists then it is unique
(because $X$ is separated and a closed subscheme of
$\mathbf{A}^1_{S'}$ that contains $\Gm$ must be the entire affine line);
we then say that ``$\lim_{t \rightarrow 0} t.x$ exists'' and denote the image
of $0 \in \mathbf{A}^1(S')$ in $X(S')$ as this limit.
A fundamental source of such examples arises as follows.

\begin{definition}
For a ring $k$ and an affine $k$-group $G$, a {\em $1$-parameter subgroup}
of $G$ is a $k$-homomorphism $\lambda:\Gm \rightarrow G$.  (We allow
that $\ker \lambda \ne 1$, and even that $\lambda = 1$, though the latter
option is not very useful.) 
\end{definition}

Any 1-parameter subgroup defines a $\Gm$-action on $G$ via $t.g = \lambda(t)g\lambda(t)^{-1}$.
 By using 
a ``weight space'' decomposition of the coordinate ring $k[G]$, a variant of the above
procedures in ${\rm{SL}}_2$ and $\GL_n$ can be carried out
for 1-parameter subgroups $\lambda:\Gm \rightarrow G$
of rather general affine groups $G$ {\em without} requiring the crutch of a
${\rm{GL}}_n$-embedding of $G$ (which is {\em not} known to exist locally
on the base in general, even for smooth affine groups over the dual numbers over a field;
cf.\,\cite[XI, 4.3, 4.6]{sga3}). 
It will also be extremely useful to consider an abstract action of $\Gm$ on $G$, not
only actions arising from conjugation against a 1-parameter subgroup, so 
in our formulation below of a vast generalization of Example \ref{sl2ex} we
treat abstract $\Gm$-actions.  The following
``dynamic'' result summarizes the main conclusions in \cite[2.1]{pred}.  It is the key to our 
approach to root groups in reductive group schemes.

\begin{theorem}\label{215}
Let $G$ be a finitely presented affine group over a ring $k$,
and consider an action $\lambda:\Gm \times G \rightarrow G$ by the $k$-group 
$\Gm$ on the $k$-group $G$.  Consider the following subfunctors of $G$
on the category of $k$-algebras: 
\begin{eqnarray*}
\underline{P}_G(\lambda)(k') &=& \{g \in G(k')\,|\,\lim_{t \rightarrow 0} \lambda(t,g) \mbox{ {\rm{exists}}}\}, \\
\underline{U}_G(\lambda)(k') &=& \{g \in \underline{P}_G(\lambda)(k')\,|\,\lim_{t \rightarrow 0} \lambda(t,g) = 1\}.
\end{eqnarray*}
Likewise, let $\underline{Z}_G(\lambda)$ be the subfunctor of points of $G$ that
commute with the $\Gm$-action $\lambda$.
\begin{enumerate}
\item These functors are unaffected by
replacing $\lambda$ with $\lambda^n$ for $n > 0$,
and they are represented by respective finitely presented closed subgroups
$P_G(\lambda)$, $U_G(\lambda)$, and $Z_G(\lambda)$ of $G$,
with $U_G(\lambda)$ normalized by $Z_G(\lambda)$.
\index{ aaapgl@$P_G(\lambda)$}
\index{ aaazgl@$Z_G(\lambda)$}
\index{ aaaugl@$U_G(\lambda)$}
\item The fibers of $U_G(\lambda) \rightarrow S$ are connected, 
and so are the fibers of $P_G(\lambda)$ and $Z_G(\lambda)$ if $G \rightarrow S$ has connected
fibers.
\item The multiplication map $Z_G(\lambda) \ltimes U_G(\lambda) \rightarrow P_G(\lambda)$
is an isomorphism.
\item Assume $G$ is smooth.  The subgroups $P_G(\lambda)$, $U_G(\lambda)$, and $Z_G(\lambda)$
are smooth and the multiplication map
$$U_G(-\lambda) \times P_G(\lambda) \rightarrow G$$
is an open immersion.  Writing $\bigoplus_{n \in \Z} \mathfrak{g}_n$
for the weight space decomposition of $\mathfrak{g}$ under
the $\Gm$-action $($so $\mathfrak{g}_n = \{v \in \mathfrak{g}\,|\,t.v = t^n v\}$$)$, 
the Lie algebras of these subgroups are
$$\mathfrak{z}_G(\lambda) = \mathfrak{g}_0 = \mathfrak{g}^{\Gm},\,\,\,
\mathfrak{u}_G(\lambda) = \mathfrak{g}_{+} := \bigoplus_{n > 0} \mathfrak{g}_n,\,\,\,
\mathfrak{p}_G(\lambda) = \mathfrak{z}_G(\lambda) \oplus \mathfrak{u}_G(\lambda).$$
Also, the fibers of $U_G(\lambda) \rightarrow S$ are unipotent.
\end{enumerate}
\end{theorem}

In terms of the theory of connected reductive groups over an algebraically closed field,
part (3) is analogous to a Levi decomposition of a parabolic subgroup 
and part (4) is analogous to
an open Bruhat cell.   When we apply 
Theorem \ref{215} to examples in which the action arises from conjugation against
a 1-parameter subgroup $\lambda:\Gm \rightarrow G$, we shall denote the
resulting closed subgroups as $P_G(\lambda)$, $U_G(\lambda)$, and $Z_G(\lambda)$.

\begin{proof}
Using the semi-direct product group $G' = G \rtimes \Gm$ defined by the given action,
the evident 1-parameter subgroup $t \mapsto (1,t)$ reduces the general case
to the special case that $\lambda$ arises
from the conjugation action against a 1-parameter subgroup.
This reduction step is explained in \cite[Rem.\,2.1.11]{pred}, so we now may
and do assume the action is conjugation against a 1-parameter subgroup,
also denoted $\lambda:\Gm \rightarrow G$.
The $\Gm$-action on $G$ yields a $\Gm$-action on
the $k$-module $k[G]$, which in turn corresponds to a $k$-linear $\Z$-grading $\oplus_{n \in \Z} k[G]_n$ of $k[G]$,
where $f(t.g) = t^n f(g)$ for $f \in k[G]_n$ (see \cite[(2.1.2)]{pred}).

It is clear from the definitions that the three subfunctors of $G$ under consideration are
subgroup functors.  Their invariance
 under passage to $\lambda^n$ with $n > 0$
is elementary; see Exercise \ref{dynex}(i) (or \cite[Rem.\,2.1.7]{pred}).
The existence of $P_G(\lambda)$ as a closed subscheme of $G$ is 
a special case of \cite[Lemma 2.1.4]{pred} (defining the closed subscheme 
by the ideal of $k[G]$ generated by the negative
weight spaces $k[G]_n$ for $n < 0$ 
relative to the $\Gm$-action on $k[G]$ through $\lambda$;
this ideal is typically larger than the $k$-linear
span of the negative weight spaces).
Existence of $Z_G(\lambda)$ and $U_G(\lambda)$ is given in
\cite[Lemma 2.1.5]{pred}, where the finite presentation property is also established
(by reduction to the case of noetherian $k$).
Explicitly, $Z_G(\lambda) = P_G(\lambda) \bigcap P_G(-\lambda)$
and $U_G(\lambda)$ is the fiber over 1 for the limit morphism
$P_G(\lambda) \rightarrow G$ defined by $g \mapsto \lim_{t \rightarrow 0} t.g$.
This settles (1).  (The proof of the existence of $P_G(\lambda)$, $Z_G(\lambda)$, and $U_G(\lambda)$
as closed subgroup schemes does not require $G$ to be finitely presented over $k$.)

Part (3) is \cite[Prop.\,2.1.8(2)]{pred},  and part (4) apart from the Lie algebra and unipotence
assertions is \cite[Prop.\,2.1.8(3)]{pred}
(which is the hardest part of the proof).  The description of the Lie algebras
in (4) is \cite[Prop.\,2.1.8(1)]{pred} (proved by a functorial calculation
with dual numbers).  The fibral unipotence for $U_G(\lambda)$ in (4) 
is part of \cite[Lemma 2.1.5]{pred} (which does not require smoothness of $G$,
once one has developed a good theory of unipotent group schemes over
a field; see \cite[XVII, 1.3, 2.1]{sga3}). 
Part (2) is \cite[Prop.\,2.1.8(4)]{pred}; the idea for proving $U_G(\lambda)$ is 
fiberwise connected is that the limiting process provides paths
$t \mapsto t.g$ linking all points $g$ of $U_G(\lambda)$ to 1, and 
for $P_G(\lambda)$ and $Z_G(\lambda)$ the fibral connectedness 
in the case of smooth $G$ follows from part (4) whereas in the general case for (2) 
it requires further work that is specific to groups over fields (e.g., the existence
of a ${\rm{GL}}_n$-embedding).
\end{proof}

\begin{example}\label{slnex}
For $G = {\rm{SL}}_n$ over any ring $k$
and $\lambda(t) = {\rm{diag}}(t^{e_1}, \dots, t^{e_n})$ for a strictly decreasing
sequence of integers $e_1 > \dots > e_n$, 
$P_G(\lambda)$ is the standard upper triangular $k$-subgroup,
$U_G(\lambda)$ is its $k$-subgroup of strictly upper triangular matrices,
and $Z_G(\lambda)$ is the $k$-subgroup of diagonal elements.   Passing
to $-\lambda$ yields the lower triangular analogues, exactly as in Example \ref{sl2ex}.
If the $e_j$'s are pairwise distinct  but not strictly monotone then
$P_G(\lambda)$ is the conjugate of the upper triangular subgroup by
a suitable permutation matrix (corresponding to rearranging the $e_j$'s to be strictly decreasing).
\end{example}

\begin{example}\label{parred}
For any connected reductive group $G$ over an algebraically
closed field $k$ and any maximal torus $T$ in $G$,
as $\lambda$ varies through the cocharacters of $T$
the resulting smooth connected subgroups $P_G(\lambda)$ of
$G$ containing $T$ are precisely the parabolic
subgroups of $G$ that contain $T$.
We refer the reader to \cite[Prop.\,2.2.9]{pred} for a proof.

Note that $P_G(\lambda) = Z_G(\lambda) \ltimes U_G(\lambda)$
with $U_G(\lambda)$ a smooth connected unipotent normal subgroup
and $Z_G(\lambda)$ the centralizer of the torus $\lambda(\Gm)$ in $G$
(so $Z_G(\lambda)$ is connected reductive).  Thus, 
 $U_G(\lambda)$ is the unipotent radical of
$P_G(\lambda)$ and $Z_G(\lambda)$ is a Levi subgroup of $P_G(\lambda)$.
\end{example}

It is immediate from the definitions that the formation of $P_G(\lambda)$, $U_G(\lambda)$,
and $Z_G(\lambda)$ commutes with any base change on $k$, and that
Theorem \ref{215} adapts to work over any base scheme $S$ (not just affine schemes).  Here are some
easy but very useful ``functorial'' properties of these subgroups.

\begin{proposition}\label{functor}
Let $(G, \lambda)$ be as in Theorem $\ref{215}$.
\begin{enumerate}
\item 
For any finitely presented closed subgroup $H$ of $G$ that is
stable under the $\Gm$-action, with the restricted action on $H$ also denoted as $\lambda$,
$$H \bigcap P_G(\lambda) = P_H(\lambda),\,\,\,
H \bigcap U_G(\lambda) = U_H(\lambda),\,\,\,
H \bigcap Z_G(\lambda) = Z_H(\lambda).$$
In particular, if $H$ and $G$ are smooth then $H \bigcap P_G(\lambda)$ is smooth
and likewise with $U_G(\lambda)$ and $Z_G(\lambda)$.
\item Let $(G',\lambda')$ be another such pair over $k$, and 
$f:G \rightarrow G'$ a $\Gm$-equivariant map. Then $f$ carries $P_G(\lambda)$ into
$P_{G'}(\lambda')$, $U_G(\lambda)$ into $U_{G'}(\lambda')$, and $Z_G(\lambda)$ into $Z_{G'}(\lambda')$.
When $f$ is flat and surjective and $G \rightarrow \Spec k$ has connected
fibers then the maps
\begin{equation}\label{puz}
P_G(\lambda) \rightarrow P_{G'}(\lambda'),\,\,\,
U_G(\lambda) \rightarrow U_{G'}(\lambda'),\,\,\,
Z_G(\lambda) \rightarrow Z_{G'}(\lambda')
\end{equation}
are surjections that are flat  between fibers over $\Spec k$.
If $f$ is a flat surjection and $G$ is $k$-smooth  with connected fibers
then the surjections 
in $(\ref{puz})$ are flat.
\end{enumerate}
\end{proposition}

\begin{proof}
The assertion in (1) is immediate from the functorial definitions of these subgroups.
The functoriality in (2) is likewise obvious, and for the surjectivity 
assertion it suffices to check on fibers over $\Spec k$.  The surjectivity problem 
when $k$ is a field is \cite[Cor.\,2.1.9]{pred}, which also gives the flatness of the surjections 
in such cases.
(Strictly speaking, \cite[Cor.\,2.1.9]{pred} considers
the case of conjugation actions against 1-parameter subgroups.  The general
case reduces to this; see \cite[Rem.\,2.1.11]{pred}.)  

Suppose $f$ is a flat surjection and $G$ is $k$-smooth, so 
the finitely presented $k$-group $G'$ in (2) is smooth as well \cite[IV$_4$, 17.7.7]{ega}. All of the subgroups of interest
in (\ref{puz})  are smooth, 
by Theorem \ref{215}(4).  Hence, in such cases the flatness of the induced surjective maps
arising from $f$ can be checked
on fibers over $\Spec k$, for which we have already noted that the flatness holds.
\end{proof}
\index{dynamic method|)}


\subsection{Root groups and coroots}\label{corootsec}

For a split maximal torus $T = D_S(M)$ in
a reductive group scheme $G \rightarrow S$ over a non-empty scheme $S$,
now we use dynamic constructions
to build ``root groups'' $U_{\pm a}$ in $G_a = Z_G(T_a) = G_{-a}$ for any root $a$ of $(G,T)$
(with $T_a = T_{-a}$ as in Lemma \ref{ss1}).
Before we prove Theorem \ref{rootgp}, we work out what it is saying for ${\rm{SL}}_2$ over any ring.

\begin{example}\label{expex} We saw in Example \ref{sl2ex} that for the standard
1-parameter subgroup $\lambda(t) = {\rm{diag}}(t, 1/t)$ in $G = {\rm{SL}}_2$, the resulting subgroups 
$U(\lambda)$ and $U(-\lambda)$ are respectively the strictly upper and strictly lower triangular
subgroups, corresponding to the roots $\pm a:T \rightrightarrows \Gm$
satisfying $a(\lambda(t)) = t^2$ and $(-a)(\lambda(t)) = t^{-2}$.   The proof
of Theorem \ref{rootgp} will show that these subgroups are respectively the images of
$\exp_a$ and $\exp_{-a}$ from Theorem \ref{rootgp}.  Via
the standard trivializations of $\mathfrak{g}_{\pm a}$ given by
$(\begin{smallmatrix} 0 & 1 \\ 0 & 0\end{smallmatrix})$
and $(\begin{smallmatrix} 0 & 0 \\ 1 & 0\end{smallmatrix})$ respectively
(to identify $\mathbf{W}(\mathfrak{g}_{\pm a})$ with $\mathbf{W}(\mathscr{O}_S) = \Add$), 
we have $\exp_a(z) = (\begin{smallmatrix} 1 & z \\ 0 & 1\end{smallmatrix})$
and $\exp_{-a}(z) = (\begin{smallmatrix} 1 & 0 \\ z  & 1\end{smallmatrix})$.
Note that the ``exponential'' terminology is reasonable: 
the nilpotent matrices $n_{+}(z) = (\begin{smallmatrix} 0 & z \\ 0 & 0 \end{smallmatrix})$
and $n_{-}(z) = (\begin{smallmatrix} 0 & 0 \\ z & 0 \end{smallmatrix})$
satisfy $n_{\pm}(z)^2 = 0$, so we
imagine that $e^{n_{\pm}(z)}$ should mean $1 + n_{\pm}(z) = \exp_{\pm a}(z)$.
\end{example}

Here is the dynamic proof of Theorem \ref{rootgp}.

\begin{proof}
By $T$-equivariance, since $T_a$ acts trivially on $\mathbf{W}(\mathfrak{g}_a)$
it follows that if $\exp_a$ is to exist then it must factor through the reductive subgroup $Z_G(T_a)$.
We may therefore replace $G$ with $Z_G(T_a)$, so by Lemma \ref{ss1}
we are reduced to the case that $G$ has geometric fibers of semisimple-rank 1
with $T_a$ central in $G$ and $\mathfrak{g} = \mathfrak{t} \bigoplus \mathfrak{g}_a \bigoplus 
\mathfrak{g}_{-a}$.

By the asserted uniqueness, compatibility with base change
will be automatic and we may work \'etale-locally on $S$ to prove the theorem. 
Choose 
a cocharacter $\lambda:\mathbf{G}_{\rm{m}} \rightarrow
T$ corresponding to an element of the dual lattice $M^{\vee}$
such that the pairing $\langle a, \lambda \rangle \in \Z$ (corresponding
to the element $a \circ \lambda \in {\rm{End}}_{S\mbox{-}{\rm{gp}}}(\mathbf{G}_{\rm{m}}) = \Z_S(S)$
that is a constant section)
lies in $\Z_{>0}$. 
Note that it may be impossible to arrange that this pairing is 1, since
$a$ may be divisible by 2 in the character group of $T$
(as happens for the long roots of ${\rm{Sp}}_{2n}$). 

The inclusion $T \subset Z_G(\lambda)$ of smooth closed subgroups of $G$ is an equality.
Indeed, it suffices to prove equality on geometric fibers over $S$, both of which are connected, 
so it is enough to compare their Lie algebras inside $\mathfrak{g} = 
\mathfrak{t} \bigoplus \mathfrak{g}_a \bigoplus \mathfrak{g}_{-a}$.
Since $\langle \pm a, \lambda \rangle \ne 0$, the description of
${\rm{Lie}}(Z_G(\lambda))$ in Theorem \ref{215}(4) implies
that this Lie algebra must coincide with $\mathfrak{t}$, as desired.

Using the indicated $T$-action on $\mathbf{W}(\mathfrak{g}_a)$, composing with $\lambda$
defines a $\Gm$-action on $\mathbf{W}(\mathfrak{g}_a)$, namely a point $c$ of $\Gm$ acts 
via multiplication by $a(\lambda(c)) = c^{\langle a, \lambda \rangle}$.  Since
$\langle a, \lambda \rangle > 0$, it follows that for $H := \mathbf{W}(\mathfrak{g}_a)$
we have $H = U_H(\lambda)$.  The $T$-equivariance requirement on $\exp_a$ implies
that (if it exists) it must be $\Gm$-equivariant, so it must carry $H$ into
$U_G(\lambda)$.  By Theorem \ref{215}(4), ${\rm{Lie}}(U_G(\lambda))$
is the ``positive'' weight space for the $\Gm$-action on $\mathfrak{g} = \mathfrak{t} \bigoplus
\mathfrak{g}_a \bigoplus \mathfrak{g}_{-a}$.  Since $\Gm$ acts on $\mathfrak{t}$ trivially
and on $\mathfrak{g}_{\pm a}$ via $t.v = t^{\langle \pm a, \lambda \rangle}v$ 
with $\langle a, \lambda \rangle > 0$, we conclude that
${\rm{Lie}}(U_G(\lambda)) = \mathfrak{g}_a$.  
Hence, if $\exp_a$ is to exist  then it must factor through an $S$-homomorphism
$\mathbf{W}(\mathfrak{g}_a) \rightarrow U_G(\lambda)$
that induces an {\em isomorphism} on Lie algebras.   But
$U_G(\lambda)$ is $S$-smooth with connected fibers, so if $\exp_a$ exists
then it must be an \'etale homomorphism {\em onto} $U_G(\lambda)$.

We can do better: such an \'etale map $\mathbf{W}(\mathfrak{g}_a) \rightarrow U_G(\lambda)$
must be an isomorphism.  Indeed, it suffices to check the isomorphism
property on geometric fibers over $S$, and both sides have geometric
fiber $\Add$ equipped with a $\Gm$-action 
inducing the {\em same} scaling action $t.v = t^{\langle a, \lambda \rangle} v$
on the Lie algebra.  But the only $\Gm$-actions on $\Add$ over
a field are $t.x = t^n x$ for $n \in \Z$, and the effect on the Lie algebra detects $n$ by the same formula.
In other words, the induced map between geometric fibers (if $\exp_a$ exists) 
must be an \'etale endomorphism of $\Add$ that is equivariant for
the action $t.x = t^{\langle a, \lambda \rangle}x$, and it is easy to check that over 
a field $F$ the only such endomorphisms are $x \mapsto cx$ for $c \in F^{\times}$,
which are visibly isomorphisms.

To summarize, if $\exp_a$ is to exist then it must be a $\Gm$-equivariant
isomorphism $\mathbf{W}(\mathfrak{g}_a) \simeq U_G(\lambda)$ (so in particular
it must be a closed immersion into $G$).  Note conversely that any such $\Gm$-equivariant
isomorphism is $T$-equivariant as a map to $G$ since $T_a$ acts trivially on both sides
and $\Gm \times T_a \rightarrow T$ defined by $(c,t) \mapsto \lambda(c)t$ is an isogeny
of tori (ensuring that $T$-equivariance is equivalent to the combination 
of $T_a$-equivariance and $\Gm$-equivariance).  Thus, for
the existence and uniqueness of $\exp_a$ it is necessary and sufficient
to prove the existence and uniqueness of a $\Gm$-equivariant isomorphism of $S$-groups
$\mathbf{W}(\mathfrak{g}_a) \simeq U_G(\lambda)$ that induces the identity map 
on Lie algebras.  Once we have $\exp_{\pm a}$ in hand, 
the open immersion claim in the theorem is immediate from Theorem \ref{215}(4)
since $T = Z_G(\lambda)$ and necessarily $\exp_{\pm a}$ carries
$\mathbf{W}(\mathfrak{g}_{\pm a})$ over to $U_G(\pm \lambda)$. 
This also gives that each $T \ltimes \mathbf{W}(\mathfrak{g}_{\pm a})$ is a closed
$S$-subgroup of 
$G$, as each is identified with $Z_G(\pm \lambda) \ltimes U_G(\pm \lambda) = P_G(\pm \lambda)$.

The uniqueness of $\exp_a$ amounts to the assertion that
$\mathbf{W}(\mathfrak{g}_a)$ has no nontrivial automorphism that is
$\Gm$-equivariant and induces the identity on the Lie algebra.  Working Zariski-locally
so that $\mathfrak{g}_a$ admits a trivialization as a line bundle, this is the assertion that
$\Add$ over $S$ admits no nontrivial automorphism that is equivariant for
the $\Gm$-action 
$t.x = t^{\langle a, \lambda \rangle} x$ and induces the identity on ${\rm{Lie}}(\Add)$.
An endomorphism of $\Add$ over a ring $k$ is precisely an additive polynomial, and
equivariance for $t.x = t^n x$ with $n \ne 0$ says precisely that the polynomial
is $x \mapsto cx$ for some $c \in k$, so the effect on the Lie algebra is multiplication by $c$.
Thus, the identity condition on the Lie algebra forces $c = 1$, as desired.

It remains to prove the existence of $\exp_a$.  The smooth $S$-group $U_G(\lambda)$ has
fibers that are connected and unipotent (see Theorem \ref{215}(2),(4))
and of dimension 1 (as the Lie algebra is the line bundle $\mathfrak{g}_a$),
so the geometric fibers of $U_G(\lambda)$ are $\Add$.  Beware
that it is not obvious that even the actual fibers of $U_G(\lambda)$
over $S$ are isomorphic to $\Add$ (let alone that this holds Zariski-locally
over $S$); the classification of forms of $\Add$ is rather subtle, even over (imperfect) fields,
because the automorphism functor
of $\Add$ is quite bad in positive characteristic (see \cite{russell}).  The key to bypassing such difficulties is
the $\Gm$-action on $U_G(\lambda)$, as we now explain.  

The uniqueness of $\exp_a$ allows us to work \'etale-locally to prove its existence, 
so we may assume that the smooth surjection $U_G(\lambda) \rightarrow S$ admits a section
$\sigma$ disjoint from the identity section.  In this case, we have:

\begin{lemma}\label{russ} The $S$-group $U_G(\lambda)$ is isomorphic to $\Add$,
via an isomorphism 
carrying its $\Gm$-action over to $t.x = t^{\langle a, \lambda \rangle} x$.
\end{lemma}

Granting this lemma, let us conclude the argument.  Fix such an isomorphism
of $S$-groups.
This induces a canonical basis of ${\rm{Lie}}(U_G(\lambda)) = \mathfrak{g}_a$,
which in turn identifies $\mathbf{W}(\mathfrak{g}_a)$ with $\Add$
carrying the $\Gm$-action over to $t.x = t^{\langle a, \lambda \rangle} x$.
Thus, visibly $\mathbf{W}(\mathfrak{g}_a)$ and $U_G(\lambda)$ are
$\Gm$-equivariantly isomorphic as $S$-groups (namely, isomorphic to $\Add$ with the indicated
$\Gm$-action).   Pick one such isomorphism, so its effect
on the Lie algebra is multiplication on $\mathfrak{g}_a$ by some global unit.
Scaling on $\mathbf{W}(\mathfrak{g}_a)$ by the reciprocal of that unit then provides
the desired $\exp_a$.

It remains to prove Lemma \ref{russ}.
By the functorial definition of $U_G(\lambda)$, the orbit map
$\Gm \rightarrow U_G(\lambda)$ defined by $t \mapsto t.\sigma$
extends to an $S$-scheme map
$$q:\mathbf{A}^1_S \rightarrow U_G(\lambda)$$
that carries 1 to $\sigma$ and is $\Gm$-equivariant
when using the {\em usual} $\Gm$-scaling action on $\mathbf{A}^1_S$ 
(as it suffices to check such equivariance on the open $\Gm$ inside 
the affine $S$-line).  Note also that $q(0) = 1$ by 
the definition of $U_G(\lambda)$. 
On fibers over a geometric point $\overline{s}$ of $S$, 
we may identify the group $U_G(\lambda)_{\overline{s}}$ with $\Add$ carrying 
the point $\sigma(\overline{s})$ over to 1,
and the $\Gm$-action on $U_G(\lambda)_{\overline{s}}$ goes over to scaling
on $\Add$ by $t^n$ for some $n \in \Z$.  Inspecting
${\rm{Lie}}(U_G(\lambda)) = \mathfrak{g}_a$ shows $n = \langle a, \lambda \rangle$.  Thus, 
$q_{\overline{s}}$ is identified with an endomorphism of the affine line over $k(\overline{s})$
that satisfies $q_{\overline{s}}(t) = t^{\langle a, \lambda \rangle}$ for
$t \in \Gm$ and hence for $t \in \mathbf{A}^1$.  

 Letting $n = \langle a, \lambda \rangle > 0$, we claim that $\mu_n$ acts trivially
 on $U_G(\lambda)$.  Indeed, the centralizer
 subgroup scheme $U_G(\lambda)^{\mu_n}$ is a {\em smooth}
 closed subgroup of $U_G(\lambda)$ (as $\mu_n$ is multiplicative type), 
 so this subgroup equals $U_G(\lambda)$ if and only if
 it does so on fibers over $S$.  This reduces the $\mu_n$-triviality claim on $U_G(\lambda)$
 to the case of geometric fibers, where it is clear from our concrete
 description ($U_G(\lambda)_{\overline{s}} = \Add$
 with $\Gm$-action $t.x = t^n x$).  
 
  It follows from the $\Gm$-equivariance of
the map $q$ that it  is invariant under the 
 natural $\mu_n$-action on the affine line.
The $n$th-power endomorphism of the affine $S$-line
is a categorical quotient by the $\mu_n$-action
in the category of $S$-affine schemes, so $q$ factors through 
an $S$-scheme map 
$$\overline{q}:\mathbf{A}^1_S \rightarrow U_G(\lambda)$$
carrying 0 to 1 that is $\Gm$-equivariant when using the action $t.x = t^n x$ on the affine line
(and the conjugation action on $U_G(\lambda)$ via $\lambda$).
On geometric fibers over $S$, our earlier calculations
with each $q_{\overline{s}}$ imply that each
$\overline{q}_{\overline{s}}$ is identified with an {\em automorphism} of
the affine line over $k(\overline{s})$ (as a scheme), so $\overline{q}$ is an isomorphism of $S$-schemes
by Lemma \ref{flatclosed}.
Thus, it remains to prove that $\overline{q}$ is an $S$-homomorphism.

We have seen above that the $\Gm$-action on $U_G(\lambda)$ makes $\mu_n$ act
trivially, and the same holds for the $\Gm$-action on the affine line over $S$ that is
the source for $q$.
Thus, the domain of $\overline{q}$ inherits
an action by the quotient $\Gm/\mu_n \simeq \Gm$
that makes the $S$-scheme isomorphism 
$\overline{q}$ identify $U_G(\lambda)$ with an $S$-group structure
on $\mathbf{A}^1_S$ that has 0 as the identity and is equivariant for the ordinary $\Gm$-scaling.
The $S$-homomorphism property for $\overline{q}$ is reduced
to checking that addition is the only such group law on the affine $S$-line.   

We may assume
$S = \Spec k$ for a ring $k$, so the abstract group law transferred from
$U_G(\lambda)$ via $\overline{q}$ is an $m \in k[x,y]$ satisfying
$m(x,0) = x$, $m(0,y) = y$, and $m(tx,ty) = tm(x,y)$ for $t \in \Gm$. 
It is clear by homogeneity considerations in $t$ that the final condition forces
$m = cx + c'y$ for some $c, c' \in k$, so the first two conditions imply $m = x+y$.
\end{proof}

\begin{definition}\label{rootgpdef}
The image $\exp_{a}(\mathbf{W}(\mathfrak{g}_{a})) \subset G$ is denoted $U_{a}$
and is called the {\em $a$-root group}.  
\end{definition}
\index{ aaaua@$U_a$}
\index{aroot group@$a$-root group}

Note that  if the line bundle $\mathfrak{g}_a$ is
trivialized (as may be done Zariski-locally on $S$) 
then $U_a$ is identified with the additive group $\mathbf{G}_{\rm{a}}$ over $S$.

\begin{remark}\label{basis} The existence of the $T$-equivariant
$\exp_a$ implies that to give an $S$-group isomorphism
$p_a:\Add \simeq U_a$
\index{ aaapa@$p_a$}
 intertwining the $T$-action on $U_a$ and the action $t.x = a(t)x$ on $\Add$ 
is precisely the same as to choose a global trivializing section $X$ of the line bundle
$\mathfrak{g}_a$ (via $p_a(z) = \exp_a(zX)$).  This follows from the faithful flatness of $a:T \rightarrow \Gm$ 
and the easy fact that the only automorphisms of the $S$-group $\Add$ that 
are equivariant for the standard $\Gm$-action are scaling by global units of $S$.
Such isomorphisms $p_a$ are called {\em parameterizations} of the root group $U_a$.
\end{remark}
\index{root group!parameterization}


With root spaces $\mathfrak{g}_a$
and root groups $U_a$ now constructed
for roots $a$ arising from triples $(G,T,M)$
over any non-empty scheme $S$, 
the remaining ingredient before
we can discuss the split semisimple-rank 1 classification over $S$ 
is the definition of coroots in the relative setting.
As we have already noted, this requires a viewpoint rather different from
the classical case over a field.  Indeed, in the classical
case coroots are defined using the classification of
semisimple groups of rank 1 (over an algebraically closed field), whereas in 
the relative setting everything gets turned upside down:
we need coroots even to state the split semisimple-rank 1 classification
over a base scheme.

In the classical theory over an algebraically closed field, it makes
sense to consider the closed subgroup 
$\langle U_a, U_{-a} \rangle$ of $G$ generated by a pair of ``opposite'' root groups, 
and one shows that this is either ${\rm{SL}}_2$ or ${\rm{PGL}}_2$.
But in the theory over rings it is unclear in what generality it makes sense to form a (smooth closed)
subgroup ``generated'' by a pair of smooth closed subgroups of a smooth affine group.
Likewise, we cannot use the alternative description 
$\mathscr{D}(Z_G(T_a))$ of $\langle U_a, U_{-a} \rangle$ since it is unclear in what generality the notion of
``derived group'' makes sense for smooth closed subgroups of
a smooth affine group over a ring. The key to our success
over $S$ is to simultaneously characterize 
the coroot $a^{\vee}:\Gm \rightarrow Z_G(T_a)$
and compatible trivializations of $\mathfrak{g}_a$ and $\mathfrak{g}_{-a}$
in intrinsic terms.  
For inspiration, once again we turn to the case of ${\rm{SL}}_2$:


\begin{example}\label{precede}
Let $G = {\rm{SL}}_2$ over a non-empty scheme $S$ and let $D$ be the diagonal split maximal torus 
$D_S(\Z)$ equipped with the standard positive root $a:D \simeq \Gm$ 
given by $\lambda(t) \mapsto t^2$ where 
$\lambda:t \mapsto {\rm{diag}}(t, 1/t)$.  This yields the standard ``open cell''
$$\Omega_a = U_{-a} \times D \times U_a = U_G(-\lambda) \times Z_G(\lambda) \times U_G(\lambda),$$
and likewise there is the other ``open cell''
$$\Omega_{-a} = U_a \times D \times U_{-a} \subset G$$
(with the groups $U_{\pm a}$ appearing in opposite
order for the multiplication).

Consider the product $U_a \times U_{-a}$ in $\Omega_{-a}$.  Its points are
$$\exp_a(z) \exp_{-a}(z') = \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 0 \\ z' & 1 \end{pmatrix} = \begin{pmatrix} 1 + zz' & z \\ z' &  1 \end{pmatrix}.$$
When does such a point lie in the other open cell $\Omega_a = U_{-a} \times D \times U_a$?
The points of $\Omega_a$ are those of the form
$$\begin{pmatrix} 1 & 0 \\ u & 1 \end{pmatrix} \begin{pmatrix} c & 0 \\ 0 & c^{-1} \end{pmatrix} 
\begin{pmatrix} 1 & u' \\  0 & 1 \end{pmatrix} = \begin{pmatrix} c  &  cu'  \\ cu & c^{-1} + cuu'  \end{pmatrix},$$
which are precisely the points of ${\rm{SL}}_2$ whose upper left entry is a unit. 
Hence, $\exp_a(z) \exp_{-a}(z')$ lies in $\Omega_a$ if and only if $1 + zz'$ is a unit
(with $z, z'$ arbitrary points of $\Add$), 
in which case $c = 1 + zz'$, $u = z'/(1 + zz')$, and $u' = z/(1 + zz')$.

Varying over points of ${\rm{SL}}_2$ valued in all $S$-schemes, 
the duality between $\mathfrak{g}_a$ and $\mathfrak{g}_{-a}$
given by multiplication in $\mathscr{O}_S$ characterizes
when $\exp_a(z)\exp_{-a}(z')$ lies in $\Omega_a$ via
the condition ``$1 + zz' \in \Gm$''.
Likewise, since 
$\exp_a(z) \exp_{-a}(z')$ equals 
$$\begin{pmatrix} 1 & 0 \\ z'/(1+zz') & 1 \end{pmatrix}
\begin{pmatrix} 1 + zz' & 0 \\ 0 & (1 +zz')^{-1} \end{pmatrix}
\begin{pmatrix} 1 & z/(1+zz') \\ 0 & 1 \end{pmatrix},$$
we see that the coroot $a^{\vee}(t) := {\rm{diag}}(t,1/t)$ is recovered
by noting that the $D$-component of $\exp_a(z)\exp_{-a}(z')$
in $\Omega_a$ is exactly $a^{\vee}(1 + zz')$.
\end{example}

The calculations with ${\rm{SL}}_2$ in Example \ref{precede} motivate:

\begin{theorem}\label{ssclass} Let $G$ be a reductive group over a scheme $S$, 
with fibers of semisimple-rank $1$.  Assume there exists  a split
maximal torus $T = D_S(M)$ in $G$
and a root $a:T \rightarrow \Gm$ arising from $M$.  
Let $U_{\pm a}$ and $\Omega_a := U_{-a} \times T \times U_a$
and $\Omega_{-a} := U_a \times T \times U_{-a}$ be the associated root groups and ``open cells'' 
in $G$ as in Theorem $\ref{rootgp}$.

There is a unique pair $(\beta_a, a^{\vee})$ consisting of an $\mathscr{O}_S$-bilinear $($hence $\Gm$-equivariant$)$ 
pairing of line bundles 
\index{coroot!existence, uniqueness}
\index{ aaaav@$a^{\vee}$}
$\beta_a:\mathfrak{g}_a \times \mathfrak{g}_{-a} \rightarrow \mathscr{O}_S$
$($denoted $(X,Y) \mapsto XY$$)$ and  an $S$-homomorphism
$a^{\vee}:\Gm \rightarrow T$ such that the following 
conditions hold:
\begin{enumerate}
\item for any $S$-scheme $S'$
and points $\exp_a(X) \in U_a(S')$ and $\exp_{-a}(Y) \in U_{-a}(S')$, the $S'$-valued point
$$\exp_a(X) \exp_{-a}(Y) \in \Omega_{-a} \subset G$$
lies in the ``open cell'' $\Omega_a$ if and only if $1 + XY$ is a unit on $S'$.
\item when this unit condition is satisfied, 
\begin{equation}\label{cellid}
\exp_a(X) \exp_{-a}(Y) = \exp_{-a}\left(\frac{Y}{1+XY}\right) a^{\vee}(1 + XY) \exp_a\left(\frac{X}{1+XY}\right)
\in \Omega_a.
\end{equation}
\end{enumerate}
In particular, the formation of this bilinear pairing and $a^{\vee}$ commute with base change on $S$.

Moreover, the pairing $(X,Y) \mapsto XY$ is a 
perfect duality, and $a \circ a^{\vee} = 2$
$($i.e., $a(a^{\vee}(c)) = c^2$ for $c \in \Gm$$)$.
\end{theorem}

This result is \cite[XX, 2.1]{sga3}, whose proof there involves elaborate calculations. 
Our proof of Theorem \ref{ssclass} will be long, but it involves very few calculations
and yields some auxiliary results that
rapidly lead to a Zariski-local version of the classification
of ``split'' reductive groups of semisimple-rank 1 
(in Theorem \ref{sslist}).  First, we make some observations.  

An interesting consequence of the duality in Theorem \ref{ssclass} is that the line bundle 
$\mathfrak{g}_a$ is globally trivial if and only if $\mathfrak{g}_{-a}$ is,
so likewise $U_a$ admits a parameterization in the sense of Remark \ref{basis} if and only if 
$U_{-a}$ does.  When such parameterizations $p_{\pm a}:\Add \simeq U_{\pm a}$
exist, we say (following \cite[XX, 2.6.1]{sga3}) that they are {\em linked} if they correspond to
dual bases for $\mathfrak{g}_a$ and $\mathfrak{g}_{-a}$; such dual bases are called
{\em linked trivializations}.   
\index{linked trivialization}
(An alternative convention, advocated by Demazure in more recent times, is to declare bases
of $\mathfrak{g}_a$ and $\mathfrak{g}_{-a}$ to be linked when 
they are negative dual to each other. This has the advantage
that the open subscheme $\Omega_a \bigcap \Omega_{-a}$
in $\Omega_a$ is defined by $XY \ne 1$ rather than $XY \ne -1$.
It thereby eliminates signs in certain equations.) 

Clearly 
for a given parameterization of $U_a$, there exists a unique
one of $U_{-a}$ to which it is linked. 
Note also that necessarily $(-a)^{\vee} = -a^{\vee}$.  Indeed, this is a known
fact in the classical theory, and in general it can be deduced from geometric fibers
over $S$. 


\begin{proof}[Proof of Theorem $\ref{ssclass}$] 
We may and do assume $S$ is non-empty.  
To prove uniqueness, we first note that 
$Z_G = \ker a$ by Corollary \ref{computez}(1). 
Concretely, inside  $T = D_S(M)$ we have $Z_G = \ker a = D_S(M/\Z a)$.
(Note that $M/\Z a$ may not be torsion-free.)

The quotient $G/Z_G$ is a reductive group scheme
in which $T/Z_G = D_S(\Z a)$ is a maximal torus
such that the induced character $\overline{a}:T/Z_G \rightarrow \Gm$ is an isomorphism.
In particular, since $\ker \overline{a} = 1$, it follows
that $G/Z_G$ has trivial schematic center (Corollary \ref{qtcenter}).
Moreover, the behavior of the open immersion (\ref{wtw}) under
$Z_G$-scaling shows that (i) the natural maps
$\mathbf{W}(\mathfrak{g}_{\pm a}) \rightrightarrows G/Z_G$ 
are isomorphisms onto closed subgroups $V_{\pm \overline{a}}$
normalized by $T/Z_G$ and (ii) the adjoint action of
$G/Z_G$ on its Lie algebra makes $T/Z_G$ have ${\rm{Lie}}(V_{\pm \overline{a}})$
as a weight space for the character $\pm \overline{a}$.  In other words, 
the quotient map
$G \rightarrow G/Z_G$ induces an isomorphism $U_{\pm a} \simeq U_{\pm \overline{a}}$
and identifies the ``open cell'' $\Omega_{\pm\overline{a}}$ as $\Omega_{\pm a}/Z_G$.

We conclude that to construct the bilinear pairing between $\mathfrak{g}_a$
and $\mathfrak{g}_{-a}$ that characterizes
the points of $U_a \times U_{-a}$  whose product in $G$ lies in the open subscheme 
$\Omega_a = U_{-a} \times T \times U_a$, it is harmless to pass to $G/Z_G$.  Likewise, for the proof
of uniqueness it is harmless to pass to $G/Z_G$ provided
that we can settle uniqueness in general over an algebraically closed
field, since any $S$-homomorphism $\Gm \rightarrow T$ is uniquely
determined by its effect on geometric fibers over $S$.  Thus, for the proof
of uniqueness it suffices to treat two cases:  $S = \Spec(k)$ for
an algebraically closed field $k$, and $Z_G  = 1$ over a general $S$.

Consider the situation over an algebraically closed field $k$. In this case
we know that $G/Z_G = {\rm{PGL}}_2$ by the classical theory, and by conjugacy of maximal tori
we can choose this identification to carry the maximal torus $T/Z_G$
over to the diagonal torus $\overline{D}$ parameterized by
$\overline{\lambda}: t \mapsto {\rm{diag}}(t,1)$.  Applying conjugation by a representative of the nontrivial
element of $W_{{\rm{PGL}}_2}(\overline{D})$ (such as 
the standard Weyl element) if necessary, we can also arrange
that $\overline{a}$ goes over to the unique root for $\overline{D}$ that satisfies 
$\overline{a}(\overline{\lambda}(t)) = t$.
In this case, existence is settled by using the calculations in Example \ref{precede}
 and composing with the degree-2 central isogeny
${\rm{SL}}_2 \rightarrow {\rm{PGL}}_2$ (e.g., we take $\overline{a}^{\vee}$
to be the composition of $t \mapsto {\rm{diag}}(t, 1/t) \in {\rm{SL}}_2$
with the central
 isogeny to ${\rm{PGL}}_2$, which is to say $\overline{a}^{\vee}(t) = {\rm{diag}}(t^2,1)$).
These calculations also imply uniqueness of the pairing of root spaces,
since we can pass
to $G/Z_G = {\rm{PGL}}_2$
and observe that any possibility for the bilinear pairing must be
a multiple of the standard one by some $c \in k$, yet the unit conditions
on $1+xy$  and $1 + cxy$ for varying $x, y \in k$ do not coincide
unless $c = 1$ since $k$ is an algebraically closed field. 

The bilinear pairing between root spaces is uniquely determined
(over the algebraically closed field $k$) 
by composing the formula (\ref{cellid}) over $k$ with projection to $G/Z_G$, so 
any possibility for the coroot $a^{\vee}:\Gm \rightarrow T$ over $k$ 
has composition with $T \rightarrow T/Z_G = \overline{D} \subset {\rm{PGL}}_2$ given by
$c \mapsto {\rm{diag}}(c,1/c) = {\rm{diag}}(c^2,1) \bmod \Gm$.  Thus, 
$a^{\vee}$ is unique up to multiplication against a cocharacter $\mu:\Gm \rightarrow Z_G$.
But any such cocharacter $\mu$ factors through the torus $(Z_G)^0_{\rm{red}}$
that is the maximal central torus in $G$, and this has finite intersection
with the connected semisimple $\mathscr{D}(G)$.  Since
(\ref{cellid}) over $k$ forces $a^{\vee}$ to be valued in $\langle U_a, U_{-a} \rangle = \mathscr{D}(G)$,
it follows that $\mu$ is trivial, so $a^{\vee}$ is also unique.
This completes the proof of existence and uniqueness over an algebraically closed
field, and in such cases the additional properties (perfectness of the bilinear pairing,
and the identity $a \circ a^{\vee} = 2$) are immediate from these calculations
(since $a$ factors through $T/Z_G$). 

Returning to the situation over a general (non-empty) base $S$, the results
over an algebraically closed field imply uniqueness of the coroot in general,
as well as perfectness of the bilinear pairing (if it exists) and the identity
$a \circ a^{\vee} = 2$ (as the latter concerns an endomorphism of
$\Gm$ and so can be checked on geometric fibers). We will next prove 
uniqueness in general, and then it will remain to address existence.  
As we have already noted, for the proof 
of uniqueness (of the bilinear pairing, as the case of the coroot is settled)
we may and do assume $Z_G = 1$.  In this case
$G$ has all geometric fibers isomorphic to ${\rm{PGL}}_2$
by the classical theory,
so the roots $\pm a:T \rightarrow \Gm$ are isomorphisms
on geometric fibers over $S$ and hence are isomorphisms over $S$.  Thus, we can apply:

\begin{proposition}\label{pgl2} Let $G \rightarrow S$ be a reductive group with trivial center
and geometric fibers of semisimple-rank $1$.
If there exists a split maximal torus $T \subset G$ then Zariski-locally
on $S$ there exists a group isomorphism
$G \simeq {\rm{PGL}}_2$.  This isomorphism
may be chosen to carry $T$ 
over to the diagonal torus.
\end{proposition}

The Zariski-local nature of this result could be improved to a unique
global isomorphism at the cost of using a relative notion of pinning and carrying out some 
preliminary arguments with the relative notion of Borel 
subgroup to prove that ${\rm{PGL}}_2$ is its own automorphism functor.
We postpone such considerations until we treat
the general Existence and Isomorphism Theorems, as Zariski-local
results will be entirely sufficient for our present purposes.

\begin{proof}
The split property of $T$ provides a weight space decomposition of
the rank-3 vector bundle $\mathfrak{g}$, and by working Zariski-locally on $S$
we may arrange that there exists a root $a:T \simeq \Gm$.
Let $\lambda:\Gm \simeq T$ be the inverse of $a$, so
$U_G(\pm \lambda) = U_{\pm a}$. 
Let $B = P_G(\lambda)$.  By Exercise \ref{dynex}(i) and the
${\rm{PGL}}_2$-variant of Example \ref{sl2ex}, on
geometric fibers this is a Borel subgroup.  
Proposition \ref{conntrans} provides
a closed normalizer subscheme $N_G(B) \subset G$.
We have not shown this normalizer to be flat,  but we claim more:
the closed immersion $B \hookrightarrow N_G(B)$ inside $G$ is an equality.
Since $B$ is flat, by Lemma \ref{flatclosed} it suffices to prove
equality on geometric fibers.  

Now consider the situation
over an algebraically closed field $k$. An elementary calculation 
with ${\rm{PGL}}_2$ over $k$ shows that $B$ and $N_G(B)$ have the same $k$-points,
so it suffices to show that $N_G(B)$ has the same Lie algebra as $B$ inside
of $\mathfrak{pgl}_2$.  By dimension considerations, this is
just a matter of ruling out the possibility that ${\rm{Lie}}(N_G(B)) = \mathfrak{pgl}_2$.
But $\mathfrak{b}$ is an ideal in ${\rm{Lie}}(N_G(B))$
and it is clearly not an ideal in $\mathfrak{pgl}_2$.  

We conclude that 
$B = N_G(B)$ as $S$-subgroups of $G$, so by 
Theorem \ref{normqt} the quotient sheaf $G/B$ is a smooth proper $S$-scheme
admitting a canonical $S$-ample line bundle. 
The formation of $G/B$ commutes
with any base change, such as passage to geometric
fibers over $S$, so these fibers are identified with
the quotient scheme of ${\rm{PGL}}_2$ modulo a Borel subgroup
over an algebraically closed field,
which is to say that the geometric fibers
$(G/B)_{\overline{s}}$ are isomorphic to $\mathbf{P}^1$.  That is, $G/B \rightarrow S$
is a smooth proper curve with connected geometric fibers of genus 0.  Moreover,
the identity section of $G \rightarrow S$ provides a section $\sigma$ to $G/B \rightarrow S$.

By standard arguments with cohomology
and base change (applied to the direct image on $S$ of the inverse
of the ideal sheaf of $\sigma$ on $G/B$, after reducing to the
case of noetherian $S$), 
Zariski-locally on $S$ there exists an isomorphism $G/B \simeq \mathbf{P}^1$
carrying the section $\sigma$  over to $\infty$.
Hence, we may assume $G/B \simeq \mathbf{P}^1_S$
carrying $1 \bmod B$ to $\infty$.  In particular,
the automorphism functor $\underline{\rm{Aut}}_{(G/B)/S}$ of $G/B$ on the category of $S$-schemes
is represented by ${\rm{PGL}}_2$ (Exercise \ref{1.1}(iv)) with the stabilizer of
$1 \bmod B$ going over to the stabilizer of $\infty \in \mathbf{P}^1$,
which is to say the standard upper triangular subgroup $B_{\infty}$ of ${\rm{PGL}}_2$.  

The left translation action of $G$ on $G/B$
defines an $S$-homomorphism
$$G \rightarrow \underline{\rm{Aut}}_{(G/B)/S} = {\rm{PGL}}_2.$$
On geometric fibers this is an isomorphism by the classical theory, so
it is an isomorphism of $S$-groups (Lemma \ref{flatclosed}).  Points of $B$ are carried into
$B_{\infty}$, and the resulting map $B \rightarrow B_{\infty}$ is an isomorphism
since it is so on geometric fibers over $S$.   Thus, the torus $T \subset B$ is
carried over to a maximal torus of ${\rm{PGL}}_2$ contained
in $B_{\infty}$.

It remains to prove that any maximal torus $T$ of ${\rm{PGL}}_2$ over $S$ that is contained
in $B_{\infty}$ can be conjugated to the diagonal torus $D = \Gm$ Zariski-locally on $S$.
Consider the smooth transporter scheme $Y = {\rm{Transp}}_{B_{\infty}}(T,D)$ over
$S$. All fibers $Y_s$ are non-empty, and $Y$ is stable under
left multiplication by $D$ in ${\rm{PGL}}_2$. We claim that this makes $Y$ a left $D$-torsor for the \'etale
topology.  Since the smooth surjection 
$Y \rightarrow S$ admits sections \'etale-locally on $S$, the torsor assertion
is equivalent to the condition that the map
$$D \times_S Y \rightarrow Y \times_S Y$$
defined by $(d, y) \mapsto (d.y, y)$ is an isomorphism.  By the smoothness
of both sides it is sufficient to check the isomorphism property on fibers over geometric
points $\overline{s}$ of $S$.
But $T_{\overline{s}}$ is $B_{\infty}(\overline{s})$-conjugate to 
$D_{\overline{s}}$ by the classical theory, so 
$Y_{\overline{s}}$ is a torsor for the {\em smooth} normalizer scheme
$N_{B_{\infty}}(D)_{\overline{s}}$ that is equal to $D_{\overline{s}}$ (via computation on geometric points).
Since $D = \Gm$, and every $\Gm$-torsor for the \'etale topology
is also a torsor for the Zariski topology (by descent theory for line bundles), 
it follows that $Y \rightarrow S$ admits sections Zariski-locally over $S$,
so the desired $B_{\infty}$-conjugation of $T$ into $D$ exists Zariski-locally on $S$.
\end{proof}


By Proposition \ref{pgl2}, for the proof of uniqueness in Theorem \ref{ssclass} 
we may assume $G = {\rm{PGL}}_2$ with $T$ the diagonal torus $\overline{D}$
parameterized by $\lambda:\Gm \simeq \overline{D}$ via $\lambda(t) = {\rm{diag}}(t,1)$.
Any root $\overline{a}:\overline{D} \rightarrow \Gm$ must be inverse to one of $\pm \lambda$
Zariski-locally on $S$, as it suffices to check this on geometric fibers
(where it follows from the classical theory).  Thus, by working Zariski-locally on $S$
and composing with conjugation by the standard Weyl element
$(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix})$ if necessary,
we can arrange that $\overline{a}$ is inverse to $\lambda$.  It follows that 
$\langle \overline{a}, \lambda \rangle = 1 > 0$, so the root group 
$U_{\overline{a}} = U_{{\rm{PGL}}_2}(\lambda)$ is the strictly upper triangular subgroup of
${\rm{PGL}}_2$ and the root group $U_{-\overline{a}}$ is the strictly lower triangular subgroup.  If
we use the standard bases of the Lie algebras of these subgroups of
${\rm{PGL}}_2$ then 
$\exp_{\pm \overline{a}}$ are the standard parameterizations of 
$U_{\pm \overline{a}}$ (see Example \ref{expex}, using 
the root ${\rm{diag}}(c,1) \mapsto c$
and composition with the central isogeny ${\rm{SL}}_2 \rightarrow {\rm{PGL}}_2$).
Thus, the calculations in ${\rm{SL}}_2$ in Example \ref{precede}
show that the standard duality between the root spaces (using
their standard bases) and the 1-parameter subgroup
$\overline{a}^{\vee}(c) = {\rm{diag}}(c^2,1) = {\rm{diag}}(c,1/c) \bmod \Gm$
satisfy the requirements.  

We have just proved existence for Theorem \ref{ssclass}
over a general base $S$ when $Z_G = 1$, and the argument gives uniqueness in such cases too.
Indeed, any possibility for the bilinear pairing must be $(X,Y) \mapsto cXY$
for some global unit $c$ on $S$, and the equivalence of the unit conditions
on $1+XY$ and $1 + cXY$ on all $S$-schemes forces $c = 1 + \xi$ for some
nilpotent $\xi$ on $S$.  Then the requirement (\ref{cellid}) 
(applied to the modified pairing $(X,Y) \mapsto cXY$) and the analogous established formula 
using the standard pairing and standard coroot force $\xi = 0$
(because $U_{-a} \times T \times U_a \simeq \Omega_a$) 
and force any possibility for the coroot to agree with the standard coroot
on any unit of the form $1 + xy$ with functions $x$ and $y$
on varying $S$-schemes.  Any unit can be expressed in this form (take $y = 1$),
so uniqueness is established in general when $Z_G = 1$.
But we already noted above that uniqueness over a general base when $Z_G = 1$
implies uniqueness in general (without restriction on $Z_G$), since
uniqueness is already known for the coroot (due to the case of an algebraically closed
ground field, which has been completely settled).

Finally, it remains prove existence without assuming $Z_G = 1$.  
%Thanks to the settled uniqueness in general, we may work
%Zariski-locally on the base so that $S = \Spec k$ for a ring $k$.
%As has already been explained, the bilinear pairing whose existence
%has been established for $G/Z_G$ does provide the expected characterization for 
%where $\Omega_a$ meets $U_a \times U_{-a} \subset \Omega_{-a}$,
%so the direct product decomposition of
%$\Omega_a$ does give an $S$-scheme map 
%$$f:\{(X,Y) \in \mathbf{A}^2_S\,|\, 1 + XY \in \Gm\} = \Spec k[X,Y]_{1+XY} \rightarrow T$$
%making (\ref{cellid}) hold when using $f(X,Y)$ in the $T$-factor on the right side.
%Applying $T$-conjugation to both sides of
%(\ref{cellid}) implies $f(a(t)X,a(t)^{-1}Y) = f(X,Y)$ for $t \in \Gm$
%whenever $1+XY \in \Gm$.  Since $f = h/(1+XY)^r$ for some $r \ge 0$ and some
%$h \in k[X,Y]$, necessarily $h$ is a polynomial in $XY$, or equivalently in $1+XY$.
%Thus, we can define an $S$-scheme map $a^{\vee}:\Gm \rightarrow T$ by the requirement
%$a^{\vee}(1+XY) = f(X,Y)$. 
%
%  In other words, 
%(\ref{cellid}) is proved except for the property that $a^{\vee}$ is 
% an $S$-homomorphism.
%Even without this homomorphism property, such a map $a^{\vee}$ is uniquely determined,
%so to prove the homomorphism property we may assume $S$ is noetherian,
%and then artin local (by the Krull intersection theorem), even with algebraically closed residue field
%(by faithfully flat base change \cite[0$_{\rm{III}}$, 10.3.1]{ega}).  We wish to reduce this problem
%to a situation for which Proposition \ref{sl2ext} can be applied to make
%everything explicit.  To that end, we now carry out some considerations
%with central extensions.
%%
%%If $G$ is a central quotient
%%$\widetilde{G}/Z$  where $\widetilde{G}$ is reductive of semisimple-rank 1
%%and $Z$ is central of multiplicative type in $\widetilde{G}$
%%then $T = \widetilde{T}/Z$ for a maximal torus $\widetilde{T}$ in $\widetilde{G}$
%%(Corollary \ref{qtcenter}), and $\widetilde{T}$ is split
%%since $S$ is artin local with algebraically closed residue field. 
%%For any cocharacter $\lambda:\Gm \rightarrow T$ satisfying
%%$\langle a, \lambda \rangle \in \Z_{>0}$ we have 
%%$U_a = U_G(\lambda)$, so replacing
%%$\lambda$ with $\lambda^n$ for some $n > 0$ is
%%harmless.  Some such $\lambda^n$ lifts to a cocharacter
%%$\widetilde{\lambda}$ of $\widetilde{T}$, so the root 
%%$a$ for $(G,T)$ uniquely lifts to a root $\widetilde{a}:\widetilde{T} \rightarrow \Gm$
%%for $(\widetilde{G}, \widetilde{T})$ (and its root group is $U_{\widetilde{G}}(\widetilde{\lambda})$).
%%In this way, we get $\Omega_a = \Omega_{\widetilde{a}}/Z$, so
%%$a^{\vee} = \widetilde{a}^{\vee} \bmod Z$, and hence it suffices 
%%to solve the problem for $(\widetilde{G}, \widetilde{T})$.
%
%%To apply these abstract considerations with central extensions, 
%
By the settled general uniqueness, we may work Zariski-locally on $S$
for existence.  Thus, we can arrange that $G/Z_G$ 
and its split maximal torus $T/Z_G$ admit an isomorphism
$$(G/Z_G, T/Z_G) \simeq ({\rm{PGL}}_2, \overline{D})$$
by Proposition \ref{pgl2}.  Consider the pullback diagram
$$\xymatrix{
1 \ar[r] & Z_G \ar@{=}[d] \ar[r] & {\widetilde{G}} \ar[r] \ar[d] & {{\rm{SL}}_2} \ar[d] \ar[r] & 1\\
1 \ar[r] & Z_G \ar[r] & G \ar[r] & {{\rm{PGL}}_2} \ar[r] & 1}$$
in which the top row is a central extension. 
By Proposition \ref{sl2ext} below, the top row uniquely splits,
so we get an isomorphism $\widetilde{G} \simeq {\rm{SL}}_2 \times Z_G$
in which the preimage $\widetilde{T}$ of $T$ goes over to $D \times Z_G$
(since the diagonal torus $D \subset {\rm{SL}}_2$ is the full preimage of the diagonal torus
$\overline{D} \subset {\rm{PGL}}_2$).
Thus, we get a cocharacter 
$$a^{\vee}:\Gm = D \rightarrow T$$
via the identification $\mu:\Gm \simeq D$ defined by $\mu:t \mapsto {\rm{diag}}(t,1/t)$.
%We shall prove that $b = a^{\vee}$, so $a^{\vee}$ is indeed a homomorphism.

Composing $a^{\vee}$ with $T \rightarrow T/Z_G = \overline{D}$ yields 
$\overline{\lambda}:t \mapsto {\rm{diag}}(t,1/t) \bmod \Gm = {\rm{diag}}(t^2,1)  = \lambda(t^2) \in {\rm{PGL}}_2$.
Thus, $$\langle a,  a^{\vee} \rangle = \langle \overline{a}, a^{\vee} \bmod Z_G
\rangle = \langle \overline{a}, \overline{\lambda} \rangle = \langle \overline{a}, 2 \lambda \rangle = 2 > 0,$$
so the standard root groups $U^{\pm} = U_{{\rm{SL}}_2}(\pm \mu)$ in ${\rm{SL}}_2$ map
into $U_G(\pm \overline{\mu}) = U_{\pm a}$ via $\widetilde{G} \rightarrow G$ 
and these maps $U^{\pm} \rightarrow U_{\pm a}$ are isomorphisms
on geometric fibers for smoothness, dimension, and unipotence reasons
because $\ker(\widetilde{G} \rightarrow G) = \mu_2$.  Hence, the maps 
$U^{\pm} \rightarrow U_{\pm a}$ are isomorphisms.
In this way, the induced map
$$U^{-} \times (D \times Z_G) \times U^{+} \rightarrow U_{-a} \times T \times U_a = \Omega_a$$
is the quotient of translation by the central $D[2] = \mu_2$, so 
it computes the full preimage of $\Omega_a$ in $\widetilde{G}$; the same holds
with the roles of $-a$ and $a$ swapped. 
Thus, our explicit knowledge of
the standard coroot and bilinear pairing of root spaces for
$({\rm{SL}}_2, D)$ (as in Example \ref{sl2ex})
imply that  $a^{\vee}:\Gm \rightarrow T$ and the bilinear
pairing constructed between $\mathfrak{g}_a$ and $\mathfrak{g}_{-a}$
satisfy the desired requirements that uniquely characterize the coroot
and bilinear pairing of root spaces.  This completes
the proof of Theorem \ref{ssclass} conditional on the (unique) splitting of central
extensions of ${\rm{SL}}_2$, provided by Proposition \ref{sl2ext} below.
\end{proof}


\subsection{Central extensions of ${\rm{SL}}_2$}\label{gabbersec}

\index{central extensions|(}
This section is devoted to proving a general splitting result
for central extensions of ${\rm{SL}}_2$ (needed to complete the proof of
Theorem \ref{ssclass}) and recording an ${\rm{SL}}_2$-variant of Proposition \ref{pgl2}.
We begin with the result on central extensions:

\begin{proposition}[Gabber]\label{sl2ext}
Let $S$ be a scheme, and $Z$ a commutative separated $S$-group.
Any fppf central extension of group sheaves
$$1 \rightarrow Z \rightarrow G' \rightarrow {\rm{SL}}_2 \rightarrow 1$$
is uniquely split as a central extension.
$($In particular, $G'$ is an $S$-scheme.$)$ 
\end{proposition}

We only need the case that $G'$ is an $S$-affine $S$-group of finite presentation
(possibly not smooth!) with $Z$ of multiplicative type.
I am grateful to Gabber for proving the result in the generality above
(my original proof via deformation theory was only for $Z$ of multiplicative type).

\begin{proof}
In $G := {\rm{SL}}_2$, define the usual parameterizations $h(t) = {\rm{diag}}(t,1/t)$ of the
 diagonal torus $D$
and $x(u) = (\begin{smallmatrix} 1 & u \\ 0 & 1 \end{smallmatrix})$
and $y(v) = (\begin{smallmatrix} 1 & 0 \\ v & 1 \end{smallmatrix})$
for the strictly upper triangular subgroup $U$ and strictly lower triangular subgroup $V$
respectively.  We first address the uniqueness of the splitting, which is to say 
(due to the centrality of $Z$ in $G'$) the
vanishing of any $S$-homomorphism $G = {\rm{SL}}_2 \rightarrow Z$.
Since
$h(t)x(u)h(t)^{-1}x(u)^{-1} = x((t^2-1)u)$
and $h(t)y(v)h(t)^{-1}y(v)^{-1} = y((t^{-2}-1)v)$,
and fppf-locally there is a unique $t$ such that $t^{\pm 2} - 1$ are units, any homomorphism $f:G \rightarrow Z$
to an $S$-separated commutative target must kill the subgroups $U$ and $V$.
But for $g(t) := y(-1/t)x(t)y(-1/t)$ (with $t \in \Gm$) the standard formula
$g(t)g(1)^{-1} = h(t)$ implies that $f$ also kills $D$, so $\ker f$ contains
the open cell $\Omega := U \times D \times V \subset G$.  Thus, 
using relative schematic density  \cite[IV$_3$, 11.10.10]{ega}
we reduce to the case over a field, where clearly $f = 1$.  This proves the uniqueness.

To build a splitting of the given central extension, 
let $D', U', V' \subset G'$ respectively denote the preimages of $D, U, V \subset G$, so
each is a central extension by $Z$ of its image in $G$.  A key point
is to verify that $D'$ is commutative. As for any central extension
of one commutative group object by another, 
the commutator of $D'$ factors through a bi-additive pairing
$$D \times D \rightarrow Z$$
whose vanishing is equivalent to the commutativity of $D'$
(see Exercise \ref{commpair}).
Hence, it suffices to show there is no nontrivial bi-additive pairing 
$\Gm \times \Gm \rightarrow Z$ into a separated commutative $S$-group.
The collection of  subgroups $\{\mu_n\}$ in $\Gm$ is relatively schematically
dense over $S$, so via $S$-separatedness of $Z$
it suffices to prove any bi-additive $\mu_n \times \Gm \rightarrow Z$ vanishes.
But $[n]:\Gm \rightarrow \Gm$ is an epimorphism of sheaves, so the vanishing is clear.

Next, we use commutativity of $D'$ to prove commutativity of $U'$ and $V'$.
By symmetry, it suffices to treat $U'$.  Note that $D'$ normalizes $U'$,
and the $G'$-action on $G'$ by conjugation factors through an action by the central quotient
$G = G'/Z$, so we get a natural action by $D$ on $U'$.  The bi-additive pairing
$$c:U \times U \rightarrow Z$$
induced by the commutator on $U'$ is clearly $D$-invariant
in the sense that $c(h.u_1,h.u_2) = c(u_1,u_2)$ for all $h \in D = \Gm$ and
$u_1, u_2 \in U = \Add$.  That is, for all $t \in \Gm$ and $u_1, u_2 \in \Add$ we have
$c(tu_1,tu_2) = c(u_1,u_2)$.  Equivalently, $c(tu_1,u_2) = c(u_1,t^{-1}u_2)$.
Consider fppf-local units $t$ such that $t+1$ is a unit and 
$t' := (t+1)^{-1} - t^{-1} - 1$ is a unit.
Bi-additivity of $c$ gives 
\begin{eqnarray*}
c(u_1,t'u_2) &=& c(u_1,(t+1)^{-1}u_2)c(u_1,t^{-1}u_2)^{-1}c(u_1,u_2)^{-1} \\ &=&
c((t+1)u_1,u_2)c(tu_1,u_2)^{-1}c(u_1,u_2)^{-1} \\
&=& c((t+1)u_1 -  tu_1 - u_1,u_2) \\
&=& 1\end{eqnarray*}
in $Z$.  As an algebraic identity it is clear that such $t'$ cover a 
relatively schematically dense open locus in $\Gm$
(namely, the locus of $u \in \Gm$ such that $u + 1 \in \Gm$), so $S$-separatedness of $Z$
then forces $c = 1$ as desired.

\begin{lemma} The $D$-equivariant quotient maps $U' \rightarrow U$ and $V' \rightarrow V$
admit unique $D$-equivariant splittings.
\end{lemma}

\begin{proof}
By symmetry it suffices to treat $U'$.  First we address the uniqueness, so then we may
work fppf-locally on $S$ to make the construction.  Uniqueness amounts to the assertion
that there are no nontrivial $D$-equivariant $S$-homomorphisms $f:U \rightarrow Z$.
The torus $D = \Gm$ acts trivially on $Z$ but acts on $U = \Add$ via $t.x(u) = x(t^2u)$, so 
$U - \{0\}$ is a single ``$D$-orbit''. Thus, relative schematic density 
of $U - \{0\}$ in $U$ and the $S$-separatedness of $Z$ then give the vanishing of any such $f$.

Now working fppf-locally on $S$, we may arrange that there exists a unit $t$ on $S$ such that $t^2 - 1$ is
also a unit, and also that the element  $h(t) \in D(S)$ admits a lift $h' \in D'(S)$.  The $h(t)$-action on $U'$
is induced by $h'$-conjugation, and as an endomorphism of the commutative
$U'$ it induces the identity on the subgroup $Z$.  Thus, the endomorphism
$\varphi:u' \mapsto h'u'{h'}^{-1} - u' = h(t).u' - u'$ of the abstract
commutative $S$-group $U'$ kills $Z$ and lies
over the endomorphism $\overline{\varphi}$ of $U = \Add$ given by $x(u) \mapsto x(t^2 u)x(u)^{-1} = x((t^2-1)u)$.
But $\overline{\varphi}$ is an automorphism, so 
$\varphi$ factors through an $S$-homomorphism $U = U'/Z \rightarrow U'$
lifting an automorphism $\overline{\varphi}$ of $U$, and by construction it is $D$-equivariant
(due to the commutativity of $D'$).  Precomposing with the inverse of $\overline{\varphi}$
then provides the desired splitting.
\end{proof}

Using the unique $D$-equivariant $S$-group isomorphisms
$U' \simeq U \times Z$ and $V' \simeq V \times Z$ that split the central extensions, we obtain
$D$-equivariant $S$-homomorphisms $x':\Add \rightarrow U'$
and $y':\Add \rightarrow V'$ lifting the respective standard parameterizations
$x$ of $U$ and $y$ of $V$.  We'll use these 
to build an $S$-homomorphism $h':\Gm \rightarrow D'$ lifting the parameterization 
$h:\Gm \simeq D$.
Define $g'(t) = y'(-1/t)x'(t)y'(-1/t)$ for $t \in \Gm$; this lifts
the point $g(t) := y(-1/t)x(t)y(-1/t) = (\begin{smallmatrix} 0 & t \\ -1/t & 0 \end{smallmatrix}) \in {\rm{SL}}_2 = G$, so
$h'(t) := g'(t)g'(1)^{-1} \in G'$ lifts $g(t)g(1)^{-1} = h(t)$ and hence is valued in $D'$.

\begin{lemma} The map $h':\Gm \rightarrow D'$ is a homomorphism lifting the parameterization $h:\Gm \simeq D$,
and $h'(s)g'(t) = g'(st)$ for all $s, t \in \Gm$.
\end{lemma}

\begin{proof}
Conjugation by $h'(s)$ on $g'(t) = y'(-1/t)x'(t)y'(-1/t)$ is the same as the action by
$h(s)$, and so by the $D$-equivariance of the construction of $x'$ and $y'$ as respective
lifts of $x$ and $y$ we have $h(s).y'(v) = y'(v/s^2)$ and $h(s).x'(u) = x'(s^2 u)$
for any $u, v \in \Add$.   Thus, $h(s).g'(t) = y'(-1/s^2t)x'(s^2t)y'(-1/s^2t) = g'(s^2 t)$,
or in other words
$$h'(s)g'(t)h'(s)^{-1} = g'(s^2t).$$
Multiplying this against the inverse of the case $t = 1$ gives
$h'(s)h'(t)h'(s)^{-1} = g'(s^2t)g'(s^2)^{-1} = h'(s^2t)h'(s^2)^{-1}$.
But $D'$ is commutative, so we obtain $h'(t) = h'(s^2t)/h'(s^2)$ for any points $s, t \in \Gm$.
This establishes that $h'$ is a homomorphism (visibly lifting $h$).
By the definition of $h'$, 
the identity $h'(st) = h'(s)h'(t)$ says 
$$g'(st)g'(1)^{-1} = h'(s) g'(t) g'(1)^{-1},$$
so $g'(st) = h'(s)g'(t)$.
\end{proof}

By going back to the definition of $g'$, the identity $h'(s)g'(t) = g'(st)$ says
$$h'(s)y'(-1/t)x'(t)y'(-1/t) = y'(-1/st)x'(st)y'(-1/st).$$
The $D$-equivariance of $y'$ gives that $h'(s)y'(-1/t) = y'(-1/s^2t)h'(s)$, so
$$y'\left(-\frac{1}{s^2t}\right)h'(s)x'(t) = y'\left(-\frac{1}{st}\right)x'(st)y'\left(-\frac{1}{st} + \frac{1}{t}\right).$$
Multiplying by $y'(-1/st)^{-1}$ on the left, we  arrive at the relation
$$y'\left(\frac{1}{s} \cdot \left(-\frac{1}{st} + \frac{1}{t}\right)\right)h'(s)x'(t) = x'(st)
y'\left(-\frac{1}{st} + \frac{1}{t}\right)$$
in $G'$ for any units $s$ and $t$, or equivalently 
we have the following analogue of (\ref{cellid})
(via the change of variables $u = st$ and $v = (s-1)/st$ making $1+uv = s$
and $u/(1+uv) = t$):
\begin{equation}\label{comrel}
x'(u)y'(v) = y'\left(\frac{v}{1+uv}\right)h'(1+uv)x'\left(\frac{u}{1+uv}\right)
\end{equation}
for $u, v \in \Add$ such that $1+uv$ and $u$ are units. 

We can establish (\ref{comrel}) without
the unit condition on $u$ by working fppf-locally, as follows.
For any points $u, v$ of $\Add$ such that $1+uv$ is a unit, fppf-locally we may 
write $u = u' + u''$ where
$u', u'', 1+u''v \in \Gm$ (as we see by treating separately the cases
when $v$ vanishes or does not vanish at a geometric point of interest), so 
by using that the commutation relation for $h'$ against $x'$ and $y'$ coincides
with that of $h$ against $x$ and $y$ (due to the $D$-equivariance underlying
the construction of $x'$ and $y'$) we get 
\begin{eqnarray*}
x'(u)y'(v) &=& x'(u')x'(u'')y'(v) \\
&\stackrel{(\ref{comrel})}{=} & x'(u')y'\left(\frac{v}{1+u''v}\right)h'(1+u''v)x'\left(\frac{u''}{1+u''v}\right)
\end{eqnarray*}
when $1+uv$ is a unit.  Continuing to assume this unit condition, a further application of
(\ref{comrel}) transforms the 4-fold product into 
$$y'\left(\frac{v}{1+uv}\right)h'\left(\frac{1+uv}{1+u''v}\right)
x'\left(\frac{u'(1+u''v)}{1+uv}\right)h'(1+u''v)x'\left(\frac{u''}{1+u''v}\right).$$
Simplifying the three middle factors, this becomes
$$y'\left(\frac{v}{1+uv}\right)h'(1+uv)x'\left(\frac{u'}{(1+u''v)(1+uv)}\right)x'\left(\frac{u''}{1+u''v}\right),$$
which is just $y'(v/(1+uv))h'(1+uv)x'(u/(1+uv))$.
Passing to inverses and negating $u$ and $v$, we conclude that 
$$y'(v)x'(u) = x'\left(\frac{u}{1+uv}\right)h'(1+uv)^{-1}y'\left(\frac{v}{1+uv}\right)$$
when $1+uv$ is a unit.

Recall that in ${\rm{SL}}_2$, $x(u)y(v)$ lies in the open cell
$U^{-} \times D \times U^+$ precisely when $1+uv$ is a unit, and the preceding calculations show that 
$x', h', y'$ satisfy the {\em same} commutation relations that govern the 
``$S$-birational'' group law on
the open cell $\Omega = U \times D \times V$
 of $G = {\rm{SL}}_2$ (but beware that $U' \times D' \times V' \rightarrow G'$ is 
not an open immersion when $Z \ne 1$, 
since $Z$ lies in all three groups $V'$, $D'$, and
$U'$).  It follows that the $S$-morphism section 
$\sigma:\Omega \rightarrow G'$ defined by
$x(u)h(s)y(v) \mapsto x'(u)h'(s)y'(v)$
for $(u,s,v) \in \Add \times \Gm \times \Add$
is ``$S$-birationally multiplicative'' in the sense that on the 
\index{sbirational mult@$S$-birational multiplicativity}
{\em fiberwise-dense} (hence relatively schematically dense \cite[IV$_3$, 11.10.10]{ega}) open locus
$m_G^{-1}(\Omega) \bigcap (\Omega \times \Omega)$ in $\Omega \times
\Omega$ consisting of points 
$(\omega_1, \omega_2)$ whose product in $G = {\rm{SL}}_2$ lies
in $\Omega$, we have $\sigma(\omega_1)\sigma(\omega_2) = \sigma(\omega_1 \omega_2)$.

To show that $\sigma$ extends (uniquely) to an $S$-homomorphism
$G \rightarrow G'$ that is the desired splitting of the given
central extension of $G$ by $Z$, we use an alternative procedure that works when
$G'$ is just a group sheaf.  Zariski-locally, every point of $G$ either lies in the open cell $\Omega$
or its translate by the point $x(1) \in U$ lies in $\Omega$, 
so as a group sheaf $G$ is generated by finite products of points
of $\Omega$.  Hence, to construct
the desired homomorphic section $G \rightarrow G'$ 
extending $\sigma$ we just have to check that if $a_1, \dots, a_n$ and $b_1,\dots, b_m$
are points of $\Omega$ (valued in some $S$-scheme)
such that $a_1 \cdots a_n = b_1 \cdots b_m$ in $G$ then
\begin{equation}\label{sigmaeq}
\sigma(a_1) \cdots \sigma(a_n) \stackrel{?}{=} \sigma(b_1) \cdots \sigma(b_m).
\end{equation}

To deduce this equality from the weaker ``$S$-birational multiplicativity''
already established for $\sigma$, observe that for such given $a_i$ and $b_j$
valued in some $S$-scheme $S'$, fppf-locally on $S'$ 
there exists a point $\omega$ of $\Omega$ such that
$\omega a_1 \dots a_i$ and $\omega b_1 \cdots b_j$ lie
in $\Omega$ for all $i, j \ge 0$.  This holds because for any 
$x = (x_1,\dots, x_r) \in G^r(S')$ the map
$X := \bigcap_i \Omega_{S'} \, x_i^{-1} \rightarrow S'$ is fppf and hence
admits sections fppf-locally on $S'$.

Letting $g = a_1 \cdots a_n = b_1 \cdots b_m$, we have
$$\omega g = (\omega a_1 \cdots a_{n-1})a_n,\,\,\,
\omega g = (\omega b_1 \cdots b_{m-1})b_m.$$
Thus, the ``$S$-birational multiplicativity'' gives that
$$\sigma(\omega g) = \sigma(\omega a_1 \cdots a_{n-1}) \sigma(a_n),\,\,\,
\sigma(\omega g) = \sigma(\omega b_1 \cdots b_{m-1})\sigma(b_m).$$
Continuing in this way (using the conditions on $\omega$),
 we get 
$$\sigma(\omega a_1 \cdots a_{n-1}) = \sigma(\omega) \sigma(a_1) \cdots 
\sigma(a_{n-1})$$
and $$\sigma(\omega b_1 \cdots b_{m-1}) = \sigma(\omega) \sigma(b_1) \cdots 
\sigma(b_{m-1}),$$
 so (\ref{sigmaeq}) holds up to a harmless left multiplication by $\sigma(\omega)$ on both sides. 
\end{proof}





%
%Every $R$-torus is a power of $\Gm$ (as
%$R$ is artin local with algebraically closed residue field), so it suffices
%to consider a central extension $G'$ of ${\rm{SL}}_2$ by $\Gm$ over $R$.
%Since $G'$ is a $\Gm$-torsor over ${\rm{SL}}_2$ for the \'etale
%topology, it is also one for the Zariski topology (by descent theory for line bundles).  We claim
%that there is a scheme-theoretic section to this torsor. 

%By thinking in terms of line bundles, we
%see that a $\Gm$-torsor over the affine $R$-group ${\rm{SL}}_2$
%has a trivializing section if and only if it does so modulo the nilpotent maximal ideal
%of $R$. Thus, we are reduced to constructing a scheme-theoretic section
%over the residue field. But the special fiber $G'_0$ has
%semisimple derived group that is a central extension of
%${\rm{SL}}_2$, and it is known in the classical theory that
%${\rm{SL}}_2$ has no such nontrivial central extensions.
%That is, $\mathscr{D}(G'_0) \rightarrow {\rm{SL}}_2$
%is an isomorphism, providing the required section over the residue field.

%Using a suitable translation by $\Gm \subset G'$, we have
%now arranged that $G' = \Gm \times {\rm{SL}}_2$
%as schemes with $1$ going over to $(1,1)$, and the group law
%is given by $(t,m)(t',m') = (tt' f(m,m'), mm')$ for a Hochschild 2-cocycle
%$$f:{\rm{SL}}_2 \times {\rm{SL}}_2 \rightarrow \Gm.$$
%The obstruction to changing the choice of scheme-theoretic section
%to $G' \rightarrow {\rm{SL}}_2$ over $R$ so that it is an $R$-homomorphism
%is precisely the vanishing or not of the Hochschild cohomology
%class $[G'] \in {\rm{H}}^2(({\rm{SL}}_2)_R,\Gm)$ represented by $f$ (but only depending
%on $G'$ as a central extension).   Thus, to prove (2) (assuming (1))
%it suffices to prove that this cohomology group vanishes.
%The preceding argument building sections by lifting from the residue field
%shows that this cohomology group classifies precisely the isomorphism
%classes of central extensions of ${\rm{SL}}_2$ by $\Gm$ over $R$
%(exactly as in ordinary group cohomology, due to the automatic
%existence of scheme-theoretic splittings to such extensions).

%We argue by induction on the length of $R$, the case of length 0
%being the group splitting using $\mathscr{D}(G'_0)$ as used above.
%In general, with length $> 0$, we may find a proper ideal
%$J$ killed by the maximal ideal, so $R/J$ has smaller length.
%Our cocycle over $R$ is a coboundary over $R/J$, and we can lift 
%the resulting 1-cochain over $R/J$ to one over $R$ because such 1-cochains
%are nothing other than units on an affine scheme (for which there
%is no obstruction to lifting through a nilpotent thickening). 
%Dividing out by the coboundary associated to such a lift,
%the obstruction to vanishing lies in 
%$${\rm{H}}^2(({\rm{SL}}_2)_k, J) = J \otimes_k {\rm{H}}^2(({\rm{SL}}_2)_k, \Ga).$$
%But it is a classical fact that for any connected
%semisimple group $G$ over an algebraically closed field, ${\rm{H}}^i(G,\Ga) = 0$
%for any $i > 0$; see \cite[II, 4.11]{jantzen}.
%\end{proof}

For later purposes, we now establish the ${\rm{SL}}_2$-analogue of
Proposition \ref{pgl2}.

\begin{proposition}\label{sl2def}
Let $G \rightarrow S$ be a reductive group scheme whose geometric 
fibers have finite center of order $2$
and semisimple-rank $1$, and let $T$ be a split maximal
torus of $G$.  Zariski-locally on
$S$ there is an $S$-group isomorphism $G \simeq {\rm{SL}}_2$,
and it can be chosen 
to identify $T$ with the diagonal torus. 
\end{proposition}

\begin{proof}
Since the center $Z_{G}$ has fibers of order 2
and is of multiplicative type (see Theorem \ref{redcenter}), 
its Cartier dual is finite \'etale of order 2.
But $(\Z/2\Z)_S$ has no nontrivial \'etale $S$-forms
since $(\Z/2\Z)^{\times} = 1$.  Hence, $Z_G \simeq \mu_2$.
Likewise, the geometric fibers $G_{\overline{s}}$ must be ${\rm{SL}}_2$
by the classical theory.  The quotient $G/Z_G$ has trivial center
(Corollary \ref{qtcenter}), and it has a split maximal torus
$T/Z_G$, so by Proposition \ref{pgl2}
there exists an isomorphism $G/Z_G \simeq {\rm{PGL}}_2$
Zariski-locally on $S$ 
that moreover carries $T/Z_G$ over to the diagonal torus. 
In view of the bijective correspondence between maximal tori in
$G$ and $G/Z_G$, as well
as in ${\rm{SL}}_2$ and ${\rm{SL}}_2/\mu_2 = {\rm{PGL}}_2$
(apply Corollary \ref{qtcenter}),
a lift of $G/Z_G \simeq {\rm{PGL}}_2$
to an isomorphism $G \simeq {\rm{SL}}_2$
(if one exists, at least Zariski-locally on $S$) must
carry $T$ to the diagonal torus.

It remains to show that Zariski-locally on $S$, any smooth 
central extension $G$ of ${\rm{PGL}}_2$
by $\mu_2$ over $S$ with connected
geometric fibers is isomorphic to ${\rm{SL}}_2$ (as a central extension). 
Using central pushout along $\mu_2 \hookrightarrow \Gm$
embeds $G$ as a closed subgroup of a central extension $G'$ of ${\rm{PGL}}_2$
by $\Gm$.  Pulling back this latter extension by the $\mu_2$-quotient map
${\rm{SL}}_2 \rightarrow {\rm{PGL}}_2$ yields
a central extension 
$$1 \rightarrow \Gm \rightarrow \widetilde{G}' \rightarrow {\rm{SL}}_2 \rightarrow 1$$
with $\widetilde{G}'$ also a central $\mu_2$-extension of $G'$.  
By Proposition \ref{sl2ext}
there exists a splitting $\widetilde{G}' = {\rm{SL}}_2 \times \Gm$, and 
Zariski-locally on $S$ this
can be arranged to carry any given maximal torus 
$\widetilde{T}'$ of $\widetilde{G}'$ over
to $D \times \Gm$ (we just have to arrange that the corresponding
isomorphism $\widetilde{G}'/Z_{\widetilde{G}'} \simeq {\rm{PGL}}_2$
carries $\widetilde{T}'/Z_{\widetilde{G}'}$ over to the diagonal torus, as can be done
Zariski-locally on $S$ by Proposition \ref{pgl2}).

On geometric 
fibers over $S$, the central subgroup $\mu = \ker(\widetilde{G}' \twoheadrightarrow G') \simeq \mu_2$ 
in $\widetilde{G}' = {\rm{SL}}_2 \times \Gm$ must
either be the center of the ${\rm{SL}}_2$-factor or the diagonally
embedded $\mu_2$ in the product ${\rm{SL}}_2 \times \Gm$.  
The first option cannot occur over any geometric point $\overline{s}$ of $S$:  it
would imply that $G'_{\overline{s}}$ is a split extension ${\rm{PGL}}_2 \times \Gm$, 
yet the smooth connected subgroup $G_{\overline{s}} \subset G'_{\overline{s}}$ is 
 semisimple, so necessarily $G_{\overline{s}} = \mathscr{D}(G'_{\overline{s}}) = {\rm{PGL}}_2$,
contradicting the hypothesis that the given central quotient
map $G_{\overline{s}} \rightarrow {\rm{PGL}}_2$ has a 
nontrivial kernel.   Thus, the central $\mu \subset Z_{\widetilde{G}'} = \mu_2 \times \mu_2$ must be the 
diagonal $S$-subgroup since this holds on geometric fibers over $S$.

We conclude that $G'$ is the pushout
${\rm{SL}}_2 \times^{\mu_2} \Gm = \GL_2$ over $S$.  Moreover,
via the bijection between the sets of maximal tori in $\widetilde{G}'$
and its central quotient $G' = \widetilde{G}'/\mu$ (Corollary \ref{qtcenter}), we see that
Zariski-locally on $S$ 
the isomorphism $G' \simeq {\rm{SL}}_2 \times^{\mu_2} \Gm = \GL_2$
can be arranged to carry any given maximal torus of $G'$ over to $D \times^{\mu_2} \Gm$,
which is the diagonal torus $\widetilde{D}$ in $\GL_2$.
Apply this to the maximal torus $T' := T \times^{\mu_2} \Gm$ inside 
$G \times^{\mu_2} \Gm = G'$.

It suffices to prove that the subgroup $G \subset G'$ 
(which meets $T'$ in $T$) coincides with the subgroup
${\rm{SL}}_2 \subset {\rm{SL}}_2 \times^{\mu_2} \Gm = \GL_2$
(which meets $T = D \times^{\mu_2} \Gm = \widetilde{D}$ in $D$).
Indeed, since the natural 
quotient map 
$G \rightarrow G/\mu_2 = G'/\Gm = {\rm{SL}}_2/\mu_2 = {\rm{PGL}}_2$
is the central quotient map $G \rightarrow G/Z_G \simeq {\rm{PGL}}_2$
arranged at the start, an equality 
$G = {\rm{SL}}_2$ inside $G'$ must respect the structures
of both sides as central extensions of ${\rm{PGL}}_2$ by $\mu_2$, so we would be done.

To relate $G$ and ${\rm{SL}}_2$ inside  $G'$, we shall use root groups. 
More specifically, we may work Zariski-locally on $S$ so as to acquire
a pair of opposite roots $\pm a$ for $(G,T)$ whose root spaces
$\mathfrak{g}_{\pm a}$ are trivial line bundles over $S$.
The torus $T'$ is generated by $T$ and the central $\Gm$ in $G'$, so
the $T$-weight spaces on $\mathfrak{g}$ are also
$T'$-weight spaces in $\mathfrak{g}'$.  Letting
$\pm a':T' \rightarrow \Gm$ be the corresponding fiberwise nontrivial $T'$-weights,
it is clear that $U_{\pm a}$ equipped with $\exp_{\pm a}$ 
satisfies the properties uniquely characterizing 
$(U_{\pm a'}, \mathfrak{g}'_{\pm a'})$.  
The identification 
$$(G',T') = (\GL_2, \widetilde{D})$$
must carry the root groups $U_{\pm a'}$ for
$(G',T')$ over to the standard root groups $U^{\pm}$ on the right side.
%Zariski-locally on $S$ (as we see via comparison over the local rings on $S$).

 We have shown that $U_{\pm a'} = U_{\pm a} \subset G$ inside $G'$,
and obviously $U^{\pm } \subset {\rm{SL}}_2$ inside $G' = \GL_2$. 
For $x(u) = (\begin{smallmatrix} 1 & u \\ 0 & 1 \end{smallmatrix}) \in U^{+}$
and $y(u) = (\begin{smallmatrix} 1 & 0 \\ u & 1 \end{smallmatrix}) \in U^{-}$
we have $$y(-1/t)x(t)y(-1/t)(y(-1)x(1)y(-1))^{-1} = {\rm{diag}}(t,1/t)$$
for any $t \in \Gm$, so the standard open cell $\Omega \subset {\rm{SL}}_2$
inside $G' = \GL_2$ lies in the closed $S$-subgroup $G \subset G'$.  By relative schematic density of
$\Omega$ in ${\rm{SL}}_2$ over $S$, it follows that ${\rm{SL}}_2 \subset G$
as closed subgroups of $G'$.  But $G$ and ${\rm{SL}}_2$ are each $S$-smooth
with connected geometric fibers of dimension 3, so
the inclusion ${\rm{SL}}_2 \subset G$ is an equality on 
geometric fibers over $S$ and hence is an equality as closed subgroups of $G'$.
\end{proof}
\index{central extensions|)}

\newpage

\subsection{Exercises}
%
%\begin{exer}\label{1.5}
% Let $G$ be a smooth group of finite type and dimension $d > 0$ over a field $k$.

%(i) Define the concept of {\em left-invariant} differential $i$-form for $i \ge 0$,
%and prove the space $\Omega^{i,\ell}_G(G)$ of such forms has dimension $d\choose{i}$.  
%Compute the 1-dimensional $\Omega^{d,\ell}_G(G)$ for ${\rm{GL}}(V)$, ${\rm{SL}}(V)$, and
%${\rm{PGL}}(V)$.

%(ii)  Using right-translation, construct a linear representation of
%$G$ on $\Omega^{d,\ell}_G(G)$;
%the associated character $\chi_G:G \rightarrow \Gm$ 
%is the {\em modulus
%character}.  Prove $\chi_G|_{Z_G} = 1$ and deduce that $\chi_G = 1$ if $G/Z_G = 
%\mathscr{D}(G/Z_G)$ or if $G$ is unipotent.  If $P$ is a parabolic subgroup
%in a connected reductive $k$-group $G$ and $U \subset P$ is
%its unipotent radical, relate $\chi_P$ on $P/U$ to the determinant of
%the adjoint action on ${\rm{Lie}}(U)$ by a Levi subgroup of $P$.
%What does this say when $P$ is the standard Borel in $G = {\rm{GL}}_n$?

%(iii) If $k$ is local (allow $\R$, $\C$) and $X$ is smooth, 
%use the $k$-analytic inverse function theorem to equip $X(k)$ with a functorial
%$k$-analytic manifold structure, and use $k$-analytic
%Change of Variables to assign a measure on $X(k)$ to a nowhere-vanishing 
%$\omega \in \Omega^{\dim X}_X(X)$.  (See \cite{serrelie}
%for relevant $k$-analytic foundations.) Relate this to Haar measures, and prove
%$\chi_G^{\pm 1}|_{G(k)}$ {\em is} the classical modulus character (left or right).
%\end{exer}

\begin{exer}\label{rootex}
Let $G$ be a reductive group scheme over a scheme $S$
and let $T$ be a split maximal torus of $G$, with a fixed isomorphism
$T \simeq D_S(M)$ for a finite free $\Z$-module $M$.
For each $s \in S$ and root $\alpha \in \Phi(G_{\overline{s}}, T_{\overline{s}})$
construct a Zariski-open neighborhood $U$ of $s$ in $S$ 
and a root $a$ of $(G_U, T_U)$ in the sense of Definition \ref{rootdef}
such that $a(\overline{s}) = \alpha$.  Prove moreover that any two such
$a$ (for the same $\alpha$) coincide on a Zariski-open neighborhood of $s$ in $S$.
\end{exer}

\begin{exer}\label{dynex}
Let $G$ be a finitely presented affine group over a ring $k$, and
choose a 1-parameter subgroup $\lambda:\Gm \rightarrow G$ over $k$.

(i) Prove that $P_G(\lambda^n) = P_G(\lambda)$ for any $n > 0$, and likewise
for $U_G(\lambda^n)$ and $Z_G(\lambda^n)$.

(ii) Suppose $k/k_0$ is a finite flat extension of noetherian rings, $G_0$ is the Weil restriction ${\rm{R}}_{k/k_0}(G)$
(an affine $k_0$-group of finite type), and $\lambda_0:\Gm \rightarrow G_0$ is
\index{Weil restriction}
the $k_0$-morphism corresponding to the $k$-homomorphism $\lambda:\Gm \rightarrow G$ via the universal
property of ${\rm{R}}_{k/k_0}$.  Prove that $\lambda_0$ is a $k_0$-homomorphism
and $P_{G_0}(\lambda_0) = {\rm{R}}_{k/k_0}(P_G(\lambda))$, and similarly for
$U_{G_0}(\lambda_0)$ and $Z_{G_0}(\lambda_0)$.
\end{exer}

\begin{exer}\label{commpair}
Prove the following fact that was used
in the proof of Proposition \ref{sl2ext}:
for any central extension $1 \rightarrow Z \rightarrow H' \rightarrow H \rightarrow 1$
of one commutative group sheaf by another (on any site),
the commutator of $H'$ factors through a bi-additive pairing
$$H \times H \rightarrow Z$$
and the vanishing of this pairing is equivalent to the commutativity of $H'$.
(This generalizes part of Exercise \ref{3.7}(iii).) 
\index{central extension!and commutator}
\end{exer}

\begin{exer}\label{4.2}
Let $A$ be a finite-dimensional associative algebra over a field $k$, 
and $\underline{A}^{\times}$ the associated $k$-group of units as in Exercise \ref{3.1}. 
Prove ${\rm{Tan}}_e(\underline{A}^{\times}) = A$ naturally, and 
that the Lie algebra structure is  $[a,a'] = aa' - a'a$.  Using 
$A = {\rm{End}}(V)$, recover $\mathfrak{gl}(V)$ without coordinates.  Use this to compute the Lie algebras
$\mathfrak{sl}(V)$, $\mathfrak{pgl}(V)$, $\mathfrak{sp}(\psi)$ (for a symplectic form $\psi$), 
$\mathfrak{gsp}(\psi)$, and $\mathfrak{so}(q)$ without coordinates.
\end{exer}

\begin{exer}\label{4.3}
Let $K$ be a degree-2 finite \'etale algebra over a field $k$ (i.e., a separable quadratic field extension or
$k \times k$, the latter called the {\em split} case), and 
let $\sigma$ be the unique nontrivial $k$-automorphism of $K$; note that
$K^{\sigma} = k$.
A {\em $\sigma$-hermitian space} is a pair $(V,h)$ consisting of a finite free $K$-module
equipped with a perfect $\sigma$-semilinear form $h:V \times V \rightarrow K$
(i.e., $h(cv,v') = c h(v,v')$, $h(v,cv') = \sigma(c) h(v,v')$, and $h(v',v) = \sigma(h(v,v'))$). 
Note that $v \mapsto h(v,v)$ is a quadratic form $q_h:V \rightarrow k$ over $k$ satisfying
$q_h(cv) = {\rm{N}}_{K/k}(c) q_h(v)$ for $c \in K$ and  $v \in V$, and $\dim_k V$ is even
(${\rm{char}}(k) = 2$ is allowed!).  (One similarly defines the notion of
a {\em $\sigma$-anti-hermitian space} by requiring $h(v',v) = -\sigma(h(v,v'))$.)

The {\em unitary group} ${\rm{U}}(h)$ over $k$  is the subgroup of
${\rm{R}}_{K/k}({\rm{GL}}(V))$ preserving $h$. 
\index{special unitary group}
\index{unitary group}
\index{ aaauh@${\rm{U}}(h)$}
\index{ aaasuh@${\rm{SU}}(h)$}
Using ${\rm{R}}_{K/k}({\rm{SL}}(V))$ gives the {\em special unitary group}
${\rm{SU}}(h)$.  Example: $V = F$ finite \'etale over $K$ 
with an involution $\sigma'$ lifting $\sigma$, 
and $h(v,v') := {\rm{Tr}}_{F/K}(v\sigma'(v'))$; e.g., CM fields $F$ and $K$, totally real $k$, 
and complex conjugations $\sigma'$ and $\sigma$.

(i) If $K = k \times k$, prove $V \simeq V_0 \times V_0^{\ast}$ with $h((v, \ell),(v',\ell')) = 
(\ell'(v),\ell(v'))$ for a $k$-vector space $V_0$.
Identify ${\rm{U}}(h)$ with ${\rm{GL}}(V_0)$ carrying ${\rm{SU}}(h)$ to ${\rm{SL}}(V_0)$.
Compute $q_h$ and prove that $q_h$ is non-degenerate.

(ii) In the non-split case prove that ${\rm{U}}(h)_K \simeq {\rm{GL}}_n$ carrying
${\rm{SU}}(h)$ to ${\rm{SL}}_n$ ($n = \dim_K V$). 
Prove ${\rm{U}}(h)$ is smooth and connected
with derived group ${\rm{SU}}(h)$ and center $\Gm$, and that $q_h$ is non-degenerate.
Compute $\mathfrak{su}(h)$.

(iii) Identify ${\rm{U}}(h)$ with a $k$-subgroup of ${\rm{SO}}(q_h)$.  Discuss the split case, 
and the case $k = \RRR$. 
\end{exer}

\begin{exer}\label{3.3}
Consider a $k$-torus $T \subset {\rm{GL}}(V)$ containing  $Z_{{\rm{GL}}(V)} =
{\mathbf{G}}_m$, with $k$ infinite. 
Let $A_T \subset {\rm{End}}(V)$ be the commutative $k$-subalgebra generated by $T(k)$. 

(i) When $k = k_s$, prove $A_T$ is a product of copies of $k$ and that the inclusion 
 $T(k) \hookrightarrow A_T^{\times}$ is an equality.

(ii) Using Galois descent and the end of  Exercise \ref{2.4}(i), prove
$(A_T)_{k_s} = A_{T_{k_s}}$, and deduce that 
$T(k) = A_T^{\times}$. Construct a natural isomorphism 
$T \simeq {\rm{R}}_{A/k}(\Gm)$, and a bijection 
between the set of  $k$-subtori in ${\rm{GL}}(V)$ containing
$Z_{{\rm{GL}}(V)}$ and the set of  \'etale finite commutative $k$-subalgebras
of ${\rm{End}}(V)$.  Generalize to {\em finite} $k$, using Galois descent
to reduce to the case just handled.
\end{exer}

\begin{exer}\label{3.4}
Let $(V,q)$ be a non-degenerate quadratic space over a field $k$ with $\dim V \ge 2$. 

(i) If $q(v)=0$ for some $v \in V - \{0\}$, prove that $v$ lies in a hyperbolic plane $H$ with 
$H \bigoplus H^{\perp} = V$.
(If ${\rm{char}}(k) = 2$ and $\dim V$ is even, work over $\overline{k}$ to show $v \not\in V^{\perp}$.)  
Use this to construct a $\Gm$ inside  ${\rm{SO}}(q)$ over $k$.

(ii) If ${\rm{SO}}(q)$ contains a $k$-subgroup $S \simeq
\Gm$, prove conversely that $q(v) = 0$ for some $v \in V - \{0\}$.
(Hint: prove that $V^S \ne V$ and compute $q(tv)$ in two ways for $t \in S$ and a nonzero $v$ in a 
weight space for a nontrivial $S$-weight.)
\end{exer}

\begin{exer}\label{gross6}
 Let $G$ be a connected semisimple group over
an algebraically closed field $k$, and let
$T$ be a maximal torus and $B$ a Borel subgroup of $G$ containing
$T$.  Let $\Delta$ be the set of simple positive roots
relative to the positive system of roots $\Phi^+ = \Phi(B,T)$.

(i) Using Corollary \ref{computez}, prove that $Z_G = 1$ (equivalently, 
${\rm{Ad}}_G$ is a closed immersion) if
and only if $\Delta$ is a basis of ${\rm{X}}(T)$.
(Do {\em not} assume the Existence and Isomorphism Theorems,
as was done in Exercise \ref{scclassical}(ii).)

(ii) Assume $G$ is adjoint, and let $\{\omega_i^{\vee}\}$ denote
the basis of ${\rm{X}}_{\ast}(T)$ dual to the basis
$\{a_i\} = \Delta$ of ${\rm{X}}(T)$. For each
subset $I \subset \Delta$, let $\lambda_I \in {\rm{X}}_{\ast}(T)$
\index{ aaalambdai@$\lambda_I$}
be the cocharacter $\sum_{a_i \in I} \omega_i^{\vee}$.
Prove that the parabolic subgroup $P_G(\lambda_I)$
coincides with the ``standard'' parabolic subgroup $P_{\Delta - I}$ containing
$B$ that arose in the proof of Proposition \ref{parpos} (so $B = P_{\emptyset} = P_G(\lambda_{\Delta})$).  
This gives a ``dynamic''
description of the parabolic subgroups of $G$ containing $B$.
(Hint:  By Proposition \ref{parpos}, it suffices to compare Lie algebras inside $\mathfrak{g}$.) 

(iii) Prove that $\rho^{\vee} := \lambda_{\Delta}$ coincides with $(1/2)\sum_{a \in \Phi^+} a^{\vee}$.
\index{ aaaro@$\rho^{\vee}$}
\index{dynamic method!parabolic subgroups}
Equivalently (by consideration of the dual root datum and Exercise \ref{dualexer}), for each $a_i \in \Delta$ 
prove $\langle \sum_{a \in \Phi^+} a, a_i^{\vee} \rangle = 2$.
(Hint: Show that $s_{a_i}$ permutes $\Phi^+ - \{a_i\}$
by using that the reflection $s_{a_i}:v \mapsto v - \langle v, a_i^{\vee} \rangle a_i$ preserves
$\Phi = \Phi^+ \coprod -\Phi^+$. 
Apply the resulting ``change of variables'' $a \mapsto s_{a_i}(a)$ to show that
$\langle \sum_{a \in \Phi^+ - \{a_i\}} a, a_i^{\vee} \rangle$ vanishes.)
\end{exer}


\newpage

\section{Split reductive groups and parabolic subgroups}\label{splitsec}


\subsection{Split groups and the open cell}\label{splitopen}

In the theory of connected reductive groups $G$ over a field $k$
(not assumed to be algebraically closed), one says that $G$ is {\em split}
if it admits a maximal $k$-torus $T \subset G$ that is $k$-split.
\index{reductive group!split}
(Keep in mind that for us, ``maximal'' means ``geometrically maximal'', as
in Definition \ref{maxtori}.  The equivalence
with other possible notions of maximality over a field, which we never use, 
rests on Remark \ref{remtorus} and Grothendieck's existence theorem
for geometrically maximal tori over any field.)
For such $(G,T)$, the weight spaces $\mathfrak{g}_a$ in $\mathfrak{g}$ for the nontrivial
weights $a$ of $T$ that occur on $\mathfrak{g}$ are all 1-dimensional (as may be inferred
from the theory over $\overline{k}$).  More specifically, each
$\mathfrak{g}_a$ is free of rank 1 over $k$ since $k$ is a field.
In the relative theory, such module-freeness for the root spaces
$\mathfrak{g}_a$ must be imposed as a condition (following \cite[XXII, 1.13]{sga3}):

\begin{definition}\label{splitdef} Let $G$ be a reductive group over a non-empty scheme $S$.  
It is {\em split} if there exists a maximal torus $T$ equipped
with an isomorphism $T \simeq D_S(M)$ for a finite free $\Z$-module $M$ such that:
\begin{enumerate}
\item the nontrivial weights $a:T \rightarrow \Gm$ that occur 
on $\mathfrak{g}$ arise from elements of $M$
(so in particular, such $a$ are roots for $(G,T)$ and are ``constant sections'' of $M_S$),
\item each root space $\mathfrak{g}_a$ is free of rank 1 over $\mathscr{O}_S$,
\item each coroot $a^{\vee}:\Gm \rightarrow T$ arises from an element of
the dual lattice $M^{\vee}$ (i.e., $a^{\vee}$ as a global section of $M^{\vee}_S$ over $S$ is a constant section).
\end{enumerate}
\end{definition}

The definition of $a^{\vee}$ is given by applying Theorem \ref{ssclass}
to the reductive subgroup $Z_G(T_a)$. 
Note that although conditions (1) and (3) are automatic
when $S$ is connected (as the global sections of $M_S$ in general 
are the locally constant $M$-valued functions on $S$, and similarly for $M^{\vee}_S$),
we do {\em not} assume $S$ is connected. The reason is that when developing
the theory of split reductive group schemes we want to work locally on $S$ and use descent theory in some
proofs, but localization on the base and (especially) descent theory often lead
to disconnected base schemes.  For this reason, we avoid the notation ``$\Phi(G,T)$''
except when $S$ is connected (e.g., $S = \Spec k$ for a field or domain $k$).

\begin{example}\label{failex}
Let $S = \Spec(k_1 \times k_2) = \Spec k_1 \coprod \Spec k_2$ for fields $k_1$ and $k_2$, 
so an $S$-group $G$ is precisely $G = G_1 \coprod G_2$ where $G_i$ is a $k_i$-group.
In the case $G_1 = {\rm{PGL}}_2 \times \Gm$ and $G_2 = \GL_2$
(with their split diagonal tori identified via $({\rm{diag}}(t,1),t') \mapsto {\rm{diag}}(tt',t')$), 
conditions (1) and (2) in Definition \ref{splitdef} are satisfied 
but condition (3) fails. 
\end{example}

\begin{lemma}\label{etsplit}
Any reductive group scheme over a non-empty scheme becomes split \'etale-locally on the base.
\end{lemma}

A basic example of this lemma is that a connected reductive group over
a field $k$ becomes split over a finite separable extension of $k$.

\begin{proof}
There exists a maximal torus $T$ \'etale-locally on the base (Corollary \ref{buildtori}), and
further \'etale localization provides an isomorphism $T \simeq D_S(M)$.  
Working Zariski-locally then makes the
weight space decomposition for $\mathfrak{g}$ under the $T$-action 
have all nontrivial $T$-weights $a$ arise from $M$, 
with $\mathfrak{g}_a$ having constant rank 1.
A final Zariski-localization (which is necessary, by Example \ref{failex}) 
makes each coroot arise from $M^{\vee}$.
\end{proof}

 There is a case when the split property is automatic
in the presence of a maximal torus:

\begin{example}\label{zsplit}
 Let $S$ be a (non-empty) connected normal noetherian scheme with trivial Picard
group and trivial \'etale fundamental group.
For example, $S$ may be $\Spec \Z$ 
or $\mathbf{A}^n_k$ for an algebraically
closed field $k$ of characteristic 0. We claim that every reductive group scheme
$G$ over $S$ admitting a maximal torus $T$ over $S$ is automatically split.
Over $\Z$ this corresponds to the fact that the action of
${\rm{Gal}}(\overline{\Q}/\Q)$ on ${\rm{X}}(T_{\overline{\Q}})$
 is unramified at all primes and hence is trivial (Minkowski).
The general case proceeds along similar lines, as follows.

By Corollary \ref{mikecor} and the hypotheses on $S$, 
%First note that by the normality of $S$, every non-empty open subset $V$ of
%a connected finite \'etale $S$-scheme is irreducible.
%Hence, the \'etale fundamental group of $V$ 
%is a quotient of the Galois group at the generic point $\eta$ of $S$.  Thus, 
tori over $S$ correspond to continuous representations $\pi_1(S, \overline{\eta})$ on discrete $\Z$-lattices.  But
$\pi_1(S, \overline{\eta}) = 1$ by hypothesis, so $T$ is split.  Hence, we may choose
an isomorphism $T \simeq D_S(M)$.  The global sections of $M_S$ are the elements
of $M$ since $S$ is connected, and likewise the global sections of $M^{\vee}_S$ are
the elements of $M^{\vee}$.  Finally, each $\mathfrak{g}_a$ for $a \ne 0$ has constant rank
(0 or 1) since $S$ is connected, and when nontrivial it is free of rank 1
since ${\rm{Pic}}(S) = 1$. 
\end{example}

\begin{remark}\label{splitrem}
The split semisimple groups over $\Z$ are called (semisimple)
{\em Chevalley groups}.  By Example \ref{zsplit},
among all semisimple $\Z$-groups these are precisely the ones that admit a 
maximal $\Z$-torus (in the sense of Definition \ref{maxtori}). 
A connected semisimple $\Q$-group admits 
a semisimple $\Z$-model if and only if 
it is split over $\Q_p$ for all $p$.  (The implication ``$\Rightarrow$'' is a consequence of
(i) the existence of a split $\Z$-form, (ii) the triviality of $\pi_1(\Spec \Z)$, and (iii) the structure
of the automorphism scheme of a semisimple group over a ring;
see \cite[Prop.\:3.9ff.]{gross}.  For the converse, we can first spread out to a semisimple
group over some $\Z[1/N]$, and then we can ``glue''
with split models over $\Z_p$ for each $p|N$ to make a semisimple $\Z$-model;
see the proof of \cite[Lemma 4.3]{gross} and references therein for further discussion of this gluing process 
over a Dedekind base.)
There are semisimple $\Z$-groups that are not split, or equivalently do not admit
a maximal torus, such as special orthogonal groups of 
even unimodular lattices; e.g., the ${\rm{E}}_8$ and  Leech lattices.
\end{remark}

The set of elements of $M$ 
that occur in condition (1) of Definition \ref{splitdef} is denoted $\Phi$ (with the 
choice of isomorphism $T \simeq D_S(M)$ understood from context), and the set of
corresponding coroots in $M^{\vee}$ is denoted $\Phi^{\vee}$.
\index{ aaafi@$\Phi$, $\Phi^{\vee}$}
We have 
$$\mathfrak{g} = \mathfrak{t} \bigoplus (\bigoplus_{a \in \Phi} \mathfrak{g}_a).$$
The subsets $\Phi \subset M - \{0\}$ and $\Phi^{\vee} \subset M^{\vee} - \{0\}$
inherit combinatorial properties from the classical theory on geometric  fibers
\cite[XXII, 1.14, 3.4]{sga3}:

\begin{proposition}\label{wfinite}
 The $4$-tuple $(M,\Phi,M^{\vee}, \Phi^{\vee})$ is a reduced root datum.
The Weyl group $W_G(T) = N_G(T)/T \subset \underline{\rm{Aut}}(M^{\vee}_S) =
{\rm{Aut}}(M^{\vee})_S$ is the constant subgroup $W(\Phi)_S$.
\end{proposition}
\index{root datum!reduced}
\index{reductive group!root datum}

\begin{proof}
The required combinatorial conditions to be a root datum can be checked on a single geometric fiber
(recall that $S \ne \emptyset$), where it follows from the classical theory.
Likewise, since $W_G(T)$ is a finite \'etale $S$-subgroup of the
(opposite group of the) automorphism scheme
of $M_S$ (as $W_G(T)$ acts faithfully on $T = D_S(M)$), 
to compare it with the constant subgroup arising from
$W(\Phi)$ we may again pass to geometric fibers and
appeal to the classical theory.
\end{proof}

We sometimes call $(G,T,M)$ 
a ``split'' group (or {\em split triple}), with the isomorphism $T \simeq D_S(M)$
and subset $\Phi \subset M$ understood to be specified.
\index{reductive group!split}
\index{split!reductive group}
\index{split!triple}
This really comes in three parts: the pair $(G,T)$, the root datum $(M, \Phi, M^{\vee}, \Phi^{\vee})$,
and the isomorphism $T \simeq D_S(M)$ that carries $\Phi$ over to roots for $(G,T)$.
Keep in mind that the axioms for a root datum uniquely determine the bijection
$a \mapsto a^{\vee}$ between roots and coroots (Remark \ref{rem324}). 


\begin{example}\label{scex} Consider a split triple $(G, T, M)$. The center $Z_G$ is $D_S(M/Q)$, 
where $Q \subset M$ is the $\Z$-span of the roots (the {\em root lattice}).  Indeed, 
\index{root lattice}
by Corollary \ref{computez}, $Z_G$ is the kernel of the adjoint action of
$T = D_S(M)$ on $\mathfrak{g}$, and by definition the nontrivial weights
for this action are the elements of $\Phi \subset M$ viewed as characters on $T$.
Hence, $D_S(M/Q) \subset Z_G$, and to prove equality we may pass
to geometric fibers, where it is clear (since all multiplicative type groups
over an algebraically closed field are split).

Since semisimplicity is equivalent to finiteness of the center, it follows
that $G$ is semisimple if and only if the elements of $\Phi$ span $M_{\Q}$ over $\Q$.
Now suppose that $G$ is semisimple.  In such cases $G$ has trivial center 
(i.e., it is {\em adjoint}) precisely when 
\index{reductive group!adjoint}
\index{adjoint!reductive group}
$\Phi$ spans $M$ over $\Z$.  Since $M$ lies inside 
the {\em weight lattice} $P$ in $M_{\Q}$ that is (by definition)
\index{weight lattice}
dual to the coroot lattice (i.e., the $\Z$-span of $\Phi^{\vee}$) in $M^{\vee}_{\Q}$, 
the center $Z_G = D_S(M/Q)$ is a quotient of
$D_S(P/Q)$.  In particular, if $M = P$ (i.e., if $M$ is as big as possible)
then the geometric fibers of $G$ admit no nontrivial central isogenous cover: 
if $M = P$ then we say $G$ is {\em simply connected}.  
\index{simply connected!semisimple group}
\index{reductive group!simply connected}
The Existence Theorem 
implies that a split semisimple group scheme is simply connected
if and only if it has no nontrivial central extension by a finite
group scheme of multiplicative type; 
see Exercise \ref{scuniv}. 
\end{example}

To make the constancy of $W_G(T)$ over $S$ in Proposition \ref{wfinite} more concrete, 
note that for each root $a$ there exists a natural map $W_{Z_G(T_a)}(T) \rightarrow W_G(T)$
that on geometric fibers computes the order-2 subgroup generated
by the involution $s_a:t \mapsto t/a^{\vee}(a(t))$ of $T = D_S(M)$
(dual to the involution $m \mapsto m - a^{\vee}(m)a$ of $M$).   Since
endomorphisms of multiplicative type $S$-groups are uniquely
determined by their effect on geometric fibers over $S$, 
we conclude that each subgroup $W_{Z_G(T_a)}(T)$ is identified with $(\Z/2\Z)_S$ 
having the unique everywhere-nontrivial section correspond
to $m \mapsto m - a^{\vee}(m)a$.   Hence, to construct elements $n_a \in N_G(T)(S)$
representing the reflections $s_a$ in the Weyl group of the root datum (as in the classical theory)
the problem is reduced to the case of 
split reductive groups with semisimple-rank 1.  In such cases
we wish to show that $N_G(T)(S) \rightarrow W_G(T)(S)$ is surjective
by exhibiting an explicit element $n_a \in N_G(T)(S)$ representing $s_a$.
This will be deduced (in Corollary \ref{wconst}) from the following classification
of split semisimple-rank 1 groups Zariski-locally on the base.
\index{semisimple-rank 1}
\index{reductive group!semisimple-rank 1}

\begin{theorem}\label{sslist}
Let $(G,T,M)$ be a split reductive group with geometric fibers of semisimple-rank $1$
over a non-empty scheme $S$. 
Up to forming a direct product against a split central torus, 
Zariski-locally on $S$ the pair
$(G,T)$ is isomorphic to exactly one of the following: 
\begin{itemize}
\item $({\rm{SL}}_2,D)$ with $D$ the diagonal torus, 
\item $({\rm{PGL}}_2, \overline{D})$
with $\overline{D}$ the diagonal torus, 
\item $(\GL_2, \widetilde{D})$ with $\widetilde{D}$ the  diagonal torus. 
\end{itemize}
\end{theorem}

We will later refine this result by constructing a unique such isomorphism globally,
subject to some additional conditions that can always be imposed in the split case
(such as compatibility with linked trivializations of the root spaces $\mathfrak{g}_{\pm a}$).
\index{linked trivialization}
This provides an explicit isomorphism $W_G(T)(S) \simeq W(\Phi)$
as in \cite[XXII, 3.4]{sga3}.

\begin{proof}
The three proposed cases are fiberwise non-isomorphic, so there are
no repetitions in the list. 
The roots $\pm a$ provide a central torus $T_a$ of relative codimension 1 in $T$,
and the classical theory on geometric fibers implies that the center $Z_G/T_a$ of $G/T_a$ is finite.
That is, each geometric fiber of $G/T_a$ is either ${\rm{SL}}_2$ or ${\rm{PGL}}_2$.
Note that $T_a$ is the split torus corresponding to the quotient of $M$ by
the saturation of $\Z a \subset M$.
Since the root datum determines the structure of the center $Z_G \subset T$
on geometric fibers, it follows that $Z_G/T_a$ must have constant fiber degree,
either 1 or 2.  We will treat the two possibilities separately.

First suppose that $G/T_a$ has center of order 2.   By Proposition \ref{sl2def},
working Zariski-locally on $S$ provides an isomorphism $G/T_a \simeq {\rm{SL}}_2$
carrying $T/T_a$ over to $D$.  Thus, $G$ is a central extension of
${\rm{SL}}_2$ by the split torus $T_a$. Applying Proposition \ref{sl2ext}, 
there exists a unique splitting $G = {\rm{SL}}_2 \times T_a$, and clearly
$T$ must then go over to $D \times T_a$. 

Next, suppose that $G/T_a$ has trivial center.  By Proposition \ref{pgl2},
working Zariski-locally on $S$ provides an isomorphism
$G/T_a \simeq {\rm{PGL}}_2$ carrying
$T/T_a$ over to $\overline{D}$. 
Pulling back along the central isogeny $q:{\rm{SL}}_2 \rightarrow {\rm{PGL}}_2$
yields a central extension $\widetilde{G}$ of ${\rm{SL}}_2$
by $T_a$ that is also a central extension of $G$ by $\mu_2$:
$$
\xymatrix{
1 \ar[r] & {T_a} \ar[r]  \ar@{=}[d] 
& {\widetilde{G}} \ar[d] \ar[r] & {{\rm{SL}}_2} \ar[d]^-{q} \ar[r] & 1 \\
1 \ar[r] & {T_a} \ar[r] & G \ar[r] & {{\rm{PGL}}_2} \ar[r] & 1
}
$$
The top row forces $\widetilde{G}$ to be a reductive group scheme,
and $G$ is a central quotient of $\widetilde{G}$ by $\mu_2$, so by Corollary \ref{qtcenter} there is
a unique maximal torus $\widetilde{T}$ of $\widetilde{G}$
satisfying $\widetilde{T}/\mu_2 = T$ inside $\widetilde{G}/\mu_2 = G$.

%We claim that the torus $\widetilde{T}$ is split Zariski-locally on $S$. 
%Since $T$ is split and $\mu_2$ has constant Cartier dual, it suffices to show that 
%any $S$-torus which is an extension of a split $S$-torus torus by the Cartier dual of a constant finite commutative
%$S$-group is split Zariski-locally on $S$.   Dually, if a locally constant abelian \'etale sheaf
%$E$ over $S$ whose stalks are finite free $\Z$-modules is an extension 
%of the constant sheaf $\underline{F}$ associated to a finite abelian group $F$
%by the constant sheaf $\underline{L}$ associated to a finite free $\Z$-module $L$ then we claim that 
%$E$ is constant Zariski-locally on $S$.   
%
%Since the stalks of $E$ are finite free of with the same rank as $L$,
%there is an \'etale cover $S' \rightarrow S$ such that $E_{S'} = \underline{N}$
%for a finite free $\Z$-module $N$ having the same rank as $L$. 
%It suffices to show that if $S$ is local then $E \simeq \underline{N}$.  By shrinking
%$S'$ around a point over the closed point of $S$, we may arrange that the inclusion
%$\underline{L}_{S'} \rightarrow E_{S'} = \underline{N}_{S'}$ arises
%from an injection $L \hookrightarrow N$ having cokernel $F$.  Hence,
%$E$ is specified by a descent datum $\theta:p_1^{\ast}(\underline{N}_{S'}) \simeq p_2^{\ast}(\underline{N}_{S'})$
%over $S'' = S' \times_S S'$ that restricts to the identity automorphism of $\underline{L}_{S''}$
%and induces the identity automorphism of the quotient $\underline{F}_{S''}$. 
%But the only automorphism of $N$ that restricts to the identity on $L$ and
%induces the identity on $F = N/L$ is the identity automorphism of $N$ (since $\Hom(F,N) = 0$),
%so by working Zariski-locally on $S''$ we see that $\theta$ is the identity automorphism of
%$\underline{N}_{S''}$.  Hence, by descent, $E \simeq \underline{N}$ as desired.
%
%We have proved that $\widetilde{T}$ splits Zariski-locally on $S$, so by Zariski localization on $S$ we may arrange
%that the exact sequence of $S$-split tori $1 \rightarrow T_a \rightarrow \widetilde{T} \rightarrow D \rightarrow 1$
%is split-exact.  Thus, 
By Proposition \ref{sl2ext}, $\widetilde{G} = {\rm{SL}}_2 \times T_a$ with
$\widetilde{T}$ going over to $D \times T_a$.
The central subgroup $\mu_2 \subset \widetilde{G} = {\rm{SL}}_2 \times T_a$
has two possibilities on fibers: it is
$\mu_2$ in the ${\rm{SL}}_2$-factor or it is a diagonally embedded
$\mu_2$ in ${\rm{SL}}_2 \times T_a$ via some  inclusion $\mu_2 \hookrightarrow T_a$.
These cases (on fibers) are distinguished by whether or not the projection to $T_a$
kills this central subgroup. 

Since a homomorphism between
multiplicative type $S$-groups is determined over a Zariski-open neighborhood of
a point $s \in S$  by its effect on $\overline{s}$-fibers, 
we conclude that Zariski-locally on $S$ either (i) $G = {\rm{PGL}}_2 \times T_a$
with $T = \overline{D} \times T_a$, or (ii) 
$G = {\rm{SL}}_2 \times^{\mu_2} T_a$
with $T = D \times^{\mu_2} T_a$ for some inclusion $\mu_2 \hookrightarrow T_a$.
Case (i) corresponds to $Z_G = T_a$ being a torus,
and the second case corresponds to $Z_G \simeq \mu_2 \times T_a$ not being a torus
since the structure of $Z_G$ is determined across all fibers by the root datum
(so its fibral isomorphism class is ``constant'').  Thus, it remains to address
the situation when $(G,T)$ falls into case (ii) Zariski-locally on $S$.
In such cases the inclusion $\mu_2 \hookrightarrow T_a$ corresponds
Zariski-locally on $S$ to an index-2 subgroup of the constant group dual to $T_a$,
so we can Zariski-locally split off this $\mu_2$ inside a $\Gm$-factor of $T_a$.
This provides a description of $G$ (Zariski-locally on $S$) as
the direct product of a split torus  against
${\rm{SL}}_2 \times^{\mu_2} \Gm = \GL_2$ 
equipped with the maximal torus $D \times^{\mu_2} \Gm = \widetilde{D}$.
\end{proof}

\begin{corollary}\label{wsec}
Let $G$ be a reductive group over a non-empty scheme $S$, 
and $T$ a maximal torus for which there exists a root $a$
$($and hence a root $-a$$)$.  Let $\mathbf{W}(\mathfrak{g}_a)^{\times}$
\index{ aaawgatimes@$\mathbf{W}(\mathfrak{g}_a)^{\times}$}
denote the  open complement of the identity section in ${\mathbf{W}}(\mathfrak{g}_a)$.
For every section $X$ of ${\mathbf{W}}(\mathfrak{g}_a)^{\times}$,
let $X^{-1}$ denote the dual section of $\mathbf{W}(\mathfrak{g}_{-a})^{\times}$.
\index{ aaaxinv@$X^{-1}$}
Define $w_a:\mathbf{W}(\mathfrak{g}_a)^{\times} \rightarrow G$ by
$$w_a(X) := \exp_a(X) \exp_{-a}(-X^{-1}) \exp_a(X).$$
\index{ aaawax@$w_a(X)$}
\begin{enumerate}
\item The values of $w_a$ lie in $N_{Z_G(T_a)}(T)$ and represent the unique everywhere
nontrivial section of $W_{Z_G(T_a)}(T) = (\Z/2\Z)_S$; i.e., 
$$t \mapsto w_a(X) t w_a(X)^{-1}$$
is the reflection $t \mapsto t/a^{\vee}(a(t))$ in $W(\Phi(G_{\overline{s}},T_{\overline{s}}))$ associated to $a_{\overline{s}}$
for all geometric points $\overline{s}$ of $S$. 
\item  For any unit $c$ on $S$ and sections $X, X'$ of $\mathbf{W}(\mathfrak{g}_a)^{\times}$,
$$w_a(cX) = a^{\vee}(c) w_a(X) = w_a(X) a^{\vee}(c)^{-1},\,\,\,\,
w_a(X)w_a(X') = a^{\vee}(-X {X'}^{-1}).$$
\item  Conjugation by $w_a(X)$ on $U_a \subset G$ is valued in $U_{-a}$ and given by
$$w_a(X) \exp_a(X') w_a(X)^{-1} = \exp_{-a}(-(X^{-1}X')X^{-1}).$$
In particular, $w_a(X) \exp_a(X) w_a(X)^{-1} = \exp_{-a}(-X^{-1})$
and the adjoint action of $w_a(X)$ on $\mathfrak{g}$
satisfies ${\rm{Ad}}_G(w_a(X))(X') = -(X^{-1}X')X^{-1}$.
\item For any $X$ we have
$w_{-a}(X^{-1}) = w_a(X)^{-1} = w_a(-X)$, 
$w_a(X) w_{-a}(Y) = a^{\vee}(XY)$, and 
$w_a(X)^2 = a^{\vee}(-1) \in T$.
\end{enumerate}
\end{corollary}

\begin{proof}
Since $T_a$ centralizes $U_{\pm a}$, it is clear that $w_a$ takes its values in
$Z_G(T_a)$.  We may therefore replace $G$ with its reductive closed subgroup
$Z_G(T_a)$ (that contains $T$ and $U_{\pm a}$) to reduce to the case that
$G$ has all geometric fibers of semisimple-rank 1.  The asserted identities
are all fppf-local on the base, so by working \'etale-locally (or fppf-locally) we can assume
that $T$ is split.  Thus, by Theorem \ref{sslist} we get an explicit
description of $(G,T)$ up to forming a direct product against a split torus. 
Such an additional central torus factor has no effect on the root groups
or the proposed relations, so we are reduced
to the three special cases in Theorem \ref{sslist}.
The third case in Theorem \ref{sslist} reduces to the first case
because in the central pushout  $\GL_2 = {\rm{SL}}_2 \times^{\mu_2} \Gm$ the subtorus $\Gm$ is central
and ${\rm{SL}}_2$ contains the root groups for $D$.

Summarizing, we are reduced to checking the special cases
${\rm{SL}}_2$ and ${\rm{PGL}}_2$ equipped with their diagonal maximal torus.
By composing with the conjugation by the standard Weyl element if necessary
(which induces inversion on the diagonal torus), we may arrange that 
$a$ is the root whose root group $U_a$ is the strictly upper triangular subgroup.
For the ${\rm{SL}}_2$-case, Example \ref{expex} makes
everything explicit.  To be precise, in this case
$$w_a(X) = \begin{pmatrix} 1 & X \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -X^{-1} & 1 \end{pmatrix}
\begin{pmatrix} 1 & X \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & X \\ -X^{-1} & 0 \end{pmatrix},$$
so for $t = {\rm{diag}}(c, 1/c)$ in part (1) we have
$w_a(X) t w_a(X)^{-1} = {\rm{diag}}(1/c,c) = t^{-1}$ and the two formulas in part (2) simply assert
$$\begin{pmatrix} 0 & cX \\ -c^{-1}X^{-1} & 0 \end{pmatrix} =
\begin{pmatrix} c & 0 \\ 0 & c^{-1} \end{pmatrix} \begin{pmatrix} 0 & X \\ -X^{-1} & 0 \end{pmatrix} =
\begin{pmatrix} 0 & X \\ -X^{-1} & 0 \end{pmatrix} \begin{pmatrix} c^{-1} & 0 \\ 0 & c \end{pmatrix},$$
$$
\begin{pmatrix} 0 & X \\ -X^{-1}& 0 \end{pmatrix} \begin{pmatrix} 0 & X' \\ -{X'}^{-1} & 0 \end{pmatrix} =
\begin{pmatrix} -X{X'}^{-1} & 0 \\ 0 & -X'X^{-1} \end{pmatrix}.$$
The identities in parts (3) and (4) are readily verified as well. (Replacing $X'$ with $tX'$ in the displayed formula
in part (3) and differentiating at $t=0$ yields the formula for ${\rm{Ad}}_G(w_a(X))(X')$ via the Chain Rule
and the definition of ${\rm{Ad}}_G$.)

The formulas in Example \ref{expex} are inherited 
by the central quotient ${\rm{PGL}}_2$ when using the diagonal
torus and associated root $\overline{a}: {\rm{diag}}(c,1) \mapsto c$
and coroot $\overline{a}^{\vee}: c \mapsto {\rm{diag}}(c^2,1) = {\rm{diag}}(c,1/c) \bmod \Gm$, so now everything is
reduced to straightforward calculations with the standard root groups, roots, and coroots
for ${\rm{SL}}_2$ and ${\rm{PGL}}_2$ equipped with their diagonal tori. 
%linked trivializations of their root spaces.  For example, 
%if $G = {\rm{SL}}_2$ and we take $X = (\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix})$
%then $w_a(cX) = (\begin{smallmatrix} 0 & c \\ -1/c & 0 \end{smallmatrix})$, so
%$w_a(X) = (\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix})$ is the standard Weyl element
%and the first identity in part (2) says
%$(\begin{smallmatrix} 1 & c \\ 0 & 1 \end{smallmatrix})
%(\begin{smallmatrix} 1 & 0 \\ -1/c & 1 \end{smallmatrix})(\begin{smallmatrix} 1 & c \\ 0 & 1 \end{smallmatrix}) = 
%(\begin{smallmatrix} c & 0 \\ 0 & 1/c \end{smallmatrix})(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}).$
\end{proof}

\begin{remark}\label{cube}
In ${\rm{SL}}_2$ we have
\begin{eqnarray*}
\begin{pmatrix} 0 & X \\ -X^{-1} & 0 \end{pmatrix}\begin{pmatrix} 1 & X \\ 0 & 1 \end{pmatrix} &=&
\begin{pmatrix} 0 & X\\ -X^{-1} & -1 \end{pmatrix},\\
\begin{pmatrix} 0 & X \\ -X^{-1} & -1 \end{pmatrix} \begin{pmatrix} -1 & -X \\X^{-1} & 0 \end{pmatrix}
&=& \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},
\end{eqnarray*}
so it follows from the proof of Corollary \ref{wsec} that 
$$(w_a(X)\exp_a(X))^3 = 1$$
for any section $X$ of $\mathbf{W}(\mathfrak{g}_a)^{\times}$; this
encodes the identity
$(\begin{smallmatrix} 0 & 1 \\ -1 & -1 \end{smallmatrix})^3 = 1$ in
${\rm{SL}}_2$.
This relation is needed when constructing homomorphisms from reductive
group schemes to other group schemes (e.g., isogenies or isomorphisms
between reductive group schemes); see Theorem \ref{bigcrit}. 
Our approach to the group-theoretic relations
among $w_{\pm a}$ and $\exp_{\pm a}$ involves reduction to 
calculations with ${\rm{SL}}_2$ and ${\rm{PGL}}_2$ (because we have already 
obtained a Zariski-local classification result).  The approach
in \cite[XX]{sga3} rests on calculations via a more indirect method.
\end{remark}


Here is the surjectivity of $N_G(T)(S) \rightarrow W_G(T)(S)$ in the split case:

\begin{corollary}\label{wconst}
Let $(G,T,M)$ be a split reductive group over a non-empty scheme $S$.
Fix linked trivializations $X_a \in \mathbf{W}(\mathfrak{g}_a)^{\times}(S)$
\index{linked trivialization}
for all $a \in \Phi$ $($so $X_{-a}$ is dual to $X_a$$)$. 
The natural map $N_G(T)(S) \rightarrow W_G(T)(S)$ is surjective,
with $n_a := w_a(X_a)$ mapping to the reflection $s_a$ in $(\ref{refl})$ and satisfying
\index{ aaana@$n_a$}
$$n_{-a} =  a^{\vee}(-1)n_a,\,\,\,n_a^2 = a^{\vee}(-1),\,\,\,n_a \exp_a(X_a) n_a^{-1} = \exp_{-a}(-X_{-a}).$$
\end{corollary}

\begin{proof}
Since $W_G(T)$ is the constant group associated to $W(\Phi)$, an element
of $W_G(T)(S)$ is a locally constant function valued in $W(\Phi)$.
Thus, by passing to the constituents of a covering of $S$ by pairwise disjoint open sets,
for the proof of surjectivity on $S$-points we can focus on constant functions.
But $W(\Phi)$ is generated by the reflections $s_a$, so
it remains to prove the assertions concerning $n_a$, which are special cases
of (3) and (4) in Corollary \ref{wsec}. 
\end{proof}

The infinitesimal version of relations among roots and root spaces 
\cite[XX, 2.10]{sga3} will not be used
in what follows, but we record it for completeness: 

\begin{corollary}\label{infroot}
Let $G$, $T$, and $a$ be as in Corollary $\ref{wsec}$.  Let $\overline{\pm a} = {\rm{Lie}}(\pm a):
\mathfrak{t} \rightarrow \mathscr{O}_S$ and define $H_a = {\rm{Lie}}(a^{\vee})(1) \in \mathfrak{t}$
using the canonical basis of ${\rm{Lie}}(\Gm)$.  Then
$$\overline{-a} = -\overline{a},\,\,\,H_{-a} = -H_a,\,\,\,\overline{a}(H_a) = 2,$$
and for all local sections $t$ of $T$ and 
$X, X' \in \mathfrak{g}_a$, $Y \in \mathfrak{g}_{-a}$, and $H \in \mathfrak{t}$, we have:
\begin{equation}
{\rm{Ad}}_G(t)(H) = H,\,\,\,
{\rm{Ad}}_G(t)(X) = a(t) X,\,\,\,
{\rm{Ad}}_G(t)(Y) = a(t)^{-1}Y,
\end{equation}
\begin{eqnarray}
{\rm{Ad}}_G(\exp_a(X))(H) = H - \overline{a}(H)X,\,\,\,\,
{\rm{Ad}}_G(\exp_a(X))(X') = X',\\
{\rm{Ad}}_G(\exp_a(X))(Y) = Y + (XY)H_a - (XY)X,
\end{eqnarray}
\begin{equation}
[H,X] = \overline{a}(H)X,\,\,\,[H,Y] = -\overline{a}(H)Y,\,\,\,
[X,Y] = (XY)H_a.
\end{equation}
\end{corollary}

\begin{proof}
As in the proof 
of Corollary \ref{wsec}, we reduce to considering the groups ${\rm{SL}}_2$ and ${\rm{PGL}}_2$ equipped
with their diagonal tori and standard linked root space trivializations, with $a$ as the standard
positive root (whose root group consists of the strictly upper triangular matrices).
In these cases the assertions are straightforward (and classical) calculations. 
\end{proof}

The following relative version of the ``open cell'' generalizes 
(\ref{wtw}) by replacing $Z_G(T_a)$ with $G$ (subject to the  hypothesis
that $(G,T)$ is split).  

\begin{theorem}\label{bigcell} Let $(G,T,M)$ be a split reductive group over
a non-empty scheme $S$.  Fix a positive system of roots $\Phi^+ \subset \Phi$.

For any enumeration $\{a_i\}$ of $\Phi^+$, the multiplication map
$\prod_i U_{a_i} \rightarrow G$ is an isomorphism onto a smooth closed
subgroup $U_{\Phi^+}$
\index{ aaaufiplus@$U_{\Phi^+}$}
 that is normalized by $T$, independent of the choice of enumeration,
and has connected unipotent fibers.  The multiplication map
$$U_{-\Phi^+} \times T \times U_{\Phi^+} \rightarrow G$$
is an isomorphism onto an open subscheme $\Omega_{\Phi^+}$ and 
\index{ aaaomegafiplus@$\Omega_{\Phi^+}$}
the semi-direct product $T \ltimes U_{\Phi^+} \rightarrow G$ is
a closed $S$-subgroup. 
\end{theorem}

The existence of $\Omega_{\Phi^+}$ is given in
\cite[XXII, 4.1.2]{sga3}, but the construction of $U_{\Phi^+}$ as a {\em closed} subgroup in
\cite[XXII, \S5]{sga3} is completely different, resting on a detailed study of
Lie algebras and smoothness properties of normalizers in $G$ for certain
subalgebras of $\mathfrak{g}$ (see \cite[XXII, 5.3.4, 5.6.5]{sga3}). 
Our proof gives
closedness results as a consequence of the
construction of $U_{\Phi^+}$ via the dynamic method, which 
builds the desired ``Borel subgroup'' and its ``unipotent radical'' without
any considerations with root groups. 

\begin{proof}
Pick $\lambda \in M^{\vee}$ so that the open half-space $\{\lambda > 0\}$ in $M_{\RRR}$
meets $\Phi$ in $\Phi^+$.  Interpreting $\lambda$ as a cocharacter $\Gm \rightarrow T$,
it makes sense to form 
the smooth closed $S$-subgroups $U_G(\lambda)$, $P_G(\lambda)$, and $Z_G(\lambda)$
with connected fibers as in Theorem \ref{215}. 
The tangent space to $Z_G(\lambda)$ coincides with
$\mathfrak{t} = \mathfrak{g}_0$ since the only $T$-weight on $\mathfrak{g}$
whose pairing with $\lambda$ vanishes is the trivial weight (as $\Phi = \Phi^+ \coprod -\Phi^+$).
Thus, the inclusion $T \subset Z_G(\lambda)$ between smooth closed $S$-subgroups
with connected fibers induces an equality on Lie algebras (inside
$\mathfrak{g}$) and hence is an equality inside $G$.  That is, $Z_G(\lambda) = T$, so 
$P_G(\lambda) = Z_G(\lambda) \ltimes U_G(\lambda) = T \ltimes U_G(\lambda)$.
The $S$-group $U_G(\lambda)$ has (connected) unipotent fibers, by Theorem \ref{215}(4), so
$P_G(\lambda)$ has (connected) solvable fibers.

By Theorem \ref{215}, the multiplication map
$$U_G(-\lambda) \times T \times U_G(\lambda) \rightarrow G$$
is an open immersion.  Note that $U_G(\lambda)$
has nothing to do with a choice of enumeration of $\Phi^+$.
Also, for $a \in \Phi$ the root group $U_a$ is normalized by $T$ and hence 
is normalized by the 
$\Gm$-action through conjugation by $\lambda$,
with $\lambda(t)$ acting on ${\rm{Lie}}(U_a) = \mathfrak{g}_a$
via scaling by $t^{\langle a, \lambda \rangle}$.  Thus, 
$U_{U_a}(\lambda)$ has Lie algebra $\mathfrak{g}_a$
if $\langle a, \lambda \rangle > 0$ (i.e., if $a \in \Phi^+$)
and has vanishing Lie algebra otherwise (i.e., if $a \in - \Phi^+$).
Since $U_{U_a}(\lambda)$ must be $S$-smooth with connected
fibers (as $U_a$ is!), this $S$-subgroup of $U_a$ vanishes when $a \in -\Phi^+$ and coincides
with $U_a$ when $a \in \Phi^+$.
In particular, $U_a \subset U_G(\lambda)$ for all $a \in \Phi^+$.

It now suffices to show that for any enumeration
$\{a_i\}$ of $\Phi^+$, the multiplication mapping
$$\prod_i U_{a_i} \rightarrow U_G(\lambda)$$
is an isomorphism of $S$-schemes.
By smoothness of both sides, it suffices to check the isomorphism property on geometric
fibers, so we may and do assume $S = \Spec k$ for an algebraically closed field $k$.
The $k$-group $P_G(\lambda) = T \ltimes U_G(\lambda)$ is connected and solvable, so 
dimension considerations with its Lie algebra imply
that it is a Borel subgroup with $U_G(\lambda)$
as its unipotent radical and that the subgroups $U_{a_i}$ must be its root groups.
In the classical theory it is proved that the unipotent radical of a Borel subgroup is
directly spanned (in any order) by its root groups, though this also
follows from general considerations using just the reducedness of
the root system; see \cite[Thm.\,3.3.11]{pred} for such an alternative proof
of direct spanning in the classical case 
(applying \cite[Thm.\,3.3.11]{pred} to the smooth connected unipotent $U_G(\lambda)$).
\end{proof}

As in the classical case, we say that $U_{\Phi^+}$ is {\em directly spanned} 
(in any order) by the $U_{a_i}$'s
for $a_i \in \Phi^+$, and we call $\Omega_{\Phi^+}$ the {\em open cell} (or {\em big cell})
associated to $\Phi^+$.
\index{directly spanned}
\index{open cell}
\index{reductive group!open cell}
The link between the root system and the commutation relations among positive root groups
 carries over as in the classical theory:

\begin{proposition}\label{commrel}
Let $(G,T,M)$ be a split reductive group over a non-empty scheme $S$. 
Pick roots $a, b \in \Phi$ such that $b \ne \pm a$.
Choose trivializations of the root spaces $\mathfrak{g}_c$
for all roots $c = ia + jb$ with integers $i, j > 0$.
Consider the associated parameterizations $p_c:\Add \simeq U_c$,
and fix an enumeration of this set of roots $c$.

The root groups $U_a$ and $U_b$ commute if there are no roots
of the form $ia + jb$ with integers $i, j > 0$, and in general the commutation relation is given by 
\begin{equation}\label{rootcomm}
(p_a(x), p_b(y)) := p_a(x) p_b(y) p_a(-x) p_b(-y) = \prod_{i, j > 0} p_{i a + j b}(C_{i, j, a, b} x^{i} y^{j})
\end{equation}
where the product is taken over all roots $ia + jb$ with $i, j > 0$
and the coefficients $C_{i,j,a,b}$ are global functions on $S$.
\end{proposition}

As in the classical case, the ``structure constants'' $C_{i,j,a,b}$ are mysterious at this stage
of the theory; a detailed study of rank-2 cases will be required to clean them up.
Note that these structure constants  depend on the choice
of ordering among the terms in the product on the right side of (\ref{rootcomm})
(and on the choice of trivializations $p_c$ of the root spaces $\mathfrak{g}_c$);
also, this product involves 
at most 6 terms (as we see by inspecting the classification of reduced rank-2 root systems). 

\begin{proof}
The classical argument via $T$-equivariance will carry over, using 
more care due to the base scheme being rather general.
Pick a positive system of roots $\Phi^+$ containing
$a$ and $b$ (as we can do since $\Phi$ is reduced and $b \ne \pm a$), and choose
an enumeration $\{c_m\}$ of $\Phi^+$ extending the choice of enumeration
of the roots of the form $ia + jb$ with $i, j > 0$.

A priori the commutator $(p_a(x), p_b(y))$ lies in
$U_{\Phi^+} = \prod_m U_{c_m}$, and we have to show that
the only factors $U_c$ which can have a nontrivial component are
for the roots $c = ia + jb$ with $i, j > 0$, and that the factor in such a component has 
the form $p_{ia + jb}(C_{i,j,a,b}x^i y^j)$.  

Consider the expression
$$(p_a(x), p_b(y)) = \prod_{c \in \Phi^+} p_c(h_c(x,y))$$
where the product on the right side is taken in the order according
to the chosen enumeration of $\Phi^+$ and where $h_c:U_a \times U_b \rightarrow U_c$
is a $T$-equivariant map of $S$-schemes.  That is, $h_c(a(t)x, b(t)y) = c(t) h_c(x,y)$
for the $S$-map $h_c:\Add \times \Add \rightarrow \Add$
given by some 2-variable polynomial Zariski-locally over $S$.
Writing $h_c = \sum_{i, j \ge 0} f_{i,j} x^i y^j$ for some
Zariski-local functions $f_{i,j}$ on $S$, 
we have $f_{i,j} a(t)^i b(t)^j = c(t) f_{i,j}$ for all $i, j$.  Setting
$y = 0$ gives $f_{i,0} = 0$ for all $i$ since $p_b(0) = 1$, 
and likewise $f_{0,j} = 0$ for all $j$.   If $i, j > 0$
then $f_{i,j}$ is killed by the character $c - (ia + jb)$ on $T = D_S(M)$
that arises from an element of $M$.  Such a character is either
trivial or fiberwise nontrivial (and hence faithfully flat onto $\Gm$), 
so $f_{i,j} = 0$ except possibly when $c = ia + jb$, in which
case such $i$ and $j$ are uniquely determined by $c$ (since
the distinct positive roots $a$ and $b$ are linearly independent).
In other words, each $h_c$ that is not identically zero is a single
monomial of some constant bi-degree $(i,j)$ such that $ia + jb \in \Phi^+$
and $i, j > 0$.  In particular, if there are no such roots
$ia + jb$ with $i, j > 0$ then $U_a$ commutes with $U_b$. 
\end{proof}

It is useful to generalize the construction of $U_{\Phi^+}$ by constructing fiberwise unipotent
smooth closed subgroups $U_{\Psi} \subset G$ directly
spanned in any order by certain subsets $\Psi \subset \Phi$.
To characterize
the $\Psi$ that we shall consider, we make a brief digression concerning 
general root systems.

Let $(V,\Phi)$ be a (possibly non-reduced) root system, with $V$ a $\Q$-vector space.
Recall that a subset $\Psi \subset \Phi$ is called {\em closed} if
\index{closed set of roots}
\index{root system!closed subset}
$a + b \in \Psi$ for any $a, b \in \Psi$ such that $a + b \in \Phi$. 
Examples of such $\Psi$ are 
$\Phi_{\lambda > q} = \{a \in \Phi\,|\,\lambda(a) > q\}$ and $\Phi_{\lambda \ge q} =
\{a \in \Phi\,|\,\lambda(a) \ge q\}$
for $\lambda \in V^{\ast}$ and $q \in \Q$,
as well as the sets of roots
$$[a,b] = \{ia + jb \in \Phi\,|\,i, j \ge 0\},\,\,\,
(a,b) = \{ia + jb \in \Phi\,|\,i, j \ge 1\},$$
$$[a,b) = \{ia + jb\,|\,i \ge 0, j \ge 1\}$$
for linearly independent $a, b \in \Phi$. 

By \cite[2.2.7]{pred}, the closed sets in $\Phi$ are precisely the subsets of the form
$\Phi \bigcap A$ for a subset $A \subset V$ that is a subsemigroup
(i.e., $a + a' \in A$ for all $a, a' \in A$; we allow $A$ to be empty).
When $\Psi = \Phi_{\lambda > q}$ we can use $A = \{v \in V\,|\,\lambda(v) > q\}$,
but when $\Psi = [a,b)$ for linearly independent $a, b \in \Phi$ there is no ``obvious'' choice for $A$.
For any closed $\Psi \subset \Phi$ there
is a unique minimal choice for $A$, namely the subsemigroup $\langle \Psi \rangle$ generated by $\Psi$
(which is empty when $\Psi$ is empty).    We are interested
in closed $\Psi$ that lie in a positive system of roots.  Such a positive system
of roots is not uniquely determined by $\Psi$, but there is 
a simple characterization for when one exists:

\begin{lemma}\label{psia} Let $(V,\Phi)$ be a root system.  For 
$\Psi \subset \Phi$, the following
are equivalent:
\begin{enumerate}
\item $\Psi$ is closed and is contained in a positive system of roots;
\item  $\Psi = \Phi \bigcap A$ for a subsemigroup $A \subset V$
such that $0 \not\in A$;
\item $\Psi$ is closed and $\Psi \bigcap -\Psi$ is empty.
\end{enumerate}
\end{lemma}

\begin{proof}
Consider a closed set $\Psi$, so $\Psi = \Phi \bigcap A$ for $A = \langle \Psi \rangle$.
The positive systems of roots in $\Phi$ are precisely the subsets $\Phi^+ = \Phi_{\lambda > 0}$
with $\lambda \in V^{\ast}$ that is nonzero on all roots.  If
$\Psi$ is contained in some $\Phi^+ = \Phi_{\lambda > 0}$ then $\langle \Psi \rangle$
lies in $\{v \in V\,|\, \lambda(v) > 0\}$, so $0 \not\in \langle \Psi \rangle$.
Thus, (1) implies (2).  The implication ``$(2) \Rightarrow (3)$'' is trivial ,
and ``$(3) \Rightarrow (1)$'' is precisely \cite[VI, \S1.7, Prop.\:22]{lie4}
(due to the characterization of positive systems of roots in $\Phi$ in terms of
Weyl chambers for $\Phi$ in $V_{\RRR}$, given by \cite[VI, \S1.7, Cor.\:1, Cor.\:2 to Prop.\:20]{lie4}). 
\end{proof}

Here is a generalization of $U_{\Phi^+}$ (inspired by \cite[XXII, 5.9.5]{sga3}).

\begin{proposition}\label{upsi}
Let $(G,T,M)$ be a split reductive group over a non-empty scheme $S$,
and let $\Psi$ be a closed set in $\Phi$ such that $\Psi \bigcap -\Psi = \emptyset$.
\begin{enumerate}
\item For any enumeration $\{a_i\}$ of $\Psi$, the multiplication map
$\prod U_{a_i} \rightarrow G$ is an isomorphism onto a smooth closed
subgroup $U_{\Psi}$. This subgroup is normalized by $T$, independent of the choice of enumeration,
and has connected unipotent fibers. 
\item Choose $\lambda \in M^{\vee}$ that is non-vanishing on $\Phi$ such
that the positive system of roots $\Phi^+ := \Phi_{\lambda > 0}$ contains $\Psi$.
For every integer $n > 0$, let $\Psi_{\ge n} = \Psi \bigcap \Phi_{\lambda \ge n}$.
The subgroups $U_{\ge n} := U_{\Psi_{\ge n}}$ are
normal in $U_{\Psi}$ and 
the multiplication map $$\prod_{a \in \Psi, \lambda(a) = n} U_a \rightarrow U_{\ge n}/U_{\ge n+1}$$
$($with the product taken in any order$)$ is an $S$-group isomorphism.
In particular, $U_{\ge n}/U_{\ge n+1}$ is a power of $\mathbf{G}_{\rm{a}}$ as an $S$-group.
\end{enumerate}
\end{proposition}

By Lemma \ref{psia}, there always exists $\lambda$ as in (2).

\begin{proof}
Choose $\lambda$ as in (2) and let $\Phi^+ = \Phi_{\lambda > 0}$.  
Since $U_{\Phi^+}$ is directly spanned in any order by the root groups $U_a$
for $a \in \Phi^+$, for any enumeration $\{a_i\}$ of $\Psi$ the multiplication map
$\prod U_{a_i} \rightarrow G$ is an isomorphism onto a smooth closed subscheme of
$U_{\Phi^+}$.  If we can prove that this closed subscheme is an $S$-subgroup for 
one choice of enumeration then for {\em any} enumeration the multiplication map is an isomorphism
onto the {\em same} closed $S$-subgroup (because a monic endomorphism of a finitely presented
scheme is necessarily an isomorphism, by \cite[IV$_4$, 17.9.6]{ega}).
Thus, to prove (1) it suffices to consider a single enumeration.
Also, once the existence of $U_{\Psi_{\ge n}}$ is proved for all $n \ge 1$,
it is immediate from (\ref{rootcomm}) in Proposition \ref{commrel} that $U_{\Psi_{\ge n}}$ is normal in $U_{\Psi}$
for all $n \ge 1$.  It is also obvious that such subgroups are normalized by $T$.

For large $m$, $\Psi_{\ge m}$ is empty.  By descending induction on $m$
we shall prove (1) for $\Psi_{\ge m}$ and then (2) for $\Psi_{\ge m}$ when using our initial
choice of $\lambda$.  Since $\Psi = \Psi_{\ge 1}$ and $\lambda$
was arbitrary, this induction will prove (1) and (2) in general.
The base of the induction (large $m$) is obvious, with $U_{\Psi_{\ge m}} = 1$ for large $m$.

Now suppose the cases $m' \ge m+1$ are settled, and consider (1) and (2) for $\Psi_{\ge m}$.
We know that to prove (1) for $\Psi_{\ge m}$ it suffices
to consider one enumeration.  We will use an enumeration that is adapted to
$\lambda$.
It is immediate from (\ref{rootcomm}) that $U_{\ge m+1}$ is normalized  by $U_a$
for all $a \in \Psi_{\ge m}$.  Likewise, 
if $a, b \in \Psi_{\ge m}$ then 
for any $S$-scheme $S'$ and $u_a \in U_a(S')$
and $u_b \in U_b(S')$ we see that  $u_a u_b u_a^{-1} u_b^{-1} \in U_{\ge m+1}(S')$
since $\lambda(ia + jb) \ge m+1$ for any $i, j \ge 1$.  
Letting $\Psi_m = \{a \in \Psi\,|\,\lambda(a) = m\}$, it follows that for any $S$-scheme $S'$
and $a, b \in \Psi_m$ the subgroups $U_a(S')$ and $U_b(S')$ in $G(S')$
commute modulo the subgroup $U_{\ge m+1}(S')$ that they normalize.
Thus, for any enumeration $\{c_i\}$ of $\Psi_m$ 
the monic multiplication map $\prod U_{c_i}(S') \times U_{\ge m+1}(S') \rightarrow G(S')$
has image that is a subgroup.  This proves (1) for $\Psi_{\ge m}$, and (2) for $n = m$ is now obvious
(by consideration of $S'$-valued points for any $S$-scheme $S'$).
\end{proof}

%
%More generally, for any subset $\Sigma \subset \Psi_m$ and enumeration
%$\{\sigma_i\}$ of $\Sigma$ we claim that the direct factor $\prod U_{\sigma_i} \times U_{\ge m+1}$ 
%of $U_{\Phi^+}$ (via multiplication in $G$) is 
%stable under multiplication.  We argue by induction on $\#\Sigma$, the case 
%of empty $\Sigma$ being obvious.  Suppose $\Sigma$ is non-empty and
%the result is known for subsets of $\Psi_m$ of strictly smaller size.
%Let $\Sigma' = \Sigma - \{\sigma_1\}$, so by induction
%$\prod_{i \ge 2} U_{\sigma_i} \times U_{\ge m+1}$ is an $S$-subgroup of $U_{\Phi^+}$.
%By (\ref{rootcomm}) this $S$-subgroup is normalized by $U_{\sigma_1}$,
%so the assertion for $\Sigma$ (and the chosen enumeration $\{\sigma_i\}$) is clear.
%This completes the proof of (1) for $\Psi_{\ge m}$, and visibly $U_{\ge m'}$
%is normal in $U_{\ge m}$ for any $m' \ge m$ (due to (\ref{rootcomm})).  Moreover, 
%the preceding argument shows that the multiplication map
%\begin{equation}\label{uuu}
%\prod_{a \in \Psi_m} U_a \rightarrow U_{\ge m}/U_{\ge m+1}
%\end{equation}
%(for any choice of enumeration of $\Psi_m$) an isomorphism of $S$-schemes.
%Using (\ref{rootcomm}), it follows that the $S$-group $U_{\ge m}/U_{\ge m+1}$
%is commutative, so (\ref{uuu}) is an isomorphism of $S$-groups. 
%This proves (2) for $\Psi_{\ge m}$.
%\end{proof}
%
%We  end this section with an application of the open cell:

%\begin{proposition}\label{imm}
%Any monic homomorphism $f:G \rightarrow H$ from a reductive group scheme to
%a separated group of finite presentation is a closed immersion.
%In particular, if $G$ is an adjoint reductive group then
%${\rm{Ad}}_G:G \rightarrow {\rm{GL}}(\mathfrak{g})$ is
%a closed immersion.
%\end{proposition}

%\begin{proof}
%Use dynamic stuff?
%\end{proof}

We end this section with applications of the open cell over a field.
(See \cite[14.10]{borelag} for an alternative approach via the structure of
automorphisms of connected semisimple groups).

\begin{proposition}\label{simpleprod} 
Let $G$ be a split nontrivial connected semisimple group over a field $k$.
The set $\{G_i\}_{i \in I}$ of minimal nontrivial normal smooth connected $k$-subgroups of $G$ is finite,
the $G_i$'s pairwise commute with each other, and the multiplication homomorphism
$\prod G_i \rightarrow G$ is a central isogeny. 
\end{proposition}

\begin{proof}
Let $T$ be a split maximal $k$-torus in $G$, and $\Phi = \Phi(G,T) \ne \emptyset$.
For each irreducible component $\Phi_i$ of $\Phi$, let $G_i$ be
the smooth connected  $k$-subgroup of $G$ generated by the root
groups $U_a$ for $a \in \Phi_i$.  For any $i' \ne i$
and roots $a' \in \Phi_{i'}$ and $a \in \Phi_i$,
$a + a' \not\in \Phi$ and we can put $a$ and $a'$ into a common positive
system of roots.  Hence, $U_a$ and $U_{a'}$ commute
(Proposition \ref{commrel}), so $G_i$ and $G_{i'}$ commute.
Since the root groups $U_a$ and $U_{-a}$ generate a subgroup containing
$a^{\vee}(\Gm)$, and the coroots generate a finite-index subgroup of ${\rm{X}}_{\ast}(T)$
(as $G$ is semisimple), the collection of all root groups generates a smooth closed
subgroup containing all
factors of the open cell in Theorem \ref{bigcell}. Hence, the $G_i$'s generate $G$, so 
each $G_i$ is normal in $G$ and the product map
$$\pi:\prod G_i \rightarrow G$$
is a surjective homomorphism.  Normality of $G_i$ in $G$ implies
that $G_i$ inherits semisimplicity from $G$.
For $i' \ne i$, the subgroups $G_i$ and $G_{i'}$ commute and are nontrivial and semisimple, so 
$G_i \ne G_{i'}$.

By induction, if $\{N_j\}$ is a finite collection
of pairwise commuting normal smooth closed  $k$-subgroups of
a smooth $k$-group $H$ of finite type 
then the multiplication homomorphism $\prod N_j \rightarrow H$
has central kernel. (This can be generalized to the setting of group sheaves,
as the interested reader can check.) Hence, $\ker \pi$ is central.
But $\prod G_i$ is semisimple, so $\pi$ is a central isogeny.

It remains to show that these $G_i$ are precisely the minimal
nontrivial normal smooth connected $k$-subgroups of $G$.   
We may and do  assume $k = \overline{k}$ since 
the formation of the $G_i$ commutes with extension of the ground field.

Let $N$ be a nontrivial normal smooth connected $k$-subgroup of $G$, so $N$ inherits
semisimplicity from $G$.  Hence, $N$ is non-commutative.  Since $G$ is generated by the pairwise commuting
subgroups $G_i$, there must be some $i$ such that the commutator subgroup $(N,G_i)$ is nontrivial.
But $(N,G_i)$ is normal in $G$ and is contained in $G_i$, so it suffices to show that
each $G_i$ is minimal as a nontrivial normal smooth connected $k$-subgroup of $G$.
Now we can assume $N$ is contained in some $G_{i_0}$
and we seek to show that $N = G_{i_0}$.  
By normality of $N$ in $G$ and Exercise \ref{5.3}(i), $S := T \bigcap N$ is a  maximal
torus in $N$ (so $S \ne 1$). 

Since $T$ is a split maximal torus of $G$ and $\pi$ is an isogeny, each $T_i := T \bigcap G_i$ is a split maximal
torus of $G_i$ and $\pi$ carries $\prod T_i$ isogenously onto $T$. 
Clearly $S = N \bigcap T_{i_0}$, and the connected reductive subgroup $N \cdot T_{i_0}$ in $G$
has maximal torus $T_{i_0}$ and derived group $N$ (as $N$ is semisimple), so $T_{i_0}$
is the almost direct product of $S$ and the maximal central torus $Z$ of $N \cdot T_{i_0}$.
Hence, the isomorphism ${\rm{X}}(T_{i_0})_{\Q} \simeq {\rm{X}}(S)_{\Q} \oplus {\rm{X}}(Z)_{\Q}$
induces a bijection $\Phi(N \cdot T_{i_0}, T_{i_0}) \simeq \Phi(N, S) \times \{0\}$.
In this way $\Phi(N,S)$ spans
a nonzero subspace of ${\rm{X}}(T_{i_0})_{\Q}$ stable under the action of
$W_{G_{i_0}}(T_{i_0}) = W(\Phi_{i_0})$. The Weyl group of 
an irreducible root system $(V,\Psi)$ acts irreducibly on $V$
\cite[VI, \S1.2, Cor.]{lie4}, so if $\Phi(G_{i_0}, T_{i_0})$
is irreducible then $\Phi(N, S)$ spans ${\rm{X}}(T_{i_0})_{\Q} = {\rm{X}}(S)_{\Q} \oplus {\rm{X}}(Z)_{\Q}$, so 
$S = T_{i_0}$ for dimension reasons.  This would force $T_{i_0} \subset N$, so the connected semisimple
$G_{i_0}/N$ would have trivial maximal torus and thus $N = G_{i_0}$ as desired.

Finally, we show that each $({\rm{X}}(T_i)_{\Q}, \Phi(G_i,T_i))$
is irreducible by relating $\Phi(G_i,T_i)$ to the irreducible component $\Phi_i$ 
of $\Phi$.
The center of a connected reductive group lies
in any maximal torus, so the direct product structure of open cells in Theorem \ref{bigcell} 
implies that a central isogeny $H' \rightarrow H$ between connected reductive $k$-groups
induces (for compatible maximal tori of $H'$ and $H$) a natural bijection between the 
collections of root groups as well as an isomorphism between
the root systems and root groups for corresponding roots
(see Exercise \ref{gross1}(i)).  More specifically, the isomorphism
${\rm{X}}(T)_{\Q} \simeq \prod {\rm{X}}(T_i)_{\Q}$
identifies $\Phi(G,T)$ with $\coprod \Phi(G_i,T_i)$.
But if $i' \ne i$ then $T_{i'}$ centralizes $G_i$
and hence centralizes all root groups of $(G_i,T_i)$,
so each $a \in \Phi_i$ kills the image of
$T_{i'}$ in $T$.  Thus, $\Phi_i \subset \Phi(G_i,T_i)$
inside ${\rm{X}}(T)_{\Q}$, so the definition of
the $\Phi_i$ as the irreducible components of $\Phi$ forces
$\Phi_i = \Phi(G_i,T_i)$ for all $i$. 
\end{proof}

%To prove that the pairwise distinct $G_i$ are 
%minimal among the nontrivial normal smooth connected subgroups $N$ of $G$, 
%we need to show that any such $N$ contains some $G_i$.
%By normality in $G$, $N$ is semisimple and $S := T \bigcap N$ is a maximal torus in $N$.  The
%root spaces in ${\rm{Lie}}(N)$ for the $S$-action are also $T$-stable
%(as $T$ centralizes $S$), so by the uniqueness of root groups in the classical theory
%\cite[13.18(4d)]{borelag} each root group for $(N,S)$ is a root group for $(G,T)$.
%Hence, for some $i$ and $a \in \Phi_i$, the root group $U_a$ lies in $N$.
%We claim that $G_i \subset N$.  The intersection $H := (N \bigcap G_i)_{\rm{red}}^0$ is a
%nontrivial normal smooth connected subgroup of $G_i$, and it suffices
%to show that $H = G_i$, so by renaming $G_i$ as $G$ we can assume $\Phi$ is irreducible
%and aim to show that $N = G$.  
%
%Since $S = N \bigcap T_{i_0}$, the set $\Phi(N, S)$ spans
%a nonzero subspace of ${\rm{X}}(T_{i_0})_{\Q}$ that is stable under the action of
%$W_{G_{i_0}}(T_{i_0}) = W(\Phi_{i_0})$.  But the Weyl group of 
%an irreducible root system $(V,\Psi)$ always acts irreducibly on $V$
%\cite[VI, \S1.2, Cor.]{lie4}, so $\Phi(N, S)$ spans ${\rm{X}}(T_{i_0})_{\Q}$
%and hence $S = T_{i_0}$.  Thus, $T_{i_0} \subset N$, so the connected semisimple
%group $G_{i_0}/N$ has trivial maximal torus; i.e., $N = G_{i_0}$. 
%\end{proof}

A connected semisimple group $H$ over a field $k$ is {\em $k$-simple}
\index{ksimple@$k$-simple} 
if $H \ne 1$ and $H$ has no nontrivial normal smooth connected proper $k$-subgroup,
and {\em absolutely simple}
\index{absolutely simple}
 if $H_K$ is $K$-simple for some (equivalently, any)
algebraically closed extension $K/k$.  For the groups $G_i$ in Proposition \ref{simpleprod}, 
if $T$ is a split maximal torus of $G$
then $T_i := T \bigcap G_i$ is a split maximal torus of $G_i$ and the proof of Proposition \ref{simpleprod}
shows that the isomorphism ${\rm{X}}(T)_{\Q} \simeq
\prod {\rm{X}}(T_i)_{\Q}$ identifies the $\Phi(G_i,T_i)$ with
the irreducible components of $\Phi$.  In particular:

\begin{corollary}\label{simplephi}
A nontrivial connected semisimple group $G$ over a field 
is absolutely simple if and only if the root system for $G_{k_s}$ is irreducible.
\end{corollary}

It follows that the $G_i$ in Proposition \ref{simpleprod} are absolutely simple.
Here is a generalization of Proposition \ref{simpleprod} beyond the split case.

\begin{theorem}[Decomposition theorem for semisimple groups]
\label{normsubgp} Let $G$ be a nontrivial connected semisimple group over a
field $k$.  The set $\{G_i\}_{i \in I}$ of minimal nontrivial normal smooth connected
$k$-subgroups of $G$ is finite, each $G_i$ is $k$-simple,
the $G_i$'s pairwise commute, and the multiplication homomorphism
$$\prod G_i \rightarrow G$$
is a central isogeny.
\index{semisimple group!$k$-simple factors|(}

For each $J \subset I$ the normal connected semisimple $k$-subgroup $G_J \subset G$ generated
by $\{G_i\}_{i \in J}$ has as its minimal nontrivial normal smooth
connected $k$-subgroups precisely the $G_i$ for $i \in J$, and 
every normal smooth connected $k$-subgroup $N \subset G$ equals $G_J$ for a unique $J$.
In particular, for each $N$ there exists a unique $N'$ that commutes with $N$ and makes
the multiplication homomorphism 
$N \times N' \rightarrow G$ a central isogeny.
\end{theorem}

See Exercise \ref{reddecomp} for the generalization 
to connected reductive $k$-groups $G$.

\begin{proof}
We first treat the case $k = k_s$, and then will deduce the general case by Galois descent.
Assuming $k = k_s$, $G$ is split and we can apply Proposition \ref{simpleprod}. 
Letting the $G_i$ be as in that result, we have proved their 
simplicity and that they pairwise commute and define a central isogeny
$\prod G_i \rightarrow G$.   Note also that each $G_J$ is semisimple, due
to normality in $G$.

For each non-empty $J$ the natural map $\prod_{i \in J} G_i \rightarrow G_J$ is a central isogeny, 
so by root system considerations (applying Proposition \ref{simpleprod}
to the split $G_J$), the set of minimal nontrivial normal smooth
connected subgroups of $G_J$ is exhausted by the $G_i$'s for $i \in J$.
To show that every $N$ has the form $G_J$ for some $J$, 
 we can assume
$N \ne 1$.  Thus, $N$ contains some $G_{i_0}$
and (by consideration of
root systems) the minimal nontrivial normal smooth connected subgroups of $\overline{G} = G/G_{i_0}$
are the images $\overline{G}_i$ of the $G_i$ for $i \ne i_0$.
By induction on dimension, 
$\overline{N} := N/G_{i_0}$ is equal to $\overline{G}_{J_0}$ for a subset $J_0 \subset I - \{i_0\}$,
so $N = G_J$ for $J = J_0 \bigcup \{i_0\}$.

Finally, we consider general $k$.  Let $\Gamma = {\rm{Gal}}(k_s/k)$.
Let $\{G'_i\}_{i \in I}$ be the set of minimal nontrivial normal smooth
connected $k_s$-subgroups of $G' = G_{k_s}$, so
$\Gamma$ naturally permutes these subgroups
and hence acts on the index set $I$.  
For each $\Gamma$-stable subset $J \subset I$,
$G'_J$ descends to a  normal smooth connected $k$-subgroup $G_J \subset G$,
and by Galois descent these $G_J$ are precisely the normal smooth connected
$k$-subgroups of $G$.  Hence, the minimal nontrivial ones
are the groups $G_J$ for $J$ a $\Gamma$-orbit in $I$.  
Since the $\Gamma$-stable subsets of $I$ are precisely the unions of
$\Gamma$-orbits, we are done.
\end{proof}

The $G_i$ in Theorem \ref{normsubgp} are
called the {\em $k$-simple factors} of $G$.  
\index{semisimple group!$k$-simple factors|)}
The formation of the
set of $G_i$'s is sensitive to extension of the ground field:

\begin{example}\label{weilsimple}  Consider the Weil restriction $G = {\rm{R}}_{k'/k}(G')$ for
\index{Weil restriction}
a finite separable extension $k'/k$ and an {\em absolutely simple} and semisimple
$k'$-group $G'$. Since $k' \otimes_k k_s$ is a product of copies of $k_s$
indexed by the set of $k$-embeddings $\sigma:k' \rightarrow k_s$, 
$G_{k_s} = \prod_{\sigma} G'_{\sigma}$ where 
$G'_{\sigma} = k_s \otimes_{\sigma, k'} G'$.  In particular,
$G$ is connected semisimple and its simple factors over $k_s$ are the $G'_{\sigma}$.
But these are permuted transitively by ${\rm{Gal}}(k_s/k)$, so 
$G$ is $k$-simple.  If $G'_{k'_s}$ has a root datum that is semisimple and simply connected
(resp.\,adjoint) then so does $G_{k_s}$.
\end{example}

Theorem \ref{normsubgp} shows that, up to central isogeny,
to classify connected semisimple groups
over a field $k$, the main case  is the $k$-simple case.
Remarkably, the $k$-simple case is always related
to the absolutely simple case over a finite separable extension via the construction in Example \ref{weilsimple}
up to a simply connected hypothesis.
We will address this more fully in Example \ref{weilfield}, as an application
of classification theorems in terms of root data.

\subsection{Parabolic subgroups and conjugacy}\label{parsec}

In the classical theory one
defines parabolic subgroups $P \subset G$ in terms of the structure of $G/P$
and uses this to infer properties such as $P = N_G(P)$ (at least
on geometric points) and the connectedness of such subgroups. 
In the version over a base scheme we will first prove 
that parabolic subgroups are their own schematic normalizers and {\em use}
that fact to construct $G/P$ as a scheme projective over the base.

\begin{definition} A {\em parabolic subgroup}
\index{parabolic subgroup}
\index{reductive group!parabolic subgroup}
of a reductive group scheme $G \rightarrow S$ is a smooth $S$-affine $S$-group $P$
equipped with a monic homomorphism
$P \rightarrow G$ such that $P_{\overline{s}}$ is parabolic
in $G_{\overline{s}}$ 
(i.e., $G_{\overline{s}}/P_{\overline{s}}$ is proper) for all $s \in S$.
\end{definition}

Note that all fibers $P_{\overline{s}}$ are connected, by the classical theory.  We do 
not require $P \rightarrow G$ to be a closed immersion, but 
it will soon be proved that this condition does hold.
Here is a natural class of examples
arising from the dynamic method in \S\ref{dynsec}.

\begin{example}\label{relparex} Let $T \subset G$ be a maximal $S$-torus,
and $\lambda:\Gm \rightarrow T$ a cocharacter.
The smooth closed $S$-subgroup $P_G(\lambda)$ is parabolic.
Indeed, its fiber at a geometric point $\overline{s}$ of $S$ is
$P_{G_{\overline{s}}}(\lambda_{\overline{s}})$,
and the classical theory implies that such subgroups are always
parabolic (see Example \ref{parred}).  
\end{example}

The dynamic description of parabolic subgroups
over an algebraically closed field (Example \ref{parred})
admits a relative formulation over any scheme:

\begin{proposition}\label{parpar} Let $G \rightarrow S$ be a reductive group scheme,
and $Q$ a parabolic subgroup of $G$.  Then $Q \rightarrow G$ is a closed
immersion, and \'etale-locally
on $S$ there exists a maximal torus $T$ of $G$ such that $T \subset Q$. If 
$G$ admits a maximal torus $T \subset Q$ and $(G,T)$ is split
then Zariski-locally on $S$ there exists $\lambda:\Gm \rightarrow T$ such that
$Q = P_G(\lambda)$.
\end{proposition}
\index{dynamic method!parabolic subgroups}

Our proof of the closedness of $Q$ in $G$ uses the dynamic method;
the proof in \cite[XXII, 5.8.5]{sga3} is rather different.  

\begin{proof}
On geometric fibers over $S$, since $Q_{\overline{s}}$
is parabolic in $G_{\overline{s}}$ we see that a maximal torus in 
$Q_{\overline{s}}$ is its own scheme-theoretic centralizer in $Q_{\overline{s}}$.
Hence, we may apply Theorem \ref{schemetori} to $Q$,
so by working \'etale-locally on $S$ (as we may do
for verifying that $Q \rightarrow G$ is a closed immersion) we obtain a maximal torus
$T$ of $Q$.  Obviously $T$ is also a maximal torus of $G$
(as this is an assertion on geometric fibers that is trivial to verify).

By further \'etale localization on $S$, we may 
suppose that $(G,T)$ arises from a split
triple $(G,T,M)$ and that $S \ne \emptyset$.  For each $s \in S$, the cocharacters of
$T_{\overline{s}}$ coincide with the cocharacters of $T_s$ over $k(s)$
since $T_s$ is split, so by Example \ref{parred}
there exists a cocharacter $\lambda_s:\Gm \rightarrow T_s$ over $k(s)$
such that $Q_{\overline{s}} = P_{G_{\overline{s}}}(\lambda_{\overline{s}}) = 
P_{G_s}(\lambda_s)_{\overline{s}}$.  Hence,
$Q_s = P_{G_s}(\lambda_s)$.  The split condition on $T$ provides
a Zariski-open neighborhood of $s$ over which $\lambda_s$ lifts
to a cocharacter $\lambda:\Gm \rightarrow T$.  Then 
we may work Zariski-locally around $s$ in $S$ to arrange that
$\lambda \in M^{\vee}$.  Clearly 
$P_G(\lambda)$ is a {\em closed} parabolic subgroup of $G$
that contains $T$ and has $s$-fiber $Q_s$.  We will prove that $Q = P_G(\lambda)$
over a Zariski-open neighborhood of $s$ in $S$.

Consider the Lie algebra $\mathfrak{q}$ of $Q$ inside 
$\mathfrak{g}$.  Although we do not yet know that
$Q$ is closed in $G$, nonetheless the inclusion
$\mathfrak{q} \hookrightarrow \mathfrak{g}$
of $\mathscr{O}_S$-modules is a subbundle because
on geometric fibers over $S$
the inclusion $Q_{\overline{s}} \hookrightarrow G_{\overline{s}}$
is a closed immersion.  Since $\mathfrak{q}$ is stable under the adjoint action
on $\mathfrak{g}$ by the split torus $T \subset Q$,
by working Zariski-locally on $S$ around $s$ we can arrange that
$\mathfrak{q}$ is a direct sum of $\mathfrak{t}$ and weight spaces
$\mathfrak{g}_a$ for some roots $a \in \Phi \subset M$.
Since $Q_{\overline{s}} = P_G(\lambda)_{\overline{s}}$, 
the $a$ which arise in this way are precisely
those that satisfy $\lambda(a) \ge 0$.  Hence, the smooth closed subgroup
$P_Q(\lambda) \subset Q$ has full Lie algebra, forcing
$P_Q(\lambda) = Q$.  

By the functoriality in Proposition \ref{functor}(2) (applied
to the $\Gm$-equivariant homomorphism $Q \rightarrow G$), the map $Q \rightarrow G$
factors through $P_G(\lambda)$.  The resulting map of smooth $S$-affine $S$-groups 
$j:Q \rightarrow P_G(\lambda)$ is an isomorphism on Lie algebras
inside $\mathfrak{g}$.  On geometric
fibers over $S$ the map $j$ induces a closed immersion between smooth connected affine groups,
so the isomorphism property on Lie algebras forces $j$ to be an isomorphism
between geometric fibers over $S$.  Hence, $j$ is an isomorphism. 
In particular, $Q$ is closed in $G$. 
\end{proof}

\begin{remark}\label{splitgt} In the Borel--Tits structure theory for connected
reductive groups over fields, 
the dynamic description $P_G(\lambda)$ of $Q$ in terms of a cocharacter
$\lambda:\Gm \rightarrow T$ over the base field
is valid without a split hypothesis on $(G,T)$ (see \cite[Prop.\,2.2.9]{pred}). 
Thus, if $S = \Spec R$ for a henselian local ring $R$ then we can remove the split
hypothesis on $T$ in Proposition \ref{parpar}.  Indeed, it suffices 
to show that any cocharacter $\lambda_0:\Gm \rightarrow T_0$ over the residue field  $k$
lifts to a cocharacter $\Gm \rightarrow T$ over $R$.
In terms of the \'etale sheaf $E$ dual to $T$,
this is precisely the  surjectivity of $E(R) \rightarrow E(k)$, which in turn is an immediate
consequence of the henselian property of $R$.  Removing the split hypothesis
over more general rings (and hence removing it from the end of Proposition \ref{parpar}
over more general schemes $S$) is rather more delicate, and we will return
to this near the end of \S\ref{leviapp}. 
\end{remark}


We now get many nice consequences, which we give in a series of corollaries.

\begin{corollary}\label{rup}
Let $G \rightarrow S$ be a reductive group scheme, and $P \subset G$ a parabolic subgroup. 
There is a unique smooth closed normal $S$-subgroup $\mathscr{R}_u(P) \subset P$
whose geometric fiber $\mathscr{R}_u(P)_{\overline{s}}$ coincides
with the unipotent radical $\mathscr{R}_u(P_{\overline{s}})$ for all $s \in S$.

The quotient $P/\mathscr{R}_u(P)$ is represented by
a reductive group scheme, and any surjective homomorphism from $P$ onto a reductive
$S$-group uniquely factors through $P/\mathscr{R}_u(P)$.
\end{corollary}

We call $\mathscr{R}_u(P)$ the {\em unipotent radical} of $P$.
\index{parabolic subgroup!unipotent radical}
\index{ aaarup@$\mathscr{R}_u(P)$}
\index{unipotent radical!of parabolic subgroup}
The proof of Corollary \ref{rup}
uses the dynamic method; an alternative is in \cite[XXII, 5.11.3, 5.11.4(ii)]{sga3}. 

\begin{proof}
In view of the uniqueness we may work \'etale-locally on $S$, so by
Proposition \ref{parpar} we may arrange that
$G$ contains a split maximal torus $T$ and 
that $P = P_G(\lambda) = Z_G(\lambda) \ltimes U_G(\lambda)$
for some cocharacter $\lambda:\Gm \rightarrow T$
(so $T \subset Z_G(\lambda) \subset P_G(\lambda) = P$).
It is clear that $U_G(\lambda)$ satisfies the requirements
to be $\mathscr{R}_u(P)$ except possibly for the uniqueness and the universal
mapping property relative to homomorphisms from $P$ onto reductive $S$-groups.

Suppose that $N \subset P$ is a smooth closed normal $S$-subgroup
such that $N_{\overline{s}} = \mathscr{R}_u(P_{\overline{s}})$ for
all $s \in S$.   Normality of $N$ in $P$ implies that
$N$ is normalized by $T$ (as $T \subset P$), so it makes
sense to form the smooth closed $S$-subgroup $U_N(\lambda)$
in $N$.  The $\overline{s}$-fiber of $U_N(\lambda)$ has
the same Lie algebra as $N$ (since $N_{\overline{s}} = \mathscr{R}_u(P_{\overline{s}}) = 
U_{G_{\overline{s}}}(\lambda_{\overline{s}})$), so
the closed immersion $U_N(\lambda) \hookrightarrow N$
between smooth $S$-groups is an isomorphism on Lie algebras
and hence an isomorphism on fibers (due to the {\em connectedness} of
each $N_{\overline{s}}$).  It follows that
$N = U_N(\lambda) \subset U_G(\lambda)$.  But by hypothesis
this inclusion between smooth closed $S$-subgroups of $G$ induces 
an equality on geometric fibers over $S$, so $N = U_G(\lambda)$.
This establishes the uniqueness of $\mathscr{R}_u(P)$.

Next, consider a surjective homomorphism $f:P \twoheadrightarrow \overline{G}$
onto a reductive $S$-group.  We want
$f$ to kill $\mathscr{R}_u(P)$.  Working \'etale-locally on $S$, by Proposition \ref{parpar} 
we may arrange that $P = P_G(\lambda)$, so
$\mathscr{R}_u(P) = U_G(\lambda)$.  By Proposition \ref{functor}(2), 
$f$ carries $U_G(\lambda)$ onto $U_{\overline{G}}(f \circ \lambda)$
(so $f$ makes $U_G(\lambda)$ an fppf cover of $U_{\overline{G}}(f \circ \lambda)$, by the fibral flatness criterion). 
But $U_G(\lambda)$ is normal in $P$, so since
$f$ and its restriction $U_G(\lambda) \rightarrow U_{\overline{G}}(f \circ \lambda)$ are fppf covers, 
it follows that $U_{\overline{G}}(f \circ \lambda)$ is
normal in $\overline{G}$.  The $S$-smooth $U_{\overline{G}}(f \circ \lambda)$
has connected unipotent fibers, so normality in
the reductive $S$-group $\overline{G}$ forces
$U_{\overline{G}}(f \circ \lambda)$ to have relative dimension 0
and therefore be trivial.  This says exactly that $f$ kills $\mathscr{R}_u(P)$.
\end{proof}

\begin{remark}\label{uarem}
As an application of Corollary \ref{rup}, we can construct
many smooth closed subgroups of $G$
directly spanned in any order by
certain collections of root groups.
This rests on the notion of {\em parabolicity}
for subsets $\Psi$ of a root system $\Phi$; see \cite[VI, \S1.7, Def.\,4]{lie4}.
\index{root system!parabolic subset}
\index{parabolic set of roots}
These are the subsets 
$$\Phi_{\lambda \ge 0} := \{a \in \Phi\,|\,\lambda(a) \ge 0\}$$
for linear forms $\lambda$ on the $\Q$-span of $\Phi$
(see \cite[Prop.\,2.2.8]{pred} for a proof), and each contains
a positive system of roots.  (See Definition \ref{parsubdef}ff.) 

Consider
subsets $\Psi \subset \Phi$ whose complement
is parabolic; i.e.,  $\Psi = \Phi_{\lambda < 0}$
for some $\lambda$ (or equivalently
$\Psi = \Phi_{\lambda > 0}$ for some $\lambda$).  An interesting example of such a 
subset for reduced $\Phi$ is $\Psi = \Phi^+ - \{a\}$ for a positive system
of roots $\Phi^+$ and a root $a$ in the base $\Delta$ of $\Phi^+$.
To see that this $\Psi$ has the asserted form, we may assume $\#\Delta > 1$ (as otherwise
$\Psi$ is empty, a trivial case). Enumerating $\Delta$ as $\{a = a_1, \dots, a_m\}$, we have
$\Psi = \Phi_{\lambda > 0}$ where $\lambda := \sum_j a_j^{\ast}$ for the basis $\{a_j^{\ast}\}$
dual to the basis $\{a_j\}$ of the $\Q$-span of $\Psi$.

For a split reductive group $(G,T,M)$
and the complement $\Psi$ of a parabolic subset in the associated root system $\Phi$, 
so $\Psi = \Phi_{\lambda > 0}$ for some $\lambda$, 
we claim that $U_G(\lambda)$  coincides with the $S$-group $U_{\Psi}$ from
Proposition \ref{upsi} (so it depends only on $\Psi$, not on the choice of $\lambda$).  
Since ${\rm{Lie}}(U_G(\lambda))$ is
spanned by the weight spaces $\mathfrak{g}_a$ for such $a$,
as is ${\rm{Lie}}(U_{\Psi})$, if there is an inclusion
$U_{\Psi} \subset U_G(\lambda)$ as smooth closed $S$-subgroups of $G$
then it must be an equality (as we may check on geometric fibers,
where smoothness and connectedness reduces the problem to the known equality of Lie algebras).
Hence, it suffices to show that $U_G(\lambda)$ contains $U_a$
for all $a \in \Phi_{\lambda > 0}$.  The explicit description of
the $T$-conjugation action on $U_a$ and the functorial
definition of $U_G(\lambda)$ imply that $U_a \subset U_G(\lambda)$
since $\langle \lambda, a \rangle > 0$.

The dependence of $U_G(\lambda)$ on $\Psi$ rather
than on $\lambda$ can be proved in another way: this $S$-group is the unipotent radical
of the parabolic subgroup $P_G(\lambda)$ containing
$T$ (in the sense of Corollary \ref{rup}), and $P_G(\lambda)$ only depends
on $\Psi$ due to Corollary \ref{parfiber}(2) below.
\end{remark}

\begin{corollary}\label{parfiber}
Let $G \rightarrow S$ be a reductive group, $s \in S$ a point,
and $P, Q \subset G$ parabolic subgroups.
\begin{enumerate}
\item If $P_{\overline{s}}$ is conjugate to
$Q_{\overline{s}}$ for some $s \in S$ then there exists an \'etale neighborhood
$U$ of $(S,s)$ such that $P_U$ is $G(U)$-conjugate to $Q_U$.
In particular, if $P$ and $Q$ are conjugate over all geometric points of $S$ then they are conjugate
\'etale-locally on $S$.
\item Assume $P$ and $Q$ contain a common maximal torus 
$T$ of $G$. If ${\rm{Lie}}(Q_{\overline{s}}) \subset {\rm{Lie}}(P_{\overline{s}})$
inside $\mathfrak{g}_{\overline{s}}$ then $Q_V \subset P_V$ 
for some Zariski-open neighborhood $V$ of $s$ in $S$.  In particular, if
${\rm{Lie}}(P_{\overline{s}}) = {\rm{Lie}}(Q_{\overline{s}})$ inside $\mathfrak{g}_{\overline{s}}$
then $P$ and $Q$ coincide over a Zariski-open neighborhood of $s$ in $S$.
\end{enumerate}
\end{corollary}
\index{conjugacy!parabolic subgroups}
\index{parabolic subgroup!conjugacy}
\index{parabolic subgroup!Lie algebra}
\index{parabolic subgroup!normalizer}
\index{normalizer!parabolic subgroup}
\index{Lie algebra!parabolic subgroup}

This result is a special case of \cite[XXII, 5.3.7, 5.3.11]{sga3}.

\begin{proof}
We first treat (2).  Since \'etale maps are open, we may
pass to an \'etale neighborhood of $(S,s)$
to split $T$.  By working Zariski-locally around $s$ we can
arrange that $(G,T)$ arises from a split triple
$(G,T,M)$.  Further Zariski localization brings us to the case 
$P = P_G(\lambda)$ and $Q = P_G(\mu)$ for cocharacters
$\lambda, \mu:\Gm \rightrightarrows T$ arising from $M^{\vee}$
(Proposition \ref{parpar}).  The containment ${\rm{Lie}}(Q_{\overline{s}}) \subset
{\rm{Lie}}(P_{\overline{s}})$
implies that $\Phi_{\mu \ge 0} \subset \Phi_{\lambda \ge 0}$
inside $\Phi$, as these are precisely the roots that appear
in the respective Lie algebras of $Q_{\overline{s}}$ and $P_{\overline{s}}$
inside $\mathfrak{g}_{\overline{s}}$.
But the containment $\Phi_{\mu \ge 0} \subset \Phi_{\lambda \ge 0}$ implies
that the smooth closed subgroup $P_Q(\lambda) \subset Q$ has full Lie algebra,
so $Q = P_Q(\lambda) \subset P_G(\lambda) = P$ inside $G$.  

Now consider (1).  By working 
\'etale-locally around $(S,s)$ we may assume
$P$ contains a split maximal torus $T$ of $G$ and that $Q$ contains a split
maximal torus $T'$ of $G$.  By Corollary \ref{etconj} we may pass
to a further \'etale neighborhood so that $T$ and $T'$ are $G(S)$-conjugate, so 
applying such conjugacy brings us to the case that $T' = T$; i.e., $P$
and $Q$ contain a common split maximal torus $T$.  Further Zariski
localization provides a split triple $(G,T,M)$.
By hypothesis there exists $g \in G(\overline{s})$ such that $gP_{\overline{s}}g^{-1} = Q_{\overline{s}}$.
Since $T_{\overline{s}}$ and $gT_{\overline{s}}g^{-1}$
are maximal tori in $Q_{\overline{s}}$, there exists
$q \in Q(\overline{s})$ such that $qgT_{\overline{s}}g^{-1}q^{-1} = T_{\overline{s}}$,
so in other words $qg \in N_G(T)(\overline{s})$ and
its class $w_0 \in W_G(T)(\overline{s})$ carries $P_{\overline{s}}$ to $Q_{\overline{s}}$.

The $S$-group $W_G(T) \rightarrow S$ is the finite constant group $W(\Phi)_S$
(Proposition \ref{wfinite}),
so $w_0$ spreads over a Zariski-open neighborhood of $(S,s)$.  Further Zariski-localization
then lifts the resulting point of $W_G(T)(S) = (N_G(T)/T)(S)$ 
to $N_G(T)(S)$ since $T$ is $S$-split (see Corollary \ref{wconst}).  Passing to such
a neighborhood yields some $n \in N_G(T)(S)$ such that $nPn^{-1}$ and $Q$
have the same $s$-fiber inside $G_s$.  But these contain $T$, so by (2)
there exists a Zariski-open neighborhood of $s$ in $S$ over which 
$nPn^{-1}$ and $Q$ coincide. 
\end{proof}

\begin{corollary}\label{parnorm}
For any parabolic subgroup $P$ in a reductive group scheme $G$,
the normalizer functor $\underline{N}_G(P)$ is represented by $P$.

The quotient sheaf $G/P$ for the \'etale topology
on the category of $S$-schemes coincides 
with the functor of subgroups of $G$ \'etale-locally conjugate
to $P$, and it is represented by a smooth proper
$S$-scheme equipped with a canonical $S$-ample line bundle.
Explicitly, if $\mathscr{P} \subset G \times (G/P)$ is the ``universal parabolic
subgroup locally conjugate to $P$'' then $\Det({\rm{Lie}}(\mathscr{P}))^{\ast}$
is $S$-ample on $G/P$.
\end{corollary}

The self-normalizer property in the 
special case $G = {\rm{PGL}}_2$ was handled in the proof of Proposition \ref{pgl2},
using the smallness of $\dim {\rm{PGL}}_2$ to give a simple argument.
The general case is proved in \cite[XXII, 5.8.5]{sga3} by a method different from the one below. 

\begin{proof}
Since the $S$-smooth $P$ is closed in $G$ and has connected fibers,
Proposition \ref{conntrans} ensures that $\underline{N}_G(P)$ is 
represented by a finitely presented closed subscheme $N_G(P)$ of $G$.
Beware that whereas normalizers
of multiplicative type subgroups in smooth affine
groups are always smooth (Proposition \ref{xi2.4bis}), in the setting of Proposition \ref{conntrans}
not even flatness of the normalizer is assured. 
Nonetheless, we do have an inclusion $P \subset N_G(P)$ as 
finitely presented closed subschemes of $G$, with $P$ flat (even smooth)
over $S$.  Hence, by Lemma \ref{flatclosed}, this inclusion is an equality if it is so on
geometric fibers over $S$.  That is, we are reduced to the classical
case $S = \Spec k$ for an algebraically closed field $k$.

By the classical theory, $P(k) = N_G(P)(k)$ inside  $G(k)$. Hence, the closed
$k$-subgroup scheme 
$N_G(P)$ in $G$ has the same dimension as $P$.  We need to establish
$P = N_G(P)$ as schemes. It  suffices to verify that $N_G(P)$ is smooth, or
equivalently (since its dimension is $\dim P$) that
${\rm{Lie}}(N_G(P)) = {\rm{Lie}}(P)$ inside  $\mathfrak{g}$.
Consider the explicit description $P = P_G(\lambda)$
for some cocharacter $\lambda:\Gm \rightarrow G$
valued in a maximal torus $T$ of $G$ that necessarily lies in $P$
(since $P_G(\lambda) \supset Z_G(\lambda) \supset Z_G(T) = T$).
If the $T$-equivariant inclusion ${\rm{Lie}}(P) \subset {\rm{Lie}}(N_G(P))$ is not an equality then 
there is some $X \in {\rm{Lie}}(N_G(P))$ not in ${\rm{Lie}}(P) = \mathfrak{g}_{\lambda \ge 0}$
that is a $T$-eigenvector for some weight $a:T \rightarrow \Gm$, so $\langle a, \lambda \rangle < 0$.
In particular, $a \ne 1$, so there is $t \in T(k)$ such that $a(t) \ne 1$. Since 
$X \in {\rm{Lie}}(N_G(P)) = \ker(N_G(P)(k[\epsilon]) \rightarrow N_G(P)(k))$, 
${\rm{Ad}}_G(h)(X) - X \in {\rm{Lie}}(P)$ for all
$h \in P(k)$.  Taking $h = t$, ${\rm{Ad}}_G(t)(X) - X = (a(t) - 1)X$ with $a(t) - 1 \in k^{\times}$.
This is a contradiction, so there is no such $X$.

The established equality $P = N_G(P)$ and Theorem \ref{normqt} imply the assertions concerning
the functorial meaning of $G/P$ and its existence as a smooth proper
$S$-scheme equipped with a canonical $S$-ample line bundle. 
\end{proof}

\begin{corollary}\label{parfunctor}
Let $G \rightarrow S$ be a reductive group scheme.
The functor on $S$-schemes
$$\underline{\rm{Par}}_{G/S}: S' \rightsquigarrow \{\mbox{parabolic subgroups of } G_{S'}\}$$
is represented by a smooth proper $S$-scheme ${\rm{Par}}_{G/S}$ equipped with the
canonical $S$-ample line bundle $\Det({\rm{Lie}}(\mathscr{P}))^{\ast}$,
where $\mathscr{P} \subset G \times {\rm{Par}}_{G/S}$
is the universal parabolic subgroup of $G$.
\end{corollary}
\index{reductive group!scheme of parabolic subgroups}
\index{ aaapargs@$\underline{\rm{Par}}_{G/S}$, ${\rm{Par}}_{G/S}$}

The existence aspect of this corollary is part of \cite[XXVI, 3.3(ii)]{sga3}.  
Over an algebraically closed field $k$ the proof shows ${\rm{Par}}_{G/k} = \coprod (G/P_i)$ where
$P_i$ varies through representatives of the finite set of conjugacy classes of parabolic
subgroups of $G$ (parameterized by the set of parabolic subsets of the root system $\Phi$
for $G$ containing a fixed positive system of roots $\Phi^+$,
or equivalently by the set of subsets of the base $\Delta$ of $\Phi^+$).
In particular, in the classical case ${\rm{Par}}_{G/k}$ is generally disconnected.

\begin{proof}
Since we aim to construct ${\rm{Par}}_{G/S}$ 
as a proper $S$-scheme equipped with a canonical $S$-ample line bundle
arising from the universal parabolic subgroup, by effective descent
in the presence of a relatively ample line bundle
it suffices to work \'etale-locally on $S$.  Thus, by
Theorem \ref{schemetori} we can assume that 
$G$ admits a split maximal torus $T$.

The isomorphism class of the fibral root system for $(G,T)$ is Zariski-locally
constant on $S$, so we can arrange that  $S \ne \emptyset$ and there exists a split 4-tuple
$(G, T, M, \Phi)$.  Choose a positive system of roots
$\Phi^+ \subset \Phi$ and let 
$\{\lambda_{j}\}_{j \in J} \subset M^{\vee}$ be a finite set of cocharacters
such that $\Phi_{\lambda_{j} \ge 0}$ varies (without repetition) through the finitely many
parabolic subsets of $\Phi$ containing $\Phi^+$.  
By Corollary \ref{parfiber}, parabolic subgroups of $G$ are precisely the subgroups conjugate to
some $P_G(\lambda_j)$ \'etale-locally on the base.
Now we apply Corollary \ref{parnorm} to every $P_G(\lambda_j)$ to conclude
that $\coprod_j (G/P_G(\lambda_j))$
represents $\underline{\rm{Par}}_{G/S}$.
(Keep in mind that the functor of points of a disjoint union $\coprod_{i \in I} X_i$
of $S$-schemes $X_i$ indexed by a set $I$ assigns to every $S$-scheme $S'$ a disjoint
union decomposition $\coprod S'_i$ of $S'$ indexed by $I$
and a point in $X_i(S'_i)$ for each $i \in I$.) 

Letting $\mathscr{P}$ denote the universal parabolic subgroup over
${\rm{Par}}_{G/S}$, 
Theorem \ref{normqt} ensures that 
the line bundle $\Det({\rm{Lie}}(\mathscr{P}))^{\ast}$ on
${\rm{Par}}_{G/S}$ is $S$-ample.
\end{proof}

\begin{definition}\label{qsdef} A {\em Borel subgroup} of a reductive group scheme $G \rightarrow S$
is a parabolic subgroup $P \subset G$ such that $P_{\overline{s}}$ is a Borel subgroup of $G_{\overline{s}}$
for all $s \in S$.  A reductive group $G \rightarrow S$ is {\em quasi-split} over
$S$ if it admits a Borel subgroup scheme over $S$.  (The relative notion of ``quasi-split''
is defined with additional requirements in \cite[XXIV, 3.9]{sga3}, especially involving the scheme of Dynkin diagrams, but
for semi-local $S$ the two notions coincide \cite[XXIV, 3.9.1]{sga3}.  For our
purposes, the definition we have given will be sufficient.)
\end{definition}
\index{reductive group!Borel subgroup}
\index{Borel subgroup}
\index{reductive group!quasi-split}
\index{quasi-split}


\begin{theorem}\label{borel}
Let $G \rightarrow S$ be a reductive group scheme. 
\begin{enumerate}
\item Let $P \subset G$ be a parabolic subgroup.  If
$P_{\overline{s}}$ is a Borel subgroup of $G_{\overline{s}}$ then $P_U$ is a Borel subgroup of $G_U$
for some open $U \subset S$ around $s$, and the open locus of $s \in S$ such that
$P_{\overline{s}}$ is a Borel subgroup is also closed.
\index{conjugacy!Borel subgroups}
\index{Borel subgroup!conjugacy}
\item Any two Borel subgroups of $G$ are conjugate \'etale-locally on $S$.
\item The functor on $S$-schemes
$$\underline{\rm{Bor}}_{G/S}: S' \rightsquigarrow \{\mbox{Borel subgroups of } G_{S'}\}$$
is represented by a smooth proper $S$-scheme ${\rm{Bor}}_{G/S}$ equipped
with the canonical $S$-ample line bundle
$\Det({\rm{Lie}}(\mathscr{B}))^{\ast}$, where
$\mathscr{B} \subset G \times {\rm{Bor}}_{G/S}$ is the universal
Borel subgroup of $G$.  
\end{enumerate}
\end{theorem}
\index{Borel subgroup!scheme of}
\index{ aaaborgs@$\underline{\rm{Bor}}_{G/S}$, ${\rm{Bor}}_{G/S}$}

The existence and properties of ${\rm{Bor}}_{G/S}$ are part of \cite[XXII, 5.8.3(i)]{sga3}.

\begin{proof}
Fiber dimension considerations for the smooth map $P \rightarrow S$ settle (1), since the isomorphism class of
the fibral root datum for $G \rightarrow S$ is locally constant over $S$.  Part (2) follows
from Corollary \ref{parfiber}. Finally, 
${\rm{Bor}}_{G/S}$ is the open and closed subscheme of ${\rm{Par}}_{G/S}$ over which
fibers of the universal parabolic subgroup are Borel subgroups (use part (1)).
\end{proof}

\begin{proposition}\label{oppborel}
Let $G \rightarrow S$ be a reductive group scheme, and $T \subset G$ a maximal torus.
If $B \subset G$ is a Borel subgroup containing $T$ then there exists
a unique Borel subgroup $B' \subset G$ satisfying $B' \bigcap B = T$.
\end{proposition}
\index{Borel subgroup!opposite}
\index{opposite Borel subgroup}

\begin{proof}
The uniqueness assertion allows us to work \'etale-locally on $S$, so we may assume
that $S \ne \emptyset$ and $(G,T)$ arises from a split triple $(G,T,M)$ such that
$B = P_G(\lambda)$ for some $\lambda \in M^{\vee}$. The
inclusion $T \subset Z_G(\lambda)$ is an equality (by checking on geometric fibers) and 
$B' := P_G(-\lambda) = Z_G(-\lambda) \ltimes U_G(-\lambda) =
T \ltimes U_G(-\lambda)$ is a Borel subgroup of $G$ containing $T$.
By Theorem \ref{215}(4), $B' \bigcap B = Z_G(\lambda) = T$. 
To establish uniqueness of $B'$, note that if $B''$ is another such Borel
subgroup then $B'' \supset T$, so
Corollary \ref{parfiber}(2) reduces uniqueness to the case
of geometric fibers over $S$, which is Proposition \ref{borpos}.
\end{proof}

In the split case, we get a Zariski-local conjugacy result for Borel subgroups:

\begin{corollary}\label{borconj} Let $(G,T,M)$ be a split reductive group over a non-empty scheme $S$,
and let $B$ be a Borel subgroup of $G$.
\begin{enumerate}
\item Every point $s \in S$ admits an open neighborhood $U$ such that
some $G(U)$-conjugate of $B_U$ contains $T_U$.
\item Any two Borel subgroups of $G$ that contain $T$ are $N_G(T)$-conjugate 
Zariski-locally over $S$.
\end{enumerate}
\end{corollary}

\begin{proof}
Let $\Phi \subset M - \{0\}$ be the root system, and $\Phi^+ \subset \Phi$
a positive system of roots, 
so $\Phi_{\lambda \ge 0} = \Phi^+$ for some $\lambda \in M^{\vee}$.
Consideration of Lie algebras shows that  the parabolic subgroup $B' := P_G(\lambda)$
containing $T$ is a Borel subgroup.
Since $B$ and $B'$ are $G$-conjugate \'etale-locally on $S$, 
${\rm{Transp}}_G(B,B')$ 
provided by Proposition \ref{conntrans} is a torsor for $N_G(B')$
in the \'etale topology, and $N_G(B') = B'$ by Corollary \ref{parnorm}.   
Thus, to prove (1) it is enough to construct Zariski-local sections for
any $B'$-torsor in the \'etale topology on $S$.  

We will construct a composition series for the $S$-group $B'$
consisting of smooth closed $S$-subgroups such that the successive quotients
are $\Add$ or $\Gm$. Since torsors for $\Add$ or $\Gm$ in the \'etale topology
are always Zariski-locally trivial, we will then get the desired Zariski-local sections
since the composition series provides a succession of exact sequences in the \'etale topology.
To build the composition series for $B'$, we use its description as $P_G(\lambda) = 
Z_G(\lambda) \ltimes U_G(\lambda)$.  The choice of $\lambda$
implies that the inclusion $T \subset Z_G(\lambda)$ is an equality, so
since $T$ is $S$-split we are reduced to considering $U_G(\lambda)$.
The desired composition series for $U_G(\lambda) = \mathscr{R}_u(P_G(\lambda))$ 
(see Corollary \ref{rup}) is given by the subgroups $U_{\Phi_{\lambda \ge n}}$
as in Proposition \ref{upsi}(2).

%will be built by 
%using the root groups $U_a$ for $a \in \Phi^+$, and will work
%for any $\lambda \in M^{\vee}$ (i.e., it is not necessary that $P_G(\lambda)$
%is a Borel subgroup).  The method
%we shall use is presented in \cite[XXII, 5.9.5]{sga3}.
%
%For each integer $n > 0$,
%let $\Phi_{\lambda \ge n}$ denote the set of $a \in \Phi$
%such that $\lambda(a) \ge n$ (so $\Phi_{\lambda \ge 1} = \Phi^+$). 
%By (\ref{rootcomm}), for distinct $a \in \Phi_{\lambda \ge n}$
%and $b \in \Phi_{\lambda \ge m}$ with $n, m > 0$, 
%all commutators $(u_a, u_b)$ for $u_a \in U_a$ and $u_b \in U_b$ are products
%among root groups $U_c$ for $c \in \Phi_{\lambda \ge n+m}$.
%For example, if $N$ is the maximal value of $\lambda$ on any roots,
%then the smooth closed subscheme $\prod_{c \in \Phi_{\lambda \ge N}} U_c$
%in $U_{\Phi^+}$ is a commutative $S$-subgroup.
%Continuing by descending induction on $n > 0$, 
%the root groups $U_c$ for $c \in \Phi_{\lambda \ge n}$
%directly span {\em in any order} a smooth closed
%$S$-subgroup $U_{\ge n}$ of $U_{\Phi^+}$
%such that $U_{\ge n+1}$ is normal in $U_{\ge n}$
%and the quotient $U_{\ge n}/U_{\ge n+1}$ is commutative
%and identified (via multiplication) with the direct product
%of the root groups $U_c \simeq \Ga$ for which $\lambda(c) = n$.
%This procedure gives a composition series of $U_G(\lambda)$
%by smooth closed $S$-subgroups such that the successive quotients
%are powers of $\Ga$.

Now we turn to the proof of (2).  By Corollary \ref{parfiber},
it suffices to show that for any $s \in S$
there exists an open neighborhood $U$
of $s$ and $n \in N_G(T)(U)$ such that $nB_U n^{-1}$ and $B'_U$ have the same $s$-fiber.
Since $T$ is split, by Corollary \ref{wconst} the map $N_G(T) \rightarrow W_G(T)$ is surjective for the Zariski topology
and by Proposition \ref{wfinite} the finite $S$-group $W_G(T)$ is constant (so it has
Zariski-local sections through any point of a fiber over $S$).
Hence, we just need to recall the fact from the classical
theory that the geometric fiber $W_G(T)_{\overline{s}} = 
W_{G_{\overline{s}}}(T_{\overline{s}})$ acts transitively on the set of
Borel subgroups of $G_{\overline{s}}$ that contain $T_{\overline{s}}$.
\end{proof}

\begin{corollary}\label{qsfinite} Let $G$ be a reductive group over a henselian local
ring $R$ with finite residue field.
Then $G$ is quasi-split 
and it becomes split over a finite \'etale extension of $R$.
\end{corollary}

See Definition \ref{qsdef} for the notion ``quasi-split'' for reductive group schemes.

\begin{proof}
By \cite[16.6]{borelag} (or Exercise \ref{langexer}), 
the special fiber $G_0$ over the finite residue field $k$ admits
a Borel subgroup $B_0$.  By Corollary \ref{buildtori},
there exists a finite (separable) extension $k'/k$ such that
$(G_0)_{k'}$ admits a split maximal torus $T'_0$.
Since ${\rm{Bor}}_{G/R}$ is $R$-smooth and $R$ is henselian local, 
any $k$-point in the special fiber lifts to an $R$-point.
Hence, $B_0$ lifts to a Borel subgroup of $G$.
The scheme ${\rm{Tor}}_{G_0/k}$ has a $k'$-point corresponding
to $T'_0$, and this may be viewed as a $k'$-point
of ${\rm{Tor}}_{G/R}$.  But ${\rm{Tor}}_{G/R}$ is a smooth
scheme over the henselian local ring $R$, so any $k'$-point must
lift to an $R'$-point, where $R \rightarrow R'$ is the local finite \'etale extension
inducing the residual extension $k'/k$.  Hence,
${\rm{Tor}}_{G/R}(R') \ne \emptyset$, so $G_{R'}$
contains an $R'$-torus that lifts the $k'$-split $T'_0$
and hence is $R'$-split too (due to the henselian property of $R'$). 
\end{proof}


Corollary \ref{qsfinite}
is very useful when $R$ is the valuation ring of a non-archimedean local field.
It says nothing about the quasi-split property for reductive groups given only over the fraction field of $R$,
and conversely Steinberg's theorem that reductive groups over the maximal unramified extension
of ${\rm{Frac}}(R)$ are quasi-split does not imply anything in the direction of the quasi-split property
over $R$ for $G$ as in the corollary.
\index{Steinberg!quasi-split reductive groups}


\subsection{Applications to derived groups and closed immersions}\label{clsec}

In the classical theory, there is a good notion of derived
group for any smooth affine group.  In the relative
theory over a scheme, a new idea is needed
to construct a satisfactory analogue
(at least in the reductive case). 
The structure of the open cell (Theorem \ref{bigcell}) and the construction 
of parabolic subgroups via cocharacters in the split case
(Proposition \ref{parpar}) will enable us to build the ``derived group''
of any reductive group scheme (as in \cite[XXII, 6.1--6.2]{sga3}):

\begin{theorem}\label{dergp} Let $G \rightarrow S$ be a reductive group scheme.
There is a unique semisimple closed normal $S$-subgroup $\mathscr{D}(G) \subset G$ such that
$G/\mathscr{D}(G)$ is a torus. Moreover, $\mathscr{D}(G)$ represents the fppf-sheafification of
the ``commutator subfunctor'' $S' \rightsquigarrow [G(S'), G(S')]$ on the category of
$S$-schemes.

In particular, the quotient map $G \rightarrow G/\mathscr{D}(G)$ is initial among all
homomorphisms from $G$ to an abelian sheaf, and the formation of
$\mathscr{D}(G)$ commutes with any base change on $S$.
\end{theorem}
\index{reductive group!derived group}
\index{derived group} 
\index{ aaadg@$\mathscr{D}(G)$}

\begin{proof}
By the asserted uniqueness, we may use \'etale descent to arrange
that $S$ is non-empty and $T = D_S(M)$ for a finite
free $\Z$-module $M$ such that $(G,T,M)$ is a split triple
encoding a root datum $(M,\Phi,M^{\vee},\Phi^{\vee})$.

Pick a positive system of roots $\Phi^+$ in $\Phi$. Consider the resulting
open cell
$$\Omega := U_{-} \times T \times U_{+} \subset G$$
where $U_{+} = U_{\Phi^+}$ and $U_{-} = U_{-\Phi^+}$. 
Let $T' \subset T$ be the split subtorus ``generated'' by the coroots; i.e.,
$T' = D_S(M/L)$ where $L \subset M$ is the saturated sublattice
that is the annihilator of the coroot lattice $\Z \Phi^{\vee} \subset M^{\vee}$.
This is the minimal subtorus of $T$ through which all coroots factor,
and for a geometric point $\overline{s}$ of $S$
the fiber $T'_{\overline{s}}$ is the subtorus of $T_{\overline{s}}$ generated by the coroots
for $(G_{\overline{s}}, T_{\overline{s}})$.  The idea is to show
that $\Omega' := U_{-} \times T' \times U_{+}$ is the open cell
for a split closed semsimple $S$-subgroup of $G$ that will be $\mathscr{D}(G)$.

Define the $S$-morphism
\begin{equation}\label{ftt}
f:\Omega \rightarrow T/T'
\end{equation} 
by $f(u_{-} t u_{+}) =  t \bmod T'$.  There is clearly
at most one $S$-homomorphism $\overline{f}:G \rightarrow T/T'$ that extends
$f$, and we will prove that $\overline{f}$ exists and is smooth
with $\mathscr{D}(G) := \ker \overline{f}$ satisfying the desired properties.
The key result to be shown is that 
the condition $u_{+} u_{-} \in \Omega'$
holds over a fiberwise-dense open subscheme $V \subset U_{+} \times U_{-}$.
Indeed, we can take such a $V$ to be the preimage under
multiplication $U_{+} \times U_{-} \rightarrow G$
of the open $V_0 \subset G$ provided by:

\begin{lemma}\label{toruslemma}  There exists an
open subscheme $V_0 \subset G$ containing
the identity section such that for all $u_{\pm} \in U_{\pm}$,
if $u_{+} u_{-} \in V_0$ then $u_{+} u_{-} \in \Omega'$.
\end{lemma}

This result is \cite[XXII, 6.16]{sga3}, whose proof rests on 
``abstract'' rank-1 calculations in \cite[XX, 3.12]{sga3}.
We provide a version of the argument that instead rests on
the explicit classification in Theorem \ref{sslist}.

\begin{proof}
Let $w_0 \in W(\Phi)$ denote the long Weyl element relative to $\Phi^+$ (i.e., the product 
\index{long Weyl element}
in any order of the reflections in the positive roots), 
so the $w_0$-action swaps $\Phi^+$ and $-\Phi^+$.  By Corollary \ref{wconst}
we may choose $n_0 \in N_G(T)(S)$ that 
is a representative for $w_0$, so $n_0$-conjugation swaps $U_{+}$ and $U_{-}$
and therefore $n_0\Omega' n_0^{-1} =  U_{+} \times T' \times U_{-}$. 
Hence, it suffices to find an open $V_{n_0} \subset G$ containing the identity section
such that $n_0\Omega' n_0^{-1} \bigcap V_{n_0} \subset \Omega'$ inside $G$ (as we may then
take $V_0 = V_{n_0}$).  We shall prove the analogous result for
any $n \in N_G(T)(S)$.  Note that this problem is Zariski-local on $S$.

Using the map $N_G(T)(S) \rightarrow W_G(T)(S)$ and the identification of
$W_G(T)$ with $W(\Phi)_S$, by working Zariski-locally on $S$ 
we may assume that the image of $n$ in $W_G(T)(S)$ is a constant
section arising from some $w \in W(\Phi)$.  For each $a$ in the set $\Delta$
of simple positive roots we pick $n_{a} \in N_G(T)(S)$ as in Corollary \ref{wconst} 
representing the simple reflection $s_{a}$ (viewed as a constant section
in $W_G(T)(S)$).  Thus, $n_a \in Z_G(T_a)$ for all $a \in \Delta$
and clearly $n = n_{a_1} \cdots n_{a_m}t$ for 
some finite sequence $\{a_j\}$ in $\Delta$ and some $t \in T(S)$.
To construct an open $V_n \subset G$ around the identity section
such that $n \Omega' n^{-1} \bigcap V_n \subset \Omega'$, 
we will argue by induction on the length $m$ of the sequence $\{a_i\}$ of simple roots that appear in
the expression for $n$.  The case $m = 0$ is trivial by taking $V_n = G$ 
($T$ normalizes $U_{\pm}$), so in general we can arrange $t = 1$.  
For $m = 1$ we will soon show that we can take $V_{n_{a}} = \Omega$
for all $a \in \Delta$.
Granting this for a moment,
when $m > 1$ we write $n = n_{a_1} n'$ and by induction may assume
that $V_{n'}$ has been found.  Then for 
the open subscheme $V_n := n_{a_1} V_{n'} n_{a_1}^{-1} \bigcap \Omega
\subset G$ we have
\begin{eqnarray*}
n\Omega'n^{-1} \cap V_n 
&=& n_{a_1}(n'\Omega'{n'}^{-1} \cap V_{n'}) n_{a_1}^{-1} \cap\Omega \\
&\subset& n_{a_1}\Omega' n_{a_1}^{-1} \cap\Omega \\
&\subset& \Omega',
\end{eqnarray*}
where the final containment follows from our temporary hypothesis that we may take
$V_{n_{a}} = \Omega$ for all $a \in \Delta$.

It remains to prove that $n_{a}\Omega' n_{a}^{-1} \bigcap \Omega \subset \Omega'$
for all $a \in \Delta$.  By applying Remark \ref{uarem}
with $\Psi = \Phi^+ - \{a\}$, the root groups $U_b$ for $b \in \Phi^+ - \{a\}$ directly span
{\em in any order} a smooth closed
$S$-subgroup $U_{\widehat{a}} \subset U_{\Phi^+} = U_{+}$ (clearly normalized by $T$).
In particular, $U_a \times U_{\widehat{a}} = U_{+}$ via multiplication.
We similarly get $U_{\widehat{-a}} \subset U_{-}$ such that
$U_{\widehat{-a}} \times U_{-a} = U_{-}$ via multiplication, so 
every point $\omega'$ of $\Omega'$ valued in an $S$-scheme $S'$ has the form
$$\omega' = g_{-} \exp_{-a}(X_{-})t' \exp_a(X_{+})g_{+}$$
where $g_{\pm} \in U_{\widehat{\pm a}}(S')$, $t' \in T'(S')$,
and $X_{\pm} \in \mathfrak{g}_{\pm a}(S')$.   The action on $\Phi$
by the reflection $s_a$ preserves $\Phi^+ - \{a\}$
(as all elements of $\Phi^+ - \{a\}$ have a positive coefficient away from $a$ somewhere
in their $\Delta$-expansion, and this property is not affected by applying $s_a$),
so $n_a$-conjugation preserves each $U_{\widehat{\pm a}}$.   Thus, 
the property $n_a \omega' n_a^{-1} \in \Omega'$ is insensitive to replacing $\omega'$
with $\exp_{-a}(X_{-})t'\exp_a(X_{+})$ (as $\Omega'$ is stable under left
multiplication by $U_{-}$ and right multiplication by $U_{+}$).  Likewise,
by definition of $\Omega$, 
the condition $n_a \omega' n_a^{-1} \in \Omega$ is insensitive to replacing
$\omega'$ with $\exp_{-a}(X_{-})t'\exp_a(X_{+})$.  
In other words, we may pass to the case $g_{\pm} = 1$.  

With $g_{\pm} = 1$, by Lemma \ref{ss1} our problem now takes place within the 
split reductive group $Z_G(T_a)$ with semisimple-rank 1 and root
system $\{\pm a\}$ relative to its split maximal torus $T$, so we
can assume $G = Z_G(T_a)$.  To be precise, since $Z_G(T_a) \cap \Omega =
U_{-a} \times T \times U_a$, it suffices to show that for
$\omega' \in (U_{-a} \times T' \times U_a)(S')$
such that $n_a \omega' n_a^{-1} \in (U_{-a} \times T \times U_a)(S')$,
the product $n_a \omega' n_a^{-1}$ lies in $(U_{-a} \times T' \times U_a)(S')$.
We may work fppf-locally on $S'$, so the $T'$-component $t'$ of $\omega'$
may be arranged to have the form $t'_a z$ for $t'_a \in a^{\vee}(\Gm)(S')$
and $z \in (T_a \cap T')(S')$.  Since $z$ is central in $Z_G(T_a)$, we can assume
$z = 1$.  Thus, $t'$ lies in the $S$-torus $a^{\vee}(\Gm)$ that is the
``$T'$'' associated to $(Z_G(T_a), T, \{\pm a\})$.  Hence, we may assume $G = Z_G(T_a)$.

If $G = G_1 \times T_1$ for a split torus $T_1$ then our problem
takes place inside $G_1$, so we may pass to $G_1$.  Thus, after Zariski-localization
on the base we
may assume that $(G,T)$ is one of the three explicit pairs
in the split semisimple-rank 1 classification in Theorem \ref{sslist} (up to direct product
against a split torus),
and conjugating by the 
standard Weyl element $w = (\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix})$
if necessary allows us
to arrange that $a$ is the standard positive root (i.e., $U_a$ is the subgroup of
strictly upper triangular matrices in ${\rm{SL}}_2$ or ${\rm{PGL}}_2$). 
In cases (1) and (2) of Theorem \ref{sslist} we have $T' = D = T$, so 
$\Omega = \Omega'$ and there is nothing to do. 
Thus, we may assume $(G,T)$ is as in case (3): $G = {\rm{SL}}_2 \times^{\mu_2} \Gm = \GL_2$ 
and $T$ is the diagonal torus. 
The torus $T'$ generated by $a^{\vee}$ is the diagonal torus $D$ in 
the subgroup ${\rm{SL}}_2$, and replacing
$n_a$ with a suitable left $T(S')$-multiple (as we may do)
allows us to take $n_a = w$. 

Now $\Omega'$ is the open subgroup of ${\rm{SL}}_2$ given by the unit condition
on the upper left entry (see Example \ref{precede}), and $\Omega = \Omega' \times^{\mu_2} \Gm$.
Thus, $n\Omega'n^{-1} \bigcap \Omega \subset {\rm{SL}}_2 \bigcap \Omega = \Omega'$
inside $G$.
\end{proof}

\medskip

For the choice of $V \subset U_{+} \times U_{-}$ built
using Lemma \ref{toruslemma}, the relatively schematically dense open
$$U_{-} \times T \times V \times T \times U_{+} \subset \Omega \times \Omega$$
lies in $m_G^{-1}(\Omega)$ and the map $f$ in (\ref{ftt}) satisfies
$f((u_1^{-} t_1 u_1^{+})(u_2^{-} t_2 u_2^{+})) = t_1 t_2 \bmod T'$. 
This ``$S$-birational multiplicativity'' for $f$ implies via the self-contained
and elementary
\index{sbirational mult@$S$-birational multiplicativity}
\cite[XVIII, 2.3(i)]{sga3} that $f$ extends uniquely to
an $S$-homomorphism $\overline{f}$.  On fibers over a geometric point $\overline{s}$ of $S$
we have a good theory of the semisimple derived group $\mathscr{D}(G_{\overline{s}})$.
In particular, $T'_{\overline{s}}$ is a maximal torus of
$\mathscr{D}(G_{\overline{s}})$ and the identification of root systems 
$\Phi(\mathscr{D}(G_{\overline{s}}),T'_{\overline{s}}) = 
\Phi(G_{\overline{s}}, T_{\overline{s}})$ via the isogeny 
$T'_{\overline{s}} \times (Z_{G_{\overline{s}}})_{\rm{red}}^0 \rightarrow T_{\overline{s}}$
identifies $\Phi^+$ with a positive system of roots
for $\Phi(\mathscr{D}(G_{\overline{s}}),T'_{\overline{s}})$. The 
corresponding open cell in $\mathscr{D}(G_{\overline{s}})$ 
is clearly $\Omega'_{\overline{s}}$.  Hence, 
$(\ker \overline{f}_{\overline{s}})^0 = \mathscr{D}(G_{\overline{s}})$,
so $f_{\overline{s}}$ factors as
$$G_{\overline{s}} \rightarrow G_{\overline{s}}/\mathscr{D}(G_{\overline{s}})
\rightarrow T_{\overline{s}}/T'_{\overline{s}},$$
where the second map is an \'etale isogeny of tori.  This isogeny is
an isomorphism since the maximal torus $T_{\overline{s}}$
maps onto the torus quotient $G_{\overline{s}}/\mathscr{D}(G_{\overline{s}})$
and $T'_{\overline{s}} \subset \mathscr{D}(G_{\overline{s}})$.  Thus, 
$\overline{f}_{\overline{s}}$ is
the quotient by the derived group of $G_{\overline{s}}$, so
$\overline{f}$ is a smooth surjection whose kernel
$\ker \overline{f}$ has $\overline{s}$-fiber $\mathscr{D}(G_{\overline{s}})$.
Thus, $\ker \overline{f}$ is a semisimple $S$-group
closed and normal in $G$; define $\mathscr{D}(G)$ to be this subgroup.

By construction, $\mathscr{D}(G)$ contains $U_{\pm}$ and $T'$ as subgroups,
with $T'$ a split maximal torus of $\mathscr{D}(G)$.  There
is a split triple $(\mathscr{D}(G), T',M')$, where $M'$ is the maximal torsion-free
quotient of $M$ that kills $(\Z \Phi^{\vee})^{\perp} \subset M$
and we let $\Phi' \subset M'$ be the image of $\Phi$.  Let
${\Phi'}^+$ be the positive system of roots corresponding
to $\Phi^+$ under the bijection $\Phi \rightarrow \Phi'$.  The direct product subfunctor
$\Omega'$ in $\Omega$ lies in $\mathscr{D}(G)$.  It is obvious that 
$\Omega'$ must be  the open cell of $\mathscr{D}(G)$ associated
to ${\Phi'}^+$. 
The commutativity of $G/\mathscr{D}(G) = T/T'$ implies
that $\mathscr{D}(G)$ contains the commutator subsheaf of $G$.
Provided that every semisimple $S$-group (such as $\mathscr{D}(G)$) coincides with its own
commutator subsheaf (for the fppf topology), the asserted uniqueness
of $\mathscr{D}(G)$ will be clear and so we will be done.

Finally, consider a semisimple $S$-group $G$.  We seek to show
that $G$ is its own commutator subsheaf.  We may
work \'etale-locally on the base, so we can assume $S \ne \emptyset$ and that $G$ is part of 
a split triple $(G,T,M)$ over $S$. Let $\Phi \subset M - \{0\}$ be the set of roots, and pick
a positive system of roots in $\Phi$.  
The associated open cell generates $G$ for the fppf topology 
(as for any open neighborhood of the identity section
of a smooth group scheme with connected fibers), 
and the map $\Gm^r \rightarrow T$ defined by the simple positive coroots
is an isogeny.  Thus, to prove that $G$ is its own commutator subsheaf (for
the fppf topology), the structure of the open cell as a direct product scheme
reduces the problem to the case of semisimple groups
with semisimple-rank 1 case, and more
specifically to the case of ${\rm{SL}}_2$ equipped
with its diagonal torus (due to Theorem \ref{sslist}).  

Once again passing to the standard open cell reduces us to some classical identities, 
as follows.  For $x(u) = (\begin{smallmatrix} 1 & u \\ 0 & 1 \end{smallmatrix})$,
$y(v) = (\begin{smallmatrix} 1 & 0 \\ v & 1 \end{smallmatrix})$,
and $h(t) = {\rm{diag}}(t,1/t)$ we have
$h(t)x(u)h(t)^{-1}x(u)^{-1} = x((t^2 - 1)u)$ and 
$h(t)y(v)h(t)^{-1}y(v)^{-1} = y((t^{-2}-1)v)$, so 
the commutator subsheaf contains both standard root groups.  These in turn generate
the diagonal torus via the identity
\begin{equation}\label{hform}
h(t) = y(-1/t)x(t)y(-1/t)(y(-1)x(1)y(-1))^{-1},
\end{equation}
so ${\rm{SL}}_2$ is indeed its own commutator subsheaf.
\end{proof}

The group $\mathscr{D}(G)$ is called the {\em derived group} of $G$.  Note that by uniqueness,
$\mathscr{D}(G) = G$ if and only if $G$ is semisimple, and by the universal property
 the formation of $\mathscr{D}(G)$ is functorial in $G$.
It follows formally that any $S$-homomorphism from a semisimple $S$-group to $G$
must factor through $\mathscr{D}(G)$.  

\begin{corollary}\label{isogcor} Let $G \rightarrow S$ be a reductive group scheme, $\mathscr{D}(G)$
its derived group.  Let $Z \subset Z_G$ be the maximal central torus of $G$, and
$T' = G/\mathscr{D}(G)$ the maximal torus quotient.  The natural $S$-homomorphisms
$f:Z \times \mathscr{D}(G) \rightarrow G$ and $h:G \rightarrow T' \times (G/Z)$
are central isogenies.
\end{corollary}
\index{central isogeny}

The result in this corollary is \cite[XXII, 6.2.4]{sga3}.
We refer the reader to Definition \ref{ceniso} for the relative notion of a central isogeny used here.

\begin{proof}
The classical theory implies that  $f_s$ and $h_s$ are central isogenies for all $s \in S$,
so we may apply Proposition \ref{central} to conclude.
(See Remark \ref{subgpM} for the existence and compatibility
with base change of a subtorus containing all others in any multiplicative 
type group, thereby providing $Z$ inside $Z_G$ compatibly with any base change.)
\end{proof}

\begin{proposition}\label{derpar}
Let $G \rightarrow S$ be a reductive group scheme, and $Z$ the maximal torus of $Z_G$.
The map $T \mapsto T \bigcap \mathscr{D}(G)$ is a bijective correspondence
between the set of maximal tori in $G$ and the set of maximal tori in $\mathscr{D}(G)$.
Conversely, if $T'$ is a maximal torus of $\mathscr{D}(G)$ then
$T' \times Z \rightarrow G$ is an isogeny onto a maximal torus $T$ of $G$,
and this reverses the bijective correspondence.

The same holds for parabolic subgroups, with the analogous procedures using
 intersection and  product against $Z$.
\end{proposition}
\index{derived group!and maximal tori}
\index{derived group!and parabolic subgroups}
\index{parabolic subgroup!of derived group}

\begin{proof}
Consider the correspondence for maximal tori.  By Example \ref{cent}, since
$\mathscr{D}(G)$ is normalized by $T$ it follows
that $T \bigcap \mathscr{D}(G)$ is smooth with
connected fibers, so it is a torus.  The classical
theory implies that this intersection is a maximal torus of $\mathscr{D}(G)$.
Likewise, the classical theory shows that if we define
$T' := T \bigcap \mathscr{D}(G)$ then the multiplication map
$Z \times T' \rightarrow T$ between tori is an isogeny.
Conversely, for a maximal torus $T'$ of $\mathscr{D}(G)$, we have 
to show that the multiplication map
$T' \times Z \rightarrow G$ is an isogeny onto a maximal torus of $G$.
This map factors through $Z_G(T')$, so it suffices
to show that the reductive group $Z_G(T')$ is a torus
and $T' \times Z \rightarrow Z_G(T')$ is an isogeny of tori.
It suffices to check these assertions on geometric fibers over $S$,
where they are well-known. 

Now we turn to the consideration of parabolic subgroups.
For any parabolic subgroup $P \subset G$, we have
$Z_G \subset P$.  Indeed, it suffices
to check this \'etale-locally on $S$, and by Proposition \ref{parpar}
we can perform such localization on $S$ so that $P = P_G(\lambda)$ for some 
$\lambda:\Gm \rightarrow G$.  Thus, $Z_G \subset Z_G(\lambda) \subset P_G(\lambda) = P$.   It follows
that $Z \subset P$, so the isogeny $Z \times \mathscr{D}(G) \rightarrow G$
implies that $P$ is uniquely determined by $P' = P \bigcap \mathscr{D}(G)$,
and $P'$ is smooth by Proposition \ref{functor}(1).
For each $s \in S$ we have $\mathscr{D}(G)_{\overline{s}}/P'_{\overline{s}} \simeq G_{\overline{s}}/P_{\overline{s}}$,
so $P'_{\overline{s}}$ is a parabolic subgroup of $\mathscr{D}(G)_{\overline{s}}$.
Hence, $P'$ is parabolic in $\mathscr{D}(G)$. 
The multiplication map $m:Z \times P' \rightarrow P$ is an isogeny on fibers,
so $m$ is a quasi-finite flat surjection, and $\ker m$ is visibly central.
But $\ker m$ is closed in the $S$-finite kernel of $Z \times \mathscr{D}(G) \rightarrow G$,
so it is $S$-finite and hence $m$ is a central isogeny.

Finally, it remains to show that every parabolic subgroup 
$Q$ of $\mathscr{D}(G)$ arises as $P \bigcap \mathscr{D}(G)$
for a parabolic subgroup $P$ of $G$.  In view of the uniqueness of
such a $P$ we may work \'etale-locally on $S$, so
by Proposition \ref{parpar}
we may arrange that $Q = P_{\mathscr{D}(G)}(\mu)$
for some $\mu:\Gm \rightarrow \mathscr{D}(G)$.
But then $P := P_G(\mu)$ is a parabolic subgroup of $G$ (Example \ref{relparex}) 
and $P \bigcap \mathscr{D}(G) = P_{\mathscr{D}(G)}(\mu) = Q$.
\end{proof}
%
%\begin{proposition}\label{kerAd}
%Let $G \rightarrow S$ be a reductive group scheme. 
%The center $Z_G$ equals the kernel of
%the adjoint representation ${\rm{Ad}}_G: G \rightarrow {\rm{GL}}(\mathfrak{g})$.
%\end{proposition}
%
%\begin{proof} The schematic center $Z_G$ is fppf over $S$ (Theorem \ref{redcenter})
%and $\ker {\rm{Ad}}_G$ is finitely presented over $S$,
%by Lemma \ref{flatclosed} the inclusion
%$Z_G \hookrightarrow \ker {\rm{Ad}}_G$ is an isomorphism
%if it is so on (geometric) fibers over $S$.  Thus, we may and do
%assume $S = \Spec(k)$ for an algebraically closed field $k$.
%Since $G/Z_G$ is perfect, a normal closed subgroup scheme of $G$ is central
%if and only if its identity component is central (see \cite[Lemma 5.3.2]{pred}
%for a self-contained proof of this elementary fact).  It therefore suffices 
%to show that $(\ker {\rm{Ad}}_G)^0$ is central in $G$.
%
%There is a finite set of maximal tori $T_1, \dots, T_n$
%such that $T_1 \times \dots \times T_n \rightarrow G$ via multiplication 
%is dominant.  This carries some  dense open
%in $\prod T_j$ onto a dense open $\Omega$ in $G$
%via a faithfully flat map.  Since 
%the surjective multiplication map $\Omega \times \Omega \rightarrow G$ is flat
%(as the multiplication map $G \times G \rightarrow G$ is flat), it follows
%that a closed subscheme of $G$ is central if it centralizes each $T_j$.
%Hence, it suffices to show that $(\ker {\rm{Ad}}_G)^0$ is contained in each
%maximal torus $T$ of $G$, or equivalently is contained in the
%schematic centralizer $Z_G(T) = T$ for all such $T$.
%
%Normality of $\ker {\rm{Ad}}_G$ in the smooth affine $k$-group $G$ implies
%the normality of its identity component in $G$.  Hence, there is 
%a $T$-action on this identity component via conjugation, and we just need
%to show that this action is trivial.
%Since $T$ is of multiplicative type, its action on a connected
%$k$-group scheme $H$ of finite type is trivial if and only its induced
%action on ${\rm{Lie}}(H)$ is trivial, by \cite[Cor.\:A.8.11]{pred} (whose proof
%simplifies significantly for the action by a torus).  Thus, we only need
%to verify that the adjoint action of $T$ on ${\rm{Lie}}(\ker {\rm{Ad}}_G)$ is trivial. 
%But ${\rm{Lie}}(\ker {\rm{Ad}}_G) = \ker({\rm{ad}}_{\mathfrak{g}})$ \cite[Prop.\:A.7.5]{pred}, so it
%suffices to prove that $\ker({\rm{ad}}_{\mathfrak{g}}) \subset {\rm{Lie}}(T)$, or
%equivalently (via $T$-weight space considerations) that
%$\ker({\rm{ad}}_{\mathfrak{g}})$ does not contain any root
%line $\mathfrak{g}_a$ for $a \in \Phi(G,T)$.
%
%For any root $a$, consider the rank-1 semisimple subgroup
%$G_a = \mathscr{D}(Z_G(T_a))$ with maximal torus $a^{\vee}(\Gm)$
%whose root groups are $U_{\pm a}$. 
%By functoriality of the adjoint representation (applied to the inclusion
%$G_a \hookrightarrow G$), if $\mathfrak{g}_a \subset \ker({\rm{ad}}_{\mathfrak{g}})$
%then the analogue holds for $(G_a, a^{\vee}(\Gm), a)$ in place of $(G, T, a)$.
%Thus, to get a contradiction we may replace $G$ with $G_a$, so 
%it suffices
%to treat the groups ${\rm{SL}}_2$ and ${\rm{PGL}}_2$,
%taking $T$ to be the diagonal torus and $a$ to correspond 
%to the upper triangular unipotent subgroup $U^{+}$.  Choose nonzero 
%$v^{\pm} \in \mathfrak{u}^{\pm}$ and $t \in \mathfrak{t}$.  In the ${\rm{SL}}_2$-case
%$[v^{+},v^{-}] \ne 0$ and in the ${\rm{PGL}}_2$-case $[v^{+},t] \ne 0$.
%\end{proof}
%For any $S$-scheme $S'$ and $g \in (\ker {\rm{Ad}}_G)(S')$,
%the Borel subgroup $gB_{S'}g^{-1}$ has Lie algebra
%${\rm{Ad}}_G(g)({\rm{Lie}}(B_{S'})) = {\rm{Lie}}(B_{S'})$.
%Since $B_{S'}$ is its own normalizer (Corollary \ref{parnorm}),
%to prove $g \in B(S')$, it suffices to
%prove that Borel subgroups are determined by their Lie subalgebras.
%This is a special case of the lemma below.
%\end{proof}
%
%\begin{lemma}\label{Pnorm} Let $P$ be a parabolic subgroup of a reductive group scheme
%$G \rightarrow S$, and let $\mathfrak{p} \subset \mathfrak{g}$ be the associated Lie subalgebra. The functorial stabilizer
%$\underline{N}_G(\mathfrak{p})$ of $\mathfrak{p}$ inside $\mathfrak{g}$ 
%via the adjoint action of $G$ is represented by $P$.
%In particular, if $P, P' \subset G$ are parabolic subgroups and $\mathfrak{p} = \mathfrak{p}'$
%inside $\mathfrak{g}$ then $P = P'$ inside $G$.
%\end{lemma}
%
%\begin{proof}
%The special case $S = \Spec(k)$ for an algebraically closed field $k$ is proved
%by considerations with the root system as follows.
%Clearly $P \subset N_G(\mathfrak{p})$ as closed $k$-subgroup schemes of $G$,
%so the smooth closed subgroup $N_G(\mathfrak{p})_{\rm{red}}$ in $G$ is parabolic
%(as it contains $P$). 
%Since parabolic subgroups of $G$ are connected, it follows that $N_G(\mathfrak{p})$ is connected.
%Hence, since $P$ is smooth, to prove $P = N_G(\mathfrak{p})$ as $k$-schemes it suffices
%to prove equality of their Lie algebras.  For any subspace $V \subset \mathfrak{g}$
%the associated stabilizer ${\rm{Stab}}_{{\rm{GL}}(\mathfrak{g})}(V)$ in ${\rm{GL}}(\mathfrak{g})$
%has Lie algebra ${\rm{Stab}}_{\mathfrak{gl}(\mathfrak{g})}(V)$ that is the space of endomorphisms of
%$\mathfrak{g}$ preserving $V$.  Thus, the fiber product description
%$$N_G(V) = G \times_{{\rm{Ad}}_G, {\rm{GL}}(\mathfrak{g})} {\rm{Stab}}_{{\rm{GL}}(\mathfrak{g})}(V)$$
%and the formula ${\rm{Lie}}({\rm{Ad}}_G) = {\rm{ad}}_{\mathfrak{g}}$ imply
%$${\rm{Lie}}(N_G(V)) = \{X \in \mathfrak{g}\,|\, [X, v] \in V \mbox{ for all } v \in V\}.$$
%
%In other words, ${\rm{Lie}}(N_G(V))$ is the normalizer $\mathfrak{n}_{\mathfrak{g}}(V)$ of $V$ in $\mathfrak{g}$ in the sense of Lie algebras.
%Thus, we have to show that the containment $\mathfrak{p} \subset \mathfrak{n}_{\mathfrak{g}}(\mathfrak{p})$
%is an equality. Let $T$ be a maximal torus of $P$, hence also maximal in $G$, 
%and let $\Phi = \Phi(G,T)$.  For a Borel subgroup $B \subset P$ containing $T$
%and $\Phi^+ = \Phi(B,T)$ and let $\Delta$ be the simple positive roots, so
%$P = P_I$ for some subset $I \subset \Delta$ (see Proposition \ref{parpos} and its proof).
%Thus, $\mathfrak{p} = \mathfrak{b} \oplus \oplus_{a \in \Phi^{-} \cap [I]} \mathfrak{g}_a$.
%Clearly $\mathfrak{n}_{\mathfrak{g}}(\mathfrak{p})$ is a $T$-stable subspace of $\mathfrak{g}$,
%so $\mathfrak{n}_{\mathfrak{g}}(\mathfrak{p})$ is the direct sum of $\mathfrak{p}$
%and some root spaces $\mathfrak{g}_a$ for $a \in \Phi^{-} \cap (\Phi - [I])$.
%It therefore suffices to show that for any $a \in \Phi^{-}$ not in $[I]$,
%$\mathfrak{g}_a$ does not normalize $\mathfrak{p}$.
%
%Consider the subgroup $G_a = \mathscr{D}(Z_G(T_a))$.
%
%% In other words, if $X \in \mathfrak{g}$
%%; see Corollary \ref{borpoor} for
%%the special case of Borel subgroups  (which is all we need above) and see
%%Exercise \ref{parlie} (and Remark \ref{parrem}) for any parabolic over $k$.
%
%In general, $\underline{N}_G(\mathfrak{p})$ is represented by a finitely presented
%closed subgroup $N_G(\mathfrak{p})$ of $G$, namely the preimage under ${\rm{Ad}}_G: G \rightarrow {\rm{GL}}(\mathfrak{g})$
%of the stabilizer
%of the subbundle $\mathfrak{p} \subset \mathfrak{g}$.  The inclusion
%$j:P \subset N_G(\mathfrak{p})$ as finitely presented closed $S$-subgroups of $G$
%induces an isomorphism between (geometric) fibers over $S$, so 
%by $S$-flatness of $P$ and the fibral flatness criterion \cite[IV$_3$, 11.3.1]{ega}
%it follows that $N_G(\mathfrak{p})$ is $S$-flat and thus 
%that $j$ is an isomorphism.
%\end{proof}

By Proposition \ref{kerAd}, if $G$ is adjoint semisimple then 
the $S$-homomorphism ${\rm{Ad}}_G:G \rightarrow {\rm{GL}}(\mathfrak{g})$
has trivial kernel.  In view of Remark \ref{immrem} it is not obvious if ${\rm{Ad}}_G$
is a closed immersion in the adjoint semisimple case over a general scheme.  In fact, 
it is always a closed immersion for such $G$, because
any monomorphism from
a {\em reductive} group scheme to a separated group of finite presentation
is a closed immersion. We will never use 
this result, but we provide a proof of it below (after some brief preparations).

For a split reductive group scheme $(G,T,M)$ and
parabolic subgroup $P = P_G(\lambda)$ with $\lambda \in M^{\vee}$, 
Proposition \ref{upsi}(2) provides a composition series $\{U_{\ge n}\}_{n \ge 1}$
for the unipotent radical $U = \mathscr{R}_u(P) = U_G(\lambda)$ 
with $U_{\ge n} = U_{\Phi_{\lambda \ge n}}$ a smooth closed subgroup directly spanned
in any order by the root groups $U_c$ for $c$ satisfying $\lambda(c) \ge n$.
Moreover, $U_{\ge n+1}$ is normal in $U_{\ge n}$ and the quotient $U_{\ge n}/U_{\ge n+1}$
is commutative and identified (as an $S$-group) with the direct product of
the root groups $U_c \simeq \Add$ with $\lambda(c) = n$. 
We shall use this general filtration of unipotent radicals of parabolic subgroups
and the dynamic method of \S\ref{dynsec} to prove: 

\begin{theorem}\label{adimm}
For a reductive group $G \rightarrow S$, any monic homomorphism
$f:G \rightarrow G'$ to a separated $S$-group of finite presentation is a closed immersion.
In particular, if $G$ is an adjoint semisimple $S$-group then
${\rm{Ad}}_G:G \rightarrow {\rm{GL}}(\mathfrak{g})$ is a closed immersion.
\end{theorem}
\index{adjoint representation!as closed immersion}
\index{dynamic method!and adjoint representation}
\index{monic homomorphism!reductive group}

Theorem \ref{adimm} is proved in another way (without the dynamic method) in \cite[XVI, 1.5(a)]{sga3}.
When $G'$ is not $S$-affine, our proof uses a difficult theorem due to Raynaud.  
Note also that if $G'$ is allowed to merely be locally of finite presentation (and separated)
then the conclusion is false; counterexamples are provided by the N\'eron lft model of
a split torus \cite[10.1/5]{neron}. Also, Example \ref{weird} gives counterexamples if reductivity of $G$
is relaxed to ``smooth affine with connected fibers'' (and $G' \rightarrow S$ is smooth and affine).

\begin{proof}
The application to ${\rm{Ad}}_G$ in the adjoint case is immediate from the rest via Proposition \ref{kerAd}.
In general, monicity means that the diagonal $\Delta_f:G \rightarrow G \times_{G'} G$
is an isomorphism, so by direct limit arguments we can reduce to the case
when $S$ is noetherian.  Since $f$ is a monomorphism, it
is a closed immersion if and only if it is proper \cite[IV$_3$, 8.11.5]{ega}.
Thus, by the valuative criterion for properness, we are reduced to checking
that if $R$ is a discrete valuation ring with fraction field $K$
and $\Spec R \rightarrow S$ is a morphism of schemes then
$G(R) = G(K) \bigcap G'(R)$ inside  $G'(K)$.  Applying
base change along $\Spec R \rightarrow S$ then reduces us to the
case $S = \Spec R$.

Next, we reduce to the case when $G'$ is affine and $R$-flat.  
(The reader who is only interested in the case of affine $G'$ can ignore this step.)
The map on generic fibers $G_K \rightarrow G'_K$ is a closed immersion,
so the schematic closure $\overline{G}$ of $G$ in $G'$ is
an $R$-flat separated subgroup of $G'$ through which $G$ factors
(since $G$ is $R$-flat).  But $\overline{G}$ is a separated flat $R$-group
of finite type with affine generic fiber, so it must be affine by a result of Raynaud
(see \cite[VI$_{\rm{B}}$, 12.10(iii), 12.10.1]{sga3} or \cite[Prop.\,3.1]{py}).
Hence, we may replace $G'$ with $\overline{G}$ to reduce to the case
that $G'$ is affine and $R$-flat.

Over a regular base of dimension $\le 1$, every flat affine
group scheme of finite type is a closed subgroup of some
${\rm{GL}}_n$.  This is easy to prove by adapting arguments
from the case when the base is a field (see Exercise \ref{glnmap}); in fact, the result is true over 
any regular affine base of dimension $\le 2$ \cite[VI$_{\rm{B}}$, 13.2]{sga3}. 
Thus, we may identify $G'$ as a closed subgroup of some ${\rm{GL}}_n$, so
we can replace $G'$ with ${\rm{GL}}_n$.  This reduces the problem to the case
that $G'$ is a reductive $S$-group, but we allow $S$ to be an arbitrary scheme
(to clarify the generality of the steps that follow). 

By working \'etale-locally on $S$, we may assume $G$ arises from a split
triple $(G,T,M)$ (and that $S$ is non-empty). Choose
$\lambda \in M^{\vee}$ not vanishing on any root, so $T = Z_G(\lambda)$ and 
$B := P_G(\lambda)$ is a Borel subgroup of $G$.
Let $U_{+} = U_G(\lambda) = \mathscr{R}_u(B)$ and $U_{-} = U_G(-\lambda) = \mathscr{R}_u(B_{-})$, 
where $B_{-} = P_G(-\lambda)$ is the opposite Borel subgroup of $G$ containing $T$
(see Proposition \ref{oppborel}).  Let $\Omega \subset G$ be the open cell
$U_{-} \times B$.

For $\lambda' = f \circ \lambda$ we likewise get smooth closed
subgroups $Z_{G'}(\lambda'), U_{G'}(\pm \lambda') \subset G'$
such that the multiplication map
$$U_{G'}(-\lambda) \times Z_{G'}(\lambda') \times U_{G'}(\lambda') \rightarrow G'$$
is an open immersion; we let $\Omega' \subset G'$ denote this
open subscheme.  Since $Z_{G'}(\lambda')$ is reductive (as it is a torus centralizer
in a reductive group scheme), by working \'etale-locally
on $S$ we can arrange that $Z_{G'}(\lambda')$ contains a split maximal torus $T'$.
Clearly $T'$ is maximal in $G'$ (by the classical theory on geometric fibers),  and 
$\lambda'$ factors through $T'$ (since $T' = Z_{G'}(T')$). Further localization
on $S$ brings us to the case that $(G',T')$ arises
from a split triple $(G',T',M')$.
Hence, for $\Psi' = \Phi'_{\lambda' > 0}$, by Remark \ref{uarem} the group $U_{G'}(\pm \lambda')$ coincides
with the subgroup $U_{\pm \Psi'}$ from Proposition \ref{upsi} 
that is directly spanned by the root groups $U_{c'}$ for 
$c' \in \pm \Psi' = \Phi'_{\pm \lambda' > 0}$.

It is harmless to work \'etale-locally on $S$ and to compose the given monomorphism
$G \rightarrow G'$ with conjugation by some element of $G'(S)$.   Thus, to reduce to the
case that $f$ carries $T$ into $T'$ it suffices (by \'etale-local conjugacy of maximal tori
in smooth relatively affine group schemes) to prove:

\begin{lemma} For any homomorphism $f:T \rightarrow H$
from a torus into a smooth relatively affine group over a scheme $S$, \'etale-locally on $S$
it factors through a maximal torus of $H$. 
\end{lemma}

\begin{proof} By replacing $H$ with $T \ltimes H$ (via the action $t.h = f(t)hf(t)^{-1}$) it suffices to show that if
$f$ is the inclusion of $T$ as a closed $S$-subgroup of $H$ then \'etale-locally on $S$
it is contained in a maximal torus of $H$.   The centralizer $Z_H(T)$ is smooth, and by the classical
theory on geometric fibers we see that its maximal tori are also maximal in $H$.  Hence, we may replace
$H$ with $Z_H(T)$ to arrange that $T$ is central.  By passing to an \'etale cover of $S$
we can assume that $Z_H(T)/T$ admits a maximal torus.  Corollary \ref{fpqcwin}(2) ensures that
the preimage of this maximal torus in $Z_H(T)$ is a torus, and by the classical theory on geometric
fibers it is a maximal torus. 
\end{proof}


Now we may and do assume $T \subset T'$.
Proposition \ref{functor}(2) gives that 
$\Omega \subset f^{-1}(\Omega')$, and the 
key point is that this containment is an {\em equality}.  To verify
this equality between open subschemes of $G$ we may pass
to geometric fibers over $S$, in which case the equality is \cite[Prop.\,2.1.8(3)]{pred}
(which has nothing to do with smoothness or reductivity).  Thus, the restriction of
$f$ over the open subscheme $\Omega' \subset G'$ is the map
$\Omega \rightarrow \Omega'$ that is the direct product of the maps
$$U_G(-\lambda) \rightarrow U_{G'}(-\lambda'),\,\,\,
T = Z_G(\lambda) \rightarrow Z_{G'}(\lambda'),\,\,\,
U_G(\lambda) \rightarrow U_{G'}(\lambda').$$
We will prove that each of these three maps is a closed immersion, so
$f$ is a closed immersion when restricted over $\Omega'$.

Since $T \rightarrow T'$ is a monic homomorphism between tori, it is a closed immersion.
To prove that $U := U_G(\lambda) \rightarrow U_{G'}(\lambda') =: U'$ is a closed
immersion, consider the filtrations $\{U_{\ge n}\}_{n \ge 1}$
and $\{U'_{\ge n}\}_{n \ge 1}$ on these as described immediately before
Theorem \ref{adimm}. 
We claim that $f$ carries $U_{\ge n}$ into $U'_{\ge n}$ for all $n$.  More specifically,
keeping in mind that we arranged $T \subset T'$ via $f$, we have:

\begin{lemma}\label{ufill} For $c \in \Phi^+$, 
$f|_{U_c}$ factors through $U_{Z_{G'}(\ker c)}(c^{\vee}) \subset G'$, 
and this closed subgroup of $G'$ is directly spanned in any order by the root groups $U'_{c'}$ for $c' \in \Phi'$ such that 
$c'|_T$ is a positive integral multiple of $c$. 
\end{lemma}

The group $Z_{G'}(\ker c)$ is smooth by Lemma \ref{centralizer} since $\ker c$ is multiplicative type, but
beware that its fibers over $S$ may not be connected (since $\ker c$ may not be a torus).

\begin{proof} Since $U_c$ is normalized by $T$ with trivial action by $\ker c$, it is 
carried into the smooth closed subgroup $Z_{G'}(\ker c)$.  It follows
that $U_c$ is carried into $U_{Z_{G'}(\ker c)}(c^{\vee})$
since $\langle c, c^{\vee} \rangle = 2 > 0$.  Likewise,
if $c' \in \Phi'$ satisfies $c'|_T = nc$ with $n \ge 1$ then $c'$ kills $\ker c$
and $\langle c', c^{\vee} \rangle = \langle nc, c^{\vee} \rangle = 2n > 0$, so
$U'_{c'}$ is contained in $U_{Z_{G'}(\ker c)}(c^{\vee})$.
It remains to show that for any choice of enumeration $\{c'_i\}$
of the set of such $c'$, the multiplication map of $S$-schemes 
\begin{equation}\label{uczc}
\prod U'_{c'_i} \rightarrow U_{Z_{G'}(\ker c)}(c^{\vee})
\end{equation}
is an isomorphism.  Since this is a map between smooth $S$-schemes,
we may pass to geometric fibers, so
$S = \Spec k$ for an algebraically closed field $k$.

Connectedness of $U_{Z_{G'}(\ker c)}(c^{\vee})$ \cite[2.1.8(4)]{pred}
 implies that it equals
$U_{Z_{G'}(\ker c)^0}(c^{\vee})$.   The group $Z_{G'}(\ker c)^0$ is smooth
since $\ker c$ is of multiplicative type, and its Lie algebra is the trivial weight 
space ${\rm{Lie}}(G')^{\ker c}$
for the linear action of the split multiplicative type group $\ker c$ 
on ${\rm{Lie}}(G')$.  Hence, the 
$T'$-weights on ${\rm{Lie}}(Z_{G'}(\ker c)^0)$ are the elements 
$c' \in \Phi'$ that are trivial on $\ker c$, which is to say $c'|_T$ is an integral multiple of $c$.  For such $c'$, 
the condition $\langle c', c^{\vee} \rangle > 0$ says exactly that $c'|_T$ is a positive
integral multiple of $c$.  It follows that (\ref{uczc}) is an isomorphism on tangent spaces
at the identity, so $U_{Z_{G'}(\ker c)}(c^{\vee})$ is generated by the $U'_{c'_i}$.
Thus, we just have to check that these root groups directly span
(in any order) a unipotent smooth connected subgroup of $G'$.
The subset $\{c'_i\} \subset \Phi'$ is closed and disjoint from its negative
(since $\langle c'_i, c^{\vee}\rangle > 0$ for all $i$), so Proposition \ref{upsi}
provides this direct spanning result. (See 
\cite[14.5(2)]{borelag} and \cite[3.3.11, 3.3.13(1)]{pred} for related direct spanning
results in the theory over a field.)
\end{proof}

By Lemma \ref{ufill}, for every $n \ge 1$ we get homomorphisms
$U_{\ge n} \rightarrow U'_{\ge n}$.  Consider the resulting homomorphisms between vector groups
$$f_n:U_{\ge n}/U_{\ge n+1} \rightarrow U'_{\ge n}/U'_{\ge n+1}.$$
We claim that each $f_n$ is a closed immersion.
By construction, $f_n$ is $\Gm$-equivariant with source and target identified with a power of
$\Add$ on which $\Gm$ acts through the $n$th-power map.  Thus, $f_n$ 
is a linear map of vector bundles, so to check if it is a closed immersion it suffices
to pass to geometric fibers and verify injectivity on Lie algebras.  For 
this purpose we may now assume that $S = \Spec k$ for an algebraically closed field $k$.

The Lie algebra of $U_{\ge n}/U_{\ge n+1}$ is the direct product
of the root groups $U_c$ for $c \in \Phi$ such that
$\lambda(c) = n$, and similarly for $U'_{\ge n}/U'_{\ge n+1}$
using $\lambda' = f \circ \lambda$. 
If ${\rm{Lie}}(f_n)$ is not injective then by the equivariance of
$f_n$ with respect to the closed immersion of tori $T \hookrightarrow T'$ it
follows that $\ker({\rm{Lie}}(f_n))$ would have to contain
some root space $\mathfrak{g}_c$
for $c \in \Phi$ satisfying 
$\lambda(c) = n$.  The vanishing of
${\rm{Lie}}(f_n)$ on $\mathfrak{g}_c$ implies
that ${\rm{Lie}}(f)$ carries $\mathfrak{g}_c$ into the span of
the root spaces $\mathfrak{g}'_{c'}$ for $c' \in \Phi'$ such that $\lambda'(c') \ge n+1$.
But the $T$-action on that span has as its weights
precisely the $T$-restrictions of these roots $c'$, so the containment of
$\mathfrak{g}_c$ in the space forces $c'|_T = c$ for at least one such $c'$.
For that $c'$ we have $n+1 \le \lambda'(c') = \lambda(c'|_T) = \lambda(c) = n$, 
a contradiction.

Returning to the relative setting over $S$, 
since the maps $U_{\ge n}/U_{\ge n+1} \rightarrow U'_{\ge n}/U'_{\ge n+1}$ are
all closed immersions, 
the map $U \rightarrow U'$ is a closed immersion by repeated applications of: 

\begin{lemma}\label{closedexact}
In a commutative diagram of short exact sequences of flat, separated, finitely presented $S$-group schemes
$$
\xymatrix{
1 \ar[r] & H'_1 \ar[d]_-{j'} \ar[r] & H_1 \ar[d]^-{j} \ar[r]^-{q_1} & H''_1 \ar[d]^-{j''} \ar[r] & 1 \\
1 \ar[r] & H'_2 \ar[r] & H_2 \ar[r]_-{q_2} & H''_2 \ar[r] & 1 }
$$
if the outer vertical maps are closed immersions then so is the middle one.  
\end{lemma}

\begin{proof}
As usual, we may reduce to the case that $S$ is noetherian.
Clearly $j$ is a monomorphism, so it suffices to prove that it is proper.  Using the valuative
criterion for properness, it suffices to show that if $R$ is a discrete valuation ring with fraction
field $K$ then the containment $H_1(R) \subset H_1(K) \bigcap H_2(R)$
is an equality.  Choose $h_1 \in H_1(K) \bigcap H_2(R)$,
so the image $h''_1$ of $h_1$ in $H''_1(K)$ lies in $H''_1(K) \bigcap H''_2(R) = H''_1(R)$.
To extend $h_1$ to an $R$-point of $H_1$ it is harmless to replace $R$ with
a flat local extension by another discrete valuation ring.  Since
$H_1 \rightarrow H''_1$ is fppf, we can choose such an extension $\widetilde{R}$ 
of $R$ so that $h''_1$ viewed as an $\widetilde{R}$-point of $H''_1$
lifts to $H_1(\widetilde{R})$.  By renaming $\widetilde{R}$ as $R$ and
multiplying $h_1$ by the inverse of an $R$-lift of $h''_1$, 
we reduce to the case that $h''_1$ is trivial, so $h_1$ arises from some $h'_1 \in H'_1(K)$.
The image $h_2 \in H_2(K)$ of $h_1$ lies in
$H_2(R) \bigcap H'_2(K) = H'_2(R)$, so 
$h'_1 \in H'_1(K) \bigcap H'_2(R) = H'_1(R)$.  Hence, $h_1 \in H_1(R)$.
\end{proof}

We have completed the proof that $U_G(\lambda) \rightarrow U_{G'}(\lambda')$
is a closed immersion.  Likewise, $U_G(-\lambda) \rightarrow U_{G'}(-\lambda')$ is a closed immersion,
so $\Omega = f^{-1}(\Omega') \rightarrow \Omega'$ is a closed immersion.  That is,
$f:G \rightarrow G'$ is a closed immersion when restricted over the open subscheme $\Omega'$
in the reductive group $G'$.  Since $(G,T)$ is split, $N_G(T)(S) \rightarrow
W_G(T)(S)$ is surjective (Corollary \ref{wconst}) and $W_G(T) = W(\Phi)_S$ 
is a finite constant $S$-group (Proposition \ref{wfinite}).  Thus, by the Bruhat decomposition on
geometric fibers (see Corollary \ref{bruhattranslate}), $G$ is covered by $N_G(T)(S)$-translates of $\Omega$.
Hence, $f$ is a closed immersion into the open union of
translates of $\Omega'$ by the image in $G'(S)$ of representatives in $N_G(T)(S)$
for the finitely many elements of $W(\Phi)$.  We conclude that 
$f$ is a (finitely presented) closed immersion into a (finitely presented) open subscheme,
or in other words it is a quasi-compact immersion.

By the valuative criterion for properness, to prove 
$f$ is a closed immersion we may assume (after limit arguments to reduce to the noetherian case) that
$S = \Spec R$ for a discrete valuation ring $R$ with fraction field $K$.  
The schematic closure $\overline{G}$ of the locally closed $G$ in $G'$
is then a closed flat $S$-subgroup with generic fiber $G_K$ and it contains
$G$ as an open subgroup.  In particular, the special fiber $\overline{G}_0$ of $\overline{G}$ 
has reductive identity component, so $\overline{G}$ is smooth.
Thus, by Proposition \ref{connfiber}, $G = \overline{G}$.  Hence, $G$ is closed in $G'$ as desired.
\end{proof}

\begin{example}\label{chevgp}
As an application (not to be used later) of the open cell and the 
closed immersion property for the adjoint representation of
a semisimple group scheme of adjoint type in Theorem \ref{adimm}, consider
such groups $G$ over $\Z$ that are split; these are the (semisimple) 
Chevalley groups of adjoint type.  Let $T$ be a split
maximal $\Z$-torus, and $\Phi = \Phi(G,T)$.  Fix a positive system of roots
$\Phi^+$ in $\Phi$, and let $\Delta \subset \Phi^+$ be the base
of simple positive roots.  For any field $k$, we claim $G(k)$ is the subgroup of 
${\rm{Aut}}(\mathfrak{g}_k)$ generated by
the elements ${\rm{Ad}}_G(\exp_{\pm a}(X))$ for $X \in \mathfrak{g}_{\pm a}$
(with $a \in \Delta$)
and the elements ${\rm{Ad}}_G(t)$ where $t \in \prod_{a \in \Delta} k^{\times}$
via the isomorphism $T \simeq \prod_{a \in \Delta} \Gm$ defined by
$t \mapsto (a(t))$ (isomorphism due to the adjoint property; see Exercise \ref{chevex}).

The groups $U_{\pm a}(k)$ generate the representative $w_a(X_a) \in N_G(T)(k)$
of $s_a \in W(\Phi)$ using any $X_a \in \mathfrak{g}_a - \{0\}$.  These reflections $s_a$ 
generate $W(\Phi)$, so conjugation by the elements $w_c(X_c)$ for $c \in \Delta$
carries the groups $U_a(k)$ for  $a \in \Delta$ to the groups $U_b(k)$
for all roots $b$.  This provides the factors $U_{\pm \Phi^+}(k)$
in the $k$-points of the open cell $\Omega$.  
The standard locally closed Bruhat cells over $\overline{k}$
are clearly defined over $k$ (using representatives for $W(\Phi)$ in $N_G(T)(k)$,
such as via the elements $w_a(X_a)$),  so the Bruhat decomposition over $\overline{k}$ implies
that $\Omega(k)$ generates $G(k)$, yielding the desired list of generators
by applying the inclusion ${\rm{Ad}}_G:G(k) \hookrightarrow {\rm{Aut}}(\mathfrak{g}_k)$.

We can go further via 
the split semisimple-rank 1 classification and Existence
Theorem over $\Z$, as follows.  
The Existence Theorem provides 
a simply connected central cover $\widetilde{G} \rightarrow G$ over $\Z$
(see Exercise \ref{scuniv}), 
so $G = \widetilde{G}/Z_{\widetilde{G}}$.
We claim that inside ${\rm{Aut}}(\mathfrak{g}_k)$, the
elements ${\rm{Ad}}_G(\exp_{\pm a}(X_a))$ for $X_a \in \mathfrak{g}_{\pm a}(k)$
(with $a \in \Delta$) generate the image of $\widetilde{G}(k)/Z_{\widetilde{G}}(k)$ in
$G(k) = (\widetilde{G}/Z_{\widetilde{G}})(k)$.

To prove this, first note that (as for any semisimple central extension of $G$ by a finite group scheme of
multiplicative type) the preimage $\widetilde{T}$ of $T$ in $\widetilde{G}$ is a split maximal torus
in $\widetilde{G}$, 
and there exists a natural identification of root systems and isomorphisms between
corresponding root groups for $(G, T)$ and $(\widetilde{G}, \widetilde{T})$ 
(Exercise \ref{scclassical}(i) on geometric fibers).   The simply connected
property for $\widetilde{G}$ implies that the simple positive
coroots are a basis of the cocharacter group of $\widetilde{T}$, so 
no coroot is divisible in the cocharacter lattice. Thus, 
each pair of opposite root groups of $\widetilde{G}_k$ generates
an ${\rm{SL}}_2$ and not a ${\rm{PGL}}_2$.   But in ${\rm{SL}}_2(k)$
the diagonal points are generated by the $k$-points of the standard root groups
(see (\ref{hform}) for a classical formula),
so the subsets $U_{\pm a}(k)$ in $G(k)$ for $a \in \Delta$ generate
the image of $\widetilde{G}(k)$ in $G(k)$.  This establishes our claim.

By using a well-chosen choice of $\Z$-basis $X_b$ of each $\mathfrak{g}_b$ for every $b \in \Phi$
(a ``Chevalley system'', as in the proof of the Existence Theorem; see Definition \ref{chevsystem}
and especially Remark \ref{chevrule}),
the Lie algebra $\mathfrak{g}$ and elements $\exp_a(X_a) \in {\rm{Aut}}(\mathfrak{g})$ can be described explicitly over $\Z$
in terms of the combinatorics of $\Phi$.  This yields an 
explicit ``universal'' formula for $\mathfrak{g}_k$ and $\exp_a(cX_a)$ for all $c \in k$,
recovering the viewpoint used in \cite{steinyale} to define ``Chevalley
groups of adjoint type'' over fields. 
\index{Steinberg!Chevalley group}
For finite $k$ and irreducible $\Phi$, this explicit list of generators inside ${\rm{Aut}}(\mathfrak{g}_k)$
is sometimes taken as the {\em definition} of the finite Chevalley groups
(rather than using the equivalent definition $\widetilde{G}(k)/Z_{\widetilde{G}}(k)$). 
The structural properties of finite Chevalley groups (especially 
simplicity of $\widetilde{G}(k)/Z_{\widetilde{G}}(k)$ for finite $k$ away from
the counterexamples ${\rm{SL}}_2(\mathbf{F}_2)$,
${\rm{SL}}_2(\mathbf{F}_3)/\mathbf{F}_3^{\times}$,
${\rm{Sp}}_4(\mathbf{F}_2) \simeq \mathfrak{S}_6$, and ${\rm{G}}_2(\mathbf{F}_2)$)
can be established via the structure
theory of split semisimple groups over general fields (using $(B,N)$-pairs).
 \end{example}

\subsection{Applications to Levi subgroups}\label{leviapp}

A further application of our study of parabolic subgroups in the relative case is an existence result for
Levi subgroups over an affine base.  Consider a finite-dimensional Lie algebra
$\mathfrak{g}$ over a field $k$ of characteristic 0.  The 
{\em radical} $\mathfrak{r}$ of $\mathfrak{g}$ is the largest solvable ideal and $\mathfrak{g}/\mathfrak{r}$
is semisimple. A {\em Levi subalgebra} of $\mathfrak{g}$ is a subalgebra $\mathfrak{s}$
\index{Levi subalgebra}
such that $\mathfrak{s} \rightarrow \mathfrak{g}/\mathfrak{r}$
is an isomorphism, or equivalently the natural map $\mathfrak{s} \ltimes \mathfrak{r} \rightarrow \mathfrak{g}$
is an isomorphism.  (In particular, $\mathfrak{s}$ is semisimple.)  By the theorem of Levi--Malcev \cite[I, \S6.8, Thm.\:5]{lie1},
Levi subalgebras exist and any two are related through the action of a $k$-point in the unipotent radical
of the linear algebraic $k$-group ${\rm{Aut}}_{\mathfrak{g}/k}$ (representing the automorphism functor of $\mathfrak{g}$ on the 
category of $k$-schemes).

Now consider a smooth affine group $G$ over a general field $k$.  (The case of most
interest will be when $G$ is a parabolic subgroup of a connected
reductive $k$-group.) A {\em Levi $k$-subgroup} of $G$ is 
a smooth closed $k$-subgroup $L \subset G$ such that $L_{\overline{k}} \rightarrow G_{\overline{k}}/\mathscr{R}_u(G_{\overline{k}})$
is an isomorphism; equivalently, $L_{\overline{k}} \ltimes \mathscr{R}_u(G_{\overline{k}}) \rightarrow G_{\overline{k}}$
is an isomorphism.  Informally, $L$ is a $k$-rational complement to the geometric unipotent radical. 
\index{Levi subgroup}
(Based on the analogy with Lie algebras,
one might consider to define Levi subgroups as complements to the geometric
radical. Experience with parabolic subgroups of connected reductive groups
shows that complements
to the geometric unipotent radical are more useful.) 

If $k$ is perfect then $\mathscr{R}_u(G_{\overline{k}})$ descends to a $k$-subgroup
$\mathscr{R}_u(G) \subset G$ and (when $L$ exists!) $L \ltimes \mathscr{R}_u(G) \rightarrow G$ is an isomorphism. 
If ${\rm{char}}(k) = p > 0$ then such an $L$ can fail to exist, even if $k$ is algebraically closed.
A counterexample is  ``${\rm{SL}}_n(W_2(k))$ as a $k$-group'' for any $n \ge 2$,
where $W_2$ denotes the ring-functor of length-2 Witt vectors. (See \cite[Prop.\:A.6.4]{pred} for a precise formulation
and proof, with ${\rm{SL}}_n$ replaced by any Chevalley group. This rests on an analysis of root 
groups relative to suitable maximal tori to reduce the fact that the natural quotient map
$W_2 \twoheadrightarrow \mathbf{G}_{\rm{a}}$ has no additive section.)

\begin{proposition}[Mostow]\label{mostowprop} If ${\rm{char}}(k) = 0$ 
 then  Levi $k$-subgroups of $G$ exist and
$(\mathscr{R}_u(G))(k)$-conjugation is transitive on the set of such $k$-subgroups. 
\end{proposition}
\index{Levi subgroup!conjugacy}
\index{conjugacy!of Levi subgroups}

\begin{proof}
More generally, consider a (possibly disconnected) reductive $k$-group $\overline{G}$, 
a unipotent $k$-group $U$, and an exact sequence of affine algebraic $k$-groups
$$1 \rightarrow U \rightarrow G \rightarrow \overline{G} \rightarrow 1.$$
We claim that this splits over $k$ as a semi-direct product, and that any two splittings are related through $u$-conjugation
for some $u \in U(k)$. 
Using a filtration of $U$ by its (characteristic) derived series, we reduce to the case where $U$ is commutative 
{\em provided} that we also show ${\rm{H}}^1(k,U)=0$ in the commutative case (so $k$-rational points conjugating one splitting
into another can be lifted through stages of the derived series of $U$ in general).  

Since ${\rm{char}}(k) = 0$ and $U$ is commutative, $U \simeq \Add^n$ for some $n$ (see Exercise \ref{unip0}).  
The endomorphism functor of $\Add$
on the category of $k$-algebras is represented by $\Add$ (i.e., the only additive polynomials over a $k$-algebra
$R$ are $rX$ for $r \in R$) since ${\rm{char}}(k)=0$, so the endomorphism functor of 
$\Add^n$ is represented by ${\rm{Mat}}_n$ and hence the automorphism functor of $\Add^n$ is
represented by ${\rm{GL}}_n$.  It follows that there is a {\em unique} linear structure on $U$ 
lifting the one on its Lie algebra, so 
this structure is compatible with extension on $k$ and equivariant for the natural action of $\overline{G} = G/U$ on the commutative
normal $k$-subgroup $U$ of $G$.  Thus,
$G$ is an extension of the possibly disconnected $\overline{G}$ by a linear representation $V$ of $\overline{G}$,
with $\overline{G}^0$ a reductive group.  The vanishing of ${\rm{H}}^1(k,V)$ is a consequence of additive Hilbert 90,
and our task is to show 
$$1 \rightarrow V \rightarrow G \rightarrow \overline{G} \rightarrow 1$$ 
splits over $k$ as a semi-direct product, with any two splittings related by 
$v$-conjugacy for some $v \in V(k)$. 

Observe that $q:G \rightarrow \overline{G}$ is a $V$-torsor for the \'etale topology 
on $\overline{G}$. Before we show 
that $q$ admits a $k$-homomorphic section, let's show that it admits a morphic section:  the underlying $V$-torsor 
(ignoring the group structure of $G$) is trivial.
More generally, for any $k$-scheme $S$ (such as $\overline{G}$)
the set of $V$-torsors over $S$ (up to isomorphism) is ${\rm{H}}^1(S_{\et}, V \otimes_k \calO_S)$, so it suffices
to prove that this cohomology group vanishes when $S$ is affine. By
choosing a $k$-basis of $V$ it suffices to treat the case $V = k$.  By descent theory for quasi-coherent sheaves, 
${\rm{H}}^1(S_{\et}, \calO_S)$ classifies the set of quasi-coherent extensions of $\calO_S$ by $\calO_S$.
Writing $S = \Spec A$, this corresponds to the set of $A$-linear extensions of $A$ by itself as an
$A$-module, and any such extension
is clearly split.  
Thus, $q$ admits a section $\sigma$ as a map of affine $k$-schemes.  

We will  modify $\sigma$ to make it a homomorphism
by studying Hochschild cohomology that imitates group cohomology
via ``algebraic cochains''.  (See \cite[III, \S3]{oesterle} and \S\ref{hochsec} for a review of
this cohomology theory.)
Consider the Hochschild cohomology 
${\rm{H}}^2(\overline{G},V)$. 
As in the classical setting of group cohomology, by Proposition \ref{obs} 
the obstruction to modifying the choice of $\sigma$ to make it a homomorphism is a 
canonically associated class in 
${\rm{H}}^2(\overline{G},V)$, and if this class vanishes then
the set of $V(k)$-conjugacy classes of such splittings is a torsor for
${\rm{H}}^1(\overline{G},V)$.  It therefore suffices to show that the higher Hochschild cohomology of $\overline{G}$ 
with coefficients in a linear representation vanishes.  The formation of such Hochschild cohomology commutes 
with extension of the ground field (Proposition \ref{bch}), so we can assume $k = \overline{k}$. 

By Lemma \ref{derhoch},
for an affine algebraic group scheme $H$ over a field $k$, its Hochschild cohomology (as a functor on the category of not necessarily finite-dimensional algebraic linear representations for the group) is the derived functor of the functor of $H$-invariants.
Consider algebraic linear representations $W$ of $H$ (i.e.,
$k$-vector spaces $W$ equipped with an $R$-linear action of $H(R)$ on $R \otimes_k W$
functorially in all $k$-algebras $R$).  By \cite[\S3.1--\S3.3]{water} any such $W$ 
 is the direct limit of its finite-dimensional 
algebraic subrepresentations, and the formation of Hochschild cohomology commutes with direct limits, 
so if $H$ has completely reducible finite-dimensional (algebraic) linear representation theory 
then the higher cohomology vanishes. Now it remains to solve Exercise \ref{1.6}(ii): if $k = \overline{k}$ with
${\rm{char}}(k)=0$ then any linear algebraic 
group $H$ over $k$ with reductive identity component has completely reducible
 finite-dimensional algebraic linear representation theory.
 
 In view of the natural isomorphism
 $\Hom_{H}(W, W') = (W' \otimes W^{\ast})^{H}$
 for finite-dimensional linear representations $W$ and $W'$ of $H$, 
 it suffices to prove that the functor of $H$-invariants is right-exact.  It suffices to separately treat $H^0$
and the finite constant $H/H^0$.
 The case of finite constant groups is settled via averaging 
 since ${\rm{char}}(k)=0$, and the connected reductive case reduces separately to the cases of split tori and connected
  semisimple groups.  The case of split tori is well-known (in any 
  characteristic), by consideration of graded modules
  as reviewed just before Proposition \ref{obs}. For a connected semisimple $k$-group $H$ and finite-dimensional algebraic
  linear representation $W$ of $H$, 
  we have naturally $W^H = W^{\mathfrak{h}}$
  via the associated Lie algebra representation on $W$ since ${\rm{char}}(k)=0$
  and $H$ is connected.  Thus, it suffices to show that $\mathfrak{h}$ is
  semisimple when $H$ is connected semisimple. 
  
  Suppose to the contrary, so 
 the radical $\mathfrak{r}$ of $\mathfrak{h}$ is nonzero.  This subspace
  of $\mathfrak{h}$ is stable under the adjoint action of $H$, so
  we can consider the weight space decomposition of $\mathfrak{r}$
  under the restriction of ${\rm{Ad}}_H$ to a maximal torus $T \subset H$.
  If $\mathfrak{r} \subset \mathfrak{h}^T = \mathfrak{t}$ then
  $\mathfrak{r}$ would lie inside intersection of the 
  $H(k)$-conjugates of $\mathfrak{t}$.  The intersection of the $H(k)$-conjugates
  of $T$ is the finite center $Z_H$ that is \'etale since ${\rm{char}}(k)=0$, and it coincides with such an intersection
  using finitely many $H(k)$-conjugates (due to the noetherian property of $G$), 
  so the $H(k)$-conjugates of $\mathfrak{t}$ have intersection ${\rm{Lie}}(Z_H) = 0$.
 Thus, $\mathfrak{r}$ contains $\mathfrak{h}_a$ for some
  $a \in \Phi(H,T)$, so it contains the line $\mathfrak{h}_{-a}$ in the ${\rm{Ad}}_H$-orbit
  of $\mathfrak{h}_a$. Thus, the solvable ideal
  $\mathfrak{r}$ contains the Lie subalgebra generated by $\mathfrak{h}_a$ and $\mathfrak{h}_{-a}$.   
  But this subalgebra is the non-solvable $\mathfrak{sl}_2$, so we have reached a contradiction.
\end{proof}

Over general fields $k$ there is an important class of smooth connected
affine $k$-groups that always admit a Levi $k$-subgroup: parabolic
$k$-subgroups in connected reductive $k$-groups.
We will use the dynamic method to prove a relative version of this result. 
(See Corollary \ref{rup} for the notion of unipotent radical in parabolic
subgroup schemes of reductive group schemes over any base.) 
To get started, we define Levi subgroups in the relative setting.

\begin{definition} Let $G \rightarrow S$ be a reductive group scheme, 
$P \subset G$ a parabolic subgroup.  A {\em Levi subgroup} of $P$
is a smooth closed $S$-subgroup $L \subset P$ such that
$L \ltimes \mathscr{R}_u(P) \rightarrow P$ is an isomorphism.
The functor $\underline{\rm{Lev}}(P)$ assigns to any $S$-scheme $S'$
the set of Levi $S'$-subgroups of $P_{S'}$.
\end{definition}
\index{Levi subgroup!of parabolic subgroup}
\index{ aaalev@$\underline{\rm{Lev}}(P)$}

If $\lambda:\Gm \rightarrow G$ is a 1-parameter subgroup 
and $P$ is the parabolic subgroup $P_G(\lambda)$ then $Z_G(\lambda)$ is a Levi subgroup
because $\mathscr{R}_u(P) = U_G(\lambda)$.  To construct Levi $S$-subgroups more generally
(at least when $S$ is affine), we shall use the 
action of $\mathscr{R}_u(P)$ on
$\underline{\rm{Lev}}(P)$.  For this purpose, it is convenient to first construct a $P$-equivariant
filtration of $\mathscr{R}_u(P)$ with vector bundle successive quotients: 

\begin{theorem}\label{dynlevi} 
There is a descending filtration $\mathscr{R}_u(P) =: U_1 \supset U_2 \supset \dots$
by $\underline{\rm{Aut}}_{P/S}$-stable smooth closed $S$-subgroups
such that 
\begin{enumerate}
\item[(i)] for all $s \in S$ we have $U_{i,s} = 1$ if $i > \dim \mathscr{R}_u(P_s)$; 
\item[(ii)] $uu'u^{-1}{u'}^{-1} \in U_{i+j}$ for any points $u \in U_i$ and $u' \in U_j$ $($valued in an $S$-scheme$)$;
\item[(iii)] each commutative $S$-group $U_i/U_{i+1}$  admits a unique $P$-equivariant $\calO_S$-linear structure
making it a vector bundle 
$($so $U_i/U_{i+1}$ is canonically identified with ${\rm{Lie}}(U_i/U_{i+1})$ respecting the actions of $P$$)$;
\item[(iv)] the formation of $\{U_i\}$ is compatible with base change on $S$ and functorial
with respect to isomorphisms in the pair $(G, P)$.
\end{enumerate}
\end{theorem}
\index{parabolic subgroup!canonical filtration of $\mathscr{R}_u(P)$}

This is \cite[XXVI, 2.1]{sga3}; the dynamic method simplifies the proof.

\begin{proof} 
We first reduce to the adjoint semisimple case. 
Let $G^{\rm{ad}} = G/Z_G$; this contains the parabolic $S$-subgroup
\index{ aaagad@$G^{\rm{ad}}$}
$P/Z_G$.  We claim that (a) the map
$q:G \twoheadrightarrow G^{\rm{ad}}$ carries $\mathscr{R}_u(P)$ isomorphically onto
$\mathscr{R}_u(P/Z_G)$ , (b) $Z_G = Z_P$.  Once these properties are proved,
$\underline{\rm{Aut}}_{P/S}$ naturally acts on $P/Z_G$ and hence
the problem for $(G, P)$ is reduced to the one for $(G^{\rm{ad}}, P/Z_G)$. 

To prove (a) and (b), we may work \'etale-locally on $S$
so that $P = P_G(\lambda)$ for some $\lambda: \Gm \rightarrow G$.
Then $P = Z_G(\lambda) \ltimes U_G(\lambda)$, so $Z_G \subset Z_G(\lambda) \subset P$
and $U_G(\lambda) = \mathscr{R}_u(P)$.  In particular, $\mathscr{R}_u(P) \cap Z_G = 1$. The
fibral isomorphism criterion (Lemma \ref{flatclosed}) and behavior
of dynamic constructions with respect to flat quotients over a field
\cite[Cor.\:2.1.9]{pred} imply
that $q$ identifies $P/Z_G$ with $P_{G^{\rm{ad}}}(q \circ \lambda)$
and carries $\mathscr{R}_u(P) = U_G(\lambda)$ isomorphically onto $U_{G^{\rm{ad}}}(q \circ \lambda) = 
\mathscr{R}_u(P/Z_G)$ (since the map $U_G(\lambda)_s \rightarrow U_{G^{\rm{ad}}}(q \circ \lambda)_s$
is faithfully flat with trivial kernel for all $s \in S$). 
This settles (a).  To prove that
$Z_P = Z_G$ we may localize on $S$ so that $P$
contains a Borel $S$-subgroup $B$ of $G$ that in turn contains a maximal torus $T$.  Since
$Z_G(T) = T$, so $Z_P \subset T$, it suffices to show
that any point of $T$ centralizing $\mathscr{R}_u(B)$ is central in $G$.
Passing to the case when $(G, T)$ is split, the $T$-action on ${\rm{Lie}}(\mathscr{R}_u(B))$
encodes all roots up to a sign. Hence, the centralizer of $\mathscr{R}_u(B)$ in $T$ is contained
in the intersection of the kernels of the roots.  But this intersection is
$Z_G$ (Corollary \ref{computez}(1)), so (b) is also proved.

Now we may and do assume $G$ is adjoint semisimple (but otherwise arbitrary).
We will make a construction satisfying the desired properties in the split case, show it is independent of all choices, and 
then use descent theory to settle the general (adjoint semisimple) case.  
Suppose $(G, T, M)$ is split and that there 
is a Borel subgroup $B \subset P$ containing
$T$; this situation can always be achieved \'etale-locally on
$S$.  By Corollary \ref{parfiber}(2)
and Zariski localization on $S$ we can arrange that there is a (unique) positive system of roots $\Phi^+$ in $\Phi \subset M$
such that $\Phi^+ = \Phi(B_s,T_s)$ for all $s \in S$.
The base $\Delta = \{a_i\}$ of $\Phi^+$
is a basis of the root lattice $\Z\Phi$ that is equal to $M$
(since $G$ is adjoint semisimple). Let $\{\omega_i^{\vee}\}$ be the dual basis of $M^{\vee} \subset {\rm{X}}_{\ast}(T)$. 
By Exercise \ref{gross6}(ii) (applied on geometric fibers)
and Corollary \ref{parfiber}(2), further Zariski localization brings us to the case that 
$P = P_G(\lambda_I)$  for a (necessarily unique) subset $I \subset \Delta$,
\index{ aaali@$\lambda_I$}
with $\lambda_I := \sum_{a_i \in I} \omega_i^{\vee}$.
The merit of this description of $P$ is that $\lambda_I$ is determined by additional group-theoretic data
$(B, T, M)$ that exist \'etale-locally on $S$. 

For each $a \in \Phi$, let $U_a \subset G$ be the corresponding root group for $(G, T, M)$.
By Remark \ref{uarem}, the $S$-group
$U := \mathscr{R}_u(P) = U_G(\lambda_I)$ is directly spanned in any order
by the $U_a$'s for $a$ such that $\langle a, \lambda_I \rangle \ge 1$.
More specifically, by Proposition \ref{upsi}, for all $n \ge 1$
the root groups $U_a$ for $a$ satisfying $\langle a, \lambda_I \rangle \ge n$
directly span (in any order) a normal smooth closed $S$-subgroup $U_{\ge n}$ of $U$
such that the successive quotients $U_{\ge n}/U_{\ge n+1}$ are commutative and the natural map
$$\prod_{\langle a, \lambda_I \rangle = n} U_a \rightarrow U_{\ge n}/U_{\ge n+1}$$
defined by multiplication in $G$ is an isomorphism. 

\begin{lemma} For all $n \ge 1$, the subgroup $U_{\ge n}$ is normal in $P$ and  the 
quotient $U_{\ge n}/U_{\ge n+1}$ has
a unique $P$-equivariant $\calO_S$-linear structure.
\end{lemma}

\begin{proof}
For all $a \in \Phi$,
the construction of root groups (see Theorem \ref{rootgp}) provides a canonical $T$-equivariant isomorphism
$\exp_a: \mathbf{W}(\mathfrak{g}_a) \simeq U_a$ (where
$\mathbf{W}(\mathscr{E})$ is the additive $S$-group scheme underlying
a vector bundle $\mathscr{E}$ on $S$).  The equivariance implies that $\exp_a$
identifies the $\Gm$-action on $U_a$ via $\lambda_I$-conjugation 
with the $\Gm$-scaling on the line bundle $\mathfrak{g}_a$ via $t^{\langle a, \lambda_I\rangle}$.
In particular, the canonical isomorphism
$$\mathbf{W}(\prod_{\langle a, \lambda_I \rangle=n} \mathfrak{g}_a) \simeq U_{\ge n}/U_{\ge n+1}$$
defines a vector bundle structure on the target under which
the $\Gm$-action via $\lambda_I$-conjugation corresponds
to multiplication by $t^n$, so 
$U_{\ge n}/U_{\ge n+1}$ is
the schematic centralizer in $U/U_{\ge n+1}$ of the action by $\mu_n \subset \Gm$ via
$\lambda_I$-conjugation.  Via {\em descending} induction on $n$, this characterizes
the subgroups $U_{\ge n}$ of $U = U_G(\lambda_I)$ solely in terms of
$\lambda_I$.  

By (\ref{rootcomm}), for all $n, m \ge 1$
and points $u \in U_{\ge n}$ and $u' \in U_{\ge m}$, we have
$uu'u^{-1}{u'}^{-1} \in U_{\ge n+m}$. Thus, conjugation by $U = U_{\ge 1}$ on the normal subgroups 
$U_{\ge n}$ and $U_{\ge n+1}$
induces the trivial action on $U_{\ge n}/U_{\ge n+1}$. 
The preceding characterization of
the $U_{\ge n}$'s in terms of $\lambda_I$ implies that $Z_G(\lambda_I)$ normalizes each $U_{\ge n}$, 
so the subgroup $P = Z_G(\lambda_I) \ltimes U$ normalizes each $U_{\ge n}$.
Likewise, the vector bundle structure constructed on $U_{\ge n}/U_{\ge n+1}$
is {\em uniquely} characterized by identifying $\lambda_I$-conjugation with scaling
against $t^n$ because the only additive automorphisms of $\mathbf{W}(\mathscr{O}^r) = \Add^r$
centralizing $t^n$-scaling for all $t \in \Gm$ are the linear automorphisms.
Hence, this vector bundle structure on $U_{\ge n}/U_{\ge n+1}$ commutes with $Z_G(\lambda_I)$-conjugation
and so more generally commutes with conjugation against $P = Z_G(\lambda_I) \ltimes U$.

The $P$-equivariant vector bundle structure just built on
each $U_{\ge n}/U_{\ge n+1}$  is {\em unique}. 
Indeed, any such structure identifies conjugation against $\lambda_I: \Gm \rightarrow T \subset P$
with a linear action of $\Gm$ that has to be scaling against $t^n$
since we can read off the action on the Lie algebra (as any {\em linear}
action of $\Gm$ on a vector bundle $\mathscr{E}$ is encoded in a weight space decomposition,
and canonically ${\rm{Lie}}(\mathbf{W}(\mathscr{E})) \simeq \mathscr{E}$ as vector bundles).
\end{proof}

The $S$-subgroups $U_i := U_{\ge i}$ for $i \ge 1$ constitute a descending filtration
with the desired properties (ii) and (iii) for $(G, P)$ except that we have only shown the $U_i$'s are
stable under $P$-conjugation rather than under the action of 
$\underline{\rm{Aut}}_{P/S}$ on $U = \mathscr{R}_u(P)$.
Suppose $(B', T', M')$ is another such triple over
$S$, so $B/U$ and $B'/U$ are Borel subgroups of the reductive $S$-group
$P/U$.  Since $P \rightarrow P/U$ is a smooth surjection, if
$S'$ is an $S$-scheme then any point in $(P/U)(S')$ lifts
to $P$ over an \'etale cover of $S'$.
Hence, by Theorem \ref{borel}(2) (applied to the reductive $P/U$) it follows that \'etale-locally on $S$ we can use
$P$-conjugation to bring $B'$ to $B$.  Once we have arranged that
$B' = B$, Theorem \ref{schemetori} (applied to $B$)
allows us to arrange by suitable $B$-conjugation over an \'etale cover
of $S$ that $T' = T$.  Zariski-locally over $S$, this torus equality identifies $M'$ with $M$.

The $U_{\ge n}$'s are normalized by $P$ 
and the preceding constructions with them are uniquely characterized via the cocharacter
$\lambda_I$ of $G$ uniquely determined by $P$ and the triple
$(B, T, M)$ with $B \subset P$, so 
the $U_{\ge n}$'s are {\em independent} of $(B, T, M)$.
Hence, by descent theory we obtain the descending filtration $\{U_i\}$
with the desired properties (ii) and (iii) in the general case, as 
the independence of all choices ensures stability under
the entire automorphism functor of $P$.  Property (iv) holds by construction,
and property (i) holds provided that the set of values $\langle \lambda_I, a \rangle \ge 1$ for $a \in \Phi^+$ 
is an interval in $\Z$ beginning at 1 (as that ensures the largest such value is at most 
$\#\Phi_{\lambda_I \ge 1} = \dim \mathscr{R}_u(P_s)$). 
Since $\langle \lambda_I, a \rangle \in \{0, 1\}$ for all $a \in \Delta$ by definition of $\lambda_I$, 
we just need to recall a general property of root systems:
any $a \in \Phi^+ - \Delta$ has the form $b + c$ with $b \in \Delta$ and $c \in \Phi^+$
(see \cite[XXI, 3.1.2]{sga3} or \cite[Ch.\:VI, \S1.6, Prop.\:19]{lie4}).
\end{proof}

\begin{proposition}\label{levit} For every maximal
torus $T \subset P$ there is a unique Levi
$S$-subgroup $L \subset P$ containing $T$.
\end{proposition}
\index{Levi subgroup!existence in parabolic}
\index{parabolic subgroup!existence of Levi}

\begin{proof}
By the uniqueness, standard limit arguments
and \'etale descent let us
assume $S$ is strictly henselian local.  Thus, 
$T$ is split and $\Phi := \Phi(G,T)$ is a root system 
in $M := \Hom_S(T, \Gm)$.  
By Corollary \ref{parfiber}
there exists $\lambda \in {\rm{X}}_{\ast}(T) = M^{\vee}$
so that $P = P_G(\lambda) = Z_G(\lambda) \ltimes U_G(\lambda)$.
Since $U_G(\lambda) = \mathscr{R}_u(P)$
and $Z_G(\lambda) \rightarrow P/\mathscr{R}_u(P)$ is an isomorphism,
so  $Z_G(\lambda)$ is a Levi $S$-subgroup, it suffices to prove
there is only one Levi subgroup $L$ of $P$ containing $T$.

The subset $\Phi(L,T)$ of the parabolic set of roots
$\Phi(P,T)$ consists of those $a \in \Phi(P,T)$ such that
$-a \in \Phi(P,T)$.  Indeed, since $L \rightarrow P/\mathscr{R}_u(P)$
is an isomorphism, it is equivalent to show
that $\Phi(\mathscr{R}_u(P),T)$ is the set of $a \in \Phi(P,T)$
such that $-a \not\in \Phi(P,T)$; this latter assertion
has nothing to do with $L$.  
Since  $\Phi = -\Phi$ and 
$$\Phi(Z_G(\lambda),T) = \{a \in \Phi\,|\, \langle a, \lambda \rangle = 0\},\,\,\,
\Phi(U_G(\lambda), T) = \{a \in \Phi\,|\, \langle a, \lambda \rangle > 0\},$$
we obtain the desired descriptions of sets of roots.
In particular, the subset $\Phi(L, T) \subset \Phi$ is determined by the pair $(P, T)$ without reference to $L$.

For all $a \in \Phi(L, T) \subset \Phi$, the root group $U_a$ for the reductive group $L$
satisfies the conditions that uniquely characterize the $a$-root group of $(G, T)$.
Hence, by consideration of the open cell of $(L, T)$ relative
to a positive system of roots in $\Phi(L, T)$, the group sheaf $L$ is generated as a group sheaf by 
$T$ and the root groups $U_a$ of $(G,T)$ for all $a \in \Phi(L, T)$.
(More generally, if $\mathscr{G} \rightarrow S$ is a smooth group scheme with connected fibers
and $\Omega \subset \mathscr{G}$ is an open subscheme with non-empty fibers over $S$
then the smooth multiplication map $\mathscr{G} \times \mathscr{G} \rightarrow \mathscr{G}$
restricts to a smooth map $\Omega \times \Omega \rightarrow \mathscr{G}$ that is surjective
since the geometric fibers $\mathscr{G}_{\overline{s}}$ are irreducible.)
This is an explicit description of $L$ in terms of data (such as $\Phi(L,T) \subset \Phi$) that depend only on $(P, T)$.
\end{proof}

\begin{corollary}\cite[XXVI, 1.8]{sga3}\label{levitors} The functor $\underline{\rm{Lev}}(P)$
of Levi subgroups of $P$, equipped with its natural
$\mathscr{R}_u(P)$-action via conjugation in $G$, is represented by an $\mathscr{R}_u(P)$-torsor.
In particular, any Levi $S$-subgroup $L$ of $P$ is its own schematic normalizer in $P$.
\end{corollary}
\index{Levi subgroup!self-normalizing}

\begin{proof}
Since $U := \mathscr{R}_u(P)$ is $S$-affine and Levi subgroups exist
\'etale-locally on $S$ (e.g., $L = Z_G(\lambda)$
when there exists $\lambda: \Gm \rightarrow G$ such that
$P = P_G(\lambda)$), it suffices to show that
the sheaf
$\underline{\rm{Lev}}(P)$ is a $U$-torsor sheaf. Using general $S$,
it suffices to show that if $L, L' \subset P$ are Levi $S$-subgroups
then $L' = uLu^{-1}$ for a unique $u \in U(S)$.
The uniqueness allows us to work \'etale-locally on $S$, so we may 
assume that $L$ and $L'$ contain
respective maximal $S$-tori $T$ and $T'$.
By further \'etale localization we may arrange that $T' = pT p^{-1}$
for some $p \in P(S)$ (see Theorem \ref{schemetori} applied to $P$).  

Writing
$p = u g$ for unique $u \in U(S)$ and $g \in L(S)$, by replacing
$T$ with $gTg^{-1}$ we may assume $T' = u T u^{-1}$.
Thus, $L'$ and $uLu^{-1}$  are Levi $S$-subgroups of $P$
containing the same maximal torus $T'$, so $L' = uLu^{-1}$
by Proposition \ref{levit}.  It remains to prove uniqueness of
$u$, which expresses the property that
$N_P(L) = L$ (since $L \ltimes U = P$).
That is, if $uxu^{-1} \in L$ for all $x \in L$
then we wish to prove $u = 1$.
Obviously $uxu^{-1}x^{-1} \in L$ for all $x \in L$,
but $u(xu^{-1}x^{-1}) \in U$, so the triviality of
$L \cap U$ implies that $u = x u x^{-1}$ for all $x \in L$.
In other words, $u \in Z_G(L)$.  But
$Z_G(L) \subset Z_G(T) = T \subset L$,
so $u \in L \cap U = 1$.
\end{proof}

\begin{remark}
If $G$ is semisimple and simply connected then 
for every Levi $S$-subgroup $L$ of a parabolic
$S$-subgroup $P$, the semisimple derived group $\mathscr{D}(L)$ is
also simply connected.  To prove this fact, which we will never use 
but is important in practice, by working 
\'etale-locally on $S$ we may assume $L = Z_G(\lambda)$
for a closed subtorus $\lambda:  \Gm \hookrightarrow T$
of a maximal $S$-torus $T \subset G$.  Then
we may apply Exercise \ref{scuniv}(iv) to conclude.
\end{remark}

The following result uses non-abelian degree-1 \v{C}ech cohomology for the \'etale topology
with group sheaves. (A geometric
interpretation of this cohomology via torsors is given in 
Exercise \ref{h1ex} for smooth affine group schemes, and when the well-known low-degree
formalism in Galois cohomology over fields is expressed in terms of the \'etale topology rather
than Galois groups then it easily adapts to construct a 6-term exact sequence of pointed sets
associated to any short exact sequence of smooth affine group schemes.)

\begin{corollary}\label{leviH1} If $S$ is affine then $P$ admits a Levi $S$-subgroup $L$
and the natural map $j_L:{\rm{H}}^1(S_{\et}, L) \rightarrow {\rm{H}}^1(S_{\et},P)$
is an isomorphism.
\end{corollary}

\begin{proof} 
Let $U = \mathscr{R}_u(P)$, so $\underline{\rm{Lev}}(P)$ is represented by a $U$-torsor
over $S$.  The existence of a Levi $S$-subgroup $L$ means exactly that this torsor
is trivial.  Hence, to find $L$ it suffices to show that {\em every} $U$-torsor over $S$ is trivial, which is
to say that ${\rm{H}}^1(S_{\et},U) = 1$.  The descending filtration $\{U_i\}$
provided by Theorem \ref{dynlevi}
reduces this to the vanishing of each ${\rm{H}}^1(S_{\et}, U_i/U_{i+1})$
for vector bundles $U_i/U_{i+1}$.  Such vanishing holds because $S$ is affine. 

Injectivity of $j_L$ is clear since the composite $L \rightarrow P \rightarrow P/U$
is an isomorphism.  For surjectivity it suffices to show that 
$f:{\rm{H}}^1(S_{\et},P) \rightarrow {\rm{H}}^1(S_{\et},P/U)$ is injective.
Since $P$ naturally acts on the short exact sequence
$$1 \rightarrow U \rightarrow P \rightarrow P/U \rightarrow 1$$
as well as on the descending filtration $\{U_i\}$ of $U$
respecting the vector bundle structure on each $U_i/U_{i+1}$, 
a representative \v{C}ech 1-cocycle $c$ for $\xi \in {\rm{H}}^1(S_{\et},P)$ 
gives descent datum throughout to built an \'etale-twisted form
$$1 \rightarrow U_c \rightarrow P_c \rightarrow (P/U)_c \rightarrow 1$$
and descending terminating filtration $\{U_{c,i}\}$ of $U_c$ consisting of smooth normal closed $S$-subgroups
such that each $U_{c,i}/U_{c,i+1}$ is a vector bundle.

The choice of $c$ provides an identification of sets
${\rm{H}}^1(S_{\et},P) \simeq {\rm{H}}^1(S_{\et},P_c)$
carrying the fiber of $f$ through $\xi$ over to the image of
${\rm{H}}^1(S_{\et}, U_c) \rightarrow {\rm{H}}^1(S_{\et},P_c)$.
Hence, it suffices to prove
${\rm{H}}^1(S_{\et}, U_c) = 1$.  
The descending
filtration $\{U_{c,i}\}$ of $U_c$ reduces this to the vanishing
of ${\rm{H}}^1(S_{\et}, U_{c,i}/U_{c,i+1})$ for all $i$.  Such vanishing holds
because $S$ is affine.
\end{proof}

To conclude our discussion of Levi subgroups, we use them to address the existence of dynamic descriptions
of parabolic subgroups in the relative setting.
First we provide motivation over a general field $k$. An ingredient in the Borel--Tits theory of relative root systems
is that {\em any} parabolic $k$-subgroup $Q$ of a connected reductive $k$-group $G$ admits
a dynamic description as $P_G(\lambda)$ for a 1-parameter $k$-subgroup
$\lambda:\Gm \rightarrow G$ (see \cite[Prop.\:2.2.9]{pred} for a proof); any such $\lambda$
is valued in $Z_G(\lambda) \subset P_G(\lambda) = Q$.  Since $Z_G(\lambda)$ is a Levi
$k$-subgroup of $Q$, the $\mathscr{R}_u(Q)(k)$-conjugacy 
of all Levi $k$-subgroups of $Q$ (Corollary \ref{levitors}) implies that {\em every} Levi $k$-subgroup $L \subset Q$
arises as $Z_G(\lambda)$ for some such $\lambda$.  

The dynamic method produces
\index{dynamic method!parabolic subgroups}
\index{parabolic subgroup!dynamic description}
parabolic subgroups $Q$ and Levi subgroups $L \subset Q$ in reductive group schemes $G$ 
over any base scheme $S$ when 1-parameter
subgroups are provided over $S$, so it is natural to ask if such pairs $(Q, L)$
 always arise in the form $(P_G(\lambda), Z_G(\lambda))$
for some $\lambda:\Gm \rightarrow G$ over $S$.  The case of connected semi-local $S$
(i.e., $S = {\rm{Spec}}(A)$ for nonzero $A$ with finitely many maximal ideals and no nontrivial idempotents)
is addressed with affirmative results in \cite[XXVI, 6.10--6.14]{sga3}.

Over any $S$, if $Q = P_G(\lambda)$ for some $\lambda:\Gm \rightarrow G$
then not only does $Q$ admit a Levi $S$-subgroup, namely $Z_G(\lambda) \subset Q$, but by Corollary \ref{levitors}
every Levi $S$-subgroup $L \subset Q$ has the form $Z_G(\mu)$ for some $\mu$
in the $\mathscr{R}_u(Q)(S)$-conjugacy class of $\lambda$.  
%Conversely, inspired by \cite[XXVI, 6.7]{sga3},
%we can show that the dynamic method explains
%the existence of Levi $S$-subgroups in general:
%
%\begin{proposition} Let $G \rightarrow S$ be a reductive group scheme, and $Q$ a parabolic $S$-subgroup of $G$.
%If $L \subset Q$ is a Levi $S$-subgroup then there exists $\lambda:\Gm \rightarrow G$ such that
%$Q = P_G(\lambda)$ and $L = Z_G(\lambda)$.
%\end{proposition}
%
%\begin{proof}
%Since $\dim P_s$ and $\dim G_s$ 
%are constant Zariski-locally on $S$, we may and do assume that these dimensions
%are constant and $\dim P_s < \dim G_s$. The conjugation action of $L$ on
%$U = \mathscr{R}_u(P)$ defines a linear representation $\rho:L \rightarrow {\rm{GL}}({\rm{Lie}}(U))$,
%so we get a character $\det(\rho):L \rightarrow \Gm$.  The center $Z_L$ of the reductive group scheme $L$ 
%is of multiplicative type, so it 
%has a maximal subtorus $Z_L^{\rm{tor}}$.  
%
%Consider the character 
%$\chi = \det(\rho)|_{Z_L^{\rm{tor}}}: Z_L^{\rm{tor}} \rightarrow \Gm$.   Using $\chi$, we will show that
%after passing to the constituents of a covering by $S$ by pairwise disjoint open and closed subschemes there is a cocharacter
%$\lambda:\Gm \rightarrow Z_L^{\rm{tor}}$ satisfying $Q = P_G(\lambda)$ and $L = Z_G(\lambda)$.
%\end{proof}
%
In \cite[\S7.3]{gillesga3} a deeper study of parabolic subgroups and their Levi subgroups 
is combined with the structure of automorphism schemes of reductive group schemes
(developed in \S\ref{autex} below) to show that for any connected $S$ the dynamic method 
produces {\em all} pairs $(Q, L)$ consisting of a 
parabolic $S$-subgroup $Q \subset G$ and Levi $S$-subgroup $L \subset Q$.

In particular, if $S$ is a connected {\em affine} scheme then every parabolic $S$-subgroup $Q$
in a reductive $S$-group $G$ admits a dynamic description because such $Q$ always admit 
a Levi $S$-subgroup (proved by non-dynamic means, such as the vanishing of cohomological obstructions 
as in the proof of Corollary \ref{leviH1}).  If we drop the affineness hypothesis then the cohomological
proof of Corollary \ref{leviH1} breaks down
and it can happen that the $\mathscr{R}_u(P)$-torsor $\underline{{\rm{Lev}}}(P)$ is nontrivial, so 
the parabolic $S$-subgroup $P$ in $G$ has {\em no} dynamic description.
I am grateful to Edixhoven for suggesting the following counterexamples.

\begin{example}\label{edixex}
Let $S$ be a scheme such that the group ${\rm{H}}^1(S,\calO_S) = {\rm{H}}^1(S_{\et},\calO_S)$ is nonzero, which is to say
that $S$ admits a nontrivial $\Add$-torsor $U$ (for the \'etale topology, or equivalently for the Zariski topology).
For example, $S$ could be a smooth proper and geometrically connected curve with positive genus
over a field $k$, or $S = \mathbf{A}^2_k - \{(0,0)\}$. (Note that $S$ is not affine.) 
We shall use $U$ to make a nontrivial $\mathbf{P}^1$-bundle $E$ over $S$
admitting a section $\sigma$ such that the automorphism scheme $G$ 
of $E$ is semisimple and the $G$-stabilizer $P$ of $\sigma$ is a parabolic $S$-subgroup with no Levi subgroup.
\index{parabolic subgroup!example with no Levi}

Let $\Add$ act on $\mathbf{P}^1_S$ via the isomorphism 
$j:x \mapsto u(x) := (\begin{smallmatrix} 1 & x \\ 0 & 1
\end{smallmatrix})$ onto the strictly upper-triangular subgroup of the $S$-group ${\rm{PGL}}_2 = {\rm{Aut}}_{\mathbf{P}^1_S/S}$.
Consider the pushout $E = U \times^{\Add} \mathbf{P}^1_S$ of $U$ along the inclusion $j$ of $\Add$ into
${\rm{PGL}}_2$; by definition, this is the quotient of $U \times \mathbf{P}^1_S$ modulo the anti-diagonal $\Add$-action
$x.(y,t) = (y+x, u(-x)(t))$.  Informally, $E$ is obtained by replacing the affine line $\mathbf{P}^1_S - \{\infty\}$ with 
$U$.  The $\mathbf{P}^1$-bundle $E \rightarrow S$ is equipped
with an evident $j(\Add)$-invariant section $\sigma \in E(S)$ such that the $S$-scheme $E - \sigma(S)$ is 
the $\Add$-torsor $U$, so
$E - \sigma(S) \rightarrow S$ has no global section.
The construction of $(E, \sigma)$ as a twisted form of $(\mathbf{P}^1, \infty)$
has no effect on the relative tangent line along the section, so the line bundle ${\rm{T}}_{\sigma}(E)$ over $S$ is globally
trivial.

Let $G$ be the automorphism $S$-scheme of $E$; it is a Zariski-form of ${\rm{PGL}}_2$ since 
the $\Add$-torsor $U$ is trivial Zariski-locally on $S$.  Let $P$ be the $G$-stabilizer
of $\sigma \in E(S)$, so $P \subset G$ is clearly a Borel subgroup.  The action of $P$
on ${\rm{T}}_{\sigma}(E)$ 
defines a character $\chi:P \rightarrow \Gm$ whose kernel is seen to be $\mathscr{R}_u(P)$
by computing Zariski-locally over $S$. Thus, to show that $P$ has no Levi $S$-subgroup it is equivalent 
to show that $\chi$ has no homomorphic section over $S$.  Suppose there is such a section
$\lambda:\Gm \rightarrow P$, so the natural map $U_G(-\lambda) - \{1\} \rightarrow (G/P) - \{1\} = E - \sigma(S)$
is an isomorphism of $S$-schemes (as we see by working locally over $S$).  In particular,
every section in $U_G(-\lambda)(S)$ meets the identity section, so $U_G(-\lambda)$ cannot be
isomorphic to $\Add$ as $S$-groups. 

The action of
$\Gm$ on $\mathfrak{g} = {\rm{Lie}}(G)$ through ${\rm{Ad}}_G \circ \lambda$ has weights $\{\pm a\}$
where $a(t) = t$, and the corresponding root groups $U_{\pm a}$
are precisely $U_G(\pm \lambda)$.  The weight space
$\mathfrak{g}_{-a}$ is identified with the line bundle ${\rm{T}}_{e}(G/P) = {\rm{T}}_{\sigma}(E)$ that is globally
trivial, so the isomorphism $\exp_{-a}: \mathbf{W}(\mathfrak{g}_{-a}) \simeq U_{-a}$ (see
Theorem \ref{rootgp}) implies that $U_G(-\lambda) \simeq \Add$ as $S$-groups, contrary
to what we saw above.  Hence, there is no such $\lambda$, so $P$ has no Levi $S$-subgroup.
\end{example}

\newpage

\subsection{Exercises}\label{exer5}

\begin{exer}\label{5.3}
Let $G$ be a smooth connected affine group over a field $k$.

(i) For a maximal $k$-torus $T$ in $G$ (see Remark \ref{remtorus})
and a smooth connected $k$-subgroup $N$ in $G$
that is normalized by $T$, 
prove that $T \bigcap N$ is a maximal $k$-torus in $N$ (e.g., smooth and connected!).
Show by example that $S \bigcap N$ can be disconnected for a non-maximal $k$-torus $S$.
Hint: first analyze $Z_G(T) \bigcap N$ using $T \ltimes N$ to reduce to the case
when $T$ is central in $G$, and then pass to $G/T$. 

(ii) Let $H$ be a smooth connected normal $k$-subgroup of $G$, and
$P$ a parabolic $k$-subgroup. 
Prove $(P_{\overline{k}} \bigcap H_{\overline{k}})_{\rm{red}}^0$ is a parabolic
subgroup of $H_{\overline{k}}$, and use Theorem \ref{parconn}
(applied to $H$) to prove $P \bigcap H$ is connected (hint: work over $\overline{k}$). 

(iii) For $H$ as in (ii), by using that $Q = N_H(Q)$ scheme-theoretically for parabolic $Q$ in $H$
(Corollary \ref{parnorm}), prove $P \bigcap H$ in (ii) is smooth and therefore parabolic in $H$.
(Hint: when $k = \overline{k}$, prove $(P \bigcap H)_{\rm{red}}^0$ is normal in $P$, hence in $P \bigcap H$.) 
In particular, prove that the scheme-theoretic
intersection $B \bigcap H$ is a Borel $k$-subgroup of $H$ for all Borel $k$-subgroups $B$ of $G$.
\end{exer}

\begin{exer}\label{reddecomp}
This exercise generalizes Theorem \ref{normsubgp} to the reductive case.
Let $G$ be a connected reductive group over a field $k$, $Z$ its maximal central $k$-torus,
and $G' = \mathscr{D}(G)$ its semisimple derived group.
Let $\{G'_i\}$ be the $k$-simple factors of $G'$.  Prove that 
they are precisely the minimal nontrivial normal smooth connected
non-central $k$-subgroups of $G$, and that the multiplication homomorphism
$$Z \times \prod G'_i \rightarrow G$$
is a central isogeny.   (Keep in mind that if $k$ is finite then
$G(k)$ is not Zariski-dense in $G$, so in general an argument is needed to prove that
the $G'_i$ are normal in $G$.) Also prove that the normal connected
semisimple $k$-subgroups of $G'$ are necessarily normal in $G$
(the converse being obvious).
\end{exer}

\begin{exer} Let $R$ be Dedekind with fraction field $K$, and $G$ a connected reductive $K$-group.
A reductive $R$-group scheme is {\em quasi-split} if it has a Borel subgroup over the base
(see Definition \ref{qsdef}).

(i) Show that $G = \mathscr{G}_K$ for a reductive group scheme $\mathscr{G}$ over a dense open
$U \subset \Spec R$.

(ii) Assume $R$ is a henselian (e.g., complete)
 discrete valuation ring and that $G = \mathscr{G}_K$ for a reductive $R$-group
$\mathscr{G}$.  Using ${\rm{Bor}}_{\mathscr{G}/R}$, 
prove that if the special fiber $\mathscr{G}_0$ is split (resp.\,quasi-split)
then so is $G$ over $K$.  What if $R$ is not assumed to be henselian?

(iii) Using (i) and (ii), show that if $G$ is a connected reductive group over a global field $F$
then $G_{F_v}$ is quasi-split for all but finitely many places $v$ of $F$.  Likewise
show that any $G$-torsor over $F$ admits an $F_v$-point
for all but finitely many $v$.
See Exercise \ref{split} for analogues with the property of
being split.
\end{exer}

\begin{exer}\label{chevex} For a split adjoint semisimple group
$(G,T,M)$ over a non-empty scheme
$S$, Example \ref{chevgp} used that any base $\Delta$ for $\Phi$ is a basis for $M$.
Using Corollary \ref{computez},
explain why $\Delta$ being a basis for $M$ {\em characterizes} the adjoint
property for $G$. 
\end{exer}

\begin{exer}\label{3.2}
This exercise develops an important special case of Exercise \ref{3.1},
the group of ``norm-1 units'' in a central simple algebra.  

(i) Linear derivations of a matrix algebra over a field are  precisely the inner derivations
(i.e., $x \mapsto yx - xy$ for some $y$); see \cite[Ch.\,3, Exer.\,30]{farb} for a proof
based on a clever application of the Skolem--Noether theorem.
Combining this with length-induction on artin local rings, prove
the Skolem--Noether theorem for ${\rm{Mat}}_n(R)$ for any artin local ring $R$
(i.e., all $R$-algebra automorphisms of
${\rm{Mat}}_n(R)$ are conjugation by a unit).  
Deduce ${\rm{PGL}}_n \simeq \underline{\rm{Aut}}_{{\rm{Mat}}_n/\Z}$.

(ii) Let $A$ be a central simple algebra with dimension $n^2$ over a field $k$.
Build an affine $k$-scheme $I$ of finite type such that naturally 
in $k$-algebras $R$, 
$$I(R) = {\rm{Isom}}_{R\mbox{-}{\rm{alg}}}(A_R, {\rm{Mat}}_n(R)).$$
Note that $I(\overline{k})$ is non-empty.
Prove $I$ is smooth by checking the infinitesimal criterion for $I_{\overline{k}}$ with the help of (i).
Deduce that $A_K \simeq {\rm{Mat}}_n(K)$ for a finite {\em separable} extension $K/k$.  

(iii) By (ii), we can choose a finite Galois extension $K/k$ and a $K$-algebra isomorphism
$\theta:A_K \simeq {\rm{Mat}}_n(K)$, and by Skolem--Noether this is unique up to conjugation by a unit.
Prove that for any choice of $\theta$, the determinant map
transfers to a multiplicative map $\underline{A}_K \rightarrow \mathbf{A}^1_K$
which is independent of $\theta$.  Deduce that it is ${\rm{Gal}}(K/k)$-equivariant,
and so descends to a multiplicative map ${\rm{Nrd}}_{A/k}:\underline{A} \rightarrow \mathbf{A}^1_k$
\index{ aaanrd@${\rm{Nrd}}_{A/k}$}
\index{reduced norm}
which ``becomes'' the determinant over {\em any} extension $F/k$ for which $A_F \simeq
{\rm{Mat}}_n(F)$.  Prove that ${\rm{Nrd}}_{A/k}^n = {\rm{N}}_{A/k}$ 
(explaining the name {\em reduced norm}
for ${\rm{Nrd}}_{A/k}$), and conclude that $\underline{A}^{\times} = 
{\rm{Nrd}}_{A/k}^{-1}(\Gm)$.  

(iv)  Let ${\rm{SL}}(A) = \ker({\rm{Nrd}}_{A/k}:\underline{A}^{\times} \rightarrow \Gm)$
(denoted ${\rm{SL}}_{m,D}$ if $A = {\rm{Mat}}_m(D)$ for a central
\index{ aaasla@${\rm{SL}}(A)$}
division algebra $D$ over $k$).  Prove that its formation
commutes with any extension of the ground field, and that it 
becomes isomorphic to ${\rm{SL}}_n$ over $\overline{k}$.  In particular,
${\rm{SL}}(A)$ is a connected semisimple $k$-group that is the derived
group of the connected reductive $\underline{A}^{\times}$.
(In contrast, $\ker {\rm{N}}_{A/k}$ is non-smooth 
whenever ${\rm{char}}(k)|n$ and is usually disconnected.)  

(v) Using the preceding constructions and Galois descent, generalize the bijective correspondence in 
Exercise \ref{3.3}(ii) to central simple algebras over any field (possibly finite).
\end{exer}

\begin{exer}\label{5.1}
This exercise builds on Exercise \ref{3.2} to prove a special case of a 
conjugacy result of Borel and Tits for maximal split tori in connected reductive groups over fields.
Let $A$ be a central simple algebra over a field $k$, $T$ a $k$-torus in 
$\underline{A}^{\times}$ containing $Z_{\underline{A}^{\times}} = 
\Gm$, and $A_T$ the corresponding \'etale commutative
$k$-subalgebra of $A$ (with $\dim_k A_T = \dim T$) as in Exercise \ref{3.3}. 

(i) Prove that ${\rm{SL}}(A)$ is $k$-anisotropic if and only if $A$ is a division algebra.

(ii) Prove that the centralizer $B_T = Z_A(A_T)$ is a semisimple $k$-algebra
with center $A_T$. 

(iii) If $T$ is $k$-split, prove $A_T \simeq k^r$ and  that the simple factors $B_i$ of $B_T$
are {\em central} simple $k$-algebras.

(iv) Assume $T$ is $k$-split.  Using (iii),  prove $T$ is maximal as a $k$-split torus in 
$\underline{A}^{\times}$ if and only if the (central!) simple factors $B_i$ of $B_T$ are {\em division 
algebras}.  

(v) Fix an isomorphism $A \simeq {\rm{End}}_D(V)$ for a right module $V$ over a central division algebra $D$, 
and consider $(T, \{B_i\})$ as in (iv), so $V$ is
a left $A$-module and $V = \prod V_i$ with {\em nonzero} left $B_i$-modules $V_i$.
If $T$ is maximal as a $k$-split torus in $\underline{A}^{\times}$, 
prove $V_i$ has rank 1 over $B_i$ and $D$, so $B_i \simeq D$.
Using $D$-bases, deduce that 
{\em all maximal $k$-split tori in $\underline{A}^{\times}$ are $\underline{A}^{\times}(k)$-conjugate}. 
\end{exer}

\begin{exer}\label{glnmap} In the proof of Theorem \ref{adimm}, we used that any flat affine
group scheme $G$ of finite type 
over a Dedekind domain $R$ occurs as a closed subgroup of some ${\rm{GL}}_n$ over $R$.

(i) Prove the analogous result over fields by adapting whatever is your favorite proof
for {\em smooth} affine groups over fields.

(ii) Make your argument in (i) work over $R$ (for flat affine groups of finite type)
by using that any finitely generated torsion-free $R$-module is projective (and hence a direct
summand of a finite free $R$-module).  
\end{exer}

\begin{exer}\label{borbij}
Let $G$ be a reductive group over a scheme $S$.
Show that if $P$ is a parabolic subgroup of $G$
then $Z_G \subset P$ and that $P \mapsto P/Z_G$ is
a bijective correspondence
between the sets of parabolic subgroups of $G$ and
of $G^{\rm{ad}} = G/Z_G$, 
\index{ aaagad@$G^{\rm{ad}}$}
\index{reductive group!scheme of parabolic subgroups} 
\index{Borel subgroup!scheme of}
with inverse given by the formation
of inverse images under the quotient map $G \rightarrow G^{\rm{ad}}$. 
Construct natural isomorphisms ${\rm{Par}}_{G/S} \simeq {\rm{Par}}_{G^{\rm{ad}}/S}$
and ${\rm{Bor}}_{G/S} \simeq {\rm{Bor}}_{G^{\rm{ad}}/S}$.
\end{exer}

\begin{exer}\label{chern}
Let $G$ be a reductive group over a non-empty scheme $S$, and 
$Z \subset Z_G$ a flat central closed subgroup scheme (so $Z$ is of multiplicative type).  
This exercise addresses splitting properties for $G/Z$ given splitting hypotheses
on $G$ and $Z$. 

(i) Prove that the smooth $S$-affine quotient $G' = G/Z$ is a reductive $S$-group,
and that if $T \subset G$ is a maximal torus (so $Z \subset Z_G \subset Z_G(T) = T$) then so is
$T' := T/Z \subset G'$.  
Give an example over a field in which $T'$ is split, $T$ is non-split, 
and $Z_G$ is not a direct factor of $T$.

(ii) Consider a split triple $(G,T,M)$, and assume
$Z$ is split, so ${\rm{X}}(Z) = \overline{M}_S$
for a quotient $\overline{M} = M/M'$ of $M$.  For each $a \in \Phi \subset M$ 
(so $a|_{Z_G} = 1$), let $a'$ denote the induced character of $T' = T/Z$. 
Prove that 
${\rm{X}}(T') = M'_S$ inside ${\rm{X}}(T) = M_S$,
and that $a' \in M'$ for all $a$.

(iii) For $(G, T, M)$ as in (ii), prove that the natural map $U_a \rightarrow U'_{a'}$ between root 
groups is an isomorphism (hint: fibral isomorphism criterion).  Deduce
that the line bundle $\mathfrak{g}'_{a'}$ on $S$ is {\em globally trivial},
and that $(G',T',M')$ is split as in Definition \ref{splitdef}
(note Example \ref{failex}). 
%In particular, a {\em central} quotient of a split reductive group modulo
%a split flat closed (central) subgroup is again split. 
%(See Exercise \ref{chern2} for the removal of the centrality hypothesis.)
\end{exer}


\begin{exer}\label{unip0}
Let $U$ be a smooth connected unipotent group over a field $k$.
If $k$ is perfect then $U$ is split (i.e., admits a composition series with successive
quotients $k$-isomorphic to $\Add$), by \cite[15.5(ii)]{borelag} or \cite[XVII, 4.1.3]{sga3}.
Now assume ${\rm{char}}(k) = 0$. 

(i) Let $U_n \subset {\rm{GL}}_n$ be the smooth connected unipotent $k$-subgroup of strictly upper-triangular matrices,
so the Lie subalgebra ${\rm{Lie}}(U_n) \subset \mathfrak{gl}_n = {\rm{Mat}}_n(k)$ consists of nilpotent matrices.
Equip ${\rm{Lie}}(U_n)$ with the ``Baker--Campbell--Hausdorff'' (BCH) group law; this law is algebraic rather than formal, 
due to uniform control on the
nilpotence.  Prove the $k$-scheme map
$\exp:{\rm{Lie}}(U_n) \rightarrow U_n$ is a
$k$-group isomorphism and that if $U \subset U_n$ is a {\em commutative} closed $k$-subgroup
then $\exp({\rm{Lie}}(U)) \subset U$ and $\exp:{\rm{Lie}}(U) \rightarrow U$ is a $k$-group isomorphism.

(ii) Let $U$ be a {\em split} unipotent $k$-group, so $U$ arises as in (i) (by 
\cite[15.4(i)]{borelag}). Equip ${\rm{Lie}}(U)$ with the BCH group law.
Prove there is a unique $k$-group isomorphism
$U \simeq {\rm{Lie}}(U)$ lifting the identity on Lie algebras. 
In particular, $U \simeq \Add^r$ when $U$ is commutative.
(This conclusion fails if ${\rm{char}}(k) > 0$ due to 
$k$-groups of truncated Witt vectors $W_r$ for $r \ge 2$, so the existence of 
linear structures in Theorem \ref{dynlevi}(iii) is remarkable in positive characteristic.)

\end{exer}

\newpage

\section{Existence, Isomorphism, and Isogeny Theorems}\label{existsec}

\subsection{Pinnings and main results} 

In \S\ref{repsec} we introduced the notion of a {\em pinning} on a triple $(G,T,B)$ over
an algebraically closed field $k$.  The purpose of that concept was to ``rigidify'' the triple
(eliminating the action of the adjoint
torus) so that passage to the root datum loses no information concerning isomorphisms.  
The 
Isomorphism Theorem for split reductive group schemes over a non-empty
scheme $S$ requires a relative version of pinnings, 
and there is a generalization (the Isogeny Theorem) that incorporates isogenies.  The purpose
of this preliminary section is to develop several concepts related to pinnings
and morphisms of root data.   At the end of this section we state
the Existence, Isomorphism, and Isogeny Theorems over any scheme $S \ne \emptyset$. 

\begin{definition} Let $(G,T,M)$ be a split reductive group over
a non-empty scheme $S$, and let $R(G,T,M) = (M,\Phi,
 M^{\vee}, \Phi^{\vee})$ be its associated root datum.
 A {\em pinning} on $(G,T,M)$ is a pair $(\Phi^+, \{X_a\}_{a \in \Delta})$
consisting of a positive system of roots $\Phi^+ \subset \Phi$ (or equivalently, a 
 base $\Delta$ of $\Phi$) and trivializing sections $X_a \in
 \mathfrak{g}_a(S)$ for each simple positive root $a \in \Delta$. 

The 5-tuple $(G,T,M,\Phi^+,\{X_a\}_{a \in \Delta})$ is a {\em pinned} split reductive $S$-group.
 \end{definition}
 \index{reductive group!pinned}
 \index{pinning}
 \index{ aaargtm@$R(G,T,M)$}
  
 Since $\Delta$ determines $\Phi^+$, we will usually write $(G,T,M,\{X_a\}_{a \in \Delta})$
and suppress the explicit  mention of $\Phi^+$.
In Exercise \ref{star} a more ``group-theoretic'' definition of pinnings 
over $S$ is given, replacing the trivializations of simple positive root spaces $\mathfrak{g}_a$
with suitable homomorphisms from ${\rm{SL}}_2$ into $\mathscr{D}(Z_G(T_a))$ for 
each $a \in \Delta$. Keep in mind that
the definition of the ``split'' property for $(G,T,M)$ in Definition \ref{splitdef} includes
the condition that the line bundles $\mathfrak{g}_a$ are free of rank 1,
so a pinning $(\Phi^+, \{X_a\}_{a \in \Delta})$ can be chosen
for any $\Phi^+ \subset \Phi$.

 
 \begin{remark}\label{borelwarn}
 It may be tempting to expect that the choice of $\Phi^+$ is ``the same''
 as a choice of Borel subgroup of $G$ containing $T$ as in the classical case,
  but that it not true when $S$ is disconnected (and we must allow
  the base scheme to be disconnected for descent theory arguments).  More precisely, 
by Proposition \ref{parpar} and Corollary \ref{parfiber}(2), 
$\Phi^+ = \Phi(B,T)$ for a unique Borel subgroup $B \subset G$ containing
 $T$, and the Borel subgroups of $G$ containing $T$ that we obtain
 by varying $\Phi^+$ are precisely those $B$ for which the subgroup
$$\Phi(B,T) \subset \Hom_{S\mbox{-}{\rm{gp}}}(T,\Gm) = \Gamma(S, {\rm{X}}(T)) = 
\Gamma(S, M_S)$$
lies inside the subgroup $M$ of ``constant sections''. 
 In particular, when $S$ is disconnected there are Borel subgroups $B'$ of $G$ containing
 $T$ that do not arise from any choice of $\Phi^+ \subset \Phi$.
  For this reason, in the relative theory we work throughout with a choice of $\Phi^+$
 rather than with a choice of $B$ (although the two viewpoints coincide when $S$ is connected,
 such as in the theory over a field, domain, or local ring). 
 \end{remark}

There is an evident notion of {\em isomorphism}
between pinned split reductive $S$-groups.  The definition of {\em isogeny} 
incorporating pinnings 
(refining the notion for smooth $S$-affine $S$-groups
as in Definition \ref{ceniso}) requires care to account for Frobenius isogenies
between root groups on geometric fibers in positive characteristic.
As motivation, consider pinned split reductive $S$-groups 
$$(G',T',M', \{X'_{a'}\}_{a' \in \Delta'}), \,\,\,\, (G,T,M,\{X_a\}_{a \in \Delta})$$
over $S$ and a quasi-finite surjective $S$-homomorphism 
$f:G' \rightarrow G$ such that $f(T') \subset T$. 
(In Proposition \ref{isogfiber} we will show that 
$f$ is necessarily finite and flat, hence an isogeny.) Note that 
$f:T' \rightarrow T$ is an isogeny, since the map 
$${\rm{X}}(f):M_S = {\rm{X}}(T) \rightarrow {\rm{X}}(T') =  M'_S$$
 induces a finite-index inclusion of lattices on geometric fibers over $S$. 

There is an open cover
$\{U_i\}$ of $S$ such that the map induced by ${\rm{X}}(f)$ on $U_i$-sections
carries $M$ into $M'$.
We now suppose (as may be achieved by working Zariski-locally on $S$)
that the map induced by ${\rm{X}}(f)$ on global sections over $S$
carries $M$ into $M'$.  In particular, 
we get an isomorphism $M_{\Q} \simeq M'_{\Q}$ between $\Q$-vector spaces.

By the classical theory on geometric fibers,
for each $a' \in \Phi'$ the root group $(U_{a'})_{\overline{s}}$ for $(G'_{\overline{s}},T'_{\overline{s}})$
is carried isogenously  by $f_{\overline{s}}$
onto the root group of $(G_{\overline{s}},T_{\overline{s}})$ for
a unique $a(s) \in \Phi$.  Since $f_{\overline{s}}$ is a (possibly non-central) isogeny, 
every root in $\Phi(G_{\overline{s}},T_{\overline{s}})$
arises in this way from a unique $a' \in \Phi'$.
Each resulting map between root groups of the $\overline{s}$-fibers
is identified with an endomorphism of $\Add$ having the form $x \mapsto c x^{q(s)}$ 
for some $c \in k(\overline{s})^{\times}$
and some integral power $q(s) \ge 1$ of the characteristic exponent of $k(s)$, 
due to the equivariance of 
$f_{\overline{s}}: (U_{a'})_{\overline{s}} \rightarrow U_{a(s)}$ with respect to 
the isogeny $T'_{\overline{s}} \rightarrow T_{\overline{s}}$.  It 
follows that ${\rm{X}}(f_{\overline{s}})(a(s)) = q(s)a'$.
Likewise, ${\rm{X}}_{\ast}(f_{\overline{s}})({a'}^{\vee}) = q(s) a(s)^{\vee}$
by the construction of coroots in the classical theory. 
The map $\Phi' \rightarrow \Phi$ defined by
$a' \mapsto a(s)$ has very weak dependence on $s$:

\begin{lemma}\label{zarroot}
For $a' \in \Phi'$, the associated function $s \mapsto a(s) \in \Phi(G_{\overline{s}},T_{\overline{s}}) = \Phi$
is Zariski-locally constant in $s$. 
\end{lemma}

\begin{proof}
We may assume that $S$ is noetherian, so every pair of distinct points
$\{s, \eta\}$ in $S$ with $s$ in the closure of $\eta$ can be dominated
by the spectrum of a discrete valuation ring.  Since our problem is
to prove a constancy result on the connected components of $S$,
and every point $s$ of $S$ is in the closure of the generic point of
each irreducible component of $S$ through $s$, by pullback
to the spectra of discrete valuation rings we may and do
assume $S = \Spec R$ for a discrete valuation ring $R$.
In this case we have to prove that $a(s) = a(\eta)$ in $\Phi$, where
$s$ is the closed point of $S$ and $\eta$ is the generic point of $S$.

Viewing $a(s)$ as an $S$-homomorphism
$T \rightarrow \mathbf{G}_m$  (i.e., a global section of
${\rm{X}}(T)$) via the inclusion $\Phi \subset M$, the saturation
of $\Z a(s)$ in $M$ defines a split
$S$-subtorus $T_{a(s)}$ of relative codimension 1 in $T$.  The 
isogeny $T' \rightarrow T$ between split $S$-tori must carry
$T'_{a'}$ into $T_{a(s)}$ since we can check this on the fibers
over a single geometric closed point of $S$ (such as $s$).  Likewise using
a geometric generic point of $S$ gives that $T'_{a'}$ is carried
into $T_{a(\eta)}$.  Hence, $a(\eta) = \pm a(s)$
(since $a(s)$ and $a(\eta)$ lie in the reduced root system $\Phi$).
%We can therefore pass to the induced map 
%$\mathscr{D}(Z_{G'}(T'_{a'})) \rightarrow \mathscr{D}(Z_G(T_{a(s)}))$
%to reduce to the case when the fibers are semisimple of rank 1. 

Consider the cocharacters $\lambda' = {a'}^{\vee} \in {M'}^{\vee}$
and $\lambda = f \circ \lambda' = q(s) a(s)^{\vee} \in M^{\vee}$.
The map $f$ carries $U_{a'} := U_{G'}({a'}^{\vee}) = U_{G'}(\lambda')$
into $U_G(\lambda) := U_G(q(s)a(s)^{\vee}) = U_G(a(s)^{\vee}) = 
U_{a(s)}$ (see Theorem \ref{215}(1) and Proposition \ref{functor}(2)), so passing
to $\overline{\eta}$-fibers gives that the elements $a(\eta)$ and $a(s)$ in $\Phi$
have the same root groups for $(G_{\overline{\eta}}, T_{\overline{\eta}})$.
Hence, $a(\eta) = a(s)$.
\end{proof}


By Lemma \ref{zarroot}, it is reasonable to impose
the additional requirement on $f$ that there exists a (necessarily unique) bijection
$d:\Phi' \rightarrow \Phi$ and a prime power $q_{a'} \ge 1$
for each $a' \in \Phi'$ so that the map $M \rightarrow M'$ induced
by ${\rm{X}}(f)$ carries $d(a')$ to $q_{a'} a'$
and the map induced by its dual ${\rm{X}}_{\ast}(f)$ carries ${a'}^{\vee}$ to $q_{a'}d(a')^{\vee}$,
where the integer $q_{a'}$ is an integral
power of the characteristic exponent of $k(s)$ for each $s \in S$. 
Indeed, the preceding discussion shows that
Zariski-locally on $S$, every quasi-finite surjection $(G',T') \rightarrow (G,T)$
satisfies these conditions.  

Since ${\rm{X}}(f)_{\Q}$ is an isomorphism 
and $d$ is injective, the conditions $d(-a') \mapsto q_{-a'}(-a') = -q_{-a'} a'$
and $d(a') \mapsto q_{a'} a'$ force $d(-a')$ and $d(a')$ to
be distinct linearly dependent elements of 
the reduced root system $\Phi$, so $d(-a') = -d(a')$.  Likewise,
$q_{-a'} = q_{a'}$.  

\begin{lemma}\label{rootisom} The quasi-finite $f$ carries the $S$-subgroup scheme
$U'_{a'} \subset G'$ into the $S$-subgroup
$U_{d(a')} \subset G$ via a homomorphism
that is $\Gm$-equivariant for respective conjugation against ${a'}^{\vee} = q_{a'} d(a')^{\vee}$
and $d(a')^{\vee}$.  Moreover, if 
$q_{a'} = p^n$ with a prime $p$ and $n > 0$ then 
$p = 0$ in $\calO_S$ $($so $(-1)^{q_{a'}} = -1$ in $\calO_S$$)$ and for all $a' \in \Phi'$ 
there exists a unique isomorphism of line bundles 
$$f_{a'}:(\mathfrak{g}'_{a'})^{\otimes q(a')} \simeq \mathfrak{g}_{d(a')}$$
such that $U'_{a'} \rightarrow U_{d(a')}$
is given by $\exp_{a'}(X') \mapsto \exp_{d(a')}(f_{a'}({X'}^{\otimes q(a')}))$
for all $X' \in \mathfrak{g}'_{a'}$.
\end{lemma}

\begin{proof}
Since $d(a') \circ f = q_{a'} a'$, $f$ carries $T'_{a'}$ into (hence onto) $T_{d(a')}$.
Thus, $f$ carries $Z_{G'}(T'_{a'})$ into $Z_G(T_{d(a')})$ via a quasi-finite surjection. 
But $f \circ {a'}^{\vee} = q_{a'}  d(a')^{\vee}$, so we can pass
to the induced map between semisimple derived groups
by working with the rank-1 split tori
${a'}^{\vee}(\Gm)$ and $d(a')^{\vee}(\Gm)$.
This brings us to the case that $G'$ and $G$ are semisimple with fibers
of rank 1.  For $\lambda' = {a'}^{\vee}$
and $\lambda = f \circ \lambda' = q_{a'}  d(a')^{\vee}$ we have
$U'_{a'} = U_{G'}({a'}^{\vee}) = U_{G'}(\lambda')$ and 
$U_{d(a')} = U_G(d(a')^{\vee}) = U_G(\lambda)$ (see Theorem \ref{215}(1)),
so $f$ carries $U'_{a'}$ into $U_{d(a')}$ by Proposition \ref{functor}(2).

By the definition of a split reductive $S$-group, 
the root spaces admit global trivializations
(as line bundles on $S$).  Choose such trivializations
for $(G,T)$ and $(G',T')$, so we get $S$-group 
isomorphisms $U'_{a'} \simeq \Add$ and $U_{d(a')} \simeq \Add$
that are respectively $T'$-equivariant and $T$-equivariant
via the respective scaling actions on $\Add$ by $a'$ and $d(a')$.
In other words, the map induced by $f$ between
the root groups becomes a surjective endomorphism $f:\Add \rightarrow \Add$ that satisfies
$$f(a'(t')x) = d(a')(f(t'))f(x) = a'(t')^{q_{a'}} f(x)$$
for points $t'$ of $T'$ and $x$ of $\Add$ over $S$.  Since $a':T' \rightarrow \Gm$
is an fppf covering, it follows that $f(ux) = u^{q_{a'}}f(x)$ for all points
$u$ of $\Gm$ and $x$ of $\Add$.  By the relative schematic density of $\Gm$ in
$\Add$ over $S$, the same identity holds with $u$ permitted to be any point of $\Add$.

Letting $q = q_{a'}$, we claim that $f(x) = c x^q$ for a unique unit $c$ on the base
(so the existence and uniqueness of $f_{a'}$ will follow).  The uniqueness
is clear, so we may work Zariski-locally on $S$ for existence.
Hence, we can assume that $f$ is given by a polynomial map
$x \mapsto c_0 + c_1 x + \dots + c_n x^n$ for some integer $n \ge 0$
and some global functions $c_0, \dots, c_n$ on the base.  
Since $f(0) = 0$ we have $c_0 = 0$, and the identity
$f(ux) = u^q f(x)$ implies that $c_j u^j = u^q c_j$ for all $j$.
If $j \ne q$ then fppf-locally on $S$ there exists a unit $u$ such that
$u^{q-j} - 1$ is a unit, so $c_j = 0$ on $S$.  Hence,
$f(x) = c x^{q}$ for some $c$ on $S$.  The maps induced by $f$ 
between root groups on geometric fibers over $S$ are isogenies, so
$c$ is nowhere-vanishing, which is to say that $c$ is a unit on $S$.

Finally, we have to show that $p = 0$ in $\calO_S$ if $q = p^n$
for a prime $p$ and $n > 0$.  The homomorphism
property for $f$ and unit property for $c$ imply
that $x \mapsto x^{q}$ is additive in $\calO_S$ (so $(-1)^q = -1$ in $\calO_S$).  
Assume $q = p^n$ for a prime $p$ and $n > 0$, so $(x + y)^{q} - x^{q} - y^{q}$
involves the monomial $x^{p^{n-1}} y^{q-p^{n-1}}$ 
with a binomial coefficient whose $p$-adic ordinal is 1
and involves the monomial $x y^{q-1}$ with a coefficient of $q$, so
the greatest common divisor of all monomial coefficients (in $\Z$) is $p$.
Hence, $p = 0$ in $\calO_S$ in such cases. 
\end{proof}

\begin{proposition}\label{weylf}
In the setting of Lemma $\ref{rootisom}$, the $\mathscr{O}_S$-linear isomorphisms
$f_{a'}$ and $f_{-a'}$ are dual relative to the canonical dualities for the pair
$\mathfrak{g}'_{a'}, \mathfrak{g}'_{-a'}$ and the pair
$\mathfrak{g}_{d(a')}, \mathfrak{g}_{-d(a')} = \mathfrak{g}_{d(-a')}$.
Moreover, if $X'$ is a  trivializing section of $\mathfrak{g}_{a'}$
and $X := f_{a'}({X'}^{\otimes q_{a'}})$
is the associated trivialization of $\mathfrak{g}_{d(a')}$ then 
$f(w_{a'}(X')) = w_a(X)$ with $a := d(a')$.
\end{proposition}

See Corollary \ref{wsec} for the definition of $w_a(X)$ for any $a \in \Phi$
and any trivializing section $X$ of $\mathfrak{g}_a$.

\begin{proof} By passing to derived groups of torus centralizers,
we can reduce to the case of groups that are fiberwise semisimple of rank 1. 
Now consider such groups, so 
we can let $q = q_{a'} = q_{-a'}$ and $a = d(a')$ (so $-a = d(-a')$).
The maps $f_{\pm a'}$ are $\mathscr{O}_S$-linear, so by using the induced map
between open cells and the unique characterization of the coroots (and the duality pairing
between root spaces in (\ref{cellid})) a straightforward calculation gives 
that if local sections $X'$ of $\mathfrak{g}'_{a'}$ and $Y'$ of $\mathfrak{g}'_{-a'}$
satisfy $1 + X'Y' \in \Gm$ then 
$$1 + f_{a'}({X'}^{\otimes q})f_{-a'}({Y'}^{\otimes q}) = (1 + X'Y')^q = 1 + (X'Y')^q.$$
By taking $Y' := u{X'}^{-1}$ for a unit $u$ such that
$1 + u$ is a unit (as may be done fppf locally on $S$), the respective trivializing sections $f_{a'}({X'}^{\otimes q})$
and 
$$f_{-a'}({Y'}^{\otimes q}) = f_{-a'}(u^q ({X'}^{-1})^{\otimes q}) = u^q f_{-a'}(({X'}^{-1})^{\otimes q})$$ 
of $\mathfrak{g}_a$ and $\mathfrak{g}_{-a}$
have pairing equal to $u^q$.  Thus, the asserted duality compatibility between $f_{a'}$ and $f_{-a'}$ holds.

By definition, $w_{a'}(X') = \exp_{a'}(X')\exp_{-a'}(-Y')\exp_{a'}(X')$
where $Y'$ is the trivialization of $\mathfrak{g}'_{-a'}$ linked to $X'$. 
The preceding discussion shows that $Y := f_{-a'}({Y'}^{\otimes q})$
\index{linked trivialization}
is the trivialization of $\mathfrak{g}_{-d(a')}$ linked to 
$X$, so (using that $(-1)^{q_{a'}} = -1$ in $\calO_S$) we have 
$$f(w_{a'}(X')) = \exp_{d(a')}(X) \exp_{d(-a')}(-Y) \exp_{d(a')}(X) = w_{d(a')}(X)$$
since $d(-a') = -d(a')$.
\end{proof}

\begin{definition}\label{defcom}
Let $(G,T,M)$ and $(G',T',M')$ 
be split reductive groups over a scheme $S \ne \emptyset$.
A quasi-finite surjection $f:(G',T') \rightarrow (G,T)$ is {\em compatible with the splittings}
if there is a homomorphism $h:M \rightarrow M'$,
bijection $d:\Phi' \rightarrow \Phi$, and function
$q:\Phi' \rightarrow \Z_{\ge 1}$ valued in prime powers such that: 
\begin{enumerate}
\item the induced map 
${\rm{X}}(f): M_S = {\rm{X}}(T) \rightarrow {\rm{X}}(T') = M'_S$
arises from $h$,
\item for all $a' \in \Phi'$, $h(d(a')) = q(a')  a'$ and $h^{\vee}({a'}^{\vee}) = q(a')  d(a')^{\vee}$,
\item if $q(a') > 1$ is a power of a  prime $p(a')$ then $S$ is a $\Z/p(a')\Z$-scheme. 
\end{enumerate}
\end{definition}

In the setting of the preceding definition, 
 $h_{\Q}$ is an isomorphism (since each $f_s$ is an isogeny) and $f$ uniquely determines
$h$, $d$, and $q$.  We will sometimes
write $f:(G',T',M') \rightarrow (G,T,M)$ to denote
that $f$ is compatible with the splittings.
In \cite[XXII, 4.2.1]{sga3} it is only required that $f$ is quasi-finite between 
the derived subgroups (or rather, this property between derived groups is a consequence of other conditions
imposed there), in which case $h_{\Q}$ may be neither injective
nor surjective when there are nontrivial central tori; we only consider quasi-finite
surjective $f$. Our interest in $f$ that 
are compatible with splittings is due to the following immediate consequence of the
preceding considerations.

\begin{proposition}\label{isogpin}
For split reductive $(G,T,M)$ and $(G',T',M')$
over a scheme $S \ne \emptyset$, any 
quasi-finite surjection $f:(G',T') \rightarrow (G,T)$ is compatible with the splittings
Zariski-locally on $S$.  If $S$ is connected
then $f$ is compatible with the splittings over $S$.
\end{proposition}

We are now led to:

\begin{definition} Let $R' = (X',\Phi',{X'}^{\vee},{\Phi'}^{\vee})$ and
$R = (X,\Phi,X^{\vee},\Phi^{\vee})$ be reduced root data,
and $p$ a prime or 1.  A {\em $p$-morphism}
$R' \rightarrow R$ is a triple $(h,d,q)$ 
\index{pmorphism@$p$-morphism, $p(S)$-morphism}
\index{ aaahdq@$(h,d,q)$}
consisting of a homomorphism $h:X \rightarrow X'$,
a bijection $d:\Phi' \rightarrow \Phi$, and a function
$q:\Phi' \rightarrow \{p^n\}_{n \ge 0}$ such that 
\begin{enumerate}
\item the induced map $h_{\Q}$ is an isomorphism
(i.e., $h$ is a finite-index injection),
\item for all $a' \in \Phi'$, $h(d(a')) = q(a')a'$ and $h^{\vee}({a'}^{\vee}) = q(a') d(a')^{\vee}$,
\end{enumerate}
For $p$-morphisms $(h,d,q):R' \rightarrow R$ and $(h',d',q'):R'' \rightarrow R'$,
the {\em composition} $R'' \rightarrow R$ is 
$(h' \circ h, d \circ d', (q \circ d') \cdot q')$.
If $S$ is a non-empty scheme, a $p$-morphism $(h,d,q)$ is called a {\em $p(S)$-morphism}
if $S$ is a $\Z/p(a')\Z$-scheme whenever $q(a') > 1$ is a power of a prime $p(a')$. 
\end{definition}

Since the root data are reduced and $q$ takes
values in $\Z_{\ge 1}$, the condition $h(d(a')) = q(a')a'$ implies that $h$ determines $d$ and $q$.
The notion of {\em isomorphism} between root data is the evident one,
and clearly a $p$-morphism that is an isomorphism must be a $1$-morphism.
Note also that for a $p$-morphism $R' \rightarrow R$, the map $h:X \rightarrow X'$
goes in the ``other'' direction, whereas the map 
$d:\Phi' \rightarrow \Phi$ goes in the ``same'' direction. 
This is motivated by the examples arising from split reductive group schemes:

\begin{example}\label{rootisog} For a non-empty scheme
$S$ and quasi-finite surjection $f:(G',T',M') \rightarrow (G,T,M)$ compatible
with the splittings, the associated triple $R(f) := (h,d,q)$ is 
a $p(S)$-morphism between the root data where either $p = 1$
or ${\rm{char}}(k(s)) = p > 1$ for all $s \in S$, and this is compatible
with composition and base change.   

By Proposition \ref{central}, a quasi-finite surjection 
$f:(G',T') \rightarrow (G,T)$ over $S$ 
is a central isogeny (in the sense of Definition \ref{ceniso})
 if and only if $\ker f_{\overline{s}} \subset T'_{\overline{s}}$ for all $s \in S$
(since $T'$ contains $Z_{G'}$ and any normal subgroup scheme
of multiplicative type in a connected group scheme over a field
is necessarily central).  We claim that $f$ is a central isogeny if and only if the 
$p(S)$-morphism $R(f)$ of root data satisfies $q(a') = 1$ for all $a' \in \Phi'$.

Assume
$q(a') > 1$ for some $a'$, so $U_{a'} \rightarrow U_{d(a')}$ has kernel $\alpha_{q(a')} \ne 1$ (cf.\:proof of
Lemma \ref{rootisom}). This cannot be contained in $T' = Z_{G'}(T')$, so
$\ker f$ is non-central in such cases.  Conversely, if $q(a') = 1$ for all $a'$ then $f$ restricts
to an isomorphism between corresponding root groups by Lemma \ref{rootisom}.  To prove that $f$ is 
a central isogeny in such cases it suffices to check on geometric fibers (Proposition \ref{central}), so we may
assume $S = \Spec k$ for an algebraically closed field $k$.  By looking at $f$ between compatible open
cells, the isomorphism condition between root groups forces $(\ker f)^0 \subset T'$, so $(\ker f)^0$ is of multiplicative type.
Thus, the normality of $(\ker f)^0$ in the smooth $k$-group $G'$ forces centrality since
$G'$ is connected and any group of multiplicative type has an \'etale automorphism scheme. By the same reasoning, it suffices
to show that $\ker f \subset T'$, so we can replace $G'$ with $G'/(\ker f)^0$ to reduce to the case that $\ker f$ is \'etale.  The normality
then again implies centrality, so we are done.
\end{example}

In view of the characterization of central isogenies between split reductive $S$-group
schemes in Example \ref{rootisog}, a $1$-morphism
$(h,d,q):R' \rightarrow R$ between reduced root data is also called a {\em central isogeny}.
\index{central isogeny!between root data}
\index{root datum!central isogeny}
The condition that $q(a') = 1$ for all $a' \in \Phi'$ says precisely that 
$h(d(a')) = a'$ and $h^{\vee}({a'}^{\vee}) = d(a')^{\vee}$ for all $a' \in \Phi'$,
or equivalently $h$ induces a bijection between the sets of roots (with inverse $d$)
and $h^{\vee}$ induces a bijection between the sets of coroots (with inverse equal to the ``dual'' of $d$).
In non-central cases, $h$ does not carry $\Phi$ into $\Phi'$. 

\begin{proposition}\label{isogfiber}
Let $f:G' \rightarrow G$ be a homomorphism between reductive group schemes.
If $f_s$ is an isogeny for all $s \in S$ then $f$ is an isogeny.
\end{proposition}

\begin{proof}
As we saw in the discussion preceding Proposition \ref{central},
$f$ is necessarily a quasi-finite flat surjection and it suffices 
to show that $\ker f$ is $S$-finite.
By limit arguments, we may assume that $S$ is noetherian.
In view of the quasi-finiteness of $f$, finiteness
is equivalent to properness.  Thus, by the valuative criterion
we can assume that $S = \Spec R$ for a  discrete
valuation ring $R$, with fraction field $K$ and residue field $k$.
We may and do assume that $R$ is strictly henselian
(so $G'$ is $S$-split). 

Since $R$ is henselian, we can apply the structure theorem for quasi-finite 
morphisms \cite[IV$_4$, 18.5.11(a),(c)]{ega}:
for any quasi-finite separated $S$-scheme $X$, 
there exists a unique open and closed subscheme $X' \subset X$
that is {\em $S$-finite} and satisfies $X'_k = X_k$.
In particular, the formation of $X'$ is functorial in $X$
and compatible with products over $S$, so if
$X$ is an $S$-group then $X'$ is an $S$-subgroup.
Consider the unique open and closed finite $S$-subgroup
$H' \subset \ker f$ with special fiber $\ker f_k$ (so $H'$ is also flat).
Clearly $\ker f$ is finite if and only if $H' = \ker f$.

We claim that $H'$ is normal in $G'$; i.e., the closed
subgroup $N_{G'}(H') \subset G'$ from Proposition \ref{xi2.4bis} is equal to $G'$. Since  $G'$ is $S$-split,
by consideration of an open cell we see that $G'(S)$ is
fiberwise dense in $G'$ (because for any field $F$, infinite subset $\Sigma \subset F$, and dense open
$\Omega \subset \mathbf{A}_F^n$, $\Omega \cap \Sigma^n$ is
Zariski-dense in $\Omega$).  Thus, 
by \cite[IV$_3$, 11.10.9]{ega}, the set of sections $G'(S)$ is relatively schematically dense in $G'$ over $S$
in the sense of \cite[IV$_3$, 11.10.8, 11.10.2, 11.10.1(d)]{ega}.  
Hence, to prove $N_{G'}(H') = G'$ it suffices to check equality on $R$-points,
which is to say that $G'(S)$ normalizes $H'$.  By the uniqueness  of
$H'$, such normalizing follows from the normality of $\ker f$ in $G'$.

A robust theory of quotients of finitely presented $S$-affine schemes modulo the free action of a 
finite locally free $S$-group scheme is developed in 
\cite[V]{sga3}; see especially \cite[V,  \S2(a), Thm.\:4.1(iv)]{sga3}.
In particular, if $\mathscr{G}$ is a finitely presented relatively affine group and $N$ is a normal closed
subgroup that is finite locally free over the base then 
the fppf quotient group sheaf $\mathscr{G}/N$ is represented by a finitely presented relatively affine group 
and $\mathscr{G} \rightarrow \mathscr{G}/N$ is fppf with kernel $N$, so it is an $N$-torsor for the fppf topology.
As a special case, $G'/H'$ exists as a reductive group scheme and $G' \rightarrow G'/H'$
is finite fppf with kernel $H'$.  We may replace $G'$ with $G'/H'$, so now $f_k$ is an isomorphism.  

Our problem is to show that $f$ is an isomorphism, and it is equivalent
to check this on the generic fiber. 
Let $T'$ be a maximal torus in $G'$, so it is split (as $R$ is strictly henselian);
fix an isomorphism
$T' \simeq D_S(M)$ for a finite
free $\Z$-module $M$. The kernel of $f|_{T'}:T' \rightarrow G$ is a quasi-finite closed
subgroup of $T'$ with trivial special fiber, and by Exercise \ref{ix6.4} any quasi-finite closed
subgroup of the torus $T'$ is finite. Thus, the $S$-group $\ker(f|_{T'})$ is $S$-finite
with trivial special fiber, so it is trivial. It follows that $f|_{T'}$ 
is a monomorphism, so it is a closed immersion since
$T'$ is of multiplicative type (Lemma \ref{ix2.5}).  Hence, we may
and do also view $T'$ as a maximal torus of $G$
and $f$ as a map $(G',T',M) \rightarrow (G,T',M)$
between reductive $S$-groups that is compatible with the splittings (using
the identity map on $M$).  
As in Example \ref{rootisog} (and the discussion preceding it),
we get an induced $p(S)$-morphism $R(f)$ between the root data.
But this map of root data can be computed
using any fiber, so working with the special fibers
implies that $R(f)$ is an isomorphism.  In particular,
$R(f)$ is central.  Passing
to the (split) generic fibers, it follows that the isogeny $f_K$
is a central (Example \ref{rootisog}), so
the degree of $f_K$ can be read off from the restriction $T'_K \rightarrow T_K$.
But $R(f_K) = R(f)$ is an isomorphism, so $f_K$ is an isomorphism.
\end{proof}



We now deduce some elementary properties of 
$p$-morphisms $(h,d,q)$ between reduced root data that were
established immediately above Lemma \ref{rootisom}
for triples $(h,d,q)$ that arise from quasi-finite surjections 
between split reductive group schemes.
Namely, we claim that always
$d(-a') = -d(a')$ and $q(-a') = q(a')$.
To prove this, first note that by the isomorphism property for $h_{\Q}$, $d(a')$ and $d(-a')$
are linearly dependent in $M$. By injectivity of $d$, we have $d(a') \ne d(-a')$.
 Hence, $d(-a') = -d(a')$ for all
$a' \in \Phi'$ (due to reducedness of the root data).   The identity $h(d(-a')) = q(-a')\cdot (-a')$ then implies
that $q(-a') = q(a')$ for all $a' \in \Phi'$.  

\begin{remark} In \cite[XXI, 6.1.1, 6.8.1]{sga3}, the notions of {\em morphism}
and {\em $p$-morphism} between reduced root data are defined
(with an integer $p \ge 1$). In the definition of a {\em morphism} there, $q$ is identically 1
and $h$ is only required to be $\Z$-linear
(so $\ker h$ may be nontrivial and $h_{\Q}$ may not be surjective);
this is intended to encode homomorphisms
between split reductive group schemes with an isogeny condition
between the derived groups but no such condition between the maximal central tori.
The $p$-morphisms in \cite[XXI, 6.8.1]{sga3} are a variant on our notion
of $p$-morphism in which $p$ is any integer $\ge 1$ 
and $h$ is only required to be $\Z$-linear.
\end{remark}

\begin{lemma}\label{simpleisog} Let $(h,d,q):R' \rightarrow R$ be a
$p$-morphism between reduced root data,
and let ${\Phi'}^{+}$ be a positive system of roots in $\Phi'$,
with $\Delta'$ its base of simple roots.  Then $\Phi^+ := d({\Phi'}^{+})$ is a positive
system of roots in $\Phi$ and $\Delta := d(\Delta')$ is its base of simple roots.
\end{lemma}

\begin{proof}
Pick a linear form $\lambda'$ on $X'_{\Q}$ such that ${\Phi'}^{+} = \Phi'_{\lambda' > 0}$,
and let $\lambda = \lambda' \circ h_{\Q}$.  
The relations $h(d(a')) = q(a')a'$ for $a' \in \Phi$ with $q(a') \in \Q_{>0}$
and the isomorphism property for $h_{\Q}$ imply that $\lambda$ is non-vanishing on $d({\Phi'})$
and that $\Phi_{\lambda > 0} = d({\Phi'}^+) =: \Phi^+$, so indeed
$\Phi^+$ is a positive system of roots.  It is likewise
clear from the isomorphism property of $h_{\Q}$
that $d(\Delta')$ is a linearly independent set whose span $\Q_{\ge 0}\cdot d(\Delta')$
over $\Q_{\ge 0}$ satisfies 
$$\Phi \subset \Q_{\ge 0} \cdot d(\Delta') \bigcup -\Q_{\ge 0} \cdot d(\Delta').$$
It then follows from elementary inductive arguments with reduced
root systems (see \cite[XXI, 3.1.5]{sga3}, which avoids
a reducedness hypothesis) that this forces $d(\Delta')$ to be the base of a positive system of
roots in $\Phi$.  But clearly $d(\Delta') \subset \Phi^+$, so  $d(\Delta')$ must be the set of
simple roots of $\Phi^+$.
\end{proof}

\begin{proposition}\label{autpin} Let $(G',T',M')$ and $(G,T,M)$
be split reductive groups over a non-empty scheme $S$, 
$f:(G',T',M') \rightarrow (G,T,M)$ an isogeny compatible with the splittings,
and $(h,d,q) := R(f)$ the associated $p(S)$-morphism between the root data.
Let ${\Phi'}^+$ be a positive system of roots in $\Phi'$, 
$\Delta'$ its base of simple roots, 
$\Phi^+ := d({\Phi'}^+)$ the associated positive system of roots
in $\Phi$, and $\Delta = d(\Delta')$ its base of simple roots.
\begin{enumerate}
\item If $B' \subset G'$ is the Borel subgroup containing $T'$ that corresponds
to ${\Phi'}^+$ and $B \subset G$ is the Borel subgroup containing $T$ that corresponds
to $\Phi^+$ then $f$ carries $B'$ into $B$.
\item For a pinning $\{X'_{a'}\}_{a'\in \Delta'}$ of $(G',T',M',\Delta')$ 
and pinning of
$(G,T,M,\Delta)$ given by the sections $X_{d(a')} = f_{a'}({X'_{a'}}^{\otimes q(a')})$,
$f$ is uniquely determined by $R(f)$ and the pinnings
$\{X'_{a'}\}_{a' \in \Delta'}$ and $\{X_a\}_{a \in \Delta}$.
In particular, if $G$ is semisimple 
then an automorphism of $(G,T,M,\{X_a\}_{a \in \Delta})$ 
whose effect on $\mathfrak{g}$ is the identity on each $X_a$ $(a \in \Delta)$
must be the identity.
\end{enumerate}
\end{proposition}

\begin{proof} Since $f$ carries $T'$ into $T$ and carries $U'_{a'}$ into $U_{d(a')}$,
part (1) is immediate from the equalities
$$B' = T' \times \prod_{a' \in {\Phi'}^{+}} U'_{a'},\,\,\,B = T \times \prod_{a \in \Phi^+} U_a$$
respectively defined in $G'$ and $G$
via multiplication (using any enumeration of the sets of positive roots).
To prove part (2), first note that for any $a' \in \Delta'$ the restriction
$f:U'_{a'} \rightarrow U_{d(a')}$ is uniquely determined because 
$$f(\exp_{a'}(c X'_{a'})) = \exp_{d(a')}(f_{a'}((c X'_{a'})^{\otimes q(a')})) = 
\exp_{d(a')}(c^{q(a')} X_{d(a')})$$
for any $c \in \Add$. 
By Proposition \ref{weylf} we have $f(w_{a'}(X'_{a'})) = w_a(X_a)$, so
$f$ is also uniquely determined on the global sections $w_{a'}(X'_{a'})$
that represent simple reflections generating the Weyl group $W(\Phi')$.

Since every $W(\Phi')$-orbit in $\Phi'$ meets $\Delta'$, it follows 
that $f$ is uniquely determined on  $U'_{a'}$ for every $a' \in \Phi'$.
The constituent $h$ in $R(f)$ determines $f:T' \rightarrow T$, so
we conclude that $f$ is uniquely determined on the open cell
$\Omega'$ of $(G',T',M',{\Phi'}^+)$.   The relative schematic density of
$\Omega'$ in $G'$ then implies that $f$ is uniquely determined.
\end{proof}

\begin{definition} Let $(G',T',M',\{X'_{a'}\}_{a' \in \Delta'})$
and $(G,T,M,\{X_a\}_{a \in \Delta})$ be pinned
split reductive groups over a non-empty scheme $S$.
An isogeny $f:(G',T',M') \rightarrow (G,T,M)$ compatible
with the splittings $($in the sense of Definition \ref{defcom}$)$ is {\em compatible with the pinnings}
\index{compatible with pinnings}
if the bijection $d:\Phi' \rightarrow \Phi$ arising from $R(f)$
carries $\Delta'$ into $\Delta$ and
$f(\exp_{a'}(X'_{a'})) = \exp_{d(a')}(X_{d(a')})$
for all $a' \in \Delta'$.
\end{definition}

For any isogeny  
$f:(G',T',M') \rightarrow (G,T,M)$ compatible with the splittings
and a pinning $\{X'_{a'}\}_{a' \in \Delta'}$
of $(G',T',M')$, we have shown
in Proposition \ref{autpin}(2)  that there exists a unique pinning $\{X_a\}_{a \in \Delta}$
of $(G,T,M)$ such that $f$ is compatible with these pinnings.
The crucial fact is that in such situations, the pinning-compatible $f$ is {\em uniquely determined}
by $R(f)$, due to Proposition \ref{autpin}(2).  In particular:

\begin{corollary}\label{faithfulroot}
Let $S$ be a non-empty scheme, and consider the category of pinned split
reductive groups $(G,T,M,\{X_a\}_{a \in \Delta})$ over $S$,
using as morphisms the isogenies  that are compatible with the splittings and pinnings.
The functor
\index{root datum!faithfulness}
\begin{equation}\label{gtmx}
(G, T, M, \{X_a\}_{a \in \Delta}) \rightsquigarrow R(G, T, M) = (M,\Phi,M^{\vee},\Phi^{\vee})
\end{equation}
into the category of root data equipped with $p(S)$-morphisms
is faithful. 

Moreover, if $(G',T')$ and $(G,T)$
are reductive $S$-groups equipped with $($possibly non-split$)$ maximal tori 
and  $f,F:(G',T') \rightrightarrows (G,T)$ are isogenies 
then $f$ and $F$ induce the same isogeny $T' \rightarrow T$
if and only if 
$f = c_{\overline{t}} \circ F$ for some $\overline{t} \in (T/Z_{G})(S)$,
where $c_{\overline{g}}$ denotes the natural action of $\overline{g} \in (G/Z_{G})(S)$
on $G$ induced by conjugation.  In such cases, $\overline{t}$ is unique.
\end{corollary}

This result is essentially the content of \cite[XXIII, 1.9.1, 1.9.2]{sga3} (except
that we record the role of the $(T/Z_G)(S)$-action when we do not require
splittings or pinnings).

\begin{proof}
The faithfulness of (\ref{gtmx}) is immediate from Proposition \ref{autpin}(2), so it remains
to address the assertion concerning the equality of $f, F:T' \rightrightarrows T$ in the absence
of splittings and pinnings.  It is clear that $c_{\overline{t}} \circ F$ and $F$ induce
the same isogeny from $T'$ to $T$ for any $\overline{t}$ (since the 
$T/Z_G$-action on $G$ is the identity on $T$), and also that
$c_{\overline{t}} \circ F$ uniquely determines $\overline{t}$  when $F$ is given (since
$F$ is faithfully flat).  To prove the existence of $\overline{t}$ when
$f|_{T'} = F|_{T'}$ in $\Hom_{S\mbox{-}{\rm{gp}}}(T',T)$, we may work 
\'etale-locally on $S$ due to the uniqueness.
Hence, we can assume that $T' = D_S(M')$ and $T = D_S(M)$
making $(G',T',M')$
and $(G,T,M)$ split 
as well as making the common isogeny $f, F:T' \rightrightarrows T$
arise from a homomorphism $h:M \rightarrow M'$ (so $f$ and $F$ are
compatible with the splittings). 

Choose a pinning $\{X'_{a'}\}_{a' \in \Delta'}$ of $(G',T',M')$.
The induced $p(S)$-morphisms $R(f), R(F):R(G',T',M') \rightrightarrows
R(G,T,M)$ have the same $h$, so they coincide; let $(h,d,q)$ denote
this common $p(S)$-morphism. We get two pinnings of
$(G,T,M)$ relative to $\Delta = d(\Delta')$, namely 
$$X_{d(a')} = f_{a'}({X'_{a'}}^{\otimes q(a')}),\,\,\,
Y_{d(a')} = F_{a'}({X'_{a'}}^{\otimes q(a')})$$
for $a' \in \Delta'$.

The action of $G$ on $G$ via conjugation factors through an action of
$G/Z_G$ on $G$.  Upon restricting this to an action of $T/Z_G$ on a root
group $U_a$ for $a \in \Phi$, we recover 
that such $a:T \rightarrow \Gm$ factors through $T/Z_G$.
Let $\overline{a}:T/Z_G \rightarrow \Gm$ denote the character thereby 
obtained from $a$; these  $\overline{a}$ are the elements of the root
system $\overline{\Phi}$ in the root datum for the split group
$(G/Z_G, T/Z_G,\overline{M})$, where $\overline{M} = \sum_{a \in \Phi} \Z a \subset M$ is the possibly
non-saturated subgroup corresponding to the quotient $T/Z_G$
of the split torus $T$ modulo the split multiplicative type
subgroup $Z_G$ (see Corollary \ref{computez}(1)). 

For each $a \in \Delta$ there exists a unique unit $u_a \in \Gm(S)$ such that
$X_a = u_a Y_a$, so the necessary and sufficient condition
for $c_{\overline{t}} \circ F$ and $f$ to agree as isogenies compatible
with splittings and pinnings 
(and hence to be equal) is that $\overline{a}(\overline{t}) = u_a$
for all $a \in \Delta$.  Hence, it is necessary and sufficient
to show that the subset $\overline{\Delta} \subset \overline{\Phi}$
corresponding to $\Delta$ is a $\Z$-basis of $\overline{M}$.
But this basis property is obvious because 
$$\overline{M} = \sum_{a \in \Phi} \Z a = \sum_{a \in \Delta} \Z a = \bigoplus_{a \in \Delta} \Z a$$
(due to the fact that $\Delta$ is a base for a positive system of roots for $\Phi$).
\end{proof}

The following theorem records the main results
proved in the rest of \S\ref{existsec}.
\index{Isogeny Theorem}
\index{Existence Theorem}

\begin{theorem}\label{mainthm} Let $S$ be a non-empty scheme.
\begin{enumerate}
\item {\rm{(Isogeny Theorem)}} 
For split reductive 
$(G',T',M')$ and $(G,T,M)$ over $S$, 
any $p(S)$-morphism $R(G',T',M') \rightarrow R(G,T,M)$
is induced by an isogeny $f:(G',T',M') \rightarrow (G,T,M)$
compatible with the splittings, and $f$ is unique up to 
the faithful action of $(T/Z_G)(S)$ on $G$.
\item {\rm{(Existence Theorem)}}  
Every root datum is isomorphic to the root datum of a split reductive $S$-group.
\end{enumerate}
\end{theorem}

The Isogeny Theorem
is a slight weakening of \cite[XXV, 1.1]{sga3} (which allows
$S$-group maps that are not isogenies between maximal central tori, and similarly
on root data). The Existence Theorem is \cite[XXV, 1.2]{sga3}.  
An immediate consequence of Theorem \ref{mainthm} 
and the arguments with the $\Z$-basis $\Delta$ of ${\rm{X}}(T/Z_G)$ 
in the proof of Corollary \ref{faithfulroot} is:

\begin{theorem}[Isomorphism Theorem]\label{isomthm} 
Let $S$ be a scheme that is non-empty. The functor
\begin{equation}\label{isomfunctor}
(G,T,M,\{X_a\}_{a \in \Delta}) \rightsquigarrow (R(G,T,M), \Delta)
\end{equation}
from pinned split reductive $S$-groups 
to 
based root data
is an equivalence of categories when using isomorphisms as morphisms.
\index{Isomorphism Theorem}

In particular, 
every split reductive $S$-group $(G,T,M)$ 
is uniquely determined up to isomorphism by its root datum,
and every isomorphism of root data $R(G',T',M') \simeq R(G,T,M)$
arises from an isomorphism
$f:(G',T',M') \simeq (G,T,M)$ 
compatible with the splittings, with $f$ 
uniquely determined up to the faithful action of $(T/Z_G)(S)$ on $G$.
\end{theorem}

In the classical theory, the Existence, Isomorphism, and Isogeny Theorems
are proved over a general algebraically closed field 
(see \cite[9.6.2, 9.6.5, 10.1.1]{springer}).  The traditional 
proof of the Existence Theorem in the classical theory builds a group from its open cell
via delicate procedures guided by the Bruhat decomposition and the structure
of the Dynkin diagram.
The approach over a general base scheme is different, because ``points''
do not  have the same geometric meaning in the relative
theory as in the classical case.
(For example, if $R$ is a nonzero ring then $\mathbf{A}^1_R$
is stratified by $Z = \{0\}$ and $U = \Gm$
but $Z(R) \bigcup U(R) = \{0\} \bigcup R^{\times} \ne R = \mathbf{A}^1_R(R)$ whenever $R$ is not a field.)
In place of arguments inspired by the Bruhat decomposition, Weil's theory of 
birational group laws will be used to 
prove the Existence Theorem over $\Z$ (from which the Existence Theorem is deduced
in general via base change).

To prove the Existence Theorem over $\Z$,  we need 
to know the Existence Theorem over {\em some} algebraically
closed field of characteristic 0, such as $\C$.  Based on such input,
we will use the full faithfulness of (\ref{isomfunctor}) and descent to prove the Existence Theorem over $\Q$
via pinnings (to rigidify structures). 
Thus, the Isogeny Theorem  will be proved before the Existence Theorem.
Split reductive groups over $\Q$ will be ``spread out'' over $\Z$ via arguments
with open cells, structure constants, and birational group laws.

We emphasize that the proof of the Isogeny Theorem over
a general (non-empty) scheme $S$ will not use the classical case as input,
and the proof of the Existence Theorem over $S$ will not
require the classical case of the Existence Theorem in positive characteristic. 
Apart from a few simplifications via the dynamic method, our treatment of the proofs
is just an exposition of the proof presented in \cite[XXIII, XXV]{sga3}.


\subsection{The Isogeny Theorem}

In this section we prove the Isogeny Theorem (i.e., Theorem \ref{mainthm}(1))
and record some consequences of the full faithfulness of (\ref{isomfunctor}). 
Fix a non-empty scheme $S$ and split triples $(G',T',M')$ and $(G,T,M)$
over $S$.  Since the root spaces 
are trivial as line bundles, these admit pinnings.
For any $p(S)$-morphism $\phi:R' \rightarrow R$ between the corresponding
root data, we seek an isogeny $f:(G',T',M') \rightarrow (G,T,M)$
compatible with the splittings such that $R(f) = \phi$.
(The uniqueness of $f$ up to the action of $(T/Z_G)(S)$ on $G$ is provided
by Corollary \ref{faithfulroot}.) 

%As in the proof of the Existence Theorem over $\C$, 
%the real work for the construction of $f$ is in the semisimple case.
%To explain this, let $Z' \subset G'$ and $Z \subset G$ denote the maximal central tori, so
%for the maximal tori $\mathscr{T}' = T' \cap \mathscr{D}(G')$
%and $\mathscr{T} = T \cap \mathscr{D}(G)$ we get 
%central isogenies $$q':(Z' \times \mathscr{D}(G'), Z' \times \mathscr{T}') \rightarrow
%(G',T'), \,\,\, q:(Z \times \mathscr{D}(G), Z \times \mathscr{T}) \rightarrow (G,T)$$ 
%(see Proposition \ref{derpar}).   The subtorus $Z \subset T$ corresponds to the quotient of $M$ by
%the saturation of $\Z \Phi$ (Corollary \ref{computez}(2)), 
%and the subtorus $T \cap \mathscr{D}(G) \subset T$ corresponds
%to the quotient 
%$$\mathscr{M} = {\rm{im}}(M \rightarrow \Hom(T,\Gm) \rightarrow \Hom(\mathscr{T},\Gm))$$
%of $M$.  (Indeed, the equality $T = D_S(M)$ induces
%a map of tori $\mathscr{T} \rightarrow D_S(\mathscr{M})$, and
%this latter map is an isomorphism because we can check it on geometric fibers over $S$.)
%Thus, $Z$ and $\mathscr{T}$ are split via explicit quotient groups of $M$,
%and we can do similarly for $Z'$ and $\mathscr{T}'$
%using $\Z \Phi'$ and an analogue $\mathscr{M}'$ of $\mathscr{M}$.

%The identification of $R(Z \times \mathscr{D}(G), Z \times \mathscr{T}) =
%R(Z, Z) \oplus R(\mathscr{D}(G), \mathscr{T})$ with
%$$(M/(\Z \Phi)_{\rm{sat}}, \emptyset, (\Z \Phi)^{\vee}, \emptyset) \oplus
%(\mathscr{M}, \Phi, \mathscr{M}^{\vee}, \Phi^{\vee})$$
%identifies $R(q)$ with the evident central isogeny of root data
%$$(M/(\Z\Phi)_{\rm{sat}}, \emptyset, (\Z\Phi)^{\perp}, \emptyset) \oplus
%(\mathscr{M},\Phi, \mathscr{M}^{\ast}, \Phi^{\vee}) \rightarrow
%(M,\Phi, M^{\vee}, \Phi^{\vee}),$$
%and similarly for $(G',T',M')$.  

To construct $f$, we require criteria for the existence of an $S$-homomorphism $f$ 
from a split reductive $S$-group $(G,T,M)$ to an $S$-group $H$ when 
the restrictions of $f$ to $T$ and its associated root groups $U_a$
are all specified. 
More precisely, suppose that
$(G,T,M,\{X_a\}_{a \in \Delta})$ is a pinned split
reductive $S$-group, and for $a \in \Delta$ let
$$n_a = w_a(X_a) = \exp_a(X_a) \exp_{-a}(-X_a^{-1}) \exp_a(X_a),$$
where $X_a^{-1}$ is the trivialization of $\mathfrak{g}_{-a}$
linked to $X_a$.  (In \cite{sga3}, the element $n_a \in N_G(T)(S)$
is denoted as $w_a$.) 
For the open cell $\Omega$
arising from $T$ and the positive system of roots $\Phi^+$
with base $\Delta$, the multiplication map $\Omega \times \Omega \rightarrow G$ is fppf.
Thus, any $S$-homomorphism $f:G \rightarrow H$ to an
$S$-group scheme $H$ is uniquely determined
by its restriction to $\Omega$, 
so 
$f$ is uniquely determined by its restrictions
$$f_T:T \rightarrow H,\,\,\,f_a:U_a \rightarrow H$$
for $a \in \Phi$.  
Since $\Phi$ is covered by the $W(\Phi)$-orbits of elements of $\Delta$,
and the elements $n_a$ represent the simple positive reflections
that generate $W(\Phi)$, instead of keeping track of the maps
$f_a$ for all $a \in \Phi^+$ it is enough to record
the maps $f_a$ for $a \in \Delta$ provided that we also
record the images $h_a = f(n_a) \in H(S)$ for
$a \in \Delta$.

Note that $N_G(T)$ is the disjoint union of translates
$n T$ for a set of elements $n \in N_G(T)(S)$
representing $W(\Phi)$, such as products
of the elements $n_a \in N_G(T)(S)$ for
$a \in \Delta$ (upon writing each $w \in W(\Phi)$ as a word
in the simple positive reflections).  Thus, a first 
step towards an existence criterion for a homomorphism $f:G \rightarrow H$
recovering given maps on $T$ and the $U_a$'s ($a \in \Delta$) 
and given values $h_a = f(n_a) \in H(S)$ 
is to settle the case when there is given a homomorphism
$f_N:N_G(T) \rightarrow H$ (instead of $f_T$ and $h_a$'s) 
and homomorphisms $f_a:U_a \rightarrow H$ for {\em all} roots $a \in \Phi$
(not just for $a \in \Delta$). 
Such a preliminary gluing criterion is provided by the following result \cite[XXIII, 2.1]{sga3}:

\begin{theorem}\label{preglue}
For $S$-homomorphisms $f_N:N_G(T) \rightarrow H$
and $f_a:U_a \rightarrow H$ for $a \in \Phi$,
there exists an $S$-homomorphism $f:G \rightarrow H$
extending $f_N$ and the maps $f_a$ if and only if the following three conditions hold:
\begin{enumerate}
\item For all $a \in \Delta$ and $b \in \Phi$, 
$$f_N(n_a) f_b(u_b) f_N(n_a)^{-1} = f_{s_a(b)}(n_a u_b n_a^{-1})$$
for all $u_b \in U_b$.
\item There exists an $S$-homomorphism
$Z_G(T_a) \rightarrow H$ extending
the triple $(f_a, f_{-a}, f_N|_{N_{Z_G(T_a)}(T)})$ 
for all $a \in \Delta$.
\item For all distinct $a, b \in \Delta$ and the subgroup $U_{[a,b]}$ directly
spanned in any order by the groups $U_{c}$
for $$c \in [a,b] := \{ia + jb \in \Phi\,|\,i, j \ge 0\},$$
\index{ aaaab@$[a,b]$}
there exists an $S$-homomorphism $U_{[a,b]} \rightarrow H$ restricting to
$f_{c}$ on $U_{c}$ for all $c \in [a,b]$. 
\end{enumerate}
\end{theorem}

\begin{remark}\label{remuab}
The existence of $U_{[a,b]}$ is a special case of Proposition \ref{upsi}.
\end{remark}

The idea of the proof of sufficiency in Theorem \ref{preglue} (necessity being obvious) 
is that since $f_N|_T$ and the $f_a$ determine what $f$ must be on
the open cell $\Omega = U_{-\Phi^+} \times T \times U_{\Phi^+}$,
and the translates $n \Omega$ by products $n$ among
the $\{n_a\}_{a \in \Delta}$ cover $G$ (Corollary \ref{bruhattranslate} on geometric fibers),
we just have to keep track of the homomorphism
property when extending $f$ across translates of $\Omega$.

By using induction on word
length in the Weyl group (relative to the simple positive reflections
$s_a$ for $a \in \Delta$), the base of the induction amounts to checking
that for {\em some} enumeration of $\Phi^+$, 
the $S$-morphism $f_U:U = U_{\Phi^+} := \prod_{a \in \Phi^+} U_a \rightarrow H$
defined by $(u_a) \mapsto \prod_{a \in \Phi^+} f_a(u_a)$
is a homomorphism. Of course, once this is proved for some choice of enumeration,
it follows that the enumeration of $\Phi^+$ does not matter,
as $U$ is directly spanned in any order by the positive root groups.
The case of $U_{-\Phi^+}$ is also needed, but this will follow formally 
from the case of $U_{\Phi^+}$ by using
a representative $n$ for the long Weyl element $w$ to swap $\Phi^+$ and $-\Phi^+$
(since $f_{w(b)}(u) = f_N(n)f_b(n^{-1}un)f_N(n)^{-1}$ for $b \in \Phi^+$ and $u \in U_{w(b)}$
due to hypothesis (1)). 

For the convenience of the reader, we now sketch the proof of
the base case for the induction (i.e., the homomorphism
property for $f_U$).  Fix a structure of ordered vector space on $M_{\Q}$
so that $\Phi^+$ is the associated positive system of roots (see the discussion
following Definition \ref{posdef}, and note that $W(\Phi)$ acts
transitively on the set of positive systems of roots in $\Phi$).
Consider the resulting enumeration 
$c_0 < \dots < c_m$ of $\Phi^+$.

\begin{lemma}\label{fumap} The map $f_U$ is an $S$-homomorphism
when it is defined using $\{c_j\}$.
\end{lemma}

This is \cite[XXIII, 2.1.4]{sga3}. 

\begin{proof} For $i \ge 1$,  consider the direct product 
scheme $U_{\ge i} := \prod_{c \ge c_i} U_c$
and the map $U_{\ge i} \rightarrow U$ of $S$-schemes defined by 
multiplication in strictly increasing order of the roots.
By Proposition \ref{upsi}, this identifies $U_{\ge i}$ with a closed $S$-subgroup of $U$
that is moreover normalized by $U_{c_{i-1}}$ when $i > 0$.
The homomorphism property for $f_U$ on $U = U_{\ge 0}$ will
be proved by descending induction:  for all $i$ we claim
that $f_U$ restricts to a homomorphism on $U_{\ge i}$.
The case $i = m$ is obvious (as the restriction to $U_{\ge m} = U_{c_m}$ is $f_{c_m}$).

In general, if the result holds for some $i > 0$ then 
since $U_{\ge i-1} = U_{c_{i-1}} \ltimes U_{\ge i}$,
it is straightforward to use the homomorphism property for $f_U$ on $U_{\ge i}$
to reduce to verifying the identities
\begin{equation}\label{fuab}
f_b(u_b)^{-1} f_a(u_a) f_b(u_b) \stackrel{?}{=} f_U(u_b^{-1}u_a u_b)
\end{equation}
for $b = c_{i-1}$, $a > c_{i-1} = b$, and all points
$u_a$ of $U_a$ and $u_b$ of $U_b$.  Note that
these identities make sense because $u_b^{-1}u_a u_b \in U_{\ge i}$
since $a \ge c_i$ and $b = c_{i-1}$.  To summarize,
we have reduced ourselves to proving that for any $a, b \in \Phi^+$
with $a > b$, the identities (\ref{fuab}) are satisfied.

For any $u_a \in U_a$ and $u_b \in U_b$, clearly 
$u_b^{-1}u_a u_b \in U_{[a,b]}$.  Thus, if $a \in \Delta$ and
$b \in [a, b_0]$ for $b_0 \in \Delta - \{a\}$ 
then the desired identities are a consequence of hypothesis (3) of Theorem \ref{preglue}.
To reduce the case of a general pair to these special cases, 
we use a result in the theory of root systems:
there exists $w \in W(\Phi)$ such that $w(a) \in \Delta$ and
$w(b) \in  [w(a),b_0]$ for some $b_0 \in \Delta$ necessarily distinct from $w(a)$;
this follows from the transitivity of the $W(\Phi)$-action on the set of
positive systems of roots and \cite[XXI, 3.5.4]{sga3}
(whose main content is 
\cite[VI, \S1.7, Cor.\,2]{lie4}, applied to the reverse
lexicographical ordering on $\Phi^+$ relative
to an enumeration of the base $\Delta$).   This 
enables us to reduce to the  settled special case $a \in \Delta$
and $b \in [a, b_0]$ with $b_0 \in \Delta - \{a\}$ 
provided that if $n \in N_G(T)(S)$
represents $w \in W(\Phi)$ 
then $$f_N(n) f_a(u_a) f_N(n)^{-1} = f_{w(a)}(n u_a n^{-1})$$
for all $a \in \Phi$ and points $u_a \in U_a$. 
The case $n = n_b$ for $b \in \Delta$ is exactly hypothesis (1)
in Theorem \ref{preglue}.  The general case is deduced
from this via induction on word length in $W(\Phi)$
relative to the simple positive reflections $\{s_c\}_{c \in \Delta}$
and applications of hypotheses (1) and (2) in Theorem \ref{preglue}.
(We need (2) because representatives $n_c \in N_G(T)(S)$ for the $s_c$'s do not
generate $N_G(T)(S)$.) 
See \cite[XXIII, 2.1.3]{sga3} for further details.
\end{proof}

Now we are ready to formulate the main criterion for constructing homomorphisms, building on
Theorem \ref{preglue}.  We will eventually obtain a criterion that reduces all difficulties
to the case of groups with semisimple-rank at most 2.  That is,
the serious computational effort will only be required with low-rank groups.
Keep in mind that we have explicitly described the split cases
with semisimple-rank 1 in Theorem \ref{sslist},
at least Zariski-locally on the base. (The intervention of Zariski-localization
can be removed from Theorem \ref{sslist} by using a pinning and 
Proposition \ref{autpin}(2), but we do not need that 
minor improvement here.) 

Let $(G,T,M,\{X_a\}_{a \in \Delta})$ be a 
pinned split reductive $S$-group, and let $H$ be an $S$-group.
For each $a \in \Delta$, let 
$n_a = w_a(X_a)$.
For $S$-homomorphisms $f_T:T \rightarrow H$
and $f_a:U_a \rightarrow H$ and elements $h_a \in H(S)$
for all $a \in \Delta$, we have seen that there
exists at most one $S$-homomorphism $f:G \rightarrow H$
such that $f|_T = f_T$ and $f|_{U_a} = f_a$ and
$f(n_a) = h_a$ for all $a \in \Delta$.  But when does $f$ exist?
There are some necessary conditions.  For example, for all
$a \in \Delta$ we must have 
\begin{equation}\label{fth}
f_T(t) f_a(u_a)f_T(t)^{-1} = f_a(t u_a t^{-1}),\,\,\,
h_a f_T(t) h_a^{-1} = f_T(s_a(t))
\end{equation}
for all points $t \in T$ and $u \in U_a$ (valued in a common $S$-scheme).

There are additional conditions imposed
by relations in the Weyl group.
More specifically, note that if $a, b \in \Delta$ then $(s_a s_b)^{m_{ab}} = 1$ in $W(\Phi)$
where $m_{ab} = m_{ba}$ is the $ab$-entry in the symmetric Cartan matrix ($m_{ab}$ is the order of $s_a s_b$
in $W(\Phi)$, so $m_{aa} =  1$). In particular, $t_{ab} := (n_a n_b)^{m_{ab}}  \in T(S)$, so 
$t_{aa} = n_a^2 = a^{\vee}(-1)$.  
This yields the further necessary conditions
\begin{equation}\label{tab}
h_a^2 = f_T(a^{\vee}(-1)),\,\,\,
(h_a h_b)^{m_{ab}} = f_T(t_{ab})
\end{equation}
for $a, b \in \Delta$ with $b \ne a$.  The relation
$(n_a \exp_a(X_a))^3 = 1$ (Remark \ref{cube}) yields the necessary condition
\begin{equation}\label{h3}
(h_a f_a(\exp_a(X_a)))^3 = 1
\end{equation}
for all $a \in \Delta$. 
Finally, if $a, b \in \Delta$ are distinct
then for a homomorphism $f_{ab}:U_{[a,b]} \rightarrow H$
extending $f_a$ on $U_a$ and $f_b$ on $U_b$
to arise from an $f:G \rightarrow H$ of the desired type, 
the following conjugation relations must hold on
the root groups $U_{c}$ for $c \in [a,b]$.  If $c \ne a$ then 
\begin{equation}\label{lastrel}
h_a f_{ab}(u_{c}) h_a^{-1} = f_{ab}(n_a u_{c} n_a^{-1}).
\end{equation}
(The right side makes sense because (i) $n_a$ conjugates $U_{c}$ to
$U_{s_a(c)}$, and (ii) $s_a(c) = c - \langle c, a^{\vee} \rangle a$ lies in $[a,b]$,
due to $s_a(c)$ having a positive $b$-coefficient with $a$ and $b$ distinct elements of 
the base $\Delta$ of $\Phi^+$.)   Likewise, if  $c \ne b$ then 
\begin{equation}\label{lastlast}
h_b f_{ab}(u_c) h_b^{-1} = f_{ab}(n_b u_c n_b^{-1}).
\end{equation}

Remarkably,  the preceding necessary conditions for the existence
of $f$, each of which only
involves subgroups $Z_G(T_a)$ of semisimple-rank 1 and 
subgroups $Z_G(T_{ab})$ of semisimple-rank 2 (with $T_{ab}$ denoting the unique torus of relative
codimension-2 in $T$ contained in $\ker a \bigcap \ker b$
for distinct $a, b \in \Phi^+$), are also sufficient.
This is \cite[XXIII, 2.3]{sga3}:

\begin{theorem}\label{bigcrit}
If the conditions $(\ref{fth})$, $(\ref{tab})$, and $(\ref{h3})$ 
hold and for all distinct $a, b \in \Delta$ there exists
a homomorphism $f_{ab}:U_{[a,b]} \rightarrow H$ extending
$f_a$ and $f_b$
and satisfying $(\ref{lastrel})$ and $(\ref{lastlast})$
then a homomorphism $f:G \rightarrow H$ exists satisfying $f|_T = f_T$, $f|_{U_a} = f_a$ for all $a \in \Delta$,
and $f(n_a) = h_a$ for all $a \in \Delta$.
\end{theorem}

\begin{proof}
The proof 
is largely a systematic (and intricate) argument with word length in Weyl groups, 
bootstrapping from the given conditions to eventually establish the hypotheses in Theorem \ref{preglue}.
This entails constructing $f_N$ and the homomorphisms
$f_a$ for {\em all} roots $a \in \Phi$ (recovering the given
homomorphisms for $a \in \Delta$).  

An elementary argument (see \cite[XXIII, 2.3.1]{sga3}) constructs
the $S$-homomorphism $f_N:N_G(T) \rightarrow H$
that extends $f_T$ and carries $n_a$ to $h_a$ for all $a \in \Delta$;
such an $f_N$ is unique since products among the $n_a$'s
represent all elements of $W(\Phi)$ (and $N_G(T)$ is covered
by the left-translates of its open and closed subgroup $T$ 
by any set of representatives of $W(\Phi)$ in $N_G(T)(S)$;
the closed $S$-subgroup $T \subset N_G(T)$ is
open because $W_G(T)$ is \'etale, by Proposition \ref{finitew}). 
The construction of well-defined maps $f_a$ for all $a \in \Phi$ is harder,
and rests on a lemma of Tits in the theory of root systems
(stated as Exercise 21 in \cite[Ch.\,VI]{lie4}, and
proved in \cite[XXI, 5.6]{sga3}).  

We refer to \cite[XXIII, 2.3.2--2.3.6]{sga3} for
the details, and sketch the main group-theoretic argument
that establishes the requirement in Theorem \ref{preglue}(2).
This requirement says that 
for each $a \in \Delta$ there exists an $S$-homomorphism $F_a:Z_G(T_a) \rightarrow H$
such that $F_a|_T = f_T$, $F_a|_{U_a} = f_a$, and $F_a(n_a) = h_a$.
(Such an $F_a$ is visibly unique if it exists, since $n_a$-conjugation swaps the opposite
root groups that appear in the open cell of $Z_G(T_a)$ relative to its split maximal torus $T$ and 
roots $\pm a$, and it satisfies $F_a|_{U_{-a}} = f_{-a}$
by {\em defining} $f_{-a}(u) = h_a^{-1}f_a(n_a u n_a^{-1})h_a$ for $u \in U_{-a}$.)  
The construction of $F_a$ in \cite[XXIII, 2.3.2]{sga3} rests
on calculations in \cite[XX, 6.2]{sga3} with an ``abstract'' split
reductive group of semisimple-rank 1. 
The explicit classification of such split groups in Theorem \ref{sslist} will now
be used to simplify those calculations.

We may replace $G$ with $Z_G(T_a)$, so our problem becomes exactly the special 
case of $G$ with semisimple-rank 1.  In particular, $\Delta = \{a\}$, so the conditions
(\ref{lastrel}) and (\ref{lastlast}) become vacuous 
and (\ref{tab}) only involves the first relation there.
The uniqueness allows us to work \'etale-locally on $S$ for existence, so
the central torus direct factor as in Theorem \ref{sslist}
can be dropped and we are reduced
to three special cases:  $(G,T)$ is
either $({\rm{SL}}_2, D)$, $({\rm{PGL}}_2,\overline{D})$,
or $(\GL_2, \widetilde{D}) = ({\rm{SL}}_2 \times^{\mu_2} \Gm, 
D \times^{\mu_2} \Gm)$.  The third case trivially reduces
to the first case (since $a^{\vee}(-1) \in \mathscr{D}(G)$ in general), and so does the second case
since the natural degree-2 central isogeny ${\rm{SL}}_2 \rightarrow {\rm{PGL}}_2$
has kernel $\mu_2 = D[2] \subset D$.  Hence,
we may and do assume that $(G,T,M) = ({\rm{SL}}_2, D, \Z)$ where the element $1 \in \Z$
goes over to the isomorphism $D \simeq \Gm$ inverse to
$c \mapsto {\rm{diag}}(c, 1/c)$, and likewise we can arrange
that $a$ is the standard positive root (i.e., $U_a$ is the strictly upper triangular subgroup
of $G = {\rm{SL}}_2$) and $X_a = (\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix}) \in \mathfrak{sl}_2$.

In this special case, the challenge is not to define $f$ on the open cell $\Omega$ (it is clear
what the unique possibility for that must be), nor how to define
$f$ on another translate of $\Omega$
that (together with $\Omega$) covers ${\rm{SL}}_2$.  The hard part is to verify
that one gets a globally well-defined $S$-morphism that is moreover a homomorphism.
The calculations to verify this in \cite[XX, 6.2]{sga3} 
are done in the absence of an explicit classification in
semisimple-rank 1, and they 
become simpler for the explicit case we need, namely $({\rm{SL}}_2, D)$
(e.g., the auxiliary parameters $u \in U_a = \Add$ and $\widetilde{u} \in U_{-a} = \Add$ there become 1).
\end{proof}


\begin{remark}
One may wonder about a finer result beyond
Theorem \ref{bigcrit} that replaces (\ref{lastrel}) and (\ref{lastlast})
with a more explicit presentation of $U_{[a,b]}$
in terms of ``generators and relations''.
This viewpoint is systematically developed in
\cite[XXIII, 2.6, 3.1.3, 3.2.8, 3.3.7, 3.4.10, 3.5]{sga3},
but it is not logically relevant to the proofs of
the Isogeny, Isomorphism, or Existence Theorems, so we will say nothing more about it.
\end{remark}

\begin{example}\label{rk1ex} We can 
make explicit what Theorem \ref{bigcrit} says concerning the construction of 
isogenies $f:(G',T',M',\{X'_{a'}\}) \rightarrow (G, T, M, \{X_a\})$
between pinned split reductive $S$-group with semisimple-rank 1
(so $\Phi(G',T') = \{\pm a'\}$ and $\Phi(G,T) = \{\pm a\}$) 
such that $f$ is compatible with the pinnings and the splittings.  
We want $f$ to restrict to an isogeny
$T' \rightarrow T$ dual to a given finite-index lattice inclusion $h:M \rightarrow M'$
satisfying $h(a) = q a'$ for a prime power $q = p^n \ge 1$
such that $p = 0$ in $\calO_S$ if $q > 1$ (duality forces $h^{\vee}({a'}^{\vee}) = q a^{\vee}$
due to being in the case of semisimple-rank 1), 
and we want $f$ to restrict to an isogeny $f_{a'}:U'_{a'} \rightarrow U_a$
given by $\exp_{a'}(cX'_{a'}) \mapsto \exp_a(c^q X_a)$
(a homomorphism when $q > 1$ because we assume 
$S$ is a $\Z/p\Z$-scheme in such cases).

We claim that for any such $h$ and $q$
there is a unique such $f$. Necessarily $R(f)$ coincides with the $p(S)$-morphism $(h,d,q)$
where $d(a') = a$ and $d(-a') = -a$, so our claim is 
exactly the Isogeny Theorem for all 
cases with semisimple-rank 1. 
(This case of the Isogeny Theorem 
is \cite[XXIII, 4.1.2]{sga3}, whose proof via \cite[XX, 4.1]{sga3} rests on extensive calculations.
The reason that we will be able to avoid those
calculations is because we have  Theorem \ref{sslist}.)

The key feature of semisimple-rank 1 is that 
the conditions in Theorem \ref{bigcrit} become very concrete in such cases. 
First of all, (\ref{tab}) says  $n_a^2 \stackrel{?}{=} f_T({a'}^{\vee}(-1))$,
and this is automatic since 
$$n_a^2 = a^{\vee}(-1) = a^{\vee}((-1)^q) = (a^{\vee})^q(-1) = h^{\vee}({a'}^{\vee})(-1) = 
f_T({a'}^{\vee}(-1))$$
(we have used that $(-1)^q = -1$ in $\calO_S$, since $p$ vanishes in $\calO_S$ when $q = p^n > 1$). 
Likewise, the second relation in (\ref{fth}) merely says that $f_T$ intertwines inversion
on $T'$ and $T$, and the first relation in (\ref{fth}) is automatic
since $h(a) = qa'$ (and $f_T = D_S(h)$).  The relations
in (\ref{lastrel}) and (\ref{lastlast}) are vacuous in cases with
semisimple-rank 1.  Finally, (\ref{h3}) is automatic since
it asserts $(w_a(X_a) \exp_a(X_a))^3 = 1$, which
always holds (see Remark \ref{cube}). 
\end{example}



The necessary and sufficient conditions in Theorem \ref{bigcrit}
 involve only reductive closed subgroups with semisimple-rank $\le 2$, 
so we immediately deduce the following crucial existence criterion
for homomorphisms \cite[XXIII, 2.4]{sga3} that is expressed
entirely in terms of closed reductive subgroups with such low semisimple-rank.

\begin{corollary}\label{rk2map} Let $(G,T,M)$ be split reductive over $S$
with semisimple-rank $\ge 2$.  Let $\Phi^+$ be a positive system
of roots in $\Phi$, and let $\Delta$ be the corresponding base of simple roots.
For each $a, b \in \Delta$, 
let $T_{ab} \subset T$ be the unique subtorus of
relative codimension-$2$ contained in $\ker a \bigcap \ker b$
when $a \ne b$ 
and let $T_{aa} = T_a$.  

For an $S$-group $H$ and given $S$-homomorphisms
$f_{ab}:Z_G(T_{ab}) \rightarrow H$
for all $a, b \in \Delta$, assume 
$f_{ab} = f_{ba}$ and 
$f_{ab}|_{Z_G(T_{a})} = f_{aa}$ for
all $a, b \in \Delta$. There is a unique $S$-homomorphism
$f:G \rightarrow H$ such that $f|_{Z_G(T_{ab})} = f_{ab}$
for all $a, b \in \Delta$.
\end{corollary}

Now we prove the Isogeny Theorem (i.e., Theorem
\ref{mainthm}(1)).  The main work is for
groups with semisimple-rank 2.

\index{Isogeny Theorem|(}
\begin{proof}[Proof of Isogeny Theorem]
Let $\Delta'$ be the base of a positive system of roots ${\Phi'}^+$ 
in $\Phi'$, so by Lemma \ref{simpleisog} the set $\Delta := d(\Delta')$
is a base for a positive system of roots $\Phi^+ = d({\Phi'}^+)$ in $\Phi$.
The triviality hypothesis on the root spaces
allows us to choose pinnings $\{X'_{a'}\}_{a' \in \Delta'}$
and $\{X_a\}_{a \in \Delta}$.
Corollary \ref{faithfulroot} shows
that $f$ is unique up to the
action of $(T/Z_G)(S)$, so it remains to prove the existence of $f$.

For the split reductive $S$-groups $G$ and $G'$, the given isogeny between their root data implies that 
their constant fibral semisimple-ranks 
are the same.  The case of semisimple-rank 0 is trivial
(as then $G$ and $G'$ are tori; i.e., $T = G$ and $T' = G'$), and the 
case of semisimple-rank 1 is Example \ref{rk1ex},
so we now assume the common semisimple-rank
of $(G',T')$ and $(G,T)$ is $\ge 2$. 

{\bf Step 1}. We reduce to the case when $G$ and $G'$ have semisimple-rank 2. 
For each pair of (possibly equal) elements $a', b' \in \Delta'$, 
let $a = d(a')$ and $b = d(b')$ in $\Delta$ and consider
the closed reductive $S$-subgroups
$Z_{G'}(T'_{a'b'}) \subset G'$ and $Z_G(T_{ab}) \subset G$
with respective maximal tori $T' = D_S(M')$ and $T = D_S(M)$.
In these $S$-subgroups the respective
sets of roots $\Phi_{a'b'}$
and $\Phi_{ab}$ lie in the respective subsets $\Phi' \subset M' - \{0\}$
and $\Phi \subset M - \{0\}$, and the root
spaces are free of rank 1 as line bundles since
they are root spaces for $(G',T')$ and $(G,T)$ respectively.
Hence, we get root data
$$R(Z_{G'}(T'_{a'b'}),T',M') = (M', \Phi'_{a'b'}, {M'}^{\vee}, {\Phi'}_{a'b'}^{\vee})$$
and $$R(Z_G(T_{ab}),T,M) = (M, \Phi_{ab}, M^{\vee}, \Phi_{ab}^{\vee})$$
equipped with positive systems of roots $\Phi'_{a'b'} \bigcap {\Phi'}^{+}$
and $\Phi_{ab} \bigcap {\Phi}^+$.

Since $a', b' \in \Delta'$ and $a, b \in \Delta$, 
the positive systems of roots 
$\Phi'_{a'b'} \bigcap {\Phi'}^{+}$ and $\Phi_{ab} \bigcap \Phi^+$
(with rank 1 when $a' = b'$ and rank 2 when $a' \ne b'$)
have as their respective bases $\{a', b'\}$ and $\{a, b\}$
when $a' \ne b'$
and $\{a'\}$ and $\{a\}$ when $a' = b'$.
Thus, the split reductive $S$-groups $Z_{G'}(T'_{a'b'})$ and $Z_G(T_{ab})$ with 
semisimple-rank $\le 2$
are also pinned (using our pinnings for $(G',T',M')$ and $(G,T,M)$).
Concretely, $\Phi'_{a'b'}$ consists of the roots that are trivial on the torus $T'_{a'b'}$,
and likewise for $\Phi_{ab}$ using
$T_{ab}$ (as we may check on geometric fibers, using the classical theory).
But ${\rm{X}}(T'_{a'b'})$ is the quotient of ${\rm{X}}(T')$ modulo the saturation
of $\Z a' + \Z b'$, and likewise for ${\rm{X}}(T_{ab})$ as a quotient of
${\rm{X}}(T)$ using $\Z a + \Z b$, so it follows from the definition of an isogeny of
reduced root data that for the given $p(S)$-morphism of root data $\phi = (h,d,q)$
the map $h:{\rm{X}}(T) \rightarrow {\rm{X}}(T')$ induces a compatible map
${\rm{X}}(T_{ab}) \rightarrow {\rm{X}}(T'_{a'b'})$.  
Hence, $\phi$ restricts to a $p(S)$-morphism of root data
$$\phi_{a'b'}:R(Z_{G'}(T'_{a'b'}),T',M') \rightarrow R(Z_G(T_{ab}),T,M)$$
for all (possibly equal) $a', b' \in \Delta'$.   Note that $\phi_{a'b'} = \phi_{b'a'}$.

Now assume that all cases of semisimple-rank $2$ are settled (as is true for
all cases with semisimple-rank 1), so we obtain isogenies 
$$f_{a'b'}:(Z_{G'}(T'_{a'b'}),T',M') \rightarrow (Z_G(T_{ab}),T,M)$$
that are compatible with the splittings and satisfy $R(f_{a'b'}) = \phi_{a'b'}$. 
By the proof of Corollary \ref{faithfulroot}, we may and do replace such an $f_{a'b'}$ 
with its composition against the action of a {\em unique}
$\overline{t} \in (T/Z_{Z_G(T_{ab})})(S)$  so that $f_{a'b'}$
is also compatible with the pinnings $\{X'_{a'}, X'_{b'}\}$
and $\{X_a, X_b\}$ (by which we mean
$\{X'_{a'}\}$ and $\{X_a\}$ when $a' = b'$).  Hence, the equality
$R(f_{a'b'}) = \phi_{a'b'} = \phi_{b'a'} = R(f_{b'a'})$
forces $f_{a'b'} = f_{b'a'}$.
Likewise, if $a' \ne b'$ then $\phi_{a'b'}$
restricts to $\phi_{a'a'}$ 
on the root datum $R(Z_{G'}(T'_{a'}),T',M')$, 
so $f_{a'b'}|_{Z_{G'}(T'_{a'})} = f_{a'a'}$.
By Corollary \ref{rk2map}, there is 
a unique isogeny $f:(G',T') \rightarrow (G,T)$ inducing all $f_{a'b'}$, so $f$ respects
the splittings  and the pinnings.
This completes the reduction of the Isogeny Theorem
to the case of semisimple-rank 2.

{\bf Step 2}.   Assume that $G$ and $G'$ have semisimple-rank 2. 
By the classification
of rank-2 reduced root systems,
there are four possibilities 
for each root system: ${\rm{A}}_1 \times {\rm{A}}_1$
(e.g., ${\rm{SL}}_2 \times {\rm{SL}}_2$), 
${\rm{A}}_2$ (e.g., ${\rm{SL}}_3$), ${\rm{B}}_2$ (e.g., ${\rm{Sp}}_4$), and ${\rm{G}}_2$.
In each of these four cases we will define a ``universal'' 
choice of trivialization of all positive root spaces
in a manner that only depends on the based root system
$(\Phi, \Delta)$ and an enumeration $\xi:\{1, 2\} \simeq \Delta$.

[Since we have not yet proved the Existence Theorem,
we may not yet know that there exists a pinned split reductive $S$-group
whose root system has type ${\rm{G}}_2$ (if we haven't constructed
${\rm{G}}_2$ already by some other means; e.g., octonion algebras).
This is not logically relevant, since at present we are only aiming
to prove that {\em if} we are given a pair of pinned split reductive
$S$-groups then we can
relate $p(S)$-morphisms between their root data to isogenies between the $S$-groups.
To prove the Isogeny Theorem we only need to consider each
of the root systems of rank 2 that {\em might} occur,
without constructing specific $S$-groups.]

Consider a pinned split reductive $S$-group
$(G,T,M,\{X_a\}_{a \in \Delta})$ whose root system has rank 2.
We use the pinning to define a specific representative
$n_a := w_a(X_a) \in N_G(T)(S)$ for the simple reflection $s_a \in W(\Phi)$
for each $a \in \Delta$.  If $c \in \Phi^+$ is a positive root
(relative to $\Delta$) then by \cite[VI, \S1.5, Prop.\:15]{lie4} there exists $w \in W(\Phi)$
such that $w^{-1}(c) \in \Delta$, so for any sign $\varepsilon_c \in \{\pm 1\}$ and any product
$n_w \in N_G(T)(S)$ among $\{n_a\}_{a \in \Delta}$
such that $n_w$ represents $w$, 
$$X_{c,n_w} := \varepsilon_c {\rm{Ad}}(n_w)(X_{w^{-1}(c)})$$ 
is a trivialization of $\mathfrak{g}_c$.
This trivialization depends on both $w$ and $n_w$, neither of
which is determined by $c$, so $X_{c,n_w}$ is not intrinsic
even if we set $\varepsilon_c = 1$
(e.g., if $S = \Spec \Z$ then there is an ambiguity from scaling by $\Z^{\times} = \{1, -1\}$).

In \cite[XXIII, 3.1--3.4]{sga3} each of the four reduced rank-2 root systems $\Phi$ is 
considered separately, along with a choice of base $\Delta$ and 
an enumeration $\xi:\{1, 2\} \simeq \Delta$ (in order of increasing
root length for ${\rm{B}}_2$ and ${\rm{G}}_2$).  In each case, 
for every $c \in \Phi^+$ an explicit choice is made for the data:
$\varepsilon_c \in \{\pm 1\}$, $w$ satisfying $w^{-1}(c) \in \Delta$, and 
$n_w$ as a product among $\{n_a\}_{a \in \Delta}$
to define a trivialization $X_c$ of $\mathfrak{g}_c$.  There is nothing 
canonical about the choices of $\varepsilon_c$, $w$, or $n_w$, but these choices are
made only depending on the based root system $(\Phi, \Delta)$ 
and enumeration $\xi:\{1, 2\} \simeq \Delta$. 
In this way, we obtain a trivialization 
$\{X_c\}_{c \in \Phi^+}$ for the positive root spaces
in each pinned split reductive group scheme $(G,T,M,\{X_a\}_{a \in \Delta})$
with semisimple-rank 2 when $\Delta$ is equipped with an
enumeration (say in order of increasing root length
for ${\rm{B}}_2$ and ${\rm{G}}_2$).  
In \cite[XXIII, 3.4.1(ii)]{sga3}, the signs $\{\varepsilon_c\}_{c \in \Phi^+}$ are
taken to be 1 except for 
${\rm{G}}_2$. 

[The enumeration $\xi:\{1, 2\} \simeq \Delta$ is most important 
for ${\rm{A}}_2$, since for ${\rm{B}}_2$ and ${\rm{G}}_2$
the root lengths give an intrinsic distinction between the two elements
of $\Delta$, whereas for ${\rm{A}}_1 \times {\rm{A}}_1$ the only positive
roots are the simple ones and their root groups commute.
The effect of the choice of enumeration for ${\rm{A}}_2$ is seen via the signs 
that break the symmetry in the formulas for $\{X_c\}_{c \in \Phi^+}$ in \cite[XXIII, 3.2.1(ii)]{sga3}
if one swaps the order of enumeration of the two simple positive roots.]

Any two isomorphisms (compatible with splittings and pinnings) 
between pinned split reductive groups of semisimple-rank 2 
that induce the same bijection between the $\Delta$'s must 
coincide on derived groups (Proposition \ref{autpin}(2)), and so 
coincide on the ``universal'' trivializations
of all positive root spaces.  The induced bijection between
the $\Delta$'s can be controlled by demanding compatibility
with the chosen enumeration of $\Delta$. 
Thus, $\{X_c\}_{c \in \Phi^+}$ is functorial with respect to isomorphisms between pinned split reductive
$S$-groups with semisimple-rank 2 when we demand that the isomorphism
be compatible with a fixed choice of enumeration of $\Delta$.  
In this sense, the above choice of $\{X_c\}_{c \in \Phi^+}$
is ``universal'' for any based reduced root system 
$(\Phi, \Delta)$ of rank 2 {\em equipped with} an enumeration of $\Delta$.

{\bf Step 3}.  The trivializations $X_c$ will now be used to unambiguously 
define ``structure constants'' (global functions on $S$, to be precise)
that encode the $S$-group law. 
Choose a pinned split reductive $S$-group $(G,T,M,\{X_a\}_{a \in \Delta})$.
For all $c \in \Phi^+$, define $p_c:\Add \simeq U_c$ via $p_c(x) = \exp_c(x X_c)$.
For any $a \in \Delta$ and $c \in \Phi^+ - \{a\}$ there exists a unique unit $u(a,c) \in \Gm(S)$ defined by
\begin{equation}\label{adgn}
{\rm{Ad}}_G(n_a)(X_c) = u(a,c)X_{s_a(c)}.
\end{equation}
(Note that $s_a(c) \in \Phi^+$
because the $\Delta$-expansion of each positive $c \ne a$ has some positive coefficient
away from $a$ and hence $s_a(c) = c - \langle c, a^{\vee}\rangle a$ does as well.)
Likewise, by introducing some universal signs in the definitions
of the coefficients in (\ref{rootcomm}) we see that for distinct $b, b' \in \Phi^+$ and roots $ib + jb' \in \Phi$
with $i, j \ge 1$ there are unique $C_{i,j,b,b'} \in \Add(S)$ 
such that 
$$p_{b'}(y)p_{b}(x) = p_b(x)p_{b'}(y) \prod_{i, j} p_{ib+jb'}(C_{i,j,b,b'} x^i y^j),$$
where the product on the right side is taken relative to the ordering
on $\Phi^+$ defined by lexicographical order relative to the chosen
enumeration of $\Delta$.  (Recall that for ${\rm{B}}_2$ and ${\rm{G}}_2$
we made the convention to enumerate $\Delta$ by putting the short root
first. This choice is implicit in the formulas in \cite[XXIII, 3.3, 3.4]{sga3}.) 

A priori, $u(a,c)$ and $C_{i,j,b,b'}$ may depend on 
$(G,T,M,\{X_a\}_{a \in \Delta})$ over $S$ and the enumeration 
$\xi:\{1, 2\} \simeq \Delta$.
We call these the {\em structure constants} for
$(G,T,M, \{X_a\}_{a \in \Delta})$.  (They are global functions on $S$).
If we already knew the Isomorphism and Existence Theorems then the following lemma 
would be immediate, and the miracle at the heart of the
Isogeny, Isomorphism, and Existence Theorems is that this lemma can be proved directly:

\begin{lemma}\label{miracle} For each based reduced root system $(\Phi,\Delta)$
of rank $2$ equipped with an enumeration of $\Delta$,
there are signs $u(a,c) \in \Z^{\times}$
and integers $C_{i,j,b,b'} \in \Z$ that 
induce the structure constants  for every pinned split reductive group
$(G,T,M,\{X_a\}_{a \in \Delta})$ with based root system
$(\Phi,\Delta)$ over any non-empty scheme $S$.
\end{lemma}

\begin{proof}
The idea of the proof is to exploit additional relations
arising from the $N_G(T)(S)$-action on the root spaces
and the $W(\Phi)$-action on $\Phi$.  For example, if
$a, b \in \Phi$ are distinct with $b \ne -a$ and if 
$n \in N_G(T)(S)$ represents $w \in W(\Phi)$ satisfying
$w(a) = b$ then for any trivializations
$X$ of $\mathfrak{g}_a$ and $Y$ of $\mathfrak{g}_b$
the unit $u \in \Gm(S)$ defined by
${\rm{Ad}}_G(n)(X) = u Y$ satisfies
\begin{equation}\label{wnrel}
n w_a(X) n^{-1} = b^{\vee}(u) w_b(Y)
\end{equation}
(where $w_a(X) = \exp_a(X) \exp_{-a}(-X^{-1}) \exp_a(X)$, and similarly for $b$ using $Y$).
An equivalent formulation is to say that $w_b(uY) = b^{\vee}(u) w_b(Y)$, which is 
the first identity in Corollary \ref{wsec}(2).

The relations (\ref{wnrel}) lead to nontrivial conditions on 
the units $u(a,c)$ in (\ref{adgn}), and for each $\Phi$ these extra relations
yield unique universal solutions
$u(a,c) \in \Z^{\times} = \{1, -1\}$ that are the same for {\em all}
$(G,T,M,\{X_a\}_{a \in \Delta})$ with root system $\Phi$ over {\em any} non-empty $S$. 
The details are in \cite[XXIII, 3.1--3.4]{sga3}, working
case-by-case depending on the rank-2 root system $\Phi$.

Relations between root groups and the $W(\Phi)$-action 
on $\Phi$ are given in \cite[XXIII, 3.1.1]{sga3} by 
applying conjugation against each $n_a = w_a(X_a)$ ($a \in \Delta$)
to the commutation relations among root groups.  This allows one to 
uniquely solve for {\em some} of the $C_{i,j,b,b'}$'s.
Miraculously, these solutions arise from constants in $\Z$ that depend
only on $(\Phi, \Delta, \xi)$ and do {\em not} depend on $S$ or
$(G,T,M,\{X_a\}_{a \in \Delta})$.
There are more commutation relations for the remaining 
unknown coefficients $C_{i,j,b,b'}$, and case-by-case arguments
in \cite[XXIII, 3.1--3.4]{sga3} depending only 
on $(\Phi, \Delta, \xi)$ yield linear equations
on the $C_{i,j,b,b'}$'s over $\Z$ that 
depend only on $(\Phi,\Delta,\xi)$
and {\em not} on $(G,T,M,\{X_a\}_{a \in \Delta})$ (or $S$).
Remarkably, these
equations over $\Z$ admit a unique solution
over any non-empty scheme $S$. The solution over $S$ arises from the unique one over $\Z$. 
\end{proof}

\begin{example}  Before we use Lemma \ref{miracle} to complete (our sketch
of) the proof of the Isogeny Theorem, we illustrate the lemma
for ${\rm{G}}_2$ (cf.\,\cite[XXIII, 3.4.1]{sga3}).
We have $\Delta = \{a_0, {b_0}\}$ with $a_0$ short and ${b_0}$ long, so 
$$\Phi^+ = \{a_0, {b_0}, a_0 + {b_0}, 2a_0 + {b_0}, 3a_0 + {b_0}, 3a_0 + 2{b_0}\}.$$ 
For any $(G,T,M,\{X_a\}_{a \in \Delta})$ 
with root system ${\rm{G}}_2$ over any non-empty scheme $S$, 
define a ``universal'' trivialization $X_c$ of $\mathfrak{g}_c$ over $S$ for every $c \in \Phi^+$ via
\begin{eqnarray*}
X_{a_0+{b_0}} &=& {\rm{Ad}}_G(n_{b_0})(X_{a_0}),\\
X_{2{a_0}+{b_0}} &=& {\rm{Ad}}_G(n_{a_0})(X_{{a_0}+{b_0}}) = {\rm{Ad}}_G(n_{a_0} n_{b_0})(X_{a_0}),\\
X_{3{a_0}+{b_0}} &=& -{\rm{Ad}}_G(n_{a_0})(X_{b_0}), \\
X_{3{a_0}+2{b_0}} &=& {\rm{Ad}}_G(n_{b_0})(X_{3{a_0}+{b_0}}) = -{\rm{Ad}}_G(n_{b_0} n_{a_0})(X_{b_0})
\end{eqnarray*}
(note the signs). 

Universality of
$u(a,c) \in \Gm(S)$ ($a \in \Delta$, $c \in \Phi^+$) is illustrated by the fact that necessarily 
$u(a_0,2a_0+b_0) = -1$, $u(a_0,3a_0+b_0) = 1$, and $u(b_0,3a_0+2b_0) = -1$ in $\Gm(S)$; that is, 
${\rm{Ad}}_G(n_{a_0})(X_{2{a_0}+{b_0}}) = -X_{{a_0}+{b_0}}$, 
${\rm{Ad}}_G(n_{a_0})(X_{3a_0+b_0}) = X_{b_0}$, 
and ${\rm{Ad}}_G(n_{b_0})(X_{3{a_0}+2{b_0}}) = -X_{3{a_0}+{b_0}}$.
Likewise, commutation relations among the parameterizations
$p_c(x) = \exp_c(x X_c)$ for the positive root groups 
involve universal constants in $\Z$ as the coefficients; e.g., 
$$p_{{a_0}+{b_0}}(y)p_{a_0}(x) = p_{a_0}(x)p_{{a_0}+{b_0}}(y) p_{2{a_0}+{b_0}}(2xy)p_{3{a_0}+{b_0}}(3x^2y) p_{3{a_0}+2{b_0}}(3xy^2),$$
\begin{equation}\label{comm3}
p_{2a_0+b_0}(y)p_{a_0}(x) = p_{a_0}(x)p_{2a_0+b_0}(y) p_{3a_0+b_0}(3xy),
\end{equation}
$$p_{3{a_0}+{b_0}}(y)p_{{b_0}}(x) = p_{b_0}(x)p_{3{a_0}+{b_0}}(y) p_{3{a_0}+2{b_0}}(-xy).$$

The universal coefficients in the commutation relations are $\pm 1$ for 
${\rm{A}}_1 \times {\rm{A}}_1$ and ${\rm{A}}_2$,
but coefficients in $\{\pm 2\}$ arise for 
${\rm{B}}_2$ and 
coefficients in $\{\pm 2, \pm 3\}$ arise for ${\rm{G}}_2$. 
An interesting consequence is that 
for ${\rm{B}}_2$ (resp.\,${\rm{G}}_2$)
there are some root groups that commute in characteristic 2 (resp.\,characteristic 3)
but not in any other characteristic. 
For example, (\ref{comm3}) gives an ``extra'' commutation among root groups
of ${\rm{G}}_2$ in characteristic 3. 
\end{example}

{\bf Step 4}. 
Returning to the (sketch of the) proof of
the Isogeny Theorem, recall that we have reduced the problem to groups of semisimple-rank 2.
Lemma \ref{miracle} implies 
that  in such cases the ``structure constants'' describing both the adjoint action of
the $n_a$'s on the positive root spaces and the commutation relations
among the positive root groups are absolute constants in $\Z$
that depend only on $(\Phi, \Delta)$ (and our enumeration of $\Delta$); they do not depend
on the base scheme $S$ or the pinned split reductive
$S$-group with root system $\Phi$.  
%This immediately yields a weak form of the Isomorphism Theorem:
%a split reductive $S$-group is determined up to isomorphism by its root datum.
%Indeed, the $S$-birational group law on the open cell is determined
%by these ``structure constants'' and the root datum
%$R = (M, \Phi, M^{\vee}, \Phi^{\vee})$ (e.g., the split maximal torus 
%is $D_S(M)$), so by the universality of these constants 
%we see that the open cell equipped with its $S$-birational group law
%only depends on $(R, \Delta)$ and the 
%enumeration of $\Delta$.

To go further and prove the Isogeny Theorem, one first
determines all $p$-morphisms among reduced semisimple root data of rank 2
(especially with $p$ a prime rather than $p = 1$); 
this is an elementary combinatorial problem
since we are only considering rank 2.  
%We have noted above
%that the $S$-birational group law is described on the open cell
%in terms of the root datum and universal structure constants in $\Z$.
Also, there is a variant on Corollary \ref{rk2map} given in \cite[XXIII, 2.5]{sga3} 
(as an immediate consequence of Theorem \ref{bigcrit}) that provides
an existence criterion for homomorphisms out of a pinned split
reductive group of semisimple-rank 2.
Combining this criterion with case-by-case arguments (depending on $\Phi$
and the associated universal structure constants in $\Z$),
one builds isogenies between pinned split reductive
$S$-groups realizing 
any $p(S)$-morphism between the reduced root data.
The details are elegantly explained in \cite[XXIII, 4.1.3--4.1.8]{sga3}.
(The hardest cases
are ${\rm{B}}_2$ over $\F_2$-schemes
with $q \in \{2^n\}_{n \ge 1}$ and 
${\rm{G}}_2$ over $\F_3$-schemes with $q \in \{3^n\}_{n \ge 1}$.)
\end{proof}
\index{Isogeny Theorem|)}

\begin{remark}
The Isogeny Theorem underlies the classification of 
``exceptional'' isogenies between connected semisimple
groups over fields.  More
specifically, in characteristic 0 all
isogenies are central, so let us
focus on connected semisimple
groups over a field $k$ with characteristic $p > 0$.
There are two evident classes of isogenies over $k$:
central isogenies and Frobenius isogenies $F_{G/k}:G \rightarrow G^{(p)}$.
It is natural to wonder if every isogeny is a composition among these.
A map factors through a Frobenius isogeny on the source
if and only if it induces the zero map on Lie algebras
(as the infinitesimal $k$-subgroups of
$G$ that have full Lie algebra are those which contain
$\ker F_{G/k}$, due to Theorem \ref{exp}), so it is equivalent to determine if there are non-central
isogenies that are nonzero on Lie algebras. This problem can be reduced to the case
$k = k_s$, so we can restrict attention to the split case,
for which the Isogeny Theorem is applicable.

By passing to simply connected central covers (Exercise \ref{scuniv}) 
and direct factors thereof, the problem becomes: when do there exist
non-central $p$-morphisms between irreducible
and reduced root data that do not factor
through ``multiplication by $p$''?  (Here we are using the Isogeny Theorem
and the fact that Frobenius isogenies in characteristic $p > 0$
correspond to ``multiplication by $p$'' on the root datum.) 
This is a purely combinatorial problem for each prime $p$,
and by considering the classification of
root systems the answer is affirmative if and only if $p \in \{2, 3\}$. 
Some explicit examples in characteristic 2 are 
classical in the theory of quadratic
forms, namely the isogenies ${\rm{SO}}_{2n+1} \rightarrow {\rm{Sp}}_{2n}$
with infinitesimal non-central commutative
kernel $\alpha_2^{2n}$ for $n \ge 1$; see \cite[Lemma 2.2]{py} for further discussion of these
isogenies.  For $n = 2$ this isogeny ${\rm{SO}}_5 \rightarrow {\rm{Sp}}_4$
gives rise to an exotic {\em endomorphism} of ${\rm{Sp}}_4$  since
${\rm{Spin}}_5 = {\rm{Sp}}_4$ (as ${\rm{B}}_2 = {\rm{C}}_2$; see
Example \ref{so5}), and this is a ``square root'' of the Frobenius
isogeny over $\F_2$.  Similarly, 
one gets an exotic endomorphism of ${\rm{F}}_4$ in characteristic 2 and 
of ${\rm{G}}_2$ in characteristic 3.  These endomorphisms
underlie the existence of the Suzuki and Ree groups
in the classification of finite simple groups. 
\end{remark}

We end this section with some interesting applications of the full faithfulness of (\ref{isomfunctor})
that is a consequence of the proved Isogeny Theorem.
(See \cite[XXIII, \S5]{sga3} for a more extensive discussion.)

\begin{proposition}\label{fpqc}
Let $G$ and $G'$ be reductive groups over a non-empty scheme $S$.
\begin{enumerate}
\item If $G$ and $G'$ are isomorphic fpqc-locally on $S$  then they are so 
\'etale-locally on $S$.
\item Assume that $G$ and $G'$ are isomorphic 
\'etale-locally on $S$, that $S$ is connected with ${\rm{Pic}}(S) = 1$, and 
that $G$ and $G'$ have respective split maximal tori $T$ and $T'$. 
The pairs $(G,T)$ and $(G',T')$ are isomorphic,
as are the triples $(G,B,T)$ and $(G',B',T')$
for Borel subgroups $B \supset T$ and $B' \supset T'$.
If $G' = G$ and $S$ is also affine $($so ${\rm{H}}^1(S, \calO_S) = 0$$)$ then these isomorphisms can be chosen
to arise from $G(S)$-conjugation.
\end{enumerate}
\end{proposition}

\begin{proof}
For (1) we may work \'etale-locally on $S$ to reach the split case
with the same root datum.  Then we can apply the full faithfulness of (\ref{isomfunctor}). 
For (2), fix isomorphisms $T \simeq D_S(M)$ and $T' \simeq D_S(M')$
for finite free $\Z$-modules $M$ and $M'$.
Since $S$ is connected, so constant sheaves on $S$ have only constant
global sections, $M = \Hom_{S\mbox{-}{\rm{gp}}}(T,\Gm)$
and similarly for $M'$ and $T'$.  Likewise, 
the root spaces for $(G,T)$ and $(G',T')$ are free of rank 1 since
${\rm{Pic}}(S) = 1$.  Thus, $(G,T,M)$ and $(G',T',M')$ are split.  In particular,
Borel subgroups $B \supset T$ and $B' \supset T'$ do exist; choose such $S$-subgroups.

The connectedness of $S$ ensures that the choices
for $B$ correspond bijectively to the positive systems of
roots in $\Phi \subset M$, and similarly for $B'$ (see Remark \ref{borelwarn}).
Hence, by choosing suitable pinnings, the full faithfulness of (\ref{isomfunctor})
provides an isomorphism $(G,B,T) \simeq (G',B',T')$.
In the special case $G' = G$ with $S$ also affine, it remains to show that 
$G(S)$ acts transitively on the set of pairs $(B,T)$.

Let $B$ be a Borel subgroup of $G$, 
so the orbit map $G \rightarrow {\rm{Bor}}_{G/S}$
through $B$ identifies ${\rm{Bor}}_{G/S}$ with $G/B$ 
by Corollary \ref{parfiber}(1) (since $N_G(B) = B$, by Corollary \ref{parnorm}). 
The $S$-group $B$ has a composition series whose successive
quotients are $\Add$ and $\Gm$, so the vanishing
of both ${\rm{H}}^1(S_{\et},\Add) = 
{\rm{H}}^1(S_{\rm{Zar}},\mathscr{O}_S)$ (as $S$ is affine) and 
${\rm{H}}^1(S_{\et},\Gm) = {\rm{H}}^1(S_{\rm{Zar}},\mathscr{O}_S^{\times})$ (as ${\rm{Pic}}(S) = 1$)
implies that ${\rm{H}}^1(S_{\et},B) = 1$ (concretely, every \'etale $B$-torsor over $S$
is split).  Hence, the $B$-torsor $G \rightarrow G/B$ induces a surjection $G(S) \rightarrow (G/B)(S)$,
so $G(S)$ acts transitively on the set of Borel $S$-subgroups of $G$.

As we saw above, any split maximal torus $T$ in $G$ 
lies in some Borel subgroup $B$.  In view of the $G(S)$-conjugacy of Borel $S$-subgroups of $G$, it remains to 
show that any two split maximal
tori $T, T' \subset G$ contained in $B$ are $B(S)$-conjugate.
Any such tori are $B$-conjugate \'etale-locally on $S$
(Proposition \ref{xi2.4bis}), so ${\rm{Transp}}_B(T,T')$
is a torsor over $S_{\et}$ for $N_B(T) = B \bigcap N_G(T) = T$ (the final equality due to
$N_G(T)/T$ being the finite constant group for $W(\Phi)$,
with $W(\Phi)$ acting simply transitively on the set of $B \supset T$
on geometric fibers over $S$).   But $T = \Gm^r$ and
${\rm{Pic}}(S) = 1$, so we are done.
\end{proof}

The Existence Theorem over $\C$ is well-known in the classical theory
(see Appendix \ref{csec}). 
As an application of the full faithfulness in (\ref{isomfunctor}), we now 
improve on the Existence Theorem over $\C$ by pushing it down to $\Q$.
This will be an ingredient in the proof of the Existence Theorem in general.

\begin{proposition}\label{existq}
For each reduced root datum $R$, there exists a split connected
reductive $\Q$-group $(G,T)$ such that $R(G,T) \simeq R$.
\end{proposition}

\begin{proof}
Choose a split connected reductive $\C$-group $(G,T,M)$ having $R$ as its root datum. 
Writing $\C = \varinjlim A_i$ for finite type $\Q$-subalgebras $A_i$,
the triple $(G,T,M)$ descends to a split reductive group scheme
$(\mathscr{G}, \mathscr{T},M)$ over $\Spec A$ for some finite type $\Q$-subalgebra
$A = A_{i_0} \subset \C$.  Its root datum is $R$, so by passing to the fiber
at a closed point we find a split triple $(G',T',M)$ over a number field
$F$ with root datum
isomorphic to $R$; fix
this isomorphism.  

By replacing $F$ with a finite extension we may assume that $F$ is Galois over $\Q$.
We will now carry out Galois descent down to $\Q$ via the crutch of a pinning.
Choose a positive system of roots $\Phi^+$ in $\Phi = \Phi(G',T')$,
and let $\Delta$ be the corresponding base.  For each $a \in \Delta$, pick
a basis $X_a$ of the $F$-line $\mathfrak{g}'_a$, so
we get a pinned split reductive group
$(G',T',M,\{X_a\}_{a \in \Delta})$. 
We have a chosen isomorphism $\phi:R(G',T',M) \simeq R$,
and for all $\gamma \in \Gamma := {\rm{Gal}}(F/\Q)$
we get another pinned split reductive $F$-group
$$(\gamma^{\ast}(G'),\gamma^{\ast}(T'), M, \{\gamma^{\ast}(X_a)\}_{a \in \Delta}))$$
via the evident identifications $\Phi(\gamma^{\ast}(G'), \gamma^{\ast}(T')) \simeq
\Phi(G',T') = \Phi \subset M - \{0\}$ and ${\rm{X}}(\gamma^{\ast}(T')) \simeq
{\rm{X}}(T') = M$ (defined by functoriality of scalar extension along
the $\Gamma$-action on $F$).   

It is easy to check that the resulting isomorphisms of root data
\begin{equation}\label{rootgal}
R(G',T',M) \simeq R(\gamma^{\ast}(G'),\gamma^{\ast}(T'), M, \{\gamma^{\ast}(X_a)\}_{a \in \Delta})
\end{equation}
satisfy the cocycle condition.
By the full faithfulness of (\ref{isomfunctor}), the isomorphism (\ref{rootgal}) arises from
a unique isomorphism between {\em pinned} split reductive $F$-groups
$$(G',T',M,\{X_a\}_{a \in \Delta}) \simeq
(\gamma^{\ast}(G'),\gamma^{\ast}(T'), M, \{\gamma^{\ast}(X_a)\}_{a \in \Delta}),$$
and the uniqueness implies that these isomorphisms
inherit the cocycle condition from that aspect of the isomorphisms of root data.
Note that these isomorphisms between pinned groups use the identity
automorphism on $M$, so they use the identity bijection on $\Delta$.
(That is, $X_a$ is carried to $\gamma^{\ast}(X_a)$.) 
Hence, by Galois descent we obtain a pinned
{\em split} reductive $\Q$-group descending $(G',T',M,\{X_a\}_{a \in \Delta})$,
and its root datum is clearly $R$.
\end{proof}


\subsection{Existence Theorem}\label{exthm}

\index{Existence Theorem|(}
Let $R = (M,\Phi,M^{\vee},\Phi^{\vee})$ 
be a reduced
root datum. By base change, to prove the Existence Theorem for $R$ (i.e., Theorem \ref{mainthm}(2))
over an arbitrary non-empty scheme $S$ 
it suffices to treat the case  $S = \Spec \Z$.  
By the following lemma, whose proof is a formal argument with root data, 
it  suffices to consider only $R$ that is semisimple
and simply connected 
(i.e., $\Phi$ spans $M_{\Q}$ over $\Q$, and $\Phi^{\vee}$ spans $M^{\vee}$ over $\Z$)
and such that the root system associated to $R$ is irreducible.  


\begin{lemma}\label{existlemma} To prove the Existence Theorem over a non-empty scheme $S$, 
it suffices to treat semisimple root data $(X,\Phi,X^{\vee},\Phi^{\vee})$
that are simply connected
and have associated root system $(X_{\Q}, \Phi)$ that is irreducible.
\end{lemma}

\begin{proof}
The idea is to use a preliminary central isogeny of root
data to separate the maximal central torus from the derived group,
and  then to  treat tori and semisimple groups separately. 
Let $R = (X,\Phi,X^{\vee},\Phi^{\vee})$ be a root datum, so
$X$ contains $\Z  \Phi \bigoplus (\Z \Phi^{\vee})^{\perp}$ with finite index,
where the annihilator $(\Z  \Phi^{\vee})^{\perp}$ in $X$ is saturated but
$\Z  \Phi$ may not be saturated.   In general, if
$L \rightarrow L'$ is an injective map between
finite free $\Z$-modules, we write $L_{\rm{sat}}$ to denote the saturation of
$L$ in $L'$ (i.e., the kernel of $L' \rightarrow (L'/L)_{\Q}$).
The natural map
$$X \rightarrow (X/(\Z  \Phi)_{\rm{sat}}) \oplus (X/(\Z  \Phi^{\vee})^{\perp}) =: X'$$
is a finite-index inclusion that carries $\Phi$ onto a subset $\Phi'$
that lies in the second summand of $X'$.
The $\Z$-dual of $X'$ is naturally identified with the direct sum
$${X'}^{\vee} = (\Z  \Phi)^{\perp} \oplus (\Z  \Phi^{\vee})_{\rm{sat}}$$
and ${\Phi'}^{\vee}$ is defined to be the image of
$\Phi^{\vee}$ under inclusion  into the second factor.  

Clearly 
$R' := (X',\Phi', {X'}^{\vee}, {\Phi'}^{\vee})$ is a reduced root datum,
and the natural isogeny
$(h,d,q):R' \rightarrow R$ is ``central'':  $q(a') = 1$ for all $a' \in \Phi'$.
If $R'$ arises from a split reductive $S$-group
$(G',T')$ then the cokernel of $h:X \rightarrow X' = {\rm{X}}(T')$
corresponds to a split finite multiplicative type $S$-subgroup $\mu \subset T'$ 
such that ${\rm{X}}(T'/\mu) = X$ inside ${\rm{X}}(T') = X'$.
In particular, the centrality of $(h,d,q)$ implies that all roots of $(G',T')$ lie
in ${\rm{X}}(T'/\mu)$, which is to say $\mu$ is a central
$S$-subgroup of $G'$ (Corollary \ref{computez}(1)).
Hence, the central quotient $G := G'/\mu$ makes sense as a 
reductive $S$-group
in which $T := T'/\mu$ is a split maximal torus. 
The inclusion $h:X \hookrightarrow X'$ carries
$\Phi$ onto $\Phi'$ (as $q$ is identically 1) and the dual map ${X'}^{\vee} \rightarrow X^{\vee}$
carries ${\Phi'}^{\vee}$ onto $\Phi^{\vee}$ (due to the unique characterization of
coroots for a root system), so
the root datum $R(G,T)$ is identified with
$(X, \Phi, X^{\vee}, \Phi^{\vee})$. 
Now it suffices to treat $R'$ instead of $R$.

Let $L = X/(\Z  \Phi^{\vee})^{\perp}$, 
$L^{\vee} = (\Z  \Phi^{\vee})_{\rm{sat}}$,
$\Psi = \Phi \bmod (\Z  \Phi^{\vee})^{\perp} \subset L$, and $\Psi^{\vee} = \Phi^{\vee} \subset L^{\vee}$, so 
$R'' := (L, \Psi, L^{\vee},\Psi^{\vee})$ is a semisimple reduced root datum and
$$R' = R'' \oplus (X/(\Z  \Phi)_{\rm{sat}}, \emptyset, (\Z  \Phi)^{\perp}, \emptyset).$$
It suffices to treat the two summands separately (as we can then form the direct product
of the corresponding split connected reductive groups).  The second summand is trivially handled 
by using the split torus with character group $X/(\Z  \Phi)_{\rm{sat}}$, so
we may now focus our attention on $R''$.  That is, we may assume that
our root datum $R$ is semisimple.  

As in (\ref{xphi}), we have $\Z \Phi \subset X \subset (\Z \Phi^{\vee})^{\ast}$.
Let $X' = (\Z \Phi^{\vee})^{\ast}$, $\Phi' = \Phi$, ${X'}^{\vee} = \Z \Phi^{\vee}$,
and ${\Phi'}^{\vee} = \Phi^{\vee}$, so
$R' = (X', \Phi', {X'}^{\vee}, {\Phi'}^{\vee})$ is a root datum that is
semisimple and simply connected.  There is an evident central isogeny of
root data $R' \rightarrow R$, so by repeating the central quotient construction above
we see that the Existence Theorem for $R$ over $S$ is reduced to 
the Existence Theorem for $R'$ over $S$.  Thus, it remains to treat the 
case of semisimple root data that are simply connected.
The equality $X^{\vee} = \Z \Phi^{\vee}$ ensures
that the decomposition of the root system into its irreducible components
is also valid at the level of the root datum.
Hence, it remains to settle the case of semisimple root data
that are simply connected and have an irreducible associated root system.
This is precisely the case that we are assuming is established.
\end{proof}


Fix a semisimple reduced root datum $R$ that is simply connected. (We will not require irreducibility for $R$.)
Proposition \ref{existq} provides
a split reductive group $(G,T,M)$ over $\Q$ with root datum $R$. This yields 
the Existence Theorem for $R$ over some $\Z[1/N]$, but $N$ might depend
on $R$.  The problem is to get the result over the entirety
of $\Spec \Z$, not ignoring any small primes.  The rest of \S\ref{exthm} is devoted to the construction of such
a split $\Z$-group by a method that works uniformly across all (simply connected
and semisimple) $R$.
In view of the classification of irreducible and reduced root systems,
it would suffice to exhibit an explicit example for each Killing--Cartan type.

For the classical types ${\rm{A}}_n$ ($n \ge 1$), ${\rm{B}}_n$ ($n \ge 3$), ${\rm{C}}_n$ ($n \ge 2$), and 
${\rm{D}}_n$ ($n \ge 4$) 
we can use the $\Z$-groups
${\rm{SL}}_{n+1}$ ($n \ge 1$), ${\rm{Spin}}_{2n+1}$ ($n \ge 3$), ${\rm{Sp}}_{2n}$
($n \ge 2$),
and ${\rm{Spin}}_{2n}$ ($n \ge 4$) respectively.  
(To make sense of spin groups over $\Z$ and not just over $\Z[1/2]$, we need
a characteristic-free viewpoint on non-degeneracy for quadratic spaces over rings.
This is discussed in Appendix \ref{sosec}.) 
Thus, the arguments that follow
are only needed to handle the exceptional types ${\rm{E}}_6$, ${\rm{E}}_7$, ${\rm{E}}_8$,
${\rm{F}}_4$, and ${\rm{G}}_2$.  Even some of these types can be settled by direct construction
(e.g., ${\rm{G}}_2$ and
${\rm{F}}_4$ can be handled
by using octonion and Jordan algebras over $\Z$).  Explicit constructions can require
special care at small primes (e.g., residue characteristic 2 
for spin groups and type ${\rm{F}}_4$, and residue characteristics 2 and 3 for 
type ${\rm{G}}_2$).  The uniform
approach below is insensitive to the peculiar demands of small primes
or of specific irreducible root systems.

Before we take up the proof of the Existence Theorem,
we need to digress and discuss the following concept:

\index{Chevalley system|(}
\begin{definition}\label{chevsystem} Let $(G,T,M)$ be a split reductive group over
a non-empty scheme $S$.  A  {\em Chevalley system} for
$(G,T,M)$ is a collection of trivializing sections $X_a \in \mathfrak{g}_a(S)$
for all $a \in \Phi$ so  that $${\rm{Ad}}_G(w_a(X_a))(X_b) = \pm X_{s_a(b)}$$
for all $a, b \in \Phi$, where the sign ambiguity is global over $S$ 
(possibly depending on $a$ and $b$) and 
$$w_c(X) := \exp_c(X) \exp_{-c}(-X^{-1}) \exp_c(X) \in N_G(T)(S)$$
for every trivializing section $X$ of $\mathfrak{g}_c$ 
and every $c \in \Phi$.
\end{definition}

The existence of a Chevalley system is vacuous for semisimple-rank 0,
and for semisimple-rank 1
we can build one by using any $X_a$ whatsoever and defining $X_{-a} := X_a^{-1}$
to be the linked trivialization of $\mathfrak{g}_{-a}$ (this works,  
since ${\rm{Ad}}_G(w_a(X))(X) = -X^{-1}$ for any trivializing
section $X$ of $\mathfrak{g}_a$; see Corollary \ref{wsec}(3)).
By setting $b = a$ and using that $s_a(a) = -a$, it likewise follows
that for any Chevalley system $\{X_a\}_{a \in \Phi}$ we must have 
$X_{-a} = \pm X_a^{-1}$ (i.e., $X_a$ and $X_{-a}$ are linked, up to a global sign depending on $a$).

\begin{example}
Chevalley systems are closely related to the notion
of a ``Chevalley basis'' for a complex semisimple Lie algebra (cf.\,\cite[25.1--25.2]{humlie}).
\index{Chevalley basis}
To explain this, consider a connected semisimple $\C$-group $G$ equipped with 
maximal torus $T$, so $\mathfrak{g} := {\rm{Lie}}(G)$ is a semisimple Lie algebra
and $\mathfrak{t} := {\rm{Lie}}(T)$ is a Cartan subalgebra.  Fix a positive system of roots
in $\Phi(G,T) = \Phi(\mathfrak{g}, \mathfrak{t})$, 
and let $\Delta$ be the corresponding set of simple roots,
so the vectors $v_a = {\rm{Lie}}(a^{\vee})(\partial_t|_{t=1})$ with $a \in \Delta$ 
are a basis of $\mathfrak{t}$. Let $X_c$ be a basis of $\mathfrak{g}_c$
for each $c \in \Phi$, so the collection of
$v_a$'s and $X_c$'s is a basis of 
$\mathfrak{g}$.  

Let's introduce the associated ``structure constants''. 
For $a \in \Delta$ and $c \in \Phi$, 
we have $[v_a, X_c] = {\rm{Ad}}_G(v_a)(X_c) = \langle c, a^{\vee}\rangle X_c$
since conjugation by $a^{\vee}(t)$ on $U_c = \Add$ acts via scaling by $c(a^{\vee}(t)) = 
t^{\langle c, a^{\vee}\rangle}$.
Consider $c, c' \in \Phi$ with $c' \ne \pm c$.  There is a positive system of
roots containing $c$ and $c'$, so by Proposition \ref{commrel}
the groups $U_c$ and $U_{c'}$ commute
if $c + c' \not\in \Phi$ (forcing $[X_c, X_{c'}] = 0$)
and otherwise $[X_c, X_{c'}] = r(c,c')X_{c+c'}$ for some $r(c,c') \in \C$.
The special feature of
$\{X_c\}_{c \in \Phi}$ being a Chevalley system is that
the numbers $r(c,c')$ are (nonzero) {\em integers} that are moreover determined
up to sign by the root system; see 
Remark \ref{chevrule}.   This provides
an explicit $\Z$-form for every complex semisimple Lie algebra,
and Chevalley used this viewpoint to construct adjoint 
split semisimple $\Z$-groups (see Theorem \ref{adimm},
as well as \cite[25.5, \S26]{humlie}). 
\end{example}

\begin{proposition}\label{chevsys}
Let $(G,T,M, \{X_a\}_{a \in \Delta})$ be a pinned split reductive
group over a non-empty scheme $S$.
There is a Chevalley system
$\{X_c\}_{c \in \Phi}$ extending the pinning,
and each $X_c$ is unique up to a global sign.
\end{proposition}

For semisimple-rank 2,
the main computations for the construction of
a Chevalley system were carried out in the proof of the Isogeny Theorem, but more work
is required even for semisimple-rank 2 (since the definition
of a Chevalley system involves the adjoint action for $w_a(X_a)$ for all $a \in \Phi$).  

\begin{proof}
Let $n_a = w_a(X_a)$
for all $a \in \Delta$. 
Since every element of $W(\Phi)$ is represented by a product
among the elements of $\{n_a\}_{a \in \Delta}$,  for any $c \in \Phi$
we can find such a product $n$ representing an element $w \in W(\Phi)$
so that $w(a) = c$ for some $a \in \Delta$ (i.e., $w^{-1}(c) \in \Delta$).  Thus, 
$X_c = \pm {\rm{Ad}}_G(n)(X_a)$ with a global sign ambiguity.
This shows the uniqueness of each $X_c$ up to a global sign.

To prove existence, we begin by running the uniqueness proof in reverse.
For each $c \in \Phi$ not in
$\Delta$, choose some $w \in W(\Phi)$ such that $w^{-1}(c) \in \Delta$.
Pick a word $a_1 \cdots a_m$ in elements $a_i$ of $\Delta$ so that
$s_{a_1} \cdots s_{a_m} = w$ in $W(\Phi)$.
For the element $n := n_{a_1} \cdots n_{a_m} \in N_G(T)(S)$, use 
the isomorphism ${\rm{Ad}}_G(n): \mathfrak{g}_{w^{-1}(c)} \simeq \mathfrak{g}_c$
to define $$X_c := {\rm{Ad}}_G(n)(X_{w^{-1}(c)}).$$
It suffices to show that $\{X_c\}_{c \in \Phi}$ is a Chevalley system.
(The definition of the $X_c$'s depends on the choice of
$w$ and the word $a_1 \cdots a_m$. However, once the proof is
done, it will follow that changing these choices affects each $X_c$ by at most a global sign.) 

For $c \in \Phi$ and $b \in \Delta$, we need to prove that 
\begin{equation}\label{adgx}
{\rm{Ad}}_G(n_b)(X_c) = \pm X_{s_b(c)}.
\end{equation}
By definition, $X_c = {\rm{Ad}}_G(n_0)(X_{a_0})$ for some $a_0 \in \Delta$
and product $n_0$ among the elements of $\{n_a\}_{a \in \Delta}$ so that
$n_0$ represents an element $w_0 \in W(\Phi)$
satisfying $w_0(a_0) = c$.  Likewise, $X_{s_b(c)} = {\rm{Ad}}_G(n_1)(X_{a_1})$
for some $a_1 \in \Delta$ and product $n_1$ among the elements of $\{n_a\}_{a \in \Delta}$
so that $n_1$ represents an element $w_1 \in W(\Phi)$
satisfying $w_1(a_1) = s_b(c) = (s_b w_0)(a_0)$.  Thus,
$n := n_0^{-1}n_b^{-1}n_1$ represents some 
$w \in W(\Phi)$ satisfying $w(a_1) = a_0$, 
and our problem is to show that ${\rm{Ad}}_G(n)(X_{a_1}) = \pm X_{a_0}$
(as then applying ${\rm{Ad}}_G(n_b n_0)$ to both sides will give
$$X_{s_b(c)} = {\rm{Ad}}_G(n_1)(X_{a_1}) = \pm {\rm{Ad}}_G(n_b)({\rm{Ad}}_G(n_0)(X_{a_0})) = 
\pm {\rm{Ad}}_G(n_b)(X_c)$$
as desired). 

Although $n = n_0^{-1}n_b^{-1}n_1$ is not written as a product among
the $\{n_a\}_{a \in \Delta}$, due to the intervention of some inversions, 
these inversions can be absorbed into the sign ambiguity in (\ref{adgx}). 
The reason is as follows.  We have $n_a^{-1} = a^{\vee}(-1)n_a$
with $a^{\vee}(-1) \in T[2]$, and for any $t \in T$ and $a' \in \Delta$ the identity
$$t n_{a'} t^{-1} = t w_{a'}(X_{a'}) t^{-1} = w_{a'}(a'(t)X_{a'}) = 
{a'}^{\vee}(a'(t)) w_{a'}(X_{a'}) = t^2 n_{a'}$$
(using Corollary \ref{wsec}(2)) implies that $T[2]$ centralizes $n_{a'}$.
In particular, for $n = n_0^{-1}n_b^{-1}n_1$
and $n_0^{\rm{opp}} := n_{a_1} \cdots n_{a_m}$
where $n_{a_1} \cdots n_{a_m} = n_0$ with $a_i \in \Delta$, 
the product $n' := n_0^{\rm{opp}} n_b n_1$ among the elements of $\{n_a\}_{a \in \Delta}$ 
lifts the {\em same} word in the involutions $s_a$ as does $n$ and we have 
$n = t n'$ with $t := \lambda(-1)$ for some
$\lambda \in M^{\vee} \in \Hom_{S\mbox{-}{\rm{gp}}}(\Gm,T)$.  Thus, 
${\rm{Ad}}_G(n) = {\rm{Ad}}_G(t) \circ {\rm{Ad}}_G(n')$, 
and the effect of ${\rm{Ad}}_G(t)$ on each $\mathfrak{g}_c$ is
scaling by $c(t) = (-1)^{\langle c, \lambda \rangle} = \pm 1$.

To summarize, we are reduced to proving a general fact
about words in the elements $n_a$:
if $a, a' \in \Delta$ and $\{a_1,\dots,a_{m}\}$ is a sequence in $\Delta$
such that $(s_{a_m} \circ \dots \circ s_{a_1})(a) = a'$
then 
$${\rm{Ad}}_G(n_{a_m} \circ \dots \circ n_{a_1})(X_a) \stackrel{?}{=} \pm X_{a'}$$
in $\mathfrak{g}(S)$, for some global sign $\pm 1$. 
Note that this equality is obvious when
the semisimple-rank is at most 1, and it is also obvious 
when the pinning extends to a Chevalley system.
Thus, to settle it for all cases with semisimple-rank 2 we just need to 
construct {\em some} Chevalley system extending
the pinning in every case.  But the rank-2 calculations in the proof
of Lemma \ref{miracle} (for the aspect concerning the units
$u(a,c)$) achieve exactly this!
To be precise, those calculations construct 
a ``positive'' Chevalley system: 
a collection of trivializations $\{X_c\}_{c \in \Phi^+}$
of the positive root spaces that
extends the pinning and satisfies
${\rm{Ad}}_G(n_a)(X_c) = \pm X_{s_a(c)}$
for any $a \in \Delta$ and $c \in \Phi - \{a\}$.
Hence, to settle the case of semisimple-rank 2
we just need to extend any ``positive'' Chevalley system
$\{X_c\}_{c \in \Phi^+}$ 
to an actual Chevalley system.  

Define $X_{-c} = X_c^{-1}$ for all $c \in \Phi^+$.
We claim that $\{X_c\}_{c \in \Phi}$
is a Chevalley system.  This amounts
to checking that ${\rm{Ad}}_G(n_a)(X_c) = \pm X_{s_a(c)}$
for all $c \in \Phi$.  The cases $(a,c)$ and $(a,-c)$ are equivalent, 
by the functoriality of duality of opposite root spaces with respect
to $n_a$-conjugation on $G$.  Hence, we may assume
$c \in \Phi^+$.  The case $c \in \Phi^+ - \{a\}$ is known by hypothesis
(as $s_a(c) \in \Phi^+$ for all $c \in \Phi^+ - \{a\}$), 
and ${\rm{Ad}}_G(n_a)(X_a) = -X_a^{-1}$ by Corollary \ref{wsec}(2).
This completes the argument for semisimple-rank 2. 

For the case of semisimple-rank $> 2$, one needs to apply 
several results in the theory of
root systems and make artful use of presentations of Weyl groups as reflection groups
to ultimately reduce to the settled case of semisimple-rank 2.
We refer to \cite[XXIII, 6.3]{sga3} for
the details, which rest on two ingredients: many of the case-by-case ``universal'' formulas established 
for pinned split reductive groups with a root system of rank 2 in \cite[XXIII, 3.1--3.4]{sga3}, 
and root system arguments from \cite[XXIII, 2.3]{sga3} that were used
in the proof of Theorem \ref{bigcrit}.
\end{proof}
\index{Chevalley system|)}

\begin{remark}\label{chevrule} 
A very useful application of the existence of Chevalley systems
is ``Chevalley's rule'' \cite[XXIII, 6.5]{sga3} that computes -- up to a sign -- universal formulas
for the structure constants in the Lie algebra of a split semisimple
group scheme $(G,T,M)$ relative to 
a Chevalley system in the Lie algebra. Explicitly, if
$\{X_c\}_{c \in \Phi}$ is a Chevalley system 
for $(G,T,M)$ then 
$[X_a,X_b] = \pm (p(a,b)+1) X_{a+b}$
whenever $a, b, a + b \in \Phi$,
where $p(a,b)$ is the greatest integer $z \ge 0$
such that $a - zb \in \Phi$. 
This result is proved by 
inspecting the universal constants
in Lemma \ref{miracle} for the commutation relations among
positive root groups in pinned split reductive groups
with semisimple-rank 2. 
\end{remark}

Returning to the proof of the Existence Theorem, we have reduced to the
task of extending a split reductive $\Q$-group $(G,T,M)$
to a split reductive $\Z$-group.
We have also seen that it suffices to treat cases
in which the root datum $R = (M, \Phi, M^{\vee}, \Phi^{\vee})$ for $(G,T,M)$ is semisimple 
and simply connected 
(i.e., $\Phi$ spans $M_{\Q}$ over $\Q$ and $\Phi^{\vee}$ spans $M^{\vee}$ over $\Z$).
%and we can assume moreover that the associated root system is irreducible.  

To construct the required split semisimple $\Z$-group
extending $(G,T,M)$, choose a base $\Delta$ of $\Phi$
and a pinning $\{X_a\}_{a \in \Delta}$ of $(G,T,M)$.
Using Proposition \ref{chevsys}, extend this to
a Chevalley system $\{X_c\}_{c \in \Phi}$.
Since $X_{-c} = \pm X_c^{-1}$ for all $c$, 
by using sign changes if necessary we may and do
arrange that $X_{-c} = X_c^{-1}$ for all $c \in \Phi$.

\begin{lemma}\label{uzlemmaprelim}
Let $T = D_S(M)$ be a split torus over a scheme $S$
and let $U$ be a smooth affine $S$-group with unipotent fibers  on which
$T$ acts.  Assume that $U$ contains a finite collection of $T$-stable $S$-subgroups
$U_i = \mathbf{W}(\mathscr{E}_i)$ $(i \in I)$ 
on which $T$ acts through nontrivial characters $a_i \in M \subset {\rm{X}}(T)$ that are pairwise linearly independent.

If the multiplication map $\prod U_i \rightarrow U$ for one enumeration of $I$ is an $S$-scheme isomorphism
then it is so for any enumeration of $I$; i.e., the $U_i$'s directly span $U$ in any order.
\end{lemma}

\begin{proof}
By the fibral isomorphism criterion it suffices to work on fibers, so we may assume $S = \Spec k$ for a field $k$.
The assertion is a special case of a general dynamical ``direct spanning in any order'' result
 in \cite[3.3.11]{pred}.
\end{proof}

Fix an enumeration $\{a_1, \dots, a_r\}$ of $\Delta$.
Use this to define the lexicographical ordering on
$M_{\Q}$, so we get an ordering $\{c_0, \dots, c_m\}$ of $\Phi^+$. 
Identify the unipotent radical $U_{+} := U_{\Phi^+}$ of
the Borel subgroup $B = T \ltimes U_{+}$ corresponding
to $\Phi^+$ with a direct product (as $\Q$-schemes) of the positive root groups $U_c$
via the chosen ordering $\{c_j\}$ on $\Phi^+$.  Let $U_{-} = U_{-\Phi^+}$.

The coroots in $\Delta^{\vee}$ are a $\Z$-basis for the cocharacter group $M^{\vee}$
of $R$, due to $R$ being simply connected, so 
we get an isomorphism $\Gm^{\Delta} \simeq T$ over $\Q$
via $(t_a)_{a \in \Delta} \mapsto
\prod a^{\vee}(t_a)$.
For each $c \in \Phi$, use $X_c$ to identify $U_c$ with $\Add$ via $p_c: x \mapsto \exp_c(x X_c)$,
so the open cell $\Omega = U_{-} \times T \times U_{+}$
in the pinned split group $(G,T,M,\{X_a\}_{a \in \Delta})$ 
over $\Q$ is identified with a product of $\Gm$'s
and $\Add$'s as a $\Q$-scheme:
\begin{equation}\label{zscheme}
\prod_{j=-m}^0 \Add \times \prod_{a \in \Delta} \Gm \times \prod_{i=0}^m \Add \simeq
U_{-} \times T \times U_{+} = \Omega
\end{equation}
via $$((x'_j), (t_a)_{a \in \Delta}, (x_i)) 
\mapsto \prod_{j=-m}^0 p_{-c_{-j}}(-x'_{j}) \cdot \prod_{a \in \Delta}
a^{\vee}(t_a) \cdot \prod_{i=0}^m p_{c_i}(x'_i)$$
in which the product description for $U_{+}$ uses the ordering on $\Phi^+$
and the one for $U_{-}$ uses the  opposite ordering on $-\Phi^+$.
(The specific choice of ordering for the product description of $U_{\pm}$ will eventually
turn out not to matter, but we need to make some definite choice at the outset.)

\begin{lemma}\label{zgp} The isomorphisms $\prod_{j=-m}^0 \Add \simeq U_{-}$ and 
$\prod_{i=0}^m \Add \simeq U_{+}$ of $\Q$-schemes as defined above carry the $\Q$-group structures
on $U_{\pm}$ over to $\Q$-group structures on $\Add^{m+1}$ that are defined over $\Z$.
\end{lemma}

\begin{proof}
Observe that via the given choice of ordering, the resulting $\Z$-structure
$\mathscr{U}_{+}$ on $U_{+}$ admits an evident action by
$\mathscr{T} := \Gm^{\Delta}$ extending the natural action on
$U_{+}$ over $\Q$.  Hence, once
$U_{+}$ is settled for the initial choice of ordering
on $\Phi^+$, by Lemma \ref{uzlemmaprelim} the same holds
for $U_{+}$ using {\em any} ordering on $\Phi^+$ to define the
identification of $U_{+}$ with the $\Q$-scheme $\Add^{m+1}$. 

The element $n = n_{a_1} \cdots n_{a_r} \in N_G(T)(\Q)$ represents
the long Weyl element $w = s_{a_1} \cdots s_{a_r} \in W(\Phi)$ relative
to $\Delta$, so $n$-conjugation carries
$U_{+}$ to $U_{-}$ but the bijection $\Phi^+ \simeq -\Phi^+$
defined by the $w$-action need not carry the ordering on $\Phi^+$ to an easily described
ordering on $-\Phi^+$ (since the bijection $\Delta \simeq -\Delta$
may be hard to understand).   Regardless,  since
$\{X_c\}$ is a Chevalley system, so  
$n p_c(x) n^{-1} = \exp_{w(c)}({\rm{Ad}}_G(n)(x X_c)) = p_{w(c)}(\pm x)$ for some universal sign
(depending only on $n$, $c$, and the choice of Chevalley system),
it follows that the result for $U_{-}$ is a formal consequence of the result for $U_{+}$
(for all enumerations of $\Phi^+$!).
Hence, we may and do now focus on the case of $U_{+}$.

Consider the $\Q$-scheme isomorphism $U_{+} \simeq \Add^{m+1}$ as
defined in (\ref{zscheme}); we shall use this to equip $U_{+}$ with a $\Z$-group structure.  Clearly 
the identity section of $U_{+}$ is defined over $\Z$,
and inversion on $U_{+}$ corresponds to reversing 
the order of multiplication and replacing each $p_c(x)$ with $p_c(-x)$,
so it suffices to check that the multiplication law on $U_{+}$ is defined over
$\Z$ (as then the inversion on $U_{+}$ is defined over $\Z$, and all of
the group scheme diagrams commute over $\Z$ since they commute over $\Q$).
In other words, we can focus on the multiplication law
and not dwell on inversion. 

The multiplication description of $U_{+}$ as a product of root groups over $\Q$ is defined relative
to the lexicographical order on $\Phi^+$ using some ordering of $\Delta$. Thus, 
for each $1 \le i \le m$, the product $U_{\ge c_i} := U_{c_i} \cdots U_{c_m}$
is a closed $\Q$-subgroup of $U$ normalized by $U_{c_{i-1}}$
(as we saw in the proof of Lemma \ref{fumap}) and this
computes the direct product structure that we have built into the $\Z$-structure.
Hence, it suffices to prove by descending induction on $i$
that the $\Q$-group law on each $U_{\ge c_i}$ is defined over $\Z$
(on the corresponding direct product of copies of $\Add$).
The base case $i = m$ is obvious, and likewise
the $\Q$-group structure on each $U_c = \Add$ is visibly defined
over $\Z$, so to carry out the induction it is enough to 
check that the conjugation action by $U_{c_{i-1}}$
on $U_{\ge c_i}$ is defined over $\Z$.  We can also assume that the semisimple-rank is at least 2
(or else there is nothing to do).

More generally, for $c, c' \in \Phi^+$ with $c < c'$,
consider the conjugation action by $p_c(x)$ on $p_{c'}(x')$,
assuming that the group law on $U_{\ge c'}$ is already known to be defined over $\Z$.  
By (\ref{rootcomm}) we have 
\begin{equation}\label{pconj}
p_c(x) p_{c'}(x') p_c(x)^{-1} = \prod_{i \ge 0, j > 0} p_{ic + jc'}(C_{i,j,c,c'} x^i {x'}^j)
\end{equation}
where the product (in $U_{+}$) is taken relative to the ordering on $\Phi^+$
and the coefficients $C_{i,j,c,c'}$ lie in $\Q$.  This product
lies in $U_{\ge c'}$, and it suffices to prove that
$C_{i,j,c,c'} \in \Z$ for all $(i,j)$. 
The first term on the right in (\ref{pconj}) is $p_{c'}(C_{0,1,c,c'} x')$ because
$c' < ic + jc'$ in $\Phi^+$ for all $i, j \ge 1$, and (\ref{rootcomm}) implies
that $C_{0,1,c,c'} = 1$.  Hence, our problem really concerns
the commutator $p_{c'}(-x')p_{c}(x)p_{c'}(-x')^{-1}p_{c}(x)^{-1}$.
We may now replace $G$ with $Z_G(T_{c,c'})$ to reduce to the
case of semisimple-rank 2 (keeping in mind that the ordering
on $\Phi^+$ is immaterial once
the full result is proved).

Now consider the case of pinned split groups of semisimple-rank 2.  The Chevalley system
extending a pinning is unique up to signs, so we can use whatever Chevalley system
we like that extends an initial choice of pinning.
By Lemma \ref{miracle}, when using the lexicographical ordering 
on $\Phi^+$ relative to {\em some} choice of 
enumeration of $\Delta$, the structure constants in the commutation
relations for the positive root groups are in $\Z$.  This establishes
the result in the semisimple-rank 2 case for some choice
of enumeration on $\Delta$, and it is sufficient to prove the result for one such choice.
\end{proof}

The evident $\Z$-scheme $\Omega_{\Z}$ extending the $\Q$-scheme on 
the left side of (\ref{zscheme}) has generic fiber 
$\Omega$, and it is a direct product
scheme $\mathscr{U}_{-} \times \mathscr{T} \times \mathscr{U}_{+}$
with $\mathscr{T} = \Gm^{\Delta}$ a split $\Z$-torus having cocharacter group 
$\Z \Phi^{\vee} = M^{\vee}$
and $\mathscr{U}_{\pm}$ a $\Z$-group extending $U_{\pm}$
(by Lemma \ref{zgp}).  For $c \in \Phi^+$ let $\widetilde{p}_c:\Add \rightarrow \mathscr{U}_{+}$
be the evident inclusion extending $p_c$ over $\Q$, and likewise using
$\mathscr{U}_{-}$ for $c \in -\Phi^+$,  so each $\widetilde{p}_c$ is a closed
immersion of $\Z$-groups (as we can check the homomorphism property over $\Q$);
let $\mathscr{U}_c$ denote its image. 
By construction the $\Z$-group $\mathscr{U}_{+}$ is directly
spanned in {\em some} order by the $\mathscr{U}_c$'s
for $c \in \Phi^+$, and similarly for $\mathscr{U}_{-}$
using $c \in -\Phi^+$. Likewise, the $T$-action on $\Omega$ extends
to a $\mathscr{T}$-action on $\Omega_{\Z}$
that normalizes $\mathscr{U}_{\pm}$,
as this amounts to some factorization assertions
for flat closed subschemes that can be checked over $\Q$;
more explicitly, $a^{\vee}(t) \widetilde{p}_c(x) a^{\vee}(t)^{-1} = \widetilde{p}_c(t^{\langle c, a^{\vee}\rangle} x)$
over $\Z$ since this holds over $\Q$.

By Lemma \ref{uzlemmaprelim}, the $\Z$-group $\mathscr{U}_{+}$ is directly spanned in any order
by the $\mathscr{U}_c$'s for $c \in \Phi^+$, and 
$\mathscr{U}_{-}$ is directly spanned in any order by the $\mathscr{U}_c$'s for $c \in -\Phi^+$.
(This direct spanning in any order is \cite[XXV, 2.5]{sga3}, and the proof there uses
a theory of Lazard from \cite[13.1]{bible} for torus
actions on unipotent groups.  We can bypass that theory because
the proof of Lemma \ref{uzlemmaprelim} rests on the dynamical results in \cite[3.3]{pred}.)

%
%\begin{proof}
%The cases of $\mathscr{U}_{+}$ and $\mathscr{U}_{-}$ are identical, so we 
%just discuss $\mathscr{U}_{+}$. 
%We have to show that for any enumeration of $\Phi^+$, the resulting
%multiplication map $\prod_{c \in \Phi^+} \mathscr{U}_c \rightarrow \mathscr{U}$
%between $\Z$-schemes is an isomorphism.  It suffices to check the isomorphism
%result on geometric fibers over $\Spec \Z$, so we are reduced 
%to the following general claim over an algebraically closed
%field $k$.  Let $U$ be a unipotent smooth connected affine $k$-group 
%directly spanned in some order by unipotent smooth closed 
%subgroups $U_i$.  Assume there is given an action on $U$ by a torus $T$
%normalizing each $U_i$ such that the 
% $T$-action on each ${\rm{Lie}}(U_i)$
%is through a nontrivial character $a_i$
%with $a_i \ne a_{i'}$ for all $i \ne i'$.
%Then we claim that $U$ is directly spanned in any order
%by the $U_i$'s.   This is a special case of a general
%criterion in \cite[3.3.11]{pred} (proved by the dynamic method) for a unipotent
%smooth affine $k$-group to be directly spanned by appropriate smooth closed subgroups
%in the presence of a sufficiently nontrivial torus action.
%\end{proof}

We conclude that the smooth affine $\Z$-groups $\mathscr{U}_{\pm}$ extending
$U_{\pm}$ can be defined using arbitrary orderings
on $\Phi^+$ and $-\Phi^+$.  
Define the ``identity section''
$\widetilde{e} \in \Omega_{\Z}(\Z)$ to correspond to the direct product of
the identity sections of $\mathscr{U}_{\pm}$ and $\mathscr{T}$
(so this extends the identity section $e \in \Omega(\Q) \subset G(\Q)$).

The $\Q$-group structure on $G$ defines a 
birational group law on $\Omega$, and we seek to extend it to 
a ``$\Z$-birational group law'' on $\Omega_{\Z}$ (see Definition \ref{Sbirgp}) 
in a manner that interacts well with the $\Z$-groups $\mathscr{T}$, $\mathscr{U}_{+}$,
and $\mathscr{U}_{-}$. 
This proceeds in several steps.  Guided by the Bruhat decomposition of
$G(\overline{\Q})$, the first step is to consider
the effect on $\Omega$ by $n_a$-conjugation for
all $a \in \Delta$.  For any $n \in N_G(T)(\Q)$,
we have $n U_c n^{-1} = U_{w(c)}$ for
$w \in W(\Phi)$ represented by $n$.   
In the special case $n = n_a$ 
for $a \in \Delta$, we have $w(c) = s_a(c) \in \Phi^+$
for all $c \in \Phi^+ - \{a\}$, whereas $w(a) = -a$.
Also, $n_a$ normalizes $T$.  

\begin{lemma}\label{lemma27} For each $a \in \Delta$,
there exists an open subscheme $\mathscr{V}_a \subset \Omega_{\Z}$
containing $\mathscr{T}$ and every $\mathscr{U}_c$
$(c \in \Phi)$ such that 
the automorphism $g \mapsto n_a g n_a^{-1}$
of $G$ carries $(\mathscr{V}_a)_{\Q}$ into $\Omega$.
Moreover, $\mathscr{V}_a$ can be chosen so that the resulting map 
$(\mathscr{V}_a)_{\Q} \rightarrow \Omega$ extends to a $\Z$-morphism
$f_a:\mathscr{V}_a \rightarrow \Omega_{\Z}$ 
restricting to an automorphism of the $\Z$-group $\mathscr{T}$ 
and carrying the $\Z$-group $\mathscr{U}_c$ isomorphically onto the $\Z$-group
$\mathscr{U}_{s_a(c)}$ for all $c \in \Phi$.
In particular, since
$\mathscr{T} \subset \mathscr{V}_a$,
 $\widetilde{e}$ factors through $\mathscr{V}_a$ and $f_a(\widetilde{e}) = \widetilde{e}$.
\end{lemma}

This result is \cite[XXV, 2.7]{sga3}.

\begin{proof}
For $c \in \Phi$, the isomorphism
$U_c \simeq U_{s_a(c)}$ defined by $n_a$-conjugation is given by 
$$p_c(x) \mapsto p_{s_a(c)}({\rm{Ad}}_G(n_a)(xX_c)) = p_{s_a(c)}(\pm xX_{s_a(c)})$$
since $\{X_c\}_{c \in \Phi}$ is a Chevalley system. 
Hence, this visibly extends to a $\Z$-group isomorphism $\mathscr{U}_c \simeq \mathscr{U}_{s_a(c)}$.
Likewise, on the $\Q$-group $T = \Gm^{\Delta} = D_{\Q}(M)$, the effect of $n_a$-conjugation is given by
the $\Z$-group automorphism 
of $\mathscr{T} = \Gm^{\Delta} = D_{\Z}(M)$ induced by $s_a$.

By Lemma \ref{uzlemmaprelim}, the $\Z$-group $\mathscr{U}_{+}$ is directly spanned in any order by 
$\{\mathscr{U}_c\}_{c \in \Phi^+}$ and the $\Z$-group
$\mathscr{U}_{-}$ is directly spanned in any order
by $\{\mathscr{U}_c\}_{c \in -\Phi^+}$. Thus, 
\begin{equation}\label{coord}
\Omega_{\Z} = \prod_{c \in \Phi^+ - \{a\}} \mathscr{U}_{-c} \times \mathscr{U}_{-a}
\times \mathscr{T} \times \mathscr{U}_a \times \prod_{c \in \Phi^+ - \{a\}} \mathscr{U}_c
\end{equation}
using some fixed choice of enumeration of
$\Phi^+ - \{a\}$ in both products.  Finally, since we arranged that $X_{-c} = X_c^{-1}$ for all $c \in \Phi$,  
$n_a$-conjugation swaps $U_a$ and $U_{-a}$
via negation on the standard coordinate of $\Add$ relative
to the parameterizations $p_a$ and $p_{-a}$ (Corollary \ref{wsec}(3)).
Thus, using the chosen enumeration of $\Phi^+ - \{a\}$ to define
the order of multiplication, $n_a$-conjugation on $\Omega$ carries
$$\prod_{c \in \Phi^+ - \{a\}} p_{-c}(x_{-c}) \cdot p_{-a}(x') \cdot t \cdot
p_a(x) \cdot \prod_{c \in \Phi^+ - \{a\}} p_c(x_c)$$
to 
$$\prod_{c \in \Phi^+ - \{a\}} p_{-s_a(c)}(\pm x_{-c}) \cdot p_{a}(-x') \cdot D_{\Z}(s_a)(t) \cdot  p_{-a}(-x) \cdot 
\prod_{c \in \Phi^+ - \{a\}} p_{s_a(c)}(\pm x_c)$$
for some universal signs. 
The terms $p_a(-x')$ and $p_{-a}(-x)$ appear in the ``wrong'' places; 
we want to swap their positions (at the cost of changing
the $\mathscr{T}$-component) so that we can make things work over $\Z$.

For $t' := D_{\Z}(s_a)(t)$ we have $(-a)(t') = a(t)$, so $p_a(-x')t' p_{-a}(-x) = 
p_a(-x')p_{-a}(-x a(t))t'$.
Since $X_{-a} = X_a^{-1}$, Theorem \ref{ssclass}(1) gives
that $p_a(-x') p_{-a}(-x a(t))$ lies in
$U_{-a} \times T \times U_a$ if and only if
$1 + x'xa(t)$ is a unit.  Under this unit hypothesis, 
(\ref{cellid}) implies that 
$p_a(-x')p_{-a}(-xa(t))t'$ equals $$p_{-a}\left(\frac{-xa(t)}{1 + x'xa(t)}\right) a^{\vee}(1 + x'xa(t))t' \,
p_a\left(\frac{-x'a(t)}{1 + x'xa(t)}\right).$$
Using the $\Z$-group law on $\mathscr{U}_{\pm}$, we conclude 
 that for the open subscheme $\mathscr{V}_a \subset \Omega_{\Z}$
defined by the unit condition $1 + x_{-a} x_{a} a(t) \in \Gm$
(using the coordinatization on $\Omega_{\Z}$ relative to the product decomposition (\ref{coord})),
$n_a$-conjugation on $\Omega$ carries
$(\mathscr{V}_a)_{\Q}$ into $\Omega$ and the resulting map
$(\mathscr{V}_a)_{\Q} \rightarrow \Omega$ extends to a $\Z$-morphism 
$f_a:\mathscr{V}_a \rightarrow \Omega_{\Z}$.

By definition it is clear that $\mathscr{V}_a$ contains
$\mathscr{T}$ and every $\mathscr{U}_c$.
The map $f_a$ carries $\mathscr{T}$ into itself
and $\mathscr{U}_c$ into $\mathscr{U}_{s_a(c)}$ for all
$c \in \Phi$ because such factorization through flat closed subschemes
can be checked over $\Q$ (where it is obvious).  Likewise,
the induced map $\mathscr{T} \rightarrow \mathscr{T}$ is a $\Z$-group
involution and the induced map $\mathscr{U}_c \rightarrow \mathscr{U}_{s_a(c)}$
is a $\Z$-homomorphism with inverse given by the induced map
$\mathscr{U}_{s_a(c)} \rightarrow \mathscr{U}_{s_a^2(c)} = \mathscr{U}_c$
up to scaling by $(-1)^{\langle c, a^{\vee} \rangle}$ since $n_a^2 = a^{\vee}(-1)$.
\end{proof}

In addition to conjugation by the representatives $n_a$ for the simple reflections $s_a \in W(\Phi)$,
we need to address conjugation by some representative
$n$ for the long Weyl element $w \in W(\Phi)$ relative to $\Delta$.
Explicitly, $w = \prod_{a \in \Delta} s_a$ in $W(\Phi)$ using multiplication taken in the order of {\em any}
enumeration of $\Delta$, but the product $\prod_{a \in \Delta} n_a \in N_G(T)(\Q)$
generally depends on the choice of enumeration of $\Delta$
(i.e., if we change the order of multiplication then the product in $N_G(T)(\Q)$ changes by
a possibly nontrivial $T(\Q)$-multiplication).  For our purposes it is only necessary to work
with {\em some} enumeration of $\Delta$, so we shall use the enumeration 
chosen earlier to define the lexicographical ordering on $M_{\Q}$
(which defined our ordering on $\Phi^+$ and $-\Phi^+$).

Let $n$ denote the resulting product $\prod_{a \in \Delta} n_a \in N_G(T)(\Q)$,
so $n$-conjugation on $G$ restricts to an automorphism of
$T$ and swaps $U_{+}$ and $U_{-}$.

\begin{lemma}\label{lemma28}
For $n$ as defined above, there
are open subschemes $\mathscr{V}, \mathscr{V}' \subset \Omega_{\Z}$
containing $\mathscr{U}_{\pm}$ and $\mathscr{T}$
such that the automorphism 
$g \mapsto n g n^{-1}$ of $G$ carries
$\mathscr{V}_{\Q}$ and $\mathscr{V}'_{\Q}$ into $\Omega$ and
the resulting maps  $\mathscr{V}_{\Q}, \mathscr{V}'_{\Q} \rightrightarrows \Omega$
extend to $\Z$-morphisms 
$$f:\mathscr{V} \rightarrow \Omega_{\Z},\,\,\,
f':\mathscr{V}' \rightarrow \Omega_{\Z}$$
satisfying the following properties: 
\begin{enumerate}
\item $f|_{\mathscr{U}_{\pm}}$ factors through a $\Z$-group morphism onto $\mathscr{U}_{\mp}$,
and similarly for $f'$;
\item $f$ and $f'$ restrict to $\Z$-group endomorphisms of $\mathscr{T}$;
\item $f' \circ f|_{f^{-1}(\mathscr{V}')}: f^{-1}(\mathscr{V}') \rightarrow \Omega_{\Z}$ is the canonical open immersion.
\end{enumerate}
In particular, $f^{-1}(\mathscr{V}')$ is fiberwise dense in $\mathscr{V}$
and  $\widetilde{e}$ factors through $\mathscr{V}$ and $\mathscr{V}'$,
with $f(\widetilde{e}) = \widetilde{e}$ and $f'(\widetilde{e}) = \widetilde{e}$.
\end{lemma}

This result is \cite[XXV, 2.8]{sga3} (except that we include some $\Z$-group compatibilities
in the statement).

\begin{proof}
By $\Z$-flatness considerations and the evident properties
on the $\Q$-fiber, once we find $\mathscr{V}$ and $\mathscr{V}'$
containing $\mathscr{T}$ and $\mathscr{U}_{\pm}$ so that $n$-conjugation carries their $\Q$-fibers into
$\Omega$ and the resulting maps $\mathscr{V}_{\Q} \rightarrow \Omega$
and $\mathscr{V}'_{\Q} \rightarrow \Omega$
extend to $\Z$-morphisms, the additional properties
in (1), (2), and (3) are immediate.
The actual construction of $\mathscr{V}$ and $\mathscr{V}'$ involves
an inductive argument on word length in $W(\Phi)$, with Lemma \ref{lemma27}
used to carry out the induction.  
The identity that makes it work is $n_a^4 = 1$ in $N_G(T)(\Q)$ for all $a \in \Delta$
(since $n_a^2 = a^{\vee}(-1)$). 
We refer the reader 
to \cite[XXV, 2.8]{sga3} for the details.  
\end{proof}

Now we bring in birational group laws.
We refer the reader to \cite[\S2.5, \S5.1--5.2]{neron} for an elegant general
discussion of $S$-rational maps and $S$-birational group laws
with smooth separated $S$-schemes, and here give just the basic definitions:

\begin{definition}\label{Sbirgp}
Let $S$ be a scheme, and $X, Y \rightrightarrows S$ be two smooth separated morphisms.
For any fiberwise-dense open subschemes $\Omega, \Omega' \subset X$
declare two $S$-morphisms $f: \Omega \rightarrow Y$ and $f':\Omega' \rightarrow Y$,
to be equivalent if $f$ and $f'$ agree on an open subset that is $S$-dense
in the sense of being fiberwise dense.
An {\em $S$-rational map} from $X$ to $Y$ is an equivalence class of such maps,
and an $S$-rational map is {\em $S$-birational} if some $($equivalently, every$)$ representative
morphism $f:\Omega \rightarrow Y$ restricts to an isomorphism between $S$-dense
open subschemes of $X$ and $Y$.

An {\em $S$-birational group law} on a smooth separated $S$-scheme $X$ is
an $S$-rational map $m$ from $X \times_S X$ to $X$ such that 
(i) the $S$-rational maps $(x,x') \mapsto (x, m(x,x'))$ and $(x,x') \mapsto (m(x,x'),x')$
from $X \times_S X$ to $X \times_S X$ are $S$-birational
$($so $m$ is $S$-dominant in the sense of carrying an $S$-dense open 
subset of $X \times_S X$ onto an $S$-dense open subset of $X$$)$
and (ii) $m$ is associative in the sense of $S$-dominant $S$-rational maps.
\end{definition}
\index{sbirational group@$S$-birational group law}
\index{srational map@$S$-rational map}

By artful use of $f$ and $f'$ from Lemma \ref{lemma28}, one can
put a $\Z$-birational group law on $\Omega_{\Z}$:

\begin{proposition}\label{birlaw}
There are open subschemes $\mathscr{V}_1 \subset \Omega_{\Z} \times \Omega_{\Z}$
and $\mathscr{V}_2 \subset \Omega_{\Z}$
such that:
\begin{enumerate}
\item $\mathscr{U}_{\pm}, \mathscr{T} \subset \mathscr{V}_2$
and $\mathscr{U}_{+} \times \mathscr{U}_{+}, \mathscr{U}_{-} \times \mathscr{U}_{-}, 
\mathscr{T} \times \mathscr{T}, \Omega_{\Z} \times \{\widetilde{e}\}, \{\widetilde{e}\} \times \Omega_{\Z}  \subset \mathscr{V}_1$, 
\item for the generic fibers
$V_j := (\mathscr{V}_j)_{\Q}$, the multiplication $m:G \times G \rightarrow G$ carries
$V_1$ into $\Omega$ and inversion $\iota:G \simeq G$ carries $V_2$ into $\Omega$,
\item the induced maps $m:V_1 \rightarrow \Omega$ and $\iota:V_2 \rightarrow \Omega$
extend to $\Z$-morphisms $m_{\Z}:\mathscr{V}_1 \rightarrow \Omega_{\Z}$
and $\iota_{\Z}:\mathscr{V}_2 \rightarrow \Omega_{\Z}$.
\end{enumerate}
Moreover, the $\Q$-birational group law $(\Omega,m)$ extends
to a $\Z$-birational group law $(\Omega_{\Z}, m_{\Z})$
with inverse $\iota_{\Z}$ and identity $\widetilde{e}$
that restricts to the $\Z$-group laws on $\mathscr{U}_{+}$, $\mathscr{U}_{-}$, and 
$\mathscr{T}$. 
\end{proposition}

This is \cite[XXV, 2.9]{sga3}, except
that the assertions concerning containments of
$\mathscr{U}_{\pm}$ and $\mathscr{T}$
as well as compatibility with their $\Z$-group structures
are not mentioned there (but 
are immediate from inspecting the construction there and
using  known identities for maps between the $\Q$-fibers).

\begin{proof}
The motivation for the construction of
$\mathscr{V}_1$ and $\mathscr{V}_2$
can be seen by considering the special case $G = {\rm{SL}}_2$.
In that special case, $\Omega_{\Z} = \Add \times \Gm \times \Add$
with the action of 
$\mathscr{T} = \Gm$ on $\mathscr{U}_{\pm} = \Add$
given by 
$t.x = t^{\pm 2}x$.  The Zariski-open condition
$1 + x'xt^2 \in \Gm$ defines a suitable  $\mathscr{V}_2 \subset \Omega_{\Z}$
for exactly the same reason that the condition $1 + x'xa(t) \in \Gm$
arose in the proof of Lemma \ref{lemma27}.
To find  a Zariski-open condition on points
$$((x'_2, t_2, x_2), (x'_1, t_1, x_1)) \in \Omega_{\Z} \times \Omega_{\Z}$$
to define $\mathscr{V}_1$, we use Theorem \ref{ssclass}(1) 
to see that the condition $1 + x_2x'_1 \in \Gm$ does the job.

The construction of
$\mathscr{V}_1$ and $\mathscr{V}_2$ in general is given in \cite[XXV, 2.9]{sga3}, modeled on the
case of ${\rm{SL}}_2$.  One uses Lemma \ref{lemma28}  to overcome the absence
in the general case of formulas as explicit as in the case of ${\rm{SL}}_2$.
The $\Z$-birational group law property amounts to an associativity 
identity that can be checked on the $\Q$-fiber,
and likewise for the inversion and identity assertions for
this birational group law.
\end{proof}

Now we are in position to apply results that promote birational group laws to group schemes.
For a scheme $S$, if $X \rightarrow S$ is a smooth surjective separated
map of finite presentation equipped with an $S$-birational group law $m$,
a {\em solution} is a smooth separated $S$-group $(X',m')$ 
of finite presentation equipped with an $S$-birational isomorphism 
between $X$ and $X'$ that is compatible
with $m$ and $m'$
(i.e., an $S$-isomorphism  $f: \Omega \simeq \Omega'$ between fiberwise dense open subschemes
$\Omega \subset X$ and $\Omega' \subset X'$
such that $m' \circ (f \times f) = f \circ m$
as $S$-rational maps from $X \times_S X$ to $X'$).
A preliminary result in the theory of $S$-birational group laws is that a solution is unique 
up to unique $S$-isomorphism (not just $S$-birationally)
if it exists \cite[5.1/3]{neron}.  This is proved
by translation arguments, using that a smooth surjective map
has many \'etale-local sections.  We emphasize that it is {\em not} required
that $X$ is open in $X'$; i.e., we allow for the possibility that
only some fiberwise dense open subscheme of $X$ appears
as an open subscheme of $X'$ (compatibly with the $S$-birational group laws).

For applications, it is useful to have a criterion to ensure
that a given $S$-birational group law $(X,m)$ occurs as an open subscheme of
an $S$-group, with no shrinking of $X$ required.  
To motivate the criterion, 
consider $S$-birational group laws that arise from fiberwise dense
open subschemes of $S$-groups. Here are some properties that such
$S$-birational group laws must satisfy:

\begin{example}\label{strgp}
Let $G \rightarrow S$ be a smooth separated $S$-group of finite presentation,
and $X \subset G$ a fiberwise dense open subscheme.  
Then $U := m_G^{-1}(X) \bigcap (X \times_S X)$ 
is the open domain of definition of the associated $S$-birational group
law on $X$, and it is {\em $X$-dense} in $X \times_S X$
\index{xdense@$X$-dense}
in the sense that $U$ is fiberwise dense relative
to both projections $X \times_S X \rightrightarrows X$. 
Indeed, for any geometric point $\overline{s}$ of $S$ and $\overline{x} \in X(\overline{s})$,
the $\overline{x}$-fibers of $U_{\overline{s}}$ under the projections
$X_{\overline{s}} \times X_{\overline{s}} \rightrightarrows X_{\overline{s}}$ are the open overlaps
$X_{\overline{s}} \bigcap (\overline{x}^{-1} \cdot X_{\overline{s}})$
and $X_{\overline{s}} \bigcap (X_{\overline{s}} \cdot \overline{x}^{-1})$
in $G_{\overline{s}}$ 
that are dense in $X_{\overline{s}}$.  

Moreover, the universal left and right ``translation'' maps
$U \rightrightarrows X \times_S X$ defined by
$u = (x_1,x_2) \mapsto (x_1, m(x_1,x_2)), 
(m(x_1,x_2), x_2)$ are open immersions 
with $X$-dense image because they
are obtained by restriction to $U \subset G \times_S G$ of the 
universal translation maps $(g,g') \mapsto (gg',g'), (g,gg')$
that are automorphisms of the $S$-scheme $G \times_S G$.
\end{example}


Motivated by Example \ref{strgp}, 
an $S$-birational group law $(X,m)$ is called {\em strict} if there exists an open subscheme
\index{strict birational group law}
$\Omega$ of the domain of definition of $m$ in $X \times_S X$ 
such that $\Omega$ is $X$-dense and the maps 
$\Omega \rightrightarrows X \times_S X$ defined by
$(x,x') \mapsto (x, m(x,x'))$ and $(x,x') \mapsto (m(x,x'),x')$
are open immersions whose respective images in $X \times_S X$
are $X$-dense.   (This is an equivalent formulation of the definition of
a ``group germ'' in \cite[XVIII, 3.1]{sga3}.)
We have just seen in Example \ref{strgp} that strictness is a necessary condition
for a solution $(X',m')$ to an $S$-birational group law
$(X,m)$ not to require any shrinking of $X$; i.e., it is necessary in order that 
$X$ be open in a solution $X'$ (as $S$-birational groups). 

Remarkably, strictness is also
sufficient for the existence of a solution $(X',m')$ containing the $(X,m)$ as a fiberwise dense 
open subscheme.  This deep result is essentially \cite[XVIII, 3.7, 3.13(iii)]{sga3}, except
that $X'$ is built there only as an fppf sheaf of groups, 
resting on the special case of strictly henselian local $S$
for which $X'$ is built as a scheme.   In fact, using \cite[Cor.\,6.3]{versal}, this construction
can be reinterpreted to obtain that $X'$ is an fppf (and hence smooth)
algebraic space group.  The construction of this solution $X'$ as a scheme
is given in \cite[5.2/3]{neron}
for the cases that $S$ is the spectrum of a separably closed field or strictly henselian
discrete valuation ring.


By \cite[6.6/1]{neron}
the algebraic space $X'$ is always a scheme, but we 
are only interested in the special case
that $S$ is Dedekind (namely, $S = \Spec \Z$). One can establish the result over such $S$ 
without any (implicit) use of algebraic spaces as follows. 
First consider the ``local'' version: $S = \Spec R$
for $R$ a field or discrete valuation ring.  By using the known
solution over a strict henselization of any such $R$,
a solution can be built over $R$ by descent arguments; see \cite[6.5/2]{neron}.
In particular, for a general Dedekind scheme $S$ 
we get a solution over the generic points
(though in the cases of interest over $\Z$ we are even given a solution $G$ over $\Q$).  This
``spreads out'' to a solution over a dense open subscheme $V$ of $S$.
By limit considerations, the general Dedekind case reduces to local
versions of the problem at the finitely many closed points of $S - V$, which
are instances of the settled local case that $R$ is a discrete valuation ring.

To summarize, the $\Z$-birational group $\Omega_{\Z}$ is fiberwise dense in a smooth finite type
and separated $\Z$-group $G_{\Z}$ once we verify:

\begin{proposition}\label{strictbir} The $\Z$-birational group law on $\Omega_{\Z}$ is strict.
\end{proposition}

\begin{proof}
Consider $\mathscr{V}_1$ as in Proposition \ref{birlaw}.
The universal left and right ``translation'' morphisms $\mathscr{V}_1 \rightrightarrows \Omega_{\Z} \times
\Omega_{\Z}$ defined by
$$(\omega, \omega') \mapsto (\omega, m_{\Z}(\omega, \omega')),\,\,\,
(m_{\Z}(\omega, \omega'), \omega')$$
are maps between smooth separated $\Z$-schemes of finite type,
and on the $\Q$-fibers they are open immersions since
$V_1$ is a non-empty open in the smooth connected $\Q$-group $G$.
Thus, these maps are birational.  We claim that these $\Z$-maps
are \'etale on an open subscheme $\mathscr{W}$ of $\mathscr{V}_1$ that
contains $\{\widetilde{e}\} \times \Omega_{\Z}$
and $\Omega_{\Z} \times \{\widetilde{e}\}$.  The case of the
universal left ``translation'' will be treated, and right ``translation'' goes
similarly.

 In view of $\Z$-smoothness
 for the source and target, it suffices to check that 
the tangent maps at $(x, \widetilde{e}(s))$ and $(\widetilde{e}(s),x)$ are isomorphisms
for every geometric point $s$ of $\Spec \Z$ and closed point $x \in \Omega_s := (\Omega_{\Z})_s$.  Restriction
to the respective open neighborhoods
$$\mathscr{V}'_{1,s} = (\mathscr{V}_1)_s \bigcap (\Omega_s \times \{\widetilde{e}(s)\}) \subset
\Omega_s,\,\,\,
\mathscr{V}''_{1,s} = (\mathscr{V}_1)_s \bigcap (\{x\} \times \Omega_s) \subset \Omega_s$$
of $x$ and $\widetilde{e}(s)$ in $\Omega_s$ gives
maps $\mathscr{V}'_{1,s}, \mathscr{V}''_{1,s} \rightrightarrows \Omega_s \times \Omega_s$
that are respectively the diagonal map and the ``left translation''
by $x$ into the slice $\{x\} \times \Omega_s$ (on an open domain containing $\widetilde{e}(s)$).
Since $\widetilde{e}$ is an identity for the $\Z$-birational group law, this latter
map at $(x,\widetilde{e}(s))$ is the canonical inclusion on the tangent space.
Hence, by using the canonical decomposition 
$${\rm{Tan}}_{(x,\widetilde{e}(s))}((\Omega_{\Z} \times \Omega_{\Z})_s) =
{\rm{Tan}}_{x}(\Omega_s) \bigoplus {\rm{Tan}}_{\widetilde{e}(s)}(\Omega_s)$$
we deduce the isomorphism property for the tangent map at $(x,\widetilde{e}(s))$ 
since $V \bigoplus V$ is the direct sum of the diagonal and $\{0\} \bigoplus V$ for any vector space $V$.
The same argument works at the points $(\widetilde{e}(s),x)$.

Pick an open subscheme $\mathscr{W} \subset 
\mathscr{V}_1$ containing $\Omega_{\Z} \times \{\widetilde{e}\}$
and $\{\widetilde{e}\} \times \Omega_{\Z}$ on which the universal left and right ``translations''
are \'etale maps to $\Omega_{\Z} \times \Omega_{\Z}$, so these
define a pair of maps 
$\mathscr{W} \rightrightarrows \Omega_{\Z} \times \Omega_{\Z}$
that are birational, separated, and quasi-finite. 
By Zariski's Main Theorem \cite[III$_1$,4.4.9]{ega}, 
 any birational, separated, and quasi-finite map between connected 
normal noetherian schemes is an open immersion.
These open immersions $\mathscr{W} \rightrightarrows \Omega_{\Z} \times \Omega_{\Z}$
have $\Omega_{\Z}$-dense images since $\{\widetilde{e}\} \times \Omega_{\Z},
\Omega_{\Z} \times \{\widetilde{e}\} \subset \mathscr{W}$ and
the fibers of $\Omega_{\Z} \rightarrow \Spec \Z$ are irreducible. 
\end{proof}

Since  $\Omega_{\Z}$ is now a fiberwise dense open subscheme of
a unique smooth and separated $\Z$-group $G_{\Z}$ of
finite type (compatibly with $\Z$-birational group laws), 
  $G_{\Z} \rightarrow \Spec \Z$ has connected fibers.
By the uniqueness of solutions to birational group laws,
the inclusion $\Omega \hookrightarrow (G_{\Z})_{\Q}$ extends
to an isomorphism $G \simeq (G_{\Z})_{\Q}$, so we can view
$G_{\Z}$ as a $\Z$-model for $G$.   
The section $\widetilde{e} \in \Omega_{\Z}(\Z) \subset G_{\Z}(\Z)$
is the identity section, as this holds over $\Q$ and equalities of
maps between separated flat $\Z$-schemes can be checked over $\Q$.
Likewise, the immersions $\mathscr{T} \rightarrow G_{\Z}$
and $\mathscr{U}_{\pm} \rightarrow G_{\Z}$
are $\Z$-homomorphisms, since we can check over $\Q$.
We wish to avoid using the
deep Theorem \ref{adimm},
so we do not yet know if these immersions are closed immersions, nor if $G_{\Z}$ is affine.

To prove that the smooth separated
finite type group $G_{\Z}$ is semisimple, 
we will first show that its geometric fibers over $\Spec \Z$ (which we know
are connected) are semisimple, and then deduce
from this that $G_{\Z}$ is affine.  The key to fibral
semisimplicity is:

\begin{lemma}\label{noncomm} For every geometric point $s$ of $\Spec \Z$
and every $c \in \Phi$, the fibral subgroups $(\mathscr{U}_c)_s$ and $(\mathscr{U}_{-c})_s$
in $(G_{\Z})_s$ generate a subgroup that contains the nontrivial torus
$c^{\vee}_s(\Gm)$.  In particular, the subgroup of $(G_{\Z})_s$ generated
by the two unipotent subgroups $(\mathscr{U}_{\pm c})_s = \Add$  is not unipotent. 
\end{lemma}

\begin{proof}
Identify $\mathscr{U}_{\pm c}$ with $\Add$ via $\widetilde{p}_{\pm c}$.
We claim that if $x, x'$ are points of $\Add$ such that $1 + x'x \in \Gm$ and
$(\widetilde{p}_c(x'), \widetilde{p}_{-c}(x)) \in \mathscr{V}_1$
then
\begin{equation}\label{pcomm}
\widetilde{p}_c(x')\widetilde{p}_{-c}(x) = 
\widetilde{p}_{-c}\left(\frac{x}{1+x'x}\right) c^{\vee}(1+x'x) \widetilde{p}_c\left(\frac{x'}{1+x'x}\right) 
\end{equation}
in the group law of $G_{\Z}$.
The two sides of (\ref{pcomm}) are scheme morphisms
$(\mathscr{U}_c \times \mathscr{U}_{-c})_{1+x'x} \rightrightarrows G_{\Z}$
using the group law on $G_{\Z}$, so to prove their equality
it suffices (by separatedness and flatness of $G_{\Z}$ over $\Z$)
to check over $\Q$.  Via the identification $(G_{\Z})_{\Q} \simeq G$, we conclude
via (\ref{cellid}).

The direct product scheme $\mathscr{U}_c \times \mathscr{U}_{-c}$
inside $\Omega_{\Z} \times \Omega_{\Z}$
meets the open neighborhood $\mathscr{V}_1$ of $(\widetilde{e}, \widetilde{e})$,
and imposing the additional condition ``$1 + xx' \in \Gm$'' defines a fiberwise dense open subscheme 
on which (\ref{pcomm}) holds.
Hence, by separatedness and smoothness considerations,
it follows that (\ref{pcomm}) holds on
the open subscheme of $\mathscr{U}_c \times \mathscr{U}_{-c} = \mathbf{A}^2_{\Z}$
where $1 + x'x \in \Gm$.   Now pass to $s$-fibers.
\end{proof}

The proof that $G_{\Z}$ is affine will use 
some  structural input on the fibers, so we first address the fibral structure:

\begin{proposition}\label{ggeom} For geometric points $s \in \Spec \Z$,
$(G_{\Z})_s$ is semisimple and $\mathscr{T}_s$ is
a maximal torus. In particular, the fibers of $G_{\Z}$ are affine.
\end{proposition}

\begin{proof}
By construction, $G_{\Z}$ contains the torus $\mathscr{T} = D_{\Z}(M)$
as an $S$-subgroup (which we have not yet shown to be closed, 
as $G_{\Z}$ is not yet shown to be affine, so we
cannot apply Lemma \ref{ix2.5} and we wish to avoid using
the deep Theorem \ref{adimm}). 
Consider the action by this torus on
$$\mathfrak{g} := {\rm{Lie}}(G_{\Z}) = {\rm{Tan}}_{\widetilde{e}}(\Omega_{\Z}) = 
{\rm{Lie}}(\mathscr{T}) \oplus \bigoplus_{c \in \Phi} {\rm{Lie}}(\mathscr{U}_{c}).$$
These direct summands are stable under the adjoint
action of $\mathscr{T}$, with ${\rm{Lie}}(\mathscr{T})$ centralized
by the action and the line subbundle ${\rm{Lie}}(\mathscr{U}_c)$ acted upon through
through the character $c \in \Phi \subset M - \{0\}$.   
It follows that under the $\mathscr{T}$-action on
$\mathfrak{g}$, the trivial weight space
is the subbundle ${\rm{Lie}}(\mathscr{T})$
and the set of nontrivial weights consists of the ``constant sections''
$c \in \Phi \subset M - \{0\}$,
with the $c$-weight space equal to ${\rm{Lie}}(\mathscr{U}_c)$ for
$c \in \Phi$.

Now pass to fibers at a geometric point $s$ of $\Spec \Z$.
The $s$-group $G_s := (G_{\Z})_s$ 
contains the $s$-subgroups $U_{c,s} := (\mathscr{U}_c)_s \simeq \Add$
and $T_s := \mathscr{T}_s = D_s(M)$ such that
$T_s$ normalizes each $U_{c,s} = \Add$
via the action $t.x = c(t)x$, so
in any commutative quotient $C_s$ of $G_s$ modulo a closed normal subgroup scheme, 
the image of $U_{c,s}$ is trivial.
Then (\ref{pcomm}) implies that for
each $c \in \Phi$ the image of $c^{\vee}(\Gm)_s$ in this quotient 
is also trivial.  But the subgroups $U_{c,s}$ and $c^{\vee}(\Gm)$
for $c \in \Phi$ generate $G_s$ (as $\Omega_{\Z}$
is a fiberwise dense open subscheme of $G_{\Z}$
whose direct product decomposition as a $\Z$-scheme is given by 
{\em multiplication} in the $\Z$-group law on $G_{\Z}$,
as we can check over $\Q$), so $C_s = 1$. 

We have proved that $G_s$ has no nontrivial commutative
quotient modulo a normal closed subgroup scheme. 
It is a theorem of Chevalley  \cite{chevold} (and Barsotti \cite{barsotti} and Rosenlicht \cite[Thm.\,16]{rosenlicht}) 
that every  smooth connected group $H$ over an algebraically closed
field  is an extension of  
an abelian variety by a connected linear algebraic group.
(See \cite{chevc} for a modern exposition of Chevalley's proof,
and \cite[\S2]{brion} and \cite{milner} for modern expositions of Rosenlicht's proof.)
In particular, if such a group $H$ has no nontrivial
commutative quotient modulo a normal closed 
subgroup scheme then $H$ is a linear algebraic group! 
Hence, $G_s$ is a connected linear algebraic
group that is equal to its own derived subgroup.
In particular, if it is reductive then it must be semisimple.

It remains to show that $G_s$ is reductive and $T_s$ is a maximal torus in $G_s$.
The inclusion $T_s \hookrightarrow Z_{G_s}(T_s)$
between {\em connected} smooth linear algebraic 
groups is an equality on Lie algebras, so it is an isomorphism.
Hence, $T_s$ is a maximal torus in $G_s$.
To establish the reductivity, we first require a dynamic
characterization of $U_{c,s}$  in terms
of the $T_s$-action on $G_s$ for each $c \in \Phi$.

\index{dynamic method!and birational groups}
Fix $c \in \Phi$. For the codimension-1
subtorus $T'_s = ((\ker c)_s^0)_{\rm{red}}$ in $T_s$, the 
smooth connected centralizer $Z_{G_s}(T'_s)$
has Lie algebra that is 
the $T'_s$-centralizer in $\mathfrak{g}_s$, which is
the span of the weight spaces for the $T_s$-weights
that are trivial on $T'_s$. Hence, 
${\rm{Lie}}(Z_{G_s}(T'_s))$ is the span
of ${\rm{Lie}}(T_s)$ and ${\rm{Lie}}(U_{\pm c,s})$
since $\Phi$ is a reduced root system in $M = {\rm{X}}(T_s)$.
Consider the smooth connected unipotent subgroup
$U_{Z_{G_s}(T'_s)}(c^{\vee})$.
This clearly has Lie algebra ${\rm{Lie}}(U_{c,s})$,
so the inclusion $U_{c,s} \subset U_{Z_G(T'_s)}(c^{\vee})$
between smooth connected linear algebraic groups
is an equality on Lie algebras.
In other words, we have established the ``dynamic'' characterization
\begin{equation}\label{ucs}
U_{c,s} = U_{Z_{G_s}(T'_s)}(c^{\vee})
\end{equation}
in terms of the $T_s$-action on $G_s$.

Via the inclusion $\exp_c:\Add = \mathscr{U}_c \hookrightarrow G_{\Z}$,
define $n_c = \exp_c(1) \in G_{\Z}(\Z)$.  By Corollary \ref{wsec}(3)
applied to $((G_{\Z})_{\Q}, T, M)$, $n_c$-conjugation carries
$(\mathscr{U}_c)_{\Q}$ into $(\mathscr{U}_{-c})_{\Q}$ and hence
carries $\mathscr{U}_c$ into $\mathscr{U}_{-c}$.  The resulting
immersion $n_c \mathscr{U}_c n_c^{-1} \subset \mathscr{U}_{-c}$
is an equality for fibral reasons, so $n_c U_{c,s} n_c^{-1} = U_{-c,s}$
for all $c$.

Now suppose that $U := \mathscr{R}_u(G_s)$ is nontrivial.
By normality in $G_s$, the nonzero subspace
${\rm{Lie}}(U)$ in $\mathfrak{g}_s$ has a weight space
decomposition relative to the $T_s$-action. 
Since $U \bigcap T_s = 1$
(due to the unipotence of $U$), passing
to Lie algebras gives ${\rm{Lie}}(U) \bigcap {\rm{Lie}}(T_s) = 0$.
Hence, all $T_s$-weights that occur on ${\rm{Lie}}(U)$
are nontrivial, so for some $c_0 \in \Phi$
the 1-dimensional $c_0$-weight space ${\rm{Lie}}(U_{c_0,s})$
is contained in ${\rm{Lie}}(U)$.

The intersection $U \bigcap Z_{G_s}(T'_s)$ is smooth
and connected since
$$T'_s \times (U \bigcap Z_{G_s}(T'_s)) = 
Z_{T'_s \ltimes U}(T'_s),$$
so by applying Proposition \ref{functor}(1) to $H = U \bigcap Z_{G_s}(T'_s)$
equipped with its action by $c_0^{\vee}(\Gm)$
we conclude via the dynamic description (\ref{ucs}) of $U_{c_0,s}$
that $U \bigcap U_{c_0,s}$ is smooth
and connected.    But ${\rm{Lie}}(U \bigcap U_{c_0,s}) = 
{\rm{Lie}}(U) \bigcap {\rm{Lie}}(U_{c_0,s}) = 
{\rm{Lie}}(U_{c_0,s})$, so the inclusion
$U \bigcap U_{c_0,s} \hookrightarrow U_{c_0,s}$
between smooth connected groups is an equality on Lie algebras.
This forces $U \bigcap U_{c_0,s} = U_{c_0,s}$, so
$U_{c_0,s} \subset U$ inside $G_s$.
Thus, the subgroup $U_{-c_0,s} = n_{c_0} U_{c_0,s} n_{c_0}^{-1}$ is 
contained in $U$ by normality of $U$ in $G_s$.  But the subgroups 
$U_{c_0,s}$ and $U_{-c_0,s}$ of $U$ generate
a non-unipotent subgroup of $G_s$ (by Lemma \ref{noncomm}),
which is absurd since $U$ is unipotent. Hence, $U = 1$. 
\end{proof}

\begin{proposition}\label{ztg} The $\Z$-group $G_{\Z}$ is affine with
the split torus $\mathscr{T}$ as a maximal torus.
\end{proposition}

\begin{proof}
Consider the adjoint action
${\rm{Ad}}_{G_{\Z}}:G_{\Z} \rightarrow \GL(\mathfrak{g})$.
The fibral semisimplicity in Proposition \ref{ggeom} implies that 
$\ker {\rm{Ad}}_{G_{\Z}}$ has finite geometric fibers (by the classical
theory), so ${\rm{Ad}}_{G_{\Z}}$ is quasi-finite.  But it is a general
fact that any quasi-finite homomorphism
between separated flat groups of finite type over
a Dedekind base is necessarily an affine morphism;
see \cite[XXV, \S4]{sga3} for the proof (which is based on a clever
translation argument).
Hence, ${\rm{Ad}}_{G_{\Z}}$ is an affine morphism, so
$G_{\Z}$ inherits affineness from $\GL(\mathfrak{g})$.

Since $G_{\Z}$ is affine, 
$\mathscr{T}$ is a closed subgroup of $G_{\Z}$ (as for any multiplicative
type subgroup of a smooth affine group, by Lemma \ref{ix2.5}). 
The maximality of this torus in geometric fibers
was proved in Proposition \ref{ggeom}.
\end{proof}

The $\Q$-fiber of $(G_{\Z},\mathscr{T},M)$ is the triple
$(G,T,M)$ whose root datum is the original
$R$ of interest, so Proposition \ref{ztg} completes the proof
of the Existence Theorem (since the root spaces for $(G_{\Z}, \mathscr{T},M)$
are free of rank 1 by construction, or because ${\rm{Pic}}(\Z) = 1$). 
\index{Existence Theorem|)}

\subsection{Applications of Existence and Isomorphism Theorems}

Chevalley groups
were originally defined to be the output of a certain explicit construction over $\Z$
given in \cite{chevbour} for any split connected semisimple $\Q$-group
descending a given connected semisimple $\C$-group. 
Turning the history around, we define 
a {\em Chevalley group} to be a reductive group scheme
$G$ over $\Z$ that admits a maximal torus $T$ over $\Z$.
By Corollary \ref{mikecor}, all tori over $\Z$ are split (as $\Spec \Z$ is normal 
and connected with no nontrivial connected finite \'etale cover). Thus, 
since any line bundle over $\Spec \Z$ is trivial, all 
Chevalley groups are necessarily split. 
These are precisely the $\Z$-groups constructed by Chevalley, at least
in the semisimple case, due to:

\begin{proposition}\label{chevunique}
A Chevalley group is determined up to isomorphism by its associated reduced
root datum, and every such root datum arises in this way.  Two Chevalley groups
are isomorphic over $\Z$ if and only if they are isomorphic over $\C$.
\end{proposition}

\begin{proof} The bijectivity assertion between sets of isomorphism classes is the combination
of the Isomorphism and Existence Theorems
over $\Z$.  By the Isomorphism Theorem, the equivalence between $\Z$-isomorphism
and $\C$-isomorphism is immediate (as root data do not ``know'' the base scheme).
\end{proof}

\begin{remark}
Let $R$ be a semisimple root datum, $\Phi$ its underlying root system, and 
$\mathfrak{g}$ a split Lie algebra over $\Q$
with root system $\Phi$. 
Chevalley initially proved the Existence Theorem over $\Z$
for adjoint $R$ by making an explicit construction inside the
automorphism algebra of a Lie algebra over $\Z$ generated by a Chevalley basis
of $\mathfrak{g}$.
\index{Chevalley basis}
From the viewpoint of \cite{sga3},
this approach ``works'' due to Theorem \ref{adimm} and Remark \ref{chevrule}.
In his 1961 Bourbaki report \cite{chevbour}, Chevalley removed the adjoint 
condition on $R$ by working with a split semisimple $\Q$-group $(G,T,M)$
having root datum $R$ rather than with the Lie algebra $\mathfrak{g}$ over $\Q$ having
root system $\Phi$. 

Chevalley's idea
was to pick a faithful representation $(V,\rho)$ of $G$ over $\Q$ and use a 
Chevalley system $\{X_a\}_{a \in \Phi}$
\index{Chevalley system}
to construct a lattice $\Lambda$ in $V$
so that the schematic closure $\mathscr{G}$ of $G$ in $\GL(\Lambda)$
has an ``open cell'' structure over $\Z$ extending one on $G$.
The fibral connectedness and semisimplicity properties of $\mathscr{G}$
were  analyzed via the open cell structure.
%(According to Tits and Verma \cite[VII, p.\,161]{essays}, this schematic closure method over $\Z$  
%does not work for every $\rho$ but the difficulties are
%eliminated by replacing $\rho$ with $\rho \bigoplus {\rm{Ad}}_G$.
%The method of Chevalley is revisited in \cite[3.9.3]{brt2} from another
%point of view. Also see \cite[\S3--\S4]{borelsem}.)  
\end{remark}
%
%\begin{remark}\label{steinex}
%The ``Chevalley groups'' defined in \cite{steinyale}
%\index{Steinberg!Chevalley group}
%are groups of $\Z$-points of Chevalley $\Z$-groups.
%To be precise, 
%if $R = (M, \Phi, M^{\vee}, \Phi^{\vee})$ is a semisimple reduced root datum 
%with root system $\Phi$ and $(G,T,M)$ is
%the corresponding Chevalley $\Z$-group then for a fixed base $\Delta$ of $\Phi$ consider 
%the subgroup $\Gamma_R \subset G(\Z)$ generated by the root
%groups $U_{\pm a}(\Z)$ for $a \in \Delta$.   This is independent of $\Delta$
%because it actually contains $U_b(\Z)$ for all $b  \in \Phi$.  Indeed, 
%for a $\Z$-basis $X_a$ of $\mathfrak{g}_a$ the element
%$w_a(X_a) \in N_G(T)(\Z)$ generated by $U_{\pm a}(\Z)$ 
%represents the reflection $s_a \in W(\Phi)$, and these $s_a$ generate $W(\Phi)$. 
%Products among such representatives therefore conjugate the groups $U_a(\Z)$ for $a \in \Delta$
%onto the groups $U_b(\Z)$ for all $b \in \Phi$ because $\Phi = W(\Phi).\Delta$.
%
%Let 
%$\widetilde{R} = (\widetilde{M}, \Phi, {\widetilde{M}}^{\vee}, \Phi^{\vee})$ be the simply connected root
%datum for $\Phi$
%(i.e., $\widetilde{M} = (\Z \Phi^{\vee})^{\ast}$), so $\Gamma_R$ is the image of
%$\Gamma_{\widetilde{R}}$ under the central isogeny $\widetilde{G} \rightarrow G$
%that carries the $a$-root group of $\widetilde{G}$
%isomorphically onto that of $G$ for all $a \in \Phi$. 
%For the split maximal torus
%preimage $\widetilde{T}$ of $T$ in $\widetilde{G}$ we have $\widetilde{T}(\Z) \subset \Gamma_{\widetilde{R}}$.
%Indeed, $\widetilde{T} = \Gm^{\Delta^{\vee}}$ 
%since ${\rm{X}}_{\ast}(\widetilde{T}) = \widetilde{M}^{\ast} = \Z \Phi^{\vee}$, 
%and likewise the split semisimple $\Z$-group $\widetilde{G}_a = \mathscr{D}(Z_{\widetilde{G}}(\widetilde{T}_a))$
%is ${\rm{SL}}_2$ (rather than ${\rm{PGL}}_2$) since $\widetilde{G}$ is simply connected.
%The formula (\ref{hform}) with $t \in \Z^{\times}$
%therefore expresses the coroot groups generating $\widetilde{T}(\Z)$ in terms of
%the root groups $U_{\pm a}(\Z)$ for $a \in \Delta$, affirming that
%$\widetilde{T}(\Z) \subset \Gamma_{\widetilde{R}}$.
%In particular, if $\widetilde{B} = \widetilde{T} \ltimes U_{\Phi^+} \subset \widetilde{G}$ is the Borel subgroup
%containing $\widetilde{T}$ that arises from the positive system of roots $\Phi^+$ generated by $\Delta$
%then $\widetilde{B}(\Z) \subset \Gamma_{\widetilde{R}}$.
%
%The role of Chevalley systems in the proof of the Existence Theorem and their relationship 
%to ``Chevalley's rule'' in Remark \ref{chevrule} imply 
%(via \cite[\S1, Thm.\,1]{steinyale} that $\Gamma_{\widetilde{R}} = \widetilde{G}(\Z)$,
%so in general $\Gamma_R$ is the image of $\widetilde{G}(\Z)$ in $G(\Z)$.
%For example, if $R$ is adjoint then $\Gamma_R = \widetilde{G}(\Z)/Z_{\widetilde{G}}(\Z)$.
%\end{remark}


\begin{example}\label{chevweyl}
Here is a useful application of Chevalley groups.  
Let $S$ be a connected non-empty scheme, and $(G,T,M)$
a split reductive group over $S$. 
Since $W_G(T) = W(\Phi)_S$, the short exact sequence of $S$-groups
\begin{equation}\label{wntext}
1 \rightarrow T \rightarrow N_G(T) \rightarrow W_G(T) \rightarrow 1
\end{equation}
induces an exact sequence of groups
$$1 \rightarrow \Hom(M,\Gm(S)) \rightarrow N_G(T)(S) \rightarrow W(\Phi) \rightarrow 1,$$
where surjectivity holds on the right because $W(\Phi)$ is generated by reflections
$s_a$ ($a \in \Phi$) that are induced by elements $w_a(X_a) \in N_G(T)(S)$
for any $\calO_S$-basis $X_a$ of $\mathfrak{g}_a$.
We claim that (\ref{wntext}) is the central pushout of an exact sequence
\index{ aaawgttilde@$\widetilde{W}_G(T)$}
\begin{equation}\label{wtits}
1 \rightarrow T[2] \rightarrow \widetilde{W}_G(T) \rightarrow W_G(T) \rightarrow 1
\end{equation}
for a finite flat $S$-subgroup $\widetilde{W}_G(T) \subset N_G(T)$ such
that $\widetilde{W}_G(T)(S)$ is carried onto the finite group $W(\Phi)$ of constant sections. 
In particular, if $S = \Spec R$ for a domain $R$ (or more generally, if $\mu_2(S)$ is finite)
 then $N_G(T)(S)$ contains
a {\em finite} subgroup mapping onto $W(\Phi)$.

The construction of (\ref{wtits}) rests on a choice of pinning,
or more specifically on a choice of $\Z$-descent of $(G,T,M)$.
In other words, it suffices to make the construction when $S = \Spec \Z$
(i.e., for Chevalley groups),
and in such cases we claim that 
the finite flat $\Z$-subgroup $\widetilde{W}_G(T) \subset N_G(T)$ is {\em canonical}; it 
does not depend on a pinning (and is called
the {\em Tits group} for $(G,T)$).  Since $\Gm(\Z) = \mu_2(\Z)$, on $\Z$-points the diagram 
(\ref{wntext}) yields a short exact sequence
$$1 \rightarrow T[2](\Z) \rightarrow N_G(T)(\Z) \rightarrow W(\Phi) \rightarrow 1,$$
so $N_G(T)(\Z)$ is {\em finite}.  
Hence, we can define $\widetilde{W}_G(T)$ to be the finite flat $\Z$-subgroup
of $N_G(T)$ generated by the $N_G(T)(\Z)$-translates of $T[2]$. 
(Translates of $T[2]$ by representatives for distinct elements of
$W(\Phi)$ are disjoint inside $N_G(T)$ due to (\ref{wntext}) over $\Z$
since $W_G(T) = W(\Phi)_{\Z}$.) 
Obviously the inclusion $\widetilde{W}_G(T)(\Z) \subset N_G(T)(\Z)$ is an equality.

Explicitly, $\widetilde{W}_G(T)(\Z)$ contains the elements
$n_a = w_a(X_a)$ for any $a \in \Phi$ and trivializing section $X_a$ of $\mathfrak{g}_a$
(well-defined up to a sign).  This same description gives the
pinning-dependent construction over a general  non-empty base $S$ (over
which unit scaling on the pinning may go beyond sign changes, thereby
making the Tits group depend on the pinning).
See \cite[\S4.6]{tits66b} and \cite[\S2.8]{tits66c} for further discussion in the simply
connected semisimple case (so that the Chevalley group is determined by the root system),
where $\widetilde{W}_G(T)(\Z)$ is called the
{\em extended Weyl group}; in \cite[IX, \S4, Exer.\,12(d)]{lie7}
there is an interpretation via compact Lie groups.
\end{example}
\index{Weyl group!extended}

Now we turn our attention to a relative version (and refinement) of
the decomposition of a connected semisimple group over a field $k$ into an ``almost direct product''
of its $k$-simple factors.  In Theorem \ref{normsubgp} we canonically described
every nontrivial connected semisimple group $G$ over a field $k$ as a central isogenous
quotient of a product  of $k$-simple semisimple subgroups $G_i$.
For the simply connected central covers $\widetilde{G}_i \rightarrow G_i$, 
each $\widetilde{G}_i$ is $k$-simple (since $G_i$ is) and the composite map
$$\prod \widetilde{G}_i \rightarrow \prod G_i \rightarrow G$$
is a central isogeny  (due to Corollary \ref{qtcenter}). Thus, this map identifies
$\prod \widetilde{G}_i$ with the simply connected central cover of $G$.

The problem of classifying all possible $G$ over $k$ is thereby largely reduced
to the case of $k$-simple $G$ that are simply connected.   We wish to explain
why the absolutely simple case (over finite separable extensions of $k$)
is the most important case.  This rests on:

\begin{proposition}\label{allsimple}
Let $G \rightarrow S$ be a fiberwise nontrivial semisimple group with simply connected fibers over
a non-empty scheme $S$. There is a finite \'etale cover
$S' \rightarrow S$ and a semisimple group $G' \rightarrow S'$ with simply connected
and absolutely simple fibers such that $G$ is $S$-isomorphic to the Weil restriction ${\rm{R}}_{S'/S}(G')$.
\index{Weil restriction}
\index{semisimple group!simply connected}

The pair $(S'/S,G')$ is uniquely determined up to unique $S$-isomorphism
in the following sense: if $(S''/S,G'')$ is another such pair then every
$S$-group isomorphism ${\rm{R}}_{S'/S}(G') \simeq {\rm{R}}_{S''/S}(G'')$
arises from a unique pair $(\alpha, f)$ consisting of 
an $S$-isomorphism $\alpha:S' \simeq S''$ and group
isomorphism $f:G' \simeq G''$ over $\alpha$.
\end{proposition}

\begin{proof}
In view of the uniqueness assertions, by \'etale descent we may work
\'etale-locally on $S$.  Thus, we can assume
that $G$ is split, say with a split maximal torus $T = D_S(M)$
whose root spaces $\mathfrak{g}_a$ are free of rank 1. 
The semisimple root datum $R(G,T,M)$ is {\em simply connected}, 
so it decomposes as a direct product $\prod_{i \in I} R_i$ of
simply connected root data $R_i$ 
whose underlying root systems are irreducible (and $I \ne \emptyset$).
By the Existence Theorem there exists a split semisimple $S$-group $(G_i, T_i, M_i)$ 
whose root datum is $R_i$.  The geometric
fibers of $G_i \rightarrow S$ are simply connected
and simple (Corollary \ref{simplephi}).  
By the Isomorphism Theorem, $G \simeq \prod G_i$.
For $S' = \coprod_{i \in I} S$ and the $S'$-group 
$G' = \coprod G_i$ we have ${\rm{R}}_{S'/S}(G') = \prod G_i = G$.

It remains to prove the asserted unique description of isomorphisms
(so then the preceding construction in the split case does indeed
settle the general case, via descent theory). Consider two pairs $(S'/S,G')$
and $(S''/S,G'')$  and an $S$-group isomorphism
$\varphi:{\rm{R}}_{S'/S}(G') \simeq {\rm{R}}_{S''/S}(G'')$.
We seek to show that $\varphi$ arises from a unique pair $(\alpha, f)$.
The uniqueness allows us to work \'etale-locally on $S$
for existence, so we can assume that $S = \Spec A$
for a strictly henselian local ring $A$.  Then $S'$ and $S''$ are each
a non-empty finite disjoint union of copies of $S$, so 
the assertion can be reformulated as follows:
if $\{G'_i\}$ and $\{G''_j\}$ are non-empty finite collections
of semisimple $A$-groups
with simply connected and absolutely simple fibers 
then any $A$-group isomorphism
$$\varphi:\prod G'_i \simeq \prod G''_j$$ arises
from a unique pair $(\alpha, \{f_i\})$ consisting of a bijection $\alpha:I \simeq J$ and 
$A$-group isomorphisms $f_i:G'_i \simeq G''_{\alpha(i)}$. 

The uniqueness of $\alpha$ is immediate from passage
to the special fiber, and then the uniqueness of $\{f_i\}$ is clear.
To prove the existence of $(\alpha, f)$, we shall use the crutch of maximal tori
over the strictly henselian local ring $A$. 
Let $T'_i \subset G'_i$ be a maximal torus, so $T' := \prod T'_i$ is a maximal torus
in $\prod G'_i$.  Over the strictly henselian local ring $A$, 
all maximal tori in a reductive $A$-group $H$ are $H(A)$-conjugate
to each other (Theorem \ref{schemetori}).  Hence, the product
construction of maximal tori in $\prod G'_i$ gives all maximal tori.
The same applies to $\prod G''_j$, so $\varphi(T') = \prod T''_j$
for maximal tori $T''_j \subset G''_j$.

We claim that the isomorphism $\varphi:\prod T'_i \simeq \prod T''_j$
arises from a bijection $\alpha:I \simeq J$
and collection of isomorphisms $h_i:T'_i \simeq T''_{\alpha(i)}$.
Since these tori are split over the local $A$, it is equivalent
to verify the assertion on the special fiber.  
But over the residue field 
we can appeal to Theorem \ref{normsubgp}
to identify the absolutely simple special fibers  $\{(G'_i)_0\}$ and $\{(G''_j)_0\}$
with the simple factors of the respective product
groups $\prod (G'_i)_0$ and $\prod (G''_j)_0$.  Hence,
$\varphi_0$ must permute these factors according
to some bijection $\alpha:I \simeq J$ and carry $(G'_i)_0$ isomorphically
onto $(G''_{\alpha(i)})_0$.  This latter isomorphism must carry
$(T'_i)_0$ isomorphically onto $(T''_{\alpha(i)})_0$, and these torus isomorphisms
(together with $\alpha$) do the job.

Having built $\alpha:I \simeq J$ and $h_i:T'_i \simeq T''_{\alpha(i)}$
compatible with $\varphi$, we can relabel the indices so that
$\varphi$ is an isomorphism of $A$-groups $\prod G'_i \simeq \prod G''_i$
carrying the $A$-subgroup
$T'_i$ isomorphically to the $A$-subgroup $T''_i$ for each $i$.  It remains to show that the $A$-subgroup 
$G'_{i_0}$ is
carried isomorphically to the $A$-subgroup $G''_{i_0}$ for each $i_0$.  
We will do this via an intrinsic characterization of $G'_{i_0}$ in terms 
of $\{T'_i\}$ and $G' = \prod G'_i$.  The centralizer of $\prod_{i \ne i_0} T'_i$
in $G'$ is $G'_{i_0} \times \prod_{i \ne i_0} T'_i$, so the derived
group of this centralizer is $G'_{i_0}$.  A similar description applies
to $G''_{i_0}$ in terms of $\{T''_i\}$ and $G'' = \prod G''_i$, so we are done.
\end{proof}


\begin{remark}
The assertions in Proposition \ref{allsimple} remain true, with the same proof, 
when ``simply connected'' is replaced by ``adjoint'' everywhere.  The key point
is that an ``adjoint'' root datum is a direct product of irreducible ones, as in the simply connected
case.  The existence of $(S'/S,G')$ can fail more generally 
(when the semisimple root data for the geometric fibers of $G$
are neither simply connected nor adjoint), as is well-known over fields.  For instance,
if $k'/k$ is a nontrivial finite separable extension then there is no such pair 
for $G = {\rm{R}}_{k'/k}({\rm{SL}}_n)/\mu_n$ for any $n > 1$.
\end{remark}


\begin{example}\label{weilfield} Let $k$ be a field and $G \ne 1$ a connected semisimple $k$-group
that is simply connected.  Proposition \ref{allsimple} provides
a canonical isomorphism $G \simeq {\rm{R}}_{k'/k}(G')$ for 
a unique pair $(k'/k, G')$ consisting of a nonzero finite \'etale $k$-algebra
$k'$ and a semisimple $k'$-group $G'$
such that all fibers of $G' \rightarrow \Spec k'$
are connected, simply connected, and absolutely simple.
Using the decomposition into factor
fields $k' = \prod k'_i$ and letting $G'_i$ denote the $k'_i$-fiber of $G'$,
we have $G \simeq \prod_i {\rm{R}}_{k'_i/k}(G'_i)$.
By Example \ref{weilsimple}, these factors are $k$-simple. 
In particular, $G$ is $k$-simple if and only if it has
the form ${\rm{R}}_{k'/k}(G')$ for a finite separable
extension $k'/k$ and a connected semisimple $k'$-group $G'$
that is {\em absolutely simple} and simply connected.

The final part of Proposition \ref{allsimple} shows that the pair $(k'/k,G')$ is canonically attached
to $G$ (not merely up to $k$-isomorphism)
in the sense that it is uniquely functorial with respect to $k$-isomorphisms in such $k$-groups $G$.
An extrinsic characterization of $k'/k$ is given in Exercise \ref{ramsplit}.
\end{example}

\begin{example} Let $G$ be a semisimple $\Z$-group with simply connected fibers.
By Proposition \ref{allsimple}, $G \simeq {\rm{R}}_{A/\Z}(G')$
for a nonzero finite \'etale $\Z$-algebra $A$ and a semisimple $A$-group $G'$
whose fibers are simply connected and absolutely simple.
By Minkowski's theorem, $A = \prod A_i$ with $A_i = \Z$. Thus, 
if $G_i$ denotes the restriction of $G'$ over $\Spec A_i = \Spec \Z$
then ${\rm{R}}_{A/\Z}(G') = \prod G_i$.  Hence,
to classify all such $G$ one loses no generality by restricting attention
to the case when $G$ has absolutely simple fibers (i.e., an irreducible root
system for the geometric fibers).

By the same  reasoning, over any connected normal noetherian scheme $S$
whatsoever, to classify semisimple $S$-groups $G$ with simply connected fibers 
one can pass to the case of $G$ with absolutely simple fibers at the cost of  replacing $S$ with 
some connected finite \'etale covers (namely, the connected components
of $S'$ as in Proposition \ref{allsimple}). 
\end{example}


\newpage

\subsection{Exercises}

%\begin{exer}\label{frob}
%This exercise works out the story behind the Frobenius isogeny, a basic example corresponding
%to a ``$p$-morphism'' of root data.  

%Let $S$ be an $\F_p$-scheme, and $f:X \rightarrow S$ an $S$-scheme.
%Let $X^{(p)} = X \times_{S,F_S} S$ be the base change by the absolute Frobenius
%map $F_S:S \rightarrow S$ ($p$th power on the coordinate ring of every affine open in $S$).
%Note that $F_S \circ f = F_X \circ f$, so it makes sense to define the 
%{\em relative Frobenius} $F_{X/S}:X \rightarrow X^{(p)}$ to be the unique $S$-map
%whose composition with the projection $X^{(p)} \rightarrow X$ is $F_X$.
%Define $X^{(p^n)}$ and $F_{X/S,n}$ similarly using $F_X^n$ and $F_S^n$ for $n > 0$.

%(i) Give a concrete description of $X^{(p^n)}$ and $F_{X/S,n}$ when $X$ is
%a closed subscheme of an affine or projective space over $S$,
%and deduce that $F_{X/S,n}$ is finite when $X$ is finite type over $S$.
%In general prove that $F_{X/S,n}$ is functorial in $X$ over $S$, compatible with base change on
%$S$, and compatible with products over $S$.  In particular, prove $F_{X/S,n}$ is a homomorphism
%when $X$ is an $S$-group (and $X^{(p^n)}$ is equipped with the evident $S$-group structure).

%(ii) Assuming $X$ is $S$-smooth with fibers of pure dimension $d$, prove that
%$F_{X/S,n}$ is finite flat of degree $p^{dn}$. Prove conversely that if
%$G$ is a {\em non-smooth} group of finite type over a field $k$ of characteristic $p$
%then $F_{G/k,n}$ is {\em not} flat (so $G/(\ker F_{G/k,n}) \rightarrow G^{(p^n)}$ is not an isomorphism).  
%Remarkably, by \cite[VII$_{\rm{A}}$, 8.3]{sga3}, 
%$G/(\ker F_{G/k,n})$ is smooth for sufficiently large $n$; why is this plausible?

%(iii) For a torus $T$ over a field $k$ of characteristic $p$, prove that $T$ and $T^{(p)}$ are
%non-canonically isomorphic.  Over any local function field $k$ of characteristic $p$,
%give an example of a connected semisimple $k$-group $G$ such that $G$ is not split but
%$G^{(p)}$ is split (hint: consider central division algebras over $k$, using the {\em functorial} description of
%${\rm{Br}}(k)$).

%(iv) Let $k$ be a field of characteristic $p > 0$, and $G$ a smooth affine $k$-group of dimension
%$d > 0$, so $F_{G/k,n}:G \rightarrow G^{(p^n)}$ is an isogeny of degree $p^{dn}$ by (ii).
%Compute $F_{G/k,n}$ for ${\rm{GL}}(V)$ and ${\rm{SO}}(q)$, and prove
%${\rm{Lie}}(F_{G/k}) = 0$ for any $G$.  Also describe
%the effect of $F_{G/k,n}$ on root data when $G$ is connected reductive with a split maximal torus $T$
%and $G^{(p^n)}$ is equipped with the split maximal torus $T^{(p^n)} = F_{G/k,n}(T)$.
%\end{exer}


\begin{exer}\label{star}
Let $(G,T,M)$ be a split semisimple group  over a non-empty scheme $S$.

(i) Choose $a \in \Phi$. Build a homomorphism
$\phi:{\rm{SL}}_2 \rightarrow Z_G(T_a)$
satisfying $\phi({\rm{diag}}(t,1/t)) = a^{\vee}(t)$.
Show any such $\phi$ is an isogeny onto
$\mathscr{D}(Z_G(T_a))$ carrying the strictly 
upper-triangular subgroup $U^+$ isomorphically onto $U_a$.
[It is insufficient to replace $Z_G(T_a)$ with $G$. Indeed, consider a simply
connected split semisimple $(G,T,M)$ having orthogonal roots $a$ and $b$ 
for which $a^{\vee} + b^{\vee}$ is a coroot, 
such as ${\rm{Sp}}_4$ with $a$ a long positive root
and $b$ the negative of the other long positive root. Writing
$c^{\vee} = a^{\vee} + b^{\vee}$, $G'_{a} := \mathscr{D}(Z_G(T_a)) =
{\rm{SL}}_2$ commutes with 
$G'_{b} = {\rm{SL}}_2$,
and the subgroup $G'_{a} \times G'_{b} \subset G$ contains $c^{\vee}(\Gm)$
as the ``diagonal'' in $a^{\vee}(\Gm) \times b^{\vee}(\Gm)$ but it meets $U_{\pm c}$ trivially.
The diagonal map $\phi:{\rm{SL}}_2 \hookrightarrow {\rm{SL}}_2 \times {\rm{SL}}_2 =
G'_{a} \times G'_{b} \subset G$
restricts to ${\rm{diag}}(t,1/t) \mapsto c^{\vee}(t)$ but 
carries $U^{+}$ ``diagonally'' into $U_{a} \times U_{b}$ rather than into $U_c$.]

Prove $\phi \mapsto {\rm{Lie}}(\overline{\phi})(\begin{smallmatrix} 0 & 1 \\ 0 & 0
\end{smallmatrix})$ is a bijection between the set of such $\phi$ and the set of
global bases $X_a$ of $\mathfrak{g}_a = {\rm{Lie}}(U_a)$, and 
that either all such $\phi$ are isomorphisms or all have the central $\mu_2$ as kernel.
(Hint: First solve the problem over a field.  Then use the self-contained
computation with open cells indicated in Example \ref{wout} to show ${\rm{SL}}_2$ and ${\rm{PGL}}_2$
 have no nontrivial $S$-automorphism that is the identity on the standard upper triangular Borel
subgroup.)

(ii) Under the dictionary in (i), we can pass from a choice of $X_a$ to a choice
of $\phi$, and then to an element
$X_{-a} := {\rm{Lie}}(\phi)(\begin{smallmatrix} 0 & 0 \\ 1 & 0 \end{smallmatrix}) \in \mathfrak{g}_{-a}(S)$.
Prove that $X_{-a}$ is a nowhere-vanishing section of $\mathfrak{g}_{-a}$,
and is the trivialization of $\mathfrak{g}_{-a}$ dual to $X_a$ via the canonical
perfect pairing between $\mathfrak{g}_a$ and $\mathfrak{g}_{-a}$ in Theorem \ref{ssclass}.
\index{linked trivialization}

(iii) Use (i) to give a group-theoretic definition of pinnings that does not mention bases for root spaces;
cf.\,\cite[A.4.12]{pred}.  
\end{exer}

%\begin{exer}\label{chern2} This exercise eliminates the centrality hypothesis
%in Exercise \ref{chern}(iii). 
%
%(i) 
%Using the Existence and Isomorphism Theorems, prove that \'etale-locally on the base 
%any reductive
%group scheme admits a closed immersion into some $\GL_n$.
%
%(ii) Prove that if $X$ and $Y$ are $S$-schemes such that
%$X \times_S Y$ is $S$-flat and $Y(S) \ne \emptyset$ then $X$ is $S$-flat.
%(Hint: a direct factor of a flat module of flat).
%
%(iii) For an affine group $G \rightarrow S$ of finite presentation
%and a multiplicative type group $H$ equipped with an action on $S$,
%prove that the coordinate ring of $Z_G(H)$
%
%(iii) Let $N$ be a closed subgroup scheme of a reductive group scheme $G$,
%and $\lambda:\Gm \rightarrow G$ a homomorphism.  Prove
%that $$
%
%Deduce via \cite[2.1.8(3)]{pred} (and \cite[2.1.11]{pred}) that if $N \subset G$
%is a finitely presented closed normal subgroup
%
%
%\begin{exer}\label{centralexer}
%Fill in the missing details at the end of Example \ref{rootisog}: prove
%that if an isogeny $f:(G',T') \rightarrow (G,T)$ between
%connected reductive groups equipped with maximal tori
%over an algebraically closed field induces
%an isomorphism between the corresponding
%root groups then $\ker f \subset Z_{G'}$.  Then state and prove
%an analogous result for an isogeny between split reductive group
%schemes over a non-empty scheme. 
%\end{exer}

\begin{exer}\label{scuniv}
Using the Existence and Isomorphism Theorems, 
the equivalence in Exercise \ref{scclassical}(ii)
among definitions for ``simply connected'' in the classical case is now proved.
A semisimple group over a non-empty scheme $S$ is {\em simply connected} when its geometric fibers
are simply connected (recovering the definition in Example \ref{scex} in the split case).
\index{reductive group!simply connected}

(i) For any semisimple $S$-group $G$, prove the existence and uniqueness of a central isogeny
$\pi:\widetilde{G} \rightarrow G$ from a semisimple $S$-group $\widetilde{G}$ that is simply connected.
By ``uniqueness'' we mean that for any two such pairs $(\widetilde{G},\pi)$ and $(\widetilde{G}',\pi')$
there exists a {\em unique} isomorphism $\widetilde{G} \simeq \widetilde{G}'$ carrying $\pi$ to $\pi'$.
We call $(\widetilde{G},\pi)$ the {\em simply connected central cover} of $G$.
\index{simply connected!central cover}

(ii) For a semisimple $S$-group $G$, prove that $\widetilde{G}/Z_{\widetilde{G}} \rightarrow G/Z_G$
is an isomorphism and $G \rightarrow G/Z_G$ is uniquely
covered by an isomorphism between the simply connected central covers of
$G$ and $G/Z_G$.  Deduce that the isomorphism classes of $\widetilde{G}$ and $G/Z_G$
determine each other.

(iii) For a simply connected semisimple $S$-group $G$, prove any central extension of
$G$ by a group $H$ of multiplicative type is uniquely split. (Hint: Reduce 
to split $H$. Any central pushout along
an inclusion of $H$ into a torus is reductive and so admits a ``derived group'' that is
semisimple.)  Deduce that any homomorphism between
semisimple $S$-groups uniquely lifts to one between 
simply connected central covers (so $(\widetilde{G},\pi)$ is uniquely functorial in $G$).

(iv) Let $T'$ be a torus in a simply connected semisimple $S$-group $G$.
Prove that $\mathscr{D}(Z_G(T'))$ is simply connected.
(Hint: pass to $S = \Spec k$ for $k = \overline{k}$ and find
$\lambda \in {\rm{X}}_{\ast}(T')$ so that $Z_G(T') = Z_G(\lambda)$. 
Pick a maximal torus $T \supset T'$
and closed Weyl chamber in ${\rm{X}}_{\ast}(T)_{\Q}$ containing $\lambda$, 
with associated bases $\Delta$ and $\Delta^{\vee}$, 
so $\langle a, \lambda \rangle \ge 0$ for all $a \in \Delta$.
Prove that $Z_G(\lambda)$ supports precisely the $T$-roots spanned over
$\Q$ by $\Delta' = \{a' \in \Delta\,|\, \langle a', \lambda \rangle = 0\}$.
Deduce that ${\rm{X}}_{\ast}(\mathscr{D}(Z_G(T')) \bigcap T)$ is spanned over $\Z$ by
${\Delta'}^{\vee}$.)
Using $T' = (\ker a)_{\rm{red}}^0$ for $a \in \Phi(G,T)$, 
show by example that the analogue with ``adjoint'' replacing
``simply connected'' is false.
\end{exer}

\begin{exer}
Is Proposition \ref{sl2ext} valid when ${\rm{SL}}_2$ is replaced by
any semisimple group scheme that is simply connected as in Exercise \ref{scuniv}?
\end{exer}

\begin{exer}\label{special}
If $G$ is a connected semisimple group
over a field $k$ and if $\mathscr{L} \rightarrow G$ is a $\Gm$-torsor with a chosen
basis $e' \in \mathscr{L}(e)$, prove that $\mathscr{L}$ admits a unique
structure of central extension of $G$ by $\Gm$ with identity $e'$.
(The proof imitates the case of abelian varieties after passing to the split case and using the triviality of
line bundles on the open cell, together with a classic result of Chevalley that 
any pointed map from a smooth connected $k$-group to $(\Gm,1)$ is a homomorphism.) 
Does this  generalize to semisimple group schemes $G$ over a non-empty scheme $S$
when a trivialization of $\mathscr{L}$ along $e \in G(S)$ is given?
\end{exer}

\begin{exer}\label{rad} Let $G$ be a connected linear algebraic group over
an algebraically closed field $k$ of characteristic 0.  Avoiding
classification theorems, prove $G$ is semisimple if and only if
${\rm{Lie}}(G)$ is semisimple.  (Hint for ``$\Rightarrow$'':  Let $\mathfrak{r}$ be a solvable Lie ideal.
By the Levi--Malcev Theorem \cite[I, 6.8, Thm.\,5]{lie1}, semisimplicity is equivalent
to the vanishing of such $\mathfrak{r}$. 
Prove $\mathfrak{r}$ is ${\rm{Ad}}_G$-stable, 
and consider its weights under the action of a maximal torus of $G$.
The details are given at the end of the proof of Proposition \ref{mostowprop}.)
\end{exer}

\begin{exer}\label{langexer} Recall Lang's Theorem \cite[16.5(i)]{borelag}:
if a connected linear algebraic group $G$ over a finite field $k$
acts on a finite type $k$-scheme $V$ such that $G(\overline{k})$
is transitive on $V(\overline{k})$ then $V(k) \ne \emptyset$.
Apply it to ${\rm{Tor}}_{G'/k}$ and ${\rm{Bor}}_{G'/k}$
for $G' = G/\mathscr{R}_u(G)$ to prove that $G$ contains a (geometrically) maximal $k$-torus
and a Borel $k$-subgroup.
\end{exer}

\begin{exer}\label{weiltor}
Let $S' \rightarrow S$ be a finite \'etale cover of schemes, $G'$ a reductive $S'$-group,
and $G := {\rm{R}}_{S'/S}(G')$. Prove that $G$ is a reductive $S$-group, and 
adapt the method of proof of Proposition \ref{allsimple} to show
that $T' \mapsto {\rm{R}}_{S'/S}(T')$ is a bijection between the sets of maximal tori of $G'$
and of $G$.  Do likewise for Borel subgroups, and use this to construct natural isomorphisms
${\rm{Tor}}_{G/S} \simeq {\rm{R}}_{S'/S}({\rm{Tor}}_{G'/S'})$ 
and ${\rm{Bor}}_{G/S} \simeq {\rm{R}}_{S'/S}({\rm{Bor}}_{G'/S'})$.
\end{exer}
\index{Weil restriction}
\index{maximal tori!and Weil restriction}
\index{Borel subgroup!and Weil restriction}

\begin{exer}\label{ramsplit}
Let $k$ be a field and $G$ a connected semisimple $k$-group
that is $k$-simple and simply connected.  By 
Example \ref{weilfield}, $G \simeq {\rm{R}}_{k'/k}(G')$ for a 
unique $(k'/k, G')$ where $k'/k$ is finite separable 
and the simply connected semisimple
$k'$-group $G'$ is absolutely simple.

(i) For an extension $L/k$, prove $G_L$ is $L$-split if and only if
$k' \otimes_k L$ is a split $L$-algebra (i.e., power of $L$)
and $G' \otimes_{k',j'} L$ is $L$-split for all $k$-embeddings $j':k' \rightarrow L$.
(Hint: Exercise \ref{weiltor}.) 

(ii) Prove that if $k$ is a global field and $v$ is a place of $k$
such that $G_{k_v}$ is $k_v$-split then $v$ is totally split in $k'/k$.
Deduce that a $k$-simple connected semisimple $k$-group that is split
at all finite places must be absolutely simple. 

(ii) Define a natural action by ${\rm{Gal}}(k_s/k)$ on the set of
$k_s$-simple factors of $G_{k_s}$, and relate $k'$ to the open subgroup
that preserves one of these factors.
\end{exer}

\newpage

\section{Automorphism scheme}\label{autex}

\subsection{Structure of automorphisms}\label{straut}
Many arithmetic properties of non-degenerate quadratic spaces $(V, q)$
of rank $\ge 3$ over number fields
are encoded in terms of the corresponding connected semisimple group
${\rm{SO}}(q)$ (or the disconnected orthogonal group ${\rm{O}}(q)$).
All such groups become isomorphic over $\overline{\Q}$, so the
problem of classifying quadratic forms can be related to the problem of
classifying connected semisimple groups with a fixed geometric isomorphism type
(equivalently, with a fixed root datum for its geometric fiber).
The classification of connected
semisimple $\Q$-groups sharing a common isomorphism type
over $\overline{\Q}$ is controlled by the structure
of automorphism groups of split connected semisimple groups over
number fields.

 To see the link between automorphism groups and classification problems,
consider the situation over a general field $k$.
Let $G$ be a connected reductive $k$-group, so 
$G_{k_s}$ admits a splitting and pinning.
This gives rise to a reduced root datum $R$ whose isomorphism class is intrinsic to $G$
(and can be computed over any separably closed extension of $k$). 
Let $G_0$ be the split connected reductive $k$-group with the same root datum
(so $G_0$ is unique up to $k$-isomorphism). 
Since $G$ and $G_0$ become isomorphic
over a finite Galois extension of $k$ (as both are split over $k_s$
with the same root datum, and hence are isomorphic over $k_s$), 
we view $G$ as a ``$k_s/k$-form'' of $G_0$.  
To classify the possibilities for $G$, we use: 

\begin{lemma}\label{gclass} Let $R$ be a reduced root datum, and $G_0$ a connected
reductive $k$-group with root datum $R$ over $k_s$.  The set of $k$-isomorphism classes of connected reductive
$k$-groups $G$  whose associated root datum over $k_s$ is isomorphic to
 $R$ is in natural bijection with the 
Galois cohomology set ${\rm{H}}^1(k_s/k, \Aut((G_0)_{k_s}))$.
\end{lemma}

\begin{proof}
Choose $G$ and a finite Galois extension $K/k$
so that there is a $K$-group isomorphism
$\varphi:G_K \simeq (G_0)_K$.  Typically this isomorphism
is not defined over $k$ (i.e., it does not descend to a $k$-group isomorphism
$G \simeq G_0$), and to measure this possible failure we examine how
$\varphi$ interacts with the canonical $K$-isomorphisms
$\gamma^{\ast}(G_K) \simeq G_K$ for $\gamma \in {\rm{Gal}}(K/k)$
that encode the $k$-descent $G$ of $G_K$, as well as the
analogous $K$-isomorphisms for $G_0$ (where $\gamma^{\ast}(X)$ denotes
the base change of a $K$-scheme $X$ through scalar extension by $\gamma:K \simeq K$).  
That is, for each $\gamma$ we get a $K$-isomorphism
$$c_{\gamma}:(G_0)_K \simeq \gamma^{\ast}((G_0)_K) \stackrel{\gamma^{\ast}(\varphi)^{-1}}{\simeq}
\gamma^{\ast}(G_K) \simeq G_K \stackrel{\varphi}{\simeq} (G_0)_K,$$
and it is straightforward to check that the cocycle condition
$$c_{\gamma'\gamma} = c_{\gamma'} \circ (\gamma'.c_{\gamma})$$
holds in $\Aut((G_0)_K)$, where $c \mapsto \gamma'.c$  denotes the natural
action of $\gamma' \in {\rm{Gal}}(K/k)$ on $\Aut((G_0)_K)$ through scalar
extension along $\gamma':K \simeq K$ (combined with the descent isomorphism
${\gamma'}^{\ast}((G_0)_K) \simeq (G_0)_K$ defined by the $k$-structure $G_0$ on 
the $K$-group $(G_0)_K$).  

The choice of $(K, \varphi)$ defines
the function $c:{\rm{Gal}}(K/k) \rightarrow \Aut((G_0)_K)$ via $\gamma \mapsto c_{\gamma}$,
and in terms of the non-abelian cohomology conventions in \cite[I, \S5.1]{serre}
this lies in the set ${\rm{Z}}^1(K/k, \Aut((G_0)_K))$ of 1-cocycles on
\index{cocycles}
${\rm{Gal}}(K/k)$ with coefficients in the ${\rm{Gal}}(K/k)$-group $\Aut((G_0)_K)$.
This 1-cocycle depends on $\varphi$, but the other choices
are precisely $a \circ \varphi$ for $a \in \Aut((G_0)_K)$, which leads to the 1-cocycle
$\gamma \mapsto a \circ c_{\gamma} \circ (\gamma.a)^{-1}$.  As we vary through all such $a$,
these cocycles vary through precisely the ones that are cohomologous to $c$.
In particular, the cohomology class $[c] \in {\rm{H}}^1(K/k, \Aut((G_0)_K))$ is
independent of $\varphi$.   Note that $c = 1$ as a function precisely when
$\varphi$ is defined over $k$, and more generally $c$ is a coboundary
$\gamma \mapsto a^{-1} \circ \gamma.a$
for $a \in \Aut((G_0)_K)$ precisely when $a \circ \varphi$ is defined over $k$.

Since Galois descent is effective for affine schemes, 
${\rm{H}}^1(K/k, \Aut((G_0)_K))$ is identified
with the pointed set of isomorphism classes of $k$-groups $G$ such that
there exists a $K$-group isomorphism $G_K \simeq (G_0)_K$.
This is a special case of 
the general formalism of ``twisted forms'' as in \cite[III]{serre}
(i.e., it has nothing to do with $G_0$ being a connected reductive $k$-group), and 
for a $k$-embedding $K \rightarrow K'$ into another finite Galois
extension of $k$ the resulting inflation map of pointed sets
$${\rm{H}}^1(K/k, \Aut((G_0)_{K})) \rightarrow {\rm{H}}^1(K'/k, \Aut((G_0)_{K'}))$$
relaxes existence
of a $K$-isomorphism (to $(G_0)_K$) to existence of a $K'$-isomorphism
(to $(G_0)_{K'}$).
Passing to the direct limit over all $K/k$ inside $k_s/k$ gives the result.
\end{proof}

\begin{remark}\label{kottrem}
If we fix a separable closure
$k_s/k$ then we can define a {\em canonical} based root datum
attached to any (possibly non-split) connected reductive $k$-group
$G$ as follows.  For each pair $(T,B)$ in $G_{k_s}$ consisting
of a (geometrically) maximal $k_s$-torus $T$ and a Borel $k_s$-subgroup $B$ that contains it,
we get an associated based root datum $R(G,T,B)$.  If
$(T',B')$ is another choice then by Proposition \ref{fpqc}(2) there exists $g \in G(k_s)$ such that
$g$-conjugation carries $(T,B)$ to $(T',B')$.  The choice of $g$ is unique up
to $N_G(B)(k_s) \bigcap N_G(T)(k_s) = B(k_s) \bigcap N_G(T)(k_s) = T(k_s)$,
so the induced isomorphism $R(G,T,B) \simeq R(G,T',B')$ of based root data is independent
of the choice of $g$.  More specifically, we get {\em canonical}
isomorphisms among all of the based root data $R(G,T,B)$ as we vary $(T,B)$,
and these isomorphisms are compatible with respect to composition.

By forming the (inverse or direct) limit along
this system of isomorphisms, we get ``the'' based root datum of $G$, to be denoted
$(R(G),\Delta)$.  This generally depends on $k_s/k$ when $G$ is not $k$-split.  That is, if 
$\gamma \in {\rm{Gal}}(k_s/k)$  and $(T,B)$ is a pair in $G_{k_s}$,
then the isomorphism $R(G_{k_s},T,B) \simeq R(G_{k_s}, \gamma^{\ast}(T),\gamma^{\ast}(B))$
defined via scalar extension along $\gamma$ and the $k$-structure $G$ on $G_{k_s}$
may not coincide with the effect of conjugation by an element of $G(k_s)$
that carries $(T,B)$ to $(\gamma^{\ast}(T),\gamma^{\ast}(B))$.  Put
in other terms, we have just defined a natural action of
$\gamma \in {\rm{Gal}}(k_s/k)$ on $(R(G),\Delta)$ via
$$(R(G),\Delta) = R(G_{k_s},T,B) \simeq R(G_{k_s}, \gamma^{\ast}(T), \gamma^{\ast}(B))
= (R(G),\Delta);$$
this is independent of the choice of $(T,B)$ and may be nontrivial.  It 
is the ``$\ast$-action'' of
${\rm{Gal}}(k_s/k)$ on ${\rm{Dyn}}(\Phi)$ that appears
in the Borel--Tits approach to classifying isotropic connected semisimple groups over fields
(modulo the ``anisotropic kernel'').

When $k_s/k$
\index{based root datum}
\index{ aaargdelta@$(R(G), \Delta)$}
is understood from context, $(R(G),\Delta)$ is called the {\em based root datum} for $G$.
By the Existence and Isomorphism Theorems, up to unique $k$-isomorphism
there exists a unique pinned split connected reductive $k$-group $(G_0,T_0,M_0,\{X_a\}_{a \in \Delta})$ 
with based root datum $(R(G),\Delta)$.
As we vary through the different choices for the base,
the resulting pinned split $k$-groups are canonically isomorphic
to each other. This all depends on $k_s/k$ except when
$G$ is $k$-split.  The dependence on $k_s/k$ can be
eliminated by working with suitable \'etale $k$-schemes,
such as the finite \'etale $k$-scheme associated
to $\Delta$ equipped with the  above action of ${\rm{Gal}}(k_s/k)$; i.e., 
the ``scheme of Dynkin diagrams'' ${\rm{Dyn}}(G)$; see Example \ref{kottscheme}.)
\index{ aaadyng@${\rm{Dyn}}(G)$}
\end{remark}

The preceding considerations can be adapted to a general base scheme.
To explain this, it is convenient to introduce some notation.

\begin{definition}
If $G \rightarrow S$ is a group scheme, 
its {\em automorphism functor} $\underline{\Aut}_{G/S}$ on
the category of $S$-schemes  is  $\underline{\Aut}_{G/S}: S' \rightsquigarrow 
{\rm{Aut}}_{S'\mbox{-}{\rm{gp}}}(G_{S'})$. A representing
object (if one exists) is denoted $\Aut_{G/S}$ and is called
the {\em automorphism scheme} of $G$.
\end{definition}
\index{ aaaautgs@$\underline{\rm{Aut}}_{G/S}$,
${\rm{Aut}}_{G/S}$}
\index{automorphism scheme}

The automorphism functor is a sheaf for the fppf (and even fpqc) topology, and 
we will see that representability and structural properties of the automorphism functor
in the case of split reductive $G$ are extremely useful in the classification of all reductive $G$.
As an example, we will show that $\Aut_{{\rm{SL}}_2/S}$
exists and is identified with ${\rm{PGL}}_2$ (see Theorem \ref{autscheme}(3)).
Note that in general $\underline{\Aut}_{G/S}(S)$ is the automorphism group 
 of $G$ in the usual sense, which we also denote as
${\rm{Aut}}(G)$ (as for any category). 

Consider a reductive group $G$ over 
a {\em connected} non-empty scheme $S$.  Connectedness of the base ensures that all geometric
fibers have the same reduced root datum $R$ (see the proof of Lemma \ref{zarroot}),
so it makes sense to define $G_0$ to be the split
reductive $S$-group with root datum $R$.  (This
defines $G_0$ uniquely up to $S$-isomorphism, by the Isomorphism Theorem.)  The $S$-groups 
$G$ and $G_0$ become isomorphic \'etale-locally on $S$,
due to the Isomorphism Theorem and the existence \'etale-locally on $S$ 
of splittings and pinnings for $G \rightarrow S$.

For any sheaf of groups $\calF$ on $S_{\et}$, we define
the set ${\rm{Z}}^1(S'/S,\calF)$ of  {\em \v{C}ech $1$-cocycles} to consist of those
$\xi \in \calF(S' \times_S S')$ such that $p_{13}^{\ast}(\xi) = p_{23}^{\ast}(\xi) p_{12}^{\ast}(\xi)$.
A pair of such 1-cocycles  $\xi$ and $\xi'$ are {\em cohomologous},
denoted $\xi \sim \xi'$, if 
there exists $g \in \calF(S')$ such that $\xi' = p_2^{\ast}(g) \xi p_1^{\ast}(g)^{-1}$.
It is straightforward to check that $\sim$ is an equivalence relation.  The quotient
set by this relation is denoted ${\rm{H}}^1(S'/S,\calF)$, and it has evident
functoriality in $S'$ over $S$.  In concrete terms,
${\rm{Z}}^1(S'/S,\calF)$ is the set of descent data relative to $S' \rightarrow S$ on the \'etale
sheaf of sets $\calF|_{S'_{\et}}$ equipped with its right $\calF$-translation action.
Thus, by effective descent for \'etale sheaves
we see that the set of 1-cocycles is naturally identified with
the set of isomorphism classes of pairs
$(\mathscr{E},\theta)$ consisting of a right $\calF$-torsor $\mathscr{E}$ on $S_{\et}$
and an element $\theta \in \mathscr{E}(S')$.

The relation $\sim$ encodes the property that two $\calF$-torsors
on $S_{\et}$ are isomorphic.  Hence, 
${\rm{H}}^1(S'/S,\calF)$ is the set of isomorphism classes of right $\calF$-torsors 
on $S_{\et}$ that admit a section over $S'$.
This interpretation shows that functoriality with respect to 
$S$-maps $S'' \rightarrow S'$ turns the pointed sets
${\rm{H}}^1(S'/S, \calF)$ into a directed system
relative to the partial order of refinement among
covers $S' \rightarrow S$ (i.e., these transition maps
do not depend on the specific $S$-maps between such covers). 
Thus, it makes sense to form the direct limit
\begin{equation}\label{h1s}
\check{\rm{H}}^1(S_{\et},\calF) :=
\varinjlim_{S'/S} {\rm{H}}^1(S'/S, \calF).
\end{equation}
\index{ aaah1sf@${\rm{H}}^1(S'/S,\calF)$, $\check{\rm{H}}^1(S_{\et},\calF)$}


\begin{example}
Let $G_0$ be a reductive
$S$-group.  The set ${\rm{Z}}^1(S'/S, {\rm{Aut}}_{G_0/S})$ is 
identified with the set of \'etale descent data on $(G_0)_{S'}$
relative to $S' \rightarrow S$, and such descent data are effective
since $(G_0)_{S'}$ is $S'$-affine. The 
equivalence relation $\sim$ encodes that 
two descent data have isomorphic $S$-descents, so 
$\check{\rm{H}}^1(S'/S, {\rm{Aut}}_{G_0/S})$ is identified
(functorially in $S'$ over $S$) with the set of $S$-isomorphism
classes of reductive $S$-groups $G$ such that $G_{S'} \simeq (G_0)_{S'}$.
The dictionary relating right ${\rm{Aut}}_{G_0/S}$-torsors over $S_{\et}$
and $S$-forms $G$ of $G_0$ is that to $G$ we associate the right 
${\rm{Aut}}_{G_0/S}$-torsor ${\rm{Isom}}(G_0,G)$
classifying group scheme isomorphisms from $G_0$ to $G$ over $S$-schemes.

For split $G_0$ with root datum $R$, 
${\rm{H}}^1(S'/S, {\rm{Aut}}_{G_0/S})$ 
is the set of isomorphism classes of reductive $S$-groups that split
over $S'$ and have geometric fibers with root datum $R$.
Hence, the set of isomorphism classes of reductive $S$-groups $G$ having geometric
fibers with root datum isomorphic to $R$ is naturally identified 
with the \'etale cohomology set
$$
{\rm{H}}^1(S_{\et},\underline{\Aut}_{G_0/S}) := 
\varinjlim_{S'/S} \mbox{\v{H}}^1(S'/S,\underline{\Aut}_{G_0/S}),
$$ 
where we vary through a cofinal set of  \'etale covers $S' \rightarrow S$.
(See Exercise \ref{moreform}(i) and Exercise \ref{h1ex}(i)) for further discussion.)
When $S$ is connected we only need to compute the root datum on a single fiber, due
to Lemma \ref{zarroot}.
\end{example}


%
%Since \'etale descent is effective for schemes affine over the base, we can
%use the same formalism as over fields to identify the set
%of isomorphism classes of $G$ having fibral root datum 
%isomorphic to $R$ 
%with the \'etale cohomology set
%\begin{equation}\label{h1s}
%{\rm{H}}^1(S_{\et},\underline{\Aut}_{G_0/S}) := 
%\varinjlim_{S'/S} \mbox{\v{H}}^1(S'/S,\underline{\Aut}_{G_0/S}),
%\end{equation}
%where we vary through (a cofinal set of) ``all'' \'etale covers $S' \rightarrow S$
%(see Exercise \ref{moreform}(i) and Exercise \ref{h1ex}(i)).
%Explicitly, the \v{C}ech-type cohomology set $\mbox{\v{H}}^1(S'/S,\underline{\Aut}_{G_0/S})$ is
%identified with the set of $S$-isomorphism
%classes of those $G$ that become split over $S'$. This interpretation explains
%why the transition maps in the direct limit (\ref{h1s}), defined by pullback along 
%$S$-maps $S'' \rightarrow S'$, form a directed system
%relative to the partial order of refinement among
%covers $S' \rightarrow S$ (i.e., these transition maps
%do not depend on the specific $S$-maps between such covers).  This is analogous to 
%the canonicity of transition maps defining \v{C}ech cohomology on ordinary topological
%spaces as a direct limit. 

The classical theory over an algebraically closed field $k$ does not address representability properties
of the automorphism functor on $k$-algebras, but it does suggest that the automorphism
group over $k$ should be viewed as having a ``geometric'' structure.
More specifically (by Proposition \ref{autdyn}, the identification 
(\ref{autg}), and Proposition \ref{pinsd}), 
if $G$ is a simply connected (and connected) semisimple group over an algebraically closed
field $k$ and $\Phi$ is its root system then $\Aut(G)$ is naturally the group of $k$-points of 
\begin{equation}\label{autvar}
(G/Z_G) \rtimes {\rm{Aut}}({\rm{Dyn}}(\Phi)),
\end{equation} where the
semi-direct product structure rests on a choice of splitting and pinning.
If we want to use (\ref{autvar})
as anything deeper than a bookkeeping device (e.g., exploit that $G/Z_G$ is connected), we should
prove representability of $\underline{\Aut}_{G/k}$ 
and not just make a construction that has the ``right'' geometric points.
This will be done in Theorem \ref{autscheme}. 

As a prelude to 
the case of a general base scheme, now consider 
the problem of describing $\underline{\Aut}_{G/k}(k) = \Aut(G)$
for a split connected
reductive group $G$ over a general field $k$.  This goes beyond the 
classical arguments related to (\ref{autvar}) over algebraically closed fields
because maps such as $G(k) \rightarrow (G/Z_G)(k)$
can fail to be surjective when $k \ne \overline{k}$
(e.g., ${\rm{SL}}_n(k) \rightarrow {\rm{PGL}}_n(k)$
for any field $k$ such that $k^{\times}$ is not $n$-divisible). 

Let $(G, T, M)$ be a split connected reductive group over
a field $k$.  The action of $G$ on itself by conjugation factors
through an action of $G/Z_G$ on $G$.  This identifies
the adjoint quotient $G/Z_G$ with a subfunctor of
the automorphism functor $\underline{\Aut}_{G/k}$,
so $(G/Z_G)(k)$ is a subgroup of $\underline{\Aut}_{G/k}(k) = \Aut(G)$.
Note that this works even when $G(k) \rightarrow (G/Z_G)(k)$ fails to be surjective,
and that it defines a normal subgroup of $\Aut(G)$.
(See Exercise \ref{gzsurj}.)  

Due to the canonicity of the associated based root datum $(R(G),\Delta)$
as in Remark \ref{kottrem}, $\Aut(G)$ naturally acts on $(R(G),\Delta)$.
Here is how it works in concrete terms, by identifying $\Aut(G)/(G/Z_G)(k)$
with $\Aut(R(G),\Delta)$.  
Pick a positive system of roots $\Phi^+$ in $\Phi = \Phi(G,T)$, with base denoted $\Delta$.
Equivalently, choose a Borel subgroup $B$ in $G$ containing $T$ (with $\Phi(B,T) = \Phi^+$). 
Let $\phi$ be an arbitrary automorphism of $G$, so $T' := \phi(T)$ is a {\em split} (geometrically) maximal 
$k$-torus of $G$ and $B' := \phi(B)$ is a Borel $k$-subgroup of
$G$ containing $T'$.  By Proposition \ref{fpqc}(2), there exists $g \in G(k)$ such that
$T' = gTg^{-1}$ and $B' = gT g^{-1}$.
Thus, by composing $\phi$ with an automorphism arising from $G(k)$
we can arrange that $\phi(T) = T$ and $\phi(\Phi^+) = \Phi^+$.
These additional requirements
are preserved under composition of $\phi$ with the action of an element $g \in (G/Z_G)(k)$
if and only if $g \in (T/Z_G)(k)$, 
since $B/Z_G = N_{G/Z_G}(B/Z_G)$ and $N_{G/Z_G}(T/Z_G) \bigcap (B/Z_G) = T/Z_G$. 

The automorphism of the based root
datum $(R(G,T), \Delta)$ induced by this $\phi$ is unaffected by
composing $\phi$ with the action of $(T/Z_G)(k)$
since the $T/Z_G$-action on $G$ is trivial on $T$.  The resulting action via $\phi$ on the set
$\Delta$ has more structure: it is an automorphism of the Dynkin diagram
${\rm{Dyn}}(\Phi)$.  This can be nontrivial:

\begin{example}\label{slaut}
Let $G = {\rm{SL}}_n$, and let $w \in G(k)$ be the anti-diagonal matrix
whose entries alternate $1, -1, 1, -1, \dots$ beginning in the upper right
(so $w w^{\top} = 1$). 
The automorphisms $g \mapsto (g^{\top})^{-1}$
and $g \mapsto w g w^{-1}$ of $G$ swap the upper triangular
and lower triangular Borel subgroups while preserving the diagonal torus $D$.  The composite automorphism
$\iota: g \mapsto w (g^{\top})^{-1} w^{-1}$ is an involution of $G$ that preserves
$D$ and the upper triangular Borel subgroup, inducing 
the involution of the Dynkin diagram when $n > 2$.
\end{example}

Returning to a general split $(G, T)$ over $k$, 
let $\{X_a\}_{a \in \Delta}$ be a pinning of $(G,T,B)$. For $X'_a := {\rm{Lie}}(\phi)(X_{\phi^{-1}(a)})$
we get another pinning $\{X'_a\}_{a \in \Delta}$.  Clearly
$X'_a = c_ a X_a$ for a unique $(c_a) \in (k^{\times})^{\Delta}$.  
\index{reductive group!pinning}
\index{pinning}
Since
$\Delta$ is a $\Z$-basis of the character group $\Z\Phi$ of 
the split maximal torus $T/Z_G \subset G/Z_G$ (Corollary \ref{computez}(1)),
we have $T/Z_G \simeq \prod_{a \in \Delta} \Gm$ via $t \bmod Z_G \mapsto (a(t))_{a \in \Delta}$.
Hence, $(T/Z_G)(k)$ acts simply transitively on the set of all pinnings
of $(G,T,B)$, so by composing $\phi$ with a {\em unique} automorphism arising
from $(T/Z_G)(k)$ we can arrange that ${\rm{Lie}}(\phi)(X_a) = X_{\phi(a)}$ for all
$a \in \Delta$ (i.e., $X'_a = X_a$ for all $a \in \Delta$).  

Let $\Theta = \Aut(R(G,T),\Delta)$ be the automorphism group of
the based root datum, so by Corollary \ref{faithfulroot} the conditions we have imposed
on $\phi$ relative to $(T, B, \{X_a\}_{a \in \Delta})$ make it determined {\em uniquely} by its image in $\Theta$.
That is, we have identified $\Aut(G)/(G/Z_G)(k)$ with a subgroup of $\Theta$.
In fact, this subgroup inclusion is an equality, or
equivalently {\em every} element of
$\Theta$ arises from a unique $\phi$ satisfying the conditions imposed relative
to $(T,B, \{X_a\}_{a \in \Delta})$.  This is exactly the content of
the precise form of the Isomorphism Theorem given in Theorem \ref{isomthm}.
In other words, the choice of pinning defines
an injective homomorphism $\Theta \hookrightarrow \Aut(G)$
with image equal to the automorphism group of
$(G,T,B, \{X_a\}_{a \in \Delta})$, so
$\Theta$ is carried to a subgroup of
$\Aut(G)$ that preserves $T$ and $B$ and maps isomorphically onto $\Aut(G)/(G/Z_G)(k)$.

We have just constructed an isomorphism of groups 
\begin{equation}\label{gzaut}
(G/Z_G)(k) \rtimes \Aut(R(G,T),\Delta) \simeq \Aut(G).
\end{equation}
This depends on the choice of pinning, as well as
the split hypothesis on $G$.  (Although 
the inclusion of $(G/Z_G)(k)$ as a normal subgroup of $\Aut(G)$
is available without any split hypothesis on $G$,
in the non-split case there may be no homomorphic section to $\Aut(G) \rightarrow
\Aut(G)/(G/Z_G)(k)$; see Example \ref{outk}.)  
Since the 
preceding discussion is compatible
with extension of the ground field, we likewise have
$(G/Z_G)(K) \rtimes \Aut(R(G,T),\Delta) \simeq \Aut(G_K) = \underline{\Aut}_{G/k}(K)$ for
any extension field $K/k$.
To summarize, we have proved:

\begin{proposition} Let $(G,T,M,\{X_a\}_{a \in \Delta})$ be a pinned
split connected reductive group over a field $k$.
The group $\Aut(G)$ is naturally an extension of
$\Aut(R(G,T),\Delta)$ by $(G/Z_G)(k)$,
and the pinning naturally splits this extension as a semi-direct product.
\end{proposition}

\begin{remark}
By Proposition \ref{autdyn}, if $(R,\Delta)$ is a semisimple based root
datum with root system $\Phi$ then 
$\Aut(R,\Delta) \subset {\rm{Aut}}({\rm{Dyn}}(\Phi))$,
with equality when $R$ is simply connected or adjoint
or when the ``fundamental group''
$(\Z\Phi^{\vee})^{\ast}/\Z\Phi$ is cyclic (which includes
all irreducible reduced root systems except ${\rm{D}}_{2n}$, $n \ge 2$). 
\end{remark}

\begin{example}\label{wout}
The root system $\Phi = {\rm{A}}_{n-1}$ ($n \ge 2$) has cyclic ``fundamental group''
$(\Z\Phi^{\vee})^{\ast}/(\Z\Phi) \simeq \Z/n\Z$ and 
$\Aut({\rm{Dyn}}(\Phi))$ is equal to $\Z/2\Z$ 
when $n > 2$ (and $\{1\}$ when $n = 2$).  Thus, 
$\Aut({\rm{SL}}_n) = \PGL_n(k) \rtimes (\Z/2\Z)$
for $n > 2$, with the factor
 $\Z/2\Z$ generated by the involution $g \mapsto (g^{\top})^{-1}$ (due to Example \ref{slaut}). 
This involution is available for ${\rm{SL}}_2$, but in that case
it is inner (arising from conjugation by
$w = (\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix})$). 

[The equality ${\rm{Aut}}_k({\rm{SL}}_2) = {\rm{PGL}}_2(k)$ for $n = 2$
can be proved in elementary terms, as follows.   Using $k$-rational conjugacy
results, it suffices to show that an automorphism $f$ of ${\rm{SL}}_2$
that is the identity on the upper triangular Borel subgroup $B^+$
is the identity on the open cell $\Omega^+$ (and hence is the identity).
Certainly $f$ preserves the opposite Borel subgroup $B^-$ relative to the diagonal torus,
and so preserves its unipotent radical $U^-$.
The effect on $U^{-}$ must be $(\begin{smallmatrix} 1 & 0 \\ x & 1 \end{smallmatrix})
\mapsto (\begin{smallmatrix} 1 & 0 \\ cx & 1 \end{smallmatrix})$ for
some $c \in k^{\times}$.  But the two standard open cells
$\Omega^+$ and $\Omega^{-}$ have intersection defined
by $1 + xy \ne 0$ inside
$\Add \times \Gm \times \Add = \Omega^+$ via
$$(y,t,x) \mapsto \begin{pmatrix} 1 & 0 \\ y & 1 \end{pmatrix}
\begin{pmatrix} t & 0 \\ 0 & 1/t \end{pmatrix}
\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix},$$
so we must have that the open loci $1 + xy \ne 0$
and $1 + cxy \ne 0$ on $\Add \times \Add$ coincide.
This forces $c = 1$, as desired.]

In Example \ref{splitex} we will compute the section
$\Z/2\Z = \Aut({\rm{Dyn}}(\Phi)) \hookrightarrow
\Aut({\rm{SL}}_n)$ associated to a ``standard'' choice
of splitting and pinning for ${\rm{SL}}_n$.
This section carries the diagram involution to
$\iota$ from Example \ref{slaut}.
In particular, the ``coordinate dependence''
of the involution $\iota$ reflects
the fact that the semi-direct product structure in (\ref{gzaut}) is not intrinsic to
$G$ or even to $(G,T)$: it 
depends on the choice of the splitting and pinning.

The same conclusions apply  to $\Aut(\PGL_n)$,
as well as to $\Aut({\rm{SL}}_n/\mu)$ for any $k$-subgroup $\mu \subset \mu_n$.
This illustrates that automorphisms of a connected
semisimple $k$-group lift uniquely to automorphisms of
the simply connected central cover (Exercise \ref{scuniv}(iii)).  
\end{example}


The preceding arguments suggest that in the split case we should  represent
$\underline{\Aut}_{G/k}$ by a smooth 
$k$-group with identity component $G/Z_G$
and constant component group $\Aut(R,\Delta)$, where
$R = R(G,T)$.
In particular, if $G$ is semisimple
then $\underline{\Aut}_{G/k}$ should be represented by a linear algebraic
$k$-group having constant component group 
that is a subgroup of the finite group ${\rm{Aut}}({\rm{Dyn}}(\Phi))$.
This can be made rather concrete: define the $G/Z_G$-group isomorphism
$$\alpha^0:(G/Z_G) \times G \rightarrow (G/Z_G) \times G$$
by passage to the $Z_G$-quotient on the
first factor relative to the map of $G$-group schemes $(g,g') \mapsto (g, gg'g^{-1})$, and use
 translation against the injection 
$\Aut(R,\Delta) \hookrightarrow \Aut(G)$ (defined by a choice of pinning) to obtain a ``universal automorphism'' 
$$\alpha:((G/Z_G) \rtimes \Aut(R,\Delta)) \times G \simeq
((G/Z_G) \rtimes \Aut(R,\Delta)) \times G$$
of group schemes over $(G/Z_G) \rtimes \Aut(R,\Delta)$.  For 
$\xi \in ((G/Z_G) \rtimes \Aut(R,\Delta))(K)$
(with a field extension $K/k$), the fiber map $\alpha_{\xi}$ is the automorphism
corresponding to $\xi$ via the analogue of
(\ref{gzaut}) for $K$-valued points. 

This construction defines a morphism of $k$-group functors
\begin{equation}\label{univaut}
(G/Z_G) \rtimes \Aut(R,\Delta) \rightarrow \underline{\Aut}_{G/k},
\end{equation}
and the classical approach would end here since
this map has been constructed to be bijective on points valued
in any extension field $K/k$ (which is sufficient
for many applications).   For problems
related to deforming automorphisms or working
over $k$-algebras that are not fields, it is useful to
go beyond field-valued points and prove that (\ref{univaut}) is an isomorphism
of functors on $k$-algebras. 

Now we turn to the general case.
Let $G$ be a reductive $S$-group, with $S \ne \emptyset$.
For the maximal central torus $Z$ and semisimple derived group $G' = \mathscr{D}(G)$,
the multiplication map $Z \times G' \rightarrow G$ is a central isogeny  
(Corollary \ref{isogcor}).
The kernel $\mu$ of this isogeny is a finite $S$-group of multiplicative type,
so an automorphism of $G$
is ``the same'' as a pair of automorphisms 
of $G'$ and $Z$ that coincide on $\mu$.  
By Exercise \ref{finiteaut}, $\underline{\Aut}_{\mu/S}$ is a finite \'etale $S$-group
and $\underline{\Aut}_{Z/S}$ is represented by a separated \'etale $S$-group
that is an \'etale form of $\GL_r(\Z)$ where $r$ is the rank of $Z$ (locally constant on $S$).
Hence, to understand 
properties of $\underline{\Aut}_{G/S}$, the real content is in the semisimple case.
Here is the main result.

\begin{theorem}\label{autscheme} Let $G$ be a reductive group over a non-empty scheme
$S$.
\begin{enumerate}
\item The functor $\underline{\Aut}_{G/S}$ is represented by a separated and smooth $S$-group
$\Aut_{G/S}$ that fits into a short exact sequence 
$$1 \rightarrow G/Z_G \rightarrow \Aut_{G/S} \rightarrow {\rm{Out}}_{G/S} \rightarrow 1$$
where: $G/Z_G$ is closed in $\Aut_{G/S}$,
\index{ aaaoutgs@${\rm{Out}}_{G/S}$}
\index{reductive group!automorphism scheme}
\index{automorphism scheme|(}
${\rm{Out}}_{G/S}$ is a separated  \'etale $S$-group that is locally constant for
the \'etale topology on $S$.  If $S$ is locally noetherian and normal then every connected
component of ${\rm{Out}}_{G/S}$ is $S$-finite.
\item 
The $S$-group ${\rm{Out}}_{G/S}$ has finite geometric fibers if and only if 
the maximal central torus of $G$ has rank $\le 1$, in which case
${\rm{Out}}_{G/S}$ is $S$-finite and $\Aut_{G/S}$ is $S$-affine.
\item For pinned split reductive $(G,T,M,\{X_a\}_{a \in \Delta})$ over $S$, 
${\rm{Out}}_{G/S}$ is identified with the constant $S$-group associated to 
$\Aut(R(G,T,M),\Delta)$.
Moreover, the pinning defines a semi-direct product
splitting of $S$-groups
$$\Aut_{G/S} \simeq (G/Z_G) \rtimes \Aut(R(G,T,M),\Delta)_S.$$
\end{enumerate}
\end{theorem}

\begin{proof}
\'Etale descent is effective for schemes that are separated and \'etale over the base.
(Indeed, one can reduce to the finite type case, which is \cite[IX, 4.1]{sga1}.
Alternatively, the descent trivially exists as an algebraic space, and 
any algebraic 
space that is separated and locally quasi-finite over a scheme is a scheme \cite[A.2]{lmb}.)
Also, any fppf group-sheaf extension $E$ of an $S$-group scheme $H$ by an fpqc $S$-affine group scheme
is necessarily representable since
$E$ is an algebraic space that is affine over $H$ and hence is a scheme.
(Algebraic spaces are not needed to prove that $E$ is a scheme; one
just has to use the effectivity of fpqc descent for schemes affine over the base.) 
Thus, for the proof of (1) apart from the final assertion it suffices to work \'etale-locally on $S$.  It is likewise
enough to work \'etale-locally on $S$ for the proof of (2). 

For the proof of the entire theorem apart from the final assertion in (1), 
it is now enough to work with a pinned split group $(G,T,M,\{X_a\}_{a \in \Delta})$.
Thus, we have the inclusion of group sheaves
$(G/Z_G) \rtimes {\rm{Aut}}(R(G,T,M),\Delta)_S \subset {\rm{Aut}}_{G/S}$
for the fppf topology on the category of $S$-schemes.  Hence, to prove that this is an equality
on $S'$-points for $S$-schemes $S'$ it suffices to work Zariski-locally on $S'$.
It suffices to treat local rings on $S'$ (as finite presentation then
permits us to ``spread out'' from local rings to Zariski-open subschemes of $S'$).

The arguments leading up to the proof of (\ref{gzaut}) were written to apply verbatim
at the level of $S'$-points over any local $S$-scheme $S'$, upon noting
that Proposition \ref{fpqc}(2) is applicable to any local scheme.
To conclude, we just need to make three observations:  (i) the 
semi-direct product structure in (3) implies
that $G/Z_G$ is closed in $\Aut_{G/S}$, 
(ii) a constant $S$-group is $S$-finite
if and only if it has finite geometric fibers,
(iii) a based root datum $(R,\Delta)$ has finite automorphism group if and only if
the underlying root datum $R = (M,\Phi,M^{\vee},\Phi^{\vee})$
satisfies $\dim M_{\Q}/\Q\Phi \le 1$ (since $\GL_1(\Z)$ is finite). 

It remains to prove the final assertion in (1).  More generally, let $S$ be
a locally noetherian scheme that is normal, and let $E \rightarrow S$ be an $S$-scheme
that becomes constant \'etale-locally on $S$.  We claim that every connected component of
$E$ is $S$-finite. We may assume that $S$ is connected, so it is irreducible. Let $C$ be a connected component of
$E$, so $C$ is normal and hence irreducible.  Thus, if $U$ is a non-empty open subscheme of $S$ then
$C_U$ is (non-empty and hence) a connected component of $E_U$.   Our problem is therefore Zariski-local on $S$, so we can
assume $S$ is affine.  We may choose an \'etale covering $\{S_i \rightarrow S\}$ 
with connected affine $S_i$ such that $E_{S_i}$ is a constant $S_i$-scheme.  The open images of the $S_i$'s cover $S$, 
and we can replace $S$ with each of those separately.  That is, we may assume there is an \'etale cover
$S' \rightarrow S$ with connected affine $S'$ such that $E_{S'}$ is a constant $S'$-scheme. The open and closed subscheme
$C_{S'}$ inside $E_{S'}$ must be a union of connected components and hence a disjoint union of copies of $S'$.  Since the map
$C \rightarrow S$ is finite if and only if $C_{S'} \rightarrow S'$ is finite,
it suffices to show that $C_{S'}$ has only finitely many connected components.  For this purpose we may pass
to the fiber over the generic point of $S'$, or equivalently work over the generic point of $S$.  Now
we may assume $S = {\rm{Spec}}(K)$ for a field $K$, so the \'etale $K$-scheme $E$ clearly has $K$-finite connected components.
\end{proof}
\index{automorphism scheme|)}

\begin{example}\label{splitex}
Let $(G,T,M,\{X_a\}_{a \in \Delta})$ be a pinned split semisimple $S$-group
whose root system $\Phi \subset M$ is irreducible and not ${\rm{D}}_{2n}$
$(n \ge 2)$.  By Proposition \ref{autdyn} we have
${\rm{Aut}}(R(G,T),\Delta) = {\rm{Aut}}({\rm{Dyn}}(\Phi))$,
so $\Aut_{G/S} = (G/Z_G) \rtimes \Aut({\rm{Dyn}}(\Phi))_S$. 

We make this explicit for $G = {\rm{SL}}_n$ with the 
diagonal torus $T = D_S(M)$ for $M = \Z^n/{\rm{diag}}(\Z)$, 
$\Delta$ corresponding to the upper triangular Borel subgroup $B$,
and the standard pinning $\{X_a\}_{a \in \Delta}$.
If $n = 2$ then the diagram is a point and so
the automorphism scheme of ${\rm{SL}}_2$ is ${\rm{SL}}_2/\mu_2 = {\rm{PGL}}_2$
with its evident action.

If $n \ge 3$ then we claim that the associated section ${\rm{Aut}}({\rm{Dyn}}(\Phi))_S \rightarrow \Aut_{{\rm{SL}}_n}$
carries the diagram involution $\varphi$ to 
the involution $\iota$ of ${\rm{SL}}_n$ from Example \ref{slaut}. 
Since $\iota$ is an involution that preserves $(T, B)$ and induces
the nontrivial involution on ${\rm{Dyn}}(\Phi)$, we just have to check
that its effect on root groups is a  permutation of the 
$X_a$'s (without the intervention of signs).  The standard root group $U_{ij}$ for $i < j$ is carried
by $\iota$ to $U_{n+1-j, n+1-i}$, and relative to the standard
parameterizations $\Add \simeq U_c$ the isomorphism
$U_{ij} \simeq U_{n+1-j,n+1-i}$ goes over to the automorphism
$x \mapsto (-1)^{1+j-i}x$.  Thus, for $j = i+1$ we get $\iota(X_a) = X_{\varphi(a)}$ as desired.
\end{example}
%
%
%For example, consider $G = {\rm{SL}}_n$ with the 
%diagonal torus $T = D_S(M)$ for $M = \Z^n/{\rm{diag}}(\Z)$, 
%$\Delta$ corresponding to the upper triangular Borel subgroup $B$,
%and the standard pinning $\{X_a\}_{a \in \Delta}$.
%We claim that the associated section ${\rm{Aut}}({\rm{Dyn}}(\Phi))_S \rightarrow \Aut_{{\rm{SL}}_n}$
%carries the diagram involution $\varphi$ to 
%the involution $\iota$ of ${\rm{SL}}_n$ from Example \ref{slaut}.  
%
%Since $\iota$ is an involution that preserves $(T, B)$ and induces
%the nontrivial involution on ${\rm{Dyn}}(\Phi)$, we just have to check
%that its effect on root groups is a  permutation of the 
%$X_a$'s.  The standard root group $U_{ij}$ for $i < j$ is carried
%by $\iota$ to $U_{n+1-j, n+1-i}$, and relative to the standard
%parameterizations $\Ga \simeq U_c$ the isomorphism
%$U_{ij} \simeq U_{n+1-j,n+1-i}$ goes over to the automorphism
%$x \mapsto (-1)^{1+j-i}x$.  Thus, for $j = i+1$ we get $\iota(X_a) = X_{\varphi(a)}$ as desired.
%\end{example}

\begin{example}\label{kottscheme}
Let $(G,T,M,\{X_a\}_{a \in \Delta})$ be a pinned split reductive $S$-group,
with $\Phi \subset M$ the set of roots.
The composite map ${\rm{Out}}_{G/S} \simeq \Aut(R(G,T,M),\Delta)_S \rightarrow
\Aut({\rm{Dyn}}(\Phi))_S$ into the constant group associated
to the finite automorphism group of the Dynkin diagram of $\Phi$ induces a map
$${\rm{H}}^1(S_{\et},\Aut_{G/S}) \rightarrow {\rm{H}}^1(S_{\et}, \Aut({\rm{Dyn}}(\Phi))_S)$$
into the pointed set of $S$-twisted forms of the Dynkin diagram.  More specifically,
if $G'$ is an $S$-form of $G$ then its class $[G']$ in
${\rm{H}}^1(S_{\et},\Aut_{G/S})$ gives rise to a finite \'etale $S$-scheme
${\rm{Dyn}}(G')$, the ``scheme of Dynkin diagrams'' for $G'$.
\index{scheme of Dynkin diagrams}
\index{Dynkin diagram}
The edges with multiplicity on geometric fibers are encoded by
a finite \'etale closed subscheme ${\rm{Edge}}(G') \subset {\rm{Dyn}}(G') \times {\rm{Dyn}}(G')$
disjoint from the diagonal (no loops) and an $S$-map ${\rm{Dyn}}(G') \rightarrow
\{1,2,3\}_S$ that assigns ``squared length''.
These $S$-schemes  ${\rm{Dyn}}(G')$
and ${\rm{Edge}}(G')$ are constant when $[G']$ arises
from ${\rm{H}}^1(S_{\et},G/Z_G)$ (i.e., inner twisting of the split form $G$).
See \cite[XXIV, \S3]{sga3} for further details.
\end{example}

If $G$ is not $k$-split then there may not be a $k$-group section to 
$\Aut_{G/k} \rightarrow {\rm{Out}}_{G/k}$, in contrast with the
split case in Theorem \ref{autscheme}(3), as even on $k$-points
this map can be non-surjective.  
Here are some examples. 

\begin{example}\label{outk}
Let $A$ be a central simple algebra of rank $n^2$ over a field $k$,
with an integer $n > 2$.  (Class field theory provides many such $A$
that are central division algebras when $k$ is a global or non-archimedean local
field.) Since $A_{k_s} \simeq {\rm{Mat}}_n(k_s)$, by the Skolem--Noether theorem the $k$-group 
$G = {\rm{SL}}(A)$ of units of reduced-norm 1 in $A$ (Exercise \ref{3.2}(iv))
is an inner form of $G_0 = {\rm{SL}}_n$.  Since
${\rm{Out}}_{G/k}$ is 
a $k$-form of the constant $k$-group ${\rm{Out}}_{G_0/k} = (\Z/2\Z)_k$ that has
no nontrivial $k$-forms (since $\Aut(\Z/2\Z) = 1$), 
${\rm{Out}}_{G/k} = (\Z/2\Z)_k$.  
We claim that the map $\Aut_{G/k} \rightarrow {\rm{Out}}_{G/k} = (\Z/2\Z)_k$ is not surjective
on $k$-points (and so has no $k$-group section) if
and only if $A$ is not 2-torsion in ${\rm{Br}}(k)$.  Equivalently, 
we claim that $G$ admits a $k$-automorphism that is not ``geometrically inner''
if and only if $A \simeq A^{\rm{opp}}$ as $k$-algebras.

To prove the implication ``$\Leftarrow$'', we first note that $G^{\rm{opp}} = {\rm{SL}}(A^{\rm{opp}})$, so a $k$-algebra isomorphism 
$A \simeq A^{\rm{opp}}$ induces a $k$-isomorphism $G \simeq G^{\rm{opp}}$ that is the identity on the common center 
(whose finite \'etale Cartier dual is geometrically cyclic order $n$).  Thus,
composing with inversion $G^{\rm{opp}} \simeq G$ yields a $k$-automorphism of $G$ that is inversion on the center.  
But since $n > 2$, inversion on the dual-to-cyclic center is nontrivial.
Hence, we have obtained a $k$-automorphism of $G$ that is not geometrically inner.  

For the converse implication ``$\Rightarrow$'',
assume that $G$ admits a $k$-automorphism $f:G \rightarrow G$ that is not geometrically inner.
We claim that $f$ acts as inversion on the finite center $Z_G$, and 
that it uniquely extends via gluing with inversion on the central $\Gm$ in $\underline{A}^{\times}$ 
to yield a $k$-automorphism of $\underline{A}^{\times}$ 
whose composition with inversion uniquely extends to a $k$-algebra isomorphism 
$A \simeq A^{\rm{opp}}$ (as desired).
In view of the uniqueness statements we may (by Galois descent) extend the ground field to $k_s$, so $A = {\rm{Mat}}_n$
and $G = {\rm{SL}}_n$.
Since $G/Z_G = \underline{A}^{\times}/\Gm$,  so $(G/Z_G)(k) = A^{\times}/k^{\times}$, 
it is harmless to precompose $f$ with the effect on $G$ of an inner automorphism of $A$.
Thus, we can focus on the case of a single $k$-automorphism of ${\rm{SL}}_n$ that is not geometrically
inner (as any two are related through composition against an inner automorphism of $A$ over $k$).  
Hence, we may assume that $f$ is the transpose-inverse automorphism of
${\rm{SL}}_n$ (here we use that $n > 2$).  This is inversion on the central $\mu_n$, which visibly uniquely glues 
with inversion on the central $\Gm$ to define a $k$-automorphism of
${\rm{GL}}_n$ (namely, transpose-inverse).  By the Zariski-density of ${\rm{GL}}_n$ in ${\rm{Mat}}_n$, its 
composition with inversion uniquely extends to an algebra anti-automorphism of ${\rm{Mat}}_n$ (namely, transpose).
\end{example}
%
%
%we claim that all elements 
%$\varphi \in \Aut_{G/k}(k) = \Aut(G)$ have trivial image in ${\rm{Out}}_{G/k}(k_s) =
%\Aut(G_{k_s})/(D_{k_s}^{\times}/k_s^{\times})$, 
%which is to say that over $k_s$ each arises 
%from conjugation against an element of $D_{k_s}^{\times}$.
%
%Fix a $k_s$-algebra isomorphism $D_{k_s} \simeq {\rm{Mat}}_n(k_s)$,
%so this defines an isomorphism $G_{k_s} \simeq (G_0)_{k_s} = {\rm{SL}}_n$
%between norm-1 unit groups.  Transport $\varphi_{k_s}$ across this isomorphism of $k_s$-groups.
%Since $\Aut_{{\rm{SL}}_n} = {\rm{PGL}}_n \rtimes (\Z/2\Z)$
%with the nontrivial class in the component group represented by $x \mapsto (x^{\top})^{-1}$
%(Example \ref{slaut}), we just have to rule out the possibility that $\varphi_{k_s}$ is transported
%across to $x \mapsto g_0 (x^{\top})^{-1} g_0^{-1}$
%for some $g_0 \in \GL_n(k_s) = {\rm{Mat}}_n(k_s)^{\times}$.  
%The key is to relate $\varphi_{k_s}$ to  a $k_s$-algebra anti-automorphism of $D_{k_s}$
%as follows.
%
%By changing the initial $k_s$-algebra isomorphism $D_{k_s} \simeq {\rm{Mat}}_n(k_s)$
%by $g_0^{-1}$-conjugation, we can get to the case $g_0 = 1$.
%Thus, for the anti-automorphism $\xi:y \mapsto \varphi(y^{-1})$ of
%${\rm{SL}}_{1,D}$ over $k$ we see that $\xi_{k_s}$ is transported over to the anti-automorphism
%$x \mapsto x^{\top}$ of ${\rm{SL}}_n$ over $k_s$.  This is the identity map on the central
%$\mu_n$ and hence extends by central gluing to an ant-automorphism
%of $\GL_n$ over $k_s$ that is the identity on the center, so likewise
%$\xi_{k_s}$ extends to an anti-automorphism of the $k_s$-group 
%$D_{k_s}^{\times}$ that is the identity
%on the central $\Gm$.  
%
%We claim that 
%the anti-automorphism $\xi$ of the Zariski-open locus of units in the affine space $D$
%extends to an anti-automorphism of the $k$-algebra $D$.  Indeed,
%it suffices to check this over $k_s$, in which case it becomes 
%the obvious analogous assertion concerning the anti-automorphism $x \mapsto x^{\top}$
%of the unit locus $\GL_n$ in the affine space
%${\rm{Mat}}_n$.  We conclude that $D$ is $k$-isomorphic to its opposite algebra,
%and hence its class $[D]$ in ${\rm{Br}}(k)$ is 2-torsion.  But global and local
%class field theory ensure that ``period equals index'' for Brauer groups of global and local fields;
%i.e., $[D]$ has order $n$. Since $n > 2$, we have a contradiction and hence
%all $k$-automorphisms of ${\rm{SL}}_{1,D}$ are (geometrically) inner.
%\end{example}

\begin{example}\label{rform}
The automorphism scheme has an application in the non-split case
over $\RRR$.  Let $G$ be an $\RRR$-anisotropic connected semisimple $\RRR$-group.
(Such $G$ correspond precisely to the connected compact Lie groups
via $G \rightsquigarrow G(\RRR)$, and $G(\RRR)$ is a maximal compact subgroup of
$G(\C)$; see Theorem \ref{anisotropic} and Proposition \ref{kcx}.)
The $\RRR$-group $\Aut_{G/\RRR}$ has identity component $G/Z_G$,
so it is also $\RRR$-anisotropic.  In \cite[III, 4.5, Ex.\,(b)]{serre}, the anisotropicity 
of $\Aut_{G/\RRR}$ is used
to recover E.\,Cartan's classification of $\RRR$-descents of a connected semisimple $\C$-group:
the set of $\RRR$-descents of $G_{\C}$ (up to isomorphism) is in bijection with the set of conjugacy classes
of involutions $\iota$ of the maximal compact subgroup $G(\RRR)$ of $G(\C)$.

This can be pushed a step further: the map $\iota \mapsto \iota_{\C}$
from conjugacy classes of involutions of $G(\RRR)$ to conjugacy
classes of involutions of the ``complexification'' $G_{\C}$ of $G(\RRR)$
is a bijection (apply \cite[III, 4.5, Thm.\,6]{serre} to the anisotropic 
$\Aut_{G/\RRR}$), so the $\RRR$-descents $H$ of 
a connected semisimple $\C$-group $\mathscr{G}$ are classified up to isomorphism
by conjugacy classes of involutions of $\mathscr{G}$.
This can be made more
concrete by using Mostow's description \cite[\S6]{mostow} of maximal compact subgroups 
of $H(\RRR)$ in terms of maximal compact subgroups of $\mathscr{G}(\C)$:
the conjugacy class of involutions of $\mathscr{G}$ corresponding to $H$ contains 
$(\theta_K)_{\C}$, where $\theta_K$ is the Cartan involution of $H$ associated
\index{Cartan involution}
to a maximal compact subgroup $K$ of $H(\RRR)$.  Equivalently, 
if $\mathscr{G}_0$ is the split $\RRR$-descent of $\mathscr{G}$
then the descent datum on $\mathscr{G}$ relative to the $\RRR$-structure $H$
corresponds to the involution $g \mapsto (\theta_K)_{\C}(\overline{g})$
of the real Lie group
$\mathscr{G}(\C) = \mathscr{G}_0(\C)$ for any maximal compact subgroup $K \subset H(\RRR)$.
\end{example}


\subsection{Cohomological approach to forms}

As an application of the structure of the automorphism scheme,
we can gain some insight into the cohomological description (in Lemma \ref{gclass}) of the
set of isomorphism classes of forms of a given connected reductive group over a field.
We begin with the situation over a field because
the relevant cohomology is Galois cohomology, which 
is more concrete than \'etale cohomology.  Later we will generalize to the case
of an arbitrary connected non-empty base scheme. 

Let $G$ be a connected reductive group over a field $k$,
so for the split $k$-form $G_0$ of $G$ the $k$-group 
$\Aut_{G/k}$ is a $k$-form of $\Aut_{G_0/k}$.
We shall explicitly describe how to build this $k$-form
as an instance of  ``inner cocycle-twisting'' from \cite[I, \S5.3]{serre}.
Fix a $k_s$-isomorphism $\varphi:G_{k_s} \simeq (G_0)_{k_s}$, so 
we get a $k_s$-isomorphism 
$$\xi:(\Aut_{G/k})_{k_s} = \Aut_{G_{k_s}/k_s} \simeq \Aut_{(G_0)_{k_s}/k_s} = (\Aut_{G_0/k_s})_{k_s}$$
via $f \mapsto \varphi \circ f \circ \varphi^{-1}$.
The obstruction to ${\rm{Gal}}(k_s/k)$-equivariance for $\xi$ is expressed by the following
identity for $\gamma \in {\rm{Gal}}(k_s/k)$:
$$\xi(\gamma^{\ast}(f)) = (\varphi \circ \gamma^{\ast}(\varphi)^{-1}) \circ
\gamma^{\ast}(\xi(f)) \circ (\varphi \circ \gamma^{\ast}(\varphi)^{-1})^{-1}.$$
Thus, for the 1-cocycle $c:\gamma \mapsto \varphi \circ \gamma^{\ast}(\varphi)^{-1}$
\index{automorphism scheme!and cocycles}
in ${\rm{Z}}^1(k_s/k, \Aut((G_0)_{k_s}))$ whose cohomology class
classifies $G$ as a form of $G_0$ in Lemma \ref{gclass}, 
the $k$-form $\Aut_{G/k}$ of $\Aut_{G_0/k}$ is
obtained through inner twisting by the 1-cocycle $c$.

Using cocycle-twisting notation as in \cite[I, \S5.3, Ex.\,2]{serre}, we obtain  an isomorphism 
\begin{equation}\label{autisom}
\Aut_{G/k} = {}_c \Aut_{G_0/k}.
\end{equation}
 This identification
depends on $c$, not just on the cohomology class
of $c$.  (Replacing $\varphi$ with $\psi \circ \varphi$ for
$\psi \in \Aut((G_0)_{k_s})$ has the effect of replacing $c$ with
a cohomologous 1-cocycle $c': \gamma \mapsto \psi \circ c_{\gamma} \circ (\gamma^{\ast}(\psi))^{-1}$,
and $\psi^{-1}$ descends to a $k$-isomorphism ${}_c \Aut_{G_0/k} \simeq {}_{c'} \Aut_{G_0/k}$
respecting the identifications with $\Aut_{G/k}$.)    
Since this twisting process is defined
through cocycles valued in inner automorphisms of the group $\Aut_{G_0/k}(k_s)$,
which in turn preserve {\em normal} $k_s$-subgroups of $(\Aut_{G_0/k})_{k_s}$, 
we have the compatible equalities of normal $k$-subgroups $G/Z_G = {}_c (G_0/Z_{G_0})$
and quotients ${\rm{Out}}_{G/k} = {}_c {\rm{Out}}_{G_0/k}$
(where we are now twisting by 
1-cocycles valued in the automorphism functors of the $k$-groups
$G_0/Z_{G_0}$ and ${\rm{Out}}_{G_0/k}$). 

In general, if $\mathscr{G}$ is a smooth affine $k$-group
then a continuous 1-cocycle $a:{\rm{Gal}}(k_s/k) \rightarrow
{\rm{Aut}}(\mathscr{G}_{k_s})$ defines 
a $k$-form ${}_a\mathscr{G}$ of $\mathscr{G}$ whose isomorphism
\index{ aaaga@${}_a\mathscr{G}$}
class as a $k$-group only depends on the class of $a$ in
${\rm{H}}^1(k_s/k, \Aut(\mathscr{G}_{k_s}))$.
\index{cocycles!and torsors}
It is an important fact that if $a$ lifts to a 1-cocycle
$\widetilde{a} \in {\rm{Z}}^1(k_s/k, \mathscr{G}(k_s))$
then a choice of such $\widetilde{a}$ defines a bijection of sets
$t_{\widetilde{a}}:{\rm{H}}^1(k_s/k,{}_a\mathscr{G}) \simeq {\rm{H}}^1(k_s/k, \mathscr{G})$
functorially in the pair $(\mathscr{G},\widetilde{a})$
and carrying the base point to the class of $\widetilde{a}$ \cite[I, \S5.3, Prop.\,35bis]{serre}.
This can  be described in terms of the more
conceptual language of torsors (say for the \'etale topology, though it is equivalent
to use the fppf topology since $\mathscr{G}$ is smooth): if $Y$ is a right $\mathscr{G}$-torsor whose
isomorphism class in ${\rm{H}}^1(k,\mathscr{G})$ is represented by
$\widetilde{a}$ then ${}_a\mathscr{G}$ is the automorphism scheme 
$\underline{\Aut}_{\mathscr{G}}(Y)$ of
the right $\mathscr{G}$-torsor $Y$ and $t_{\widetilde{a}}^{-1}$ carries the class of
a right $\mathscr{G}$-torsor $X$ to the class 
of the right $\underline{\Aut}_{\mathscr{G}}(Y)$-torsor $\underline{{\rm{Isom}}}(X,Y)$.  See
\cite[I, 5.3]{serre}  or \cite[App.\,B]{conrad} for further details.

It is important to distinguish between the images in
${\rm{H}}^1(k_s/k, \Aut(\mathscr{G}_{k_s}))$ of
${\rm{H}}^1(k,\mathscr{G})$ and ${\rm{H}}^1(k,\mathscr{G}/Z_{\mathscr{G}})$.
The $k$-groups ${}_a\mathscr{G}$ arising
from $a$ in the image of ${\rm{H}}^1(k,\mathscr{G}/Z_{\mathscr{G}})$
are called {\em inner forms} of $\mathscr{G}$ (because
$\mathscr{G}/Z_{\mathscr{G}}$ is considered to be the scheme of ``inner
automorphisms'' of $\mathscr{G}$), and the inner forms
\index{inner form}
\index{pure inner form}
arising from $a$ in the image of ${\rm{H}}^1(k,\mathscr{G})$ are
called {\em pure inner forms}.  In general, many
inner forms are not pure inner forms, and
the map ${\rm{H}}^1(k, \mathscr{G}) \rightarrow
{\rm{H}}^1(k,\mathscr{G}/Z_{\mathscr{G}})$ is neither injective
nor surjective.  Pure inner forms play an important role in
local harmonic analysis and the local Langlands correspondence;
see \cite[\S2]{gross2} for some examples with classical groups.  We have seen above 
that passage to a pure inner
form does not change the degree-1 Galois cohomology (as a set).
The same is not true for passage to general inner forms:

\begin{example} For $n > 1$ we have ${\rm{H}}^1(k,{\rm{SL}}_n) = 1$ \cite[X, \S1, Cor.]{serrelocal}. Thus, 
${\rm{SL}}_n$ has no nontrivial pure inner forms.
The quotient ${\rm{SL}}_n/Z_{{\rm{SL}}_n}$ is ${\rm{PGL}}_n$, 
and ${\rm{H}}^1(k,{\rm{PGL}}_n)$ classifies 
rank-$n^2$ central simple $k$-algebras $A$.
The associated inner forms of ${\rm{SL}}_n$ are
the $k$-groups ${\rm{SL}}(A)$
(see Exercise \ref{3.2}(iv)). 

For a rank-$n^2$ central
division algebra $D$ over $k$, we claim that the $k$-form 
${\rm{SL}}_{1,D} := {\rm{SL}}(D)$ of ${\rm{SL}}_n$ can have {\em nontrivial} degree-1
Galois cohomology (in which case ${\rm{H}}^1(k,{\rm{SL}}_{1,D})$ has no
bijection with ${\rm{H}}^1(k, {\rm{SL}}_n)$).  
To see this, consider the exact sequence of $k$-groups
$$1 \rightarrow {\rm{SL}}_{1,D} \rightarrow \underline{D}^{\times} \stackrel{{\rm{Nrd}}}{\rightarrow}
\Gm \rightarrow 1.$$
The pointed set ${\rm{H}}^1(k,\underline{D}^{\times})$ is trivial:  it classifies \'etale sheaves
of rank-$1$ left modules over the quasi-coherent sheaf $D$
on $\Spec k$, and the only such object up to isomorphism is $D$ 
(due to effectivity of \'etale descent for quasi-coherent sheaves and the freeness
of finitely generated $D$-modules).  Thus, ${\rm{H}}^1(k,{\rm{SL}}_{1,D})$
is trivial if and only if the reduced norm map $\nu_D:D^{\times} \rightarrow k^{\times}$
is surjective.  Already with $n = 2$, among quaternion division algebras
over $\Q$ one finds many examples
where such surjectivity fails.  For example, if
$D_{\RRR} \simeq \mathbf{H}$ (there are many such $D$,
by class field theory) then $\nu_D$ does not hit
$\Q_{<0}$.  (The surjectivity of the reduced norm map for all $D$ over $k$ 
is closely related to $k$ being a ${\rm{C}}_r$-field with $r \le 2$; see  \cite[II, 4.5]{serre}.)
\end{example}

Returning to (\ref{autisom}), 
twisting against the $\Aut_{G_0/k}(k_s)$-valued
1-cocycle $c$ and using (\ref{autisom}) defines a bijection of sets
$${\rm{H}}^1(k, \Aut_{G/k}) \simeq {\rm{H}}^1(k,\Aut_{G_0/k}).$$
By construction this carries the trivial point to the class of $G$ in ${\rm{H}}^1(k, \Aut_{G_0/k})$, 
and it induces a compatible bijection
${\rm{H}}^1(k,{\rm{Out}}_{G/k}) \simeq {\rm{H}}^1(k,{\rm{Out}}_{G_0/k})$
(viewing $c$ with values in the automorphism functor
of ${\rm{Out}}_{G_0/k} = \Aut(R,\Delta)_k$). 
That is, we have a commutative diagram of exact sequences of pointed sets 
\begin{align}
\begin{split}
\label{h1comm}
\xymatrix{
{\rm{H}}^1(k,G/Z_G)  \ar[r] &
 {\rm{H}}^1(k,\Aut_{G/k}) \ar[d]^-{\simeq} \ar[r] &
{\rm{H}}^1(k, {\rm{Out}}_{G/k}) \ar[d]^-{\simeq} \\
{\rm{H}}^1(k,G_0/Z_{G_0}) \ar[r] & {\rm{H}}^1(k,\Aut_{G_0/k}) \ar[r] & {\rm{H}}^1(k,{\rm{Out}}_{G_0/k})}
\end{split}
\end{align}
We do {\em not} claim to construct a map between the leftmost terms in
(\ref{h1comm}) 
making a commutative square. 
But the commutativity and exactness
imply that the fiber of ${\rm{H}}^1(k,\Aut_{G_0/k}) \rightarrow {\rm{H}}^1(k,{\rm{Out}}_{G_0/k})$
through $[G]$ is identified with the image of
${\rm{H}}^1(k,G/Z_G) \rightarrow {\rm{H}}^1(k,\Aut_{G/k})$;
cf.\,\cite[I, \S5.5, Prop.\,39, Cor.\,2]{serre}. 
The image of ${\rm{H}}^1(k,G/Z_G)$ in
${\rm{H}}^1(k,\Aut_{G/k})$ consists of 
the $k$-forms of $G$ obtained by twisting by 1-cocycles valued in 
$(G/Z_G)(k_s)$, which we called
the {\em inner forms} of $G$.  We have proved:

\begin{proposition}\label{innclass}
The fibers of ${\rm{H}}^1(k,\Aut_{G_0/k}) \rightarrow {\rm{H}}^1(k,{\rm{Out}}_{G_0/k})$
are precisely the classes that are inner forms of each other.
\end{proposition}

The commutativity of (\ref{h1comm}) also gives that
${\rm{H}}^1(k,\Aut_{G/k}) \rightarrow {\rm{H}}^1(k,{\rm{Out}}_{G/k})$
is {\em surjective}, since the analogous surjectivity holds for $G_0$
(due to a $k$-group section to
$\Aut_{G_0/k} \rightarrow {\rm{Out}}_{G_0/k}$ defined by a choice
of pinning). 

If $k$ is finite then 
 ${\rm{H}}^1(k,G/Z_{G})$ is a single point, due to Lang's Theorem \cite[16.5(i)]{borelag}.
In equivalent terms, the fibers of
$\Aut_{G/k} \rightarrow {\rm{Out}}_{G/k}$ over $k$-points
are $G/Z_G$-torsors and hence all have a $k$-point (as $G/Z_G$ is a smooth connected
$k$-group).  Let's
push this case further:

\begin{example}\label{finiteh1} Let $k$ be a finite field, and let
$(G_0,T_0,M)$ be a split connected reductive $k$-group.
Let $R$ be the root datum and $\Delta$ a base of
the associated root system, so $\Theta := \Aut(R,\Delta)$ is identified
with the constant component group ${\rm{Out}}_{G_0/k}$
of $\Aut_{G_0/k}$.   The natural surjective
map ${\rm{H}}^1(k,\Aut_{G_0/k}) \rightarrow {\rm{H}}^1(k,\Theta_k)$
is bijective because the fiber through each class $[G]$
is an image of the set ${\rm{H}}^1(k,G/Z_G)$ that consists of a single point.

Since $\Theta_k$ is a constant group and ${\rm{Gal}}(k_s/k) = \widehat{\Z}$
with generator given by ${\rm{Frob}}_k$, the cohomology set 
${\rm{H}}^1(k,\Theta_k)$ consists of conjugacy classes
of elements $\theta \in \Theta = \Aut(R,\Delta)$.
Hence, the isomorphism classes of $k$-forms of $G_0$ are classified by 
the conjugacy classes in $\Theta = \Aut(R,\Delta)$.  In more concrete
terms, if we fix a pinning $\{X_a\}_{a \in \Delta}$ and
identify $\Theta = \Aut(R,\Delta)$ with $\Aut(G_0,T_0,M,\{X_a\}_{a \in \Delta})$
via Theorem \ref{isomthm} then to each $\theta \in \Theta$
we associate the $k$-form $G_{\theta}$ of $G_0$ obtained by replacing 
the ${\rm{Frob}}_k$-action on $G_0(k_s)$ with the usual action followed
by the action of $\theta$ on $G_0$ via the pinning.

For example, if $G_0$ is semisimple and either simply connected or adjoint (or $\Phi$ is irreducible
with cyclic fundamental group) then the set of isomorphism classes of $k$-forms of $G_0$
is in bijective correspondence with the set of conjugacy classes in ${\rm{Aut}}({\rm{Dyn}}(\Phi))$.
Since a connected semisimple $k$-group is split if and only if its
simply connected central cover is split, we conclude that a connected semisimple
$k$-group with irreducible root system $\Phi$ over $k_s$
is necessarily split whenever ${\rm{Dyn}}(\Phi)$ has no nontrivial automorphisms.
This holds for types ${\rm{A}}_1$, ${\rm{B}}$, ${\rm{C}}$, ${\rm{E}}_7$, ${\rm{E}}_8$,
${\rm{F}}_4$, and ${\rm{G}}_2$.  (For type ${\rm{A}}_1$, this expresses
the fact that $\mathbf{P}^1$ has no nontrivial forms over the finite field $k$.) 

Among the simply connected
semisimple $k$-groups of a fixed type equal to either ${\rm{A}}_n$ with $n \ge 2$,
${\rm{E}}_6$, or ${\rm{D}}_n$ with $n \ge 5$, there is exactly one
nontrivial $k$-form since $\Aut({\rm{Dyn}}(\Phi)) = \Z/2\Z$
for such $\Phi$.  For example, in 
type ${\rm{A}}_n$ ($n \ge 2$) a non-split simply connected
form is ${\rm{SU}}_{n+1}(k'/k)$, where $k'/k$ is a quadratic
extension. With ${\rm{D}}_4$ there 
are {\em two} nontrivial $k$-forms since the group ${\rm{Aut}}({\rm{Dyn}}({\rm{D}}_4)) = \mathfrak{S}_3$ has
two nontrivial conjugacy classes.
\end{example}

\begin{remark}
Let $S = \Spec R$ for a henselian local ring $R$ with finite residue field
$k$ (e.g., the valuation ring of a non-archimedean local field).
The classification in Example \ref{finiteh1}
via conjugacy classes in $\Theta$ works over $S$ because 
$\pi_1(S) = \pi_1(\Spec k)$ and every smooth surjection
$X \rightarrow S$ acquires a section over the connected
finite \'etale cover $S' \rightarrow S$ corresponding
to any finite extension $k'/k$ such that $X_k(k') \ne \emptyset$.
\end{remark}

Consider a reductive group $G$ over a non-empty scheme $S$.
The {\em inner forms} of $G$ are the forms of $G$ that correspond to the image 
\index{inner form}
of ${\rm{H}}^1(S_{\et}, G/Z_G)$ in ${\rm{H}}^1(S_{\et},\Aut_{G/S})$.
If all geometric fibers of $G$ have the same root datum $R$ (as occurs when $S$ is connected)
and if $G_0$ denotes the split reductive $S$-group with root datum $R$,
then the inner forms of $G$ correspond to the fiber through the class $[G]$ of $G$
under the map
$${\rm{H}}^1(S_{\et}, \Aut_{G_0/S}) \rightarrow {\rm{H}}^1(S_{\et}, {\rm{Out}}_{G_0/S}).$$
Indeed, when $S = \Spec k$ for a field $k$ this is Proposition \ref{innclass},
and a variant of the method of proof works in general (Exercise \ref{innform}). 


%
%See Exercise \ref{qsplit} for the generalization to reductive groups over a connected scheme.
%
%\begin{proof}
%We have already proved the existence of a quasi-split inner form $G'$,
%and our construction of $G'$ provides more properties:
%it has a Borel $k$-subgroup $B'$ containing
%a (geometrically) maximal torus $T'$ such that
%$(G',T',B')$ is a twist of a split triple
%$(G_0,T_0,B_0)$ by a class in ${\rm{H}}^1(k,\Theta)$,
%where $\Theta = \Aut(R,\Delta)$ is the automorphism group
%of the based root datum relative to a choice
%of pinning $\{X_a\}_{a \in \Delta}$ (which embeds $\Theta$ into the automorphism group
%of $(G_0,T_0,B_0)$).  The section to $\Aut_{G_0/k} \rightarrow
%{\rm{Out}}_{G_0/k}$ defined by the pinning carries
%${\rm{Out}}_{G_0/k}$ isomorphically onto the constant $k$-subgroup $\Theta_k
%\subset \Aut_{G_0/k}$.  Hence, this $k$-subgroup is stable under
%the twisting process, so it defines
%a $k$-subgroup of $\Aut_{G'/k}$  that maps isomorphically onto
%${\rm{Out}}_{G'/k}$. 
%
%It remains to prove the uniqueness (up to $k$-isomorphism) of
%the quasi-split inner form.  To prove that $G'$ is the only such inner form of $G$,
%we will use some additional properties of $(G',B',T')$.  
%
%
%
%Likewise, the torus $T'/Z_{G'}$
%in the adjoint quotient $G'/Z_{G'}$ is a twist of $T_0/Z_{G_0} = D_k(\Z \Phi)$ by
%a cocycle valued in the image of $\Theta$ in ${\rm{Aut}}(\Delta)$, 
%so if we decompose the $\Z$-basis $\Delta$ of $\Z \Phi$
%into 
%
%
%assume that $G$ is quasi-split and our problem is to show that $G$ 
%has no nontrivial quasi-split inner form and that
%$\Aut_{G/k} \rightarrow {\rm{Out}}_{G/k}$ admits a $k$-group section.
%
%To check that a quasi-split inner form $G'$ of $G$ must be $k$-isomorphic to $G$,
%pick Borel $k$-subgroups $B' \subset G'$ and $B \subset G$, 
%and choose a $k_s$-isomorphism $f:G'_{k_s} \simeq G_{k_s}$ such that
%the associated 1-cocycle $c:{\rm{Gal}}(k_s/k) \rightarrow \Aut(G_{k_s})$
%is valued in $(G/Z_G)(k_s)$.  It is harmless to compose $f$ with conjugation by
%an element of $G(k_s)$, so we can arrange that $f(B'_{k_s}) = B_{k_s}$.
%Hence, $c$ is valued in the subgroup of $G/Z_G$ whose action on $G$
%preserves $B$, which is to say that it is valued in $N_{G/Z_G}(B/Z_G) = B/Z_G$.
%In other words, $G'$ is classified by the image of a class in
%${\rm{H}}^1(k, B/Z_G)$.  Uniqueness is reduced to proving
%that ${\rm{H}}^1(k,B/Z_G) = 1$.  
%This is a general vanishing result for Borel
%$k$-subgroups in {\em adjoint} (connected) semisimple $k$-groups
%(such as $G/Z_G$), as we now explain. 
%
%Since Borel subgroups of reductive group schemes are conjugate \'etale-locally on the base, 
%and \'etale-locally on the base there exist split maximal tori
%as well as Borel subgroups containing them, it follows
%that any Borel subgroup acquires a maximal torus \'etale-locally on the base.
%Hence, $B_{k_s}$ contains a maximal $k_s$-torus $T'$.
%Let $U = \mathscr{R}_u(B)$ in the sense of Corollary \ref{rup},
%so $U_{k_s}$ is directly spanned in any order by 
%the root groups associated to $\Phi(B_{k_s},T')$.
%In particular, $U_{k_s}$ admits a composition series over $k_s$ whose successive quotients are $\Ga$
%(use the $k_s$-subgroups $(U_{k_s})_{\ge i}$ as in the proof of Lemma \ref{fumap}), so 
%$U_{k_s}$ is $k_s$-split in the sense of \cite[15.1]{borelag}. 
%Thus, $U$ is split over $k$: if ${\rm{char}}(k) > 0$ then this follows from being $k_s$-split,
%due to 
% Tits' results on unipotent groups (see \cite[B.3.4]{pred}), and if ${\rm{char}}(k) = 0$ (or $k$ is perfect)
%then it is \cite[15.5(ii)]{borelag} or \cite[XVII, 4.1.3]{sga3}. 
% Hence, ${\rm{H}}^1(k,U) = 1$ and the quotient $T = B/U$ is a torus such that 
% $T \ltimes U \simeq B$ and the map 
%${\rm{H}}^1(k,B/Z_G) \rightarrow {\rm{H}}^1(k,T/Z_G)$ has trivial kernel.
%It therefore suffices to prove ${\rm{H}}^1(k,T/Z_G) = 1$.  We will prove
%that $T/Z_G = {\rm{R}}_{k'/k}(\Gm)$ for a finite \'etale $k$-algebra $k'$,
%so by Shapiro's Lemma ${\rm{H}}^1(k,T/Z_G) = {\rm{H}}^1(k',\Gm) = 1$ as desired.
%(Note that $T/Z_G$ lifts to a maximal $k$-torus in $B/Z_G$, so the proposed
%description of $T/Z_G$ is really an assertion concerning the structure
%of maximal $k$-tori of Borel $k$-subgroups $B'$ in quasi-split
%{\em adjoint} connected semisimple $k$-groups $G'$; any such torus
%is naturally $k$-isomorphic to $B'/\mathscr{R}_u(B')$.)
%
%To determine the structure of $T/Z_G$, consider how to build $(G,B)$ via cocycle twisting
%from the split case.  Letting $G_0$ be the $k$-split form of $G$, for a Borel $k$-subgroup
%$B_0 \subset G_0$ containing a split (geometrically) maximal $k$-torus
%$T_0$ we can pick a $k_s$-isomorphism $f:G_{k_s} \simeq (G_0)_{k_s}$
%such that $f(B_{k_s}) = (B_0)_{k_s}$.  
%The resulting 1-cocycle $c$ is valued in 
%the group of $k_s$-automorphisms of $(G_0)_{k_s}$
%that preserve $(B_0)_{k_s}$.  This subgroup of
%$$\Aut((G_0)_{k_s}) = (G_0/Z_{G_0})_{k_s} \rtimes \Theta_{k_s}$$
%is exactly $(B_0/Z_{G_0})_{k_s} \rtimes \Theta_{k_s}$
%because the section $\iota_0:\Theta_k \hookrightarrow \Aut_{G_0/k}$ 
%in the split case lands in the subgroup of automorphisms that preserve $B_0$ (and even $T_0$).
%For $U_0 = \mathscr{R}_u(B_0)$, clearly
%$f(U_{k_s}) = (U_0)_{k_s}$.
%Passing to the commutative torus quotients $T = B/U$ and $T_0 = B_0/U_0$, it follows
%that $T$ is the twist of $T_0$ by the action of a cocycle valued in the constant $k$-group
%$\Theta_k = \Aut(R,\Delta)$.
%The same then holds for $T/Z_G$ as a twist of $T_0/Z_{G_0}$.
%
%Since $T_0/Z_{G_0}$ is $k$-split with character group $\Z \Phi$
%having $\Z$-basis $\Delta$,  $T_0/Z_{G_0} \simeq \prod_{a \in \Delta} \Gm$
%via $t_0 \bmod Z_{G_0} \mapsto (a(t_0))_{a \in \Delta}$.  The action on
%$T_0/Z_{G_0} = D_k(\Z\Phi)$ by elements of $\Theta$ is through
%the image of $\Theta$ in the subgroup $\Aut({\rm{Dyn}}(\Phi))$ of the permutation
%group of $\Delta$, so the Galois-module structure on
%the geometric character lattice of the corresponding twist $T/Z_G$ is an induced
%module over $\Z$ defined in terms of the ${\rm{Gal}}(k_s/k)$-orbits
%in $\Delta$.  Thus, $T/Z_G$ is the product of
%tori ${\rm{R}}_{k_i/k}(\Gm)$ for subfields $k_i \subset k_s$ over $k$
%associated to the stabilizers of elements of $\Delta$ under the ${\rm{Gal}}(k_s/k)$-action.
%This completes the proof of uniqueness (up to $k$-isomorphism)
%for the quasi-split inner form.
%
%Consider the problem of constructing a section to
%$\Aut_{G/k} \rightarrow {\rm{Out}}_{G/k}$ when $G$ is quasi-split.
%Using the cocycle $c$ and section $\iota_0$ introduced in the preceding argument, 
%composing $\iota_0$
%with the projection of $c$ to a 1-cocycle $\overline{c}$ valued in $\Theta_k$ 
%yields a 1-cocycle $c' = \iota_0 \circ \overline{c}$
%valued in $\Aut((G_0)_{k_s})$ whose image in ${\rm{H}}^1(k_s/k,\Theta_k)$
%coincides with the image of the class of $c$.  Thus, 
%the $k$-group $G'$ defined by $c'$ is a quasi-split inner form of $G$.
%By uniqueness, $G \simeq G'$ as $k$-groups.  Hence, $G$ is obtained
%from $G_0$ through twisting against a 1-cocycle valued
%in $\Theta_k$.  But such twisting visibly preserves the semi-direct product structure
%on the automorphism scheme over $k_s$, 
%so it provides a $k$-subgroup of $\Aut_{G/k}$ that maps isomorphically onto
%${\rm{Out}}_{G/k}$. 
%\end{proof}

\begin{example}\label{slinner}
The inner forms of ${\rm{SL}}_n$ over a field $k$ are the $k$-groups ${\rm{SL}}(A)$
for central simple $k$-algebras $A$ of rank $n^2$  (see Exercise \ref{3.2}(iv)
for ${\rm{SL}}(A)$).
Indeed, ${\rm{SL}}(A)$ is the image under
$f:{\rm{H}}^1(k,{\rm{PGL}}_n) \rightarrow {\rm{H}}^1(k,\Aut_{{\rm{SL}}_n})$
of the class of $A$ in
${\rm{H}}^1(k,{\rm{PGL}}_n) \subset {\rm{Br}}(k)[n]$.  Since $\ker f = 1$, 
so ${\rm{SL}}(A) \not\simeq {\rm{SL}}_n$ when $A$ is not split,
the uniqueness in Proposition \ref{qsplitk} below implies 
that ${\rm{SL}}(A)$ is not quasi-split over $k$ when $A$ is not a matrix algebra over $k$.
\end{example}

An automorphism $\phi  \in \Aut_{G/S}(S')$ is {\em inner}
\index{inner automorphism}
if it arises from $(G/Z_G)(S')$ (such an automorphism may not arise
from $G(S')$!).  Since the $S$-group $\Aut_{G/S}/(G/Z_G) = {\rm{Out}}_{G/S}$ is \'etale and separated 
over $S$, so an element of ${\rm{Out}}_{G/S}(S)$ is trivial if and only if
it is so on geometric fibers over $S$, 
an automorphism of $G$ is inner if and only if it is so on geometric
fibers over $S$. 
Here is an interesting application: 

\begin{proposition} Let $G \rightarrow S$ be a reductive group scheme, and $H \rightarrow S$
a group scheme with connected fibers.  Any action of $H$ on $G$ must be through inner automorphisms;
i.e., it is the composition of a unique $S$-homomorphism $H \rightarrow G/Z_G$
followed by the natural action of $G/Z_G$ on $G$.
\end{proposition}

\begin{proof} To give an action of $H$ on $G$ is to give an $S$-homomorphism
$H \rightarrow \Aut_{G/S}$, so we just have to show that the composite map
$H \rightarrow {\rm{Out}}_{G/S}$ is trivial.  Since ${\rm{Out}}_{G/S}$ is locally constant over $S$,
it suffices to work on geometric fibers over $S$, where the result is clear
since each fiber $H_s$ is connected.
\end{proof}

We now discuss the classification of forms in the semisimple case over
a general connected non-empty scheme $S$.  Fix a semisimple reduced root datum $R$,
and let $G_0$ be the split semisimple $S$-group with root datum $R$.
For any semisimple $S$-group $G$ whose geometric fibers
have root datum $R$, the simply connected
central cover $\widetilde{G}$ is a form of $\widetilde{G}_0$
with root datum $R^{\rm{sc}}$ equal to the simply connected ``cover'' of $R$
(using $M = (\Z \Phi^{\vee})^{\ast}$).  Likewise, 
the adjoint quotient $G/Z_G$ is a form of
$G_0/Z_{G_0}$ with root datum $R^{\rm{ad}}$
equal to the adjoint ``quotient'' of $R$ (using $M = \Z \Phi$).   The kernel
of $\widetilde{G}_0 \rightarrow G_0/Z_{G_0} = \widetilde{G}_0/Z_{\widetilde{G}_0}$
is the finite multiplicative type group $Z_{\widetilde{G}_0} = D_S(\Pi_0)$
where $\Pi_0$ is the ``fundamental group'' $(\Z \Phi^{\vee})^{\ast}/\Z\Phi$.
The kernel $\mu = \ker(\widetilde{G} \rightarrow G) \subset Z_{\widetilde{G}}$
is a form of $\mu_0 := \ker(\widetilde{G}_0 \rightarrow G_0) \subset D_S(\Pi_0)$.

The problem of classifying the possibilities for $G$ falls into two parts:
classify the forms of $\widetilde{G}_0$, and then for
each such form determine if the twisting process applied
to $D_S(\Pi_0)$ preserves the subgroup $\mu_0$.
This second part is always affirmative when
$\Pi_0$ is cyclic, such as for irreducible $\Phi$
not of type ${\rm{D}}_{2n}$.  
Also, preservation of $\mu_0$ can be studied on a single geometric fiber
and is always a purely combinatorial problem since
the action of the $S$-subgroup $G_0/Z_{G_0} = \widetilde{G}_0/Z_{\widetilde{G}_0}
 \subset \Aut_{\widetilde{G}_0/S}$ has no effect
on the center of $\widetilde{G}_0$. 

It follows that we lose little of the real content of the classification problem in the semisimple case 
by focusing on simply connected $G$, so now consider such $G$.  
\index{semisimple group!cohomological classification|(}
The root datum decomposes
as a direct product according to the irreducible components of
the root system $\Phi$.  Setting aside the combinatorial problem of
permutations of irreducible components of $\Phi$ in the twisting process (handled
in practice using Weil restriction through a suitable finite \'etale covering), we likewise
lose little generality by assuming that $\Phi$ is irreducible.   Then 
the automorphism group of the based root datum coincides with the automorphism
group of the Dynkin diagram (i.e., no problems arise for
${\rm{D}}_{2n}$ as in Example \ref{d2n}), so we obtain:

\begin{corollary}\label{corform} Let $\Phi$ be an irreducible reduced root system,
and $S$ a connected non-empty scheme.  The set of isomorphism classes
of simply connected and semisimple $S$-groups with root system $\Phi$
on geometric fibers is in natural bijection with
\begin{equation}\label{h1ets}
{\rm{H}}^1(S_{\et}, (G_0/Z_{G_0}) \rtimes \Aut({\rm{Dyn}}(\Phi))_S),
\end{equation}
where $(G_0,T_0,M,\{X_a\}_{a \in \Delta})$ is the pinned
split simply connected and semisimple $S$-group with root system $\Phi$.
\end{corollary}

The cohomological description of forms via (\ref{h1ets}) is useful in multiple ways.
Firstly, if we make many constructions of forms then the cohomological
viewpoint can be helpful for proving that all possibilities have been exhausted.
Secondly, cohomology provides an efficient mechanism for understanding the conceptual meaning
of invariants that enter into a classification theorem, such as
auxiliary Galois extensions that occur in a construction (e.g., 
the local invariants that arise in the Hasse--Minkowski theorem
for non-degenerate quadratic forms over global fields). 
%For example, by working systematically with cohomology having coefficients
%in orthogonal and special orthogonal groups (and using special cases of
%the Hasse Principle), one can give a uniform
%non-inductive and characteristic-free proof of the Hasse--Minkowski
%theorem over all global fields, without any special complications in characteristic 2
%(cf.\,Example \ref{hassemink}). 

Here is an example that illustrates the usefulness of the fact that
the automorphism scheme of a semisimple group scheme is smooth and affine.

\begin{example}\label{gilleex}
Let $F$ be a global field, and $\Sigma$ a non-empty finite set of places of $F$ containing
the archimedean places. The \'etale cohomology set ${\rm{H}}^1(\calO_{F,\Sigma},\mathscr{G})$ is
finite for any smooth affine $\calO_{F,\Sigma}$-group $\mathscr{G}$ with reductive fibers 
such that the order of the component group of $\mathscr{G}_{\overline{F}}$
is not divisible by ${\rm{char}}(F)$ \cite[5.1, \S7]{gille}. 
(The hypothesis of reductive fibers can easily be removed when ${\rm{char}}(F) = 0$, but 
to do so when ${\rm{char}}(F) > 0$ requires a local-global finiteness result \cite[1.3.3(i)]{conrad}
which rests on the structure theory of pseudo-reductive groups.)
In particular, if $G_0$ is a split semisimple 
$\calO_{F,\Sigma}$-group whose root system $\Phi$
is irreducible then ${\rm{H}}^1(\calO_{F,\Sigma}, \Aut_{G_0/\calO_{F,\Sigma}})$ is finite 
provided that ${\rm{char}}(F)$ does not divide the order of ${\rm{Aut}}({\rm{Dyn}}(\Phi))$.
By inspecting the list of Dynkin diagrams, the restriction on ${\rm{char}}(F)$ only arises
when ${\rm{char}}(F) \in \{2, 3\}$.  

[This restriction on the characteristic is genuine:
when $G_0$ is simply connected and its connected Dynkin diagram has an automorphism of
order $p = {\rm{char}}(F) \in \{2, 3\}$ then the infinitely many degree-$p$ Artin--Schreier extensions of 
$F$ unramified outside $\Sigma$ give rise to infinitely many pairwise non-isomorphic 
$\calO_{F,\Sigma}$-forms of $G_0$. There is a similar infinitude
phenomenon over local function fields of characteristic 2 or 3. 
Examples include special unitary groups 
associated to quadratic Galois extensions $F'/F$ in characteristic 2.] 

We conclude that as long as ${\rm{char}}(F) \ne 2, 3$, up to isomorphism there are only {\em finitely many} 
semisimple $\calO_{F,\Sigma}$-groups with a given irreducible root datum over $\overline{F}$.
Beware that this is {\em not} saying anything about the number of
(isomorphism classes of) connected semisimple $F$-groups arising as generic fibers
of $\calO_{F,\Sigma}$-forms of $G_0$. 
Indeed, since ${\rm{Aut}}_{G_0/\calO_{F,\Sigma}}$ is not $\calO_{F,\Sigma}$-proper,
if $\mathscr{G}$ and $\mathscr{G}'$ are $\calO_{F,\Sigma}$-forms
of $G_0$ then the Isom-scheme ${\rm{Isom}}(\mathscr{G}, \mathscr{G}')$
is not $\calO_{F,\Sigma}$-proper and hence may have
an $F$-point that does not extend to an $\calO_{F,\Sigma}$-point.   This
is illustrated in the next example. 
\end{example}


\begin{example}\label{nodyn} Let $\Phi$ be an irreducible and reduced root system such that 
$\Aut({\rm{Dyn}}(\Phi)) = 1$ (i.e., types ${\rm{A}}_1$, ${\rm{B}}$, ${\rm{C}}$, 
${\rm{E}}_7$, ${\rm{E}}_8$, ${\rm{F}}_4$, ${\rm{G}}_2$). Let
$G_0$ be the split simply connected semisimple $S$-group with root system $\Phi$.
The cohomology set in Corollary
\ref{corform} is ${\rm{H}}^1(S_{\et},G_0/Z_{G_0})$, so we get an exact sequence of pointed sets
$${\rm{H}}^1(S,Z_{G_0}) \rightarrow 
{\rm{H}}^1(S,G_0) \rightarrow {\rm{H}}^1(S,\Aut_{G_0/S}) \rightarrow {\rm{H}}^2(S,Z_{G_0})$$
(using the fppf topology if $Z_{G_0}$ is not smooth; see Exercise \ref{fppf} for
the effect on ${\rm{H}}^1(S,G_0)$ and ${\rm{H}}^1(S,\Aut_{G_0/S})$
when passing from the \'etale topology to the fppf topology). 
For example, if $S = \Spec \Z$ then $Z_{G_0}$ is a product of various $\mu_n$'s and 
${\rm{H}}^2(\Z,\mu_n) = {\rm{H}}^2(\Z,\Gm)[n]$ 
(using fppf cohomology). But ${\rm{H}}^2(\Z,\Gm)$ is the same
whether we use fppf or \'etale topologies \cite[11.7]{brauer3}, so 
since $\Z$ has only one archimedean place, the 
group ${\rm{H}}^2(\Z,\Gm)$ vanishes by global class field theory (see \cite[III, Ex.\,2.22(f)]{milne}).
Likewise, since ${\rm{Pic}}(\Z) = 1$ and $\Z^{\times} = \{\pm 1\}$, 
we have ${\rm{H}}^1(\Z,Z_{G_0}) = \Pi_0/(2)$ and 
$${\rm{H}}^1(\Z,\Aut_{G_0/S}) = {\rm{H}}^1(\Z,G_0)/{\rm{H}}^1(\Z,Z_{G_0})$$
(the right side denotes the pointed quotient set arising from the translation action on $G_0$ by its
central subgroup scheme $Z_{G_0}$). 
One can give a Weil-style adelic description of
${\rm{H}}^1(\Z,G_0)$ when partitioned according to the isomorphism class
of the $\Q$-fiber; 
see \cite[Rem.\:7.1]{gross} and references therein for 
$\Z$-models with an $\RRR$-anisotropic $\Q$-fiber. 

The case $S = \Spec K$ for a global or non-archimedean local field $K$ works out nicely: 
since $G_0$ is simply connected, 
${\rm{H}}^1(K,G_0)$ is rather small and entirely understood
(by work of Kneser, Harder, Bruhat--Tits, and Chernousov),
and the contribution from ${\rm{H}}^2(K,Z_{G_0}) \subset {\rm{Br}}(K)$ is well-understood. 
This illustrates the general fact that the classification
of connected semisimple groups over a field is intimately tied up with
the Galois cohomological properties of the field (e.g., the structure
of the Brauer group).   
\end{example}

To prove more definitive classification results
over a field, especially over arithmetically interesting
fields, an entirely different approach is required. One has to use the finer structure theory 
of Borel--Tits  that involves rational
conjugacy for maximal $k$-split tori and minimal
parabolic $k$-subgroups, relative root systems, and the``$\ast$-action'' of ${\rm{Gal}}(k_s/k)$
on the Dynkin diagram (see Remark \ref{kottrem}).  This gives
a classification of $k$-groups ``modulo the $k$-anisotropic groups'' (whose structure depends
on special features of $k$). 

As we noted in Remark \ref{kottrem}, the $\ast$-action of ${\rm{Gal}}(k_s/k)$ on
the diagram can be refined to an action on the based root datum $(R(G),\Delta)$.
If $(G_0,T_0,M_0,\{X_a\}_{a \in \Delta_0})$ denotes the pinned split connected reductive $k$-group
with $(R(G_0),\Delta_0) = (R(G),\Delta)$ then 
the homomorphism 
$${\rm{Gal}}(k_s/k) \rightarrow {\rm{Aut}}((R(G),\Delta)) = {\rm{Out}}_{G_0/k}(k)$$
defines a class in ${\rm{H}}^1(k, {\rm{Out}}_{G_0/k})$.
Using Proposition \ref{innclass},
the image of this class under the natural section to ${\rm{H}}^1(k,\Aut_{G_0/k}) \rightarrow 
{\rm{H}}^1(k,{\rm{Out}}_{G_0/k})$ is the quasi-split inner form of $G$,
in view of the proof of uniqueness of this inner form in Proposition \ref{qsplitk}. 
See the tables at the end of \cite{tits66} and \cite{springer} for
the classification of $k$-forms for general as well as special fields $k$.
(Examples without this extra technology 
are also discussed in \cite[III, 1.4, 2.2--2.3, 3.2]{serre}
and \cite[III, App.\,2, \S3]{serre}.)  See \cite{petrov} for a relativization over 
arbitrary semilocal rings.

\begin{example}\label{suexample}
The semi-direct product structure of $\Aut_{G_0/S}$
in Theorem \ref{autscheme}(3) is generally destroyed under passage
to a form of $G_0$ that is not quasi-split
(Example \ref{outk}), and this creates difficulties in any attempt to explicitly describe
the degree-1 cohomology.  For example, 
if $G_0 = {\rm{SL}}_n$ over a field $k$ with $n > 2$, the exact sequence
$$
1 \rightarrow {\rm{PGL}}_n \rightarrow \Aut_{{\rm{SL}}_n/k} \rightarrow \Z/2\Z \rightarrow 1
$$
induces an exact sequence of pointed sets
\begin{equation}\label{slpgl}
{\rm{H}}^1(k,{\rm{PGL}}_n) \rightarrow {\rm{H}}^1(k, \Aut_{{\rm{SL}}_n/k}) \rightarrow
{\rm{H}}^1(k,\Z/2\Z).
\end{equation}
The map on the right must be surjective, since the automorphism scheme 
splits as a semi-direct product over $k$, and likewise
the map on the left has trivial kernel (as a map of pointed sets)
though it is not generally injective (Example \ref{outk}).
In Example \ref{slinner} we addressed 
the image of ${\rm{H}}^1(k,{\rm{PGL}}_n)$.

For any form $G$ of ${\rm{SL}}_n$ there is an associated class in
${\rm{H}}^1(k,\Z/2\Z)$ via (\ref{slpgl}). This class is trivial precisely when 
$G$ comes from ${\rm{H}}^1(k, {\rm{PGL}}_n)$, which is to say that it is one of
the norm-1 unit groups as in Example \ref{slinner}. Let us focus on the case when the
class in ${\rm{H}}^1(k,\Z/2\Z)$ is nontrivial, so it corresponds to 
a quadratic separable extension field $k'/k$. 
The fiber in ${\rm{H}}^1(k,\Aut_{{\rm{SL}}_n/k})$
over the class $[k']$ of $k'/k$ in ${\rm{H}}^1(k,\Z/2\Z)$
contains a unique quasi-split form $G$.  We wish to describe it explicitly
(and then its inner forms will exhaust the entire fiber, by Proposition \ref{innclass}).
The group $G_{k'}$ is identified with ${\rm{SL}}_n$ in such a way that for any 
commutative $k$-algebra $A$, the subgroup 
$$G(A) \subset
G(k' \otimes_k A) = {\rm{SL}}_n(k' \otimes_k A) = {\rm{R}}_{k'/k}({\rm{SL}}_n)(A)$$ 
consists of the points $g' \in {\rm{SL}}_n(k' \otimes_k A)$
satisfying $g' = \iota(\overline{g}')$ where
$\iota$ is as in Example \ref{splitex}
and $c' \mapsto \overline{c}'$ is the  nontrivial $k$-automorphism of $k'$.
Since $\iota$ is an involution, for $w$ as in Example \ref{slaut} the condition
$g' = \iota(\overline{g}')$ says exactly 
$$\overline{g}' = \iota(g') = w ({g'}^{\top})^{-1} w^{-1} = w^{-1} ({g'}^{\top})^{-1} w$$
(the final equality uses that $w^{-1} = \pm w$).  This means 
that $g'$ preserves the non-degenerate $(-1)^{n+1}$-hermitian pairing 
$h:{k'}^n \times {k'}^n \rightarrow k'$ defined by
$h(x',y') = {x'}^{\top}w \overline{y}'$, so
$G = {\rm{SU}}(h)$ (see Exercise \ref{4.3}).  Note that $h$ has an isotropic subspace of the 
maximal possible dimension, $\lfloor n/2 \rfloor$, due to the 
matrix for $w$.   

The $k$-group $G' = {\rm{SU}}_n(k'/k)$ defined using the hermitian 
$H(x',y') = {x'}^{\top} \overline{y}'$ on ${k'}^n$ is an inner form of
$G$.  Detecting when $G'$ is distinct from $G$ (equivalently: quasi-split)
is an arithmetic problem.  For example, if $k$ is finite then $G' \simeq G$
(all connected semisimple groups over finite fields are
quasi-split) whereas if $k = \RRR$ then $G' \not\simeq G$ (since $G'(\RRR)$ is compact). 
\end{example}
\index{semisimple group!cohomological classification|)}

It is a very useful fact (for the classification of forms
and other purposes) that over any non-empty
scheme $S$ every reductive $S$-group $G$ has an inner form that is quasi-split (see Definition \ref{qsdef}).
Put another way, every $G$ can be obtained
from a quasi-split reductive $S$-group via inner twisting:

\begin{proposition}\label{qsexist} Let $S$ be a non-empty scheme, $(G_0,T_0,M)$ a split reductive $S$-group
with root datum $R$, and $G$ a reductive $S$-group whose geometric fibers have root datum
$R$.  There exists a quasi-split inner form $G'$ of $G$
such that $\Aut_{G'/S} \rightarrow {\rm{Out}}_{G'/S}$
has an $S$-group section, some Borel subgroup $B' \subset G'$
contains a maximal torus $T'$ of $G'$, and $T'/Z_{G'} \simeq {\rm{R}}_{S'/S}(\Gm)$
for a finite \'etale cover $S' \rightarrow S$.
\end{proposition}
\index{quasi-split inner form}

\begin{proof}
Let $\Delta$ be the base of a positive system of roots
$\Phi^+ \subset \Phi$, and let $B_0$ be the Borel subgroup containing $T_0$
that corresponds to $\Phi^+$.
Define $\Theta = \Aut(R,\Delta)$.  Choose
a pinning $\{X_a\}_{a \in \Delta}$; this defines an $S$-group section to
$\Aut_{G_0/S} \rightarrow {\rm{Out}}_{G_0/S} \simeq \Theta_S$ which (by construction)
lands inside the subgroup scheme of automorphisms of
$G_0$ that {\em preserve} $T_0$ and $B_0$.  

Consider the cohomology class $[G] \in {\rm{H}}^1(S_{\et},\Aut_{G_0/S})$ classifying
$G$.   If $\xi$ denotes its image in ${\rm{H}}^1(S_{\et},\Theta_S)$, then the section
$\Theta \rightarrow \Aut(G_0)$ defined by the pinning carries $\xi$ to a class
$[G'] \in {\rm{H}}^1(S_{\et},\Aut_{G_0/S})$ in the fiber 
over $\xi$.  By Exercise \ref{innform}, all classes in the same fiber of
the map ${\rm{H}}^1(S_{\et},\Aut_{G_0/S}) \rightarrow
{\rm{H}}^1(S_{\et},{\rm{Out}}_{G_0/S})$ are inner forms of
each other, so $G'$ is an inner form of $G$.
Since the class of $G'$ in ${\rm{H}}^1(S_{\et}, \Aut_{G_0/S})$ is represented
by a 1-cocycle with values in the subgroup of
$\Aut_{G_0/S}$ that preserves $T_0$ and $B_0$, 
$G'$ admits a maximal torus $T'$ and Borel subgroup $B'$ containing $T'$.
This cocycle is valued in $\Theta_S$, so 
the twisting process that constructs $\Aut_{G'/S}$ 
as a form of $\Aut_{G_0/S}$ also twists $\Theta_S$
into an $S$-subgroup of $\Aut_{G'/S}$ 
that maps isomorphically onto the twist ${\rm{Out}}_{G'/S}$ of
${\rm{Out}}_{G_0/S}$.

To describe the maximal torus $T'/Z_{G'}$ in the adjoint quotient $G'/Z_{G'}$, we may
pass to the case when $G_0$ is of adjoint type (due to the functoriality of the formation of
adjoint central quotients). Thus, 
the twisting process is done against a cocycle representing some 
$\xi \in {\rm{H}}^1(S_{\et}, \Theta_S)$ for 
$\Theta = {\rm{Aut}}({\rm{Dyn}}(\Phi)) \subset {\rm{Aut}}(\Delta)$.
Since $T_0 = \prod_{a \in \Delta} \Gm$ via $t \mapsto (a(t))$
(as $G_0$ is adjoint), we have $T' = {\rm{R}}_{S'/S}(\Gm)$ where $S' \rightarrow S$
is the twist of $S \times \Delta$ corresponding
to the image of $\xi$ in ${\rm{H}}^1(S_{\et}, {\rm{Aut}}(\Delta))$. 
\end{proof}

%Here is another quasi-split inner form, with
%additional properties for a Borel subgroup:
%
%\begin{proposition}\label{qsplitex}
%Let $S$ be a non-empty scheme 
%and $G$ a reductive $S$-group.
%There is a quasi-split inner form $G'$ of $G$ containing a Borel subgroup $B'$ 
%satisfying: 
%\begin{enumerate}
%\item the unipotent radical $U' = \mathscr{R}_u(B') \simeq \mathscr{R}_u(B'/Z_{G'})$
%admits a composition series $U' = U'_0 \supset U'_1 \supset \dots \supset U'_N = 1$
%by smooth closed normal subgroups of $B'$ such
%that each $U'_n/U'_{n+1}$ is a vector group;
%\item for the torus $T' = B'/U'$,
%the natural map $Z_{G'} \rightarrow T'$ is a closed immersion and
%$T'/Z_{G'} \simeq {\rm{R}}_{S'/S}(\Gm)$
%for a finite \'etale cover $S' \rightarrow S$;
%\item $T'$ lifts to a maximal torus in $B'$.
%\end{enumerate}
%\end{proposition}
%
%This result is treated separately from Proposition \ref{qsexist} due to difficulties 
%with the outer automorphism group when passing from reductive groups to adjoint semisimple groups. 
%
%\begin{proof}
%In (2), the map $Z_{G'} \rightarrow T'$ between
%multiplicative type groups is a closed immersion because it can be checked on
%geometric fibers (where $Z_{G'} \bigcap U'$ must be trivial). 
%To construct $G'$, we reduce to treating the adjoint semisimple $G^{\rm{ad}} = G/Z_G$ in place of $G$ as follows.
%%Let $Z$ be the maximal central torus in $G$,
%%so $\mu := Z \bigcap \mathscr{D}(G)$ is a finite $S$-group
%%of multiplicative type and $G = Z \times^{\mu} \mathscr{D}(G)$
%%(Corollary \ref{isogcor}).  In particular, the natural map
%%$\mathscr{D}(G) \rightarrow G^{\rm{ad}} := G/Z_G$ is a central isogeny that
%%identifies $G/Z_G$ with the adjoint quotient of 
%%$\mathscr{D}(G)$ (Proposition \ref{central}). 
%%This identification of adjoint quotients is compatible
%Via the natural map $\Aut_{G/S} \rightarrow \Aut_{G^{\rm{ad}}/S}$,
%the induced map 
%$${\rm{H}}^1(S_{\et}, \Aut_{G/S}) \rightarrow {\rm{H}}^1(S_{\et}, \Aut_{G^{\rm{ad}}/S})$$
%carries the set of inner forms
%of $G$ onto the set of inner forms of $G^{\rm{ad}}$. 
%More specifically, any inner form of $G^{\rm{ad}}$ is the adjoint
%quotient of an inner form of $G$.
%%Suppose there exists an inner form $H$ of $G^{\rm{ad}}$ admitting
%%a Borel subgroup $B$ satisfying the desired properties. 
%%Let $\xi \in {\rm{H}}^1(S_{\et}, G^{\rm{ad}})$ be the class of $H$.
%%The inner twisting by $\xi$ on $\mathscr{D}(G)$ has no effect on the central
%%subgroup $\mu$, so
%%$\mu$ is identified with a central subgroup of $H$.
%%The central pushout $G' = Z \times^{\mu} H$ is the
%%inner form of $G$ obtained from $\xi$, and $\mathscr{D}(G') = H$.
%%Thus, $B' := Z \times^{\mu} B$ is a Borel subgroup of $G'$,
%%$B'/Z_{G'} = B/Z_H$,  and $U := \mathscr{R}_u(B)$ is
%%identified with $\mathscr{R}_u(B')$ (so $T' := B'/U' = (Z \times^{\mu} B)/U = Z \times^{\mu} T$ for the maximal
%%torus $T \subset B$ lifting $T'/Z_{G'} \subset B'/Z_{G'} = B/Z_H$, 
%%and hence $T'/Z_{G'} = T/Z_H$). Likewise, by Proposition \ref{derpar} the set of 
%%maximal tori in $\mathscr{D}(G)$ corresponds bijectively to the set of fmaximal
%%tori in $G$. It is therefore enough to 
%%consider $\mathscr{D}(G)$ in place of $G$, so we may and do now assume
%%that $G$ is semisimple.  
%There is a bijective correspondence between the set of Borel subgroups
%of a reductive group and the set of Borel subgroups of its adjoint quotient (Exercise \ref{borbij}),
%and likewise for sets of maximal tori. Thus, we may replace
%$G$ with $G^{\rm{ad}}$ to make $G$ adjoint. 
%
%We may assume that all fibers $G_s$ are nontrivial
%(as the locus of trivial fibers is open and closed in $S$). 
%By the adjoint version of Proposition \ref{allsimple},
%$G \simeq {\rm{R}}_{\mathscr{S}/S}(\mathscr{G})$ for
%a finite \'etale cover $\mathscr{S} \rightarrow S$ and
%an adoint semisimple $\mathscr{S}$-group $\mathscr{G}$
%with absolutely simple fibers.  Weil restriction through a finite \'etale cover 
%carries inner forms to inner forms, 
%Borel subgroups to Borel subgroups, vector groups to vector groups,
%and preserves short exactness.   We may assume that
%the geometric
%fibers of $G$ all have the same root datum (as the locus for each isomorphism type
%is open and closed in $S$), so we may pass
%to $\mathscr{G} \rightarrow \mathscr{S}$ 
%in order that the geometric fibers of $G \rightarrow S$ have a
%common root datum $R$ with an irreducible root system. 
%Let $\Phi$ be this root system.
%
%Let $(G_0, T_0,M_0, \{X_a\}_{a \in \Delta})$ 
%be a pinned split adjoint semisimple $S$-group with root datum $R$.
%Let $B_0$ be the Borel subgroup corresponding to $\Delta$,
%and $\Phi^+$ the positive system of roots with base $\Delta$.
%Applying the proof of Proposition \ref{qsexist}
%in this situation, we get a triple 
%$(G',B',T')$ with $G'$ a quasi-split
%inner form of $G$. We claim that this satisfies all of the asserted properties.
%Note that $T' \rightarrow B'/\mathscr{R}_u(B')$ is an isomorphism, as we check on geometric
%fibers over $S$. 
%
%The twisting process is done against a cocycle representing some 
%$\xi \in {\rm{H}}^1(S_{\et}, \Theta_S)$ for 
%$\Theta = {\rm{Aut}}({\rm{Dyn}}(\Phi))$.  The group $\Theta$ is a subgroup of ${\rm{Aut}}(\Delta)$
%since $G_0$ is semisimple. 
%Since $T_0 = \prod_{a \in \Delta} \Gm$ via $t \mapsto (a(t))$
%(as $G_0$ is adjoint), we have $T' = {\rm{R}}_{S'/S}(\Gm)$ where $S' \rightarrow S$
%is the twist of $S \times \Delta$ corresponding
%to the image of $\xi$ in ${\rm{H}}^1(S_{\et}, {\rm{Aut}}(\Delta))$. 
%
%To describe $\mathscr{R}_u(B')$, let $\lambda_0 = \sum_{c \in \Phi^+} c^{\vee}: \Gm \rightarrow T_0$. 
%For all $a \in \Delta$ we have $\langle a, \lambda_0 \rangle = 2 > 0$ (Exercise \ref{gross6}(iii)),
%so $\Phi_{\lambda_0 > 0} = \Phi^+$.  Hence, $P_{G_0}(\lambda_0) = B_0$
%(as we can check on geometric fibers, by Corollary \ref{parfiber}(2)), so $U_{G_0}(\lambda_0) = 
%\mathscr{R}_u(B_0)$.  The cocharacter $\lambda_0$ 
%of $T_0$ is invariant under the action of
%$\Theta$ on $G_0$ defined via the pinning, 
%so $\xi$-twisting carries $\lambda_0$ over to a {\em fiberwise nontrivial} cocharacter
%$\lambda:\Gm \rightarrow T'$ such that $U_G(\lambda) = U' := \mathscr{R}_u(B')$.
%For $n \ge 1$, let $\Phi_{\ge n}$ denote the set of
%$c \in \Phi^+$ such that $\lambda_0(c) \ge n$.
%By Proposition \ref{upsi}, the root groups $U_c \subset U_0$
%for $c \in \Phi_{\ge n}$ directly span (in any order) 
%a smooth closed subgroup $U_{\ge n}$ in $U_0$
%that is  normalized
%by all positive root groups (hence by $B_0$)
%and the quotient $U_{\ge n}/U_{\ge n+1} \simeq \prod_{\lambda_0(c)=n} U_c$ thereby inherits a trivial
%$U_0$-action via $U_0$-conjugation.  Under the resulting action of $T_0 = B_0/U_0$,
%the $\Gm$-action on $U_{\ge n}/U_{\ge n+1}$ through $\lambda_0$ is via $n$th-power scaling
%on each $U_c \simeq \Add$ (using any trivialization of $\mathfrak{g}_c$). 
%This $\Gm$-action encodes the vector group structure on $U_{\ge n}/U_{\ge n+1}$
%(via the action of $\Gm/\mu_n = \Gm$) since the unit $S$-group of the ring scheme $\Add$ 
%is relatively schematically dense. 
%
%The twisting
%process carries $\lambda_0$ to $\lambda$
%and so carries each $U_{\ge n}$ over
%to a smooth closed subgroup $U'_n \subset U'$ that is normal in $B'$. 
%Each $U'_n/U'_{n+1}$ is commutative with a trivial $U'$-action
%and an induced action by $T' = B'/U'$
%that provides a $\Gm$-action through $\lambda:\Gm \rightarrow T'$.
%By descent, this action makes $\mu_n$ act trivially,
%and the resulting action of $\Gm/\mu_n = \Gm$ on $U'_n/U'_{n+1}$
%is a $\Gm$-equivariant \'etale twist of $U_{\ge n}/U_{\ge n+1}$
%with its natural linear vector group structure.  As above, the linear structure
%on a vector group is encoded in the $\Gm$-action.  Thus, 
%by \'etale descent for vector bundles 
%each $U'_n/U'_{n+1}$ is a vector group over $S$.
%\end{proof}

When $S$ is semi-local, we have a uniqueness result:

\begin{proposition}\label{qsplitk}
For semi-local $S$, up to isomorphism every reductive $S$-group $G$ admits a unique quasi-split
inner form $G'$. 
\end{proposition}

\begin{proof}
We may assume $G$ admits
a Borel subgroup $B$ satisfying the properties
in Proposition \ref{qsexist}. 
Letting $G'$ be a quasi-split inner form of $G$, with
$B'$ a Borel subgroup, we seek to prove that $G' \simeq G$.
Consider a class $\xi \in {\rm{H}}^1(S_{\et},G/Z_G)$
which twists $G$ into $G'$.  Since Borel subgroups are conjugate \'etale-locally on the base
(Corollary \ref{borconj}), we can arrange
that $\xi$ is valued in $N_{G/Z_G}(B/Z_G) = B/Z_G$.
Hence, it suffices to prove that ${\rm{H}}^1(S_{\et}, B/Z_G) = 1$. 

By Theorem \ref{dynlevi}, the unipotent radical $U = \mathscr{R}_u(B/Z_G) \simeq \mathscr{R}_u(B)$
has a composition series whose successive quotients are
vector groups, and the torus $(B/Z_G)/U$ has the form
${\rm{R}}_{S'/S}(\Gm)$ for a finite \'etale cover $S' \rightarrow S$
(so $S'$ is semi-local).
The vanishing of ${\rm{H}}^1(S_{\et}, B/Z_G)$ is therefore
reduced to the vanishing of ${\rm{H}}^1(S_{\et}, \mathscr{V})$
for vector groups $\mathscr{V}$ and the vanishing of 
${\rm{H}}^1(S_{\et}, {\rm{R}}_{S'/S}(\Gm))$.
The higher vector group cohomology vanishes
because it agrees with the Zariski cohomology and $S$ is affine.
The Weil restriction through $S'/S$ is a finite pushforward
for the \'etale sites, so by a degenerating Leray spectral sequence
$${\rm{H}}^1(S_{\et}, {\rm{R}}_{S'/S}(\Gm)) \simeq {\rm{H}}^1(S'_{\et}, \Gm) = {\rm{Pic}}(S')$$
(the final equality by descent theory).  Since $S'$ is semi-local, ${\rm{Pic}}(S') = 1$.
\end{proof}

\begin{remark}
The only role of semi-locality for the affine $S$ in the proof of Proposition \ref{qsplitk} is to ensure
that every finite \'etale cover of $S$ has trivial Picard group.  Thus, the conclusion
also holds (with the same proof) for $S = \Spec \Z$ and $S = \mathbf{A}^n_k$
for a field $k$ of characteristic 0.
\end{remark}

As a final application of our work with the degree-1 cohomology of automorphism schemes, we establish a 
useful result in the structure theory of reductive groups over local fields.  
Let $K$ be a non-archimedean local field, and $R$ its ring of integers.  Let $G$ be a reductive group scheme over $S = \Spec R$.
Bruhat--Tits theory shows that the compact open subgroup $G(R)$ in $G(K)$ is maximal as a compact
subgroup; these are called {\em hyperspecial} maximal compact subgroups of $G(K)$.
\index{hyperspecial maximal compact subgroup|(}
Such a subgroup of the topological group $G(K)$ is defined in terms of a specified reductive $R$-group model of
the $K$-group $G_K$, and so one may wonder if there are several such $R$-models of $G_K$
for which the associated compact open subgroups are not related through $G(K)$-conjugacy. 

There are generally several $G(K)$-conjugacy classes of such subgroups, due to the fact that
$G(K) \rightarrow G^{\rm{ad}}(K)$ may not be surjective, where $G^{\rm{ad}} := G/Z_G$ is the adjoint central quotient of $G$.
\index{ aaagad@$G^{\rm{ad}}$}
To understand  this, consider an automorphism $f:G_K \simeq G_K$
and the (maximal) compact open subgroup $f(G(R))$ in $G(K)$.  Is this $G(K)$-conjugate to $G(R)$?  By Theorem \ref{autscheme}, 
${\rm{Aut}}_{G/R}$ has open and closed relative identity component $G^{\rm{ad}}$
with ${\rm{Out}}_{G/R} := {\rm{Aut}}_{G/R}/G^{\rm{ad}}$ a disjoint union of finite \'etale $R$-schemes.
In particular, ${\rm{Out}}_{G/R}(R) = {\rm{Out}}_{G/R}(K)$. 
The map ${\rm{Aut}}(G) \rightarrow {\rm{Out}}_{G/R}(R)$ is surjective
because the obstruction to surjectivity lies in the cohomology set
${\rm{H}}^1(S_{\et},G^{\rm{ad}})$ that vanishes (due to Lang's theorem over the finite residue
field and the smoothness of $G^{\rm{ad}}$-torsors over the henselian local $R$). 

 Thus, the image of $f \in {\rm{Aut}}(G_K)$ in ${\rm{Out}}_{G_K/K}(K) =
{\rm{Out}}_{G/R}(R)$ 
lifts to ${\rm{Aut}}(G)$, so at the cost of pre-composing $f$ with an $R$-automorphism of $G$ (which is harmless) we may arrange
that $f$ has trivial image in ${\rm{Out}}_{G_K/K}(K)$; i.e., $f$ arises from the action of a point $\overline{g} \in G^{\rm{ad}}(K)$.
Whether or not $f(G(R))$ is $G(K)$-conjugate to $G(R)$ depends only on the image of $\overline{g}$ in 
$G(K)\backslash G^{\rm{ad}}(K)/G^{\rm{ad}}(R)$.  In this split case the obstruction to $G(K)$-conjugacy can be
made rather explicit:

\begin{proposition}\label{gad} If $G$ is $R$-split then 
the ``$\overline{g}$-conjugate'' of $G(R)$ inside $G(K)$ is {\em not} $G(K)$-conjugate
to $G(R)$ whenever $\overline{g} \not\in G(K) G^{\rm{ad}}(R)$ inside $G^{\rm{ad}}(K)$. 
Equivalently, $$G^{\rm{ad}}(K)/G(K)G^{\rm{ad}}(R)$$ labels the set of $G(K)$-conjugacy classes
within the $G^{\rm{ad}}(K)$-orbit of $G(R)$.
\end{proposition}

\begin{proof}
Assume $f(G(R)) = h G(R) h^{-1}$ for some $h \in G(K)$. Letting $\overline{h} :=  h \bmod (Z_G)_K$, we see that the action of 
$\overline{h}^{-1}\overline{g} \in G^{\rm{ad}}(K)$ carries $G(R)$ onto itself. 
Letting $T$ be a split maximal $R$-torus of $G^{\rm{ad}}$, the Iwasawa decomposition (from the split case of  Bruhat--Tits theory,
due to Iwahori and Matsumoto) says 
$G^{\rm{ad}}(K) = G^{\rm{ad}}(R)T(K)G^{\rm{ad}}(R)$, so 
$$\overline{h}^{-1}\overline{g} = \gamma t \gamma'$$
for $\gamma, \gamma' \in G^{\rm{ad}}(R)$ and $t \in T(K)$.
Thus, $t$-conjugation on $G(K)$ carries $G(R)$ onto itself.

For each root $b \in \Phi(G^{\rm{ad}},T)$, the root group scheme $U_b \simeq \mathbf{G}_{\rm{a}}$ over $R$ is preserved
by the $T$-action. More precisely, scaling by $b(t) \in K^{\times}$ on $U_b(K) = K$ carries $U_b(R) = R$ onto itself, so 
$b(t) \in R^{\times}$.  The roots span the cocharacter group of the $R$-split $T$ 
since $G^{\rm{ad}}$ is of adjoint type, so $t \in T(R)$ and hence $\overline{h}^{-1}\overline{g} \in G^{\rm{ad}}(R)$. In other words,
$\overline{g} \in G(K) G^{\rm{ad}}(R)$.
\end{proof}

\begin{example} Consider $G = {\rm{SL}}_n$ with $n > 1$, so
$$G^{\rm{ad}}(K)/G(K)G^{\rm{ad}}(R) = K^{\times}/(K^{\times})^nR^{\times} = \Z/n\Z.$$
In this case the Iwasawa decomposition used in the proof of the 
preceding proposition is entirely elementary,
and the proposition provides (at least) $n$ distinct conjugacy classes of hyperspecial maximal compact subgroups of
${\rm{SL}}_n(K)$.  In concrete terms, consider the $R$-group ${\rm{SL}}_n$ identified
as an $R$-structure of the $K$-group ${\rm{SL}}_n$  by composing the natural generic fiber identification
with conjugation by elements of ${\rm{PGL}}_n(K)$ whose ``determinants''
in $K^{\times}/(K^{\times})^n$ have ``valuations mod $n$'' that vary through all
classes in $\Z/n\Z$.  The subgroups of ${\rm{SL}}_n(K)$ arising from the
$R$-points of these various $R$-structures are
pairwise {\em non-conjugate}, but by construction
they are all related to each other through the action on
${\rm{SL}}_n(K)$ by the subgroup ${\rm{PGL}}_n(K)$ of ${\rm{Aut}}({\rm{SL}}_n)$.
\end{example}
\index{hyperspecial maximal compact subgroup|)}

In general, without an $R$-split hypothesis on $G$, 
we shall next prove that the failure of hyperspecial maximal compact subgroups to be $G(K)$-conjugate is entirely explained
by the gap between $G^{\rm{ad}}(K)$ and $G(K)$. We do not make this more precise (i.e., prove a version of
Proposition \ref{gad} without a split hypothesis), 
as that gets involved with the general Iwasawa decomposition for $G(K)$ and so requires more substantial
input from Bruhat--Tits theory when $G$ is not $R$-split. 

Let $G$ and $G'$ be reductive $R$-groups. For an isomorphism $f:G_K \simeq G'_K$ between their generic fibers
(so $G(R)$ and $G'(R)$ may be viewed as subgroups of $G(K)$), what is the obstruction
to extending $f$ to an $R$-group isomorphism $G \simeq G'$?  Obviously we wish to permit
at least the ambiguity of $G(K)$-conjugation on the source (permitting the same on the target is superfluous, 
as $f$ induces a bijection between $K$-points), and
the preceding considerations suggest that it is more natural to permit the ambiguity of the action of $G^{\rm{ad}}(K)$ on the
source.   We aim to prove:

\begin{theorem} By composing $f$ with the action of a suitable element of $G^{\rm{ad}}(K)$, 
it extends to an $R$-group isomorphism $G \simeq G'$.
Equivalently, the natural map 
$${\rm{Isom}}(G,G') \rightarrow {\rm{Isom}}(G_K,G'_K)/G^{\rm{ad}}(K)$$
is surjective.  In particular, $G$ and $G'$ are abstractly isomorphic.
\end{theorem}

\begin{proof}
The key point is first to prove the final assertion in the theorem: the $R$-groups $G$ and $G'$ are abstractly isomorphic.
Since $G$ and $G'$ are $R$-forms for the \'etale topology, the $R$-isomorphism class of $G'$ is classified by
an element in the pointed set ${\rm{H}}^1(R_{\et},{\rm{Aut}}_{G/R})$, where the automorphism scheme
${\rm{Aut}}_{G/R}$ fits into a short exact sequence (for the \'etale topology)
$$1 \rightarrow G^{\rm{ad}} \rightarrow {\rm{Aut}}_{G/R} \rightarrow {\rm{Out}}_{G/R} \rightarrow 1.$$
The $R$-scheme ${\rm{Out}}_{G/R}$ becomes constant over a finite \'etale cover of $R$, so every 
${\rm{Out}}_{G/R}$-torsor $E$ for the \'etale topology over $R$
becomes constant over an \'etale cover of $R$ and hence is a disjoint union of
finite \'etale $R$-schemes (by the argument given in the proof of the final assertion in Theorem \ref{autscheme}(1)).
Thus, $E(R) = E(K)$, so the natural restriction map ${\rm{H}}^1(R_{\et},{\rm{Out}}_{G/R}) \rightarrow {\rm{H}}^1(K_{\et},
{\rm{Out}}_{G_K/K})$
has trivial kernel. 

The class $[G'] \in {\rm{H}}^1(R_{\et},{\rm{Aut}}_{G/R})$ has trivial image in
${\rm{H}}^1(K_{\et},{\rm{Aut}}_{G_K/K})$, so its image in ${\rm{H}}^1(R_{\et},{\rm{Out}}_{G/R})$ is also trivial
due to the commutative diagram
$$\xymatrix{
{{\rm{H}}^1(R_{\et},{\rm{Aut}}_{G/R})} \ar[d] \ar[r] & {{\rm{H}}^1(R_{\et},{\rm{Out}}_{G/R})} \ar[d] \\
{{\rm{H}}^1(K_{\et},{\rm{Aut}}_{G_K/K})} \ar[r] & {{\rm{H}}^1(K_{\et},{\rm{Out}}_{G_K/K})}}$$
It follows that $[G']$ lies in the image of ${\rm{H}}^1(R_{\et},G^{\rm{ad}})$.  But this latter
cohomology set is trivial by Lang's theorem and smoothness considerations (since $R$ is henselian with finite residue field).
Hence, $G$ and $G'$ are $R$-isomorphic.

Now we can rename $G'$ as $G$ and recast our problem in terms of $K$-automorphisms of $G_K$:  we claim that 
$${\rm{Aut}}(G) G^{\rm{ad}}(K) = {\rm{Aut}}(G_K).$$
Pick $f \in {\rm{Aut}}(G_K)$, so its image in ${\rm{Out}}_{G_K/K}(K) = {\rm{Out}}_{G/R}(R)$ lifts
to some $F \in {\rm{Aut}}(G)$.  Thus, $F_K^{-1} \circ f$ has trivial image in ${\rm{Out}}_{G_K/K}(K)$
and hence arises from $G^{\rm{ad}}(K)$.  This says $f \in {\rm{Aut}}(G)G^{\rm{ad}}(K)$.  
\end{proof}

As an illustration, we conclude that the number of ${\rm{SL}}_n(K)$-conjugacy classes of hyperspecial maximal compact
subgroups in ${\rm{SL}}_n(K)$ (relative to the $K$-group structure ${\rm{SL}}_n$) is {\em exactly} $n$,
with ${\rm{PGL}}_n(K)$ acting transitively on this set.  
More generally, it follows from the theorem that 
 if $G$ is a split reductive $R$-group then the set of $G(K)$-conjugacy classes of
hyperspecial maximal compact subgroups in $G(K)$ is acted upon transitively by $G^{\rm{ad}}(K)$,
with $G^{\rm{ad}}(K)/G(K)G^{\rm{ad}}(R)$ naturally labeling the set of these
$G(K)$-conjugacy classes (by considering the $G^{\rm{ad}}(K)$-orbit of the hyperspecial maximal compact 
subgroup $G(R) \subset G(K)$).
\newpage 

\subsection{Exercises}

\begin{exer}\label{finiteaut}
Let $T$ be a group of multiplicative type over a scheme $S$.
Prove that the automorphism functor $\underline{\Aut}_{T/S}$ on $S$-schemes 
is represented by a separated \'etale $S$-group, and that if
$T = D_S(M)$ for a finitely generated $\Z$-module $M$
then this functor is represented by the constant $S$-group associated to
${\rm{Aut}}(M)^{\rm{opp}}$. 
\index{multiplicative type!automorphism scheme of}

Deduce that if $T$ is normal in an $S$-group $G$ with connected
fibers then $T$ is central in $G$. (If $G \rightarrow S$ is
smooth and affine then Theorem \ref{xi5.3bis} yields another proof of this fact.)
\end{exer}

\begin{exer}\label{gzsurj} Let $G$ be a smooth group scheme over 
a field $k$.  

(i) Prove that the image of $G(k)$ in $(G/Z_G)(k)$
is a normal subgroup and that the quotient by this image is 
a subgroup of the commutative fppf cohomology group
${\rm{H}}^1(k,Z_G)$ (so $(G/Z_G)(k)/G(k)$ is {\em commutative}).
 Make this explicit when $G = {\rm{SL}}_n$,
 and see Exercise \ref{fppf}.
 
 (ii) The $G$-action on itself through conjugation factors through
 an action by $G/Z_G$ on $G$.  Explain how
 this identifies $(G/Z_G)(k)$  with a subgroup of $\Aut(G)$
 even when $(G/Z_G)(k)$ contains points not lifting to $G(k)$.
 Make it explicit for $G = {\rm{SL}}_n$ via ${\rm{SL}}_n \hookrightarrow 
{\rm{GL}}_n$.
 
 (iii) Show that the $k$-automorphisms of $G$ arising 
 from $(G/Z_G)(k)$ are precisely the $k$-automorphisms which become inner
 on $G(\overline{k})$. Deduce that $(G/Z_G)(k)$ is normal in
 $\Aut(G)$, and over every separably closed
 field $k \ne \overline{k}$ give an example in which the action on $G(k)$ by 
 some element of $(G/Z_G)(k)$ is not inner in the sense of abstract group theory. 
 \end{exer}

\begin{exer}\label{moreform}
Let $G$ be a reductive group over a connected scheme $S$,
with $R$ the common root datum of the geometric fibers.

(i) Explain in terms of descent theory why the set of isomorphism classes of reductive $S$-groups 
with root datum $R$ on geometric fibers is in
natural bijection with the set 
${\rm{H}}^1(S_{\et},\Aut_{G/S})$.  The classes arising
from ${\rm{H}}^1(S_{\et},G/Z_G)$ are called {\em inner forms} of $G$.
\index{inner form}

(ii) Let $G_0$ be the unique split reductive $S$-group with root datum $R$.
Prove that the set of (isomorphism classes of) inner forms of $G$ is identified
with the fiber through the class of $G$ for 
$h:{\rm{H}}^1(S_{\et},\Aut_{G_0/S}) \rightarrow {\rm{H}}^1(S_{\et}, {\rm{Out}}_{G_0/S}) =
{\rm{H}}^1(S_{\et},\Theta_S)$,
where $\Theta = \Aut(R(G_0,T_0),\Delta)$. 

(iii) Show ${\rm{H}}^1(S_{\et}, G_0^{\rm{ad}}) \rightarrow
{\rm{H}}^1(S_{\et}, {\rm{Aut}}_{G_0/S})$ has trivial kernel  but via Example \ref{outk} it is {\em not} injective
for $G_0 = {\rm{SL}}_n$ over a field $k$ admitting 
a rank-$n^2$ central simple algebra not in ${\rm{Br}}(k)[2]$. 

(iv)  For $n \ge 2$, prove the forms of ${\rm{SO}}_{2n+1}$ (type ${\rm{B}}_n$) over $S$ 
are  the groups ${\rm{SO}}(q)$ for non-degenerate $(V,q)$ of rank $2n+1$ over $S$
(see Proposition \ref{projsim} for a refinement).
What happens for ${\rm{SO}}_{2n}$ with $n \ge 4$ (type ${\rm{D}}_n$)?
\end{exer}


\begin{exer}\label{fppf}
Let $S$ be scheme, and $G$ a smooth $S$-affine $S$-group.
Use descent theory to prove that the natural map ${\rm{H}}^1(S_{\et},G) \rightarrow
{\rm{H}}^1(S_{\rm{fppf}},G)$ is bijective.  If you are familiar
with algebraic spaces, prove the same result without the affineness restriction on $G$.
(For a remarkable generalization to higher cohomology in the commutative case,
see \cite[11.7]{brauer3}.)
\end{exer}

%
%\begin{exer}
%Let $(V,q)$ be a non-degenerate quadratic space over a global field $k$
%(${\rm{char}}(k) = 2$ allowed).
%Using the structure of ${\rm{O}}_n/{\rm{SO}}_n$ depending on the parity of $n \ge 2$,
%as well as the simply connected central cover
%$$1 \rightarrow \mu_2 \rightarrow {\rm{Spin}}_n \rightarrow {\rm{SO}}_n \rightarrow 1$$
%and the equality ${\rm{H}}^2(k,\mu_2) = {\rm{Br}}(k)[2]$,
%give a characteristic-free definition of global and local invariants
%that characterize $(V,q)$ up to isomorphism over $k$.
%The {\em Hasse Principle} (due to Kneser, Harder, and Chernousov)
%for absolutely simple and connected semisimple groups over global fields
%implies that any such $\xi$ is trivial!
%\end{exer}

\begin{exer}\label{split}
Let $G$ be a connected semisimple $k$-group for a global field $k$.

(i) If $G$ is an inner form of a $k$-split group, prove
$G_{k_v}$ is split for all but finitely many $v$.

(ii) If $G$ is {\em not} an inner form of a $k$-split group, prove
$G_{k_v}$ is non-split for a set of $v$ with positive Dirichlet density.
Make this explicit for ${\rm{SU}}_n(k'/k)$ defined as
in Example \ref{suexample}. 
\end{exer}


\begin{exer} Let $(G,T,M)$ be split semisimple 
over a connected non-empty scheme $S$. 
For each positive system of roots $\Phi^+ \subset \Phi$  (with base $\Delta$)
and choice of pinning $\{X_a\}_{a \in \Delta}$,
let $\{X_{-a}\}_{a \in \Delta}$ be the linked pinning for 
$(G,T,-\Phi^+)$ (i.e., $X_{-a} = X_a^{-1}$ is the dual trivialization of
$\mathfrak{g}_{-a}$). 

(i) Use the Isomorphism Theorem and the functoriality of duality between
$\mathfrak{g}_a$ and $\mathfrak{g}_{-a}$ to construct an {\em involution} 
of $G$ that restricts to inversion on $T$ and swaps
$\Phi^+$ and $-\Phi^+$.  Prove that up to a $(G/Z_G)(S)$-conjugation, this involution of $G$ 
is independent of
the choice of $T$, $\Phi^+$, and pinning (hint: Proposition \ref{fpqc}(2) and Corollary \ref{computez}(1)).
Deduce the existence of a canonical 
$(G/Z_G)(S)$-conjugacy class of involutions of $G$ (called {\em Chevalley involutions}).
What are the Chevalley involutions of ${\rm{SL}}_n$?  Of ${\rm{Sp}}_{2n}$? 
\index{Chevalley involution}

(ii) Prove the Chevalley involutions are inner (i.e., arise from
$(G/Z_G)(S)$) precisely when the long
element of $W(\Phi)$ (relative to a choice of $\Phi^+$) acts via negation on $M$.

(iii) Via the dictionary in Example \ref{rform}, show that if $S = \Spec \RRR$
then the Chevalley involutions of $G$ are the Cartan involutions.
See \cite{vogan} for more on Chevalley involutions.
\end{exer}
\index{Cartan involution}

\begin{exer}\label{maxtb}
This exercise considers
groups with no maximal torus or no Borel subgroup.

(i) Prove that if $S$ is a connected normal noetherian scheme with $\pi_1(S)  = 1$ 
and if $V$ is a vector bundle on $S$ of rank $r > 1$ that is not
a direct sum of line bundles then ${\rm{SL}}(V)$ is a semisimple $S$-group
with no maximal $S$-torus.  Show that an example of such $(S,V)$ is $S_0 = \mathbf{P}^2_k$ 
for any separably closed field $k$ and $V_0 = {\rm{Tan}}_{S_0/k}$. (Hint:
$\dim V_0(S_0) = 8$.)

(ii) Let $K = \Q(\sqrt{5})$, and let $D$ be the quaternion division $K$-algebra 
ramified at precisely the two archimedean places.   Use gluing methods
as in Remark \ref{splitrem} to extend 
the anisotropic $K$-group $G = {\rm{SL}}(D)$ to a semisimple $\calO_K$-group $\mathscr{G}$.
Prove that $\mathscr{G}$ contains no maximal $\calO_K$-torus. (Hint: show that $K$ has no quadratic
extension unramified at all finite places.) 

(iii)  Let $(L,q)$ be a positive-definite even unimodular lattice with rank $\ge 3$.
Prove that the semisimple $\Z$-group ${\rm{SO}}(q)$ has no maximal $\Z$-torus. 

(iv) Over $\RRR$ or any non-archimedean local field $k$, prove that a 
nontrivial connected semisimple $k$-group $G$ has no Borel $k$-subgroup if $G(k)$ is compact.
Verify the compactness for $G = {\rm{SL}}(D)$ 
with any central division algebra $D$ over $k$.
\end{exer}

\begin{exer}\label{innform}
Prove that Proposition \ref{innclass} remains valid over any non-empty scheme $S$. 
\end{exer}

\begin{exer}\label{splitexz} 
Let $G$ be reductive over $S \ne \emptyset$ with every $G_{\overline{s}}$ 
having root datum $R$.  Choose a base $\Delta$ of $R$.
Let $(G_0,T_0,M)$ be split reductive over $S$ with associated root datum $R$, and let
$B_0 \subset G_0$ the Borel subgroup containing $T_0$ that
corresponds to the choice of $\Delta$. 

(i) Using Proposition \ref{fpqc} and Corollary \ref{parnorm}, show
${\rm{Aut}}_{(G_0,B_0)/S} = (B_0/Z_{G_0}) \rtimes {\rm{Aut}}(R,\Delta)$. 

(ii) If $B \subset G$ is a Borel subgroup, classify
$(G,B)$ up to isomorphism by a class in the cohomology set ${\rm{H}}^1(S_{\et},(B_0/Z_{G_0}) \rtimes {\rm{Aut}}(R,\Delta))$ 
and prove that $G$ splits if it is an inner form of $G_0$ and ${\rm{H}}^1(S,\calO_S) = 0$ and ${\rm{Pic}}(S) = 1$.

(iii) Using the properness of ${\rm{Bor}}_{G/S}$, 
prove that if $S$ is connected and Dedekind then ${\rm{Bor}}_{G/S}(S) \ne \emptyset$ 
when the generic fiber $G_{\eta}$ is split (hint: valuative criterion).

(iv) If $R$ is Dedekind with
fraction field $K$ and ${\rm{Pic}}(R) = 1$, show that 
a reductive $R$-group $G$ is split if $G_K$ is split. 
\end{exer}

\begin{exer}\label{zartwist} This exercise demonstrates the necessity of the triviality 
of the Picard group in Exercise \ref{splitexz}(iv).  Let $R$ be a domain with fraction field $K$.

(i) Prove that the automorphism functor of the $\Z$-scheme ${\rm{Mat}}_n$ (viewed as a functor valued
in associative rings) is represented by an affine $\Z$-group scheme ${\rm{Aut}}_{{\rm{Mat}}_n/\Z}$ of finite type,
and show that the resulting natural map of $\Z$-groups
$f:{\rm{PGL}}_n \rightarrow {\rm{Aut}}_{{\rm{Mat}}_n/\Z}$ is 
bijective on artinian points (hint: Exercise \ref{3.2}(i)). Deduce that $f$ is an isomorphism.

(ii) Using (i) and Theorem \ref{autscheme}(3), the set ${\rm{H}}^1(\Spec(R)_{\et}, {\rm{PGL}}_2)$
parameterizes $R$-forms of ${\rm{SL}}_2$ as well as $R$-forms of ${\rm{Mat}}_2$.
For an $R$-form $A$ of ${\rm{Mat}}_2$ (i.e., a rank-4 Azumaya algebra over $R$),
show that the cohomology class of $A$ coincides with that of the $R$-group 
${\rm{SL}}_1(A)$ of units of reduced norm 1 (as an $R$-form of ${\rm{SL}}_2$).
For rank-2 vectors bundles $V$ and $W$ over $R$, deduce
that ${\rm{SL}}(V) \simeq {\rm{SL}}(W)$ as $R$-groups if and only if ${\rm{End}}(V) \simeq
{\rm{End}}(W)$ as $R$-algebras.

(iii) Let $V$ and $W$ be vector bundles over $R$ with the same finite rank $n > 0$.
Show that the natural map of Zariski sheaves of sets
$$q:\mathscr{I}\!som(V,W) \rightarrow \mathscr{I}\!som(\mathscr{E}nd(V), \mathscr{E}nd(W))$$
over $\Spec R$ (defined by $\varphi \mapsto (T \mapsto \varphi \circ T \circ \varphi^{-1})$) 
corresponds to  pushout of right torsors along ${\rm{GL}}(V) \rightarrow {\rm{PGL}}(V)$.
Deduce that $q$ is a $\mathbf{G}_m$-torsor for the Zariski topology via the $\mathbf{G}_m$-action on $W$
(which is invisible on $\mathscr{E}nd(W)$!).

(iv) Let $f:{\rm{End}}_R(V) \simeq {\rm{End}}_R(W)$ be an $R$-algebra isomorphism,
with $V$ and $W$ as in (iii). 
Viewing $f$ in $\Gamma({\rm{Spec}}(R), \mathscr{I}\!som(\mathscr{E}nd(V),\mathscr{E}nd(W)))$, 
use the $\mathbf{G}_m$-torsor $q^{-1}(f)$ over $\Spec(R)$ to construct an invertible $R$-module
$L$ such that $f$ is induced by an isomorphism $V \simeq L \otimes_R W$. 

(v) Using (ii) and (iv), for an  invertible $R$-module $J$ such that 
${\rm{SL}}(R \oplus J) \simeq {\rm{SL}}_2$ as $R$-groups show that $J \simeq L^{\otimes 2}$
for an invertible $R$-module $L$. 
(Note that ${\rm{SL}}(R \oplus J)$ has an evident split maximal $R$-torus,
but its root spaces are not globally free when $J$ is not.)  Deduce that if $K$ is a number field with even class number
then ${\rm{SL}}_{2,K}$ extends to a semisimple $\mathscr{O}_K$-group that is not isomorphic to ${\rm{SL}}_{2,\mathscr{O}_K}$
but splits Zariski-locally over $\calO_K$.
\end{exer}

%
%Using the uniqueness of the quasi-split inner form over a local base
%(Proposition \ref{qsplitk}) and the properness of the scheme of
%Borel subgroups, prove that if $R$ is a discrete valuation ring with fraction field $K$
%then a reductive $R$-group $G$ is split if $G_K$ is split (equivalently, 
%${\rm{H}}^1(R,{\rm{Aut}}_{G_0/R}) \rightarrow {\rm{H}}^1(K,{\rm{Aut}}_{G_0/K})$
%has trivial kernel, where $G_0$ is a split reductive $R$-group).
%\end{exer}

%\begin{exer}\label{qsplit} Let $G$ be a reductive group over a non-empty scheme $S$.
%It is {\em quasi-split} if it contains a Borel $S$-subgroup. 
%
%(i) Prove that $G$ is quasi-split if and only if its adjoint quotient
%$G/Z_G$ is quasi-split, and likewise for the simply connected
%central cover $\widetilde{G}$ when $G$ is semisimple.
%
%
%(ii) If $G$ is quasi-split, with $B \subset G$ a Borel subgroup, prove
%that the subfunctor $$\underline{\Aut}_{(G,B)/S} \subset \underline{\Aut}_{G/S}$$
%of automorphisms preserving $B$ is represented by a smooth closed $S$-subgroup
%$\Aut_{(G,B)/S} \subset \Aut_{G/S}$
%that meets $G/Z_G$ in $B/Z_G$ and has the same ``relative component group'' ${\rm{Out}}_{G/S}$
%as $\Aut_{G/S}$. 
%
%(iii) Assume $S$ is connected,
%and let $(G_0,T_0,M)$ be a split connected reductive $S$-group that is a form of $G$.
%Let $B_0 \subset G_0$ be a Borel $S$-subgroup containing $T_0$.
%Use Exercise \ref{moreform}(ii) and the description of the ``relative component group''
%of $\Aut_{(G_0,B_0)/S}$ in (ii) to deduce that $G$ admits a quasi-split inner form.
%Find a quasi-split inner form $G'$ of $G$ so that
%$\Aut_{G'/S} \rightarrow {\rm{Out}}_{G'/S}$ has an $S$-group section and there exists
%a Borel subgroup $B'$
%containing a maximal $S$-torus $T'$ of $G'$
%and admitting a filtration for
%$\mathscr{R}_u(B')$ with vector groups over $S$ as the successive quotients.
%
%(iv) Assume $S$ is connected and affine, and that
%${\rm{Pic}}(S') = 1$ for all finite \'etale covers $S'$ of $S$.
%(This holds for local $S$, and $S = \Spec \Z$).  Prove that the quasi-split inner form is 
%unique up to $S$-isomorphism. 
%\end{exer}

\newpage


\begin{appendix}


\section{Grothendieck's theorem on tori}\label{torusapp}

\subsection{Motivation and definitions} 
In the early days of the theory of linear algebraic groups, the ground field was
assumed to be algebraically closed (as in work of Chevalley).  
The needs of Lie theory, number theory, and finite group theory (such as finite simple groups
of Lie type) led to the development (independently by Borel, Satake, and Tits) of a theory 
of connected reductive groups
over any perfect field (using Galois-theoretic techniques to deduce results from
the algebraically closed case).  Problems over local and global
function fields motivated the elimination of the perfectness assumption
by Borel and Tits \cite{boreltits}.  The initial breakthrough that
made it possible to work over an arbitrary field is the following
result \cite[XIV, 1.1]{sga3}:

\begin{theorem}[Grothendieck]\label{grthm} Any smooth connected affine group $G$ over a field $k$
contains a $k$-torus $T$ such that $T_{\overline{k}}$ is maximal
in $G_{\overline{k}}$.
\end{theorem}
\index{maximal tori}
\index{Grothendieck's theorem on tori}

The hardest case of the proof is when $k$ is imperfect, and it was
for this purpose that Grothendieck's scheme-theoretic ideas were essential, at first.  
(In \cite[XIV, 1.5(d)]{sga3} there is a second scheme-theoretic proof for
infinite $k$, using the scheme of maximal tori for general smooth connected
affine groups over a field (Exercise \ref{maxfield}); this 
is unirational \cite[XIV, 6.1]{sga3}, and unirational varieties over infinite fields have rational points.)
Borel and Springer eliminated the use of schemes via Lie-theoretic methods
(see \cite[18.2(i)]{borelag}); this 
amounts to working with certain infinitesimal group schemes in disguise, as we shall see.
The aim of this appendix is to give a scheme-theoretic interpretation of their argument.

\begin{remark}\label{remtorus}
As an application of Theorem \ref{grthm}, we now show
via torus-centralizer arguments
that if $K/k$ is any extension field
and if $T \subset G$ is a $k$-torus not contained in a strictly larger one then 
$T_K$ is not contained in a strictly
larger $K$-torus of $G_K$.  In particular, taking $K=\overline{k}$, 
it follows that over a field $k$, ``maximality'' for $k$-tori in the geometric sense
of Definition \ref{maxtori} is the same as maximality in the $k$-rational sense
of containment of $k$-tori; i.e., there is
no ambiguity about the meaning of the phrase ``maximal $k$-torus'', 
and all such tori have the same dimension (due to conjugacy over $\overline{k}$).
(One can also consider the same problem over artin local $S$; see \cite[XIV, 1.4]{sga3}.)
 The common dimension of the maximal $k$-tori is sometimes called the {\em reductive rank} of $G$
 because it coincides with the same invariant  for the reductive quotient
 $G_{\overline{k}}/\mathscr{R}_u(G_{\overline{k}})$.

The equivalence of maximality for $T$ over
$k$ and for $T_K$ over $K$ was mentioned 
in the Introduction, and we now deduce it from Theorem \ref{grthm}. 
For any field extension $K/k$, a torus of $G_K$ containing $T_K$ 
must lie in the closed subgroup scheme $Z_{G_K}(T_K) = Z_G(T)_K$,
but $Z_G(T)$ is also smooth (see Corollary \ref{zgood} or Lemma \ref{centralizer}), so we may replace
$G$ with $Z_G(T)$ to reduce to the special case that $T$ is {\em central} in $G$.
Then we can pass to the affine quotient $k$-group $G/T$ 
to arrive at the case $T = 1$.  

A smooth connected affine group over an algebraically closed
field contains no nontrivial torus if and only if it is unipotent, so 
we are reduced to the following:  if a smooth connected affine group $G$ over a field $k$
contains no nontrivial $k$-torus then must it be unipotent?
The general problem, for an arbitrary smooth connected affine
$k$-group, is the existence of {\em one}
$k$-torus that is maximal over $\overline{k}$; i.e., 
a $k$-torus that is maximal in the sense of Definition \ref{maxtori}.
This is exactly Theorem \ref{grthm}.  
It therefore follows from this theorem that if the only such $k$-torus is the trivial one
then the group must be unipotent, as desired.  
\end{remark}


\begin{remark} 
 Beware that if $k\neq k_s$ then typically there are \textit{many} $G(k)$-conjugacy classes of 
 maximal $k$-tori.  For example, 
 if $G = {\rm{GL}}_n$ then by Exercise \ref{3.3}(i) the set of maximal $k$-tori 
in $G$ is in bijective correspondence with the set of maximal 
finite \'etale commutative $k$-subalgebras of ${\rm{Mat}}_n(k)$.  In particular, 
two maximal $k$-tori are $G(k)$-conjugate if and only if the corresponding
maximal finite \'etale commutative $k$-subalgebras of
${\rm{Mat}}_n(k)$ are ${\rm{GL}}_n(k)$-conjugate.  Hence, if such $k$-subalgebras 
are not abstractly $k$-isomorphic then the corresponding maximal $k$-tori are
not $G(k)$-conjugate.  For example, non-isomorphic degree-$n$ finite
separable extension fields of $k$ yield such algebras.  
\end{remark}

\begin{remark} By the classical theory, 
$G_{\overline{k}}$ has no nontrivial tori if and only if $G_{\overline{k}}$ is unipotent.  Hence, 
Grothendieck's theorem implies that every smooth connected affine
$k$-group is {\em either} unipotent or contains a nontrivial $k$-torus.  If all $k$-tori in $G$ are {\em central}
then for a maximal $k$-torus $T$ the quotient $G/T$ is unipotent (as $(G/T)_{\overline{k}} = G_{\overline{k}}/T_{\overline{k}}$
contains no nontrivial torus).  Hence, in such cases $G$ is solvable.  Thus, in the non-solvable case
there are always $k$-tori $S$ whose scheme-theoretic 
centralizer $Z_G(S)$ (which is smooth, by Corollary \ref{zgood}, 
and connected by Theorem \ref{torithm}(1)) has lower dimension than $G$.  This is useful for
induction arguments based on dimension. 
\end{remark}

\begin{definition}\label{cartandef}
For a maximal $k$-torus $T$ in a smooth connected affine $k$-group $G$,
the associated {\em Cartan $k$-subgroup} $C\subset G$
is $C=Z_G(T)$, the scheme-theoretic centralizer. 
\end{definition}

Cartan $k$-subgroups are always smooth (Corollary \ref{zgood}) 
and connected (by the classical theory).
Since $T$ is central in its Cartan $C$, it follows that $T$ is the \textit{unique} maximal $k$-torus in $C$.
(Indeed, if there exists another then the 
$k$-subgroup it generates along with the central $T \subset C$ would be a bigger $k$-torus.)
We have $C_{\overline{k}}=Z_{G_{\overline{k}}}(T_{\overline{k}})$
since the formation of scheme-theoretic centralizers commutes with base change,
and over $\overline{k}$ all maximal tori are conjugate. Hence, 
over $\overline{k}$ the Cartan subgroups 
are conjugate, so the dimension of a Cartan $k$-subgroup
is both independent of the choice of Cartan $k$-subgroup
and invariant under extension of the ground
field.  This dimension is called the \textit{nilpotent rank} of $G$ in 
\cite[XII, 1.0]{sga3}, and the {\em rank} of $G$ in \cite[12.2]{borelag}.
For a connected
reductive group, the Cartan subgroups are precisely the maximal tori.

\begin{remark}
It is a difficult theorem that for any smooth connected
affine group $G$ over any field $k$, among 
all $k$-\textit{split} tori in $G$ the maximal ones (with respect to inclusion) 
are \textit{rationally} conjugate, i.e. conjugate under $G(k)$.
This is \cite[20.9(i)]{borelag} for reductive $G$. 
The general case was announced without proof by Borel and Tits,
and is proved in \cite[C.2.3]{pred}. 
The dimension of a maximal {\em split} $k$-torus is thus also an invariant, called the 
$k$-\textit{rank} of $G$ (and is of much interest in the reductive case).  
 \end{remark}

\subsection{Start of proof of main result}\label{startmain}
For the proof of Theorem \ref{grthm}, we proceed by induction on $\dim G$, the case
$\dim G \le 1$ being trivial.  Thus, we now assume Theorem \ref{grthm}
is known in all dimensions $< \dim G$. We will largely focus on the case when $k$ is \textit{infinite}, which ensures
that the subset $k^n\subset \mathbf{A}^n_{\overline{k}}$ is Zariski-dense,
and thus  $\mathfrak{g}=\operatorname{Lie}(G)$
is Zariski-dense in $\mathfrak{g}_{\overline{k}}$.
The case of finite $k$ requires a completely different argument;
see Exercise \ref{langexer}.

We first treat the case when $G_{\overline{k}}$ has a \textit{central} maximal torus $S$.
(The argument in this case will work over all $k$, even finite fields.) 
Since all maximal tori are $G(\overline{k})$-conjugate, 
there exists a \textit{unique} maximal $\overline{k}$-torus $S\subset G_{\overline{k}}$. 
Our problem is to produce one defined over $k$.  This is rather elementary over perfect fields via
Galois descent, but here is a uniform
method based on group schemes that applies over all fields (and the technique will
be useful later). 

Let $Z=Z_G^0$, the identity component of the scheme-theoretic center of $G$.
(See Proposition \ref{zrep}, Lemma \ref{centralizer}, or Exercise \ref{2.7}(ii) for the existence of $Z_G$.) 
Since the formation of the center and its identity component commute with 
extension of the ground field (see Exercise \ref{geomconn}), 
we have $S\subset (Z_{\overline{k}})_{\mathrm{red}}$ 
as a maximal torus in the smooth commutative affine $\overline{k}$-group
$(Z_{\overline{k}})_{\mathrm{red}}$.
By the structure of smooth connected commutative affine $\overline{k}$-groups, 
it follows
that $(Z_{\overline{k}})_{\mathrm{red}}=S\times U$
for a smooth connected unipotent $\overline{k}$-group $U$.
For any $n$ not divisible by $\operatorname{char}(k)$,
consider the torsion subgroup $Z[n]$.
This is a commutative affine $k$-group of finite type. The derivative of $[n]:Z\rightarrow Z$ at 0 
is $n:\operatorname{Lie}(Z)\rightarrow\operatorname{Lie}(Z)$
(as for any commutative $k$-group scheme), so 
$\operatorname{Lie}(Z[n])$ is killed by $n \in k^{\times}$.
Thus $\operatorname{Lie}(Z[n])= 0$, so $Z[n]$ is finite \'etale over $k$.

This implies that 
\[Z[n]_{\overline{k}}=Z_{\overline{k}}[n]
\supset (Z_{\overline{k}})_{\mathrm{red}}[n]
\supset Z[n]_{\overline{k}},\]
so $Z[n]_{\overline{k}}=(Z_{\overline{k}})_{\mathrm{red}}[n]$.
Since $U$ is unipotent, $U[n] = 0$. 
Hence, $Z[n]_{\overline{k}}=S[n]$.

Set $H=(\overline{\bigcup_n Z[n]})^0\subset G$,
where the union is taken over $n$ not divisible by ${\rm{char}}(k)$. This 
is a connected closed $k$-subgroup scheme of $G$.

\begin{lemma}\label{tordescent}
The $k$-group $H$ is a torus descending $S$.
\end{lemma}

\begin{proof}
By Galois descent, the formation of $H$ commutes with scalar extension to
$k_s$, so we can assume $k = k_s$.  Hence, the finite \'etale groups $Z[n]$ are constant, so
$H$ is the identity component of the Zariski closure of a set of $k$-points.
It follows that the formation of $H$ 
commutes with any further extension of the ground field, so 
$$H_{\overline{k}}=(\overline{\bigcup_{n} Z_{\overline{k}}[n]})^0
=(\overline{\bigcup_{n} S[n]})^0=S$$
where the final equality uses that in any $k$-torus, the collection of $n$-torsion subgroups
for $n$ not divisible by ${\rm{char}}(k)$ is dense (as
we see by working over $\overline{k}$ and checking for $\Gm$ by hand). 
\end{proof}

Now we turn to the hard case, when $G_{\overline{k}}$ does not have a central maximal torus.
In particular, there must exist a non-central 
\begin{equation}\label{noncen}
S=\Gm \hookrightarrow G_{\overline{k}}.
\end{equation} We will handle
these cases via induction on $\dim G$. (The case $\dim G = 1$ is
trivial.) 

\begin{lemma}\label{2.2lem} It suffices to prove $G$ contains a nontrivial $k$-torus $M$.
\end{lemma}

\begin{proof}
Suppose there exists a nontrivial $k$-torus $M\subset G$.
Consider $Z_G(M)$, which is a smooth connected $k$-subgroup of $G$.
The maximal tori of $Z_G(M)_{\overline{k}}=Z_{G_{\overline{k}}}(M_{\overline{k}})$
must have the same dimension as those of $G_{\overline{k}}$, as can be seen by considering
one containing $M_{\overline{k}}$.
Thus, if we can find a $k$-torus in $Z_G(M)$ that remains maximal as such after extension of the
ground field to 
$\overline{k}$ then the $\overline{k}$-fiber of
such a torus must also be maximal in $G_{\overline{k}}$ for dimension reasons.
Hence, it suffices to prove Theorem \ref{grthm} for $Z_G(M)$.

Consider $Z_G(M)/M$. Since $M$ was assumed to be nontrivial,
this has strictly smaller dimension (even if $Z_G(M) = G$, which might have happened).
Hence, by dimension induction, there exists a $k$-torus $\overline{T}\subset Z_G(M)/M$ 
which is geometrically maximal.
Let $T$ be the scheme-theoretic preimage of $\overline{T}$ in $Z_G(M)$.
Since $M$ is smooth and connected, the quotient map $Z_G(M)\rightarrow Z_G(M)/M$ is smooth,
so $T$ is a smooth connected closed $k$-subgroup of $G$.
It sits in a short exact sequence of $k$-groups 
\[1\rightarrow M\rightarrow T\rightarrow \overline{T}\rightarrow 1.\]
Since $M$ and $\overline{T}$ are tori and $T$ is smooth and connected,
by the structure theory for connected solvable $\overline{k}$-groups it follows that $T$ is a torus.
The quotient $T_{\overline{k}}/M = \overline{T}_{\overline{k}}$
is a maximal torus in $(Z_G(M)/M)_{\overline{k}}$, so $T_{\overline{k}}$ is a maximal
torus in $Z_G(M)_{\overline{k}}$.  Hence, $T_{\overline{k}}$ is also maximal as a torus
in $G_{\overline{k}}$. 
\end{proof}

Now we need to find such an $M$. The idea for infinite
$k$ is to use $\operatorname{Lie}(S) = \mathfrak{gl}_1\subset \mathfrak{g}_{\overline{k}}$
(with $S$ as in (\ref{noncen})) 
and the Zariski-density 
of $\mathfrak{g}$ in $\mathfrak{g}_{\overline{k}}$ (infinite $k$!) 
to create a nonzero $X\in \mathfrak{g}$ that is ``semisimple'' and 
such that the $k$-subgroup $Z_G(X)^0 \subset G$ is a lower-dimensional smooth subgroup
in which the maximal $\overline{k}$-tori are maximal in $G_{\overline{k}}$.  
(We define $Z_G(X)$ to be the schematic $G$-stabilizer of $X$ under ${\rm{Ad}}_G$.) Then 
a geometrically maximal $k$-torus in $Z_G(X)^0$ will do the job.
(Below we will define what we mean by ``semisimple'' for elements
of $\mathfrak{g}_{\overline{k}}$ relative to $G_{\overline{k}}$.  This is a Lie-theoretic version of
Jordan decomposition for linear algebraic groups.) 
The motivation is that whereas it is hard to construct tori over $k$, it is much easier to 
use Zariski-density arguments in $\mathfrak{g}_{\overline{k}}$ to
create semisimple elements in $\mathfrak{g}$.  Those will serve as a substitute for
tori to carry out a centralizer trick and apply dimension induction. 

\begin{remark}
Beware that non-centrality of $S$ in $G_{\overline{k}}$ does not
imply non-centrality of ${\rm{Lie}}(S)$ in $\mathfrak{g}_{\overline{k}}$
when ${\rm{char}}(k) > 0$.
For example, if $S$ is the diagonal torus in ${\rm{SL}}_2$
and ${\rm{char}}(k) = 2$ then ${\rm{Lie}}(S)$ coincides
with the Lie algebra of the central $\mu_2$, so it is
central in $\mathfrak{sl}_2(k)$ (as is also clear by inspection).
\end{remark}

\subsection{The case of infinite $k$}\label{infk}

Now we assume $k$ is infinite, but otherwise arbitrary.  
Consider the following hypothesis:
\begin{equation*}\label{hypstar}\tag{$\star$}
\text{  there exists a non-central semisimple element $X\in \mathfrak{g}$.}
\end{equation*}
To make sense of this, we now define the concept 
``non-central, semisimple'' in $\mathfrak{g}$.  The definition of ``semisimple'' will involve
$G$.  This is not surprising, since  the same 1-dimensional Lie algebra $k$ arises
for both $\mathbf{G}_a$ and $\Gm$, and in the first
case we want to declare all elements of the Lie algebra to be nilpotent
(since unipotent subgroups of ${\rm{GL}}_N$ have all elements in their Lie algebra nilpotent inside
$\mathfrak{gl}_N$, by  the Lie--Kolchin theorem) and in the latter case we want to declare all 
elements of the Lie algebra to be semisimple.

We now briefly digress for a review of Lie algebras of 
smooth linear algebraic groups $G$ over general fields $k$. 
The \textit{center} of a Lie algebra $\mathfrak{g}$ is the kernel
of the adjoint action
\[\operatorname{ad}:\mathfrak{g}\rightarrow\operatorname{End}(\mathfrak{g}), X\mapsto [X,-].\]
In \cite[\S4.1--\S4.4]{borelag}, a general ``Jordan decomposition'' is constructed
as follows in $\mathfrak{g}_{\overline{k}}$, with $\mathfrak{g} = {\rm{Lie}}(G)$.  Choose a closed
$k$-subgroup inclusion $G \hookrightarrow {\rm{GL}}_N$, 
and consider the resulting inclusion of
Lie algebras $\mathfrak{g} \hookrightarrow \mathfrak{gl}_N$ over $k$.
For any $X \in \mathfrak{g}_{\overline{k}}$ we have 
an additive Jordan decomposition $X = X_s + X_n$ in $\mathfrak{gl}_N(\overline{k}) =
{\rm{Mat}}_N(\overline{k})$.  In particular,  $[X_s,X_n]=0$.  The elements 
$X_s, X_n \in \mathfrak{gl}_N({\overline{k}})$ lie in $\mathfrak{g}_{\overline{k}}$
and as such are independent of the chosen embedding $G \hookrightarrow {\rm{GL}}_N$
(proved similarly to the construction of Jordan decomposition in $G(\overline{k})$). 
Likewise, the decomposition $X = X_s + X_n$ is functorial in $G$ (not $\mathfrak{g}$!), 
so $\operatorname{ad}(X_s)=\operatorname{ad}(X)_s$ and 
$\operatorname{ad}(X_n)=\operatorname{ad}(X)_n$.

\begin{definition}\label{ssdef}
An element $X \in \mathfrak{g}$ is 
{\em semisimple} if $X = X_s$ and {\em nilpotent} if $X = X_n$.
\end{definition}

\begin{remark} Note that we are {\em not} claiming that 
${\rm{ad}}(X)$ alone detects the semisimplicity or nilpotence,
nor that the definition is being made intrinsically to $\mathfrak{g}$.
The definitions of semisimplicity and nilpotence involve the $k$-group $G$. 
By definition, these concepts are preserved
under passage from $\mathfrak{g}$ to $\mathfrak{g}_{\overline{k}}$
(and as with algebraic groups, the Jordan components of
$X \in \mathfrak{g}$ are generally only rational over the perfect closure of $k$). 
One can develop versions of these concepts intrinsically to the Lie algebra,
but we do not discuss it; see \cite[V.7.2]{seligman}. 
\end{remark}

If $p=\operatorname{char}(k)>0$, then upon choosing a faithful representation
$\rho:G \hookrightarrow {\rm{GL}}_N$, the resulting inclusion 
$\mathfrak{g}\hookrightarrow\mathfrak{gl}_N$ makes 
 the $p$-power map $A \mapsto A^p$ 
 on $\mathfrak{gl}_N = {\rm{Mat}}_N(k)$ ({\em not} the $p$-power map on matrix entries) induce  
 the structure of a \textit{$p$-Lie algebra}
 \index{pLie algebra@$p$-Lie algebra}
 \index{ aaaxp@$X^{[p]}$}
on $\mathfrak{g}$.  This is 
a map of sets $\mathfrak{g} \rightarrow \mathfrak{g}$, denoted 
$X\mapsto X^{[p]}$, that satisfies $(cX)^{[p]} = c^p X^{[p]}$, 
$\operatorname{ad}(X^{[p]})=\operatorname{ad}(X)^p$ (a computation in
$\mathfrak{gl}_N$), and a certain identity for
$(X+Y)^{[p]} - X^{[p]} - Y^{[p]}$.  It has the intrinsic description 
$D\mapsto D^p$ from the space of left-invariant derivations to itself
(so it is independent of $\rho$), and is functorial in $G$. 
For further details on $p$-Lie algebras, see \cite[\S3.1]{borelag}, 
\cite[A.7]{pred} (especially \cite[A.7.13]{pred}), and
\cite[VII$_{\rm{A}}$, \S5]{sga3};
e.g., the interaction of $p$-Lie algebra structures
and scalar extension is addressed in \cite[VII$_{\rm{A}}$, 5.3.3bis]{sga3}. 


\begin{remark}\label{nilx}
In characteristic $p > 0$, if $X\in \mathfrak{g}$ is nilpotent then $X^{[p^r]}=0$ for $r\gg 0$.
This is very important below, and follows from a computation in the special case
of $\mathfrak{gl}_N$. 
\end{remark}

Returning to our original problem over infinite $k$, 
let us verify hypothesis \eqref{hypstar} in characteristic zero.
The non-central $S= \Gm \hookrightarrow G_{\overline{k}}$ gives
an action of $S$ on $\mathfrak{g}_{\overline{k}}$ (via 
the adjoint action of $G_{\overline{k}}$ on $\mathfrak{g}_{\overline{k}}$),  
and this decomposes as a direct sum of weight spaces 
$$\mathfrak{g}_{\overline{k}}=\bigoplus \mathfrak{g}_{\chi_i}.$$
The $S$-action is described by the weights $n_i$, where $\chi_i(t)=t^{n_i}$.

\begin{lemma}\label{wt}
There is at least one nontrivial weight.
\end{lemma}

\begin{proof}
The centralizer $Z_{G_{\overline{k}}}(S)$ is a smooth (connected) subgroup of $G_{\overline{k}}$,
and ${\rm{Lie}}(Z_{G_{\overline{k}}}(S)) = \mathfrak{g}_{\overline{k}}^S$  by Proposition \ref{zrep}.
Thus, if $S$ acts trivially on $\mathfrak{g}_{\overline{k}}$ then $Z_{G_{\overline{k}}}(S)$ has Lie algebra with full dimension, forcing
$Z_{G_{\overline{k}}}(S) = G_{\overline{k}}$ by smoothness, connectedness, and dimension reasons.
This says that $S$ is central in $G_{\overline{k}}$, which is contrary to our hypotheses on $S$. 
\end{proof}

If we choose a $\overline{k}$-basis $Y$
for $\operatorname{Lie}(S)$ then the element 
$Y \in \mathfrak{g}_{\overline{k}}$ is semisimple
since any $G_{\overline{k}} \hookrightarrow {\rm{GL}}_N$ carries
$S$ into a torus and hence carries ${\rm{Lie}}(S)$ onto a semisimple
subalgebra of $\mathfrak{gl}_N = {\rm{Mat}}_N(k)$. 
By Lemma \ref{wt}, some weight is nonzero.
Thus, in characteristic zero (or more generally if $\operatorname{char}(k)\nmid n_i$ for some $i$)
we know moreover that $\operatorname{ad}(Y)$ is nonzero.
Hence, $Y$ is semisimple and in characteristic 0 is {\em non-central}. 

This does not establish \eqref{hypstar} when ${\rm{char}}(k) = 0$, 
since $Y \in \mathfrak{g}_{\overline{k}}$ and we seek a non-central
semisimple element of $\mathfrak{g}$.
To remedy this, consider the characteristic polynomial $f(X,t)$
of $\operatorname{ad}(X)$ for ``generic'' $X\in \mathfrak{g}$,
as a polynomial in $k[\mathfrak{g}^{\ast}][t]$.
Working in $\overline{k}[\mathfrak{g}^{\ast}][t]
=\overline{k}[\mathfrak{g}^{\ast}_{\overline{k}}][t]$,
the existence of the non-central semisimple element $Y$ as 
established above (when ${\rm{char}}(k) = 0$) 
shows that $f(X,t)\neq t^{\dim \mathfrak{g}}$.
In other words, there are lower-order (in $t$) coefficients in $k[\mathfrak{g}^{\ast}]$ 
which are nonzero as functions on $\mathfrak{g}_{\overline{k}}$.
The subset $\mathfrak{g}\subset \mathfrak{g}_{\overline{k}}$ is
Zariski-dense (as $k$ is infinite), so there exists $X\in \mathfrak{g}$ such that
$f(X,t)\in k[t]$ is not equal to $t^{\dim \mathfrak{g}}$.
In particular, $\operatorname{ad}(X)$ is not nilpotent, so $\operatorname{ad}(X)_s$ is nonzero.
Since $\operatorname{ad}(X_s)=\operatorname{ad}(X)_s\neq 0$, 
$X_s$ is noncentral and semisimple in 
$\mathfrak{g}_{\overline{k}}$.  When $k$ is perfect, such as a field of characteristic 0,
the Jordan decomposition is rational over the ground field.  Thus, $X_s$ satisfies
the requirements in 
\eqref{hypstar}.


\subsection{Hypothesis \eqref{hypstar} for $G$ implies the existence
of a nontrivial $k$-torus}

Now we assume there exists $X\in \mathfrak{g}$ that is noncentral and semisimple.
We will show (for infinite $k$ of any characteristic) that there exists a smooth closed $k$-subgroup
$G'\subset G$ \textit{which is a proper subgroup}
(and hence $\dim G'<\dim G$) 
such that $\mathfrak{g}' := \operatorname{Lie}(G')$ contains
a nonzero semisimple element of $\mathfrak{g}$.
This implies that $G'_{\overline{k}}$ is not unipotent
(for if it were unipotent then its Lie algebra would be nilpotent inside $\mathfrak{g}_{\overline{k}}$).
By dimension induction, $G'$ contains a geometrically maximal $k$-torus.
Since $G'_{\overline{k}}$ is not unipotent,
this means $G'$ (and hence $G$!) contains a \textit{nontrivial} $k$-torus, which is
all we need  (by Lemma \ref{2.2lem}). 

Granting (\ref{hypstar}), it is  very easy to finish the proof, as follows. 
Consider the scheme-theoretic stabilizer $Z_G(X)$ of $X$
(for the action $\operatorname{Ad}_G:G \rightarrow {\rm{GL}}(\mathfrak{g})$).
By Cartier's theorem in characteristic 0,
or the semisimplicity of $X$
and \cite[9.1]{borelag} in any characteristic, $Z_G(X)$ is smooth. We must have $Z_G(X) \ne G$.
Indeed, assume $Z_G(X) = G$, so ${\rm{Ad}}_G(g)(X) = X$ for all $g \in G$. 
By differentiating, $\operatorname{ad}(X)=0$ on $\mathfrak{g}$. 
But $X$ is non-central in $\mathfrak{g}$, so this is a contradiction.
Thus $Z_G(X)$ is a smooth subgroup of $G$ distinct from $G$,
and its Lie algebra contains the nonzero semisimple $X$.  This does the job as required
above, so we are done in characteristic 0, and are
also done in characteristic $p > 0$ if the specific $G$ under consideration satisfies
\eqref{hypstar}.

For the remainder of the proof, assume ${\rm{char}}(k) = p > 0$. 
We will make essential use of $p$-Lie algebras, due to two
facts:  (i) $p$-Lie subalgebras $\mathfrak{h}\subset \mathfrak{g}$
are in functorial bijection with
infinitesimal $k$-subgroup schemes $H\subset G$ that have
height $\le 1$ (meaning $a^p=0$ for all nilpotent functions $a$ on $H$) 
via $H \mapsto {\rm{Lie}}(H)$, 
and (ii) $\mathfrak{h}$ is commutative if and only if $H$ is commutative.
\index{pLie algebra@$p$-Lie algebra}
The idea behind the proofs of these facts 
is to imitate classical
Lie-theoretic arguments by using Taylor series truncated in degrees $< p$.
This makes it possible to reconstruct
$H$ from ${\rm{Lie}}(H)$ via the 
functor $\mathfrak{h} \rightsquigarrow \Spec U_p(\mathfrak{h})^{\ast}$,
where $U_p(\mathfrak{h})$ denotes the {\em restricted universal enveloping 
algebra} (carrying the $p$-operation on $\mathfrak{h}$ over to the
$p$-power map on the associative algebra $U_p(\mathfrak{h})$;
see \cite[VII$_{\rm{A}}$, 5.3]{sga3}). 
A precise statement is in \cite[VII$_{\rm{A}}$, 7.2. 7.4]{sga3}: 

\begin{theorem}\label{exp} 
For a commutative $\mathbf{F}_p$-algebra $B$, the $p$-Lie algebra functor
$H \rightsquigarrow {\rm{Lie}}_p(H)$ is an equivalence between the category
of finite locally free $B$-group schemes whose augmentation ideal is killed
by the $p$-power map and the category of finite locally free $p$-Lie algebras over $B$.

In particular, if $k$ is a field of characteristic $p > 0$ and $G$ is a $k$-group scheme of finite type,
then for any $H$ of height $\le 1$ the $p$-Lie algebra functor defines a bijection 
\begin{eqnarray*}
\Hom_k(H,G) = \Hom_k(H,\ker F_{G/k}) &\simeq& \Hom({\rm{Lie}}_p(H), {\rm{Lie}}_p(\ker F_{G/k}))
\\ &=& 
\Hom({\rm{Lie}}_p(H), {\rm{Lie}}_p(G)).
\end{eqnarray*}
\end{theorem}

In this result, $F_{G/k}:G \rightarrow G^{(p)}$ denotes the relative
\index{relative Frobenius morphism}
\index{ aaafxkn@$F_{X/k,n}, F_{X/k}, F_{G/k}$}
Frobenius morphism, discussed in Exercise \ref{frobex} over fields and in \cite[A.3]{pred}
over $\F_p$-algebras. 
(For ${\rm{GL}}_n$ it is the $p$-power map on matrix entries, and in general it is functorial in $G$.) 
By Nakayama's Lemma, a map $f:H \rightarrow G$ from an
infinitesimal $H$ is a closed immersion if
and only if ${\rm{Lie}}(f)$ is injective. 

\begin{remark} The proof of Theorem \ref{exp} rests
on general arguments with
$p$-Lie algebras in \cite[VII$_{\rm{A}}$, \S4-\S5]{sga3}, 
and a key ingredient is that the natural 
identification of $\mathfrak{h}$
with ${\rm{Tan}}_e(\Spec U_p(\mathfrak{h})^{\ast})$
 respects the $p$-Lie algebra structures.
This compatibility rests on a functorial description of the $p$-Lie algebra structure
arising from a group scheme (and the explicit description of $U_p(\mathfrak{h})$).
Such a functorial description is proved in \cite[A.7.5, A.7.13]{pred}
(and is proved in related but more abstract terms in \cite[VII$_{\rm{A}}$, \S6]{sga3}). 


In the special case of {\em commutative} $k$-groups whose augmentation ideal
is killed by the $p$-power map, an elementary
proof of the equivalence with finite-dimensional {\em commutative} $p$-Lie algebras 
over $k$ is given in the proof of the unique Theorem in \cite[\S14]{mumford} via a method
which works over any field (even though \cite{mumford} always
assumes the ground field is algebraically closed).
But beware that the commutative case is not enough for us, since we need the final
bijection among Hom's in Theorem \ref{exp}, and that rests on using the $k$-group scheme
$\ker F_{G/k}$ which is generally non-commutative.
\end{remark}

As an illustration of the usefulness of Theorem \ref{exp},
we now give an alternative  construction of 
a nontrivial $k$-torus when ${\rm{char}}(k) = p > 0$ and  (\ref{hypstar}) holds, 
bypassing the smoothness of $Z_G(X)$.  This also provides 
an opportunity to present some arguments that will be useful in our treatment
of the cases when (\ref{hypstar}) fails.

Let $\mathfrak{h}=\operatorname{Span}_k(\{X^{[p^i]}\}_{i \ge 0})$.
This is manifestly closed under the map $v\mapsto v^{[p]}$.
Moreover, the $X^{[p^i]}$ all commute with one another.
(Proof: use an 
embedding $\mathfrak{g}\hookrightarrow\mathfrak{gl}_n$ arising from a $k$-group inclusion of
$G$ into ${\rm{GL}}_n$, and the description of
the $p$-operation on $\mathfrak{gl}_n = {\rm{Mat}}_n(k)$ as 
$A \mapsto A^p$.) 
Thus $\mathfrak{h}$ is a commutative $p$-Lie subalgebra of $\mathfrak{g}$.
A linear combination of commuting semisimple operators
is semisimple. Moreover the $p$th power of a nonzero semisimple operator
is {\em nonzero}. Hence, $v\mapsto v^{[p]}$ has trivial kernel on $\mathfrak{h}$.
It is a general fact in Frobenius-semilinear algebra that 
if $V$ is a finite-dimensional vector space
over a perfect field $F$ of characteristic $p$ and if $\phi:V \rightarrow V$ is
a Frobenius-semilinear endomorphism then there exists a unique decomposition
$V = V_{\rm{ss}} \bigoplus V_{\rm{n}}$ such that 
$\phi$ is nilpotent on $V_{\rm{n}}$ and there is a basis
of ``$\phi$-fixed vectors'' ($\phi(v) = v$) for $(V_{\rm{ss}})_{\overline{F}}$.
We will only need this over an algebraically
closed field, in which case it is proved 
in the Corollary at the end of \cite[\S14]{mumford}.

\begin{lemma}\label{smoothz}
The scheme-theoretic centralizer $Z_G(\mathfrak{h}) \subset G$ of $\mathfrak{h} \subset \mathfrak{g}$
under ${\rm{Ad}}_G$ is smooth.
\end{lemma}
\begin{proof}
We may assume $k=\overline{k}$,
as smoothness can be detected over $\overline{k}$ and the formation
of scheme theoretic centralizers commutes with base change.
Now using Theorem \ref{exp}, let $H \subset G$ be
the infinitesimal $k$-subgroup scheme of $\ker F_{G/k}$ whose Lie algebra is $\mathfrak{h} \subset \mathfrak{g}$.

As observed above, $\mathfrak{h}$ splits as a direct sum of $(\cdot)^{[p]}$-eigenlines,
\[\mathfrak{h}=\bigoplus kX_i,\qquad X_i^{[p]}= X_i.\]
Thus, $H$ is a power of the order-$p$ infinitesimal commutative $k$-subgroup
 corresponding to the $p$-Lie algebra $kv$ with 
$v^{[p]}=v$.  But there are only {\em two} 1-dimensional
$p$-Lie algebras over $k$: the one with $v^{[p]} = 0$ and
the one with $v^{[p]} = v$ for some $k$-basis $v$.
(Indeed, if $v^{[p]} = c v$ for some $c \in k^{\times}$ then 
by replacing $v$ with $w = av$ where $a^{p-1} = c$ we get $w^{[p]} = w$.)
Hence, there are exactly two commutative infinitesimal order-$p$ groups over an algebraically closed field,
so the non-isomorphic $\mu_p$ and $\alpha_p$ must be these two possibilities. 
That is, $H$ is a power of either $\mu_p$ or $\alpha_p$.

We claim that $H$ is a power of $\mu_p$.  To prove this,
we will use the $p$-Lie algebra structure.  The embeddings
$\alpha_p\hookrightarrow\mathbf{G}_a$ and 
$\mu_p\hookrightarrow\Gm$ induce isomorphisms on $p$-Lie algebras, and 
the nonzero invariant derivations 
on $\mathbf{G}_a$ and  $\Gm$ are given by 
$\partial_t$ and $t \partial_t$ respectively. 
Taking $p$-th powers of these derivations computes 
the $(\cdot)^{[p]}$-map on them, and clearly $\partial_t^p = 0$ and 
$(t \partial_t)^p = t \partial_t$. Hence, the $p$-operation
on $\alpha_p$ vanishes and on $\mu_p$ is non-vanishing.
Thus, the condition $v^{[p]}=v$ forces $H=\mu_p^N$ for some $N$.

By Lemma \ref{centralizer}, $Z_G(H)$ is smooth.
To conclude the proof, it will suffice to show that the evident inclusion
$Z_G(H) \subset Z_G(\mathfrak{h})$ as $k$-subgroup schemes of $G$ is an equality.
Theorem \ref{exp} provides more:
if $R$ is any $k$-algebra, then the $p$-Lie functor defines a 
bijective correspondence between the sets of $R$-group maps
$H_R \rightarrow G_R$ and $p$-Lie algebra maps
$\mathfrak{h}_R \rightarrow \mathfrak{g}_R$.  Hence, by Yoneda's lemma,
$Z_G(H)=Z_G(\mathfrak{h})$ 
because to check this equality of $k$-subgroup schemes of $G$ 
it suffices to compare $R$-points
for arbitrary $k$-algebras $R$. 
\end{proof}

As in the characteristic zero case, since $\mathfrak{h}$ contains noncentral elements of $\mathfrak{g}$, it follows
that $Z_G(\mathfrak{h}) \ne G$. 
And as we saw above, this guarantees the existence of a nontrivial $k$-torus
in $G$, by dimension induction applied to the smooth identity component $Z_G(\mathfrak{h})^0$
that is {\em nontrivial} (since it contains the infinitesimal $H \ne 1$)

We have already completed the proof of Theorem \ref{grthm} in characteristic zero,
since \eqref{hypstar} always holds in characteristic 0,
and more generally we have completed it over any $k$ whatsoever 
for $G$ that satisfy (\ref{hypstar})  when
the conclusion of Theorem \ref{grthm} is known over $k$ in all lower dimensions
(as we may always assume, since we argue by induction on $\dim G$). 

\subsection{The case $\operatorname{char}(k)=p>0$ and \eqref{hypstar} fails}\label{fail}

Now the idea is to find a central infinitesimal $k$-subgroup $M\subset G$
such that $G/M$ satisfies \eqref{hypstar}.  We will lift the result from $G/M$
back to $G$ when such an
$M$ exists, and if no such $M$ exists
then we will use a different method to find a nontrivial $k$-torus in $G$.

\begin{lemma}\label{ssexist}
Regardless of whether or not \eqref{hypstar} holds $($but still
assuming, as we have been, that $G_{\overline{k}}$ has a noncentral $\Gm$$)$,
there exists a nonzero semisimple element $X\in \mathfrak{g}$.
\end{lemma}
\begin{proof}
 Arguing as at the end of \S\ref{infk}, 
 and \textit{using the infinitude of the field $k$}, 
 there exists $X_0\in \mathfrak{g}$ such that $\operatorname{ad}(X_0)$ is not nilpotent.
Consider the additive Jordan decomposition
$X_0=(X_0)_s+(X_0)_n$ in $\mathfrak{g}_{\overline{k}}$ 
as a sum of commuting semisimple and nilpotent elements. These components
of $X_0$ are defined over the perfect closure of $k$, by Galois descent. 
For $r \gg 0$
we see that
$X:=X_0^{[p^r]}=((X_0)_s)^{[p^r]}$.  This 
is nonzero and semisimple, and if $r \gg 0$ then $X \in \mathfrak{g}$
(since $(X_0)_s$ is rational over the perfect closure of $k$). 
\end{proof}

Obviously if \eqref{hypstar} fails for $G$ 
then every semisimple element of $\mathfrak{g}$ is central.
Assume this is the case.
Set $$\mathfrak{m}= \operatorname{Span}_k(\text{all semisimple }X\in \mathfrak{g})\subset \mathfrak{g};$$
this is nonzero due to Lemma \ref{ssexist}. 
Since all semisimple elements of $\mathfrak{g}$
are central, $\mathfrak{m}$ is a  commutative Lie subalgebra of $\mathfrak{g}$.
The $p$th power of a semisimple element of ${\rm{Mat}}_N(\overline{k})$ is semisimple,
so $\mathfrak{m}$ is $(\cdot)^{[p]}$-stable.
Thus, $\mathfrak{m}$ is a $p$-Lie subalgebra, so 
we can exponentiate it to an infinitesimal commutative subgroup $M\subset \ker F_{G/k}$
by Theorem \ref{exp}.
A linear combination of commuting semisimple elements in
${\rm{Mat}}_N(\overline{k})$ is semisimple, so 
$\mathfrak{m}$ consists only of semisimple elements. This 
implies that $(\cdot)^{[p]}$ has vanishing kernel on $\mathfrak{m}_{\overline{k}}$.
Thus, as in the proof of
Lemma \ref{smoothz}, $M_{\overline{k}}=\mu_p^N$ for some $N > 0$. 

\begin{lemma}\label{centralm}
The $k$-subgroup scheme $M$ in $G$ is central.  
\end{lemma}

\begin{proof}
Let $V\subset \mathfrak{g}_{k_s}$
be the $k_s$-span of all semisimple central elements of $\mathfrak{g}_{k_s}$.
Clearly  we have $\mathfrak{m}_{k_s}\subset V$.
Let $\Gamma=\operatorname{Gal}(k_s/k)$.
Since $V$ is $\Gamma$-stable, by Galois descent we have
$V=(V^\Gamma)_{k_s}$.
Since $V^\Gamma\subset \mathfrak{m}$, 
this gives $V=\mathfrak{m}_{k_s}$.
By inspection, it is clear that $V$ is stable
under the action of $G(k_s)$ on $\mathfrak{g}_{k_s}$.  But $G(k_s)$ is Zariski-dense in $G_{k_s}$, so 
$G_{k_s}$ preserves $V=\mathfrak{m}_{k_s}\subset \mathfrak{g}_{k_s}$
under the adjoint action.  
Hence, $G$ preserves $\mathfrak{m}$, so 
$M$ is normal in $G$.
Thus, it is central by Theorem \ref{xi5.3bis} or Exercise \ref{finiteaut}.
\end{proof}

Now consider the central purely inseparable $k$-isogeny $\pi:G \rightarrow G' := G/M$.
Note that $G'$ is smooth and connected of the same dimension as
$G$, and even contains a non-central
torus $\pi_{\overline{k}}(S)$ over $\overline{k}$ (as $\pi$ is bijective on $\overline{k}$-points). 
Does $G'$ satisfy (\ref{hypstar})?   If it does not,
then we can run through the same procedure all over again
to get a nontrivial central $M' \subset G'$ such $M'_{\overline{k}} \simeq \mu_p^{N'}$,
and can then consider the composite purely inseparable $k$-isogeny
$$G \rightarrow G/M = G' \rightarrow G'/M'.$$
This is not so bad, since the kernel $E$ of this composite map is necessarily 
an infinitesimal multiplicative type subgroup,
by the following lemma, so it is {\em central} in $G$
due to normality and Theorem \ref{xi5.3bis} (or Exercise \ref{finiteaut}):

\begin{lemma}\label{mstuff}  If 
$$1 \rightarrow M' \rightarrow E \rightarrow M \rightarrow 1$$
is a short exact sequence of finite $k$-group schemes with $M$ and $M'$
multiplicative infinitesimal $k$-groups then so is $E$; in particular, 
$E$ is commutative.
\end{lemma}

\begin{proof}
We may assume $k = \overline{k}$.  
The infinitesimal nature of
$M$ and $M'$ implies that $E(k) = 1$, so $E$ is infinitesimal (hence connected).
The normality of $M'$ in $E$ implies that the conjugation action of $E$ on $M'$
is classified by a $k$-group homomorphism from the connected 
$E$ to the \'etale automorphism group scheme of $M'$.
This classifying map must be trivial, so $M'$ is {\em central} in $E$. 
Since $M = E/M'$ is commutative, 
the functorial commutator $E \times E \rightarrow E$  
factors through a $k$-scheme morphism
$$[\cdot,\cdot]:M \times M = (E/M') \times (E/M') \rightarrow M'$$
which is seen to be bi-additive by thinking about $M = E/M'$ 
in terms of fppf quotient sheaves.  In other words, this bi-additive
pairing corresponds (in two ways!) to a $k$-group homomorphism
$$M \rightarrow \underline{\rm{Hom}}(M,M'),$$
where the target is the affine finite type $k$-scheme classifying group scheme homomorphisms
(over $k$-algebras).  By Cartier duality,  this Hom-scheme is \'etale, so 
the map to it from $M$ must be {\em trivial}.  This shows that $E$ has trivial
commutator, so $E$ is commutative. 
\index{Cartier duality}

With commutativity of $E$ established, we apply Cartier duality
$\mathbf{D}(\cdot)$  to our original
short exact sequence.  This duality operation is contravariant and preserves exact sequences
(since it is order-preserving and carries right-exact sequences to left-exact sequences), 
so we get an exact sequence
$$1 \rightarrow \mathbf{D}(M) \rightarrow \mathbf{D}(E) \rightarrow \mathbf{D}(M') \rightarrow 1.$$
The outer terms are finite constant groups of $p$-power order, so the middle one must be too.
Hence, $E = \mathbf{D}(\mathbf{D}(E))$ is  multiplicative, as desired.
\end{proof}

Returning to our setup of interest, by Lemma \ref{mstuff} the composite isogeny
$$G \rightarrow G/M = G' \rightarrow G'/M'$$
is a quotient by a central multiplicative
infinitesimal $k$-group.  
Now we're in position to wrap things up in positive characteristic (when $k$ is infinite,
arguing by induction on $\dim G$).   

First, we handle the
case when the above process keeps going on {\em forever}. 
This provides a {\em strictly increasing} sequence $M_1 \subset M_2 \subset \dots$
of central multiplicative infinitesimal $k$-subgroups of $G$. 
This is all happening inside the $k$-subgroup scheme $Z_G$, so
it forces $Z_G$ to not be finite (as otherwise there would be an upper
bound on the $k$-dimensions of the coordinate rings of the $M_j$'s). 
Since $(Z_G)^0_{\overline{k}}/((Z_G)^0_{\overline{k}})_{\rm{red}}$
is a finite infinitesimal group scheme,  for large enough $j$ the map
to this from $(M_j)_{\overline{k}}$ must have nontrivial kernel.
In other words, the smooth connected commutative group
$((Z_G)^0_{\overline{k}})_{\rm{red}}$ contains a nontrivial
infinitesimal subgroup that is multiplcative.  The
group $((Z_G)^0_{\overline{k}})_{\rm{red}}$ therefore cannot be unipotent (since 
a smooth unipotent group cannot contain $\mu_p$),
so it must contain a nontrivial torus!   We conclude
by the same argument with $Z_G[n]$'s as in \S\ref{startmain} (using
$n$ not divisible by ${\rm{char}}(k) = p$) that $Z_G$ contains
a nontrivial $k$-torus, so we win. 

There remains the more interesting case when the preceding process does eventually stop,
so we wind up with a central quotient map
$$G \rightarrow G/M$$
by a multiplicative infinitesimal $k$-subgroup $M$
such that $G/M$ satisfies (\ref{hypstar}); beware
that now $M_{\overline{k}}$ is merely a product of several $\mu_{p^{n_i}}$'s, not
necessarily a power of $\mu_p$.   We therefore
get a nontrivial $k$-torus $T$ in $G/M$, so if
$E \subset G$ denotes its scheme-theoretic preimage then there exists a short exact sequence
of $k$-group schemes
$$1 \rightarrow M \rightarrow E \rightarrow T \rightarrow 1$$
with $M$ central in $E$.  We will be done (for infinite $k$) if
any such $E$ contains a nontrivial $k$-torus.   This is the content of the following lemma.

\begin{lemma}
For any field $k$ of characteristic $p > 0$ and short exact sequence of $k$-groups 
$$1 \rightarrow M \rightarrow E \rightarrow T \rightarrow 1$$
with a central multiplicative infinitesimal $k$-subgroup $M$ in $E$ 
and a nontrivial $k$-torus $T$, there exists a nontrivial $k$-torus in $E$.
\end{lemma}

\begin{proof}
Certainly $E_{\overline{k}}$ is connected, since $T$ and $M$ are connected.
The commutator map on $E$ factors through a bi-additive pairing
$T \times T \rightarrow M$ which is trivial
since $T$ is smooth and $M$ is infinitesimal. 
Hence, $E$ is commutative.  The map $E_{\overline{k}} \rightarrow T_{\overline{k}}$ is
bijective on $\overline{k}$-points, so
$(E_{\overline{k}})_{\rm{red}}$ is a nontrivial smooth connected
commutative $\overline{k}$-group.  It is therefore a direct product
of a torus and a smooth connected unipotent group, and the unipotent part
must be trivial (since $T_{\overline{k}}$ is a torus).  Hence,
$(E_{\overline{k}})_{\rm{red}}$ is a nontrivial torus. 
Since $E$ is commutative,
the identity component of the Zariski-closure of the $k$-subgroup schemes
$E[n]$ for $n$ not divisible by $p$ is a $k$-torus $T'$ in $E$ such that 
$T'_{\overline{k}} \rightarrow (E_{\overline{k}})_{\rm{red}}$ is surjective, so $T' \ne 1$. 
\end{proof}

\newpage

 \section{Groups of multiplicative type}\label{multsec}
 
 \subsection{Basic definitions and properties}\label{basicmult}
   For a scheme $S$ and finitely generated $\Z$-module 
 $M$, we define $D_S(M)$ to be the $S$-group $\Spec(\calO_S[M])$
 \index{ aaadsm@$D_S(M)$}
 representing the functor 
 $\underline{\rm{Hom}}_{S\mbox{-}{\rm{gp}}}(M_S,
\mathbf{G}_{\rm{m}})$ of characters of the constant $S$-group $M_S$.
(This is denoted $D(M)_S$ in \cite[I, 5.1]{oesterle}; it is the base change to $S$
of the analogous group scheme over $\Spec \Z$.  See \cite[I, 5.2]{oesterle} for
the proof that it represents the functor of characters of $M_S$.)  

\begin{definition}\label{ourmult} A group scheme $G \rightarrow S$ is {\em of multiplicative type} if 
\index{multiplicative type}
 there is an fppf covering $\{S_i\}$ of $S$ such that $G_{S_i} \simeq D_{S_i}(M_i)$
for a finitely generated abelian group $M_i$ for each $i$.
\end{definition}

By fppf descent, a multiplicative type $S$-group $G$ is faithfully flat and finitely presented over $S$. Such a $G$ is 
{\em split} (or {\em diagonalizable}) over $S$ if $G \simeq D_S(M)$ for some $M$; see \cite[I]{oesterle} for the 
theory of such groups, and \cite[I, 5.2]{oesterle} for the anti-equivalence
between the categories of split $S$-groups of multiplicative type and constant commutative $S$-groups 
with finitely generated geometric fibers. 
\index{multiplicative type!split}
\index{split!multiplicative type group}
\index{diagonalizable group}

In \cite{oesterle} the basic theory of multiplicative type groups $G \rightarrow S$ is developed
under weaker conditions (following \cite{sga3}):  $M$ is not required to be finitely generated,
$G$ is fpqc and affine over $S$
but may not be of finite presentation, and 
diagonalizability is required only fpqc-locally on $S$.  The proofs of all results
in \cite{oesterle} that we cite below 
work verbatim under our finiteness restrictions, 
due to our insistence on fppf-local diagonalizability in the definition.  (In Corollary \ref{fpqcwin}(1) 
we show that fpqc-local diagonalizability recovers Definition \ref{ourmult} for fppf group schemes. Thus,
our multiplicative type groups are precisely those of \cite{oesterle} and \cite{sga3}
with finite type structural morphism; we never use this.) 

\begin{remark}\label{subgpM} In \cite[I, 5.4]{oesterle} it is noted that for any fppf closed $S$-subgroup $H$ of $D_S(M)$ there is
a unique partition $\{S_N\}_{N \subset M}$ of $S$ into pairwise disjoint open and closed subschemes $S_N$
indexed by the subgroups $N \subset M$, with $H|_{S_N} = D_S(M/N)$ inside $D_S(M)$.
The special case $S = \Spec k$ for a field $k$ is addressed there ($M$ may not be
torsion-free, so $D_k(M)$ may be disconnected or non-smooth), and the general case is addressed
in \cite[II, \S1.5, Rem.\:3, 4]{oesterle}.  A direct
proof over fields under our finiteness hypotheses on $M$ is given in Exercise \ref{closedmult}, and 
to settle the case of arbitrary $S$ we may use
``spreading out'' considerations (under our finiteness hypotheses)
to reduce to $S$ that is local, and even noetherian.  This case is deduced 
from the field case in the proof of Corollary \ref{moreflat} (which provides a generalization
for all $S$-groups of multiplicative type). 

An interesting consequence of this description of all such $H$ is the general fact that if
$G \rightarrow S$ is a group scheme of multiplicative type then there is a closed subtorus $T \subset G$
that is maximal in the sense that it (i) contains all closed subtori of $G$ and  (ii) retains
this property after {\em arbitrary} base change.  Such a $T$ is unique if it exists, so by fppf descent
it suffices to treat the case $G = D_S(M)$ for a finitely generated abelian group $M$.
Let $M' = M/M_{\rm{tor}}$ denote the maximal torision-free quotient of $M$, and define
$T = D_S(M') \subset D_S(M) = G$.  Clearly it suffices to show that every closed
subtorus of $G$ is contained in $T$, and we may assume $S$ is non-empty.  Working Zariski-locally on $S$,
the above description reduces our task to considering closed subtori of the form $D_S(M/N)$
for a subgroup $N$ in $M$.  For any such $N$, the torus property for $D_S(M/N)$ forces
$M/N$ to be torsion-free (since $S \ne \emptyset$).  Hence, $M/N$ is dominated by $M'$ as quotients of $M$, so we are done.
\end{remark}

We refer the reader to \cite[II, 2.1]{oesterle} for several notions of ``local triviality''
for multiplicative type groups (isotriviality, quasi-isotriviality, etc.) 
 For many applications, it is important
 that multiplicative type groups are split \'etale-locally on the base (quasi-isotriviality).  
 \index{quasi-isotriviality}
 \index{multiplicative type!quasi-isotrivial}
 This will be proved in Proposition \ref{fppfet} and rests on the following lemma.

\begin{lemma}\label{ix2.5} Let $G$ be an $S$-affine $S$-group scheme of
finite presentation and $H$ an $S$-group of multiplicative type.
Any monic homomorphism $j:H \rightarrow G$ 
is necessarily a {\em closed immersion}.
\end{lemma}
\index{monic homomorphism!multiplicative type}

This lemma  is useful in constructions with fiberwise maximal tori in reductive group schemes, and 
eliminates ambiguity about the meaning of ``subgroup of multiplicative type'' 
for homomorphisms from groups of multiplicative type: 
working sheaf-theoretically (or in terms of group functors)
with monomorphisms is equivalent to working 
algebro-geometrically with closed immersions.   

\begin{proof}
This result is \cite[IX, 2.5]{sga3} without finite presentation hypotheses,
and it is also a special case of \cite[VIII, 7.13(b)]{sga3} (relaxing affineness to separatedness).
Here is an alternative direct argument under our hypotheses.

We may pass to the case of noetherian $S$, and it suffices
to show that $j$ is proper (since a finitely presented proper monomorphism is a closed
immersion \cite[IV$_3$, 8.11.5]{ega}).  By the valuative criterion for properness, 
we may express the problem in terms of points valued in a discrete valuation ring
$R$ and its fraction field $K$ over $S$, so we may reduce to the case that $S = \Spec R$
and it is harmless to make a (necessarily faithfully flat) local injective base change $R \rightarrow R'$
to another discrete valuation ring.  Thus, we can assume that $R$ is henselian. We claim that such an $R'$
may be found so that $H_{R'}$ is split.  Pick an fppf cover
$S' \rightarrow S$ that splits $H$.  As for any fppf cover of an affine
scheme, there is an affine flat quasi-finite surjection $S'' \rightarrow S$ 
admitting a map $S'' \rightarrow S'$ over $S$ \cite[IV$_4$, 17.16.2]{ega}.  
Since $R$ is henselian local, by \cite[IV$_4$, 18.5.11(c)]{ega}
the affine flat quasi-finite cover $S''$ of $S$ contains
an open and closed local subscheme that is $S$-finite.
This subscheme is finite flat over $S$ and non-empty,
so by the Krull--Akizuki theorem \cite[Thm.\:11.7]{crt}
the normalization of its underlying reduced scheme has the form $\Spec R'$ for a discrete
valuation ring $R'$. Moreover, by design $H_{R'}$ is split, so by renaming $R'$ as $R$, 
now $H$ is split.

Letting $k$ be the residue field of $R$, the maps $j_K$ and $j_k$ are monomorphisms
between affine finite type group schemes over a field, so they are closed immersions (apply 
Remark \ref{classicalmonic} on geometric generic and special fibers).
Let $H'$ be the schematic closure of $H_K$ in $G$.  This is an $R$-flat closed
subgroup since $R$ is Dedekind, and it is commutative since the $K$-group $H'_K = H_K$ is commutative.
We may replace $G$ with $H'$, so now $G$ is commutative.
For any $n \ge 1$, the monomorphism $H[n] \rightarrow G$ is proper since $H[n]$ is finite,
so it is a closed immersion. 

 It is harmless to pass to $G/H[n]$ and $H/H[n]$.  (See \cite[VIII, 5.1; IX, 2.3]{sga3} for 
a discussion of the existence and properties of quotients by 
the free action of a group of multiplicative type on a finitely presented relatively affine
scheme.  This is simpler than the general theory of quotients by the free action of a finite locally free
group scheme as developed in \cite[V, \S2(a), Thm.\:4.1]{sga3}.)  More specifically, for 
$n$ divisible by the order of
the torsion subgroup of the finitely generated abelian group that is
``dual'' to the split $H$, 
geometric fibers of $H/H[n]$ are smooth (i.e., tori).  Thus, by passing to
$G/H[n]$ and $H/H[n]$ we may assume $H \simeq \Gm^r$ for
some $r \ge 0$.

We can assume that $k$ is algebraically closed and $H \ne 1$.
Let $T = \Gm$ be the first factor of $H = \Gm^r$
and let $T'$ be the closure of $T_K$ in $G$.  If we can prove that $T' = T$ then we may
pass to the quotient by $T$ and induct on $r$.  Thus, we may assume
$H = \Gm$, so $G$ also has 1-dimensional fibers.  In particular,
the closed immersion $j_k$ must identify $H_k$ with $(G_k)^0_{\rm{red}}$.
For $N > 0$ relatively prime to the order of the finite group scheme $G_k/H_k$,
a snake lemma argument shows that $N:G_k \rightarrow G_k$ is a quotient map with
kernel $H_k[N]$, so it is flat.  Hence, by the fibral flatness criterion, $N:G \rightarrow G$ is flat,
so $G[N]$ is flat and hence $G[N] = H[N]$ as closed subschemes of
$G$ due to the equality of their generic fibers in $G_K = H_K$.

The translation action by $H = \Gm$ on
$G$ defines a $\Z$-grading $\bigoplus_{n \in \Z} A_n$ of the coordinate ring $A$ of $G$
\cite[III, 1.5]{oesterle}. 
(Explicitly, this grading extends
the natural one on the coordinate ring $A_K$ of $G_K = H_K = \Gm$.)
The quotient map $A \twoheadrightarrow \calO(G[N])$ between flat $R$-modules is injective
on each $A_n$ because it is so over $K$ (as $G_K = \Gm$).
It follows that each $A_n$ is {\em finitely generated} over $R$, and thus
is finite free of rank 1 because we can compute the rank over $K$.
The map $A \rightarrow \calO(H) = R[t,1/t]$ respects
the $\Z$-gradings and induces a surjection on special
fibers (as $j_k$ is a closed immersion), so the induced maps between the rank-1 graded
parts are isomorphisms.  Hence, $A \rightarrow \calO(H)$ is an isomorphism.
\end{proof}

\begin{remark}\label{immrem}
Beware that even over a discrete valuation ring, there are examples
of monic homomorphisms $f$ between smooth affine groups with
connected fibers such that $f$ is not an immersion! 
\index{monic homomorphism}
See \cite[XVI, 1.1(c)]{sga3} for some explicit examples
(and \cite[VIII, \S7]{sga3} for further discussion).
The contrast with the case $S = \Spec k$ for
a field $k$ is that in such cases homomorphisms between smooth
affine $S$-groups are always faithfully flat onto a closed image
(due to the ``locally closed'' property of $G$-orbits over fields, which has
no good analogue in comparable generality in the relative case).
\end{remark}

\begin{lemma}\label{fieldsplit}  Let $k$ be a field.  A $k$-group $H$ of multiplicative
type splits over a finite separable extension of $k$.
\end{lemma}
\index{multiplicative type!splitting field}

\begin{proof}
By direct limit considerations, it
suffices to prove that $H_{k_s}$ is split.  Thus, we may  assume
$k = k_s$ and aim to prove that $H \simeq D_k(M)$ for a finitely generated
abelian group $M$. Since any fppf cover of
$\Spec k$ acquires a rational point over a finite extension $k'/k$,
there is a finitely generated abelian group $M$
such that $H$ and $D_k(M)$ become isomorphic over a finite extension $k'$ of $k$.

The functor $I = \underline{{\rm{Isom}}}(H,D_k(M))$ is
an fppf sheaf on the category of $k$-schemes, so
it is an $\Aut(M)$-torsor because this can be checked upon restriction
to the category of $k'$-schemes.  More specifically, the
restriction of $I$ over $k'$ is represented by a constant $k'$-scheme,
and that constant scheme is equipped with an evident descent
datum (arising from $I$) relative to $k'/k$.   Although $I_{k'}$
is generally not affine, nor even quasi-compact, the descent to $k$ is
easily checked to be effective
because (i) fppf descent is effective
for affine schemes and (ii) in our descent problem the ``equivalence classes'' inside $I_{k'}$ are open and
closed subschemes that are $k$-{\em finite}.
Hence, $I$ is represented by a $k$-scheme
that is non-empty and \'etale (as it becomes constant and non-empty over $k'$),
so $I$ is constant (since $k$ is separably closed).
\end{proof}


\index{Hochschild cohomology|(}
\subsection{Hochschild cohomology}\label{hochsec}
The key to the infinitesimal properties of groups of multiplicative type
is the vanishing of their higher Hochschild cohomology over an affine base.
A general introduction to Hochschild cohomology ${\rm{H}}^i(G,\calF)$
for flat affine group schemes $G \rightarrow S$ acting linearly on quasi-coherent $\calO_S$-modules $\calF$
over an affine $S$ is given in \cite[III, \S3]{oesterle}.  
Hochschild cohomology is a scheme-theoretic version of ordinary group cohomology, and 
is explained (with proofs) in \cite[III, \S3]{oesterle}. Some applications below require
a more ``functorial'' description of Hochschild cohomology, so we now review the 
basic setup using a variation on the formulation given in \cite[III,\S3]{oesterle}.

Let $M$ be a commutative group functor on the category of
$S$-schemes, equipped with an action by an $S$-group scheme $G$. A case of much interest
is the functor $S' \rightsquigarrow \Gamma(S',\calF_{S'})$ arising
from a {\em quasi-coherent $G$-module},
by which we mean a quasi-coherent $\calO_S$-module $\calF$ equipped with a linear $G$-action;
see \cite[III, 1.2]{oesterle}. 
(This amounts to an $\calO_{S'}$-linear action of $G(S')$ on $\calF_{S'}$
functorially in $S'$. By consideration of 
the ``universal point'' ${\rm{id}}_G:G \rightarrow G$, it is equivalent to 
specifying an $\calO_G$-linear automorphism of $\calF_G$.)  For $n \ge 0$, let 
${\rm{C}}^n(G,M)$ be the abelian group of natural transformations of set-valued functors 
$c:G^n \rightarrow M$ on the category of $S$-schemes
(i.e., compatible systems of maps of sets $G(T)^n \rightarrow M(T)$, or equivalently 
the abelian group $M(G^n)$). 
For example, if $S = \Spec A$ is affine and $G = \Spec(B)$ is affine
and $\calF = \widetilde{V}$ for an $A$-linear $B$-comodule $V$
then the element $v \otimes b_1 \otimes \dots \otimes b_n \in V \otimes_A B^{\otimes n} =
\Gamma(G^n, \calF_{G^n})$ corresponds
to $c:G^n \rightarrow \calF$ defined functorially by $(g_1,\dots,g_n) \mapsto (\prod g_i^{\ast}(b_i))v_{S'}$
on $S'$-valued points for any $S$-scheme $S'$.

The groups ${\rm{C}}^n(G,M) = M(G^n)$ form a complex ${\rm{C}}^{\bullet}(G,M)$ in the habitual manner, by defining
\index{ aaacgm@${\rm{C}}^n(G,M)$}
$({\rm{d}}c)(g_0,\dots,g_n)$ to be
$$g_0.c(g_1,\dots,g_n) + \sum_{i=1}^n (-1)^i c(g_0,\dots,g_{i-1}g_i,\dots,g_n) + 
(-1)^{n+1}c(g_0,\dots,g_{n-1}).$$
We define the Hochschild cohomology
\index{ aaahgm@${\rm{H}}^n(G,M)$}
$${\rm{H}}^n(G,M) = {\rm{H}}^n({\rm{C}}^{\bullet}(G,M)).$$
For example, ${\rm{H}}^0(G,M)$ is the group $M(S)^G$ of 
$m \in M(S)$ that are $G$-invariant in the sense that
$g.m_{S'} = m_{S'}$ in $M(S')$ for any $S' \rightarrow S$ and $g \in G(S')$.
More concretely, $G$-invariance means that the pullback $m_{G} \in M(G)$ is invariant
under the universal point ${\rm{id}}_G \in G(G)$ (not to be confused
with the $G$-pullback of the identity section $e \in G(S)$).

\begin{example}\label{hochimm}
Let  $j:S_0 \hookrightarrow S$ be a closed immersion of schemes. For any quasi-coherent sheaf
$\mathscr{F}_0$ on $S_0$ equipped with a $G_0$-action, 
and the associated quasi-coherent sheaf $\mathscr{F} := j_{\ast}(\mathscr{F}_0)$ on $S$ with 
$G$-action via $G \rightarrow j_{\ast}(G_0)$, naturally 
${\rm{H}}^{\ast}(G, \mathscr{F}) \simeq {\rm{H}}^{\ast}(G_0, \mathscr{F}_0)$. 
\end{example}

\begin{proposition}\label{bch} If $f:S' = \Spec A' \rightarrow \Spec A = S$ is a flat map of affine schemes and
$G$ is $S$-affine then for any quasi-coherent $G$-module 
$\calF$ the natural map
$A' \otimes_A {\rm{H}}^n(G,\calF) \rightarrow {\rm{H}}^n(G_{A'}, \calF_{A'})$
is an isomorphism for all $n \ge 0$.
\end{proposition}
\index{Hochschild cohomology!and flat base change}

\begin{proof} Each $G^n$ is affine, so  the natural map of complexes
$$A' \otimes_A {\rm{C}}^{\bullet}(G,\calF) \rightarrow {\rm{C}}^{\bullet}(G_{S'}, \calF_{S'})$$
is identified in degree $n$ with the natural map $A' \otimes_A V_{G^n} \rightarrow
(A' \otimes_A V)_{G_{A'}^n}$ where $\widetilde{V} = \calF$. This map is visibly an isomorphism.  Passing
to homology in degree $n$ recovers the map of interest as an isomorphism
because the exact functor $A' \otimes_A (\cdot)$ commutes with the formation of homology
in the evident manner.
\end{proof}

Assume $G$ is $S$-affine and $S$-{\em{flat}}, and that $S$ is also affine (so each $G^n$ is affine and $S$-flat).
The functor $\calF \rightsquigarrow {\rm{C}}^n(G,\calF)$ on quasi-coherent $G$-modules
is exact for each $n$, so 
${\rm{H}}^{\bullet}(G,\calF)$ is $\delta$-functorial in such $\calF$ via the snake lemma.
We shall prove that the category of
quasi-coherent $G$-modules has enough 
injectives and the derived functor of $\calF \rightsquigarrow \calF(S)^G$ on this category 
is ${\rm{H}}^{\bullet}(G,\cdot)$.

For $S$-affine $G$, 
there is a right adjoint ``Ind'' to the forgetful functor from quasi-coherent $G$-modules 
to quasi-coherent $\calO_S$-modules.  Explicitly, for any $S$-scheme $S'$
and quasi-coherent $\calO_S$-module $\calG$, 
${\rm{Ind}}(\calG)(S') := \Gamma(G_{S'}, \calG_{G_{S'}})$ equipped
with the left $G(S')$-action $(g.f)(x) = f(xg)$ for points $x$ of $G_{S'}$.
In other words, as in \cite[III, 3.2, Ex.]{oesterle}, 
${\rm{Ind}}(\calG)$ is the quasi-coherent pushforward of
$\calG_G$ along the affine map $G \rightarrow S$, equipped with a natural $\calO_S$-linear $G$-action.
The $G$-action amounts to an $\calO_G$-linear automorphism of the pullback 
of ${\rm{Ind}}(\calG)$ along $G \rightarrow S$.
By general nonsense, ${\rm{Ind}}$ carries injectives to injectives, and monomorphisms
to monomorphisms.  Moreover, the adjunction morphism
$\calF \rightarrow {\rm{Ind}}(\calF)$ is given by $m \mapsto (g \mapsto g.m)$; it has trivial kernel
since $f \mapsto f(1)$ is a retraction.  Thus, the category of quasi-coherent $G$-modules
on $S$ has enough injectives (since the category of quasi-coherent $\calO_S$-modules does;
recall that $S$ is affine).

\begin{lemma}\label{derhoch} For affine $S$ and $S$-flat $S$-affine
$G$, the $\delta$-functor ${\rm{H}}^{\bullet}(G,\cdot)$ on quasi-coherent $G$-modules is
the right derived functor of  $\calF \rightsquigarrow \calF(S)^G$. 
\end{lemma}
\index{Hochschild cohomology!as derived functor}

\begin{proof}
An elegant argument in \cite[III, 3.2, Rem.]{oesterle} (inspired
by the proof of acyclicity of induced modules for ordinary group cohomology)
shows that ${\rm{H}}^i(G,{\rm{Ind}}(\calG)) = 0$ for any 
quasi-coherent $\calO_S$-module $\calG$ and $i > 0$.
For any injective $\calF$ in the category of 
quasi-coherent $G$-modules, the inclusion
$\calF \rightarrow {\rm{Ind}}(\calF)$ splits
off $\calF$ as a $G$-equivariant direct summand. But
${\rm{Ind}}(\calF)$ is acyclic for Hochschild cohomology, so $\calF$ is as well. 
We conclude that ${\rm{H}}^{\bullet}(G,\cdot)$ 
is erasable on the category of quasi-coherent $G$-modules, so on this category it is 
the desired right derived functor. 
\end{proof}

\begin{remark}\label{multvanish}
A consequence of Lemma \ref{derhoch} (as in \cite[III, 3.3]{oesterle}) is that for any 
{\em multiplicative type} group $G$ over an affine $S = \Spec A$,
${\rm{H}}^i(G,\cdot) = 0$ on quasi-coherent objects for $i > 0$. 
Indeed, by Proposition \ref{bch} it suffices
to prove this after a faithfully flat affine base change on $A$, so
we may assume $G = D_S(M)$. Thus, quasi-coherent
$G$-modules are ``the same'' as $M$-graded quasi-coherent
$\calO_S$-modules $\calF = \bigoplus_{m \in M} \calF_m$ \cite[III, 1.5]{oesterle}, so 
${\rm{H}}^0(G,\calF) = \calF_0(S)$. This is exact in such $\calF$, so 
its higher derived functors vanish. 
\end{remark}
\index{multiplicative type!vanishing Hochschild cohomology}

An important application of Hochschild cohomology is the construction
of obstructions to deformations of homomorphisms between group schemes.
This is inspired by the use of low-degree group cohomology to classify
group extensions, and comes out as follows.

\begin{proposition}\label{obs}  Consider a scheme $S$ and short exact sequence of 
group sheaves 
$$1 \rightarrow M \rightarrow G \rightarrow G' \rightarrow 1$$
for the fpqc topology, with $M$ commutative.  Use this exact sequence
to make $G'$ act on $M$ via the left $G$-conjugation action  on $M$.

Let $H$ be an $S$-group scheme, and fix an $S$-homomorphism
$f':H \rightarrow G'$.  Make $H$ act on $M$ through composition with
$f'$.  Assume that $f'$ admits a lifting to an $H$-valued point of $G$.

There is a canonically associated class
$c(f') \in {\rm{H}}^2(H, M)$ that vanishes if and only if
$f'$ lifts to an $S$-homomorphism $f:H \rightarrow G$.
If such an $f$ exists, the set of such lifts taken up to conjugation
by $M(S)$ on $G$ is a principal homogeneous space for
the group ${\rm{H}}^1(H, M)$.
\end{proposition}

The result holds with the same proof using any topology for which representable
functors are sheaves (e.g., Zariski, \'etale, fppf).
See \cite[III, 1.2.2]{sga3} for further generality.

\begin{proof}
Fix an $f \in G(H)$ lifting $f'$.
The obstruction to $f$ being a homomorphism is the vanishing of the
map of $c = c_f \in M(H \times H)$ 
defined by $c(h_0,h_1) = f(h_0 h_1) f(h_1)^{-1}f(h_0)^{-1}$.
The action of any $h_0$ on $c(h_1,h_2)$ is induced by 
$f(h_0)$-conjugation, so
\begin{eqnarray*}
h_0.c(h_1,h_2) &=& f(h_0)(f(h_1h_2)f(h_2)^{-1}f(h_1)^{-1})f(h_0)^{-1}\\
&=& c(h_0,h_1h_2)^{-1} c(h_0h_1,h_2)c(h_0,h_1)\\
&=& c(h_0 h_1,h_2) - c(h_0,h_1h_2) + c(h_0,h_1).
\end{eqnarray*}
In other words, the element $c_f \in M(H \times H)$ is a Hochschild 2-cocycle
(where $M$ is equipped with its natural $H$-action through $f'$).
Fixing one choice of $f$, all choices have exactly the form
$m \cdot f: h \mapsto m(h) \cdot f(h)$ for $m \in M(H)$, 
and the value on $(h_0, h_1)$ for the associated 2-cocycle $c_{m \cdot f}$  is
\begin{eqnarray*}
m(h_0h_1) f(h_0h_1)f(h_1)^{-1}f(h_0)^{-1}(f(h_0)m(h_1)f(h_0)^{-1})^{-1}m(h_0)^{-1} =\\
m(h_0h_1) + c_f(h_0,h_1) - h_0.m(h_1) - m(h_0).
\end{eqnarray*}
Thus, the class of $c_f$ in ${\rm{H}}^2(H,M)$ 
only depends on $f'$ and not the choice of $f$,
and as we vary through all $f$ this 2-cocycle exhausts
exactly the members of its cohomology class.
Thus, this class vanishes if and only if 
we can choose $f$ so that $c_f = 0$, which is
to say that $f$ is an $S$-homomorphism.

Assume that an $S$-homomorphism $f$ lifting $f'$ exists.
Fix one such choice of $f$. 
The preceding calculation shows that the possible choices for $f$ as an $S$-homomorphism 
are precisely $h \mapsto m(h)f(h)$ where
$m$ is a Hochschild 1-cocycle on $H$ with values in $M$.
Applying conjugation to such an $f$ by some $m_0 \in M(S)$
replaces $f$ with the lifting
$$h \mapsto m_0 \cdot f(h) \cdot m_0^{-1} = m_0 \cdot (f(h) m_0^{-1} f(h)^{-1}) \cdot f(h) = 
(m_0 - h.m_0) \cdot f(h)$$
due to the definition of the $H$-action on $M$ (through the $G'$-action on $M$ induced by $G$-conjugation).
It follows that ${\rm{H}}^1(H,M)$ acts simply transitively on the 
set of $M(S)$-conjugacy classes of $S$-homomorphisms
$f$ lifting $f'$.
\end{proof}

\begin{corollary}\label{h2obs}
Let $G \rightarrow S$ be a smooth group over an affine scheme $S = \Spec A$,
and let $H$ be an affine $S$-group.
Let $J$ be a square-zero ideal in $A$, $A_0 = A/J$, $S_0 = \Spec A_0$,
$G_0 = G \bmod J$, and $H_0 = H \bmod J$.  Fix an $S_0$-homomorphism
$f_0:H_0 \rightarrow G_0$, and let $H_0$
act on ${\rm{Lie}}(G_0)$ via ${\rm{Ad}}_{G_0} \circ f_0$.

There is a canonically associated
class $c(f_0) \in {\rm{H}}^2(H_0, {\rm{Lie}}(G_0) \otimes J)$
whose vanishing is necessary and sufficient for $f_0$
to lift to an $S$-homomorphism $f:H \rightarrow G$.
If such an $f$ exists, the set of such lifts taken up to conjugation by 
$\ker(G(S) \rightarrow G(S_0))$ is a principal homogeneous space for
the group ${\rm{H}}^1(H_0, {\rm{Lie}}(G_0) \otimes J)$.
\end{corollary}

See \cite[III, 2.2, 2.3]{sga3} for generalizations.  
By Remark \ref{multvanish}, $c(f_0) = 0$ if $H$ is of multiplicative type.
\index{deformation theory!homomorphism}
\index{deformation theory!multiplicative type subgroup}

\begin{proof}
Let $i:S_0 \rightarrow S$ be the canonical closed immersion, so
$i_{\ast}(G_0)$ is the group functor $S' \rightsquigarrow G_0(S'_0)$
on $S$-schemes.  An $S_0$-homomorphism $f_0:H_0 \rightarrow G_0$ corresponds
to an $S$-group functor homomorphism $f'_0:H \rightarrow i_{\ast}(G_0)$.
An $S$-homomorphism $f:H \rightarrow G$ lifts $f'_0$ if and only if 
$f \bmod J = f_0$. Thus, we focus on lifting $f'_0$.

By the smoothness of $G$, the natural homomorphism $q:G \rightarrow i_{\ast}(G_0)$ is surjective for the Zariski topology
(and hence for any finer topology).  More specifically,
since $G$ is smooth and $H_0 \hookrightarrow H$ is
defined by a square-zero ideal on the affine scheme $H$, $f_0$ admits a lifting $f'_0$ through
$q$ as a scheme morphism.
It also follows from the smoothness 
of $G$ that $\ker q$ is the group functor on $S$-schemes associated to the quasi-coherent 
$\calO_{S_0}$-module ${\rm{Lie}}(G_0) \otimes J$.
(This uses that $J \calO_G = J \otimes_{A_0} \calO_{G_0}$, a consequence of 
the $A$-flatness of $G$.)
The conjugation action by $G$ on the commutative $\ker q = {\rm{Lie}}(G_0) \otimes J$
factors through an action by $i_{\ast}(G_0)$, and this ``is'' 
the adjoint action of $G_0$ on ${\rm{Lie}}(G_0)$.  
Thus, Proposition \ref{obs} applies to the 
 exact sequence 
$$1 \rightarrow {\rm{Lie}}(G_0) \otimes J \rightarrow G \rightarrow i_{\ast}(G_0) \rightarrow 1.$$
Quasi-coherence of the kernel implies 
${\rm{H}}^i(H,{\rm{Lie}}(G_0) \otimes J) = {\rm{H}}^i(H_0, {\rm{Lie}}(G_0) \otimes J)$
via Example \ref{hochimm}.
\end{proof}
\index{Hochschild cohomology|)}

Via induction, the preceding corollary immediately yields:

\begin{corollary}\label{rigidhom}
Let $S = \Spec(A)$ and $S_0 = \Spec(A/J)$ for an ideal $J$ of $A$ such that $J^{n+1}=0$ for some $n \ge 0$.
An $S$-homomorphism $f:H \rightarrow G$ from a multiplicative type $S$-group $H$ to an arbitrary $S$-group scheme $G$
is trivial if its restriction $f_0$ over $S_0$ is trivial.
\end{corollary}
\index{multiplicative type!rigidity}
\index{rigidity of homomorphisms}

This rigidity property is \cite[IX, Cor.\:3.5]{sga3}.

\subsection{Deformation theory}\label{defthy}
To relativize results established over fields, it is important to have ``fibral criteria''
for properties of morphisms (such as flatness, smoothness, etc.) as well
as deformation-theoretic results concerning the obstructions to lifting problems.  
In the direction of fibral criteria, we often need the ``fibral isomorphism criterion'':
\index{fibral isomorphism criterion}

\begin{lemma}\label{flatclosed} Let $h:Y \rightarrow Y'$ be a map between
locally finitely presented schemes over a scheme $S$, and assume that $Y$ is $S$-flat.
If $h_s$ is an isomorphism for all $s \in S$ then $h$ is an isomorphism.
\end{lemma}

\begin{proof}
This is part of \cite[IV$_4$, 17.9.5]{ega} (or see Exercise \ref{flatexer}). 
\end{proof}


\begin{theorem}\label{torlift}
Let $(A, \mathfrak{m})$ be a complete local noetherian ring with residue field $k$,
$S = \Spec A$, $G$ an affine $S$-group of finite type,
and $H$ an $S$-group of multiplicative type that splits over a finite \'etale cover of $S$. Let
$S_n = \Spec A/\mathfrak{m}^{n+1}$.  

\begin{enumerate}
\item The natural map
\begin{equation}\label{limmap}
\Hom_{S\mbox{-}{\rm{gp}}}(H,G) \rightarrow \invlim \Hom_{S_n\mbox{-}{\rm{gp}}}(H_{S_n}, G_{S_n})
\end{equation}
is bijective.  
\item If $G$ is $S$-flat and the special fiber $G_0$ is of multiplicative type then the map
$$\Hom_{S\mbox{-}{\rm{gp}}}(H,G) \rightarrow \Hom_{k\mbox{-}{\rm{gp}}}(H_0,G_0)$$
is bijective and any isomorphism $j_0:H_0 \simeq G_0$ uniquely lifts
to an open and closed immersion $j:H \rightarrow G$.
\end{enumerate}
\end{theorem}

The splitting hypothesis on $H$ is temporary in the sense that it will be shown to always hold (see
Proposition \ref{fppfet}).  Also, the flatness hypothesis on $G$ in (2) cannot be dropped: for an integer $d > 1$ let $H = (\Z/d\Z)_R$
over a discrete valuation ring $R$ such that $d \in R^{\times}$ and let
$G$ be the reduced closed complement of the open non-identity locus in the generic fiber.

\begin{proof} 
Let $S' = \Spec A'$ be a finite \'etale cover of $S$ that splits $H$.  We may assume
$S'$ is connected and Galois over $S$, so $A'$ is a complete local noetherian ring
with maximal ideal $\mathfrak{m}A'$ and descent from $S'$ to $S$ can be expressed in terms of
actions by $\Gamma = {\rm{Aut}}(S'/S)$.  The same $\Gamma$ works
for descent from $S'_n$ to $S_n$.  Thus, if the analogue of (\ref{limmap}) over $S'$
is bijective then $\Gamma$-equivariance considerations show that (\ref{limmap}) is bijective.
The same holds for part (2). 
We therefore may and do assume $H = D_S(M)$ for a finitely generated abelian group $M$.
Upon expressing the problem in (1) in terms of
maps of Hopf algebras, it is solved by a clever use of the interaction between
completions and tensor products beyond the module-finite setting. See the proof
of \cite[IX, 7.1]{sga3} for this important calculation.

We now prove part (2).  In such cases, by (1) 
the bijectivity of ``passage to the special fiber'' on homomorphisms is reduced to showing
that for all $n \ge 1$ the reduction map
$$\Hom_{S_n\mbox{-}{\rm{gp}}}(D_{S_n}(M),G_{S_n}) \rightarrow \Hom_{k\mbox{-}{\rm{gp}}}(D_k(M),G_0)$$
is bijective.  That is, upon renaming $S$ as $S_n$, we may assume
$A$ is an artin local ring.  By descent we may replace $A$ with a finite
\'etale extension so that $G_0 \simeq D_k(N)$ for a finitely generated abelian group $N$.
Since $G$ is an {\em infinitesimal} flat deformation of $D_k(N)$, by the deformation theory
of split multiplicative type groups we claim that $G$ must be of multiplicative type and even 
that $G \simeq D_S(N)$.  

To be precise, by induction on $n$ 
it suffices to show that if $J$ is a square-zero ideal in a ring $R$
and $G = \Spec A$ is an  fppf affine $R$-group such that
over $R_0 = R/J$ there is a group isomorphism
$f_0:G_0 := G \bmod J \simeq D_{R_0}(N)$ for a finitely generated abelian group $N$
then $f_0$ lifts to an $R$-group isomorphism
$f:G \simeq D_R(N)$.  Since ${\rm{H}}^2(G_0,\cdot) = 0$ 
on quasi-coherent $G_0$-modules (as $G_0$ is of multiplicative type),
by Corollary \ref{h2obs} the map $f_0$ lifts to an
$S$-homomorphism $f$.  But any such lift must be an isomorphism because
$f_0$ is an isomorphism and the source and target of $f$ are fppf over $S$.
(A more explicit construction of $f$ is given in 
\cite[IV, \S1]{oesterle}, directly building an obstruction in a degree-2 Hochschild 
cohomology group for the multiplicative type $G_0$.)
The desired bijectivity in (2) now follows via duality
for diagonalizable groups (of finite type).

Finally, with general complete local noetherian $A$, it remains to show that if $G_0$ is of multiplicative type 
and $G$ is flat then for any $j_0:D_k(M) \simeq G_0$ the unique
$S$-homomorphism $j:D_S(M) \rightarrow G$ lifting $j_0$ is an open and closed immersion. 
The preceding argument over $S_n$'s shows that $j_n := j \bmod \mathfrak{m}^{n+1}$
is an isomorphism for all $n \ge 0$, so the map induced by $j$ between formal completions
along the identity is flat 
(by \cite[22.3(1)$\Leftrightarrow$(5)]{crt}).  
Thus, $j$ is flat near the identity section.   But $j_s$ must be flat for all $s \in S$ since
a homomorphism between finite type groups over a field is flat
if it is flat near the identity (use translation considerations on a geometric fiber),
so by the fibral flatness criterion $j$ is flat.

Consider the kernel $K = \ker j$, a flat closed $S$-subgroup of $D_S(M)$
with $K_0 = 0$.  It suffices to prove $K = 0$.  Indeed, then 
$j$ will be a monomorphism and so even a closed immersion
(Lemma \ref{ix2.5}), and a flat closed immersion between noetherian schemes is
an open immersion.  To prove that the flat $S$-group $K$ is trivial,
observe that each fiber $K_s$ ($s \in S$) is of multiplicative type, due to being
a closed subgroup scheme of $D_s(M)$ (see Exercise \ref{closedmult}).
For $n \ge 1$, the $n$-torsion $K[n]$ is a finite $S$-group scheme (as it is closed
in $D_S(M)[n] = D_S(M/nM)$) and it has special fiber $K[n]_0 = K_0[n] = 0$,
so by Nakayama's Lemma we have $K[n] = 0$.  Hence, for each $s \in S$
the torsion $K_s[n]$ vanishes for all $n \ge 1$, so the multiplicative type group $K_s$
over $k(s)$ vanishes \cite[II, 3.2]{oesterle}.  In other words, the identity section $e:S \rightarrow K$ is an isomorphism
on fibers over $S$, so it is an isomorphism by the fibral isomorphism criterion
(Lemma \ref{flatclosed}).
\end{proof}


\begin{corollary}\label{moreflat}
Let $S$ be a scheme and $H'$ an $S$-group of multiplicative type.
Any fppf closed subgroup $H \subset H'$ is of multiplicative type.
\end{corollary}
\index{multiplicative type!flat closed subgroup of}

\begin{proof}
Passing to an fppf cover, we may assume that $H' = D_S(M)$ for a finitely generated abelian group $M$.
We may reduce to the case when $S = \Spec A$ for
a ring $A$ that is noetherian, and then local; let $k$ be the residue field.
Exercise \ref{closedmult} provides a subgroup $N \subset M$ such that
$H_k = D_k(M/N)$ inside $D_k(M)$.  We claim that $H = D_S(M/N)$ inside $D_S(M)$.
It suffices to check this equality of closed subschemes after the fpqc base change to $\Spec \widehat{A}$, so we may assume
$A$ is complete.  Thus, by Theorem \ref{torlift}
the isomorphism $D_k(M/N) \simeq H_k$ uniquely lifts to an abstract $S$-homomorphism
$j:D_S(M/N) \rightarrow H$  that is moreover an open and closed immersion.
The composition of $j$ with the inclusion $H \hookrightarrow H' = D_S(M)$
is an $S$-homomorphism $D_S(M/N) \rightarrow D_S(M)$ that
reduces to the canonical inclusion over the closed point, so by duality
for diagonalizable groups it is the canonical inclusion over the entire connected base $S$.
In other words, $j$ is a containment inside $H' = D_S(M)$; i.e., 
$H$ contains $D_S(M/N)$ as an open and closed subscheme inside $D_S(M)$.  
Now we can pass to the quotients by $D_S(M/N)$ to reduce to the case
that $H_k$ is the trivial $k$-group and the identity section of
$H$ is an open and closed immersion. In this case we will prove
that $H$ is the trivial $S$-group.  Since the fppf group scheme $H \rightarrow S$ has
an identity section that is an open and closed immersion, its fibers are \'etale, so 
$H$ is $S$-\'etale.  

We claim that $H \rightarrow S$ is killed by some
integer $n > 0$. Since $S$ is noetherian and $H \rightarrow S$ is quasi-finite, there is an $n > 0$ 
that is a multiple of all fiber-degrees for $H \rightarrow S$.  Thus, $n$ kills
each finite \'etale fiber group $H_s$.   
Since $\Delta_{H/S}:H \rightarrow H \times_S H$ is an open and closed immersion
(as $H \rightarrow S$ is \'etale and separated), the pullback of $\Delta_{H/S}$ under
$([n], 0):H \rightrightarrows H \times_S H$ is an open and closed subscheme of $H$.
But this open and closed subscheme has been seen to contain all fibers, so it coincides with $H$.
Hence, $n$ kills the $S$-group $H$.

We conclude that the {\em closed} subgroup $H \subset H'$ is contained
in the $S$-finite $H'[n]$, so the $S$-\'etale $H$ is $S$-finite and therefore $H \rightarrow S$
has constant fiber rank (by connectedness of $S$). This rank must be 1, due to triviality of $H_k$.
But the identity section $e:S \rightarrow H$ is an open and closed immersion, so it is surjective
and therefore an isomorphism as desired.
\end{proof}


\begin{proposition}\label{fppfet}
Let $H \rightarrow S$ be an $S$-group scheme that
becomes diagonalizable $($of finite type$)$ 
fppf-locally on $S$.  Then $H$ is diagonalizable
\'etale-locally on $S$; i.e., $H$ is quasi-isotrivial. 
More specifically, 
the functor $H \rightsquigarrow \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,\Gm)$
is an anti-equivalence between the category
of $S$-groups of multiplicative type and the category
of locally constant abelian \'etale sheaves on $S$ 
whose geometric fibers are finitely generated abelian groups.

If $S = \Spec A$ for a henselian local $A$ then
$H$ splits over a finite \'etale cover of $S$.
\end{proposition}
\index{multiplicative type!\'etale-local splitting}
\index{quasi-isotriviality}

This result is used very often, generally without comment.
For example, it implies that the splitting hypothesis on $H$ 
in Theorem \ref{torlift} is always satisfied. 

\begin{proof}
The final assertion concerning local henselian $S$ is a consequence
of the rest because any \'etale cover of such an $S$ has a refinement
that is finite \'etale over $S$ (due to the equivalent characterizations
of henselian local schemes in \cite[IV$_4$, 18.5.11(a),(c)]{ega}).

Now using general $S$, the group $H$ is commutative of finite presentation
(by fppf descent).  By standard limit arguments (including the descent of
quasi-compact fppf coverings through limits in the base), we may assume
$S = \Spec A$ is local noetherian,
and even strictly henselian.  Consider the special fiber $H_s$,
a group scheme of multiplicative type over the separably closed $k(s)$.  By Lemma \ref{fieldsplit},
there is an isomorphism $j_s:H_s \simeq D_s(M)$ for a finitely generated abelian group $M$.
Let $\widehat{S} = \Spec \widehat{A}$.  The map $j_s$ lifts to
an open and closed immersion of $\widehat{S}$-groups $j:\mathscr{H} = D_{\widehat{S}}(M) \hookrightarrow \widehat{H} :=
H_{\widehat{S}}$ (see Theorem \ref{torlift}). 
We will prove that $j$ is an isomorphism, and 
then descend it to an isomorphism $D_S(M) \simeq H$.

The fppf-local hypothesis on $H$ is preserved by base change, so there 
is an fppf cover $S' \rightarrow \widehat{S}$ such
that $H_{S'} \simeq D_{S'}(M')$ for a finitely generated abelian group $M'$.
Localize $S'$ at a point over $s$, so $S' \rightarrow \widehat{S}$
is a local flat map (hence fpqc).  The map $j_{S'}:D_{S'}(M) \rightarrow D_{S'}(M')$
must arise from a map $u:M' \rightarrow M$ since $M$ and $M'$
are finitely generated (and $S'$ is connected), and passage to the special fiber implies
that $u$ is an isomorphism (since $D_s(u_s) = j_s$).  Hence, $j_{S'} = D_{S'}(u)$ is an isomorphism,
so $j$ is an isomorphism (by fpqc descent).

Any descent of $j$ to an $S$-homomorphism $D_S(M) \rightarrow H$ is necessarily an isomorphism (by fpqc descent),
so it suffices to prove that the natural map
$$\Hom_{S\mbox{-}{\rm{gp}}}(D_S(M),H) \rightarrow \Hom_{\widehat{S}\mbox{-}{\rm{gp}}}(D_{\widehat{S}}(M),H_{\widehat{S}})$$
is bijective.  Injectivity is clear, and for surjectivity we will use fpqc descent for morphisms
(inspired by the proof of \cite[X, 4.3]{sga3}).

Since $H$ splits over an fppf covering of $S$, 
the map $n:H \rightarrow H$ is finite flat for each $n \ge 1$ because this can be checked over an fppf covering where
$H$ becomes diagonalizable.  Hence, each $H[n]$ is a commutative {\em finite} 
$S$-group of multiplicative type with \'etale Cartier dual.
Consider a map $\widehat{f}:D_{\widehat{S}}(M) \rightarrow H_{\widehat{S}}$.
For each $n \ge 1$, the induced map $\widehat{f}_n$ between $n$-torsion
subgroups uniquely descends to an $S$-homomorphism
$f_n:D_S(M)[n] \rightarrow H[n]$ because we can apply
Cartier duality and use that $E \rightsquigarrow E_{\widehat{S}}$ is an equivalence between 
the categories of finite \'etale schemes over $S$ and $\widehat{S}$
(as $A$ is henselian local).  To
descend $\widehat{f}$ to an $S$-homomorphism, by
fpqc descent it is equivalent to check the equality of the pullback maps
$$p_1^{\ast}(\widehat{f}), p_2^{\ast}(\widehat{f}):
D_{\widehat{S} \times_S \widehat{S}}(M) \rightrightarrows H_{\widehat{S} \times_S \widehat{S}}.$$

A homomorphism from a multiplicative type
group to a separated group scheme is determined
by its restrictions to the $n$-torsion subgroups for all $n \ge 1$
\cite[II, 3.2]{oesterle}.
By applying this over the (typically non-noetherian!) base
$\widehat{S} \times_S \widehat{S}$, the desired equality
$p_1^{\ast}(\widehat{f}) = p_2^{\ast}(\widehat{f})$ is reduced
to the same with $D_S(M)$ replaced by $D_S(M)[n] = D_S(M/nM)$ 
for every $n \ge 1$.  Thus, we may assume that $M$ is
killed by some $n \ge 1$.  Since $\widehat{f}_n$ descends, 
we are done.
\end{proof}

 
\begin{corollary}\label{biglift}
Let $S = \Spec A$ for a complete local noetherian ring $(A,\mathfrak{m})$ with residue field $k$.
Let $G$ be a smooth affine $S$-group and $H$ an $S$-group of multiplicative type.
Any homomorphism $f_0:H_0 \rightarrow G_0$ between special fibers
lifts to an $S$-homomorphism $f:H \rightarrow G$, and if 
$f':H \rightarrow G$ is another such lift then $f$ and $f'$ are conjugate
under $\ker(G(S) \rightarrow G(k))$.

Moreover, if $f_0$ is a closed immersion then so is any such $f$.
\end{corollary}
\index{multiplicative type!deformation theory}
\index{deformation theory!homomorphism}

This corollary is a special case of \cite[IX, 7.3]{sga3}.

\begin{proof} Since $G$ is affine, the natural map
$G(S) \rightarrow \invlim G(S_n)$ is bijective.
The smoothness of $G$ implies
that $G(S_{n+1}) \rightarrow G(S_n)$ is surjective for
all $n \ge 0$.  In view of the bijectivity of (\ref{limmap})
(which is applicable, due to Proposition \ref{fppfet}),
to prove the existence of $f$ and its uniqueness up
to $G(S)$-conjugacy it suffices to show that for each $n \ge 1$ the maps
$$\Hom_{S_{n+1}\mbox{-}{\rm{gp}}}(H_{S_{n+1}}, G_{S_{n+1}}) \rightarrow
\Hom_{S_{n}\mbox{-}{\rm{gp}}}(H_{S_{n}}, G_{S_{n}})$$
are surjective and each (necessarily non-empty)  fiber is a single orbit
under conjugation by $\ker(G(S_{n+1}) \rightarrow G(S_n))$.  

Let $J_n = \mathfrak{m}^{n+1}/\mathfrak{m}^{n+2}$.
Since $G$ is $S$-smooth, $H$ is $S$-affine, 
and $S_n \hookrightarrow S_{n+1}$ is defined by the square-zero ideal $J_n$, 
by Corollary \ref{h2obs} the obstructions to surjectivity lie in 
degree-2 Hochschild cohomology for $H_{S_n}$ with coefficients
in a quasi-coherent $H_{S_n}$-module over $S_n$.
Since $H_{S_n}$ is of multiplicative type, this cohomology vanishes (Remark \ref{multvanish}). 
Applying Corollary \ref{h2obs} once more, 
the obstruction to the transitivity of the conjugation action of $\ker(G(S_{n+1}) \rightarrow G(S_n))$ on the 
non-empty set of homomorphisms $H_{S_{n+1}} \rightarrow G_{S_{n+1}}$ 
lifting a given homomorphism $f_n:H_{S_n} \rightarrow G_{S_n}$ lies 
in a degree-1 Hochschild cohomology group for $H_{S_n}$ with quasi-coherent
coefficients, so again the obstruction vanishes.

Finally, we assume $f_0$ is a closed immersion and aim to show that $f$ is a closed immersion.
By Lemma \ref{ix2.5}, it suffices to show that $\ker f = 1$.  Applying Nakayama's
Lemma to the augmentation ideal of the kernel (viewed as a finitely generated module over
the coordinate ring of $\ker f$), it suffices to check that
$\ker(f_s) = 1$ for all $s \in S$.  The case when $s$ is the closed point is our
hypothesis, so assume $s$ is not the closed point.  By \cite[II, 7.1.7]{ega}, there is a complete discrete valuation
ring $R$ and a map $\Spec(R) \rightarrow S$ carrying the closed point to the closed
point and the generic point to $s$.  Via base change along such maps, we
may assume $A$ is a discrete valuation ring, say with residue field $k$ and fraction field $K$.  
We need to prove that $\ker(f_K) = 1$.

Let $H' \subset H$ be the schematic closure of $\ker(f_K)$ in $H$.  This is an $A$-flat
closed subgroup scheme of $H$, so $H'$ is of multiplicative
type by Corollary \ref{moreflat}.  By $A$-flatness of $H'$, the map $f|_{H'}:H' \rightarrow G$ vanishes since
it vanishes over $K$. Thus, $H' \subset \ker f$.  But $\ker(f_k) = 1$, so $H'_k = 1$.
Hence, $H' = 1$ since $H'$ is of multiplicative type, so 
$\ker(f_K) = 1$ as desired.
\end{proof}

As an illustration, 
if $G = \GL_n$ and $H_0 = \Gm$ then Corollary \ref{biglift} just says 
that a decomposition of $k(s)^n$ into a direct sum of
subspaces can be lifted to a decomposition of $A^n$
into a direct sum of finite free submodules.

 \begin{corollary}\label{mikecor}
 Let $S$ be a normal scheme. Every $S$-group $H \rightarrow S$  of multiplicative
 type is locally isotrivial: there is a Zariski-open cover $\{U_i\}$ of $S$ such that 
 each $H|_{U_i}$ is isotrivial $($i.e., splits over a finite \'etale cover of $U_i$$)$.
 If $S$ is irreducible $($e.g., connected and locally noetherian$)$ then
 $H \rightarrow S$ is isotrivial.
 \index{isotriviality}
 \index{multiplicative type!pi@$\pi_1$-module}
  
 In particular, for irreducible normal $S$ and a geometric point $\overline{s}$ of $S$, 
the functor  $H \rightsquigarrow \underline{\Hom}_{S\mbox{-}{\rm{gp}}}(H,\Gm)_{\overline{s}}$
 is an anti-equivalence from the category of multiplicative type $S$-groups 
to the category of discrete $\pi_1(S,\overline{s})$-modules
 that are finitely generated as abelian groups.
 \end{corollary}
 
 As an example, for connected locally noetherian normal $S$, the category of $S$-tori is anti-equivalent to the category of
discrete  $\pi_1(S,\overline{s})$-representations on $\Z$-lattices (generalizing the classical
 case $S = \Spec k$ for a field $k$).  For instance, all $\Z$-tori are split
 because $\pi_1(\Spec \Z) = 1$ (Minkowski). 
Although a connected normal scheme is irreducible
 in the locally noetherian case, irreducibility can fail in the non-noetherian affine case;
 see Exercise \ref{oferex}.
  
 \begin{proof}
 We call an \'etale sheaf on a scheme {\em isotrivial} if it becomes constant over a finite \'etale cover.
 By Proposition \ref{fppfet}, our task is to prove that if $\calF$ is a 
 finite type locally constant abelian \'etale sheaf on
a normal scheme $S$ then: $\calF$ is isotrivial if $S$ is irreducible, 
 and in general $\calF$ is isotrivial Zariski-locally on $S$. Each $\Spec \calO_{S,s}$ is normal and irreducible,
 and any finite \'etale cover of $\Spec \calO_{S,s}$ spreads out to a finite \'etale cover
 of an open neighborhood of $s$ in $S$. Thus, 
 by the local constancy and finite type hypotheses on $\calF$, the isotriviality Zariski-locally on $S$
is reduced to the isotriviaity for each $\calF|_{\Spec \calO_{S,s}}$.

Now we may and do assume that $S$ is irreducible.  
Every connected finite \'etale $S$-scheme
$S'$ is also irreducible.  Indeed, the generic points of $S'$ lie over the unique generic point of the irreducible $S$, 
so there are only finitely many of them. Hence,
there are only finitely many irreducible components of $S'$. These components are pairwise disjoint (since $S'$ is normal),
so by {\em finiteness} each is open and closed.  By connectedness, $S'$ must therefore be irreducible.

  Since \'etale maps are open for the Zariski topology, 
 and $\calF$ becomes constant over the constituents of
 an \'etale cover of $S$,  the isomorphism type of the geometric stalk $\calF_{\overline{s}}$ is locally constant in $s$
 for the Zariski topology on
 $S$. Thus, by connectedness of $S$, there exists a finitely generated abelian group $M$ so that 
 $\calF_{S'} \simeq M_{S'}$ for some \'etale cover $S' \rightarrow S$.
For each $n \ge 1$, we have $(\calF/n\calF)_{S'} \simeq \underline{M/nM}_{S'}$.
 The constant $S'$-group $(M/nM)_{S'}$ is finite \'etale (especially {\em affine}) over $S'$, so the descent datum on
 it relative to $S' \rightarrow S$ arising from $\calF/n\calF$ is effective.  Hence, $\calF/n\calF$ is represented by
 a finite \'etale $S$-group $G_n \rightarrow S$.  Since a finite \'etale map has open and closed image,
 the connectedness of $S$ and constancy of the fiber degree of a finite \'etale $S$-scheme $E$
 forces any such $E$ to ``disconnect'' at most finitely many times; i.e., $E$ is a disjoint union
 of finitely many {\em connected} finite \'etale $S$-schemes.  By choosing a single connected
Galois finite \'etale cover of $S$ that dominates all connected components of such an $E$,
we can split $E$ using a connected finite \'etale cover $S' \rightarrow S$.  
 
  Fix $n \ge 3$ divisible by the exponent of $M_{\rm{tor}}$
 and apply base change to a connected finite \'etale cover $S' \rightarrow S$ that splits
$G_n$,  so $G_n$ is a  constant $S$-group.  We shall prove that $\calF$ is constant.
  First consider the special case that
$S = \Spec k$ for a field $k$.  The category of \'etale abelian sheaves
on $S$ is the category of discrete ${\rm{Gal}}(k_s/k)$-modules, so  $\calF$ is identified with an action on $M$ by
a finite Galois group ${\rm{Gal}}(k'/k)$.  To prove that $\calF$ is constant we need to prove the
triviality of this action. 
The constancy of $\calF/n\calF$ implies that the ${\rm{Gal}}(k'/k)$-action 
factors through a finite subgroup of $\Gamma = \ker({\rm{Aut}}(M) \rightarrow {\rm{Aut}}(M/nM))$,
so it suffices to prove that $\Gamma$ is torsion-free.

 Recall the classical fact that 
 for any integer $d > 0$, the kernel of $\GL_d(\Z) \rightarrow \GL_d(\Z/n\Z)$ is torsion-free
 for $n \ge 3$.  (This is most efficiently proved by observing that a torsion element
 in the kernel has eigenvalues that are roots of unity, and considering $p$-adic logarithms
 for $p|n$.)  Since $M' := M/M_{\rm{tor}} \simeq \Z^d$ for some $d$,
 and $M'/nM'$ is a quotient of $M/nM$, any finite-order $\gamma \in \Gamma$
 is trivial on $M'$; i.e., $\gamma(m) = m + h(m)$ for some $h:M \rightarrow M_{\rm{tor}}$.
 The hypothesis $\gamma \equiv {\rm{id}} \bmod nM$ implies that $h$ is valued in $nM \bigcap M_{\rm{tor}} \subset
 (nM)_{\rm{tor}}$. Non-canonically $M \simeq M' \oplus M_{\rm{tor}}$, so $nM$ is torsion-free.  This forces
 $h = 0$, so $\gamma = {\rm{id}}$ as desired.
  For general $S$, we conclude that 
 $\calF_s$ over $\Spec k(s)$ is constant for each $s \in S$.

Next consider the case $S = \Spec A$ for an integrally closed {\em local} domain $A$. 
Let $K$ be the fraction field of $A$. Fix an isomorphism $f_{\eta}:M_K \simeq \calF_K$.
Let $S' \rightarrow S$ be an \'etale cover such that $M_{S'} \simeq \calF|_{S'}$. We may assume $S' = \Spec A'$ is affine,
so $A'$ has only finitely many minimal primes (due to the finiteness of $S'_K$). 
The finitely many irreducible components of $S'$ are pairwise disjoint, so each is open. Hence, 
there are only finitely many connected components of $S'$ and they are irreducible. At least one of these has non-empty special fiber,
and so must cover $S$ (as its open image in the {\em local} $S$ contains the closed point and so is full).
Hence, we may assume $A'$ is a domain, so $S'_K$ is the generic point $\eta'$ of $S'$.
By the connectedness of $S'$ and constancy of $\calF_{S'}$, 
$(f_{\eta})_{\eta'}$ uniquely extends to an $S'$-isomorphism $\theta:M_{S'} \simeq \calF_{S'}$.
For $S'' = S' \times_S S'$, the descent datum $p_1^{\ast}(\calF_{S'}) \simeq p_2^{\ast}(\calF_{S'})$
is identified via $\theta$ with an $S''$-group isomorphism $\varphi:M_{S''} \simeq M_{S''}$
whose restriction over $S''_K = \Spec(K' \otimes_K K')$ is the {\em identity} map (due to the descent of
$\theta_{S'_K} = \theta_{\eta'}$ to $f_{\eta}$).
But $S''$ is a disjoint union of finitely many connected components, each of which
meets $S''_K$, so $\varphi$ is the identity.  Thus, 
$f_{\eta}$ extends to an isomorphism $M_S \simeq \calF$.

Finally, we treat the general irreducible case.  Fix an isomorphism 
$f_{\eta}:M_{\eta} \simeq \calF_{\eta}$.  Applying the preceding over the local rings of $S$ implies that 
$\calF$ is constant Zariski-locally on $S$.  Let $\{U_i\}$ be a covering of the irreducible $S$
by non-empty open subschemes so that there are isomorphisms $f_i:M_{U_i} \simeq \calF|_{U_i}$. 
Each $U_i$ is connected, so we may uniquely choose each $f_i$ such that $(f_i)_{\eta} = f_{\eta}$.
The overlaps $U_i \bigcap U_j$ are irreducible with generic point $\eta$, so 
$f_i$ and $f_j$ coincide over $U_i \bigcap U_j$. 
Hence, the $f_i$ glue to an isomorphism $M_S \simeq \calF$.
  \end{proof}
 
 \begin{remark} A scheme $S$ is {\em unibranched} if $\Spec \calO_{S,s}^{\rm{h}}$ is
 \index{unibranched!and isotriviality}
 irreducible for all $s \in S$.   For irreducible $S$,  it is equivalent that $S_{\rm{red}}$ has normalization
 $S' \rightarrow S_{\rm{red}}$ that is radiciel (in which case the integral surjective morphism $S' \rightarrow S$ is
 radiciel).  Pullback along a radiciel integral surjection defines an equivalence between \'etale sites
 (by \cite[VIII, 1.1]{sga4}, which reduces to the finitely presented case treated in \cite[IX, 4.10]{sga1}), so 
 by the anti-equivalence in Proposition \ref{fppfet} we see that 
 Corollary \ref{mikecor} is valid with ``normal'' replaced by ``unibranched''. 
This fact is also noted at the end of \cite[II, 2.1]{oesterle}. (See \cite[IV, \S2]{oesterle} for
an elegant general discussion of the ``topological invariance'' of the theory of multiplicative type groups
without finiteness hypotheses.)
 \end{remark}

In Corollary \ref{moreflat} we saw that any fppf closed subgroup scheme of a multiplicative type group scheme
is of multiplicative type, so in particular every fppf closed subgroup scheme of a torus is of multiplicative type.
Using the anti-equivalence between group schemes of multiplicative type and locally constant \'etale abelian sheaves
with finitely generated stalks (Proposition \ref{fppfet}), we obtain a converse result:

\begin{proposition}\label{torusembed}
Let $H \rightarrow S$ be a group scheme of multiplicative type.
There exists an $S$-torus $T$ such that $H$ is a closed $S$-subgroup of $T$.
\end{proposition}

\begin{proof}
As a preliminary step, we check that $H$ is uniquely an extension of a {\em finite} $S$-group of multiplicative type
by an $S$-torus.  Via the anti-equivalence in Proposition \ref{fppfet}, under which tori correspond
to locally constant finitely generated \'etale abelian sheaves with torsion-free stalks, it is equivalent to show that
any locally constant finitely generated \'etale abelian sheaf $\calF$ on $S_{\et}$ contains a unique locally constant 
finitely generated \'etale abelian
subsheaf $\calF' \subset \calF$ such that the stalks of $\calF'$ are torsion groups and the stalks of
$\calF/\calF'$ are torsion-free.  In view of the local constancy condition, it is clear
that the subsheaf $\calF_{\rm{tor}}$ of locally torsion sections of $\calF$ is the unique possibility for $\calF'$ and that it works.

Now consider the unique short exact sequence of fppf $S$-groups
\begin{equation}\label{tsequence}
0 \rightarrow T \rightarrow H \rightarrow H' \rightarrow 0
\end{equation}
with $T$ an $S$-torus and $H'$ finite over $S$ (necessarily of multiplicative type).
The isomorphism class of stalks $H'_{\overline{s}}$ at geometric points $\overline{s}$ of $S$ is Zariski-locally constant on $S$, so 
by passing to the constituents of a disjoint union decomposition of 
$S$ we may assume the isomorphism class of $H'_{\overline{s}}$ is the same for all 
$\overline{s}$.  Hence, $H'$ is killed by an integer $n > 0$.

By the snake lemma applied to the multiplication-by-$n$ endomorphism of (\ref{tsequence})
and the fppf surjectivity of $n:T \rightarrow T$,  we obtain a short exact sequence of finite multiplicative type $S$-groups
\begin{equation}\label{thh}
0 \rightarrow T[n] \rightarrow H[n] \rightarrow H' \rightarrow 0.
\end{equation}
If $S' \rightarrow S$ is a finite \'etale cover and $j':H_{S'} \hookrightarrow T'$ is an inclusion into an $S'$-torus 
then $H$ is an $S$-subgroup of an $S$-torus:  we compose the
canonical inclusion $H \hookrightarrow {\rm{R}}_{S'/S}(H_{S'})$
with the inclusion ${\rm{R}}_{S'/S}(j')$ of ${\rm{R}}_{S'/S}(H_{S'})$ into the $S$-group
${\rm{R}}_{S'/S}(T')$ that is an $S$-torus (as $S' \rightarrow S$ is finite \'etale).
Thus, it is harmless to make a base change to a finite \'etale cover $S'$ of $S$ (which we promptly rename as $S$)
so that the terms in (\ref{thh}) have constant Cartier dual.  

For any two finite $\Z/n\Z$-modules $M$ and $M'$, any homomorphism 
$f:M_S \rightarrow M'_S$ between the associated constant $S$-groups
defines a disjoint-union decomposition $S = \coprod S_{\phi}$ indexed by
the elements $\phi$ of the finite group $\Hom(M,M')$ via the condition that  $f|_{S_{\phi}}$ arises from $\phi$.
Thus, by passing to the constituents of such a decomposition we may assume that (\ref{thh})
arises from applying Cartier duality to a short exact sequence
$$0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$$
of finite $\Z/n\Z$-modules in which $M''$ is {\em free} over $\Z/n\Z$.
This latter short exact sequence splits, so (\ref{thh}) also splits and hence
$H'$ lifts to an $S$-subgroup of
$H[n]  \subset H$. It follows that $H \simeq T \times H'$, so we
may replace $H$ with $H'$ to reduce to the case when $H$ is $S$-finite.
Passing to a further finite \'etale cover of $S$ reduces us to the case when
$H$ has constant Cartier dual, so $H \simeq D_S(M)$ for a finite abelian group $M$.
By choosing a surjection $F \twoheadrightarrow M$ with $F$ a finitely generated free $\Z$-module
we get an $S$-group inclusion of $H = D_S(M)$ into the $S$-group $D_S(F)$ that is an $S$-torus.
\end{proof}

\subsection{Fibral criteria}

For a finite type group scheme $G_0$ over a field $k$ and any extension field
$K/k$, $G_0$ is of multiplicative type if and only if $(G_0)_K$ is.  Indeed,
this is a problem involving compatible algebraic closures of $k$ and $K$,
so it suffices to prove that $G_0 \simeq D_k(M)$ for a finitely generated abelian group $M$
if and only if $(G_0)_K \simeq D_K(M)$.  But $D_K(M) = (D_k(M))_K$, so the equivalence
is clear (as $G_0$ and $D_k(M)$ are finite type over $k$); see the proof of
Proposition \ref{maxrem} for an illustration of the general ``spreading out and specialization''
technique for descending results from an algebraically closed field to an algebraically closed subfield. 
We conclude that when we consider whether or not the fibers of
an fppf $S$-affine group scheme $G \rightarrow S$ are of multiplicative type, 
it does not matter if we consider the actual fibers $G_s$
or associated geometric fibers $G_{\overline{s}}$ for geometric points
$\overline{s}$ of $S$.

\begin{theorem}\label{flatmult}
Let $S$ be a scheme and $H$ an fppf $S$-affine $S$-group.  Then 
$H$ is of multiplicative type if and only if its geometric fibers are of multiplicative
type and the order of $H_s[n]$ is locally constant in $s$ for each $n \ge 1$.  
The fibral torsion condition can be dropped if $H \rightarrow S$ has connected fibers;
e.g., tori.  
\end{theorem}
\index{multiplicative type!fibral criterion}

Before we prove Theorem \ref{flatmult}, we note that the torsion-order hypothesis 
holds if we assume that the ``type'' of each $H_s$ (i.e., the isomorphism class of the character group
of the geometric fiber at $s$) is locally constant in $s$.  Local constancy of the ``type'' is used in the formulation of the fibral criterion in 
\cite[X, 4.8]{sga3}.  In the absence of a fibral connectedness condition it is necessary to impose
some local constancy hypothesis on the type or at least its torsion levels, even if we assume $H$ is commutative. 

For example,
suppose $S = \Spec R$ for a discrete valuation ring $R$ and $d > 1$ is an integer
not divisible be the residue characteristic. Consider the fppf affine $R$-group $H$ 
obtained by removing the closed non-identity locus from the special fiber of
the constant group $(\Z/d\Z)_R$. This is quasi-finite flat and not finite flat (due to jumping
of fiber-degree), so it is not of multiplicative type but its fibers are of multiplicative type. 

\begin{proof}  
The necessity is clear, and for the proof of sufficiency we know that each fiber $H_s$ is of multiplicative
type since we have already noted that the ``algebraic'' geometric
fibers $H_{\overline{s}}$ (i.e., using an algebraic closure of $k(s)$)
must be of multiplicative type. 

For any $s \in S$, any \'etale cover of $\Spec \calO_{S,s}$ admits an affine refinement
and an affine \'etale cover of $\Spec \calO_{S,s}$ spreads out to an \'etale cover over an open neighborhood of $s$ in $S$.
Hence, since multiplicative type groups split over
an \'etale cover, it suffices to treat the case when $S = \Spec A$ for a local ring $A$.   We may also
assume that $A$ is strictly henselian (since a strict henselization of $A$ is a directed union
of local-\'etale extensions of $A$).  Let $k$ be the separably
closed residue field of $A$.  By Lemma \ref{fieldsplit} applied to the special fiber, 
there is an isomorphism $j_k:D_k(M) \simeq H_k$ for a finitely generated abelian group $M$. 

We claim that $j_k$ lifts to an $A$-homomorphism $j:D_A(M) \rightarrow H$ that is an open
and closed immersion.  Granting this, we prove that $j$ is
an isomorphism as follows.  If $H \rightarrow S$ has connected
fibers then $j$ is fiberwise surjective and hence an isomorphism (as it is an open immersion).
Suppose instead that the order of $H_s[n]$ is locally constant in $s \in S$ for each $n \ge 1$.
By the connectedness of $S$ this order must be constant, so 
$D_s(M)[n]$ and $H_s[n]$ have the same order for all $s \in S$
due to comparison of orders of the special fibers.  Thus, 
the open and closed immersion $j_s:D_s(M) \hookrightarrow H_s$ between multiplicative
type groups at each $s \in S$ is an isomorphism on $n$-torsion for all $n \ge 1$,
so $j_s$ is an isomorphism \cite[II, 3.2]{oesterle}.  The open immersion $j$ is therefore surjective, so it is an isomorphism. 

To construct $j$ lifting $j_k$, we only use the weaker hypothesis
that $H_k$ is of multiplicative type.  
The reason for weakening the hypothesis is that if we express $A$ as a directed union of 
strictly henselian local {\em noetherian} subrings $\{A_i\}$ (with local inclusion maps) then a descent of
$H$ to an fppf affine $A_i$-group $H_i$ for large $i$ inherits the multiplicative type
hypothesis for its special fiber but the same for other fibers seems hard to control in the limit process.
In this way, we may assume that $A$ is also noetherian. 
By Theorem \ref{torlift}, the isomorphism
$j_k:D_k(M) \simeq H_k$ uniquely lifts to an open and closed immersion
$$\widehat{j}:(D_A(M))_{\widehat{A}} = D_{\widehat{A}}(M) \hookrightarrow H_{\widehat{A}}.$$
In particular, $\widehat{j}$ is an isomorphism between infinitesimal special fibers 
(so $H$ has commutative infinitesimal fibers). 
Using fpqc descent for morphisms as at the end of the proof
of Proposition \ref{fppfet}, we shall prove that $\widehat{j}$ descends to an $A$-morphism $j:D_A(M) \rightarrow H$;
such a descent is necessarily a homomorphism and an open and closed immersion (thereby establishing what we need),
by general results on fpqc descent for properties of morphisms.  

To perform descent for $\widehat{j}$, it suffices to show that the two pullback homomorphisms
$$p_1^{\ast}(\widehat{j}), p_2^{\ast}(\widehat{j}):D_A(M)_{\widehat{A} \otimes_A \widehat{A}} \rightrightarrows
H_{\widehat{A} \otimes_A \widehat{A}}$$
over the (typically non-noetherian) ring $\widehat{A} \otimes_A \widehat{A}$ coincide. 
Arguing via \cite[II, 3.2]{oesterle} as near the end of the proof of Proposition \ref{fppfet},
it suffices to check equality on $n$-torsion for each $n \ge 1$, which is exactly descent for the restriction
$\widehat{j}_n$ of $\widehat{j}$ to $D_A(M/nM)_{\widehat{A}}$ for each $n \ge 1$.  

The scheme morphism $f_n:H \rightarrow H$ given by $h \mapsto h^n$ is quasi-finite flat
(as may be checked on fibers, using that each $H_s$ is of multiplicative type) 
and a homomorphism on infinitesimal special fibers, so $H[n] := f_n^{-1}(1)$ is a quasi-finite flat
closed subscheme of $H$ that is a subgroup scheme on infinitesimal special fibers.  
By the structure theorem for quasi-finite separated morphisms
\cite[IV$_4$, 18.5.11(a),(c)]{ega}, any quasi-finite separated
scheme over a henselian local base $Z$ is uniquely the disjoint union of
a {\em $Z$-finite} open and closed subscheme and an open and closed subscheme with
empty special fiber. Let $H[n]_{\rm{fin}}$ denote the resulting ``finite part'' of $H[n]$.  
This closed subscheme of $H$ is a subgroup scheme on infinitesimal special fibers,
so it is a subgroup scheme of $H$ (as the preimage of $X := H[n]_{\rm{fin}}$
under $X \times X \hookrightarrow H \times H \rightarrow H$
contains all infinitesimal special fibers of the $A$-finite $X \times X$ and hence
exhausts $X \times X$).
%
%
%This is a {\em subgroup} scheme of $H[n]$ and hence of $H$ as a special case of:  
%
%\begin{lemma}\label{fingp}
%Let $A$ be a henselian local ring, and $X$ a quasi-finite separated $A$-group scheme.
%The formation of the finite part $X_{\rm{fin}}$ is functorial in $X$ and compatible
%with the formation of direct products in $X$.  In particular, if $G$ is a finite $A$-group scheme
%then $G_{\rm{fin}}$ is an $A$-subgroup scheme of $G$.
%\end{lemma}
%
%\begin{proof} 
%The open and closed subscheme $X_{\rm{fin}}$ of $X$ is uniquely characterized by
%the properties that it is $A$-finite and contains the entire special fiber of $X$.
%Consider another quasi-finite separated $A$-scheme $Y$ and an $A$-morphism
%$f:X \rightarrow Y$.  The preimage $f^{-1}(Y_{\rm{fin}})$ is an open and closed subscheme of $X$
%containing the entire special fiber of $X$, so 
%$(f^{-1}(Y_{\rm{fin}}))_{\rm{fin}}$ makes sense as an open and closed
%subscheme of $X$ that satisfies
%the properties characterizing $X_{\rm{fin}}$.  Thus, 
%$f$ carries $X_{\rm{fin}}$ into $Y_{\rm{fin}}$.
%Since the open and closed subscheme $X_{\rm{fin}} \times Y_{\rm{fin}}$ in $X \times Y$
%is finite and contains the entire special fiber, it coincides with the finite part of $X \times Y$.
%\end{proof}

The finite flat group schemes $H[n]_{\rm{fin}}$ are {\em commutative}
because we may check on the infinitesimal special fibers (where even $H$ becomes commutative).
Their Cartier duals are \'etale, as  this may be checked on the special fiber (due to the openness of
the \'etale locus, or by more direct arguments). 
Moreover, since the formation of the ``finite part'' commutes with local henselian base change, we have 
$(H[n]_{\rm{fin}})_{\widehat{A}} = (H_{\widehat{A}}[n])_{\rm{fin}}$. 
Since $D_A(M/nM)$ is $A$-finite, 
$\widehat{j}_n$ factors through $(H_{\widehat{A}}[n])_{\rm{fin}} = (H[n]_{\rm{fin}})_{\widehat{A}}$.
But $D_A(M/nM)$ and $H[n]_{\rm{fin}}$ are finite flat commutative $A$-groups whose
Cartier duals are \'etale (and hence {\em constant}), so any homomorphism
between their special fibers uniquely lifts.  The same holds over
$\widehat{A}$, so comparison through the common special fibers over $A$ and $\widehat{A}$
implies that $\widehat{j}_n$ descends to an $A$-homomorphism $D_A(M/nM) \rightarrow H[n]_{\rm{fin}} \subset H$
for each $n \ge 1$. 
This completes the proof that $\widehat{j}$ descends, and hence the proof that $H$ is of multiplicative type.
\end{proof}

\begin{corollary}\label{fpqcwin}
Let $S$ be a scheme, $H \rightarrow S$ an $S$-affine fppf group scheme.
\begin{enumerate}
\item If $H$ becomes multiplicative type
 fpqc-locally on $S$ then it is of multiplicative type.
\item Consider a short exact sequence
$$1 \rightarrow H' \rightarrow H \rightarrow H'' \rightarrow 1$$
of fppf $S$-affine $S$-groups with $H'$ and $H''$ of multiplicative type.
If each $H_s$ is either connected or commutative then 
$H$ is of multiplicative type. 
\end{enumerate}
\end{corollary}
\index{multiplicative type!extensions by}

Assertion (1) shows that the notion of ``multiplicative type'' used in \cite{sga3} (with fpqc-local triviality)
coincides with the notion that we are using (with fppf-local triviality) in the finite type case. 
Also, there must be some fibral hypothesis on $H$ in (2), since the finite constant group
over a field of characteristic 0 associated to a non-commutative solvable group of order $p^3$ for a prime $p$
is an extension of one multiplicative type group by another.

\begin{proof}
The validity of (1) is immediate from Theorem \ref{flatmult}: 
the fpqc-locality hypothesis ensures that $H$ is commutative and each $H_s$ is of multiplicative
type, so $H[n]$ is quasi-finite over $S$ for all $n \ge 1$.
For each $n \ge 1$, the locus of points $s \in S$ where $H_s[n]$ has a given order is open 
since Zariski-locally this holds over an fpqc cover (and fpqc maps are topologically quotient maps).
Hence, we have local constancy in $s$ for
the order of $H_s[n]$ for each $n \ge 1$. 

For the proof of (2) we may assume $S = \Spec A$ for a strictly
henselian local noetherian ring $A$ at the cost of only knowing
that the special fiber $H_k$ of $H$ (rather than every fiber $H_s$) is connected or commutative.
Now $H' = D_S(M')$ and $H'' = D_S(M'')$
for finitely generated abelian groups $M'$ and $M''$ (Proposition \ref{fppfet}).
Granting that the special fiber $H_k$ is of multiplicative type, we may conclude as follows.
By (1), we may assume $A$ is complete. 
The $k$-group $H_k$ has the form $D_k(M)$ for an abelian group $M$ that is an extension of $M'$ by $M''$
(dual to the given exact sequence on special fibers).  By Theorem \ref{torlift}(2), the isomorphism
$D_k(M) \simeq H_k$
uniquely lifts to a homomorphism $j:D_S(M) \rightarrow H$
that is moreover an open and closed immersion. 
The composition of $j$ with the canonical homomorphism $D_S(M') \hookrightarrow D_S(M)$
must be the inclusion $D_S(M') =  H' \hookrightarrow H$ because
the two maps agree on special fibers (and we can appeal to
the uniqueness for lifting in Theorem \ref{torlift}). 
In other words, the open and closed subgroup $D_S(M) \subset H$ contains $H' = D_S(M')$.
This forces $D_S(M) = H$ because passing to quotients by $D_S(M')$ gives a chain of inclusions
$$D_S(M)/D_S(M') \subset H/D_S(M') \subset H'' = D_S(M'')$$
whose composition is induced by the identification of $M'$ with $M/M''$.

It remains to prove (2) when $S = \Spec k$ for a field $k$, and we may assume $k$ is algebraically closed.
First we show that if $H$ is connected (so $H''$ is also connected) then $H$ must be commutative.  The automorphism functor
of $H'$ is represented by a constant $k$-group (via duality for diagonalizable finite type $k$-groups), 
so the conjugation action on $H'$ by the connected $H$ must be trivial.
That is, $H'$ is central in $H$.  The commutator morphism for $H$ therefore factors through
a bi-additive pairing $\beta:H'' \times H'' \rightarrow H'$ 
since for points $a_1, a_2, b$ of $H$ (valued in an $S$-scheme)
the centrality of a commutator $a_2 b a_2^{-1}b^{-1} \in H'$ in $H$ implies 
$$(a_1 b a_1^{-1}b^{-1})(a_2 b a_2^{-1} b^{-1}) = a_1(a_2 b a_2^{-1}b^{-1}) b a_1^{-1}b^{-1} =
(a_1 a_2)b(a_1 a_2)^{-1}b^{-1}.$$

The vanishing of $\beta$ is equivalent 
to the commutativity of $H$.  To prove that $\beta$ vanishes it suffices
to prove that the only homomorphism of group functors $f:H'' \rightarrow
\underline{\Hom}_{k\mbox{-}{\rm{gp}}}(H'',H')$ is the trivial one.  This Hom-functor
is represented by a constant $k$-group since $H''$ and $H'$ are diagonalizable finite
type $k$-groups and $H''$ is connected, so $f$ must vanish.

Now working with commutative $H$ in general, note that
the subgroup $T := H_{\rm{red}}^0$ is smooth and connected
without $\Add$ as a subgroup, so it is a torus by the classical theory.  Let $T'' \subset {H''}_{\rm{red}}^0$
be the image of $T$ and let $G' = H' \bigcap T$, so $1 \rightarrow H'/G' \rightarrow H/T \rightarrow H''/T'' 
\rightarrow 1$ is a short exact sequence of finite commutative $k$-group schemes.  Both
finite quotients $H''/T''$ and $H'/G'$ are of multiplicative type (since $H'$ and $H''$ are; use Exercise \ref{closedmult}),
so by Cartier duality for finite commutative $k$-groups we see that $H/T$ has {\em \'etale} Cartier dual.
Double duality then implies that $H/T$ is of multiplicative type.   In other words, $H$ is a commutative
affine finite type extension of $D_k(M)$ by a torus $T$ for a finite abelian group $M$.   It suffices to show
that any such extension splits, which is to say that the abelian group ${\rm{Ext}}^1_k(D_k(M), T)$
of isomorphism classes of such $k$-group extensions vanishes.  Bi-additivity of this Ext-functor
reduces the problem to the case $T = \Gm$ and $M = \Z/n\Z$ for an integer $n \ge 1$.

If $E$ is a commutative affine finite type $k$-group extension of $\mu_n$ by $\Gm$, the fppf surjectivity of $[n]:\Gm \rightarrow \Gm$
implies (via the snake lemma in the abelian category of commutative fppf group sheaves over $k$) 
that the sequence 
\begin{equation}\label{emn}
1 \rightarrow \mu_n \rightarrow E[n] \rightarrow \mu_n \rightarrow 1
\end{equation}
is {\em short exact} for the fppf topology. It suffices to show that this splits.  Applying Cartier duality to (\ref{emn}),
the dual of $E[n]$ must be \'etale, so it is the constant $k$-group
corresponding to an {\em $n$-torsion} finite abelian group that is an extension
of $\Z/n\Z$ by $\Z/n\Z$. Any such exact sequence of $\Z/n\Z$-modules splits, so 
by double duality the sequence (\ref{emn}) also splits.
\end{proof}
 


\newpage

\section{Orthogonal group schemes}\label{sosec}


\subsection{Basic definitions and smoothness results}\label{oversec}

Let $V$ be a vector bundle of constant rank $n \ge 1$ over a 
scheme $S$, and let $L$ be a line bundle on $S$.  A {\em quadratic form} 
\index{quadratic form}
$q:V \rightarrow L$ is a map of sheaves of sets such that 
$q(cv) = c^2 q(v)$ 
and the symmetric map $B_q:V \times V \rightarrow L$ defined by
$$B_q(x,y) = q(x+y) - q(x) - q(y)$$
is $\calO_S$-bilinear.  We call any such $(V,L,q)$ a {\em line bundle-valued
quadratic form}.  Using local trivializations of $V$ and $L$, 
$q(x) = \sum_{i \le j} a_{ij} x_i x_j$. There is an evident notion 
of base change for line bundle-valued quadratic forms.  

Assume $q$ is fiberwise  non-zero over $S$, so the zero scheme 
$(q = 0) \subset \mathbf{P}(V^{\ast})$ (which is well-posed without assuming
$L$ to be trivial) is an $S$-flat hypersurface with fibers of dimension $n-2$
(understood to be empty when $n = 1$).
By Exercise \ref{1.4} (and trivial considerations when $n = 1$), this is $S$-smooth
precisely when for each $s \in S$ one of the following holds: (i) $B_{q_s}$ is non-degenerate and either 
${\rm{char}}(k(s)) \ne 2$ or
${\rm{char}}(k(s)) = 2$ with $n$ even, (ii) the defect $\delta_{q_s} := \dim V_s^{\perp}$ is $1$, $q_s|_{V_s^{\perp}} \ne 0$, and 
${\rm{char}}(k(s)) = 2$ with $n$ odd. 
(By Exercise \ref{1.4}, $\delta_{q_s} \equiv \dim V_s \bmod 2$ when
${\rm{char}}(k(s)) = 2$.) 
 In such cases we say $(V,L,q)$ is {\em non-degenerate}
 (the terminology {\em ordinary} is used
 in \cite[XII, \S1]{sga7}). Case (ii) is the ``defect-1'' case at $s$.
 \index{quadratic form!non-degenerate}
 \index{non-degenerate quadratic form}
 
 A {\em quadratic space} is a pair $(V,q)$ where $q:V \rightarrow \calO_S$ 
 \index{quadratic space}
 is a non-degenerate $\calO_S$-valued quadratic form.  (When we need to consider
 pairs $(V,q)$ with $q$ possibly not non-degenerate, we may call $(V,q)$
 a {\em possibly degenerate} quadratic space.)  The evident notion of isomorphism among the non-degenerate 
 line bundle-valued quadratic forms
 $(V,\calO_S,q)$ corresponds to the classical notion of {\em similarity} (compatibility up to a unit scaling on the form)
 for quadratic spaces.   We will usually work with quadratic spaces
 rather than non-degenerate line bundle-valued quadratic forms below, at least once we begin needing
 to consider Clifford algebras, due to difficulties with Clifford
constructions for line bundle-valued $q$ when there is not a given trivialization of $L$. 
 
 \begin{remark}\label{ndrem}  
Our notion of ``non-degeneracy'' is frequently called ``regularity'' or ``semi-regularity'' (especially
for odd $n$ when 2 is not a unit on $S$); see \cite[IV, 3.1]{knus}.
In the study of quadratic forms $q$ over a domain $A$, such as the ring of
integers in a number field or a discrete valuation ring, the phrase ``non-degenerate''
is often used to mean ``non-degenerate over the fraction field''.  Indeed, non-degeneracy over $A$
in the sense defined above is rather more restrictive, since  for even $n$ it says that 
the discriminant is a global unit and for odd $n$ it says that the ``half-discriminant'' (see Proposition \ref{halfprop})
is a global unit.
Non-degenerate examples over $\Z$
(in our restrictive sense) include the quadratic spaces arising from even unimodular lattices, such as 
the ${\rm{E}}_8$ and Leech lattices. 
  \end{remark}

 
 For a non-degenerate line bundle-valued quadratic form $(V,L,q)$, clearly the functor 
$$S' \rightsquigarrow  \{g \in {\rm{GL}}(V_{S'})\,|\,q_{S'}(gx) = q_{S'}(x) \mbox{ for all } x \in V_{S'}\}$$
on $S$-schemes is represented by a 
finitely presented closed $S$-subgroup ${\rm{O}}(q)$ of ${\rm{GL}}(V)$.  We call it the {\em orthogonal group} of $(V,L,q)$.
\index{ aaaoq@${\rm{O}}(q)$}
\index{ aaasoqp@${\rm{SO}}'(q)$}
\index{orthogonal group}
\index{quadratic form!orthogonal group}
This has bad properties without a non-degeneracy hypothesis, and is $\mu_2$ if $n = 1$. 
Define the {\em naive special orthogonal group} to be 
$${\rm{SO}}'(q) := \ker(\Det:{\rm{O}}(q) \rightarrow \Gm)$$
(so ${\rm{SO}}'(q) = 1$ if $n = 1$); we say ``naive'' because this is the wrong notion
for non-degenerate $(V,L,q)$ when $n$ is even
and 2 is not a unit on $S$.
The {\em special orthogonal group} ${\rm{SO}}(q)$ will be defined shortly
in a characteristic-free way, using 
Clifford algebras when $n$ is even.
(The distinction between
 even and odd $n$ when defining ${\rm{SO}}(q)$ in terms of ${\rm{O}}(q)$ is natural, because
 we will see that ${\rm{O}}(q)/{\rm{SO}}(q)$ is  $\mu_2$ for odd
 $n$ but $(\Z/2\Z)_S$ for even $n$. Also, if $n \ge 3$ then ${\rm{SO}}(q)_{\overline{s}}$ will be 
 connected semisimple of type
 ${\rm{B}}_m$ for $n = 2m+1$ and type ${\rm{D}}_m$ for $n = 2m$.)
     
 \begin{definition}
Let $S = \Spec \Z$.
The {\em standard split} 
quadratic form $q_n$ on $V = \Z^n$ is as follows, depending
\index{quadratic space!split}
\index{split!quadratic space}
 on the parity of $n \ge 1$:
 \begin{equation}\label{stdsp}
q_{2m} = \sum_{i=1}^m x_{2i-1} x_{2i},\,\,\,\,
 q_{2m+1} = x_0^2 + \sum_{i=1}^m x_{2i-1} x_{2i}
 \end{equation}
 (so $q_1 = x_0^2$). 
 We define ${\rm{O}}_n = {\rm{O}}(q_n)$ and ${\rm{SO}}'_n = {\rm{SO}}'(q_n)$.
 \end{definition}
 \index{ aaaon@${\rm{O}}_n$}
 \index{ aaasonp@${\rm{SO}}'_n$}
 \index{ aaaqn@$q_n$}
  
 It is elementary to check that $(\Z^n, q_n)$ is non-degenerate.   We do not define a notion of
 ``split'' for general line bundle-valued non-degenerate $q$ because for odd $n$ this turns out not to
 be an interesting concept except essentially for cases when $L$ is trivial.
 
 \begin{remark}
 In some references (e.g., \cite[11.16]{borelarith}) the quadratic forms
 $$x_1 x_{2m} + \dots + x_m x_{m+1},\,\,\,
 x_0 x_{2m} + \dots + x_{m-1} x_{m+1} + x_m^2$$
 are preferred over (\ref{stdsp}), since for such $q$ the split group
 ${\rm{SO}}(q)$ admits a Borel subgroup contained in the upper triangular Borel subgroup of
 ${\rm{SL}}_n$.
 \end{remark}
   
  There is a convenient characterization of non-degeneracy 
  for $(V,L,q)$ when $L$ and $V$ are globally free, using the unit condition on values of a 
 polynomial associated to $q$ (depending on the parity of $n$), as follows.  
 Suppose there is a chosen isomorphism $L \simeq \calO_S$ and ordered $\calO_S$-basis $\mathbf{e} = \{e_1,\dots,e_n\}$ of $V$
 (as may be arranged by Zariski-localization on $S$). Let $[B_q]_{\mathbf{e}}$ be the matrix
 $(B_q(e_i,e_j))$ that computes $B_q$ relative to $\mathbf{e}$.  The determinant ${\rm{disc}}_{\mathbf{e}}(q) = \Det([B_q]_{\mathbf{e}})$ 
 is the {\em discriminant}
 \index{discriminant}
 \index{quadratic space!discriminant}
 of $q$ relative to $\mathbf{e}$ (and the chosen trivialization of $L$). 
 If $\mathbf{e}'$ is a second ordered $\calO_S$-basis of $V$ then ${\rm{disc}}_{\mathbf{e}'}(q) = u^2
 {\rm{disc}}_{\mathbf{e}}(q)$ for the unit $u$ given by the determinant of the  matrix 
that converts $\mathbf{e}'$-coordinates into 
 $\mathbf{e}$-coordinates.   In more intrinsic terms, ${\rm{disc}}_{\mathbf{e}}(q)$ computes
 the induced linear map $\wedge^n(V) \rightarrow \wedge^n(V^{\ast})$ arising from the linear map $V \rightarrow V^{\ast}$
defined by  $v \mapsto B_q(v,\cdot) = B_q(\cdot,v)$ (when using the bases $e_1 \wedge \dots \wedge e_n$
 and $e_1^{\ast} \wedge \dots \wedge e_n^{\ast}$).
  The condition ${\rm{disc}}_{\mathbf{e}}(q) \in \Gm(S)$ is independent of the choice of $\mathbf{e}$,
 and it expresses exactly the property that $B_q$ is a perfect pairing on $V$.  Hence, if 
 $n$ is even {\em or} if $n$ is odd and $2$ is a unit on $S$ then the condition ${\rm{disc}}_{\mathbf{e}}(q) \in \Gm$
 is equivalent to the non-degeneracy of $q$.  
 
 To handle the case of odd $n$ in a characteristic-free way,
 it is convenient to introduce a modification of the discriminant that
  was independently discovered by Grothendieck and M.\,Kneser.  This involves a ``universal'' construction:
 
 \begin{proposition}\label{halfprop} Let $n \ge 1$ be odd and $Q = \sum_{i \le j} A_{ij} x_i x_j \in \Z[A_{ij}][x_1,\dots,x_n]$ 
 the universal quadratic form
 in $n$ variables.  
 \begin{enumerate}
 \item The polynomial ${\rm{disc}}'(Q) := (1/2){\rm{disc}}(Q) \in (1/2)\Z[A_{ij}]$ lies in $\Z[A_{ij}]$.
 \index{half-discriminant}
  \item Over the ring $\Z[A_{ij}, C_{hk}][1/\Det(C_{hk})]$ consider the universal linear change of
 coordinates $x_h = \sum_k C_{kh} x'_k$ dual to the universal change of basis
 $e_h = \sum_k C_{hk} e'_k$.  The quadratic form $Q'$ in $x'_1,\dots,x'_n$ obtained from $Q(x_1,\dots,x_n)$
 satisfies ${\rm{disc}}'(Q') = {\rm{disc}}'(Q) \Det(C_{hk})^2$.
 \item If $R$ is any commutative ring
 and $q = \sum_{i \le j} a_{ij} x_i x_j$ is a quadratic form in $n$ variables over $R$ then 
 ${\rm{disc}}'(q) := ({\rm{disc}}'(Q))(a_{ij})$ changes by a unit square under linear change of variables
 and it lies in $R^{\times}$ if and only if $q$ is non-degenerate over $\Spec R$.
 \end{enumerate}
  \end{proposition}
 
We call ${\rm{disc}}'(q)$ the {\em half-discriminant} of $q$ (since when $2 \in R^{\times}$, it is
$(1/2) {\rm{disc}}(q)$).  Computation of the half-discriminant when $2 \not\in R^{\times}$ (especially when 2 is
a zero-divisor in $R$) requires
lifting $q$ to a ring in which 2 is not a zero-divisor (e.g., if $R = \F_2$ then
we can work with a lift of $q$ over $\Z$), but it will nonetheless be theoretically useful.
The non-degeneracy of $q_{2m+1}$ in characteristic 2 shows that for odd $n$ the universal half-discriminant in characteristic 2 is
not identically zero. 

  \begin{proof}
  For (1) it suffices to show that over the fraction field $\F_2(A_{ij})$
 of $\Z[A_{ij}]/(2)$ the quadratic form $Q$ has vanishing discriminant. Over any field $k$ of characteristic 2,
 a quadratic form $q$ on a finite-dimensional vector space $V$ satisfies $\delta_q \equiv \dim V \bmod 2$
 (Exercise \ref{1.4}),
 so the defect $\delta_q$ is positive when $\dim V$ is odd.  Since $\delta_q > 0$ precisely when
 ${\rm{disc}}(q) = 0$, part (1) is proved.  Part (2) is obvious, since we can multiply both sides by 2 to
 reduce to the known case of the usual discriminant.
 
 To prove part (3), note that the unit square aspect follows
 from (2).  Thus, we may assume $R = k$ is an algebraically closed field and  it is harmless
 to apply a linear change of variables.  The case ${\rm{char}}(k) \ne 2$
 is trivial, so we can assume ${\rm{char}}(k) = 2$.  First we show that
 ${\rm{disc}}'(q) \in k^{\times}$ if $q$ is non-degenerate. For such $q$, by Exercise \ref{1.4}(ii)
 we may apply a linear change of variables so that $q = x_0^2 + \sum_{i=1}^m x_{2i-1} x_{2i}$ with $n = 2m+1$.
 This arises by scalar extension from the quadratic form $q_n$ over $\Z$ given by the same formula,
 so ${\rm{disc}}'(q)$ is the image of ${\rm{disc}}'(q_n) \in \Z$.  Clearly ${\rm{disc}}(q_n) = 2(-1)^m$, so 
 ${\rm{disc}}'(q_n) = (-1)^m$.  Hence, ${\rm{disc}}'(q) = (-1)^m \in k^{\times}$. 
 
 For the converse, we assume $q$ is degenerate and shall prove that ${\rm{disc}}'(q) = 0$ in $k$.
 In accordance with the definition of ${\rm{disc}}'(q)$, we can compute it by working with
 a lift of $q$ over the ring $W(k)$ of Witt vectors (in which 2 is not a zero divisor).  It is harmless
 to first apply a preliminary linear change of variables over $k$.  
 Consider the defect space $V^{\perp}$ for the alternating $B_q$.  This has odd dimension
 (since $n$ is odd and $B_q$ induces a symplectic form on $V/V^{\perp}$), 
 and $q|_{V^{\perp}}$ is the square of a linear form (since $k$ is algebraically closed
 of characteristic 2).  The degeneracy of $q$ implies that either $\dim V^{\perp} \ge 3$
 or $\dim V^{\perp} = 1$ with $q|_{V^{\perp}} = 0$.  Either way, $q|_{V^{\perp}}$ can be expressed
 in terms of fewer than $\dim V^{\perp}$ variables relative to a suitable basis, so 
$q$  can be expressed in fewer than $n$ variables after a linear change of coordinates. Hence, there is a 
lift $\widetilde{q}$ of $q$ to a quadratic form over $W(k)$ that can be expressed in fewer than
$n$ variables after a linear change of coordinates over $W(k)$, so the discriminant ${\rm{disc}}(\widetilde{q})$
attached to $(W(k)^n, \widetilde{q})$ vanishes (as we may check over the field $W(k)[1/2]$ of characteristic 0). 
Since ${\rm{disc}}'(q)$ is the reduction of ${\rm{disc}}'(\widetilde{q}) = (1/2){\rm{disc}}(\widetilde{q}) = 0$,
we are done.  
  \end{proof}
 
 It is important to note that the discriminant and half-discriminant are attached to a
 line bundle-valued quadratic form $(V,L,q)$ equipped 
 with global bases of $V$ and $L$, and not to a ``bare'' degree-2 homogeneous polynomial.  For example,
 to define the discriminant (or half-discriminant) of the quadratic form $x_0^2 + x_1 x_2$ over $\Z$,
 it is necessary to specify whether this is viewed as a quadratic form on $\Z^3$ or $\Z^n$ for some $n > 3$.
 The convenience of the half-discriminant is demonstrated by its role in the proof of:
 
 \index{orthogonal group!smoothness and dimension}
 \begin{theorem}\label{yusmooth} Let $(V,L,q)$ be a non-degenerate line bundle-valued quadratic
 form with $V$ of rank $n \ge 1$
 over a scheme $S$.  The $S$-group ${\rm{O}}(q)$ is smooth if and only if either $n$ is even 
 or $n$ is odd with $2$ a unit on $S$, and the $S$-group ${\rm{SO}}'(q)$ is smooth
if either $n$ is odd or $n$ is even with $2$ a unit on $S$.
These smooth groups have fibers of dimension $n(n-1)/2$.
 \end{theorem}
  
If $n$ is even then over fields $k$ of characteristic 2 the map $\Det:{\rm{O}}(q) \rightarrow \mu_2$
is identifically 1 due to smoothness of ${\rm{O}}(q)$, so ${\rm{SO}}'(q) = {\rm{O}}(q)$ in such cases.
  
 \begin{proof} We use the following {\em smoothness criterion}:  if $X$ and $Y$ are $S$-schemes locally
 of finite presentation such that $X$ is $S$-flat, then an $S$-morphism $f:X \rightarrow Y$ is smooth
 if and only if $f_s:X_s \rightarrow Y_s$ is smooth for all $s \in S$.  To prove the criterion
 it is only necessary to show that $f$ is flat when each $f_s$ is flat, and this follows
 from the $S$-flatness of $X$ (as part of the fibral flatness criterion \cite[IV$_3$, 11.3.10]{ega}).
 
 {\bf Step 1}. First we treat the case of orthogonal group schemes. 
 The $S$-scheme ${\rm{Quad}}(V,L)$ of $L$-valued quadratic forms on $V$ is represented by
 a smooth $S$-scheme that is an affine space of relative dimension $n(n+1)/2$ Zariski-locally
 over $S$, and the subfunctor of such forms that are non-degenerate is represented
 by an open subscheme $Y \subset {\rm{Quad}}(V,L)$ given Zariski-locally over $S$ by the non-vanishing
 of the discriminant or half-discriminant depending on the parity of $n$.  Thus, $Y \rightarrow S$
 is smooth and surjective (since $q_n$ is non-degenerate over any field). 
 There is an evident right action of $X := \GL(V)$ on $Y$ over $S$ via $(Q,g) \mapsto Q \circ g$,
 and the orbit map $f: X \rightarrow Y$ through $q \in Y(S)$ is surjective
 because over any algebraically closed field $k$ the non-degenerate quadratic forms on $k^n$
 are a single $\GL_n(k)$-orbit (Exercise \ref{1.4}(ii),(iii)).  The fiber of $f$ over a geometric
 point of $Y$ is the orthogonal group scheme for the corresponding quadratic form, so 
 by the smoothness criterion we may assume $S = \Spec k$ for an algebraically closed
 field $k$.  
 
 Now $X$ and $Y$ are $k$-smooth and irreducible with respective dimensions
 $n^2$ and $n(n+1)/2$, so the smoothness of $f$ is equivalent to surjectivity of the 
 maps ${\rm{Tan}}_g(X) \rightarrow {\rm{Tan}}_{q \circ g}(Y)$ for $g \in X(k)$.
 By homogeneity it is equivalent to verify such surjectivity for a single $g$,
 such as $g = 1$, and this in turn is equivalent to ${\rm{Tan}}_1(f)$ having
 kernel of dimension $n^2 - n(n+1)/2 = n(n-1)/2$. But $f(1) = q$ and $f^{-1}(q) = {\rm{O}}(q)$, so 
 the kernel of ${\rm{Tan}}_1(f)$
 is ${\rm{Tan}}_1({\rm{O}}(q))$.  Thus, the case of orthogonal
 group schemes is reduced to showing that for any
 quadratic form $q:V \rightarrow k$, $\dim {\rm{Tan}}_1({\rm{O}}(q)) = 
n(n-1)/2$ precisely when $B_q$ is non-degenerate. 
 
 Computing with the algebra $k[\epsilon]$ of dual numbers,
 the subspace $${\rm{Tan}}_1({\rm{O}}(q)) \subset {\rm{Tan}}_1(\GL(V)) = {\rm{End}}(V)$$
 consists of $T:V \rightarrow V$ such that $q((1 + \epsilon T)(v)) = q(v)$ on $V_{k[\epsilon]}$.
 Since $q(v + \epsilon T(v)) = q(v) + \epsilon B_q(\overline{v}, T(\overline{v}))$ for
 $\overline{v} = v \bmod \epsilon$, the necessary and sufficient condition on $T$
 is that $B_q(x,T(x)) = 0$ for all $x \in V$, which is to say that
the bilinear form $B_q(\cdot, T(\cdot))$ on $V$ is alternating. 
Let $V' = V/V^{\perp}$, so $B_q$ induces a non-degenerate bilinear form
$B'_q$ on $V'$. 
When $B_q(v,T(w))$ is alternating it is skew-symmetric, 
so in such cases $T$ must preserve $V^{\perp}$ (because $B_q(v,T(w)) = 0$
for $v \in V^{\perp}$ and any $w \in V$).   

Thus,
${\rm{Tan}}_1({\rm{O}}(q))$ is the space of $T$ that preserve $V^{\perp}$
and whose induced endomorphism $T'$ of $V'$ makes $B'_q(v',T'(w'))$ alternating.
Since $T'$ determines $T$ up to precisely translation by
$\Hom(V,V^{\perp})$, by the non-degeneracy of $B'_q$ on $V'$ we obtain a short exact sequence
$$0 \rightarrow \Hom(V,V^{\perp}) \rightarrow {\rm{Tan}}_1({\rm{O}}(q)) \rightarrow
{\rm{Alt}}^2(V/V^{\perp}) \rightarrow 0$$
(the second map is $T \mapsto B'_q(\cdot,T'(\cdot))$). 
Letting $\delta = \dim V^{\perp}$ denote the defect, we find that
$$\dim {\rm{Tan}}_1({\rm{O}}(q)) = n \delta + (n-\delta)(n-\delta-1)/2 = n(n-1)/2 +  (\delta^2 + \delta)/2.$$
This coincides with $n(n-1)/2$ precisely when $\delta = 0$.
 
 {\bf Step 2}. 
 Now consider the case of ${\rm{SO}}'(q)$.  By Zariski-localization on $S$, we may assume $L = \calO_S$ and
may choose an ordered basis of $V$. For any quadratic form $Q$ on $V_{S'}$ for an $S$-scheme $S'$, relative to the chosen basis 
 let $D(Q)$ denote ${\rm{disc}}(Q)$ when $n$ is even and ${\rm{disc}}'(Q)$ when $n$ is odd.
 For any $S$-scheme $S'$ and any quadratic form $Q$ on $V_{S'}$ 
 we have $D(Q(gx)) = (\Det g)^2 D(Q(x))$ for $g \in \GL(V_{S'})$.  Indeed, this is obvious when $n$ is even 
 and follows from reduction to the $\Z$-flat universal case
  for the half-discriminant when $n$ is odd.   In particular, 
  orthogonal group schemes for non-degenerate line bundle-valued quadratic forms 
  $(V,L,q)$ lie in $\Det^{-1}(\mu_2)$.  
  Since the determinant map ${\rm{O}}(q) \rightarrow \mu_2$ restricts to the identity on
  the central $\mu_2$ when $n$ is odd, we get ${\rm{O}}(q) = \mu_2 \times {\rm{SO}}'(q)$
  for odd $n$.  This settles the case of odd $n$ when $2$ is a unit, but below we will give
  a characteristic-free argument for odd $n$ that does not ignore characteristic 2. 
  
  Let $X' = {\rm{SL}}(V)$ and let $Y'$ be the scheme of non-degenerate $L$-valued quadratic forms 
  $Q$ on $V$ such that $D(Q) = D(q)$, so $X'$ acts on $Y'$.  Let $f':X' \rightarrow Y'$ be the orbit
  map through $q \in Y'(S)$.   Note that ${\rm{SO}}'(q) = {f'}^{-1}(q)$, so this is $S$-smooth {\em provided} that $f'$ is smooth.
    We claim that $f'$ is surjective. It suffices
  to treat the case $S = \Spec k$ for an algebraically closed
  field $k$ and to work with $k$-points.  By Exercise \ref{1.4}, Proposition \ref{halfprop}(3), 
  and the behavior of $D(Q)$ under a linear change of coordinates on $V$, 
  any non-degenerate $Q$ on $k^n$ with $D(Q) = D(q)$ has the form $Q = q \circ g$ for
  some $g \in \GL_n(k)$ satisfying $(\Det g)^2 = 1$.  Writing $n = 2m$ or $n = 2m+1$,
  $q = q_n \circ \gamma$ with $D(q) = (\Det \gamma)^2(-1)^m$
  and $Q = q_n \circ \gamma'$ with $D(Q) = (\Det \gamma')^2(-1)^m$. Hence, $\Det \gamma' = \pm \Det \gamma$, so 
 it suffices to check that ${\rm{O}}(Q)(k)$ contains an element with determinant $-1$. 
  This is obvious by direct inspection of $q_n$ depending on the parity of $n$
  (see Exercise \ref{1.4}(ii),(iii)). 

   Since $X'$ is $S$-flat (even smooth), $f'$ is smooth if it is so between geometric fibers over $S$.  
   We shall now show that $f'$ is smooth
   when $S = \Spec k$ for an algebraically closed field $k$ provided that either $n$ is odd 
   or $n$ is even with ${\rm{char}}(k) \ne 2$.    
   It is harmless to scale $q$ by some $c \in k^{\times}$, so we may now assume $D(q) = D(q_n)$.  Thus, 
 $q_n$ lies in the level set of $q$, so by surjectivity of the orbit map $f'$ through $q$
 it suffices for the proof of smoothness of $f'$ to assume $q = q_n$. 
 Since $f'$ is either everywhere smooth or nowhere smooth (by homogeneity),
 we shall check smoothness holds  at $g = 1 \in {\rm{SL}}_n(k) = X'(k)$.  
  
  {\bf Step 3}. Assume $n$ is odd.  We
   claim that the level set $Y'$ is smooth (so smoothness of $f'$ is equivalent to surjectivity on tangent spaces).
By transitivity of the ${\rm{SL}}(V)$-action,
this amounts to checking that the equation $D(Q) = D(q_n)$ defining $Y'$ in the smooth scheme $Y$ of
 all non-degenerate quadratic forms on $V$ is non-constant to first order at $q_n$.  That is, relative
 to suitable linear coordinates on $V$, we claim the polynomial $D(q_n + \epsilon Q) - D(q_n)$ 
 in a varying $Q \in {\rm{Quad}}(k^n)$ is nonzero.  For $Q = c x_0^2$ we have $D(q_n + \epsilon Q) - D(q_n) = \pm \epsilon c$
 by base change from the universal case over $\Z[C]$ (the sign depends
 on the parity of $(n-1)/2$). This proves the smoothness of $Y'$. 
 
 Since $X' = {\rm{SL}}(V)$ is a smooth hypersurface in $X = {\rm{GL}}(V)$, the difference
 in tangent space dimensions at $1 \in X'(k)$ and $q_n \in Y'(k)$ is $n^2 - n(n+1)/2 = n(n-1)/2$.  Thus,
 surjectivity of $f'$ on tangent spaces is equivalent to the kernel ${\rm{Tan}}_1({\rm{SO}}'(q))$ of
 ${\rm{Tan}}_1(f')$ having dimension $n(n-1)/2$.  But  
 ${\rm{O}}(q) = \mu_2 \times {\rm{SO}}'(q)$ due to the oddness of $n$, so we want ${\rm{Tan}}_1({\rm{O}}(q))$ to have
 dimension $\dim {\rm{Tan}}_1(\mu_2) + n(n-1)/2$; i.e., $n(n-1)/2$ when
 ${\rm{char}}(k) \ne 2$ and $1 + n(n-1)/2$ in the defect-1 case in characteristic 2.  These
 were both established in the analysis of smoothness for orthogonal group schemes. 
 
 Suppose $n = 2m$ is even and ${\rm{char}}(k) \ne 2$.  
 We can diagonalize $q$ as a sum of squares of all variables, so 
 the smoothness proof for $Y'$ when $n$ is odd carries over. 
The surjectivity on tangent spaces can also be proved exactly
 as for odd $n$, so $f'$ is smooth as desired.
% 
% Consider 
% characteristic 2.  We claim that the polynomial function ${\rm{disc}}(q_n + \epsilon Q) = \det([B_{q_n}] + \epsilon [B_Q])$ of 
% $Q = \sum_{i \le j} a_{ij} x_i x_j \in {\rm{Quad}}(k^n)$
% equals the constant ${\rm{disc}}(q_n) = (-1)^m$ in $k[\epsilon]$.   The $jj$-entry of $[B_{q_n}] + \epsilon [B_Q]$
% is $2a_{jj} \epsilon = 0$ for $1 \le j \le n = 2m$, the 
% $(2i-1,2i)$ and $(2i,2i-1)$-entries are $1 + a_{2i-1,2i}\epsilon$ for $1 \le i \le m$, and all other off-diagonal entries
% are of the form $a_{ij}\epsilon$.  Thus, by inspection, all terms in the sum of products that defines
% the discriminant as a determinant are divisible by $\epsilon^2 = 0$ except for the term $(-1)^m \prod_{i=1}^m (1 + 
% a_{2i-1,2i}\epsilon)^2$.
% Since ${\rm{char}}(k) = 2$, this term is equal to $(-1)^m = {\rm{disc}}(q_n)$. 
% 
% Such nowhere smoothness would force all fibers of $f'$ to be non-smooth (since $X'$ is smooth), 
% which is to say that ${\rm{SO}}'(q)$ is non-smooth in characteristic 2 when $n$ is even. 
%
   \end{proof} 
   
    An alternative proof of the smoothness of orthogonal group schemes
 for even $n$ is to use Lemma \ref{stdq} below to pass to the case of $q_n$, for
 which a direct counting of equations does the job.  This alternative method is worked out in \cite[III, \S5, 2.3]{dg},
 based on an equation-counting smoothness criterion in \cite[II, \S5, 2.7]{dg}. 
 We have chosen to use the preceding argument based on another smoothness criterion because 
 it treats even and odd $n$ on an equal  footing (and adapts to other groups;
 see \cite{ganyu}).
   
 \subsection{Clifford algebras and special orthogonal groups}\label{cliffso}
 
Let $q:V \rightarrow \calO_S$ be a possibly degenerate quadratic space, with $V$ of
rank $n \ge 1$. 
The {\em Clifford algebra} 
${\rm{C}}(V,q)$ is the quotient of the tensor algebra of
$V$ by the relations $x^{\otimes 2} = q(x)$
\index{Clifford algebra}
\index{ aaacvq@${\rm{C}}(V,q)$}
for local sections $x$ of $V$.   This has a natural 
$\Z/2\Z$-grading (as a direct sum of an ``even'' part
and ``odd'' part) via the $\Z$-grading on the tensor algebra.
By considering expansions relative to a local basis
of $V$ we see that ${\rm{C}}(V,q)$ is a finitely generated
$\calO_S$-module.  For $q = 0$, this is the exterior algebra of $V$. 
We are primarly interested in non-degenerate $q$.

There are versions of the Clifford construction
for (possibly degenerate) line bundle-valued $q$, but numerous complications arise;
see \cite[1.8]{auel}, \cite{bk}, and \cite[\S4]{ps} for further discussion. 
An alternative reference on Clifford algebras 
and related quadratic invariants is 
\cite[IV]{knus}, at least for an affine base.  

 \begin{lemma}\label{stdq}
 If $(V,q)$ is a quadratic space of rank $n \ge 1$ over a scheme $S$ 
then it  is isomorphic to $(\mathscr{O}_S^n, q_n)$ fppf-locally on $S$.
 If $n$ is even or $2$ is a unit on $S$ then it suffices to use the \'etale topology
 rather than the fppf topology.
 \end{lemma}
  
  Keep in mind that a ``quadratic space'' is understood to be non-degenerate 
  (in the fiberwise sense) unless we say otherwise.
  
 \begin{proof}
 In \cite[XII, Prop.\,1.2]{sga7} the smoothness
 of $(q=0)$ is used to prove the following variant by induction on $n$:  \'etale-locally on $S$,
 $q$ becomes isomorphic to $q_n$ when $n$ is even
 and to $u x_0^2 + q_{2m}$ for some unit $u$ when $n = 2m+1$ is odd. 
Once the induction is finished, we are done when
$n$ is even and we need to extract a square root of $u$ when $n$ is odd.
This requires working fppf-locally for odd $n$ when
2 is not a unit on the base. 
 \end{proof}

 
   Lemma \ref{stdq} is useful for reducing problems with general (non-degenerate) 
 quadratic spaces to the case of $q_n$ over $\Z$, as we shall now see.
 In what follows, we fix a  quadratic space $(V,q)$ of rank $n \ge 1$ over a scheme $S$. 

\index{Clifford algebra!algebraic properties|(}
\begin{proposition}\label{cliffstr} Assume $n$ is even. The $\calO_S$-algebra 
${\rm{C}}(V,q)$ and its even part ${\rm{C}}_0(V,q)$
are respectively isomorphic, fppf-locally on $S$, to
${\rm{Mat}}_{2^{n/2}}(\calO)$ and a product
of two copies of ${\rm{Mat}}_{2^{(n/2)-1}}(\calO)$, 
with the left ${\rm{C}}_0(V,q)$-module ${\rm{C}}_1(V,q)$ free of rank $1$ 
Zariski-locally on $S$. 
\index{ aaacnvq@${\rm{C}}_0(V,q)$, ${\rm{C}}_1(V,q)$}

In particular, ${\rm{C}}(V,q)$
and the ${\rm{C}}_j(V,q)$ are vector bundles 
and the quasi-coherent centers 
of ${\rm{C}}(V,q)$ and ${\rm{C}}_0(V,q)$ are respectively equal to 
$\mathscr{O}_S$ and a rank-$2$ finite \'etale $\mathscr{O}_S$-algebra $Z_q$.
\index{ aaazq@$Z_q$}
Moreover, the natural map $V \rightarrow {\rm{C}}(V,q)$ is a subbundle inclusion
and ${\rm{C}}_0(V,q)$ is the centralizer of $Z_q$ in ${\rm{C}}(V,q)$. 
\end{proposition}

It follows that ${\rm{C}}(V,q)$ is an Azumaya algebra over $S$
(and ${\rm{C}}_0(V,q)$ is an Azumaya algebra over a degree-2 finite \'etale cover of $S$).
The notion of Azumaya algebra will only arise in Example \ref{so4}.

\begin{proof} By Lemma \ref{stdq} we may assume that $V$ admits a basis $\{e_i\}$ identifying $q$ with $q_n$. 
 In ${\rm{C}}(V,q)$ we have $v'v = -vv' + B_q(v,v')$ for $v, v' \in V$,
so ${\rm{C}}(V,q)$ is spanned by the $2^n$ products $e_J = e_{j_1} \cdots e_{j_h}$
for subsets $J = \{j_1,\dots,j_h\} \subset \{1,\dots,n\}$
(with $j_1 < \dots < j_h$, and
$e_{\emptyset} = 1$ for $h = 0$).  Thus, if we construct a surjection from ${\rm{C}}(V,q)$ onto
${\rm{Mat}}_{2^{n/2}}(\calO)$ then it must be an isomorphism (and the $e_J$ must be a basis
of ${\rm{C}}(V,q)$). 

 Since $n$ is even, there are complementary isotropic free subbundles
$W, W' \subset V$ of rank $n/2$, in perfect duality via $B_q$;
e.g., $W = {\rm{span}}\{e_{2i-1}\}_{1 \le i \le n/2}$
and $W' = {\rm{span}}\{e_{2i}\}_{1 \le i \le n/2}$. 
Let $A := {\rm{C}}(W,q|_W) = {\rm{C}}(W,0) = 
\wedge^{\bullet}(W)$, with even and odd parts $A_{+} = \bigoplus \wedge^{2j}(W)$
and $A_{-} = \bigoplus \wedge^{2j+1}(W)$, so $A$ is a vector bundle of rank $2^{n/2}$.
The endomorphism algebra $\mathscr{E}nd(A)$ has a $\Z/2\Z$-grading:
an endomorphism of $A = A_{+} \bigoplus A_{-}$ is {\em even}
if it carries $A_{\pm}$ into $A_{\pm}$ and {\em odd}
if it carries $A_{\pm}$ into $A_{\mp}$.
We will construct a surjective algebra homomorphism
$\rho:{\rm{C}}(V,q) \twoheadrightarrow \mathscr{E}nd(A) \simeq {\rm{Mat}}_{2^{n/2}}(\calO)$
respecting the $\Z/2\Z$-grading, so  $\rho$ is an isomorphism that 
carries ${\rm{C}}_0(V,q)$ onto  $\mathscr{E}nd(A_{+}) \times \mathscr{E}nd(A_{-})$
and carries ${\rm{C}}_1(V,q)$ onto
$\mathscr{H}om(A_{+}, A_{-}) \oplus \mathscr{H}om(A_{-},A_{+})$, completing
the proof since $A_{+}$ and $A_{-}$
visibly have the {\em same} rank, as one sees via the binomial expansion of $0 = (1-1)^{n/2}$.
(In \cite{ko} there is given a generalization of the isomorphism $\rho$ for certain non-split $(V,q)$.
Also, there is a simpler direct proof of the invertibility of
${\rm{C}}_1(V,q)$ as a left ${\rm{C}}_0(V,q)$-module, namely via right multiplication
by a local section of $V$ on which $q$ is unit-valued; this is the argument used in
\cite[IV, 7.5.2]{knus}.) 

To build $\rho$, by the universal property of the Clifford algebra we just need to define
a linear map $L:W \bigoplus W' = V \rightarrow \mathscr{E}nd(A)$ such that
$L(v) \circ L(v)$ is multiplication by $q(v)$.  
For $w \in W$ and $w' \in W'$, define $L(w) = w \wedge (\cdot)$ and define 
$L(w')$ to be the contraction operator 
$$\delta_{w'}:w_1 \wedge \dots \wedge w_j \mapsto  \sum_{i=1}^j (-1)^{i-1}  B_q(w_i,w')
w_1 \wedge \dots \widehat{w_i} \dots \wedge w_j$$
(the unique anti-derivation of $A$ coinciding with $B_q(\cdot,w')$ on $W \subset A_{-}$).
Induction on $j$ gives $\delta_{w'} \circ \delta_{w'} = 0$,
and clearly $L(w) \circ L(w) = 0$, so $$L(w+w') \circ L(w+w') = L(w) \circ L(w') + L(w') \circ L(w).$$
This is multiplication by $B_q(w,w') = q(w+w')$ because 
\begin{eqnarray*}
(L(w') \circ L(w))(x) = \delta_{w'}(w \wedge x) &=& \delta_{w'}(w)x - w \wedge \delta_{w'}(x) \\
&=& B_q(w,w')x - (L(w) \circ L(w'))(x).
\end{eqnarray*}
The resulting map of algebras $\rho:{\rm{C}}(V,q) \rightarrow \mathscr{E}nd(A)$ respects
the $\Z/2\Z$-gradings since $w \wedge (\cdot)$ and $\delta_{w'}$ are odd endomorphisms.
To prove $\rho$ is surjective it suffices to check on fibers. The
maps $\rho_s$  are isomorphisms by the classical theory over fields (see the proof of
\cite[II.2.1]{chevquad}, or \cite[IV, 2.1.1]{knus}).
\end{proof}

\begin{remark}\label{evencenter}
In the special case $q = q_{2m}$, the computation of the center $Z_q$ of ${\rm{C}}_0(V,q)$
as $\calO_S \times \calO_S$ via 
an explicit description of a fiberwise nontrivial idempotent $z$ is given in
\cite[III, 5.2.4]{dg}.  (Generalizations for
non-split $q$ are given in 
\cite[IV, 2.3.1, 4.8.5]{knus}.)  We now give such a calculation.  Consider the standard basis
$\{e_1,\dots,e_{2m}\}$, so $\{e_{2i-1}, e_{2i}\}$ for $1 \le i \le m$
is a collection of pairwise orthogonal bases of standard hyperbolic planes.
To compute $Z_q$ in terms of products among the $e_i$'s,
we may and do work over $\Z$. 

In ${\rm{C}}_0(V,q)$, for each such pair $\{e,e'\} = \{e_{2i-1}, e_{2i}\}$ we have
\begin{equation}\label{qee}
1 = q(e + e') = (e+e')^2 = e^2 + (ee' + e'e) + {e'}^2 = ee' + e'e,
\end{equation}
so $(ee')^2 = e(e'e)e' = e(1-ee')e' = ee'$.  Hence, the elements $w_i = e_{2i-1}e_{2i}$
pairwise commute (signs cancel in pairs) and $(1 - 2w_i)^2 = 1$, so the product $w := \prod_{i=1}^m (1 - 2w_i)$
satisfies $w^2 = 1$.  If we define $w'_i = e_{2i} e_{2i-1} = 1 - w_i$ then 
$1 - 2w'_i = -(1-2w_i)$, so $w' := \prod_{i=1}^m (1 - 2w'_i)$ is equal to $(-1)^m w$.
Direct calculation shows that $e_j$ commutes with $1 - 2w_i$ when
$j \not\in \{2i-1,2i\}$ whereas $e_j$ anti-commutes with $1 - 2w_i$
if $j \in \{2i-1, 2i\}$, so $e_j$ anti-commutes with $w$.  Hence, all 
$e_j e_{j'}$ with $j < j'$ commute with $w$, so  $w$ is central in ${\rm{C}}_0(V,q)$.

Over $\Z[1/2]$ we define 
$$z = (1/2)(1-w) = \sum_i w_i - 2 \sum_{i < i'} w_i w_{i'} + 4 \sum_{i < i' < i''} w_i w_{i'} w_{i''} + \dots + (-2)^{m-1} \prod_{i=1}^m w_i.$$
By inspection, $z$ lies in ${\rm{C}}_0(V,q)$ over $\Z$ and modulo every prime it is distinct from 0 and 1.  Hence,
$\Z \oplus \Z z$ is a direct summand of ${\rm{C}}_0(V,q)$ that is a central subalgebra, so it is $Z_q$.
Moreover, by computing over $\Z[1/2]$ we see that $z^2 = z$, so $z$ is a fiberwise nontrivial idempotent of $Z_q$
(i.e., $Z_q = \Z \cdot z \times \Z \cdot (1-z)$).  Note that for $z' := (1/2)(1-w')$
we have $z' = z$ when $m$ is even and $z' = 1-z$ when $m$ is odd.
Also, $ze_j + e_j z = e_j$ for all $j$, so $zv = v(1-z)$ for all $v \in V$.
Thus, if $\sigma$ denotes the nontrivial involution of $Z_q$ (given by swapping the idempotents $z$ and $1-z$)
then $\zeta v = v \sigma(\zeta)$ for all $v \in V$ and $\zeta \in Z_q$.  

A straightforward calculation that we omit shows that if $m \ge 2$ then
the expansion of $e_1 z$ and $e_2 z$ in the basis of products of $e_i$'s
contains a term of the form $e_1 e_3 e_4$ and $-e_1 e_3 e_4$ respectively.
Thus, if $m \ge 2$ then an element $a + bz \in Z_q$ (with $a, b \in \calO_S$) satisfies $V\cdot (a+bz) \subset V$
inside ${\rm{C}}(V,q)$ if and only if $b = 0$.   Hence, by fppf descent from
the split case we conclude in general (without any split hypothesis on $(V,q)$) that if $m \ge 2$ 
and $\zeta \in Z_q^{\times}$ then 
$\zeta V \zeta^{-1} = V$ if and only if $\sigma(\zeta)/\zeta \in \calO_S^{\times}$, or equivalently
$\zeta^2 \in \calO_S^{\times}$.  That is, for $m \ge 2$ the 
normalizer of $V$ in the $S$-torus ${\rm{R}}_{Z_q/S}(\Gm)$ 
via its conjugation action on ${\rm{C}}(V,q)$
is the subgroup of points $\zeta$ such that $\zeta^2 \in \Gm$.
In contrast, if $m = 1$ then the entire group ${\rm{R}}_{Z_q/S}(\Gm)$ preserves $V$,
either by computations in the split case or the observation that when $V$ has rank 2
the subbundle inclusions $V \subset {\rm{C}}_1(V,q)$
and $Z_q \subset {\rm{C}}_0(V,q)$ are equalities for rank reasons. 
\end{remark}

The structure of ${\rm{C}}(V,q)$ and ${\rm{C}}_0(V,q)$ for odd $n$ is opposite that for
even $n$:

\begin{proposition}\label{cliffodd}
Assume $n = 2m+1$ is odd.  The even part ${\rm{C}}_0(V,q)$ is isomorphic, fppf-locally on $S$,
to ${\rm{Mat}}_{2^m}(\calO)$. The center $Z_q$ of ${\rm{C}}(V,q)$ is a $\Z/2\Z$-graded finite locally free $\calO_S$-module
\index{ aaazq@$Z_q$}
of rank $2$ whose degree-$0$ part 
$Z^0_q$ is $\calO_S$ and whose degree-$1$ part $Z_q^1$ is an invertible sheaf
that satisfies $Z_q^1 \otimes Z_q^1 \simeq Z_q^0 = \calO_S$ via multiplication
$($so $Z_q$ is locally generated as an $\calO_S$-algebra 
by the square root of a unit, and hence over $S[1/2]$ it is finite \'etale
of rank $2$$)$.

The natural $\Z/2\Z$-graded multiplication map $Z_q \otimes_{\calO_S} {\rm{C}}_0(V,q) \rightarrow {\rm{C}}(V,q)$ is an isomorphism.
In particular, ${\rm{C}}(V,q)$ is a locally free $\calO_S$-module of rank $2^n$ that is isomorphic
to ${\rm{Mat}}_{2^{m}}(Z_q)$ as a $Z_q$-algebra fppf-locally on $S$. Moreover, 
$V \rightarrow {\rm{C}}(V,q)$ is a subbundle inclusion.
\end{proposition}

This result implies that ${\rm{C}}_0(V,q)$ is an Azumaya algebra over $S$ whereas
${\rm{C}}(V,q)$ is an Azumaya algebra over a degree-2 finite fppf cover of $S$.

\begin{proof}
First consider the case $n = 1$ (i.e., $m = 0$), so by Zariski localization
$q = u x^2$ for a unit $u$ on $S = \Spec R$.  Then ${\rm{C}}(V,q)  = R[z]/(z^2 - u) = R \oplus Rz$
with degree-0 part $R$ and degree-1 part $Rz$.  This settles all of the assertions in this case,
so now we may and do assume $n \ge 3$ (i.e., $m \ge 1$). 

The assertions imply that the center
is a local direct summand, so via an fppf base change and 
Lemma \ref{stdq} it suffices to treat the case $q = q_n$ over any $S$. This is the orthogonal
direct sum of $x_0^2$ and $q_{2m}$, and we can use the known results
for rank $2m \ge 2$ from Proposition \ref{cliffstr}.  More specifically, 
if $(V',q')$ and $(V'',q'')$ are (possibly degenerate) quadratic spaces over a scheme $S$ 
then the Clifford algebra of their orthogonal sum $(V' \bigoplus V'', q' \perp q'')$
is naturally isomorphic as a $\Z/2\Z$-graded algebra to the ``super-graded tensor product''
${\rm{C}}(V',q') \otimes' {\rm{C}}(V'',q'')$ which is the ordinary tensor product module
equipped with the algebra structure defined by the requirement $(1 \otimes a'')(a' \otimes 1) =
(-1)^{{\rm{deg}}(a'){\rm{deg}}(a'')}(a' \otimes a'')$ for homogeneous $a'$ and $a''$.
(See \cite[IV, 1.3.1]{knus}.) 
Hence, ${\rm{C}}(q_n) = {\rm{C}}(x_0^2) \otimes' {\rm{C}}(q_{2m})$, so 
${\rm{C}}(q_n)$ is a free $\calO_S$-module of rank $2^{1+2m} = 2^n$.
In particular, the even and odd parts of ${\rm{C}}(V,q) = {\rm{C}}(q_n)$
are locally free of finite rank and $V \rightarrow {\rm{C}}(V,q)$ is a subbundle inclusion.

Multiplication by the standard basis vector $e_0$ swaps
even and odd parts of ${\rm{C}}(V,q)$ (since $e_0^2 = q(e_0) = 1$), so each part 
has rank $2^{n-1} = 2^{2m}$.  For any $w$ in the span $W$ of $\{e_1,\dots,e_{2m}\}$
we have $(e_0 w)(e_0w) = -e_0^2 w^2 = -q_{2m}(w)$.  Thus, 
$w \mapsto e_0 w \in {\rm{C}}_0(q_n)$ extends to a homomorphism
$f:{\rm{C}}(W,-q_{2m}) \rightarrow {\rm{C}}_0(q_n)$.  
The map $f$ is fiberwise injective since geometric fibers of ${\rm{C}}(W,-q_{2m})$
are simple algebras with rank $2^{2m}$ (in fact, ${\rm{Mat}}_{2^m}$),
so $f$ is an isomorphism on fibers over $S$.
The algebra ${\rm{C}}(W,-q_{2m})$ is a locally free module of rank $2^{2m}$
(in fact, fppf-locally on the base it is ${\rm{Mat}}_{2^m}(\calO)$, by
Proposition \ref{cliffstr}), so $f$ is an isomorphism of algebras.   In particular,
${\rm{C}}_0(V,q)$ has the asserted matrix algebra structure fppf-locally on $S$.

Consider the element $z  = e_0 \prod_{i=1}^m (1 - 2e_{2i-1} e_{2i}) \in {\rm{C}}_1(V,q)$.
Computing as in Remark \ref{evencenter}, if $j > 0$ then $e_j$ anti-commutes
with $\prod_{i=1}^m (1 - 2 e_{2i-1}e_{2i})$ by (\ref{qee}), yet such $e_j$ also anti-commutes
with $e_0$, so $e_j$ commutes with $z$.  It is likewise clear that
$z$ commutes with $e_0$ (since the anti-commutation of
$e_0$ with both $e_{2i-1}$ and $e_{2i}$ implies
that $e_0$ commutes with $e_{2i-1} e_{2i}$ for all $i$), 
so $z$ commutes with every $e_j$  ($j \ge 0$).
Likewise, $z^2 = 1$.
Thus, $\{1, z\}$ spans a $\Z/2\Z$-graded central subalgebra $Z$ of ${\rm{C}}(V,q)$ that is a subbundle of rank 2
with odd part spanned by $z$, so $Z^1 \otimes Z^1 \simeq \calO_S$ via multiplication. 
Since $z$ is a unit, the natural $\Z/2\Z$-graded algebra map $Z \otimes {\rm{C}}_0(V,q) \rightarrow {\rm{C}}(V,q)$
defined by multiplication is fiberwise injective and hence (for rank reasons) an isomorphism.
But ${\rm{C}}_0(V,q)$ has trivial center, so $Z$ is the center of ${\rm{C}}(V,q)$.
\end{proof}
\index{Clifford algebra!algebraic properties|)}

Assume $n$ is even.
The natural action of 
${\rm{O}}(q)$ on ${\rm{C}}(V,q)$ preserves the $\Z/2\Z$-grading and hence
induces an action of ${\rm{O}}(q)$ on ${\rm{C}}_0(V,q)$, so we obtain an action of ${\rm{O}}(q)$ 
on the finite \'etale center $Z_q$
of ${\rm{C}}_0(V,q)$.  The automorphism scheme $\underline{\rm{Aut}}_{Z_q/\calO_S}$ is
uniquely isomorphic to $(\Z/2\Z)_S$ since $Z_q$ is finite \'etale 
of rank 2 over $\calO_S$. Thus, for even $n$ we get a homomorphism
\index{ aaadq@${\rm{D}}_q$}
\begin{equation}\label{Dq}
D_q:{\rm{O}}(q) \rightarrow (\Z/2\Z)_S
\end{equation}
compatible with isomorphisms in the quadratic space $(V,q)$, and its
formation commutes with any base change on $S$.
This is the {\em Dickson invariant}, 
\index{Dickson invariant}
and it is discussed in detail in \cite[IV, \S5]{knus}.
(The Dickson invariant for even $n$ 
goes back to Dickson \cite[p.\,206]{dickson} over finite fields,
and Arf \cite{arf} over general fields of characteristic 2.)
By Remark \ref{evencenter}, $D_q(-1) = 0$.

\index{Clifford algebra!line bundle variant|(}
\begin{remark}\label{dicksontriple}
We have defined the $S$-group ${\rm{O}}(q) \subset {\rm{GL}}(V)$
for any non-degenerate line bundle-valued quadratic form 
$(V,L,q)$, but the definition of the Dickson invariant $D_q$ involves the Clifford algebra
${\rm{C}}(V,q)$ that we have only defined and studied for quadratic forms
rather than for general line bundle-valued $q$. 

Below we explain 
how to define the Dickson invariant $D_q: {\rm{O}}(q) \rightarrow (\Z/2\Z)_S$
for any non-degenerate $(V,L,q)$ by working Zariski-locally on $S$
to trivialize $L$.  The main point, going back to the thesis of
W.\,Bichsel, is that we can always define
what should be the ``even Clifford algebra'' ${\rm{C}}_0(V,L,q)$ 
equipped with its ${\rm{O}}(q)$-action 
for line bundle-valued $q$, despite the lack of a definition of ${\rm{C}}(V,L,q)$
for such general $q$ (cf. \cite[1.8]{auel} and references therein).  
We will also define 
the ``odd Clifford module'' ${\rm{C}}_1(V,L,q)$ as an ${\rm{O}}(q)$-equivariant left ${\rm{C}}_0(V,L,q)$-module
that is Zariski-locally (on $S$) free of rank 1 
and naturally contains
$V$ as an ${\rm{O}}(q)$-equivariant subbundle (similarly to the case $L = \calO_S$
as in Proposition \ref{cliffstr}).

The construction of ${\rm{C}}_0(V,L,q)$, explained
below, will be compatible
with base change on $S$ and naturally recover the usual even Clifford algebra 
when $q$ takes values in the trivial line bundle
(and similarly for ${\rm{C}}_1(V,L,q)$ encoding the left
module structure over the even part).  Granting this, the 
center $Z_q$ of ${\rm{C}}_0(V,L,q)$ is therefore a finite \'etale
$\calO_S$-algebra of degree 2 on which ${\rm{O}}(q)$ naturally acts
(cf. \cite[Def.\,1.12]{auel} and \cite[Thm.\,3.7(2)]{bk}).
Such a general $Z_q$ is often called
the {\em discriminant algebra} (or {\em discriminant extension}).
(If $n = 2$ it recovers the classical notion of discriminant, by Proposition \ref{rzq}; we do not use this.) 
The resulting homomorphism
${\rm{O}}(q) \rightarrow \underline{\rm{Aut}}_{Z_q/\calO_S} = (\Z/2\Z)_S$
defines $D_q$ in general.  (In particular, $D_q = D_{uq}$
for units $u$ and quadratic spaces $(V,q)$, via the natural
equality ${\rm{O}}(q) = {\rm{O}}(uq)$ inside ${\rm{GL}}(V)$.) 

To build ${\rm{C}}_0(V,L,q)$, 
we show that for any quadratic space $(V,q)$ and $u \in \Gm(S)$,
there is a natural $\calO_S$-algebra isomorphism $h_{u,q}: {\rm{C}}_0(V,q) \simeq {\rm{C}}_0(V,uq)$ 
that satisfies the following
properties: it is compatible with base change on $S$, 
it respects the actions by the common subgroup
${\rm{O}}(q) = {\rm{O}}(uq)$ inside ${\rm{GL}}(V)$, and it is multiplicative in $u$
in the sense that $h_{u',uq} \circ h_{u,q} = h_{u'u,q}$ for units $u$ and $u'$.
Once such isomorphisms are in hand, we can globally build ${\rm{C}}_0(V,L,q)$ 
equipped with its ${\rm{O}}(q)$-action for  general non-degenerate line bundle-valued quadratic forms via Zariski-gluing.

The existence of such isomorphisms $h_{u,q}$ is provided
over an affine base by \cite[IV, 7.1.1, 7.1.2]{knus} via direct work that avoids base change.
Here is an alternative construction
via descent theory.  Consider an fppf cover $S'$ of $S$ such that the pullback unit $u' = u|_{S'}$ has the form
$u' = a^2$ for a unit $a$ on $S'$.  Let $(V',q') = (V,q)_{S'}$, so there is 
an isomorphism of quadratic spaces $(V',q') \simeq (V',u'q')$ defined by $v' \mapsto a^{-1}v'$.
This induces a graded isomorphism of Clifford algebras $f'_a:{\rm{C}}(V',q') \simeq {\rm{C}}(V',u'q')$
that is clearly equivariant for the actions of the common subgroup ${\rm{O}}(q') = {\rm{O}}(u'q')
\subset {\rm{GL}}(V')$.  Changing the choice of $a$ amounts to multiplying $a$ 
by some $\zeta \in \mu_2(S')$, and $f'_{\zeta a} = f'_a \circ [\zeta]$
where $[\zeta]$ is the graded automorphism of ${\rm{C}}(V',q')$ induced by the automorphism of
$(V',q')$ defined by $v' \mapsto \zeta v'$.  Since $\zeta^2 = 1$, $[\zeta]$ induces
the identity on ${\rm{C}}_0(V',q')$, so the restriction of $f'_a$ to the even parts
is {\em independent} of the choice of $a$.  Consequently, denoting
this isomorphism on the even parts as $f'$, we see that the pullbacks ${\rm{pr}}_1^{\ast}(f')$
and ${\rm{pr}}_2^{\ast}(f')$ over $S'' = S' \times_S S'$ coincide, so $f'$ descends
to an algebra isomorphism $h_{u,q}:{\rm{C}}_0(V,q) \simeq {\rm{C}}_0(V,uq)$.
By fppf descent from $S'$, $h_{u,q}$ is equivariant for the actions of ${\rm{O}}(q) = {\rm{O}}(uq)$,
compatible with base change on $S$, and multiplicative in $u$.

To construct the ${\rm{O}}(q)$-equivariant left ${\rm{C}}_0(V,L,q)$-module
${\rm{C}}_1(V,L,q)$, we need to modify the procedure 
used for the even part because 
the restriction of $f'_a$ to the odd part generally depends on the choice of square root $a$ of $u$.
More precisely, for a point $\zeta$ of $\mu_2$, the effect of $[\zeta]$
on the odd part of ${\rm{C}}(V,q)$ for $\calO_S$-valued $q$
is multiplication by $\zeta$ rather than the identity map.  
Thus, letting $m_{a}$ denote multiplication by the $\calO_{S'}^{\times}$-valued $a$, 
we replace $f'_a$ with $F_a := m_{a} \circ f'_a = f'_a \circ m_{a}$. Clearly
$F_{\zeta a} = \zeta^2 F_a = F_a$ for any $\mu_2$-valued $\zeta$, so $F_a$ depends only on
$a^2 = u$ and hence by using $F_a$ we can carry out the descent and verify 
the desired  properties.  (For $L = \calO_S$ the effect of $F_a$ on the subbundle $V \subset {\rm{C}}_1(V,q)$
is multiplication by $a \cdot 1/a = 1$, so for general $L$ we naturally find $V$ as an ${\rm{O}}(q)$-equivariant subbundle
of ${\rm{C}}_1(V,L,q)$.)
\end{remark}
\index{Clifford algebra!line bundle variant|)}

To analyze properties of the Dickson invariant for even $n$, it is convenient
to introduce a certain closed subgroup of the $S$-group ${\rm{C}}(V,q)^{\times}$ of units
of the Clifford algebra.  The following definition for even $n$
will later be generalized to odd $n$, for which some additional complications arise.

\begin{definition}\label{gpineven}
For even $n$, the {\em Clifford group} ${\rm{GPin}}(q)$ is the closed $S$-subgroup scheme
\begin{equation}\label{gpin}
{\rm{GPin}}(q) := \{ u \in {\rm{C}}(V,q)^{\times}\,|\,uVu^{-1} = V\}
\end{equation}
of units of the Clifford algebra that normalize $V$ inside ${\rm{C}}(V,q)$.
\end{definition}
\index{Clifford group}
\index{ aaagpinq@${\rm{GPin}}(q)$}

\begin{remark}\label{ZqGPin} Assume $n = 2m$ is even.  The explicit computations with $Z_q$ in the split case
in Remark \ref{evencenter} show that if $n \ge 4$ then ${\rm{GPin}}(q) \bigcap {\rm{R}}_{Z_q/S}(\Gm)$
is the subgroup of points of the rank-2 torus ${\rm{R}}_{Z_q/S}(\Gm)$  whose square lies in $\Gm$
whereas if $n = 2$ then ${\rm{R}}_{Z_q/S}(\Gm) \subset {\rm{GPin}}(q)$.
The case $n \ge 4$ with $S = \Spec k$ for a field $k$ is \cite[III, 13.16]{bookinv}.
\end{remark}

Consider $(V,q)$ with even rank $n$. Since the center of 
${\rm{C}}(V,q)$ is $\calO_S$, an elementary calculation (given in \cite[II.3.2]{chevquad} over fields,
but working verbatim over any $S$) shows
that every point of ${\rm{GPin}}(q)$ is ``locally homogeneous'': Zariski-locally on the base, it is in either the even or
odd part of ${\rm{C}}(V,q)$.   (This is false for odd $n$, in view of the
structure of $Z_q$ in Proposition \ref{cliffodd}.) Thus, the $S$-group
${\rm{GPin}}(q)$ agrees with the ``Clifford group'' as defined in
\cite[IV, 6.1]{knus}.  For odd $n$ we will have to force local homogeneity into 
the definition of the Clifford group in order to get the right notion for such $n$; 
see \S\ref{spinsec}.  

There is a natural action by ${\rm{GPin}}(q)$ on $V$ via conjugation,
and the resulting homomorphism ${\rm{GPin}}(q) \rightarrow {\rm{GL}}(V)$
lands in ${\rm{O}}(q)$ because
for $v \in V$ and $u \in {\rm{GPin}}(q)$ we have 
$q(uvu^{-1}) = (uvu^{-1})^2 = uv^2 u^{-1} = q(v)$
in ${\rm{C}}(V,q)$.  However, this action has a drawback:
the intervention of an unpleasant sign for the conjugation action by the
dense open non-vanishing locus $U = \{q \ne 0\} \subset V$.
To be precise, any $u \in U$ satisfies
$u^2 = q(u) \in \mathbf{G}_{\rm{m}}$ in ${\rm{C}}(V,q)$, so $u \in {\rm{C}}(V,q)^{\times}$
and $uv  + vu = B_q(u,v)$ for any $v \in V$. Hence, 
\begin{equation}\label{negref}
uvu^{-1} = -v + (B_q(u,v)/q(u))u \in V,
\end{equation}
 so $U \subset {\rm{GPin}}(q)$ but the conjugation
action on $V$ by any $u \in U$ is the {\em negative} of reflection
through $u$ relative to $q$.  Thus, following Atiyah--Bott--Shapiro \cite[\S3]{abs}
(and \cite[IV, 6.1]{knus}), we define a representation
$$\pi_q:{\rm{GPin}}(q) \rightarrow {\rm{GL}}(V)$$
by 
\begin{equation}\label{piq}
\pi_q(u)(v) = (-1)^{{\rm{deg}}_q(u)} uvu^{-1}
\end{equation}
where ${\rm{deg}}_q:{\rm{GPin}}(q) \rightarrow (\Z/2\Z)_S$
is the restriction of the degree on the subsheaf 
${\rm{C}}(V,q)_{\rm{lh}}$ of locally homogenous sections of
the $\Z/2\Z$-graded Clifford algebra ${\rm{C}}(V,q)$.
\index{ aaadegq@${\rm{deg}}_q$}
\index{ aaacvqlh@${\rm{C}}(V,q)_{\rm{lh}}$}

In other words, $\pi_q$ is the twist of the conjugation action on $V$
against the $S$-homomorphism ${\rm{GPin}}(q) \rightarrow \mu_2$
defined by $u \mapsto (-1)^{{\rm{deg}}_q(u)}$.  Since
quadratic forms are $\mu_2$-invariant, the representation $\pi_q$
lands in ${\rm{O}}(q)$ as well, so $\pi_q$ extends to an
action of the Clifford group ${\rm{GPin}}(q)$ on the entire $\Z/2\Z$-graded algebra
${\rm{C}}(V,q)$.  This action coincides with
the ordinary conjugation action when restricted to the intersection of ${\rm{GPin}}(q)$ with the even subalgebra 
${\rm{C}}_0(V,q)$.  (The representation $u: x \mapsto (-1)^{\deg_q(u)}uxu^{-1}$
on
${\rm{C}}(V,q)$ by the group 
${\rm{C}}(V,q)^{\times}_{\rm{lh}}$ of locally homogeneous units
is denoted as $u \mapsto i_u$ in \cite[IV, 6.1]{knus}. We will not use it.) 

By Remark \ref{evencenter}, $Z_q \subset {\rm{C}}_0(V,q)$. Thus, 
the ${\rm{GPin}}(q)$-action on ${\rm{C}}(V,q)$
via $\pi_q$ on $V$ restricts to ordinary conjugation on $Z_q$.  
But ${\rm{C}}_0(V,q)$
is the centralizer of $Z_q$ in ${\rm{C}}(V,q)$ (Proposition \ref{cliffstr}), so 
$D_q \circ \pi_q:{\rm{GPin}}(q) \rightarrow (\Z/2\Z)_S$ computes the restriction
to ${\rm{GPin}}(q)$ of the $\Z/2\Z$-grading of ${\rm{C}}(V,q)$ (for even $n$). 

 \begin{proposition}\label{disconnok}
Assume $n$ is even.  The map 
$\pi_q:{\rm{GPin}}(q) \rightarrow {\rm{O}}(q)$ is a smooth surjection
with kernel $\mathbf{G}_{\rm{m}}$, and 
$D_q:{\rm{O}}(q) \rightarrow (\Z/2\Z)_S$
 is a smooth surjection.
 In particular, the $S$-affine $S$-group ${\rm{GPin}}(q)$ is smooth.
  \end{proposition}
   
%Our proof of  the surjectivity of $\pi$ will be indirect (using a relative Skolem--Noether theorem).
%Constructive proofs are given in
% \cite[IV, 5.2.2]{knus} and \cite[III, 5/2.4]{dg}
% (and also see Remark \ref{dflat}). 
   
 \begin{proof}
 Since ${\rm{O}}(q)$ is smooth (as $n$ is even; see Theorem \ref{yusmooth}), to prove
  $D_q$ is a smooth surjection it suffices to check
 surjectivity on geometric fibers.  
 We have already noted that $D_q \circ \pi_q$ computes the degree on locally
 homogeneous sections of ${\rm{GPin}}(q)$, so for the assertion concerning
 $D_q$ it suffices to check that on geometric fibers ${\rm{GPin}}(q)$ 
 does not consist entirely of even elements.  But 
 as we saw above, the Zariski-dense open locus
 $\{q \ne 0\} \subset V$ viewed in the odd part of
 ${\rm{C}}(V,q)$ consists of units that lie
 in the Clifford group ${\rm{GPin}}(q)$.  
 
 Now we turn to the assertion that $\pi_q$ is a smooth surjection
 with kernel $\mathbf{G}_{\rm{m}}$.  The kernel of $D_q \circ \pi_q$
 consists of the points of
 ${\rm{GPin}}(q)$ in the even part of ${\rm{C}}(V,q)$,
 and this even part acts on $V$ through ordinary conjugation
 under $\pi_q$, so $\ker \pi_q$ is the intersection of
 ${\rm{GPin}}(q)$ with the part of
 ${\rm{C}}_0(V,q)$ that centralizes $V$ inside ${\rm{C}}(V,q)$.
 But $V$ generates ${\rm{C}}(V,q)$ as an algebra, so
 the centralizer of $V$ inside ${\rm{C}}(V,q)$ is the center
 of ${\rm{C}}(V,q)$.  This center is $\calO_S$ since $n$ is even, so
 $\ker \pi_q = \mathbf{G}_{\rm{m}}$.  By smoothness
 of this kernel, to show $\pi_q$ is a smooth surjection
 it suffices to prove $\pi_q$ is surjective fppf-locally on the base.
 
 By applying a preliminary base change on $S$
 and renaming the new base as $S$, it suffices to show that
 for any $g \in {\rm{O}}(q)(S)$, fppf-locally on $S$ there exists a point $u$ of
 ${\rm{GPin}}(q)$ satisfying $\pi_q(u) = g$.
 Define the sign $\varepsilon = (-1)^{D_q(g)}$ that is Zariski-locally constant on $S$. 
 Consider the automorphism $[ \varepsilon g]$
 of the $\calO_S$-algebra ${\rm{C}}(V,q)$ induced by $ \varepsilon g \in {\rm{O}}(q)(S)$.
 We claim that this automorphism is inner, fppf-locally (even Zariski-locally) on $S$.
Since the quotient $S$-group
 ${\rm{C}}(V,q)^{\times}/\mathbf{G}_{\rm{m}}$
is a subfunctor of the automorphism scheme of the algebra
 ${\rm{C}}(V,q)$ via conjugation, it suffices to show that this quotient
 coincides with the automorphism scheme.  The problem is fppf-local on $S$, 
so we can assume $q = q_{2m}$,
 in which case ${\rm{C}}(V,q)$ is a matrix algebra over $\calO_S$.
But for any $N \ge 1$, the natural map of finite type $\Z$-groups 
${\rm{PGL}}_N \rightarrow \underline{\rm{Aut}}({\rm{Mat}}_N)$
is an isomorphism (``relative Skolem--Noether'') by Exercise \ref{3.2}(i) applied
on artinian points, 
so ${\rm{C}}(V,q)^{\times}/\mathbf{G}_{\rm{m}} = \underline{\rm{Aut}}({\rm{C}}(V,q))$. 

Since $\mathbf{G}_{\rm{m}}$-torsors for the fppf topology are automatically
torsors for the Zariski topology, we may now arrange by Zariski localization on $S$ that 
there exists $u \in {\rm{C}}(V,q)^{\times}$ such that
$[ \varepsilon g](x) = uxu^{-1}$ for all $x \in {\rm{C}}(V,q)$.  Setting $x = v \in V$
gives $[ \varepsilon g](x) =  \varepsilon g(v)$, so $u \in {\rm{GPin}}(q)$
and $\pi_q(u) = (-1)^{\deg_q(u)} \varepsilon g$.  But $\deg_q = D_q \circ \pi_q$
and $D_q(-1) = 0$, so $\deg_q(u) = D_q(g)$.  Hence, $\varepsilon = (-1)^{\deg_q(u)}$, so
$\pi_q(u) = g$. 
\end{proof}

\begin{remark}\label{halfspin}
Our preceding study of the structure of Clifford algebras
provides representations of the $S$-group of units ${\rm{C}}_0(V,q)^{\times}$
that underlies the half-spin and spin representations of spin groups
(see Remark \ref{buildspin}).
\index{spin \& half-spin representations}

Consider $(V,q)$ with even rank $n \ge 2$, and suppose there are
complementary isotropic subbundles $W, W'$ of rank $n/2$.  These
are in perfect duality via $B_q$ and hence Zariski-locally on $S$ can be put into the form
that was considered in the proof of Proposition \ref{cliffstr}.  In that proof we showed for
$A_{+} := \oplus \wedge^{2j}(W)$ and $A_{-} := \oplus \wedge^{2j+1}(W)$
that naturally ${\rm{C}}_0(V,q) \simeq \mathscr{E}nd(A_{+}) \times \mathscr{E}nd(A_{-})$.
Hence, each of $A_{\pm}$ are equipped with a natural representation
of the $S$-group ${\rm{C}}_0(V,q)^{\times}$.   The same argument identifies
${\rm{C}}(V,q)$ with $\mathscr{E}nd(A_{+} \oplus A_{-})$. 

Suppose instead that $V$ has  odd rank $n \ge 1$ and that
$V$ admits a pair of isotropic subbundles $W$ and $W'$ of rank $(n-1)/2$
in perfect duality under $B_q$.  Non-degeneracy on fibers implies
(via Zariski-local considerations over $S$) that
$L := W^{\perp} \cap {W'}^{\perp}$ is a line subbundle of $V$
on whose local generators the values of $q$ are units
and for which $L \oplus W \oplus W' = V$. By Zariski-localizing to acquire a trivialization
$e_0$ of $L$, the proof of Proposition \ref{cliffodd}
shows that $x \mapsto e_0 x$ defines an isomorphism
${\rm{C}}(W \oplus W', \lambda q) \simeq {\rm{C}}_0(V,q)$
for $\lambda := q(e_0) \in \calO(S)^{\times}$.   The preceding treatment of even rank identifies
${\rm{C}}(W \oplus W', q)$ with $\mathscr{E}nd(A)$ where the vector bundle 
$A$ is the exterior algebra of $W$.  Thus, this provides
a representation of the $S$-group ${\rm{C}}_0(V,q)^{\times}$ on $A$.
\end{remark}

 
 We can finally define special orthogonal groups, depending on the parity of $n$
 (and using (\ref{Dq}) and Remark \ref{dicksontriple}).

\begin{definition}\label{soqdef}
Let $(V,L,q)$ be a non-degenerate line bundle-valued quadratic form
with $V$ of rank $n \ge 1$ over a scheme $S$.
The {\em special orthogonal group}
\index{special orthogonal group}
\index{ aaasoq@${\rm{SO}}(q)$}
${\rm{SO}}(q)$ is ${\rm{SO}}'(q) = \ker(\Det|_{{\rm{O}}(q)})$ when $n$ is odd 
and $\ker D_q$ when $n$ is even (with ${\rm{D}}_q$ as in (\ref{Dq})). 
For any $n \ge 1$, ${\rm{SO}}_n := {\rm{SO}}(q_n)$.
\index{ aaason@${\rm{SO}}_n$}
\end{definition}

By definition, ${\rm{SO}}(q)$ is a closed subgroup of ${\rm{O}}(q)$,
and it is also an open subscheme of ${\rm{O}}(q)$ when $n$ is even.
(In contrast, ${\rm{SO}}_{2m+1}$ is not an open subscheme of ${\rm{O}}_{2m+1}$
over $\Z$ because ${\rm{O}}(q) = {\rm{SO}}(q) \times \mu_2$ for odd $n$
via the central $\mu_2 \subset \GL(V)$, and over $\Spec \Z$ the identity section of $\mu_2$
is not an open immersion.) 
 
 The group ${\rm{SO}}'(q)$ is not of any real interest when $n$ is even and 2 is not a unit
 on the base (and we will show
 that it coincides with ${\rm{SO}}(q)$ in all other cases).  The only reason
 we defined ${\rm{SO}}'(q)$ for all $n$
 is because it is the first thing that comes to mind when trying
 to generalize the theory over $\Z[1/2]$ to work over $\Z$.  We will see
 that ${\rm{SO}}'_{2m}$ is not $\Z_{(2)}$-flat.  (Example:
Consider $m = 1$ and $S = \Spec \Z_{(2)}$.  We
 have ${\rm{O}}_2 = \Gm \rtimes (\Z/2\Z)$ using inversion on $\Gm$ for the semi-direct product, 
 and ${\rm{SO}}_2 = \Gm$, whereas
 ${\rm{SO}}'_2$ is the reduced closed subscheme of ${\rm{O}}_2$ obtained
 by removing the open non-identity component in the generic fiber.)

Here are the main properties we shall prove for the ``good'' groups associated to 
quadratic spaces $(V,q)$.
 
 \index{orthogonal group!center}
 \index{special orthogonal group!properties|(}
 \begin{theorem}\label{goodgp}  The group ${\rm{SO}}(q)$ is smooth with connected fibers of
 dimension $n(n-1)/2$.  In particular, ${\rm{SO}}(cq) = {\rm{SO}}(q)$ for
 $c \in \calO(S)^{\times}$. The functorial
 center of ${\rm{SO}}(q)$ is trivial for odd $n$ and is the central $\mu_2 \subset {\rm{O}}(q)$
 for even $n > 2$.  
  For $n \ge 3$, the functorial center of ${\rm{O}}(q)$ is the central $\mu_2$.
%whereas the functorial center of ${\rm{SO}}'(q)$ coincides with that of
%${\rm{SO}}(q)$. 
 \end{theorem}
 
 The smoothness and relative dimension aspects are immediate
 from Theorem \ref{yusmooth} since ${\rm{SO}}(q) = {\rm{SO}}'(q)$ for odd $n$
 and ${\rm{SO}}(q)$ is an open and closed subgroup of ${\rm{O}}(q)$
for even $n$ (as the Dickson invariant $D_q:{\rm{O}}(q) \rightarrow (\Z/2\Z)_S$ is a smooth surjection
 for such $n$, by Proposition \ref{disconnok}).  
The problem is to analyze the
fibral connectedness (so ${\rm{SO}}(cq) = {\rm{SO}}(q)$) and the center.
%We will also show that the functorial center of ${\rm{SO}}'(q)$ coincides with that of ${\rm{SO}}(q)$
%for $n \ge 3$.
 
 \begin{remark}\label{parityrem}
 Via the Dickson invariant 
 $D_q$, for even $n$ we have ${\rm{O}}(q)/{\rm{SO}}(q) = (\Z/2\Z)_S$, so Theorem \ref{goodgp} implies that 
 $\#\pi_0({\rm{O}}(q)_s) = 2$ for all $s \in S$. 
In contrast, for odd $n$ multiplication against the central $\mu_2 \subset
 {\rm{O}}(q)$ defines an isomorphism $\mu_2 \times {\rm{SO}}(q) \simeq {\rm{O}}(q)$
 because $\Det:{\rm{O}}(q) \rightarrow \Gm$
 factors through $\mu_2$ (due to the half-discriminant, as we saw in the proof of
 Theorem \ref{yusmooth}).  Thus, if $n$ is odd then ${\rm{O}}(q)$ is fppf over $S$
with ${\rm{O}}(q)/{\rm{SO}}(q) = \mu_2$ $($so ${\rm{O}}(q)_s$ is connected and non-smooth when 
${\rm{char}}(k(s)) = 2$$)$. 
\end{remark}

We first analyze even rank, and then we analyze odd rank.  In each case, fibers in residue characteristic 2
are treated by a special argument.

\subsection{Connectedness and center}\label{connZ}

Let $(V,L,q)$ be a non-degenerate line bundle-valued quadratic form with $V$ of even rank $n \ge 2$. 
We first seek to understand the connectedness properties of the
fibers of ${\rm{SO}}(q) \rightarrow S$.

%
%One subtlety is that
%since we do not yet know $S$-flatness for ${\rm{O}}(q)$ (which will be obtained from smoothness
%results for ${\rm{SO}}(q)$),
%we do not know if there exists a plentiful supply of sections 
%to ${\rm{O}}(q) \rightarrow S$ after fppf-local base change on
%$S$.  In particular, it is not evident whether smoothness near the identity section
%is sufficient to deduce smoothness of the entire group scheme (as that intuition over a field
%is based on the ability to do translations after a ground field extension, which is always faithfully flat).
%
%As a concrete example, consider the reduced
%closed complement $H$ of the open non-identity point
%in the generic fiber of the constant group $(\Z/2\Z)_R$ over
%a discrete valuation ring $R$.  The $R$-group $H$ 
%is a disjoint union of the identity section and an additional
%rational point in the special fiber, so $H$ is affine and also $R$-smooth near the identity section with
%constant fibers but it is not $R$-flat (and so not $R$-smooth). 
%To circumvent this problem, we will use a ``global'' smoothness criterion in terms of equation-counting:
%
%\begin{lemma}\label{smcrit} Let $R$ be a ring and $G$ a smooth affine $R$-group.
%Let $G' \hookrightarrow G$ be a closed subgroup scheme whose
%defining ideal admits $c$ global generators $f_1,\dots,f_c$ and whose
%fibers $G'_s$ satisfy $\dim {\rm{Tan}}_{e(s)}(G'_s) = \dim {\rm{Tan}}_{e(s)}(G_s) - c$.
%Then $G'$ is $R$-smooth.
%\end{lemma}
%
%This result is \cite[II, \S5, 2.7]{dg} (stated for $G = \GL(V)$, with a proof that works for any $G$).
%
%\begin{proof}
%Each fiber $G'_s$ has codimension at most $c$ in $G_s$,
%so 
%$$\dim G'_s \ge \dim G_s - c = \dim {\rm{Tan}}_{e(s)}(G_s) - c = \dim {\rm{Tan}}_{e(s)}(G'_s).$$
%Thus, the $k(s)$-group $G'_s$ is smooth for all $s \in S$ (due to the homogeneity of
%the geometric fiber $G'_{\overline{s}}$).
%Since an open subset of $G'$ that contains all closed points of all fibers must be the entire space,
%by openness of the smooth locus it suffices to prove that $G'$ is smooth at each point
%$g'$ that is closed in its fiber $G'_s$.  
%
%Choose $g'$, so the established $k(s)$-smoothness of $G'_s$ 
%implies that the local ring $\O_{G'_s,g'} = \O_{G_s,g'}/((f_1)_s,\dots,(f_c)_s)$
%is regular with dimension $\dim G'_s = \dim G_s - c = \dim \O_{G_s,g'} - c$. 
%Hence, in the regular local ring $\O_{G_s,g'}$ the sequence $\{(f_j)_s\}$ in the maximal ideal is 
%part of a regular system of parameters.  Since $G_s$ and $G'_s$ are $k(s)$-smooth, by
%computing at a point over $g'$ on geometric fibers over $s$ we see
%that the elements 
%$${\rm{d}}(f_j)(g') \in \Omega^1_{G'/S,g'}/\mathfrak{m}_{g'} = 
%\Omega^1_{G'_s/k(s),g'} \otimes_{\O_{G'_s,g'}} k(g')$$
%are $k(g')$-linearly independent.   It then follows from the Jacobian criterion
%for smoothness of a closed subscheme of a smooth scheme \cite[2.2/7]{neron} that 
%$G'$ is $R$-smooth.
%\end{proof}
%
%\begin{lemma}\label{lieo} Inside $\mathscr{E}nd(V) = {\rm{Tan}}_e({\rm{GL}}(V))$, we have
%$${\rm{Tan}}_e({\rm{O}}(q)) = 
%\{T \in \mathscr{E}nd(V)\,|\,B_q(v,Tw) \mbox{\rm{ is alternating}}\}.$$
%\end{lemma}
%
%This lemma makes no hypothesis on the parity of $n$.
%
%\begin{proof}
%We may assume $S = \Spec k$ for a ring $k$. 
%In terms of dual numbers,
%${\rm{Tan}}_e({\rm{O}}(q))$ 
%is the space of linear endomorphisms $T$ of $V$ such that $1 + \epsilon T$
%preserves $q_{k[\epsilon]}$ on $V_{k[\epsilon]}$.
%For any $x \in V_{k[\epsilon]}$ 
%with reduction $x_0 \in V$, clearly $\epsilon T(x) = \epsilon T(x_0)$, so 
%$$q_{k[\epsilon]}(x + \epsilon T(x_0)) = 
%q_{k[\epsilon]}(x) + \epsilon B_q(x_0, T(x_0))$$
%since $\epsilon^2 = 0$.  Thus, the necessary and sufficient condition on
%$T$ is that the bilinear form $B_q(v,Tw)$ vanishes on the diagonal, which is
%to say that it is alternating.  
%\end{proof}
%
%\begin{proposition}\label{evensmooth} Assume $n$ is even. The $S$-group ${\rm{O}}(q)$ is smooth
%of relative dimension $n(n-1)/2$.
%In particular, the open and closed subgroup ${\rm{SO}}(q)$ is smooth and 
%the surjective Dickson invariant $D_q:{\rm{O}}(q) \rightarrow (\Z/2\Z)_S$
%is smooth, identifying $(\Z/2\Z)_S$ with ${\rm{O}}(q)/{\rm{SO}}(q)$. 
%\end{proposition}
%
%\begin{proof}
%Once smoothness of ${\rm{O}}(q)$ is proved, the Dickson invariant 
%must be smooth since it is visibly smooth over fibers over $S$.
%The other assertions are then clear as well.
%The smoothness of ${\rm{O}}(q)$ is fppf-local on the base,
%so by Lemma \ref{stdq} it suffices to treat $q = q_n$ over $\Z$ (or over any affine base).
%By a permutation of the variables, we may equivalently assume
%$q = \sum_{i=1}^m x_i x_{i+m}$ where $2m = n$.
%To prove the smoothness for this $q$, we will use the criterion in Lemma \ref{smcrit}.
%
%We express $n \times n$ matrices in the block form
%$(\begin{smallmatrix} A & B \\ C & D \end{smallmatrix})$
%where $A, B, C, D$ are $m \times m$ matrices, and we likewise
%express $(x_1,\dots,x_n)$ as a pair $(x,y)$ where $x$ and $y$ are ordered $m$-tuples.
%Thus, our quadratic form is $q(x,y) = \vec{y}^{\top}\vec{x}$
%where $\vec{y}$ and $\vec{x}$ are ``column vectors'' (i.e., $m \times 1$ matrices).
%For any $M = (\begin{smallmatrix} A & B \\ C & D \end{smallmatrix})$
%we have $M(x,y) = (Ax + By, Cx + Dy)$, so 
%$$q(M(x,y)) = \vec{x}^{\top}C^{\top}A\vec{x} + \vec{y}^{\top} D^{\top}B \vec{y} +
%\vec{y}^{\top}(D^{\top}A + B^{\top}C) \vec{x}.$$
%Hence,  $M \in {\rm{O}}(q)$ if and only if 
%$D^{\top}A + B^{\top}C = 1_m$ and the matrices
%$C^{\top}A$ and $D^{\top}B$ are alternating (in the sense
%that the associated bilinear forms in $m$ variables that they define
%are alternating; i.e., vanish on pairs $(x,x)$).  
%
%The alternating condition
%on an $m \times m$ matrix amounts to $m + m(m-1)/2 = m(m+1)/2$ equations in the matrix entries,
%so the alternating conditions on $C^{\top}A$ and $D^{\top}B$
%amount to $m(m+1)$ equations in the matrix entries of $A, B, C, D$.  The condition
%$D^{\top}A + B^{\top}C = 1_m$ amounts to $m^2$ equations in the matrix entries of $A, B, C, D$, 
%so the closed subscheme ${\rm{O}}(q) \subset \GL_n$
%is defined by an ideal generated by
%$m^2 + m(m+1) = m(2m+1)$ elements.  
%Thus, by Lemma \ref{smcrit}, to prove ${\rm{O}}(q)$ is smooth we just need to check that
%over an algebraically closed field $k$, ${\rm{Tan}}_e({\rm{O}}(q))$ has
%codimension $m(2m+1)$ in $\mathfrak{gl}_{2m}(k)$.
%By Lemma \ref{lieo}, an
%element $M \in \mathfrak{gl}_{2m}(k)$ lies in
%${\rm{Tan}}_e({\rm{O}}(q))$ if and only if the matrix
%$(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}) M = 
%(\begin{smallmatrix} C & D \\ A & B \end{smallmatrix})$ is alternating.
%This says that $B$ and $C$ are alternating and $D = -A^{\top}$,
%which amounts to $m(2m+1)$ (linear) equations on
%$\mathfrak{gl}_{2m}(k)$.  We 
%conclude that ${\rm{O}}(q)$ is smooth with relative dimension 
%$(2m)^2 - m(2m+1) = 2m^2 - m = n(n-1)/2$.
%\end{proof}
%

\begin{proposition}\label{conneven} 
If $n$ is even then ${\rm{SO}}(q) \rightarrow S$ has connected fibers.
\end{proposition}

\begin{proof}
We proceed by induction on the even $n$, and we
can assume that $S = \Spec k$ for an algebraically closed field $k$.
Without loss of generality, $q = q_n$.  In view of the surjectivity
of the Dickson invariant, it is equivalent to show that ${\rm{O}}(q)$
has exactly 2 (equivalently, at most 2) connected components.  Since $q_2 = xy$, clearly
${\rm{O}}_2 = \Gm \coprod \Gm \iota$ for
$\iota = (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix})$.
Now assume $n \ge 4$ and that the result is known for $n - 2$.

Since $q$ is not a square (as $n > 1$),
it is straightforward to check that the smooth affine hypersurface $H = \{q = 1\}$ is irreducible.
The points in $H(k)$ correspond to isometric embeddings 
$(k, x^2) \hookrightarrow (V,q)$. 
% (Note that $(k, x^2)$ is degenerate when ${\rm{char}}(k) = 2$.) 
By Witt's extension theorem
\cite[I.4.1]{chevquad}, if $Q:W \rightarrow K$ is a quadratic form on a finite-dimensional
vector space over a field $K$ and $B_Q$ is non-degenerate
(so $\dim W$ is even when ${\rm{char}}(K) = 2$) then 
${\rm{O}}(Q)(K)$ acts transitively on the set of isometric embeddings of a fixed
(possibly degenerate) quadratic space into $(W,Q)$.
Hence, ${\rm{O}}(q)(k)$ acts transitively on $H(k)$, so 
the orbit map ${\rm{O}}(q) \rightarrow H$ through
$e_1 + e_2 \in H(k)$ is surjective with stabilizer
$G' := {\rm{Stab}}_{e_1 + e_2}({\rm{O}}(q))$
that preserves the orthogonal complement
$V' := (e_1+e_2)^{\perp} = k(e_1 - e_2) + \sum_{i \ge 3} k e_i$.  

By Theorem \ref{yusmooth}, $\dim {\rm{O}}(q) = \dim {\rm{SO}}'(q) = n(n-1)/2$, so 
$$\dim G' = \dim {\rm{O}}(q) - \dim H = n(n-1)/2 - (n-1) =  \dim {\rm{O}}_{n-1} = \dim {\rm{Sp}}_{n-2},$$
(for the final equality, which we will use when ${\rm{char}}(k)=2$, note that $n$ is even).
Since $H$ is connected, the identity component ${\rm{O}}(q)^0$ also acts transitively on $H$. Hence, 
$\#\pi_0({\rm{O}}(q)) \le \#\pi_0(G')$, so 
it suffices to show that $G'$ has at most 2 connected components.

The kernel of the action map $G' \rightarrow \GL(V')$ consists of
$g' \in \GL(V)$ fixing $e_1 + e_2$ and $\{e_1 - e_2, e_3, \dots,e_n\}$  pointwise 
and preserving $q$.  Thus, if ${\rm{char}}(k) \ne 2$ then $g'=1$, 
and if ${\rm{char}}(k)=2$ then preserving $(\sum_{i \ge 3} k e_i)^{\perp} = ke_1 + ke_2$
and stabilizing $e_1 + e_2$ and $q$ implies $\ker(G' \rightarrow {\rm{GL}}(V')) = \Z/2\Z$, 
generated by the automorphism of $V$ swapping $e_1$ and $e_2$
and fixing $e_3, \dots, e_n$.  For this reason, we shall argue separately depending on whether
or not ${\rm{char}}(k) = 2$.

Suppose ${\rm{char}}(k) \ne 2$, so $\{e_1 + e_2, e_1 - e_2\}$ is an orthogonal basis of $ke_1  + ke_2 = W^{\perp}$
where $W := \sum_{i \ge 3} ke_i$, with $q(e_1 \pm e_2) = \pm 1$.  The restriction $q' := q|_{V'}$ is given by the formula 
$q(c(e_1 - e_2) + w) = -c^2 + q(w)$ for $w \in W$, and the inclusion $G' \hookrightarrow {\rm{GL}}(V')$ 
has image exactly ${\rm{O}}(q')$ since relative to the basis $\{e_1 + e_2, e_1 - e_2, e_3, \dots, e_n\}$ of $V$ 
we identify $(V,q)$ with the orthogonal direct sum of $(k(e_1 + e_2), x^2)$ and $(V',q')$.  

Since $B_{q'}$
is non-degenerate (as ${\rm{char}}(k) \ne 2$), 
Witt's extension theorem is applicable to the hypersurface $H' = \{q' = -1\}$ in $V'$ that contains
$e_1 - e_2$
and is irreducible (as $n-1 > 1$), so ${G'}^0$ acts transitively on $H'$.  Thus, $\#\pi_0(G')$ is at most
the number of connected components of the stabilizer of $e_1 - e_2$ in $G' = {\rm{O}}(q')$.
But since ${\rm{char}}(k) \ne 2$, this stabilizer is the orthogonal group of the $q'$-orthogonal
space in $V'$ to $e_1 - e_2$. This orthogonal group is identified with ${\rm{O}}(q'|_W) = {\rm{O}}(q|_W)$.
Since $q|_W = q_{n-2}$, we conclude by induction
when ${\rm{char}}(k) \ne 2$.

Now assume ${\rm{char}}(k) = 2$, so $e_1 - e_2$ spans the defect
line $\ell$ of the non-degenerate quadratic space $(V',q')$ of dimension $n-1$.  The action of $G'$ on $V'$ preserves $q'$, so its
induced action on $V'$ defines an action on $V'/\ell$ preserving the induced
symmetric bilinear form $\overline{B}_{q'}$ that is alternating
since ${\rm{char}}(k)=2$.   Since the line $\ell$ is the defect space ${V'}^{\perp}$,
$\overline{B}_{q'}$ is non-degenerate and hence symplectic.
The induced homomorphism 
$$h:{\rm{O}}(q') \rightarrow {\rm{Sp}}(V'/\ell,\overline{B}_{q'}) \simeq {\rm{Sp}}_{n-2}$$
 to a symplectic group has kernel
that is seen by calculation to be $\mu_2 \ltimes \alpha_2^{n-1}$, where $\mu_2$ acts on
the Frobenius kernel $\alpha_2^{n-1} \subset \mathbf{G}_{\rm{a}}^{n-1}$ by the usual diagonal scaling action.
But $\dim {\rm{O}}_{n-1} = \dim {\rm{Sp}}_{n-2}$ and symplectic groups over fields are {\em connected} (proved 
by a fibration argument using lower-dimensional symplectic spaces, or see \cite[23.3]{borelag} for another
proof), so $h$ is surjective for dimension reasons.

We saw above that the restriction map
$G' \rightarrow {\rm{GL}}(V')$ lands in ${\rm{O}}(q')$ and has kernel $\Z/2\Z$,
so since the restriction map ${\rm{O}}(q') \rightarrow {\rm{Sp}}(\overline{B}_{q'})$
in characteristic 2 has infinitesimal kernel, the
composite homomorphism $f:G' \rightarrow {\rm{Sp}}(\overline{B}_{q'}) = {\rm{Sp}}_{n-2}$ has finite kernel
with 2 geometric points.  But $\dim G' = \dim {\rm{Sp}}_{n-2}$
and symplectic groups are connected, so $f$ must be surjective for dimension reasons
and $\#\pi_0(G') \le 2$. 
\end{proof}
%
%
%Such automorphisms must preserve
%the orthogonal complement $k e_1 + k e_2$ of the span of $\{e_3,\dots,e_n\}$,
%so it is an elementary calculation that the scheme-theoretic
%kernel of $G' \rightarrow \GL(V')$ is trivial.  
%Let $W = (ke_1 + k e_2)^{\perp} = \sum_{i \ge 3} k e_i$. For any $w \in W$
%we have $q(ce_2 + w) = c^2 q(e_2) + B_q(ce_2,w) + q(w) = q(w)$, so
%relative to the ordered basis $\{e_2,\dots,e_n\}$ of $V'$
%the map $G' \rightarrow \GL(V')$ is an isomorphism onto the subgroup of $(n-1) \times (n-1)$ matrices
%whose left column is $(1,0,\dots,0)$, top row has arbitrary entries beyond the initial 1,
%and lower right $(n-2) \times (n-2)$ block is in ${\rm{O}}(q|_W)$.
%In other words, $G'$ is an extension of ${\rm{O}}(q|_W)$ by $\Ga^{n-2}$.
%By induction we know that ${\rm{O}}(q|_W)$ has exactly 2 connected components,
%so the same holds for $G'$.  
%\end{proof}

\begin{corollary}\label{deteven}
Assume $n$ is even.   Let $f_S:(\Z/2\Z)_S \rightarrow \mu_2$ be the unique
$S$-homomorphism satisfying $f(1) = -1$.  The determinant map $\Det: {\rm{O}}(q) \rightarrow \mu_2$
coincides with $f_S \circ D_q$.  In particular, $\Det$
kills ${\rm{SO}}(q)$.

The inclusion ${\rm{SO}}(q) \subset \ker(\Det) = {\rm{SO}}'(q)$
is an equality over $S[1/2]$,
and ${\rm{SO}}'(q) \hookrightarrow {\rm{O}}(q)$
is an equality on fibers at points in characteristic $2$.
\end{corollary}

\begin{proof}
By Lemma \ref{stdq}, we may pass to the case $q = q_n$ over $\Z$.
The equality $\Det = f_{\Z} \circ D_{q_n}$ can then be checked over $\Q$, where it is 
immediate from the connectedness of ${\rm{SO}}_n$ over $\Q$ and the nontriviality of $\Det$
on ${\rm{O}}_n$ over $\Q$.  Since $f_{\Z[1/2]}$ is an isomorphism, we get the equality of
${\rm{SO}}_n$ and ${\rm{SO}}'_n$ over $\Z[1/2]$.
Over $\F_2$, the smooth group ${\rm{O}}_n$ must be killed by
the determinant map into the infinitesimal $\mu_2$,
so ${\rm{SO}}'_n = {\rm{O}}_n$ over $\F_2$. 
\end{proof}

\begin{remark}\label{dflat}
Assume $n$ is even.  Consider the element $g \in {\rm{O}}_n(\Z)$
that swaps $e_1$ and $e_2$
while leaving the other $e_i$ invariant. The section
$D_q(g)$ of the constant $\Z$-group $\Z/2\Z$ is equal to $1 \bmod 2$
since it suffices to check this on a single geometric fiber
(and at any fiber away from characteristic 2 it is clear, 
as ${\rm{SO}}_n$ coincides with ${\rm{SO}}'_n$
over $\Z[1/2]$).
Thus, the Dickson invariant $D_q: {\rm{O}}(q) \rightarrow (\Z/2\Z)_S$ splits
as a semi-direct product when $q = q_n$.

The induced map 
${\rm{H}}^1(S_{\et}, {\rm{O}}_n) \rightarrow 
{\rm{H}}^1(S_{\et},\Z/2\Z)$ assigns to every non-degenerate $(V,q)$ of rank $n$ over $S$
(taken up to isomorphism)
a degree-2 finite \'etale cover of $S$.  Consideration
of \'etale \v{C}ech 1-cocycles and the definition of $D_q$ shows that this double cover
corresponds to the quadratic \'etale center $Z_q$ of ${\rm{C}}_0(V,L,q)$. 
If $S$ is a $\Z[1/2]$-scheme (so $(\Z/2\Z)_S = \mu_2$)
and ${\rm{Pic}}(S) = 1$ then it recovers the class in 
${\rm{H}}^1(S, \mu_2) = \Gm(S)/\Gm(S)^2$ of a unit $c$ such that 
the induced quadratic form on the top exterior power $\Det V$ of $V$
is locally $cx^2$.  
(Concretely, if $V$ is globally free then this is represented by 
$\Det [B_q]$, where $[B_q]$ 
is the matrix of $B_q$ relative to a basis for $V$; see \cite[IV, 4.1.1, 5.3.2]{knus} for affine $S$.)
If $S$ is an $\F_2$-scheme then it recovers the {\em pseudo-discriminant},
also called the {\em Arf invariant} when $S = \Spec k$ for a field $k/\F_2$;
see \cite[IV, 4.7]{knus} for an explicit formula when $S$ is affine and $V$ is globally free.
\end{remark}




\begin{remark}\label{naive}
For even $n$, the $\Z$-group
${\rm{SO}}'_n$ turns out to be reduced but not $\Z$-flat (due to problems at the prime 2).
The failure of flatness is a consequence of the more precise observation that 
the open and closed subscheme ${\rm{SO}}_n \hookrightarrow {\rm{SO}}'_n$
has complement
equal to the non-identity component of $({\rm{O}}_n)_{\F_2}$.
To prove these assertions (which we will never use),
first note that Corollary \ref{deteven} gives the result over $\Z[1/2]$,
as well as the topological description of the $\F_2$-fiber.
It remains to show that ${\rm{SO}}'_n$ is reduced. 

It is harmless to pass to the quotient by the {\em smooth}
normal subgroup ${\rm{SO}}_n$ (since reducedness asecends through smooth surjections), so under the 
identification of ${\rm{O}}_n/{\rm{SO}}_n$ with the constant
group $\Z/2\Z$ via the Dickson invariant 
we see that $G := {\rm{SO}}'_n/{\rm{SO}}_n$ is identified
with the kernel of the unique homomorphism of $\Z$-groups
$f:\Z/2\Z \rightarrow \mu_2$ sending $1$ to $-1$.
As a map 
from the $\Z$-group $(\Z/2\Z)_{\Z} = \Spec \Z[t]/(t^2 - t)$ to $\mu_2 = \Spec \Z[\zeta]/(\zeta^2 - 1)$, 
 it is given by $\zeta - 1 \mapsto -2t$ on coordinate rings, 
so the kernel $G$ is $\Spec \Z[t]/(-2t, t^2 - t)$.
This is the disjoint union of the identity section
and a single $\F_2$-point in the $\F_2$-fiber.
\end{remark}

Assume $n$ is odd, so ${\rm{SO}}(q) = {\rm{SO}}'(q)$.
This is smooth by Theorem \ref{yusmooth}.
As  in Remark \ref{parityrem}, by consideration of 
the half-discriminant, the morphism $\Det:{\rm{O}}(q) \rightarrow \Gm$
factors through $\mu_2$. Thus, by the oddness of $n$, the determinant splits
off the central $\mu_2 \subset {\rm{O}}(q)$ as a direct factor:
${\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$.  
Hence, ${\rm{SO}}(q) \rightarrow S$ has fibers of dimension
$n(n-1)/2$ by Theorem \ref{yusmooth}.
Fibral connectedness will be proved by induction on the odd $n$:

%
%
%By Lemma \ref{stdq}, to prove smoothness of ${\rm{SO}}(q)$
%for odd $n$ in general, it suffices to treat the case $q = q_n$ over $\Z$.
%We use a smoothness 
%criterion over a Dedekind base that upgrades fibral smoothness to
%relative smoothness in the presence of a {\em global} hypothesis
%of fibral connectedness:
%
%\begin{proposition}\label{gsmooth}
% Let $S$ be a Dedekind scheme, and $G$ an $S$-group of finite type such that 
%all fibers $G_s$ are smooth of the same dimension.  Then $G$ contains
%a unique smooth open subgroup $G^0$ whose $s$-fiber is $(G_s)^0$ for all $s \in S$.
%In particular, $G$ is smooth if its fibers are connected.
%\end{proposition}
%
%\begin{proof}
%We may assume $S$ is connected, say with generic point $\eta$.
%The smooth open subgroup $G_{\eta}^0 \subset G_{\eta}$
%then ``spreads out'' over a dense open $U \subset S$ to a smooth open subgroup 
%of $G_U$ with connected fibers.  This solves the problem over $U$, 
%and to handle the remaining finitely many closed points in $S - U$
%we may assume that $S = \Spec R$ for a discrete valuation ring $R$, say with fraction field $K$.
%We may and do remove the closed union of the non-identity components of
%the special fiber, so $G$ has connected special fiber.
%
%Let $\mathscr{G}$ denote the schematic closure in $G$ of the generic fiber $G_K$,
%so $\mathscr{G}_K =  G_K$.
%The $R$-flat $\mathscr{G} \times \mathscr{G}$ is the schematic closure
%of its generic fiber $G_K \times G_K$, so it follows that the $R$-flat 
%$\mathscr{G}$ is an $R$-subgroup of $G$. 
%This is a flat closed subscheme of $G$ with constant fiber dimension (by flatness), so
%the closed immersion $\mathscr{G}_0 \hookrightarrow G_0$ between special fibers 
%must be an isomorphism, as $G_0$ is smooth and connected
%and $\dim \mathscr{G}_0 = \dim \mathscr{G}_K = \dim G_K = \dim G_0$.
%Thus, $\mathscr{G}_0$ is smooth (as is $\mathscr{G}_K = G_K$), 
%so $\mathscr{G}$ is smooth. 
%
%The closed immersion $\mathscr{G} \hookrightarrow G$ 
%is an isomorphism on fibers, so by flatness of $\mathscr{G}$ it is an isomorphism, 
%due to Lemma \ref{flatclosed}.
%\end{proof}
%
%\begin{remark} The final assertion in Proposition \ref{gsmooth}
%is valid more generally: if $G \rightarrow S$ is a finite type
%group over any reduced noetherian scheme $S$ and if the fibers $G_s$ are smooth
%and connected of the same dimension then $G$ is smooth.
%Indeed, the problem is to verify flatness,
%and by the ``valuative criterion for flatness'' over a reduced noetherian base
%\cite[IV$_3$, 11.8.1]{ega} it suffices to check this after base change
%to discrete valuation rings, to which Proposition \ref{gsmooth} applies.
%See \cite[VI$_{\rm{B}}$, 4.4]{sga3} for a further generalization.
%\end{remark}
%
%The role of identity components in Proposition \ref{gsmooth} cannot be dropped.
%For a quasi-finite example, consider the constant
%group $(\Z/d\Z)_R$ over a discrete
%valuation ring $R$ with $d > 1$. This
%contains a reduced closed subgroup $G^{(d)}$ given by the reduced closed 
%complement of 
%the open non-identity points in the generic fiber.  The $R$-group $G^{(d)}$ 
%has \'etale fibers
%but  is not flat over $R$ (the non-identity points of the special fiber are open).
%A more interesting example is ${\rm{SO}}'_{2m}$ over $\Z_{(2)}$
%(which is the pushout of $G^{(2)}$ along the identity section $\Spec \Z_{(2)}
%\hookrightarrow {\rm{SO}}_{2m}$; see the proof of Corollary \ref{naive}).
%
%
%To apply Proposition \ref{gsmooth} to ${\rm{SO}}(q)$ when $n$ is odd,
%we need to verify three things in the theory over an algebraically
%closed field:  connectedness, smoothness, and dimension depending only on $n$.
%We first address the connectedness and dimension aspects by
%a fibration argument in the spirit of the proof of connectedness for
%${\rm{SO}}_{2m}$:

\begin{proposition}\label{smodd} Let $(V,q)$ be a 
quadratic space over a field $k$, with $n = \dim V$ odd.  
The group ${\rm{SO}}(q)$ is connected. 
\end{proposition}

\begin{proof}
We may assume $k$ is algebraically closed and
$q = q_n$.  The case $n = 1$ is trivial, so
we assume $n \ge 3$.  We treat characteristic 2 separately from other
characteristics, due to the appearance of the defect space $V^{\perp} = k e_0$
in characteristic 2.

First assume ${\rm{char}}(k) \ne 2$, so the symmetric bilinear form $B_q$ is non-degenerate
and $\mu_2 = (\Z/2\Z)_k$. 
Since $\mu_2 \times {\rm{SO}}(q) = {\rm{O}}(q)$, 
${\rm{O}}(q)$ has at least 2 connected components.
It has exactly 2 such components if and only if ${\rm{SO}}(q)$ is connected.
Since $n > 1$, the hypersurface $H = \{q = 1\}$ is irreducible,
and exactly as in
the proof of Proposition \ref{conneven}
we may apply Witt's extension theorem (valid for odd $n$ since ${\rm{char}}(k) \ne 2$)
to deduce that the action of ${\rm{O}}(q)$ on $H$ is transitive.
The orthogonal complement $V'$ of $e_0$ is spanned
by $\{e_1,\dots,e_{n-1}\}$ since ${\rm{char}}(k) \ne 2$, and 
it is preserved by ${\rm{Stab}}_{e_0}({\rm{O}}(q))$.
It is straightforward to check that the action of this stabilizer
on $V'$ defines an isomorphism onto ${\rm{O}}(q|_{V'}) \simeq
{\rm{O}}_{n-1}$.  This group has 2 connected components by Proposition
\ref{conneven}, so by irreducibility of $H$ it follows that 
${\rm{O}}(q)$ has 
 at most 2 connected
components (hence exactly 2 such components).   This
settles
the case ${\rm{char}}(k) \ne 2$.

Now assume ${\rm{char}}(k) = 2$.  Since ${\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$,
the connectedness of ${\rm{SO}}(q)$ is equivalent to the connectedness
of ${\rm{O}}(q)$.  The non-vanishing defect space
obstructs induction using the action on $H$, so instead we use the 
quotient $\overline{V} := V/V^{\perp} = V/ke_0$ by the defect line $V^{\perp}$ rather
than use a hyperplane as above.  As in the proof of Proposition \ref{conneven},
$B_q$ on $V$ induces a symplectic form $\overline{B}_q$ on $\overline{V}$, yielding a
natural map
${\rm{O}}(q) \rightarrow {\rm{Sp}}(\overline{V},\overline{B}_q) \simeq {\rm{Sp}}_{n-1}$
that is surjective with infinitesimal kernel, so the connectedness of symplectic groups
implies the connectedness of ${\rm{O}}(q)$.
\end{proof}
%
%The kernel is seen by calculation to be $\mu_2 \ltimes \alpha_2^{n-1}$
%(along the top row of matrices, with $\mu_2$ in the upper left), where $\mu_2$ acts on
%the Frobenius kernel $\alpha_2^{n-1} \subset \mathbf{G}_{\rm{a}}^{n-1}$ by the usual diagonal scaling action.
%Indeed, rather explicitly, since a point $T$ of ${\rm{O}}(q)$ must 
%restrict to an automorphism of the quadratic space $V^{\perp} = (ke_0, x_0^2)$, 
%it has the block form
%$$T = \begin{pmatrix} \zeta & \alpha & \alpha' \\ 0 & A & B \\ 0 & C & D \end{pmatrix}$$
%for $m \times m$ matrices $A, B, C, D$, a point
%$\zeta$ of $\mu_2$, and ordered $m$-tuples $\alpha$ and $\alpha'$.
%Writing a typical ordered $(2m+1)$-tuple as $(x_0, x, x')$ for ordered
%$m$-tuples $x$ and $x'$, we see that
%$$q(T(x_0, x, x')) =  x_0^2 + \langle \alpha, x \rangle^2 + \langle \alpha', x' \rangle^2 +
%B'_q(Ax + Bx', Cx + Dx'),$$
%where $\langle \cdot, \cdot \rangle$ is the standard bilinear form
%$(w,z) \mapsto \sum w_j z_j$.  Setting this equal to $q(x_0,x,x')$
%then imposes equations on $\alpha, \alpha', A,B,C,D$ that define
%the closed subscheme ${\rm{O}}(q) \subset \GL_n$.  This not only implies
%the asserted description of $\ker h$, but also shows that $h$ is surjective.
%
%We have shown that $h$ is surjective with infinitesimal fibers, so it
%suffices to prove the connectedness of symplectic groups over fields.
%This is proved by an inductive fibration argument
%(using lower-dimensional symplectic spaces).
%\end{proof}
%
%Now we can pass to the relative case and establish smoothness too:
%
%\begin{proposition} Assume $n$ is odd.  The map ${\rm{SO}}(q) \rightarrow S$ is smooth
%with connected fibers of dimension $n(n-1)/2$.
%\end{proposition}
%
%\begin{proof}
%By Lemma \ref{stdq}, we may assume $q = q_n$ over $S = \Spec \Z$.
%The case $n = 1$ is trivial, so we assume $n = 2m + 1$ with $m \ge 1$.
%All fibers are connected of dimension $n(n-1)/2$
%(Proposition \ref{smodd}), so to prove smoothness  we may apply 
%Proposition \ref{gsmooth} to reduce to proving
%fibral smoothness.  In other words, we wish to show
%that over a field $k$, the tangent space
%${\rm{Tan}}_e({\rm{SO}}(q))$ has dimension $n(n-1)/2$.
%
%First assume ${\rm{char}}(k) \ne 2$, so the equality 
%$\mu_2 \times {\rm{SO}}(q) = {\rm{O}}(q)$ implies
%${\rm{Tan}}_e({\rm{SO}}(q)) = 
%{\rm{Tan}}_e({\rm{O}}(q))$.   This latter tangent space
%was identified in the proof of Theorem \ref{yusmooth}: it is the space
%of linear endomorphisms $T$ of $V$ such that $B_q(v,Tw)$
%is alternating.  But $B_q$ is non-degenerate
%since ${\rm{char}}(k) \ne 2$, so $T \mapsto B_q(\cdot, T(\cdot))$
%identifies ${\rm{Tan}}_e({\rm{O}}(q))$
%with the space ${\rm{Alt}}^2(V)$ of alternating bilinear forms on $V$.
%This is the dual of $\wedge^2(V)$, so it has dimension $n(n-1)/2$,
%as desired.
%
%Now assume ${\rm{char}}(k) = 2$.  Let $V' = V/V^{\perp}$,
%and let $B'_q$ the induced non-degenerate
%alternating form on $V'$.  Since
%$\mu_2 \times {\rm{SO}}(q) = {\rm{O}}(q)$ 
%and ${\rm{Tan}}_e(\mu_2)$ is 1-dimensional, 
%it is equivalent to show that ${\rm{Tan}}_e({\rm{O}}(q))$
%has dimension $1 + n(n-1)/2$.  We will construct a short exact sequence
%$$0 \rightarrow \Hom(V,V^{\perp}) \rightarrow
%{\rm{Tan}}_e({\rm{O}}(q)) \rightarrow {\rm{Alt}}^2(V/V^{\perp}) \rightarrow 0,$$
%from which we will get the desired dimension count
%$$n + (n-1)(n-2)/2 = m(2m-1) + (2m+1) = 2m^2 + m + 1 = m(2m+1)+1 = 1 + n(n-1)/2.$$
%To construct the exact sequence, we will compute using dual numbers as in
%the proof of Lemma \ref{lieo}.  
%Using notation as in that calculation, since the alternating property
%for $B_q(v,Tw)$ implies skew-symmetry
%and hence symmetry (as ${\rm{char}}(k) = 2$), $T$ must preserve the defect line 
%$V^{\perp}$ (as $B_q(v,Tw) = 0$ for $v \in V^{\perp}$ and any $w$, and
%$B_q(v,Tw)$ is symmetric for general $v, w \in V$).
%Thus, ${\rm{Tan}}_e({\rm{O}}(q))$ consists of
%those $T$ which preserve $V^{\perp}$ and whose induced
%endomorphism $T'$ of $V' = V/V^{\perp}$ makes $B'_q(v',T'w')$ alternating.
%By non-degeneracy of $B'_q$, every bilinear form on
%$V'$ is $B'_q(v',Lw')$ for a unique endomorphism $L$ of $V'$, 
%so the vector space ${\rm{Tan}}_e({\rm{O}}(q))$ fits into the asserted exact sequence
%since ${\rm{Hom}}(V,V^{\perp})$ is precisely the ambiguity in $T$ when $T'$ is given.
%
%Smoothness has now been proved in the general relative setting,
%and it remains to prove that the natural homomorphism
%$f:\mu_2 \times {\rm{SO}}(q) \rightarrow {\rm{O}}(q)$
%is an isomorphism.  The map $f_s$ between fibers over any $s \in S$ is
%an isomorphism (Proposition \ref{smodd}),
%and the source of $f$ is $S$-flat (as ${\rm{SO}}(q)$ is even $S$-smooth).
%Thus, $f$ is an isomorphism due to the fibral
%isomorphism criterion in Lemma \ref{flatclosed}.
%\end{proof}
%
%\begin{corollary}\label{odddet}  Assume $n$ is odd.
%The map $\det: {\rm{O}}(q) \rightarrow \Gm$
%factors through $\mu_2$,
%and its kernel is ${\rm{SO}}(q)$.
%In particular, $\det$ identifies ${\rm{O}}(q)/{\rm{SO}}(q)$ with $\mu_2$.
%\end{corollary}
%
%\begin{proof}
%Since ${\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$ via multiplication
%and the determinant on the central $\mu_2$ is the inclusion $\mu_2 \hookrightarrow \Gm$
%(as $n$ is odd), we are done.
%\end{proof}

\begin{remark}\label{remblah}
Assume $n$ is odd.  As in the proof of Proposition \ref{conneven}, 
if ${\rm{char}}(k)=2$ then 
there is a surjective homomorphism
$h:{\rm{O}}(q) \rightarrow {\rm{Sp}}(\overline{V},\overline{B}_q)$ with 
$(\ker h)_{\overline{k}} = \mu_2 \ltimes \alpha_2^{n-1}$. 
This kernel meets the kernel ${\rm{SO}}(q)$ of the determinant map
on ${\rm{O}}(q)$ in $\alpha_2^{n-1}$ over $\overline{k}$, so by smoothness of
${\rm{SO}}(q)$ we obtain a purely inseparable isogeny
${\rm{SO}}(q) \rightarrow {\rm{Sp}}(\overline{V},\overline{B}_q)$
with kernel that is a form of $\alpha_2^{n-1}$
(and hence is isomorphic to $\alpha_2^{n-1}$, as
$\alpha_p^N$ has automorphism scheme ${\rm{GL}}_N$). 
This ``unipotent isogeny'' is a source of many phenomena related
to algebraic groups in characteristic 2 (e.g., see \cite[A.3]{map}). 

Special orthogonal groups in $2m+1$ variables are type ${\rm{B}}_m$ (see Proposition \ref{typeBD}) and
symplectic groups in $2m$ variables are type ${\rm{C}}_m$; these
types are distinct for $m \ge 3$ (and they coincide for $m = 1, 2$; see
Example \ref{so3} and
Example \ref{so5} respectively.)  In characteristics distinct from 2 and 3 
there are no isogenies between (absolutely simple) connected
semisimple groups of different types.  In characteristic 2 we have just built ``exceptional'' 
isogenies between ${\rm{B}}_m$ and ${\rm{C}}_m$ for all $m \ge 3$.
See \cite[XXI, 7.5]{sga3} for further details. 
\end{remark}


\begin{remark}\label{altso} For even $n$, by definition ${\rm{SO}}(q)$ is the kernel of the action
of ${\rm{O}}(q)$ on the degree-2 finite \'etale center $Z_q$ of ${\rm{C}}_0(V,L,q)$.
For odd $n$ and $\calO_S$-valued $q$ there is a similar description of ${\rm{SO}}(q)$:  it is the kernel
of the action of ${\rm{O}}(q)$ on the center $Z_q$ of the entire Clifford algebra ${\rm{C}}(V,q)$.
(Triviality of the action on an appropriate commutative rank-2 subalgebra
of the Clifford algebra is the unified definition of ${\rm{SO}}(q)$ for all $n$ 
in \cite[Ch.\,IV, \S5]{knus}.)

Since ${\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$ for odd $n$,
and $\mu_2$ acts by ordinary scaling on the line bundle $Z_q^1 = Z_q \cap {\rm{C}}_1(V,q)$
(immediate from the explicit description of $Z_q$ 
in the proof of Proposition \ref{cliffodd} for $q = q_n$, to which the general case may be reduced),
to justify this description of ${\rm{SO}}(q)$ for odd $n$ 
it suffices to check triviality of the ${\rm{SO}}(q)$-action on $Z_q$.  By working fppf-locally on $S$ we may assume $q = q_n$,
so it is enough to treat $q_n$ over $\Z$.  But ${\rm{SO}}_n$ is $\Z$-flat, so to verify triviality of the $\Z$-homomorphism
from ${\rm{SO}}_n$ into the automorphism scheme of the rank-2 algebra $Z_q$ it suffices to to work
over $\Z[1/2]$.  Now $Z_q$ is a quadratic \'etale algebra, so its automorphism scheme is $\Z/2\Z$, which admits no nontrivial
homomorphism from a smooth group scheme with connected fibers.
\end{remark}


%
%
%
%Since we have not yet proved smoothness (or even flatness) properties
%for ${\rm{O}}(q)$ or ${\rm{SO}}'(q)$, we do not yet know if their Lie
%algebras are vector bundles or 
%commute with base change on $S$.  In particular, we cannot yet use fibral
%methods to study their Lie algebras in the relative setting.
%Our first step in the study of orthogonal groups
%is to determine the tangent spaces ${\rm{Tan}}_e({\rm{SO}}'(q))$
%and  ${\rm{Tan}}_e({\rm{O}}(q))$ inside 
%${\rm{Tan}}_e({\rm{GL}}(V)) = {\rm{End}}(V)$ via computations with dual numbers over $S$: 
%
%\begin{lemma}\label{tanform} Inside ${\rm{End}}(V) = {\rm{Tan}}_e({\rm{GL}}(V))$, we have
%$${\rm{Tan}}_e({\rm{O}}(q)) = 
%\{T \in {\rm{End}}(V)\,|\,B_q(v,Tw) \mbox{\rm{ is alternating}}\}$$
%and
%$${\rm{Tan}}_e({\rm{SO}}'(q)) = \{T \in {\rm{End}}(V)^{{\rm{Tr}}=0}\,|\,
%B_q(v,Tw) \mbox{\rm{ is alternating}}\}.$$
%If $S$ is a $\Z[1/2]$-scheme then these tangent spaces coincide and moreover 
%${\rm{SO}}'(q)$ is an open subscheme of ${\rm{O}}(q)$.
%\end{lemma}
%
%\begin{proof}
%We may assume $S = \Spec k$ for a ring $k$. 
%In terms of dual numbers,
%${\rm{Tan}}_e({\rm{O}}(q))$ 
%is the space of linear endomorphisms $T$ of $V$ such that $1 + \epsilon T$
%preserves $q_{k[\epsilon]}$ on $V_{k[\epsilon]}$, and
%${\rm{Tan}}_e({\rm{SO}}'(q))$ is the subspace of such $T$ for which
%$\det(1+\epsilon T) = 1$.   This latter condition says exactly ${\rm{Tr}}(T) = 0$.
%
%For any $x \in V_{k[\epsilon]}$ 
%with reduction $x_0 \in V$, clearly $\epsilon T(x) = \epsilon T(x_0)$, so 
%$$q_{k[\epsilon]}(x + \epsilon T(x_0)) = 
%q_{k[\epsilon]}(x) + \epsilon B_q(x_0, T(x_0))$$
%since $\epsilon^2 = 0$.  Thus, the necessary and sufficient condition on
%$T$ is that the bilinear form $B_q(v,Tw)$ vanishes on the diagonal, which is
%to say that it is alternating.  
%
%Any point $g$ of (the functor) ${\rm{O}}(q)$ preserves $B_q$, so when $B_q$
%is non-degenerate (i.e., we avoid fibers $(V_s,q_s)$ in the defect-1 case) then
%the action of $g$ is identified with the inverse of its dual action.  Passing to determinants,
%this gives $\det(g)^2 = 1$ for such $g$.  In other words, away from the 
%closed locus of defect-1 fibers, 
%the map $\det:{\rm{O}}(q) \rightarrow \Gm$ factors through $\mu_2$.
%Provided that $k$ is a $\Z[1/2]$-algebra, we can identify $\mu_2$ with the
%finite constant group $\Z/2\Z$, and so in such cases the kernel ${\rm{SO}}'(q)$ is an open subscheme
%since the identity section in the $k$-group $\Z/2\Z$ is an open subscheme.
%\end{proof}
%
%\begin{lemma}\label{tan}  If $B_q$ is fiberwise non-degenerate
%$($i.e., not in the defect-$1$ case$)$, 
%then ${\rm{Tan}}_e({\rm{O}}(q))$ is identified with 
%the space ${\rm{Alt}}^2(V)$ of alternating
%bilinear forms on $V$.  In particular, it is a vector bundle of rank $n(n-1)/2$
%in such cases.
%
%At points $s \in S$ for which $(V_s,q_s)$ is in the defect-$1$ case, there is a short exact sequence
%$$0 \rightarrow {\rm{Hom}}(V_s,V_s^{\perp}) \rightarrow
%{\rm{Tan}}_{e(s)}({\rm{O}}(q_s)) \rightarrow {\rm{Alt}}^2(V_s/V_s^{\perp}) \rightarrow 0$$
%and $\dim {\rm{Tan}}_{e(s)}({\rm{O}}(q_s)) = 1 + n(n-1)/2$ whereas
%$\dim {\rm{Tan}}_{e(s)}({\rm{SO}}'(q_s)) = n(n-1)/2$. 
%\end{lemma}
%
%This lemma does not address ${\rm{Tan}}_{e(s)}({\rm{SO}}'(q_s))$ for $s$ of characteristic 2 
%away from the defect-1 case (i.e., for even $n$).  We will later see that in such cases
%${\rm{O}}(q_s) = {\rm{SO}}'(q_s)$.  
%
%\begin{proof}
%If $B_q$ is fiberwise non-degenerate then $T \mapsto B_q(\cdot, T(\cdot))$ is an isomorphism
%of ${\rm{End}}(V)$ onto the space of bilinear forms on $V$.
%Thus, the description of the tangent space away from the defect-1 case
%follows from Lemma \ref{tanform}
%
%Now consider the defect-1 case (with $n = 2m+1$ for $m \ge 1$) over a field $k$ of characteristic 2.
%The alternating property implies skew symmetry
%and hence symmetry (!), so $T$ must preserve the defect line 
%$V^{\perp}$ (as $B_q(v,Tw) = 0$ for $v \in V^{\perp}$ and any $w$, and
%$B_q(v,Tw)$ is symmetric for general $v, w \in V$).
%Thus, ${\rm{Tan}}_e({\rm{O}}(q))$ consists of
%those $T$ which preserve $V^{\perp}$ and whose induced
%endomorphism $T'$ of $V'$ makes $B'_q(v',T'w')$ alternating.
%By non-degeneracy of $B'_q$, every bilinear form on
%$V'$ is $B'_q(v',Sw')$ for a unique endomorphism $S$ of $V'$, 
%so the vector space ${\rm{Tan}}_e({\rm{O}}(q))$ fits into the asserted exact sequence
%since ${\rm{Hom}}(V,V^{\perp})$ is precisely the ambiguity in $T$ when $T'$ is given.
%Using the exact sequence, we compute
%$$\dim {\rm{Tan}}_e({\rm{O}}(q)) = (n-1)(n-2)/2 + n = m(2m-1) + (2m+1) = 2m^2 + m + 1 = m(2m+1)+1.$$
%But $n(n-1)/2 = (2m+1)m$, so we get the desired dimension formula.
%
%Finally, we show that ${\rm{Tan}}_e({\rm{SO}}'(q))$ has codimension 1 in
%the defect-1 case (over a field $k$ of characteristic 2).  
%The description in Lemma \ref{tanform} shows that it has codimension at
%most 1, so we just have to exhibit some element of
%${\rm{Tan}}_e({\rm{O}}(q))$ not in ${\rm{Tan}}_e({\rm{SO}}'(q))$.  
%It suffices to treat the standard form (relative to some basis $\{e_i\}$) 
%$$q = x_0^2 + \sum_{i=1}^m x_i x_{m+i}.$$
%Note that $B_q(x,y)$ does not involve $x_0$ or $y_0$, so the linear map 
%$T:V \rightarrow V$ which kills $e_i$ for $i > 0$ and sends $e_0$ to itself
%makes $B_q(v,Tw)$ identically zero yet ${\rm{Tr}}(T) = 1$.
%(More conceptually, $1 + \epsilon T$ spans the tangent line 
%to the infinitesimal $k$-group $\mu_2$ embedded into 
%${\rm{O}}(q)$ via action on $e_0$.) 
%\end{proof}
%
%We now take up smoothness results.  Due to the possibility of 
%deforming away from defect-1 in mixed characteristic with residue characteristic 2,
%it will simplify matters to treat smoothness over fields
%now and to return to the general case later (in Corollary \ref{sorel}).
%
%\begin{theorem}\label{soo}  Let $k$ be a field.
%The $k$-group ${\rm{SO}}'(q)$ is smooth of dimension 
%$n(n-1)/2$.  If ${\rm{char}}(k) \ne 2$ then 
%${\rm{O}}(q)$ is also smooth of dimension $n(n-1)/2$.
%If ${\rm{char}}(k) = 2$ then: 
%\begin{enumerate}
%\item when $n$ is even,
%${\rm{O}}(q) = {\rm{SO}}'(q)$. 
%\item when $n$ is odd,  $\det:{\rm{O}}(q) \rightarrow \Gm$ factors through
%$\mu_2$ and 
%${\rm{O}}(q) \rightarrow \mu_2$ admits a central section via the central
%$\mu_2$ in ${\rm{GL}}(V)$, so 
%${\rm{O}}(q) = {\rm{SO}}'(q) \times \mu_2$. 
%\end{enumerate}
%In particular, if ${\rm{char}}(k) = 2$ then 
%${\rm{O}}(q)_{\rm{red}} = {\rm{SO}}'(q)$. 
%\end{theorem}
%
%\begin{proof}
%If ${\rm{char}}(k) \ne 2$ then we just need to prove the smoothness of
%${\rm{O}}(q)$ (since we have already computed
%the dimension of the tangent spaces at the identity, and have shown that
%${\rm{SO}}'(q)$ is an open subscheme of ${\rm{O}}(q)$ in such cases).
%
%We will prove the smoothness of ${\rm{O}}(q)$ away from the defect-1 case in general
%(this includes characteristic 2 when $n$ is even, for which we have
%seen already that the tangent space at the identity has dimension
%$n(n-1)/2$).  Finer arguments will be needed to handle the defect-1 case 
%in characteristic 2 and to pass
%between ${\rm{O}}(q)$ and ${\rm{SO}}'(q)$ in the other cases in characteristic 2. 
%
%\medskip\noindent
%{\sc Step 1}. According to the infinitesimal 
%criterion, smoothness of
%${\rm{O}}(q)$ is precisely the condition that 
%for any finite local $k$-algebra $R$ with residue field $k$,
%and any nonzero $\epsilon \in \mathfrak{m}_R$ such that $\epsilon \mathfrak{m}_R = 0$,
%the map ${\rm{O}}(q)(R) \rightarrow {\rm{O}}(q)(R/(\epsilon))$ is surjective.
%Letting $\overline{R} = R/(\epsilon)$, we choose any
%$\overline{T} \in {\rm{O}}(q)(\overline{R})$ and wish to lift it to an element of
%${\rm{O}}(q)(R)$ away from the defect-1 case. 
% In other words, $\overline{T}$ is an $\overline{R}$-linear automorphism
%of $V_{\overline{R}}$ preserving $q_{\overline{R}}$ and we wish to lift it to an $R$-linear
%automorphism of $V_R$ preserving $q_R$, at least away from the defect-1 case.  
%In the defect-1 case we will require 
%$\overline{T} \in {\rm{SO}}'(q)(R)$, which is to say 
%that we also assume $\det \overline{T} = 1$.  
%
%Let $T \in {\rm{Aut}}_R(V_R)$
%be any $R$-linear automorphism of $V_R$ lifting $\overline{T}$
%(as we know can always be found, by lifting matrix entries and using nilpotence of
%$\epsilon$ to check that $\det T \in R^{\times}$).
%In the defect-1 case, by smoothness of ${\rm{SL}}(V)$ 
%we may and do also arrange that $\det T = 1$.
%In all cases 
%$q_R(Tx) \equiv q_R(x) \bmod \epsilon R$ for all $x \in V_R$,
%and we seek a choice of $T$ such that $q(Tx) = q(x)$ for all $x \in V_R$.
%Note that 
%$q$ is a {\em non-linear} object and when ${\rm{char}}(k) = 2$
%we cannot replace its study with that of a ``linear'' object such as the bilinear $B_q$.
%Thus, we really have to work with all $x \in V_R$ and not just $x$ coming from $V$.
%Fortunately, it will turn out that the key calculation will only require using elements of $V$.  
%
%\medskip\noindent
%{\sc Step 2}.  
%We have
% $q_R(Tx) = q_R(x) + \epsilon h(x)$ for a  unique function $h:V_R \rightarrow R/\mathfrak{m}_R = k$
%(recall that $\epsilon \mathfrak{m}_R = 0$ and $\epsilon \ne 0$). We now check that
%$h$ factors through the reduction map $V_R \twoheadrightarrow V$ followed by
%a $k$-valued quadratic form on $V$.  For any $c \in R$, 
%$$\epsilon h(cx) = q_R(T(cx)) - q_R(cx) = c^2(q_R(Tx) - q_R(x)) = 
%c^2 \epsilon h(x) = \epsilon \cdot c^2 h(x)$$
%(where $c^2$ on the right is to be viewed in the residue field $k$ of $R$). 
%In particular, if $c \in \mathfrak{m}_R$ then $h(cx) = 0$.  Likewise, for any $x, y \in V_R$, 
%$$\epsilon h(x+y) = q_R(T(x+y)) - q_R(x+y) = \epsilon(h(x) + h(y)) + (B_q)_R(Tx,Ty) - B(x,y).$$
%Thus, $h(x+y) - h(x) - h(y)$ as a function from $V \times V$ to $k$ is bilinear over
%the reduction map $R \twoheadrightarrow k$ since this can be checked after
%multiplying against $\epsilon$.  
%
%As a special case, if
%$y \in \mathfrak{m}_R V$ then $h(x+y) - h(x) - h(y) = 0$.  As an even more special case,
%the restriction of $h$ to $\mathfrak{m}_R V_R$ is
%an additive map into $k$, so by the homogeneity of $h$ as discussed above it follows
%that $h(y) = 0$ for any $y \in \mathfrak{m}_R$.  Hence, $h(x+y) = h(x)$ for any
%$x \in V_R$ and $y \in \mathfrak{m}_R V_R$, so indeed $h$ factors through the
%reduction map $V_R \twoheadrightarrow V$.  To summarize, we have proved that
%$$q_R(Tx) = q_R(x) + \epsilon Q_T(x_0)$$
%for all $x \in V_R$ with a quadratic form $Q_T:V \rightarrow k$ (depending on $T$), where
%$x_0 \in V$ is the reduction of $x$.  
%
%The problem is to find a choice of $T$ such that $Q_T = 0$.
%Having made an initial choice of $T$ lifting $\overline{T}$ as an $R$-linear automorphism of $V_R$,
%the set of all possible lifts is exactly $\{T(1 + \epsilon S)\}$ for $S \in {\rm{End}}(V)$,
%subject to the extra condition ${\rm{Tr}}(S) = 0$ in the defect-1 case.
%We seek a choice of $S$ such that $Q_{T(1 + \epsilon S)} = 0$, so let's compute
%$Q_{T(1+\epsilon S)}$ in terms of $Q_T$ and $S$ (and the common reduction $T_0 \in
%{\rm{O}}(q)(k)$ of $T$ and $\overline{T}$).  Since $q_R$ is a quadratic form over $R$
%and $\epsilon^2 = 0$, for any $x \in V_R$ we have 
%$$q_R((T(1 + \epsilon S))(x)) = q_R(Tx) + \epsilon (B_q)_R(Tx, TSx) =
%q_R(x) + \epsilon(Q_T(x_0) + B_q(T_0 x_0, T_0S x_0)).$$
%Since the map $T_0 \in {\rm{O}}(q)(k)$
%preserves $q$, it also preserves $B_q$. 
%Thus, it is necessary and sufficient to find
%$S$ so that $B_q(v,S(v)) = Q_T(v)$ for all $v \in V$.  
%
%\medskip\noindent
%{\sc Step 3}.  
%Away from the defect-1 case it will turn out to not be necessary to know anything about
%$Q_T$ beyond that it is a quadratic form (which is fortunate, since
%we do not know much about $T$).
%To handle the defect-1 case, we need to verify an extra fact:  
%$T$ can be chosen so that $V^{\perp}$ is contained
%in the defect space for $Q_T$. To find such a $T$, note that
%by the definition of $Q_T$, the associated bilinear form
%$B_{Q_T}$ on $V$ satisfies 
%$$\epsilon B_{Q_T}(v,w) = (B_q)_R(Tx,Ty) - (B_q)_R(x,y)$$
%for any $x, y \in V_R$ lifting $v, w \in V$ respectively.
%If $w \in V^{\perp}$ then we want this to vanish. For such $w$ 
%we can choose $y \in V^{\perp}_R$, so $(B_q)_R(x,y) = 0$.
%(If we allowed $k$ to be a ring such as $\Z/4\Z$ then 
%$V^{\perp}_R$ would be problematic.) 
%Provided that $T(V^{\perp}_R) \subset V^{\perp}_R$, we would similarly
%have $(B_q)_R(Tx,Ty) = 0$, so $V^{\perp}$ would be in the defect
%space of $B_{Q_T}$ for such $T$.  
%
%To arrange that
%$T$ preserves the ``line'' $V^{\perp}_R$ in $V_R$, 
%recall first that up to now $T$ was an arbitrary $R$-linear endomorphism of
%$V_R$ lifting $\overline{T} \in {\rm{SO}}'(q)(\overline{R})$ and having determinant 1.  Thus, if 
%$\overline{T}$ preserves $V^{\perp}_{\overline{R}}$ then we can 
%certainly choose $T$ to preserve $V^{\perp}_R$ and have determinant 1.
%(This just expresses the concrete fact that the subgroup of ${\rm{SL}}(V)$ stabilizing a line $L$ is
%smooth, which is easily checked by inspection.) 
%It therefore suffices (for the purpose of choosing a better $T$ in the
%defect-1 case) to check that $V^{\perp}_{\overline{R}}$ is intrinsically
%determined by $q_{\overline{R}}$ (as then any element of ${\rm{O}}(q)(\overline{R}) = 
%{\rm{Aut}}(V_{\overline{R}}, q_{\overline{R}})$ must preserve it, such as $\overline{T}$). 
%Since $B_q$ induces a non-degenerate bilinear form on
%$V/V^{\perp}$, the same holds after extension of scalars to
%$\overline{R}$.  Hence, 
%$$V^{\perp}_{\overline{R}} = \{x \in V_{\overline{R}}\,|\,B_{q_{\overline{R}}}(x,\cdot) = 0\},$$
%so we have an intrinsic characterization as desired. 
%
%With the preceding refinement in the defect-1 case settled, it now suffices to prove rather generally that
%if $Q:V \rightarrow k$ is {\em any} quadratic form
%and if in the defect-1 case (for $B_q$) we also require that its bilinear form $B_Q$ has defect space
%containing the defect line $V^{\perp}$ for $B_q$, then 
%$Q(v) = B_q(v,Sv)$ for some $S \in {\rm{End}}(V)$
%and we can moreover arrange ${\rm{Tr}}(S) = 0$ in the defect-1 case.  This is 
%a linear algebra problem entirely about the bilinear form $B_q$,
%and it is  solved by a dimension argument, as follows.
%
%\medskip\noindent
%{\sc Step 4}.  Consider the $k$-linear map
%$L:{\rm{End}}(V) \rightarrow {\rm{Quad}}(V)$  from the space
%of linear endomorphisms to the space of quadratic forms, 
%defined by $$S \mapsto (v \mapsto B_q(v,Sv)).$$
%We need to show that $L$ is surjective away from the defect-1
%case, and the defect-1 case will require a more refined
%analysis (since $Q$ satisfies an extra
%condition in such cases).  
%
%Observe that 
%$\dim {\rm{End}}(V) = n^2$ and $\dim {\rm{Quad}}(V) = n(n+1)/2$
%(evident from the coordinate description of quadratic
%forms: $q(a_1,\dots,a_n) = \sum_{i \le j} c_{ij} a_i a_j$), so
%they differ by $n(n-1)/2$.  The kernel of $L$ consists of all $S$
%for which the bilinear form $B_q(v,Sw)$ on $V$ is alternating
%(i.e., vanishes on the diagonal), and the space of all alternating
%bilinear forms on $V$ has dimension $n(n-1)/2$.
%Provided that $B_q$ is {\em non-degenerate}, {\em every}
%bilinear form on $V$ is given by $B_q(v,Sw)$ for a unique $S$,
%so in particular $\ker L$ is identified with the space of alternating
%bilinear forms on $V$ in such cases.  Thus, for dimension reasons
%we are done with the proof of smoothness of ${\rm{O}}(q)$ when $B_q$ is non-degenerate. 
%This establishes smoothness for
%${\rm{SO}}'(q)$ away from characteristic 2.  When 
%$B_q$ is non-degenerate and ${\rm{char}}(k) = 2$,
%it gives something interesting:  the determinant defines a $k$-homomorphism
%$\det:{\rm{O}}(q) \rightarrow \mu_2$ (since $B_q$ is non-degenerate), 
%but the $k$-group $\mu_2$ is infinitesimal
%so by smoothness of
%${\rm{O}}(q)$ this determinant homomorphism must be trivial.  
%Hence, ${\rm{SO}}'(q) = {\rm{O}}(q)$ when ${\rm{char}}(k) = 2$ and
%$n$ is even.  
%
%\medskip\noindent
%{\sc Step 5}.  It remains 
%to consider the defect-1 case, so
%suppose ${\rm{char}}(k) = 2$ and $\dim V = 2m+1$ with $m \ge 1$.
%We will prove that ${\rm{SO}}'(q)$ is smooth, and then relate it to ${\rm{O}}(q)$. 
%Let $V^{\perp}$ be the defect line and
%$V' = V/V^{\perp}$ the quotient space of dimension $n-1 = 2m$
%on which $B_q$ induces a {\em non-degenerate} bilinear form
%$B'_q$, and let ${\rm{Quad}}'(V)$ be the space of quadratic
%forms $Q$ on $V$ for which $B_Q$ contains $V^{\perp}$
%in its defect space (i.e., $B_Q$ factors through a bilinear form on
%$V/V^{\perp} = V'$).   For example, $Q(v) := B_q(v,Sv)$
%is such a quadratic form for any $S$ which preserves 
%$V^{\perp}$, so $L$ restricts to a linear map
%$L':{\rm{End}}'(V) \rightarrow {\rm{Quad}}'(V)$
%where 
%$${\rm{End}}'(V) = \{S \in {\rm{End}}(V)\,|\,S(V^{\perp}) \subset V^{\perp}, 
%{\rm{Tr}}(S) = 0\}.$$
%To complete the proof of smoothness of
%${\rm{SO}}'(q)$, it suffices to prove:
%
%
%\begin{lemma} The map $L'$ is surjective.  
%\end{lemma}
%
%\begin{proof}
%We will again use dimension considerations. The kernel of $L'$
%consists of all $S$ such that the bilinear form
%$B_q(v,Sw)$ is alternating.  The condition $S \in {\rm{End}}'(V)$ ensures that
%this bilinear form on $V$ factors through the bilinear form
%$B'_q(v',S'w')$ on $V'$, where $S':V' \rightarrow V'$ is the linear map induced by $S$.  
%If $\lambda$ is the scaling factor for $S$ on the line $V^{\perp}$ then
%$$0 = {\rm{Tr}}(S) = \lambda + {\rm{Tr}}(S'),$$
%so the effect of $S$ on $V^{\perp}$ is determined by $S'$.
%Also, 
%non-degeneracy of $B'_q$ on $V'$ implies that {\em every}
%bilinear form on $V'$ arises this way for a unique $S'$.
%Note  that every linear endomorphism $S'$ of $V'$ can be lifted to a linear
%endomorphism of $V$ preserving $V^{\perp}$, and that by changing its effect on $V^{\perp}$
%we can even arrange for the lift to have vanishing trace. Thus, the 
%vanishing trace condition on $S$ imposes
%no constraint on $S'$. 
%
%The space of alternating forms on $V'$ has dimension $2m(2m-1)/2 = m(2m-1)$,
%and $S'$ determines $S$ up to the space ${\rm{Hom}}(V/V^{\perp},V^{\perp})$ 
%of dimension $n-1 = 2m$.  (We have $V/V^{\perp}$ rather than $V$ because $S|_{V^{\perp}}$
%is scaling by $-{\rm{Tr}}(S')$.) Thus, 
%$\dim \ker L' = 2m + m(2m-1) = 2m^2 +m$.  Hence, 
%$$\dim({\rm{image}}(L')) = n^2 - 2m^2 - m = (2m+1)^2 - 2m^2  - m = 2m^2 + 3m+1.$$
%The condition on a quadratic form $Q \in {\rm{Quad}}(V)$ that it lies in
%${\rm{Quad}}'(V)$ is that $Q(v+w) = Q(v) + Q(w)$ for all $v \in V$ and $w \in V^{\perp}$.
%In other words, if we choose a basis $\{e_i\}$  of $V$ such that $e_1$ spans 
%the line $V^{\perp}$ then the condition on $Q(x_1,\dots,x_n) = \sum_{i \le j} c_{ij} x_i x_j$
%is that the only term which can involve $x_1$ is $x_1^2$; i.e., $c_{1j} = 0$ for $j \ge 2$.
%This says that ${\rm{Quad}}'(V)$ has codimension $n-1 = 2m$ in ${\rm{Quad}}(V)$, so
%$$\dim {\rm{Quad}}'(V) = n(n+1)/2 - 2m = (2m+1)(m+1) - 2m = 2m^2 + 3m + 1.$$
%The equality of dimensions does the job. 
%\end{proof}
%
%{\sc Step 6}.  With the smoothness of ${\rm{SO}}'(q)$ now settled in the defect-1 case,
%we turn to the relationship with ${\rm{O}}(q)$ in such cases.
%The central $\mu_2$ in ${\rm{GL}}(V)$ clearly lies
%in ${\rm{O}}(q)$ and the restriction of the determinant to this $\mu_2$
%is the natural closed immersion of $\mu_2$ into
%$\Gm$ since $\dim V$ is odd. 
%We get the desired direct product decomposition
%provided that the determinant on ${\rm{O}}(q)$ factors through $\mu_2$.
%
%For any $k$-algebra $R$ and $g \in {\rm{O}}(q)(R)$, the $g$-action on $V_R$
%must preserve $(B_q)_R = B_{q_R}$, so it also preserves the
%``defect line'' $(V_R)^{\perp} = (V^{\perp})_R$, as we have seen above.  
%The linear automorphism of $V^{\perp}_R$ induced by $g$ must be scaling
%by a unit $z \in R^{\times}$, so  preservation of $q^{\perp}_R = c x^2$
%with $c \in k^{\times}$ forces $z^2 = 1$; i.e., $z \in \mu_2(R)$.  
%(This is the first time we use that $q|_{V^{\perp}} \ne 0$!) 
%Thus, to compute $\det g$ we can use the $g$-equivariance of the exact sequence
%$$0 \rightarrow V^{\perp}_R \rightarrow V_R \rightarrow (V/V^{\perp})_R \rightarrow 0$$
%to get $\det(g) = z \det(g')$ where $g' \in {\rm{GL}}(V/V^{\perp})(R)$ is the automorphism
%induced by $g$.  But $g'$ preserves the (scalar extension to $R$ of the) {\em non-degenerate}
%quadratic form $q'$ induced by $q$ on $V' := V/V^{\perp}$, which is to say
%$g' \in {\rm{O}}(q')(R)$.  Since $\dim V'$ is even we know that
%${\rm{O}}(q') = {\rm{SO}}'(q')$, so $\det g' = 1$.
%Thus, $\det(g) = z$ is a point of $\mu_2$.  
%\end{proof}

%We end this section with a really fantastic application of the smoothness of orthogonal groups
%(away from the defect-1 case in characteristic 2).  By Exercise \ref{1.4}, 
%if $(V,q)$ is a non-degenerate quadratic space of dimension $n \ge 2$ over
%any field $k$, then over $\overline{k}$ it becomes isomorphic to the National Bureau of Standards
%form $q_n$.  This isomorphism therefore occurs over a finite extension $k'/k$, and it is natural to
%ask if we can do better (for imperfect $k$):  can we arrange this extension to be {\em separable}?
%Put another way, if $k = k_s$ then is $q \simeq q_n$ over $k$?

%\begin{proposition}\label{isomtrick}
%Let $(V,q)$ be a non-degenerate quadratic space of dimension $n \ge 2$ over a field $k$.
%There is a finite separable extension such that $(V_{k'}, q_{k'})$ is isomorphic to $({k'}^n, q_n)$
%except possibly in the defect-$1$ case.  In such cases, 
%if $q|_{V^{\perp}} = cx^2$ then such a separable $k'/k$ exists if and only if $c$ is a square in $k^{\times}$.
%\end{proposition}

%Note that the criterion in the defect-1 case makes sense, since $c \in k^{\times}$ 
%and it is well-defined up to square multiple
%in $k^{\times}$.

%\begin{proof}
%For any two quadratic space $(V,q)$ and $(V',q')$ of the same rank $n$ over $k$,
%consider the functor on $k$-algebras 
%$$\underline{\rm{Isom}}(q,q'):R \rightsquigarrow {\rm{Isom}}_R((V_R,q_R), (V'_R, q'_R)).$$
%By working with quadratic forms as homogeneous polynomials, it is clear
%that this functor is represented by an affine $k$-scheme of finite type,
%denoted ${\rm{Isom}}(q,q')$.  If both are non-degenerate, then by  Exercise \ref{1.4} we 
%see that ${\rm{Isom}}(q,q')(\overline{k})$ is {\em non-empty}.  Moreover,
%we claim that ${\rm{Isom}}(q,q')$ is {\em smooth} away from the defect-1 case.
%To check this we can extend scalars to $\overline{k}$ so that $k$ is
%algebraically closed, and then use a choice of isomorphism
%of $q'$ with $q$ to convert the scheme into the {\em automorphism scheme}
%${\rm{O}}(q)$ of $q$.  But we have proved above that this is smooth away
%from the defect-1 case!  By 
%non-empty smooth schemes over a field always have rational points
%over a finite separable extension, so that provides the desired $k'/k$ away from
%the defect-1 case.

%Now suppose $(V,q)$ is in the defect-1 case, so ${\rm{char}}(k) = 2$.
%In this case ${\rm{Isom}}(q,q_n)_{\overline{k}} \simeq {\rm{O}}_n$ is {\em not} smooth. 
%Fix a basis $v_0$ of $V^{\perp}$, so $q|_{V^{\perp}} = cx^2$ for some $c \in k^{\times}$.
%Note that $c \bmod (k^{\times})^2$ is independent of the basis, and making
%a separable algebraic extension in characteristic 2 has no effect on whether
%or not $c$ is a square.  

%Assume that $c$ is a square, so by scaling $v_0$ we can arrange that $c = 1$.
%For any $R$-linear isomorphism $V_R \simeq R^n$ that carries $q_R$ to
%$q_n$, there is an induced isomorphism
%$V^{\perp}_R \simeq R e_0$ between the defect modules.  It therefore
%makes sense (as a Zariski-closed condition on ${\rm{Isom}}(q,q_n)$)
%to demand that this isomorphism carries $v_0$ to $e_0$.
%Let $I$ denote the resulting closed subscheme.  
%We have that $I(\overline{k})$ is non-empty precisely because we arranged $c = 1$
%(check!), so it follows by a variant on the preceding argument in the other cases that 
%$I_{\overline{k}} \simeq {\rm{SO}}_n$ as $\overline{k}$-schemes.
%Thus, $I$ is smooth (and non-empty!), so we win in these cases. 

%Finally, assume (still in the defect-1 case) that there is an isomorphism between the quadratic spaces
%over some finite separable extension $k'/k$. Restricting it to the defect lines provides 
%a square root of $c$ in $k'$.  Thus, $c$ becomes a square in a separable extension of
%$k$, so $c$ is a square in $k$ since ${\rm{char}}(k) = 2$. 
%\end{proof}
%
%\subsection{Relative properties of naive special orthogonal groups}
%
%In this section we compute
%the component group for geometric fibers of ${\rm{SO}}'(q)$, and use
%that to improve our earlier fibral smoothness results 
%for ${\rm{SO}}'(q)$ to relative smoothness results. We also
%compute the functorial center of ${\rm{SO}}'(q)$ over any base scheme. 
%
%\begin{theorem}\label{connthm}  Let $(V,q)$ be a non-degenerate quadratic space
%of dimension $n \ge 2$ over a 
%field $k$.  The $k$-group ${\rm{SO}}'(q)$ is connected except when
%${\rm{char}}(k) = 2$ and $n$ is even. 
%If ${\rm{char}}(k) = 2$ and $n$ is even then the $k$-group ${\rm{SO}}'(q) = {\rm{O}}(q)$ 
%has component group of order $2$. 
%\end{theorem}
%
%There is a classical proof using the Cayley transform, but it does not work
%in characteristic 2.  We give another argument, inspired by the Gram-Schimdt process,
%which is characteristic-free. 
%
%\begin{proof}
%The case $n = 2$ is trivial.
%Thus, we can assume $n \ge 3$.
%The idea of the proof is to proceed by descending induction in 2-fold steps
%(from $n$ down to $n - 2$) by using an algebro-geometric substitute for
%a dynamic viewpoint on the Gramm--Schmidt process: 
%we can assume $k = \overline{k}$, $V = k^n$, and $q = q_n$,
%and we will interpolate in 1-parameter families of ordered bases from the standard basis $\{e_i\}$ to
%the basis of columns $\{v_i\}$ of a fixed $g \in {\rm{SO}}'_n(k)$.
%A further argument will modify this into
%a family of orthogonal bases at the expense of some permutations on
%the $v_i$'s.  The ``special'' orthogonal property for
%$g$ will then be used to almost eliminate the ambiguity introduced by such permutations.
%
%Depending on the parity of the initial $n$, we will then
% be reduced to determining if $g \in ({\rm{SO}}'_n)^0(k)$ 
%for $n \in \{3, 4\}$ and a specific $g$ in each case.   
%We will show that certain homomorphisms
%$${\rm{PGL}}_2 \rightarrow {\rm{SO}}'_3,\,\,\,\,{\rm{SL}}_2 \times {\rm{SL}}_2 
%\rightarrow {\rm{SO}}'_4$$
%hit the specific $g$'s we consider in the special cases for $n \in \{3, 4\}$ except 
%in characteristic 2 with $n = 4$ (corresponding
%to the general case of characteristic 2 with even $n$), 
%so the connectedness of ${\rm{PGL}}_2$ and
%${\rm{SL}}_2 \times {\rm{SL}}_2$ will then force $g$ to be in the identity component
%as required except for possibly 
%characteristic 2 with even $n$.  Up to that point, the argument will be characteristic-free. 
%
%Now we begin the argument. 
%Let $S$ be the quasi-affine scheme of ordered bases of $V$; this is a Zariski-open locus in
%the affine space attached to $V^n$, corresponding to the non-vanishing of a determinant
%polynomial. 
%(In differential geometry, this is the {\em Steifel manifold}.)
%Over $S$ there is the {\em universal basis}:  an ordered
%$n$-tuple $\{\sigma_i\}$ of sections of $\mathscr{O}_S^n$ whose specialization at
%each geometric point is an ordered basis of a standard Euclidean $n$-space
%and which has the evident universal property.  For example, there is a distinguished
%point $\xi_0 \in S(k)$ corresponding to the standard ordered basis $\{e_i\}$ and another point
%$\xi_1 \in S(k)$ corresponding to the ordered basis $\{v_i\}$ of columns of $g$.
%
%To construct an irreducible curve in
%${\rm{SO}}'_n$ linking $g$ to the identity, we first consider a punctured
%affine line $L$ joining $\xi_0$ to $\xi_1$ in $S$.
%(Such a punctured line makes sense since $S$ is open in
%the affine space of all ordered $n$-tuples in $V$.)   
%For the pullback frame $\{w_i\} = \{\sigma_i|_L\}$ of $\mathscr{O}_L^n$,
%we have $w_i(\xi_0) = e_i$ and $w_i(\xi_1) = v_i$ in $k^n$.
%Concretely, $L = \Spec A$ for a localization $A = k[t][1/h]$
%and $\{w_i\}$ is an ordered basis of $A^n$, and if we make a linear change of variable on
%$t$ we can even say that $\xi_0$ corresponds to $t =0$ and $\xi_1$ corresponds to $t = 1$.
%(Do not attach any significance to this coordinatization, 
%since we will soon replace $A$ with some finite covers that are generally not punctured affine lines.)
%
%In order to handle the case of even and odd $n$ by a uniform argument, it will be convenient to
%use the indexing $\{w_1,\dots,w_n\}$ (and {\em not}
%$\{w_0, \dots, w_{2m}\}$ when $n = 2m+1$). 
%Consider the rank-2 subbundle $\mathscr{H}$ in $\mathscr{O}_L^n$ with basis
%$\{w_{n-1}, w_n\}$, which is to say the direct summand $H$ of $A^n$ spanned by $w_{n-1}$ and $w_n$.
%Then $(H,q)$ is a quadratic space over $A$, and its fibers at $\xi_0$ and $\xi_1$ are non-degenerate
%(with fiber $(k^2, xy)$).  Thus, ${\rm{disc}}(q) \in A$ is non-vanishing at the $\xi_i$'s, so by localizing
% more we can assume that $(H,q)$ is non-degenerate in the sense of Remark \ref{ndrem}.
%The pointwise identities $q_{\xi_i} = xy$ do not generally hold over the entirety of $\Spec A$, but
%we can pass to a suitable cover to get to this case.  It is convenient to express this process more
%generally to simplify the subsequent induction:
%
%\begin{lemma}\label{basislem} Let $C = \Spec A$ be a smooth irreducible affine curve over $k = \overline{k}$
%and let $\{w_i\}$ be an ordered basis of $A^n$ with $n \ge 3$.  Let $\{\xi_0, \dots, \xi_r\}$ be a non-empty
%finite set of $k$-points of $C$ such that $q := q_n$ on $k^n$ retains its standard form
%in the ordered basis $\{w_i(\xi_j)\}$ for each $j$ $($i.e., for each $j$,
% the $n \times n$ matrix with columns $w_i(\xi_j)$ lies in ${\rm{O}}_n(k)$$)$.
% Let $H = A w_{n-1} \bigoplus A w_n$.  
% 
% There is a dense open
% $U = \Spec(A[1/a]) \subset C$ containing every $\xi_j$, a finite surjection 
% $\Spec(A') = C' \twoheadrightarrow U$, an ordered basis $\{w'_{n-1}, w'_n\}$
% of $H' = H|_{C'} = A' \otimes_A H$, and a point $\xi'_j \in C'(k)$ over $\xi_j$ for each $j$ such that:
% \begin{enumerate}
% \item $w'_{n-1}(\xi'_0) = w_{n-1}(\xi_0)$ and $w'_n(\xi'_0) = w_n(\xi_0)$ inside  $k^n$,
% \item for $1 \le j \le r$, $\{w'_{n-1}(\xi'_j), w'_n(\xi'_j)\}$ coincides with
% $\{w_{n-1}(\xi_j), w_n(\xi_j)\}$ as unordered pairs in $H(\xi_j) = H'(\xi'_j) \subset k^n$,
% \item $q|_{H'}$ has the form $x_{n-1} x_n$ in the ordered basis $\{w'_{n-1}, w'_n\}$;
% that is, $q(w'_{n-1}) = q(w'_n) = 0$ and $B_q(w'_{n-1}, w'_n) = 1$.
% \end{enumerate}
% \end{lemma}
% 
% Note that $(w'_{n-1}, w'_n)$ and $(w_{n-1}, w_n)_{A'}$ are related by an element of
% ${\rm{GL}}_2(A')$ since they are both $A$-bases of $H' = H_{A'}$.   The idea of the proof
% is simple, but the details make it a bit long to carry out.  
% 
% \begin{proof}
% Consider the quadratic space $(H,q|_H)$ over $A$.  This is non-degenerate at each
% $\xi_j$, so by replacing $C$ with a suitable affine open subscheme $U$ containing
% the $\xi_j$'s (i.e., invert a certain discriminant) we can assume that $(H,q|_H)$ is non-degenerate.
% Thus, using the ordered basis  $(w_{n-1}, w_n)$ of
% $H$, the closed subscheme $(q|_H = 0)$ makes sense in $\mathbf{P}^1_A = \mathbf{P}^1 \times C$
% and by local calculation over $C$ we see that it is finite flat of degree 2 over $C$.
% Hence, each irreducible component of $(q|_H = 0)$ is an irreducible curve mapping onto
% the smooth curve $C$ by a necessarily flat finite map.  
% 
%The fiber over each $\xi_j$ in $(q|_H = 0) \subset \mathbf{P}^1 \times C$ 
%consists of two  distinct reduced points $[1,0]$ and $[0,1]$ in $\mathbf{P}^1_k$.  
%Let  $Z$ be an irreducible component of $(q|_H = 0)$ passing through $[1,0]$ in the fiber
%over $\xi_0$, and let $C' = \Spec A'$ be the normalization of $Z$.  Pick
%$\xi'_0 \in C'(k)$ over $[1,0] \in Z_{\xi_0}$. By construction, the pullback
%$(q|_{H'} = 0)$ in $\mathbf{P}^1_{A'}$ has a section corresponding to the map
%$C' \rightarrow (q|_H = 0)$, and by the universal property of
%$\mathbf{P}^1$ this section corresponds to a rank-1 local direct summand
%$\mathscr{L}$ in $H' := H_{A'}$ on which $q$ vanishes.  The choice of $Z$ and $\xi'_0$ ensures
%that the fiber $\mathscr{L}(\xi'_0) \subset H'(\xi'_0) = H(\xi_0)$ is the line $k w_n(\xi_0)$. 
%Let $\xi'_j$ be a point of $C'(k)$ over $\xi_j$ (there are at most two choices).  
%For $1 \le j \le r$, the line $\mathscr{L}(\xi'_j)$ in $H'(\xi'_j) = H(\xi_j)$
%is in the zero locus of $q|_{H(\xi_j)}$, and by hypothesis this
%latter quadratic form is $x_n x_{n-1}$ in the ordered basis $\{w_{n-1}(\xi_j), w_n(\xi_j)\}$.
%Hence, $\mathscr{L}(\xi'_j)$ is the line through one of $w_{n-1}(\xi_j)$ or $w_n(\xi_j)$
%(but we do not know which one).  
%
%By weak approximation in Dedekind domains (such as $A'$) and the fact that $A \rightarrow A'$
%is a finite extension of Dedekind domains, by localizing a bit more on $A$ away from the
%$\xi_j$'s we can arrange that
%$\mathscr{L}$ is a free $A'$-module, say with a basis denoted $v'_n$.  By suitable unit scaling
%with the help of the Chinese Remainder Theorem in $A'$ after a bit more localization we
%can assume that $v'_n(\xi'_0) = w_n(\xi_0)$ and that for $1 \le j \le r$
%the specialization $v'_n(\xi'_j)$ lies in $\{w_{n-1}(\xi_j), w_n(\xi_j)\}$
%(i.e., there is no $k^{\times}$-scalar discrepancy, just a permutation problem).  
%
%
%Writing $v'_n = \alpha w_n + \beta w_{n-1}$ with $\alpha, \beta \in A'$,
%we have $\alpha A' + \beta A' = A'$ with $(\alpha(\xi'_0), \beta(\xi'_0)) = (1,0)$
%and $\{\alpha(\xi'_j), \beta(\xi'_j)\} = \{0,1\}$ as unordered pairs for $j > 0$.  The linear form 
%$\beta x_n - \alpha x_{n-1}$ which is fiberwise nonzero over $C'$
%has zero locus contained in $(q|_{H'} = 0)$, so since $A'[x_{n-1}, x_n]/(\beta x_n - \alpha x_{n-1})$ 
%is reduced
%it follows that $\beta x_n - \alpha x_{n-1}$ {\em divides} $q|_{H'}$.  That is:
%$$q|_{H'} = (\beta x_n - \alpha x_{n-1})(\delta x_n + \gamma x_{n-1})$$
%in $A'[x_{n-1}, x_n]$
%At each $\xi_j$ we have assumed that $q|_{H}$ specializes to
%$x_n x_{n-1}$ relative to the linear coordinates for the ordered basis $\{w_{n-1}(\xi_j), w_n(\xi_j)\}$, so
%by specializing over each $\xi_j$ we see that the cross-term coefficient
%$\beta \gamma - \alpha \delta$ specializes to 1.   Thus, localizing a bit more
%on $C$ allows us to assume that this cross-term coefficient is a unit on $C'$.
%Thus, we can make a change of basis of $H'$ via the matrix
%$\left(\begin{smallmatrix} \gamma & \alpha \\ \delta & \beta \end{smallmatrix}\right)$
%to get a new ordered basis $\{v''_{n-1}, v''_n\}$ of $H'$ such that
%$v''_n(\xi'_j) = v'_n(\xi'_j)$ for all $j$ (so this is $w_n(\xi_0)$ for $j = 0$
%and is one of $w_n(\xi_j)$ or $w_{n-1}(\xi_j)$ for $j > 0$).  In
%this new basis $q|_{H'}$ acquires the form
%$y_{n-1} (u y_{n} + v y_{n-1})$, where $\{y_{n-1}, y_n\}$ is the dual basis
%to $\{v''_{n-1}, v''_n\}$ and $u$ is a {\em unit} due to the non-degeneracy of
%$q|_{H'}$.  (Remember that the condition $y_{n-1} = 0$ cuts out the line spanned
%by $v''_n$!)  Thus, replacing $v''_{n-1}$ with $v v''_n - u v''_{n-1}$ 
%brings us to a basis $\{w'_{n-1}, w'_n\}$ of $H'$ relative to which
%$q|_{H'} = y_{n-1} y_n$ and $w'_n$ has the desired specialization properties
%at each $\xi'_j$.   
%
%How does $w'_{n-1}$ specialize at the points $\xi'_j$? Since in this new frame
%$q|_{H'}$ specializes to the product of the linear coordinates, whereas
%in other original frame $\{w_{n-1}, w_n\}$ it is assumed to do the same
%relative to the linear coordinates dual to this original specialized frame, by
%unique factorization in $k[x,y]$ it follows that $w'_{n-1}(\xi'_j)$ must specialize
%to a $k^{\times}$-multiple of whichever of
%$w_{n-1}(\xi_j)$ or $w_n(\xi_j)$ is {\em not} equal to $w'_n(\xi'_j)$.  
%But $B_q(w''_{n-1}, w''_n) = 1$ (as we can compute $B_q$ using any linear coordinate
%system, and $q|_{H'} = y_{n-1} y_n$ in the frame $\{w'_{n-1}, w'_n\}$), so
%the mystery multiplier is 1 for each $j$.
% \end{proof}
%
%Applying the Lemma once, we get a smooth irreducible curve $C' = \Spec A'$ finite
%flat over an open in $C$ around $\xi_0$ and $\xi_1$, equipped
%with points $\xi'_j \in C'(k)$ over $\xi_j$, such that 
%after applying a suitable element of ${\rm{GL}}_2(A')$
%we get an ordered basis $\{w'_{n-1}, w'_n\}$ of $H' = H_{A'}$
%relative to which $q|_{H'}$ acquires the form $x'_{n-1} x'_n$
%and 
%$$(w'_{n-1}(\xi'_0), w'_n(\xi'_0)) = (e_{n-1}, e_n),\,\,\,\,
%\{w'_{n-1}(\xi'_1), w'_n(\xi'_1)\} = \{v_{n-1}, v_n\}.$$
%The orthogonal space ${H'}^{\perp}$ satisfies
%$H' \bigoplus {H'}^{\perp}$ since $q|_{H'}$ is non-degenerate,
%and the isomorphism ${H'}^{\perp} \simeq {A'}^n/H'$
%provides an ordered basis of ${H'}^{\perp}$ corresponding
%to the induced ordered basis $\{w'_1,\dots,w'_{n-2}\}$ of ${A'}^n/H'$.
%Note that ${H'}^{\perp}$ equipped with its ordered basis
%specializes at $\xi'_0$ to the span of $\{e_1,\dots,e_{n-2}\}$
%with that ordered basis, and at $\xi'_1$ it specializes to the span
%of $\{v_1,\dots,v_{n-2}\}$ equipped with that ordered basis
%(since $g \in {\rm{O}}_n(k)$).    Thus, we can then repeat
%the process if $n > 3$ by applying the Lemma to
%$(C', \xi'_0, \xi'_1, {H'}^{\perp})$ and the ordered basis of
%${H'}^{\perp}$ just constructed, and continue in this way $[n/2]$ times.
%
%In the end, we arrive at a smooth irreducible affine curve $C = \Spec A$
%equipped with a pair of points
%$\xi_0, \xi_1 \in C(k)$ and an ordered basis $\{w_1,\dots,w_n\}$ of $A^n$ such that 
%it specializes to $\{e_i\}$ at $\xi_0$, satisfies
%$$\{w_{n-2i}(\xi_1), w_{n-2i-1}(\xi_1)\} = \{v_{n-2i}, v_{n-2i-1}\}$$
%as unordered pairs for all $0 \le i \le [(n-1)/2]$, and the expression for
%$q_n$ in this basis is $q_n$ except that for odd $n$ the $x_1^2$-term
%is  $u x_1^2$ for some unknown unit $u$.   Of course, $u(\xi_0) = u(\xi_1) = 1$,
%but in general we may have $u \ne 1$ in $A$.  By passing to the normalization
%of an irreducible component of $A[t]/(t^2 - u)$ we may reduce to the case
%when $u = v^2$ for some unit $v$.  Observe that $v(\xi_0), v(\xi_1) \in \mu_2(k)$,
%and so if we replace $v$ with $-v$ if necessary then we can arrange that
%$v(\xi_0) = 1$.  Hence, if $n$ is odd we may replace $w_1$ with $v w_1$
%to get to the case that $u = 1$, $w_1(\xi_0) = e_1$, and $w_1(\xi_1) = \pm v_1$.
%
%We conclude that the matrix $M \in {\rm{Mat}}_n(A)$ with $i$th column $w_i$
%lies in ${\rm{O}}_n(A)$ and satisfies $M(\xi_0) = 1$, and moreover 
%$M(\xi_1)$ coincides with $g$ up to possibly negating the leftmost column when
%$n$ is odd and permuting the $(n-2i)$th and $(n-2i-1)$th columns for
%$0 \le i \le [(n-1)/2]$.  The determinant $\det M$ can be viewed as a $k$-morphism
%$\Spec A \rightarrow \mu_2$ carrying $\xi_0$ to 1, so by arguing separately
%when ${\rm{char}}(k) = 2$ (using that $\Spec A$ is smooth) we see that this
%map must be the constant map to 1.    In other words, we have 
%$M \in {\rm{SO}}'_n(A) = ({\rm{SO}}'_n)^0(A)$, where the final equality holds because 
%$\Spec A$ is connected and $M(\xi_0) = 1$.  Hence, we have proved that 
%$g' := M(\xi_1) \in ({\rm{SO}}'_n)^0(k)$,
%where $g'$ is obtained by $g$ by possibly negating the left column (if $n$ is odd) and permuting
%some pairs of columns as indicated above.   
%
%We can replace $g$ with $g {g'}^{-1}$, so
%this brings us to the special case when $g$ is of a very special form depending on the parity of
%$n$.  If $n$ is even then $g$ is built from $2 \times 2$ blocks along the diagonal
%that are each either the identity or $\left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right)$.
%If $n$ is odd then we have a similar description for the lower-right $(n-1) \times (n-1)$ part,
%and there is an entry of $\pm 1$ in the upper left (and 0's elsewhere along the first row and first column). 
%But $\det g = 1$, so we get an important constraint when ${\rm{char}}(k) \ne 2$:  if $n$ is even then 
%the number of non-identity $2 \times 2$ blocks is even, and if $n$ is odd then the
%number of such blocks is even precisely when the upper left entry is $1$.  
%On the other hand, if ${\rm{char}}(k) = 2$ then there are no such constraints:  all such special elements
%have determinant 1.
%
%For example, if ${\rm{char}}(k) \ne 2$ and $n = 3$ and the upper left entry is 1 then $g = 1$, so for $n = 3$
%with ${\rm{char}}(k) \ne 2$
%we only have to consider the case when the upper left entry is $-1$.  In characteristic 2 with $n = 3$
%the upper left entry is automatically equal to 1 since $-1 = 1$ in such cases.  
%In general we are thereby reduced to two very special cases:
%$$g_0 = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \in
%{\rm{SO}}'_3(k),\,\,\,\,
%g_0 = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}
%\in {\rm{SO}}'_4(k)$$
%in any characteristic, as well as the exceptional case
%$$g'_0 = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
%\in {\rm{SO}}'_4(k)$$
%when ${\rm{char}}(k) = 2$ (with ${g'_0}^2 = 1$ by inspection).  
%For this final case, once we show that $g_0$ (not $g'_0$!) is
%in the identity component we can conclude 
%that the $k$-group ${\rm{O}}_4 = {\rm{SO}}'_4$ has at most two connected components
%(with $g'_0$ representing the nontrivial component if one exists). But we know 
%by Proposition \ref{disconnok}
%that ${\rm{O}}_{2m}$ has at least two components
%in characteristic 2.   This would then settle the connectedness assertion
%for even $n \ge 4$ when ${\rm{char}}(k) = 2$. 
%
%First we treat the case of $g_0 \in {\rm{SO}}'_4$ (in any characteristic).
%It is straightforward to check that 
%the 4-dimensional quadratic space $({\rm{Mat}}_2, \det)$ is non-degenerate;
%up to a sign, it is exactly $q_4$.  By definition of ${\rm{SL}}_2$,
%the left and right multiplication actions by ${\rm{SL}}_2$ on ${\rm{Mat}}_2$
%preserve this quadratic form.  Since these actions commute with each other, they define
%a natural homomorphism
%\begin{equation}\label{sl2sl2}
%{\rm{SL}}_2 \times {\rm{SL}}_2 \rightarrow 
% {\rm{SO}}'({\rm{Mat}}_2, \det)^0 = ({\rm{SO}}'_4)^0
% \end{equation}
%defined via $(g,g').x = gx{g'}^{-1}$, where the identification with
%${\rm{SO}}'_4$ rests on the linear isomorphism of ${\rm{Mat}}_2$ with affine 4-space via
%$(a,d,-b,c)$ to carry $\det$ over to $q_4$.  (The map (\ref{sl2sl2}) arises
%in Example \ref{so4}.) When 
%$g_0$ is thereby identified with an operation on the 4-dimensional ${\rm{Mat}}_2$, its
%matrix relative to the standard basis of ${\rm{Mat}}_2$ is the conjugate $g_0$ by
%${\rm{diag}}(1,1,-1,1)$ (which negates the lower-right $2 \times 2$ block in $g_0$).
%It is then easy to check that for $w = \left(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right) \in
%{\rm{SL}}_2(k)$, the point $(w, w^{-1})$ is carried over to $g_0$. 
%This completes the proof for even $n$.
%
%Finally, we treat ${\rm{SO}}'_3$ and the specific $g_0$ as above.  The ``conjugation'' action of
%${\rm{PGL}}_2 = \GL_2/\Gm$ on $\mathfrak{sl}_2 = {\rm{Mat}}_2^{{\rm{Tr}}=0}$ 
%leaves
%the determinant invariant, and a straightforward calculation with the actions
%on $(\begin{smallmatrix} 1 & y \\ 0 & -1 \end{smallmatrix})$
%and $(\begin{smallmatrix} 1 & 0 \\ z & -1 \end{smallmatrix})$
%shows that this action is scheme-theoretically faithful.  More specifically, 
%the coordinatization of
%$\mathfrak{sl}_2$ via $\left(\begin{smallmatrix} x & y \\ z & -x \end{smallmatrix}\right)$.
%identifies $\det$ with $-q_3$ (the sign not affecting the associated orthogonal group), 
%so we get a homomorphism 
%$${\rm{PGL}}_2 \rightarrow {\rm{SO}}'(\mathfrak{sl}_2, \det)^0 \simeq ({\rm{SO}}'_3)^0$$
%which is easily checked to have trivial scheme-theoretic kernel.  
%(This map arises in Example \ref{so3}.) 
%Thus,  it suffices to show that the image of
%this homomorphism contains $g_0$.  The point $w = 
%\left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right)   \in {\rm{PGL}}_2(k)$
%does the job since
%$$w \begin{pmatrix} x & y \\ z & -x \end{pmatrix} w^{-1} =
%\begin{pmatrix} -x & z \\ y & x \end{pmatrix}.$$
%This settles the case of odd $n$.  (In characteristic 2 the connectedness for odd $n$ 
%can also be proved in another way; see the discussion following Proposition \ref{oddconn}.)
%\end{proof}
%
%As an application of the fibral connectedness for ${\rm{SO}}'(q)$ (apart
%from characteristic 2 when $n$ is even), we shall next establish relative smoothness
%results as a consequence of smoothness properties on fibers in Theorem \ref{soo}.
%It may seem surprising that we use global notions such as fibral connectedness 
%in our proof of relative smoothness.  
%For odd $n$,
%a local analysis using the infinitesimal smoothness criterion seems to get mired in difficulties 
%over base rings such as $\Z/4\Z$ or $\Z_2$ 
%in which 2 is neither a unit nor zero, due to the failure of
%the defect subsheaf $V^{\perp} = \ker(V \rightarrow V^{\ast})$
%(consisting of $v$ such that $B_q(v,\cdot) = 0$)
%to be a subbundle of $V$ as we ``deform away from defect-1''.
%To bypass such difficulties, we will use Lemma \ref{stdq}
% to reduce smoothness problems to the consideration of
%the $\Z$-groups ${\rm{O}}_n$ and ${\rm{SO}}'_n$.  The advantage of passing to 
%a Dedekind base is that we can apply: 
%
%\begin{proposition}\label{gsmooth}
% Let $S$ be a Dedekind scheme, and $G$ an $S$-group of finite type such that 
%all fibers $G_s$ are smooth of the same dimension.  Then $G$ contains
%a unique smooth open subgroup $G^0$ whose $s$-fiber is $(G_s)^0$ for all $s \in S$.
%In particular, $G$ is smooth if its fibers are connected.
%\end{proposition}
%
%\begin{proof}
%We may assume $S$ is connected, say with generic point $\eta$.
%The smooth open subgroup $G_{\eta}^0 \subset G_{\eta}$
%then ``spreads out'' over a dense open $U \subset S$ to a smooth open subgroup 
%of $G_U$ with connected fibers.  This solves the problem over $U$, 
%and to handle the remaining finitely many closed points in $S - U$
%we may assume that $S = \Spec R$ for a discrete valuation ring $R$, say with fraction field $K$.
%We may and do remove the closed union of the non-identity components of
%the special fiber, so $G$ has connected special fiber.
%
%Let $\mathscr{G}$ denote the schematic closure in $G$ of the generic fiber $G_K$,
%so $\mathscr{G}_K =  G_K$.
%The $R$-flat $\mathscr{G} \times \mathscr{G}$ is the schematic closure
%of its generic fiber $G_K \times G_K$, so it follows that the $R$-flat 
%$\mathscr{G}$ is an $R$-subgroup of $G$. 
%This is a flat closed subscheme of $G$ with constant fiber dimension (by flatness), so
%the closed immersion $\mathscr{G}_0 \hookrightarrow G_0$ between special fibers 
%must be an isomorphism, as $G_0$ is smooth and connected
%and $\dim \mathscr{G}_0 = \dim \mathscr{G}_K = \dim G_K = \dim G_0$.
%Thus, $\mathscr{G}_0$ is smooth (as is $\mathscr{G}_K = G_K$), 
%so $\mathscr{G}$ is smooth. 
%
%The closed immersion $\mathscr{G} \hookrightarrow G$ 
%is an isomorphism on fibers, so by flatness of $\mathscr{G}$ it is an isomorphism, 
%due to Lemma \ref{flatclosed}.
%\end{proof}
%
%\begin{remark} The final assertion in Proposition \ref{gsmooth}
%is valid more generally: if $G \rightarrow S$ is a finite type
%group over any reduced noetherian scheme $S$ and if the fibers $G_s$ are smooth
%and connected of the same dimension then $G$ is smooth.
%Indeed, the problem is to verify flatness,
%and by the ``valuative criterion for flatness'' over a reduced noetherian base
%\cite[IV$_3$, 11.8.1]{ega} it suffices to check this after base change
%to discrete valuation rings, to which Proposition \ref{gsmooth} applies.
%See \cite[VI$_{\rm{B}}$, 4.4]{sga3} for a further generalization.
%\end{remark}
%
%The role of identity components in Proposition \ref{gsmooth} cannot be dropped.
%For a quasi-finite example, consider the constant
%group $(\Z/d\Z)_R$ over a discrete
%valuation ring $R$ with $d > 1$. This
%contains a reduced closed subgroup $G^{(d)}$ given by the reduced closed 
%complement of 
%the open non-identity points in the generic fiber.  The $R$-group $G^{(d)}$ 
%has \'etale fibers
%but  is not flat over $R$ (the non-identity points of the special fiber are open).
%A more interesting example is ${\rm{SO}}'_{2m}$ over $\Z_{(2)}$
%(which is the pushout of $G^{(2)}$ along the identity section $\Spec \Z_{(2)}
%\hookrightarrow {\rm{SO}}_{2m}$; see Remark \ref{225}).
%
%\begin{corollary}\label{sorel}
%Let $(V,q)$ be a non-degenerate quadratic space of rank $n \ge 2$ over a 
%scheme $S$.
%The $S$-group ${\rm{SO}}'(q)$ is smooth of relative
%dimension $n(n-1)/2$ if $n$ is odd or if $S$ is a $\Z[1/2]$-scheme 
%or an $\F_2$-scheme.
%\end{corollary}
%
%\begin{proof}
%We may work fppf-locally on $S$, so without loss of generality $q = q_n$.
%By base change, we may assume $S$ is the spectrum of $\Z$, $\Z[1/2]$, or $\F_2$.  
%Theorem \ref{soo} implies that all fibers are smooth of dimension $n(n-1)/2$,
%and Theorem \ref{connthm} gives connectedness of all fibers
%except the $\F_2$-fiber when $n$ is even.  Hence, 
%by Proposition \ref{gsmooth}, we get $\Z$-smoothness when $n$ is odd
%and $\Z[1/2]$-smoothness when $n$ is even.
%For even $n$, smoothness also holds over $\F_2$ by Theorem \ref{soo}.
%\end{proof}
%
%\begin{remark}\label{225}
%The avoidance of even $n$ in mixed characteristic with residue
%characteristic 2 in Corollary \ref{sorel}
%cannot be eliminated.  Consider $q_{2m}$ over $\Z_{(2)}$.
%The open and closed subgroup
%${\rm{SO}}_{2m} \subset {\rm{SO}}'_{2m}$ has open complement that consists of the
%non-identity component in the special fiber (see 
%Proposition \ref{sosmooth} and Theorem \ref{soo}(1)).  Thus, 
%${\rm{SO}}'_{2m} \rightarrow \Spec \Z_{(2)}$ is not an open map.
%\end{remark}

%We begin with a lemma that is useful when $\dim V$ is odd (especially for characteristic 2).

%\begin{lemma}\label{mu2}
%Let $k$ be a local ring, and $\mu_2$ the kernel of squaring on
%$\Gm$ over $k$.  
%\begin{enumerate}
%\item The only automorphism of
%$\mu_2$ as a $k$-group is the identity. 
%\item If $k$ is a field and $M$ is a finite commutative $k$-group scheme that becomes
%isomorphic to $\mu_2$ over $\overline{k}$ then
%$M$ is uniquely isomorphic to $\mu_2$ over $k$.
%\end{enumerate}
%\end{lemma}

%This lemma is nontrivial only when ${\rm{char}}(k) = 2$, as otherwise
%$\mu_2 = \Z/2\Z$ consists of two copies of $\Spec k$. 

%\begin{proof}
%The proof of (1) is an elementary calculation left to the reader.
%(It is true more generally for $\mu_p$ with $(\Z/p\Z)^{\times}$
%as the automorphism group, but once $p > 2$
%it should really be proved using some theory, such as Cartier duality.)
%For the proof of (2), the uniqueness follows from (1), and for
%the existence we can use Galois descent
%(in view of the uniqueness!) to reduce to the case $k = k_s$.
%For the existence in this case one argues very similarly to the case of tori
%as in Exercise 2 of HW2. 
%\end{proof}

The remaining task for 
${\rm{SO}}(q)$ and ${\rm{O}}(q)$ is 
to determine the functorial center if $n \ge 3$ (as the case $n \le 2$ is easy to 
analyze directly). For odd $n \ge 3$, the central $\mu_2$ in
${\rm{O}}(q)$ has trivial intersection with
${\rm{SO}}'(q)$, and hence with ${\rm{SO}}(q)$.
If $n > 2$ is even then the central $\mu_2$ is contained
in ${\rm{SO}}'(q)$, and we claim that it also
lies in ${\rm{SO}}(q)$.  In other words, for even $n$ 
we claim that the Dickson invariant 
$D_q:{\rm{O}}(q) \rightarrow (\Z/2\Z)_S$ kills
the central $\mu_2$.  It suffices
to treat the case of $q = q_n$ over $\Z$, in which case
we just need to show that the {\em only}
homomorphism of $\Z$-groups $\mu_2 \rightarrow \Z/2\Z$ is the trivial one.
By $\Z$-flatness, to prove such triviality it suffices
to check after localization to $\Z_{(2)}$.
But over the local base $\Spec \Z_{(2)}$ the scheme $\mu_2$ is connected
and thus it must be killed by a homomorphism into a constant group.

\begin{proposition}\label{findcenter} Assume $n \ge 3$.
The functorial center of ${\rm{SO}}(q)$ is represented by
$\mu_2$ in the central $\Gm \subset {\rm{GL}}(V)$ when $n$ is even,
and it is trivial when $n$ is odd.
\end{proposition} 

\begin{proof}
By Lemma \ref{stdq}, it suffices to treat $q_n$
over $S = \Spec k$ for any ring $k$.
We will use a method similar to 
the treatment of $Z_{{\rm{Sp}}_{2n}}$ in Exercise \ref{2.1}:  exhibit a specific 
torus $T$ that we show to be its own centralizer in $G := {\rm{SO}}'(q)$
(so $T$ is its own centralizer in ${\rm{SO}}(q)$) 
and then we will look for the center inside this $T$. 
We will also show that the functorial center of ${\rm{SO}}(q)$ coincides with that of $G$.


Suppose $n = 2m$, so relative to some ordered basis $\{e_1, e'_1, \dots,
e_m, e'_m\}$ we have 
$q = \sum_{i=1}^m x_i x'_i$.    In this case we identify ${\rm{GL}}_1^m$
with a $k$-subgroup $T$ of ${\rm{SO}}'(q)$ via
$$j:(t_1, \dots, t_m) \rightarrow (t_1, 1/t_1, \dots, t_m, 1/t_m).$$
The action by $T$ on $k^n = k^{2m}$ has each standard basis line
as a weight space for a collection of $2m$ fiberwise distinct characters over $\Spec k$.
Hence, $Z_{{\rm{GL}}_n}(T)$ is the diagonal torus in
${\rm{GL}}_n$, so clearly $Z_{{\rm{O}}(q_n)}(T) = T$
and hence $Z_G(T) = T$.

Next, assume $n = 2m+1$ for $m \ge 1$.  Pick a basis
$\{e_0, e_1, e'_1,\dots, e_m, e'_m\}$ relative to which
$$q = x_0^2 + \sum_{i=1}^m x_i x'_i.$$
If we define $T$ in the same way (using the span of $e_1, e'_1, \dots, e_m,  e'_m$)
then the same analysis gives the same result: $T$ is its own scheme-theoretic
centralizer in ${\rm{SO}}'(q)$.  The point is that there is no difficulty
created by $e_0$ because we are requiring the determinant to be 1.
(If we try the same argument with ${\rm{O}}(q)$ then the centralizer of $T$  is 
$\mu_2 \times T$.)  

We are now in position to identify
the center of ${\rm{SO}}'(q)$ for general $n \ge 3$.  First we assume $n \ge 4$ (i.e., $m \ge 2$).
 In terms of the ordered bases as above, consider
the automorphism $g_i$ obtained by swapping the ordered pairs 
$(e_i, e'_i)$ and $(e'_1, e_1)$ for $1 < i \le m$.  
(Such $i$ exist precisely because $m \ge 2$.)  These automorphisms $g_i$ lie in
${\rm{SO}}'(q)$ since the determinant is
$(-1) \cdot (-1) = 1$, and a point of $T$ centralizes $g_i$ if and only if
$t_1 = t_i$.  Letting $i$ vary, we conclude
that the center of ${\rm{SO}}'(q)$ is contained in the ``scalar'' subgroup
$\Gm  \hookrightarrow T$ given by $t_1 = \dots = t_m$.
This obviously holds when $n = 3$ as well. 

Letting $\lambda$ denote the common value of the $t_j$, to constrain it further we 
consider more points of ${\rm{SO}}'(q)$ that it must centralize. 
First assume $m \ge 2$.  
Consider the automorphism $f$ of $V$ which  acts on the plane
$ke_i \bigoplus k e'_i$ by the matrix
$w = \left(\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right)$
for exactly {\em two} values $i_0, i_1 \in \{1, \dots, m\}$
(and leaves all other basis vectors invariant), so
$\Det f = 1$.  
Clearly $f$ preserves $q$, so $f$ lies in
${\rm{SO}}'(q)(k)$.  But $f$-conjugation of $t \in T$ viewed in
${\rm{SO}}'(q)$ (or ${\rm{GL}}(V)$) swaps the entries $t_i$ and $1/t_i$ for
$i \in \{i_0, i_1\}$.  Thus,  the centralizing property
forces $\lambda \in \mu_2$, so if $n \ge 4$ is even then $Z_{{\rm{SO}}'(q)}$ is contained in  
the central $\mu_2$ in ${\rm{GL}}(V)$. 
This inclusion is an equality for even $n \ge 4$, since 
the central $\mu_2 \subset \GL(V)$ is contained 
in ${\rm{SO}}'(q)$ for even $n$.
If instead $n \ge 4$ is odd then ${\rm{SO}}'(q) = {\rm{SO}}(q)$ by definition and we have shown that
its center viewed inside ${\rm{GL}}_n$ lies in the subgroup $\mu_2 \hookrightarrow \Gm^n$ defined by the inclusion
$\zeta \mapsto (1, \zeta, \zeta, \dots \zeta)$.   The automorphism
$$(x_0, x_1, x_2, x_3, \dots, x_n) \mapsto (x_0 + x_1, x_1, -2x_0 -x_1 + x_2, x_3, \dots, x_n)$$
arising from ${\rm{SO}}(x_0^2 + x_1 x_2)$ lies in ${\rm{SO}}(q)$ and centralizing this forces $\zeta = 1$, so the center is trivial. 
This completes our analysis for odd $n \ge 5$.  

Next, we prove that 
$Z_{{\rm{SO}}_3} = 1$.   The action 
of ${\rm{PGL}}_2$ on $\mathfrak{sl}_2$ via conjugation 
defines an isomorphism 
${\rm{PGL}}_2 \simeq {\rm{SO}}_3$; see
the self-contained calculations in Example \ref{so3}.
By Exercise \ref{2.1}(ii) the scheme-theoretic
center of ${\rm{PGL}}_r$ is trivial for any $r \ge 2$
(and for ${\rm{PGL}}_2$ it can be verified by direct calculation), so ${\rm{SO}}_3$
has trivial center.

We have settled the case of odd $n \ge 3$, and for even $n \ge 4$ we have proved
that ${\rm{SO}}'(q)$ has functorial center $\mu_2$ that also lies in ${\rm{SO}}(q)$.    It remains to show,
assuming $n \ge 4$ is even, 
that the functorial center of ${\rm{SO}}(q)$ is no larger than this $\mu_2$.
We may and do assume $q = q_n$.
The torus $T$ constructed above in ${\rm{SO}}'_n$ 
lies in the open and closed subgroup ${\rm{SO}}_n$
for topological reasons, and $Z_{{\rm{SO}}_n}(T) = T$
since $T$ has been shown to be its own centralizer in ${\rm{SO}}'_n$.
Thus, it suffices to show that the central $\mu_2$ is the kernel
of the adjoint action of $T$ on ${\rm{Lie}}({\rm{SO}}_n) = {\rm{Lie}}({\rm{O}}_n)$.
The determination of the weight space decomposition for $T$ acting on 
${\rm{Lie}}({\rm{O}}_n)$ for even $n$ is classical, so the kernel is seen to be
the diagonal $\mu_2$ for such $n$.
\end{proof}

\begin{corollary}\label{centero} For $n \ge 2$, the functorial center of ${\rm{O}}(q)$
is represented by the central $\mu_2$.
\end{corollary}

\begin{proof}
The case $n = 2$ is handled directly, so assume $n \ge 3$. 
If $n$ is odd then the identification ${\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$
yields the result since ${\rm{SO}}(q)$ has trivial functorial center for such $n$.
Now suppose that $n$ is even.   In this case the open and closed subgroup 
${\rm{SO}}(q)$ contains the central $\mu_2$ as its functorial center.
To prove that $\mu_2$ is the functorial center 
of ${\rm{O}}(q)$ we again pass to the case $q = q_n$. 
 It suffices to check that the diagonal torus $T$ in ${\rm{SO}}_n$
is its own centralizer in ${\rm{O}}_n$, and this was shown in the proof
of Proposition \ref{findcenter}.  
\end{proof}

\begin{proposition}\label{typeBD}
For $m \ge 1$, the smooth affine $\Z$-group ${\rm{SO}}_{2m+1}$
is adjoint semisimple and it contains a split maximal torus $T \subset {\rm{SO}}_{2m+1}$ 
defined by 
$$(t_1,\dots,t_m) \mapsto {\rm{diag}}(1,t_1,1/t_1,\dots,t_m,1/t_m).$$
The root system $\Phi({\rm{SO}}_{2m+1},T)$ is ${\rm{B}}_m$.

For $m \ge 2$, the smooth affine $\Z$-group ${\rm{SO}}_{2m}$
is semisimple  and it contains a split maximal torus $T \subset {\rm{SO}}_{2m}$
defined by
$$(t_1,\dots,t_m) \mapsto {\rm{diag}}(t_1,1/t_1,\dots,t_m,1/t_m).$$
The diagonal $\mu_2 \subset T$ is the schematic center of ${\rm{SO}}_{2m}$,
and $\Phi({\rm{SO}}_{2m},T)$ is ${\rm{D}}_m$.
\end{proposition}

We use the convention that ${\rm{B}}_1 = {\rm{A}}_1$ and ${\rm{D}}_2 = {\rm{A}}_1 \times
{\rm{A}}_1$. 

\begin{proof}
Let $n = 2m+1$ and $2m$ in these respective cases, so $n \ge 3$ and we are studying
${\rm{SO}}_n \subset {\rm{GL}}_n$.  The smoothness of
${\rm{SO}}_n$ follows from Theorem \ref{yusmooth}
(as we have noted immediately after the statement of Theorem \ref{goodgp}), 
and the fibral connectedness is Proposition \ref{conneven} for even $n$
and Proposition \ref{smodd} for odd $n$.
The structure of the center is given by Proposition \ref{findcenter}.
Clearly $T$ is a split torus in ${\rm{SO}}_n$, and 
its maximality on geometric fibers was shown in the proof of Proposition \ref{findcenter}.

The remaining problem is to show that over an algebraically closed
field $k$ of any characteristic (including characteristic 2), the smooth connected affine 
group ${\rm{SO}}_n$ is semisimple
with the asserted type for its root system.  The cases $n = 3, 4$ can be 
handled by direct arguments (given in a self-contained manner in Examples \ref{so3} and \ref{so4}),
so we may restrict attention to $n \ge 5$ (i.e., $m \ge 2$ for odd $n$ and $m \ge 3$ for even $n$). 
Since smoothness
and connectedness are known, as is the dimension, 
it is straightforward to directly compute the weight
space decomposition for $T$ on ${\rm{Lie}}({\rm{SO}}_n)$
and so to verify reductivity
by the general technique in Exercise \ref{redexer}(i), 
using constructions given in Exercise \ref{vuex}.  This method
also shows that ${\rm{X}}(T)_{\Q}$ is spanned by the roots, so
${\rm{SO}}_n$ is semisimple, and an inspection of the roots
shows that the root system is of the desired type (depending on the parity of $n \ge 5$).
These calculations are left to the reader. 
\end{proof}
 \index{special orthogonal group!properties|)}

As an application of the
basic properties of orthogonal and special orthogonal group schemes, 
we now prepare to prove an interesting fact in the {\em global} theory of quadratic forms.
Observe that for any scheme $S$, the group ${\rm{Pic}}(S)$
acts on the set of isomorphism classes of non-degenerate
line bundle-valued quadratic forms $(V,L,q)$ 
with $V$ of a fixed rank $n \ge 1$: the class of a line bundle $L'$ on $S$
carries the isomorphism class of $q:V \rightarrow L$ to the isomorphism class
of $q_{L'}: V \otimes L' \rightarrow L \otimes {L'}^{\otimes 2}$
(defined by $v \otimes \ell' \mapsto q(v) \otimes {\ell'}^{\otimes 2}$). 
We seek to understand the orbits under this action,
called {\em projective similarity classes}. 
In the special case ${\rm{Pic}}(S) = 1$ we can use the language of quadratic spaces $(V,q)$
and this becomes the consideration of similarity classes: 
$(V',q')$ is in the same {\em similarity class} as $(V,q)$
\index{quadratic space!similarity class}
if there is a linear isomorphism $f:V' \simeq V$ such that $q \circ f = u q'$ for a unit $u$ on $S$.
(Projective similarity classes can be interpreted
via algebras with involution; see \cite[3.1]{auel} for literature references.)

We wish to classify  projective similarity classes in terms of another invariant. 
If ${\rm{Pic}}(S) = 1$ this amounts to classifying
similarity classes of quadratic spaces over $S$, and over a field $k$ it is an interesting
arithmetic problem (different from classifying isomorphism classes of quadratic
spaces over $k$ when $k^{\times}$ contains non-squares).
Under the evident identification ${\rm{GL}}(V \otimes L') = {\rm{GL}}(V)$
it is easy to check that ${\rm{SO}}(q_{L'}) = {\rm{SO}}(q)$,
so the isomorphism class of ${\rm{SO}}(q)$ is the same across
all members of a projective similarity class. Our aim (achieved
in Proposition \ref{projsim}) is to show that
if $n \ne 2$ then the isomorphism class of ${\rm{SO}}(q)$
{\em determines} the projective similarity class of $(V,L,q)$; the case
$n = 2$ exhibits more subtle behavior, as we shall see.

To analyze the general problem, it is convenient to introduce an appropriate group:
the {\em orthogonal similitude group}
\index{orthogonal group!similitude variant}
\index{ aaagoq@${\rm{GO}}(q)$}
$${\rm{GO}}(q) \subset {\rm{GL}}(V)$$ is 
the closed subgroup of linear automorphisms of $V$ preserving
$q:V \rightarrow L$ up to an automorphism of $L$. (The automorphism
of $L$ is uniquely determined since $q$ is fiberwise non-zero.)   
The group
${\rm{GO}}(q)$ contains ${\rm{O}}(q)$ and the central $\Gm$, with 
${\rm{GO}}(q)/\Gm = {\rm{O}}(q)/\mu_2$ as fppf quotient sheaves.  This $S$-affine quotient
group is denoted ${\rm{PGO}}(q)$ and is called the {\em projective similitude group}. 
\index{orthogonal group!projective similitude variant}
\index{ aaapgoq@${\rm{PGO}}(q)$}

Inside ${\rm{GL}}(V)$, the intersection of  $\mathbf{G}_{\rm{m}}$
and ${\rm{SO}}(q)$ is trivial if $n$ is odd and is $\mu_2$ if $n$ is even
(by Theorem \ref{goodgp} if $n \ne 2$ and by inspection if $n = 2$), so
the fppf subgroup sheaf of ${\rm{GO}}(q)$
generated by ${\rm{SO}}(q)$ and $\mathbf{G}_{\rm{m}}$
is identified with $\mathbf{G}_{\rm{m}} \times {\rm{SO}}(q)$
for odd $n$ and with $(\mathbf{G}_{\rm{m}} \times {\rm{SO}}(q))/\mu_2$ for even $n$;
this smooth $S$-subgroup of ${\rm{GO}}(q)$ is denoted ${\rm{GSO}}(q)$.
By Theorem \ref{goodgp}, for
all $n \ge 3$ the group ${\rm{GSO}}(q)$ is smooth with
\index{ aaagsoq@${\rm{GSO}}(q)$}
center $\mathbf{G}_{\rm{m}}$ and the quotient ${\rm{GSO}}(q)/\mathbf{G}_{\rm{m}}$ is the adjoint semisimple 
${\rm{SO}}(q)/Z_{{\rm{SO}}(q)}$ (so ${\rm{GSO}}(q)/\Gm$ is sometimes denoted ${\rm{PGSO}}(q)$). 
\index{ aaapgsoq@${\rm{PGSO}}(q)$}


For odd $n$ the equality ${\rm{SO}}(q) \times \mu_2 = {\rm{O}}(q)$ implies
${\rm{GSO}}(q) = {\rm{GO}}(q)$. For even $n$, ${\rm{GSO}}(q)$ is an open and closed
normal $S$-subgroup of ${\rm{GO}}(q)$ satisfying ${\rm{GO}}(q)/{\rm{GSO}}(q) = {\rm{O}}(q)/{\rm{SO}}(q) = 
(\Z/2\Z)_S$.  This follows immediately from the description of ${\rm{PGO}}(q)$
as ${\rm{O}}(q)/\mu_2$ and the observation (for even $n$)
that the central $\mu_2$ in ${\rm{O}}(q)$ lies in ${\rm{SO}}(q)$.
Put another way, for even  $n$, the isomorphism of fppf group sheaves ${\rm{GO}}(q)/{\rm{GSO}}(q) \simeq (\Z/2\Z)_S$
defines a quotient map of $S$-group schemes 
\begin{equation}\label{Gdickson}
{\rm{GD}}_q:{\rm{GO}}(q) \rightarrow (\Z/2\Z)_S
\end{equation} whose open and closed
kernel is ${\rm{GSO}}(q)$.

\begin{remark}\label{bettergso} For even $n$, the quotient map ${\rm{GD}}_q$ in (\ref{Gdickson})
clearly extends $D_q:{\rm{O}}(q) \rightarrow (\Z/2\Z)_S$.
We now give a ``Clifford'' construction of this map on ${\rm{GO}}(q)$.
 The main point is that the ${\rm{O}}(q)$-action on the algebra 
${\rm{C}}_0(V,L,q)$ from Remark \ref{dicksontriple} (which exists even in 
the absence of ${\rm{C}}(V,L,q)$ for line bundle-valued $q$) naturally extends to 
a ${\rm{GO}}(q)$-action, and similarly on the left
${\rm{C}}_0(V,L,q)$-module ${\rm{C}}_1(V,L,q)$
(extending the natural ${\rm{GO}}(q)$-action on the subbundle
$V \subset {\rm{C}}_1(V,L,q)$).

To build this ${\rm{GO}}(q)$-action, first observe that if $g \in {\rm{GO}}(q)$
has action on $V$ that intertwines with the action on $L$ by a unit $u$ then 
$g$ induces an isomorphism $(V,q,L) \simeq (V,uq,L)$ that is the {\em identity} on $L$.
Thus, if $L$ is trivial then $g$ induces an isomorphism ${\rm{C}}(V,q) \simeq {\rm{C}}(V,uq)$,
so if moreover $u = a^2$  then composing this isomorphism with the isomorphism ${\rm{C}}(V,uq) \simeq
{\rm{C}}(V,q)$ defined by $v \mapsto av$ yields
a composite isomorphism $[g]:{\rm{C}}_0(V,q) \simeq {\rm{C}}_0(V,q)$
that is independent of $a$ and hence {\em multiplicative} in such $g$.  Feeding this into
the descent procedure used in Remark \ref{dicksontriple} (to handle the possibility that 
$u$ is not a square on $S$ and $L$ may not be
globally trivial), we thereby obtain a natural ${\rm{GO}}(q)$-action on ${\rm{C}}_0(V,L,q)$
without any triviality requirement on $L$. Similarly we get a compatible
${\rm{GO}}(q)$-action on the ${\rm{C}}_0(V,L,q)$-module ${\rm{C}}_1(V,L,q)$
that restricts to the usual action on the subbundle $V$.  

These actions visibly extend
the ${\rm{O}}(q)$-actions, so in particular we get an action of 
${\rm{GO}}(q)$ on the center $Z_q$ of ${\rm{C}}_0(V,L,q)$ that extends 
the ${\rm{O}}(q)$-action on $Z_q$.  Thus, we obtain an $S$-homomorphism 
${\rm{GO}}(q) \rightarrow {\rm{Aut}}_{Z_q/S} = (\Z/2\Z)_S$
extending the Dickson invariant $D_q$ on ${\rm{O}}(q)$.
This recovers (\ref{Gdickson}) because 
${\rm{GSO}}(q)$ has no nontrivial $S$-homomorphism to $(\Z/2\Z)_S$ for fibral connectedness reasons
and $(\Z/2\Z)_S$ has no nontrivial automorphism.
A consequence of the identification ${\rm{GSO}}(q) = \ker {\rm{GD}}_q$ is that 
${\rm{GSO}}(q)$ acts $Z_q$-linearly on ${\rm{C}}_1(V,L,q)$. 
\end{remark}

\begin{lemma}\label{pgoq} The $S$-affine $S$-groups ${\rm{GO}}(q)$ and ${\rm{PGO}}(q)$ are smooth,
with geometric fibers that are connected when $n$ is odd
and have two connected components when $n$ is even. In general, if
$n \ge 1$ is odd then 
${\rm{PGO}}(q)$ coincides with the adjoint group ${\rm{SO}}(q)$ and if $n \ne 2$ is even then ${\rm{PGO}}(q)$
is an extension of $(\Z/2\Z)_S$ by the adjoint quotient ${\rm{SO}}(q)/\mu_2 = {\rm{PGSO}}(q)$. 
\end{lemma}

\begin{proof}
Since ${\rm{GO}}(q)$ is an fppf $\mathbf{G}_{\rm{m}}$-torsor over ${\rm{PGO}}(q)$,
it suffices to study ${\rm{PGO}}(q) = {\rm{O}}(q)/\mu_2$.
If $n$ is odd then this is ${\rm{SO}}(q)$ since the determinant
on ${\rm{O}}(q)$  splits off the central $\mu_2$ as a direct
factor for such $n$, and if $n$ is even then ${\rm{O}}(q)/\mu_2$ is an extension
of $(\Z/2\Z)_S$ by ${\rm{SO}}(q)/\mu_2$. Theorem \ref{goodgp} provides 
the required properties of ${\rm{SO}}(q)$ for $n \ge 3$ to complete the proof.
\end{proof}

Our interest  in the orthogonal similitude group is twofold.  First, 
for the smooth group ${\rm{GO}}_n = {\rm{GO}}(q_n)$, the \'etale
\index{ aaagon@${\rm{GO}}_n$}
cohomology set ${\rm{H}}^1(S_{\et}, {\rm{GO}}_n)$
is the set of isomorphism classes of rank-$n$ non-degenerate line bundle-valued quadratic forms over $S$.
To prove this, observe that any $q$ is an fppf form of $q_n$ (by Lemma \ref{stdq}),
so $(V,L,q)$ is an fppf form of $(\calO_S^n, \calO_S, q_n)$.  Thus, the $S$-scheme 
${\rm{Isom}}((\calO_S^n,\calO_S,q_n), (V,L,q))$ is an fppf right ${\rm{GO}}_n$-torsor over
$S$, and by the smoothness of ${\rm{GO}}_n$ its torsors
for the fppf topology are actually trivialized \'etale-locally on the base
(i.e., the Isom-scheme inherits smoothness from ${\rm{GO}}_n$, so
it admits sections \'etale-locally on $S$).  The assignment of this Isom-scheme
therefore defines the desired bijection (since ${\rm{GO}}_n$ represents
the automorphism functor of $(\calO_S^n, \calO_S, q_n)$). 
In \cite{auel} this bijection is used to systematically
transfer properties of ${\rm{GO}}(q)$ (such as short exact sequences) into global
structural results concerning quadratic forms valued in line bundles.

The second reason for our interest in orthogonal similitude groups 
is that the central subgroup $\mathbf{G}_{\rm{m}}$ inside ${\rm{GO}}_n$
induces an action by ${\rm{H}}^1(S_{\et},\mathbf{G}_{\rm{m}}) = {\rm{Pic}}(S)$
on ${\rm{H}}^1(S_{\et}, {\rm{GO}}_n)$ that is precisely
the natural twisting action of ${\rm{Pic}}(S)$ on  the set of isomorphism classes of rank-$n$
non-degenerate line bundle-valued quadratic forms over $S$.
Thus, these orbits are the projective similarity classes, so we seek to classify
the orbits of the ${\rm{H}}^1(S_{\et},\mathbf{G}_{\rm{m}})$-action on
${\rm{H}}^1(S_{\et}, {\rm{GO}}_n)$.  This is a useful viewpoint because of 
an interesting interpretation of the quotient group 
${\rm{PGO}}(q) = {\rm{GO}}(q)/\mathbf{G}_{\rm{m}}$ that is due to Dieudonn\'e 
(over fields with characteristic not equal to 2) and which we now recall.

Since ${\rm{GO}}(q)$ is generated for the fppf topology by
${\rm{O}}(q)$ and the central subgroup $\Gm$, the normality of
${\rm{SO}}(q)$ in ${\rm{O}}(q)$ implies that ${\rm{SO}}(q)$ is normal in ${\rm{GO}}(q)$.
The conjugation action of ${\rm{GO}}(q)$ on its normal subgroup ${\rm{SO}}(q)$ defines a 
homomorphism of group functors
${\rm{GO}}(q) \rightarrow \underline{\rm{Aut}}_{{\rm{SO}}(q)/S}$.
Since the maximal central torus in
${\rm{SO}}(q)$ is trivial when $n \ge 3$
and has rank 1 when $n = 2$,
by Theorem \ref{autscheme} the automorphism functor
of ${\rm{SO}}(q)$ is represented by a smooth $S$-affine $S$-group ${\rm{Aut}}_{{\rm{SO}}(q)/S}$  that is
an extension of a finite \'etale group by the adjoint quotient 
${\rm{SO}}(q)/Z_{{\rm{SO}}(q)}$.  A geometric fiber of this \'etale group
is the outer automorphism group ${\rm{Out}}({\rm{SO}}_n)$ of ${\rm{SO}}_n$.
Since the central $\mathbf{G}_{\rm{m}}$ in ${\rm{GO}}(q)$ acts trivially on
${\rm{SO}}(q)$, we arrive at an $S$-homomorphism
$$h_q:{\rm{PGO}}(q) \rightarrow {\rm{Aut}}_{{\rm{SO}}(q)/S}.$$

\begin{lemma}[Dieudonn\'e]\label{hqisom} If $n \ne 2$ then   $h_q$ is an isomorphism. 
\end{lemma}

See \cite{classique} for a classical treatment away from characteristic 2,
and \cite[VI, 26.12, 26.15, 26.17]{bookinv} for a treatment over all fields.
(Strictly speaking, for even $n \ge 4$,
\cite{bookinv} considers automorphisms of the adjoint quotient ${\rm{SO}}(q)/\mu_2$ rather than
of ${\rm{SO}}(q)$ itself.  This has no effect on the automorphism group,
due to the structure of the outer automorphism group for even $n \ge 4$, as we shall see in the proof below.)

\begin{proof}
Since $h_q$ is a map between smooth $S$-affine $S$-groups, it suffices
to prove the isomorphism property on geometric fibers.
Hence, we may assume $S = \Spec(k)$ for an algebraically closed field $k$.
In particular, $q = q_n$. The case $n = 1$ is trivial, so assume $n \ge 3$.

First consider odd $n \ge 3$, so 
${\rm{SO}}(q)$ is semisimple with trivial center 
and ${\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$.  The assertion in this case
is that ${\rm{SO}}(q)$ is its own automorphism scheme.  Such an equality holds
for any connected semisimple $k$-group with trivial center and no nontrivial
diagram automorphisms (by Theorem \ref{autscheme}).
For $m \ge 1$ we know that ${\rm{SO}}_{2m+1}$ has type ${\rm{B}}_m$
and trivial center (see Proposition \ref{findcenter}), and by inspection
(treating $m = 1$ separately) 
the ${\rm{B}}_m$ diagram has no nontrivial diagram automorphisms. 

Now assume $n$ is even, so $n = 2m$ with
$m \ge 2$ and ${\rm{O}}(q)/{\rm{SO}}(q) = \Z/2\Z$.  
The group ${\rm{SO}}_n$
is semisimple of type ${\rm{D}}_m$
(with ${\rm{D}}_2 = {\rm{A}}_1 \times {\rm{A}}_1$),
its center $Z_{{\rm{SO}}(q)}$ is equal to 
the central $\mu_2$ inside ${\rm{O}}(q)$ (by Proposition \ref{findcenter}), and ${\rm{O}}(q)/\mu_2$ is
an extension of ${\rm{O}}(q)/{\rm{SO}}(q) = \Z/2\Z$ by 
${\rm{SO}}(q)/Z_{{\rm{SO}}(q)}$.  By Theorem \ref{autscheme}, $h_q$ identifies
${\rm{SO}}(q)/Z_{{\rm{SO}}(q)}$ with ${\rm{Aut}}_{{\rm{SO}}(q)/k}^0$
and moreover the component group of ${\rm{Aut}}_{{\rm{SO}}(q)/k}$ (which is visibly
${\rm{Out}}({\rm{SO}}_{2m})$) is identified
with the automorphism group of the based root datum attached to ${\rm{SO}}_{2m}$.

Any point in ${\rm{O}}(q)(k) - {\rm{SO}}(q)(k)$
acts on ${\rm{SO}}(q)(k)$ by a non-inner
automorphism (since $Z_{{\rm{O}}(q)} = \mu_2 \subset {\rm{SO}}(q)$; see Corollary \ref{centero}),
so $\ker h_q = 1$ and our problem is to show that 
$\#{\rm{Out}}({\rm{SO}}_{2m}) \le 2$. 
But ${\rm{Out}}({\rm{SO}}_{2m})$ is the automorphism
group of the based root datum for the semisimple group ${\rm{SO}}_{2m}$,
so it is a subgroup 
of the automorphism group $\Gamma_m$ of the ${\rm{D}}_m$ diagram.
By inspection $\#\Gamma_m  = 2$ for all $m \ge 2$ except for $m = 4$,
so we are done except if $m = 4$.

Finally, consider the case $m = 4$, so $\Gamma_m \simeq \mathfrak{S}_3$ has order 6.
We just have to rule out the possibility
that the action of the entire group $\Gamma_4$ on the pinned simply connected
central cover $G$ of ${\rm{SO}}_8$ descends to an action on ${\rm{SO}}_8$.
For any $m \ge 2$, the center of ${\rm{SO}}_{2m}$ has order 2 and the fundamental group $\Pi_m$ of 
the ${\rm{D}}_m$ root system has order 4, so 
the simply connected central cover of ${\rm{SO}}_{2m}$ has degree $4/2 = 2$ 
over ${\rm{SO}}_{2m}$.  Hence,  $\ker(G \twoheadrightarrow {\rm{SO}}_8)$
is a subgroup of $Z_G$ of order 2 and we just have to show that the $\Gamma_4$-action
on $Z_G$ does not preserve this subgroup.  
The Cartier dual of $Z_G$ is the fundamental group $\Pi_m$ that 
is $\Z/2 \times \Z/2\Z$ for even $m$.  Thus, it suffices
to observe by inspection that the action by $\Gamma_4 = \mathfrak{S}_3$ on the
2-dimensional $\F_2$-vector space $\Pi_4$ is transitive
on the set of three $\F_2$-lines, so it does not preserve any of these lines.
\end{proof}

The isomorphism in Lemma \ref{hqisom} is the key input into the proof of our desired result away from rank 2:

\begin{proposition}\label{projsim} Let $(V,L,q)$ be a non-degenerate line bundle-valued quadratic form with $V$  
of rank $n \ne 2$ over a scheme $S$.
The projective similarity class of $(V,L,q)$ 
is determined by the isomorphism class of the $S$-group ${\rm{SO}}(q)$.
\end{proposition}

\begin{proof}
If $n = 1$ then ${\rm{SO}}(q) = 1$
and so we need to prove that there is a single projective similiarity class.
Since $V$ is a line bundle, we can twist
by its dual to arrive at a non-degenerate line bundle-valued quadratic form $(\calO_S,L',q')$.
The non-degeneracy implies that $q'(1)$ is a trivializing section of $L'$,
under which $q'$ becomes $x \mapsto x^2$.  Hence, for $n = 1$ 
there is indeed only one projective similarity class. 

Now we assume $n \ge 3$.
It is straightforward to check that the induced map
$${\rm{H}}^1(S_{\et},{\rm{GO}}_n) \rightarrow {\rm{H}}^1(S_{\et}, {\rm{Aut}}_{{\rm{SO}}_n/S})$$
carries the ${\rm{Pic}}(S)$-orbit of $(V,L,q)$ to the isomorphism class of the \'etale form
${\rm{SO}}(q)$ of ${\rm{SO}}_n$. Thus, our problem is to show that 
the fibers of this map are precisely the ${\rm{Pic}}(S)$-orbits. 

By Lemma \ref{hqisom}, since $n \ge 3$ we obtain an exact sequence of smooth $S$-affine $S$-groups 
$$1 \rightarrow \mathbf{G}_{\rm{m}} \rightarrow {\rm{GO}}(q) \rightarrow
{\rm{Aut}}_{{\rm{SO}}(q)/S} \rightarrow 1.$$
Consider the induced map of pointed sets
$$f_q:{\rm{H}}^1(S_{\et},{\rm{GO}}(q)) \rightarrow {\rm{H}}^1(S_{\et},{\rm{Aut}}_{{\rm{SO}}(q)/S}).$$
As we saw in the discussion preceding Lemma \ref{hqisom},
for any rank-$n$ non-degenerate line bundle-valued quadratic form $(V',L',q')$ over $S$,
the scheme ${\rm{Isom}}(q',q)$ of isomorphisms 
from $(V',L',q')$ to $(V,L,q)$ is an \'etale left ${\rm{GO}}(q)$-torsor
whose isomorphism class over $S$ determines
the isomorphism  class of $(V',L',q')$.  Hence, the source of
$f_q$ is the set of isomorphism classes of such $(V',L',q')$ of rank $n$ 
over $S$.  Likewise, the target of $f_q$
is the set of \'etale forms of ${\rm{SO}}(q)$ as an $S$-group.

The map $f_q$ carries the \'etale 
${\rm{GO}}(q)$-torsor ${\rm{Isom}}(q',q)$ to the 
class of ${\rm{SO}}(q')$ as an \'etale form of ${\rm{SO}}(q)$.
This proves that $\ker f_q$ is the set of isomorphism classes of those $(V',L',q')$
for which ${\rm{SO}}(q') \simeq {\rm{SO}}(q)$ as $S$-groups.
But $\ker f_q$ is the image of ${\rm{H}}^1(S_{\rm{fppf}},\mathbf{G}_{\rm{m}}) = {\rm{Pic}}(S)$
induced by the central inclusion $\mathbf{G}_{\rm{m}} \rightarrow {\rm{GO}}(q)$.
This image is the ${\rm{Pic}}(S)$-orbit of (the distinguished point) 
$(V,L,q)$ for the natural action
of ${\rm{Pic}}(S)$ on the set of isomorphism classes of line bundle-valued quadratic forms, 
which is to say the projective similarity class of $(V,L,q)$, so we are done.
\end{proof}

We conclude \S\ref{connZ} by considering the remaining case $n = 2$,
which exhibits entirely different behavior. 
In particular, we will see that usually the isomorphism class of ${\rm{SO}}(q)$
badly fails to determine the projective similarity class of $(V,L,q)$
(except when $S$ is local).
By inspection (see Example \ref{so2}), 
${\rm{GO}}_2 = (\Z/2\Z)_S \ltimes {\rm{GSO}}_2$ 
with ${\rm{GSO}}_2$ a torus of rank 2 
that contains two split rank-1 subtori
having intersection $\mu_2$: the torus ${\rm{SO}}_2$
and the ``scalar'' torus $\mathbf{G}_{\rm{m}} \subset {\rm{GL}}_2$. 
%
%In particular, the scalar $\mathbf{G}_{\rm{m}}$ splits off as a direct
%factor of ${\rm{GSO}}_2$ (as for any split subtorus of a split torus over a connected
%normal noetherian base, such as over $\Spec \Z$). 
%Clearly ${\rm{Aut}}_{{\rm{SO}}_2/S} = (\Z/2\Z)_S$ and the action map
%$h_{q_2}:{\rm{GO}}_2 \rightarrow {\rm{Aut}}_{{\rm{SO}}_2/S} = (\Z/2\Z)_S$
%is the quotient map modulo ${\rm{GSO}}_2$.  This is very far from an isomorphism,
%in contrast with Lemma \ref{hqisom}.  However, if ${\rm{Pic}}(S) = 1$
%then ${\rm{H}}^1(h_{q_2})$ {\em is} an isomorphism
%even though $h_{q_2}$ is not (this injectivity when ${\rm{Pic}}(S) = 1$ requires some
%care, since ${\rm{GO}}_2$ is not commutative), so the similitude class of a non-degenerate
%binary quadratic form $(V,q)$ over $S$ is determined by 
%the isomorphism class of ${\rm{SO}}(q)$ when ${\rm{Pic}}(S) = 1$.
%For example, if $S = \Spec(R)$ for a local ring $R$
%(such as a field) then Proposition \ref{projsim} is also valid over $S$ when $n = 2$.
%(A nice alternative interpretation using discriminants
%when ${\rm{Pic}}(S) = 1$ is given by the self-contained Proposition \ref{rzq}.)
Our description of ${\rm{GO}}_2$ shows that in general if
$n = 2$ then the following properties hold: ${\rm{GSO}}(q)$ is a torus of rank 2
that coincides with ${\rm{SO}}(q) \times^{\mu_2} \mathbf{G}_{\rm{m}}$, the group 
${\rm{SO}}(q)$ is a rank-1 torus whose automorphism scheme is uniquely
isomorphic to $(\Z/2\Z)_S$, the quotient ${\rm{PGSO}}(q)$ of ${\rm{GSO}}(q)$ is also a rank-1 torus, 
and the map ${\rm{GO}}(q) \rightarrow {\rm{Aut}}_{{\rm{SO}}(q)/S} = (\Z/2\Z)_S$
induced by conjugation 
is the quotient modulo ${\rm{GSO}}(q)$.  In particular, it coincides with the enhanced Dickson invariant
${\rm{GD}}_q$ from (\ref{Gdickson}) and Remark \ref{bettergso} (for $n=2$).

\index{special orthogonal group!binary case|(}
\begin{proposition}\label{rzq} Let $(V,L,q)$ be non-degenerate of rank $2$.
\begin{enumerate}
\item There is a natural isomorphism ${\rm{GSO}}(q) \simeq {\rm{R}}_{Z_q/S}(\mathbf{G}_{\rm{m}})$
extending the natural inclusion on $\mathbf{G}_{\rm{m}}$ and 
under which ${\rm{SO}}(q)$ is identified with the group of norm-$1$ units. 
Moreover, the coordinate ring over $\calO_S$
of the finite \'etale zero scheme $(q=0)  \subset \mathbf{P}(V^{\ast})$
is naturally isomorphic to $Z_q$.
\item The class 
of ${\rm{SO}}(q)$ in ${\rm{H}}^1(S_{\et}, {\rm{Aut}}_{{\rm{SO}}_2/S}) =
{\rm{H}}^1(S_{\et},\Z/2\Z)$ corresponds to
the \'etale double cover $Z_q$.  In particular, for non-degenerate $(V,L,q)$ and $(V',L',q')$
of rank $2$, ${\rm{SO}}(q') \simeq {\rm{SO}}(q)$ if and only if 
$Z_q \simeq Z_{q'}$. 
\end{enumerate}
\end{proposition}

This result is due to Kneser \cite[\S6, Prop.\,2]{kneser}
(also see \cite[6.1]{bk}, \cite[5.2]{auel}).
Note that the existence of an abstract $S$-group isomorphism as in (1) 
is not Zariski-local on $S$. 
Thus, the case of general $S$ does not formally follow from the case of affine $S$.

\begin{proof}
Granting (1), let us deduce (2).
The second assertion in (2) reduces immediately to the first.
For the first assertion in (2) we use the canonical identification of ${\rm{SO}}(q)$ as a norm-1
torus in (1) to reduce to showing that a degree-2 finite \'etale cover $E \rightarrow S$
is uniquely determined up to unique isomorphism by the norm-1
subtorus $T_E$ inside ${\rm{R}}_{E/S}(\mathbf{G}_{\rm{m}})$.  More precisely,
if $E' \rightarrow S$ is another such cover then we claim that the natural map
${\rm{Isom}}_S(E',E) \rightarrow {\rm{Isom}}_{S\mbox{-}{\rm{gp}}}(T_{E'},T_E)$
is bijective and that every rank-1 torus $T$ over $S$ arises in the form $T_E$ for some
$E$.  For $E = S \coprod S$
the torus $T_E$ is the ``$\Gm$-hyperbola'' of points $(t,1/t)$ inside $\Gm^2$. 
 By descent theory, it suffices to prove the bijectivity result for isomorphisms
when $E$ and $E'$ are split covers.   Thus, 
upon identifying the hyperbola 
$T_{S \coprod S}$ with $\Gm$ via projection to the first factor, it suffices to check that the natural map
${\rm{Aut}}_S(S \coprod S) \rightarrow {\rm{Aut}}_{S\mbox{-}{\rm{gp}}}(\Gm)$ is bijective.
By ``spreading out'' arguments we may assume $S$ is local, so 
both automorphism groups have size 2 and the bijectivity is obvious.  

To prove the first assertion in (1), note that since $V$ has rank 2, the subbundle inclusion
$V \hookrightarrow {\rm{C}}_1(V,L,q)$ is an isomorphism.
Likewise, the $\calO_S$-algebra ${\rm{C}}_0(V,L,q)$ has rank 2
and so coincides with its quadratic \'etale center $Z_q$.
Left multiplication by $Z_q = {\rm{C}}_0(V,L,q)$
must preserve ${\rm{C}}_1(V,L,q) = V$ and thereby makes $V$ into
a $Z_q$-module.  As such, $V$ is an invertible $Z_q$-module
due to the general invertibility (Zariski-locally on $S$) of ${\rm{C}}_1(V,L,q)$ as a left
${\rm{C}}_0(V,L,q)$ for any even rank (see Proposition \ref{cliffstr}).
In Remark \ref{bettergso} we defined
compatible actions of ${\rm{GO}}(q)$ on 
${\rm{C}}_0(V,L,q)$ and on the left
${\rm{C}}_0(V,L,q)$-module ${\rm{C}}_1(V,L,q)$
(extending the natural action on $V \subset {\rm{C}}_1(V,L,q)$), so
in our rank-2 setting this recovers the usual action of
${\rm{GO}}(q)$ on $V$ and shows that it is semilinear over
an action on the $\calO_S$-algebra $Z_q$.  But
the subgroup ${\rm{GSO}}(q)$ acts trivially on $Z_q$
(as we saw in Remark \ref{bettergso} for any even rank),
so the natural ${\rm{GSO}}(q)$-action on $V$
is linear over the invertible $Z_q$-module structure.

Letting $T_q$ denote the
rank-2 torus ${\rm{R}}_{Z_q/S}(\mathbf{G}_{\rm{m}})$,
the multiplication action by $Z_q$ on 
$V$ defines a closed immersion 
$$T_q := {\rm{R}}_{Z_q/S}(\mathbf{G}_{\rm{m}}) \rightarrow {\rm{GL}}(V)$$
extending the natural inclusion on $\Gm$.  This identifies $T_q$
with the functor of $Z_q$-linear automorphisms of $V$, so the
$Z_q$-linearity of the natural ${\rm{GSO}}(q)$-action on $V$ thereby shows that 
${\rm{GSO}}(q) \subset T_q$ as 
closed $S$-tori inside ${\rm{GL}}(V)$
containing the ``scalar'' torus $\Gm$.
The tori ${\rm{GSO}}(q)$ and $T_q$ have rank 2, so
fibral considerations show that ${\rm{GSO}}(q) = T_q$ inside
${\rm{GL}}(V)$.

To show that this identification of
${\rm{GSO}}(q)$ with $T_q$ identifies ${\rm{SO}}(q)$ with the norm-1 torus inside $T_q$,
we claim more specifically that the norm map $T_q \rightarrow \Gm$ is identified
with the restriction to ${\rm{GSO}}(q)$ of the similitude character
${\rm{GO}}(q) \rightarrow \Gm$ (giving the $\Gm$-scaling action of ${\rm{GO}}(q)$ on $L$
that intertwines through $q$ with the action on $V$). 
To prove this equality of characters of an $S$-torus
it suffices to work on geometric fibers over $S$, so we can assume
$S = \Spec(k)$ for an algebraically closed field $k$ and $q = q_2$.
(See \cite[V, 2.5.2]{knus} for a direct argument over any ring.)
Thus, $V = k^2$ with standard ordered basis $\{e, e'\}$, $L = k$, 
and $q(x,y) = xy$.
By Remark \ref{evencenter} (with $m = 1$),
$${\rm{C}}(q_2) = k \oplus (k e \oplus k e') \oplus k ee'$$
with $ee' + e'e = 1$ and $e^2 = 0 = {e'}^2$. 
Clearly $Z_q = {\rm{C}}_0(V,q_2) = k \oplus k ee'$
with $ee'$ and $e'e$ orthogonal idempotents in $Z_q$ whose sum is 1.
Since 
$$ee'(e) = e(1 - ee') = e,\,\,\,
ee'(e') = 0,$$
the left multiplication action by $Z_q$ on $V = k e \oplus k e'$
carries $ee'$ to $\left(\begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix}\right)$.  
This action therefore defines 
an isomorphism of $Z_q$ onto the diagonal \'etale subalgebra of
${\rm{Mat}}_2(k)$, whose unit group is exactly ${\rm{GSO}}_2$.
The norm character on the group $T_q(k) = Z_q^{\times}$ of diagonal
elements in ${\rm{GL}}_2(k)$ is the restriction of the determinant on ${\rm{GL}}_2$,
and the determinant also clearly restricts
to the homothety character on $k^{\times} \cdot {\rm{O}}_2(k) = {\rm{GO}}_2(k)$.

Returning to the relative setting over a general base $S$,
it remains to naturally identify $Z_q$ with the coordinate ring of the finite \'etale zero scheme
$E_q$ of $q$ in ${\mathbf{P}}(V^{\ast})$.  This is a problem of identifying degree-2 finite \'etale covers of $S$,
so as we saw in the reduction of (2) to (1) it is enough to identify the torus of norm-1 units
on $E_q$ with the torus of norm-1 units in $Z_q$.  This latter torus has already been identified with
${\rm{SO}}(q)$, so we just need to construct a natural isomorphism between
${\rm{SO}}(q)$ and the $S$-group of norm-1 units on $E_q$.  By \'etale descent it
suffices to construct such an isomorphism when $q$ is split {\em provided}
that the isomorphism  is natural in the sense that it is compatible with base
change and functorial with respect to isomorphisms
in $(V,q)$. Now we may assume $q = xy$ for fiberwise independent linear forms $x, y$ on $V$,
and we need to naturally identify ${\rm{SO}}(q)$ with the group of norm-1 units on $E_q$. 

By working over the local rings of $S$ and using unique factorization over the residue field
at the closed point, it is easy to verify that any two such factorizations of $q$
are related Zariski-locally on $S$ through a combination of swapping $x$ and $y$
as well as multiplying them by reciprocal units.  In particular, the unordered pair of complementary line subbundles
$\ell^{\ast} = \calO_S x$ and ${\ell'}^{\ast} = \calO_S y$ in $V^{\ast}$ is intrinsic, so likewise
for the associated unordered dual pair of line subbundles $\ell = \calO_S x^{\ast}$
and $\ell' = \calO_S y^{\ast}$ in $V$. 
The zero-scheme $E_q$ is the union of the disjoint sections $0 := (x = 0) = \mathbf{P}({\ell'}^{\ast})$ 
and $\infty := (y = 0) = {\mathbf{P}}(\ell^{\ast})$
in ${\mathbf{P}}(V)$, so a unit $u$ on $E_q$ amounts to a pair of units $u(0),  u(\infty)$ on $S$
and the condition that $u$ is a norm-1 unit is that $u(0)u(\infty) = 1$. 
Consider the linear automorphism $[u]$ of $V$ that is multiplication
by $u(0)$ on $\ell'$ and multiplication by $u(\infty)$ on $\ell$.  The formation of $[u]$ is obviously functorial
with respect to isomorphisms in the pair $(V,q)$ 
(given that $q$ is split as above), and the map $u \mapsto [u]$ is visibly an isomorphism
from the unit group of $E_q$ onto ${\rm{GSO}}(q)$ carrying the norm-1 unit group onto
${\rm{SO}}(q)$.
\end{proof}

The exact sequence 
\begin{equation}\label{gsoexact}
1 \rightarrow \Gm \rightarrow {\rm{GSO}}(q) \rightarrow {\rm{PGSO}}(q) \rightarrow 1
\end{equation}
of $S$-tori induces an exact sequence of commutative groups
$$\dots \stackrel{\delta_1}{\rightarrow} {\rm{Pic}}(S) \rightarrow {\rm{H}}^1(S_{\et},{\rm{GSO}}(q)) \rightarrow
{\rm{H}}^1(S_{\et}, {\rm{PGSO}}(q)) \stackrel{\delta_2}{\rightarrow} {\rm{Br}}(S),
$$
where ${\rm{Br}}(S) := {\rm{H}}^2(S_{\et},\Gm)$ is the cohomological
Brauer group.  (The image of $\delta_2$ lands in the subset
of classes represented by a rank-4 Azumaya algebra, due to the compatibility of (\ref{gsoexact})
with the analogous exact sequence that expresses ${\rm{GL}}(V)$ as a central extension
of ${\rm{PGL}}(V)$ by $\Gm$.) 
By Proposition \ref{rzq}(1) and the exactness of finite pushforward
for the \'etale topology, $\delta_2$  has image
$\ker({\rm{Br}}(S) \rightarrow {\rm{Br}}(Z_q))$.
Thus, in general there is an exact sequence
$$0 \rightarrow {\rm{H}}^1(S_{\et},{\rm{GSO}}(q))/{\rm{Pic}}(S) \rightarrow
{\rm{H}}^1(S_{\et},{\rm{PGSO}}(q)) \rightarrow {\rm{Br}}(S) \rightarrow {\rm{Br}}(Z_q).$$
By Proposition \ref{rzq}(2) and the explicit description of ${\rm{GD}}_q$ immediately above Proposition \ref{rzq}, 
the pointed set $\ker {\rm{H}}^1({\rm{GD}}_q)$ classifies isomorphism classes
of $(V',L',q')$ whose associated special orthogonal group is 
isomorphic to ${\rm{SO}}(q)$.  There is a natural surjection of pointed sets 
${\rm{H}}^1(S_{\et}, {\rm{GSO}}(q)) \twoheadrightarrow \ker {\rm{H}}^1({\rm{GD}}_q)$
that intertwines the ${\rm{Pic}}(S)$-action on the source
with projective similarity on the target, so 
the set of projective similarity classes in $\ker {\rm{H}}^1({\rm{GD}}_q)$
is a quotient of the subgroup
$${\rm{H}}^1(S_{\et},{\rm{GSO}}(q))/{\rm{Pic}}(S) \subset {\rm{H}}^1(S_{\et}, {\rm{PGSO}}(q)).$$
By Proposition \ref{rzq}(1), ${\rm{H}}^1(S_{\et}, {\rm{GSO}}(q))$ is
naturally identified with ${\rm{Pic}}(Z_q)$, which vanishes
when $S$ is local (as then $Z_q$ is affine and semi-local).
Thus, we obtain:

\begin{corollary} Assume $S$ is local.  The set $\ker {\rm{H}}^1({\rm{GD}}_q)$ consists
of a single projective similarity class for any $(V,L,q)$ with $V$ of rank $2$.
In particular, the similitude class of a binary quadratic form $(V,q)$ over $S$
is determined by the isomorphism class of ${\rm{SO}}(q)$,
or equivalently by the isomorphism class of the discriminant scheme
$(q = 0) \subset \mathbf{P}(V^{\ast})$.
\end{corollary}
\index{special orthogonal group!binary case|)}

This corollary shows that Proposition \ref{projsim} is also valid for $n = 2$ when
$S = \Spec(R)$ for a local ring $R$ (e.g., any field). 

An interesting situation over general $S$ is when 
(\ref{gsoexact}) is split-exact, as we shall soon see.
Split-exactness occurs over local $S$ if ${\rm{GSO}}(q)$ is a split torus,
since an inclusion between split tori always splits off globally as a direct factor
over local $S$ (with a split torus complement too),
as we see via duality from the elementary analogue for a surjection
$\Z_S^N \twoheadrightarrow \Z_S^{N'}$ between constant $S_{\et}$-groups.
Here is a characterization in rank 2 for when ${\rm{GSO}}(q)$ is a split torus:

\begin{lemma} Assume $V$ has rank $2$. The $S$-torus ${\rm{GSO}}(q)$ is split if and only if
$Z_q$ is globally split as a quadratic \'etale $\calO_S$-algebra.
\end{lemma}

Note that $Z_q$ can be globally split 
even when $V$ not globally free, so generally $q$ cannot be identified with $q_2$
when $Z_q$ is split.

\begin{proof}
By Proposition \ref{rzq}(1), the problem is to show that the quadratic \'etale $\calO_S$-algebra
$Z_q$ is globally split if and only if the $S$-torus of units
${\rm{R}}_{Z_q/S}(\Gm)$ is $S$-split.   (If $S$ is not normal noetherian 
then it is {\em not} sufficient for the subtorus of norm-1 units to be
$S$-split, as there can be non-split $S$-tori $T$ that are an extension of $\Gm$ by $\Gm$.
Indeed, it suffices to build a nontrivial $\Z$-torsor $E$ over $S$ and then use
the subgroup of upper triangular unipotent elements in ${\rm{GL}}_2(\Z)$ to build
such a $T$ using $E$.  
The nodal cubic is an irreducible non-normal noetherian $S$ admitting a nontrivial
$\Z$-torsor.)

The isomorphism class of the $\calO_S$-algebra $Z_q$ corresponds
to an element of the pointed set ${\rm{H}}^1(S_{\et},\Z/2\Z) = {\rm{H}}^1(S_{\et},\Z^{\times})$, 
and the isomorphism class of the $S$-torus ${\rm{GSO}}(q)$ 
corresponds to an element of the pointed set ${\rm{H}}^1(S_{\et},{\rm{GL}}_2(\Z))$.
The determinant map ${\rm{GL}}_2(\Z) \rightarrow \Z^{\times}$
has a section $\sigma$ given by 
$$-1 \mapsto \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},$$
and it is straightforward to check that the map
$${\rm{H}}^1(\sigma): {\rm{H}}^1(S_{\et},\Z^{\times}) \rightarrow {\rm{H}}^1(S_{\et},{\rm{GL}}_2(\Z))$$
carries the class of $Z_q$ to the class of ${\rm{R}}_{Z_q/S}(\Gm)$.
Thus, we may conclude by noting that ${\rm{H}}^1(\sigma)$ has trivial kernel
(since ${\rm{H}}^1(\Det)$ provides a retraction).
\end{proof}

Now assume (with $n=2$) that (\ref{gsoexact}) is split-exact (e.g., ${\rm{GSO}}(q)$ is split), so the connecting maps $\delta_1$
and $\delta_2$ vanish.  This yields an isomorphism
$${\rm{Pic}}(Z_q)/{\rm{Pic}}(S) = 
{\rm{H}}^1(S_{\et},{\rm{GSO}}(q))/{\rm{Pic}}(S) \simeq {\rm{H}}^1(S_{\et}, {\rm{PGSO}}(q)),$$
so the pointed set ${\rm{H}}^1(S_{\et},{\rm{PGSO}}(q))$ parameterizes
(possibly with some repetitions when ${\rm{Pic}}(S) \ne 1$) 
the set of projective similarity classes among the set of isomorphism
classes of $(V',L',q')$ for which ${\rm{SO}}(q') \simeq {\rm{SO}}(q)$.
Typically there are many projective similarity classes
among such $(V',L',q')$, in contrast with the case $n \ne 2$.

Overall, for non-local $S$ it is hard to give a simple interpretation of when
two non-degenerate binary quadratic forms $(V,L,q)$ and $(V',L',q')$ lie in the same projective similarity class.
However, by Proposition \ref{rzq}(2), 
there is a nice interpretation of when $(V,L,q)$ and $(V',L',q')$ have 
isomorphic special orthogonal groups (replacing the answer via projective
similarity classes for rank $\ne 2$): it is equivalent to the discriminant algebras $Z_q$ and $Z'_q$
being isomorphic. Understanding the global geometry of the double covers of $S$
associated to these quadratic \'etale algebras is a rather nontrivial problem 
when these covers are not split. 


%
%\subsection{Properties of special orthogonal groups}\label{spo}
%
%The group ${\rm{SO}}'(q)$ is now well understood in the relative setting in terms of its
%smoothness properties, its center, and its fibral component groups.
%We shall use these properties to analyze ${\rm{O}}(q)$
%and ${\rm{SO}}(q)$. The case of odd $n$ is immediate from
%our work with ${\rm{SO}}'(q)$ since for such $n$ we have
%${\rm{SO}}(q) = {\rm{SO}}'(q)$ by definition:
%
%\begin{proposition}\label{oddconn}
%Assume $n$ is odd.  The $S$-group
%${\rm{SO}}(q)$ is smooth with connected fibers
%and it has trivial scheme-theoretic center.
%The $S$-group ${\rm{O}}(q)$ is
%naturally isomorphic to $\mu_2 \times {\rm{SO}}(q)$.
%In particular, ${\rm{O}}(q)$ is $S$-flat
%with functorial center $\mu_2$ and over $S[1/2]$
%it is smooth with fibral component groups of order $2$. 
%\end{proposition}
%
%\begin{proof}
%Corollary \ref{sorel} and Theorem \ref{connthm} provide the smoothness
%and fibral connectedness properties, and by Proposition \ref{findcenter}
%the center of ${\rm{SO}}(q)$  is trivial. Theorem \ref{soo}(2)
%gives the natural isomorphism $\mu_2 \times {\rm{SO}}(q) = {\rm{O}}(q)$.
%\end{proof}
% 
% A consequence of Proposition \ref{oddconn} is that if $n$ is odd then
% ${\rm{O}}(q)$ is {\em connected} over fields of characteristic 2.  
% We now briefly digress to explain a much simpler proof of this connectedness,
% without going through Theorem \ref{connthm}, by using 
%a link between orthogonal groups and symplectic groups
%in characteristic 2.
%Observe that over $\F_2$-algebras, $B_q$ is alternating:
%  $$B_q(v,v) = q(2v) - 2 q(v) = 0.$$ 
%   If $n$ is even and $S = \Spec k$ for a field $k/\F_2$ then $(V,B_q)$ is a symplectic space over $k$
%   (i.e., the alternating $B_q$ is non-degenerate), so
%${\rm{O}}(q) \subset {\rm{Sp}}(V,B_q)$ in such cases.  (This inclusion has codimension $n$
%because $\dim {\rm{Sp}}(V,B_q) = n(n+1)/2$ whereas $\dim {\rm{O}}(q) =
%\dim {\rm{SO}}'(q) = n(n-1)/2$.)  Of greater interest is the defect-1 case
%over a field, which is
%to say $n$ is odd and $S = \Spec k$ with a field $k/\F_2$.  
%
%In such defect-1 cases, on $V' = V/V^{\perp}$
%the alternating bilinear form $B_q$ induces a non-degenerate alternating bilinear form
%$B'_q$, so we get a homomorphism $h:{\rm{O}}(q) \rightarrow
%{\rm{Sp}}(V',B'_q)$.  The dimensions match, since $\dim V' = n-1$.
%In fact, for $q = q_n$ the kernel seen by direct calculation to be $\alpha_2^{n-1} \rtimes \mu_2$
%(along the top row of matrices, with $\mu_2$ in the upper left), where $\mu_2$ acts on
%the Frobenius kernel $\alpha_2 \subset \mathbf{G}_a$ by the usual scaling action.
%Hence, $h$ is surjective with infinitesimal kernel, so 
%in characteristic 2 with odd $n$ we deduce
%the connectedness of ${\rm{O}}(q)$ from the connectedness of symplectic groups.
%
%The central $\mu_2$ (from scalar action) lies in
%${\rm{O}}(q)$ and has trivial scheme-theoretic intersection with ${\rm{SO}}(q)$ when $n$ is odd,
%and the resulting homomorphism $\mu_2 \times {\rm{SO}}(q) \rightarrow {\rm{O}}(q)$ is an 
%isomorphism (Theorem \ref{soo}(2)). 
%Restricting $h$ to ${\rm{SO}}(q)$ then gives a surjective
%homomorphism ${\rm{SO}}(q) \rightarrow {\rm{Sp}}(V',B'_q)$ with 
%infinitesimal geometric 
%kernel $\alpha_2^{n-1}$ when $n$ is odd; this 
%``unipotent isogeny'' is a source of many weird phenomena related
%to algebraic groups in characteristic 2. 
%
%\begin{remark}\label{remblah} Orthogonal groups in $2m+1$ variables are type ${\rm{B}}_m$ and
%symplectic groups in $2m$ variables are type ${\rm{C}}_m$, and these
%types are distinct for $m \ge 3$. (For $m = 1$ and $m = 2$ the types coincide; see
%Example \ref{so3} and
%Example \ref{so5} respectively.)  In characteristics distinct from 2 and 3 
%there are no isogenies between (absolutely simple) connected
%semisimple groups of different types.  In characteristic 2 we have just seen that such exceptional
%isogenies exist between the distinct types ${\rm{B}}_m$ and ${\rm{C}}_m$ for all $m \ge 3$.
%See \cite[XXI, 7.5]{sga3} for further details. 
%\end{remark}
%
%  
% \begin{proposition}\label{sosmooth} Assume $n$ is even. The $S$-group ${\rm{SO}}(q)$ is smooth
% with connected fibers and 
% ${\rm{SO}}(q) \subset {\rm{SO}}'(q)$
%with equality over $S[1/2]$.
% \end{proposition}
%  
% \begin{proof}
%By Corollary \ref{sorel} and Theorem \ref{connthm},
%over $S[1/2]$ the closed $S$-subgroup ${\rm{SO}}'(q) \subset {\rm{O}}(q)$ 
%is smooth with connected fibers whereas
%at points in characteristic 2 its fibers are smooth with 
%two geometric components. 
%We claim that ${\rm{SO}}(q)$ is contained
%in ${\rm{SO}}'(q)$. 
%It suffices to check fibers at geometric points $s$ of $S$ since ${\rm{SO}}(q)$ is
%an open subscheme of ${\rm{O}}(q)$.
%
%On fibers in characteristic 2 the assertion is immediate from Theorem \ref{soo}(1).
%On the geometric fibers away from characteristic 2 the determinant 
%map ${\rm{O}}(q_s) \rightarrow \Gm$ is valued in the subgroup $\mu_2 \subset \Gm$
%that is finite \'etale, so the known connectedness and smoothness of the scheme-theoretic kernel
%$${\rm{SO}}'(q_s) = \ker(\det:{\rm{O}}(q_s) \rightarrow \Gm)$$
%implies that ${\rm{O}}(q_s)$ is smooth with identity component
%${\rm{SO}}'(q_s)$ of index 2.  Hence, the open and closed
%subgroup scheme ${\rm{SO}}(q_s)$ in ${\rm{O}}(q_s)$ must
%coincide with ${\rm{SO}}'(q_s)$ for such $s$.
%
%We conclude that ${\rm{SO}}(q)$ is an open and closed subscheme of 
%${\rm{SO}}'(q)$.  Over $S[1/2]$ the open and closed immersion
%${\rm{SO}}(q) \hookrightarrow {\rm{SO}}'(q)$ must be an equality,
%since ${\rm{SO}}'(q)$ has connected fibers over $S[1/2]$.
%In particular, ${\rm{SO}}(q)$ is smooth over $S[1/2]$. 
%On fibers at geometric points $s$ in characteristic 2,
%${\rm{SO}}(q_s)$ is the kernel of a nontrivial homomorphism
%$D_{q_s}:{\rm{O}}(q_s) \rightarrow \Z/2\Z$
%and ${\rm{O}}(q_s) = {\rm{SO}}'(q_s)$ is smooth with two geometric components.
%Hence, ${\rm{SO}}(q_s)$ is connected and smooth for such $s$.
%
%The affine $S$-group ${\rm{SO}}(q)$ has 
%smooth connected fibers (and is smooth over $S[1/2]$).  To prove that ${\rm{SO}}(q)$ is
%$S$-smooth, we use Lemma \ref{stdq} to reduce to
%the case of ${\rm{SO}}_n$ over the Dedekind base $\Spec \Z$, in which case
%Proposition \ref{gsmooth} may be applied. 
%\end{proof}
%
%\begin{corollary}\label{dflat}
%Assume $n$ is even.  The $S$-group ${\rm{O}}(q)$ is smooth
%and the surjective map $D_q:{\rm{O}}(q) \rightarrow (\Z/2\Z)_S$
%is smooth.  This identifies $(\Z/Z\Z)_S$ 
%as a representing object for the quotient sheaf ${\rm{O}}(q)/{\rm{SO}}(q)$.
%For $q = q_n$, $D_q$ splits as a semi-direct product.
% \end{corollary}
%
%\begin{proof}
%By Lemma \ref{stdq} we may reduce to the case
%of $q = q_n$ over $\Spec \Z$. Consider the element $g \in {\rm{O}}_n(\Z)$
%that swaps $x_1$ and $x_2$
%while leaving all other $x_i$'s invariant. The section
%$D_q(g)$ of the constant $\Z$-group $\Z/2\Z$ is equal to $1 \bmod 2$
%since it suffices to check this on a single geometric fiber,
%and at any fiber away from characteristic 2 it is clear
%(as ${\rm{SO}}_n$ coincides with ${\rm{SO}}'_n$
%on such fibers, and $\det g = -1 \ne 1$).  Thus, the $\Z$-group 
%${\rm{O}}_n$ is covered by the smooth disjoint open subschemes ${\rm{SO}}_n$
%and its $g$-translate, so ${\rm{O}}_n$ is $\Z$-smooth.
%The remaining assertions are likewise now clear.
%\end{proof}
%
%\begin{remark}\label{dickson} For even $n$
%and $q = q_n$, Corollary \ref{dflat} provides a map 
%${\rm{H}}^1(S_{{\rm{fppf}}}, {\rm{O}}_n) \rightarrow {\rm{H}}^1(S_{{\rm{fppf}}}, \Z/2\Z) =
%{\rm{H}}^1(S_{\et},\Z/2\Z)$ that assigns to every non-degenerate $(V,q)$ of rank $n$ over $S$
%(taken up to isomorphism)
%a degree-2 finite \'etale cover of $S$.  This is the {\em Dickson invariant}
%of $(V,q)$.  If $S$ is a $\Z[1/2]$-scheme (so $(\Z/2\Z)_S = \mu_2$)
%then it recovers the {\em discriminant} viewed in $\Gm(S)/\Gm(S)^2$.
%If $S$ is an $\F_2$-scheme then it recovers the {\em pseudo-discriminant},
%also called the {\em Arf invariant} when $S = \Spec k$ for a field $k/\F_2$. 
%\end{remark}
%
%
%  
%The behavior of orthogonal groups in characteristic 2
%depends on the parity of $n$ because ${\rm{O}}(q)/{\rm{SO}}(q)$ 
%turns out to be the $S$-group $\mu_2$ when $n$ is odd and $(\Z/2\Z)_S$ when $n$ is even: 
% 
% \begin{proposition}\label{soqt}
% If $n$ is odd then ${\rm{O}}(q) = {\rm{SO}}(q) \times \mu_2$ via multiplication, where
% $\mu_2$ is the central $2$-torsion in ${\rm{GL}}(V)$. 
% If $n$ is even then there is a canonical short exact sequence of $S$-group schemes
% $$1 \rightarrow {\rm{SO}}(q) \rightarrow {\rm{O}}(q) \stackrel{\delta}{\rightarrow} \Z/2\Z \rightarrow 1,$$
%where $\delta$ is the map from Proposition {\rm{\ref{disconnok}}}.
% \end{proposition}
% 
% \begin{proof}
%The determinant on
%${\rm{O}}(q)$ has restriction to the central $\mu_2$ that is trivial when $n$ is even
%and is the identity map when $n$ is odd.  Thus, for odd $n$ the
%multiplication homomorphism 
% $$m:{\rm{SO}}(q) \times \mu_2 \rightarrow {\rm{O}}(q)$$
% has trivial kernel.  By Theorem \ref{soo}(2) this map 
%is an isomorphism on $s$-fibers for all $s$, 
%and the source group is flat, due to the smoothness of ${\rm{SO}}(q)$.
%Hence, the fibral 
%isomorphism criterion in Lemma \ref{flatclosed} implies that 
%$m$ is always an isomorphism when $n$ is odd.
%
%Now consider even $n > 0$.  
%The kernel of $\delta:{\rm{O}}(q) \rightarrow (\Z/2\Z)_S$
%is an open and closed subgroup of ${\rm{O}}(q)$, 
%so ${\rm{SO}}(q)$ must be killed
%by $\delta$ (as ${\rm{SO}}(q)$ has connected fibers). 
%We need to prove that the closed immersion
%${\rm{SO}}(q) \hookrightarrow \ker \delta$ is an isomorphism.
%By Theorem \ref{soo}, ${\rm{O}}(q)$ has smooth fibers
%(even in characteristic 2), so the same holds for its open
%and closed subscheme $\ker \delta$.
%Likewise, by construction (and properties
%of ${\rm{SO}}'(q)$ summarized
%above, and to be proved later) ${\rm{SO}}(q)$ has smooth connected fibers. 
%By Lemma \ref{stdq}, we can pass to the case $q = q_n$ over $\Spec \Z$.
%
%Over $\Z[1/2]$ we have ${\rm{SO}}(q) = {\rm{SO}}'(q)$,
%so on fibers at $s \in \Spec \Z[1/2]$
%$${\rm{SO}}(q_s) = \ker(\det:{\rm{O}}(q_s) \rightarrow \mu_2 = \Z/2\Z).$$
%This forces ${\rm{SO}}(q_s) \hookrightarrow \ker \delta_s$
%to be an inclusion between open and closed subgroups of ${\rm{O}}(q_s)$
%with index 2 on geometric points, so ${\rm{SO}}(q)_s = \ker \delta_s$
%for such $s$.  Also, ${\rm{O}}(q)_{\F_2} = {\rm{SO}}'(q)_{\F_2}$ is smooth
%with two geometric components,  so
%${\rm{SO}})(q)_{\F_2} := {\rm{O}}(q)^0_{\F_2} = \ker \delta_{\F_2}$
%since $\delta_{\F_2} \ne 0$.
%Hence, ${\rm{SO}}(q) \hookrightarrow \ker \delta$ is
%a closed immersion between affine $\Z$-groups of finite type
%with smooth fibers, and the map induces an isomorphism between
%fibers over $\Spec \Z$.  
%Since ${\rm{SO}}(q)$ is $\Z$-smooth,
%the fibral isomorphism criterion (Lemma \ref{flatclosed}) gives
%that ${\rm{SO}}(q) \simeq \ker \delta$.
%\end{proof}
%
%
%\begin{example}\label{hassemink} The non-degenerate quadratic spaces $(V,q)$ of rank $n \ge 2$ over
%a field $k$ are classified by ${\rm{H}}^1(k,{\rm{O}}_n)$ (using the fppf topology if
%${\rm{char}}(k) = 2$ and $n$ is odd).  Suppose $n$ is even, so 
%$\delta_n:{\rm{O}}_n \twoheadrightarrow
%(\Z/2\Z)_k$ defines an invariant  $\delta_n([q]) \in {\rm{H}}^1(k,\Z/2\Z)$.
%Fix a section to $\delta_n$, so for any $c \in {\rm{H}}^1(k,\Z/2\Z)$ we get
%a cocycle in ${\rm{Z}}^1(k,{\rm{O}}_n)$ against which we can twist $q_n$
%to get a quadratic form $Q_c$.
%If ${\rm{char}}(k) \ne 2$ (so $(\Z/2\Z)_k = \mu_2$) then $\delta_n([q]) \in k^{\times}/(k^{\times})^2$ and is
%the {\em discriminant} of $q$.  If ${\rm{char}}(k) = 2$  then $\delta_n([q]) \in
%k/\wp(k)$ and is the {\em Arf invariant} (or {\em pseudo-discriminant}) of $q$.
%Either way, $q$ is classified by $n$, $d(q) := \delta_n([q]) \in {\rm{H}}^1(k,\Z/2\Z)$,
%and a class in ${\rm{H}}^1(k,{\rm{SO}}(Q_{d(q)}))$.
%But ${\rm{Spin}}(Q)$ is a central extension of ${\rm{SO}}(Q)$ by $\mu_2$, so
%we get a connecting map ${\rm{H}}^1(k,{\rm{SO}}(Q)) \rightarrow {\rm{H}}^2(k,\mu_2) = {\rm{Br}}(k)[2]$.
%Hence, $q$ is classified by $n$, $d(q)$, and a class in ${\rm{Br}}(k)[2]$ up to twisting
%by a degree-1 class valued in a spin group. If $k$ is a global field with no real places
%then by the Hasse Principle for simply connected groups the degree-1 cohomology of 
%spin groups vanishes, in which case $q$ is classified by
%$n$, $\delta_n([q])$, and a class in ${\rm{H}}^2(k,\mu_2) =
%{\rm{Br}}(k)[2]$ (which encodes local invariants). This recovers the Hasse--Minkowski theorem
%for such $k$.  A refinement incorporates $k$ with real places and odd $n$.
%\end{example}

\subsection{Spin groups and related group schemes}\label{spinsec}

Let $(V,q)$ be a non-degenerate quadratic space of rank 
$n \ge 1$ (so $q$ is valued in $\calO_S$). 
We shall use unit groups of Clifford algebras to ``explicitly'' construct a 
degree-2 central extension ${\rm{Spin}}(q)$ of ${\rm{SO}}(q)$
by $\mu_2$ that is the simply connected central cover when $n \ge 3$.
For even $n$ the Clifford algebra has center $\calO_S$ (Proposition \ref{cliffstr}) 
whereas for odd $n$ the center is 
a finite locally free $\calO_S$-algebra of rank 2 that is non-\'etale in characteristic 2 (Proposition \ref{cliffodd}).  
Thus, we shall first consider even $n$.  In such cases we will use the Clifford group 
${\rm{GPin}}(q)$ from Definition \ref{gpineven} to construct
${\rm{Spin}}(q) \rightarrow {\rm{SO}}(q)$ by relativizing 
arguments over fields in \cite[II.3.5]{chevquad}. 
(In \cite[II.3.5]{chevquad}, spin groups are called {\em reduced Clifford groups}.) 

Assume $n = 2m \ge 2$ is {\em even}. 
The $S$-group ${\rm{GPin}}(q)$ is a central extension of ${\rm{O}}(q)$ by $\Gm$ (Proposition \ref{disconnok}), so  
${\rm{GSpin}}(q) := {\rm{GPin}}(q)^0$ is a central extension of ${\rm{SO}}(q)$ by $\Gm$ and
\index{ aaagspinq@${\rm{GSpin}}(q)$}
\begin{equation}\label{spinqt}
{\rm{GPin}}(q)/{\rm{GSpin}}(q) \simeq {\rm{O}}(q)/{\rm{SO}}(q) = (\Z/2\Z)_S.
\end{equation}
In particular, ${\rm{GSpin}}(q)$ is open and closed in ${\rm{GPin}}(q)$,
so it is $S$-smooth and $S$-affine.   Since ${\rm{GSpin}}(q)$ is an extension of
${\rm{SO}}(q)$ by $\Gm$, 
its fibers are connected reductive. Thus, ${\rm{GSpin}}(q)$ is a reductive $S$-group.

Via the link between $D_q$ in (\ref{Dq}) and the $\Z/2\Z$-grading
on ${\rm{GPin}}(q)$ as noted immediately before Proposition \ref{disconnok}, 
we see that ${\rm{GSpin}}(q) = {\rm{C}}_0(V,q) \bigcap {\rm{GPin}}(q)$.
For this reason, ${\rm{GSpin}}(q)$ is called the {\em even Clifford group}. 
\index{Clifford group!even}
If $Z_q$ denotes the quadratic \'etale center of
${\rm{C}}_0(V,q)$ then by
Remark \ref{ZqGPin} the $S$-group ${\rm{GSpin}}(q)$ contains
the $S$-torus ${\rm{R}}_{Z_q/S}(\Gm)$  when $n = 2$ whereas if $n \ge 4$ then 
${\rm{GSpin}}(q) \bigcap {\rm{R}}_{Z_q/S}(\Gm)$
is the subgroup of points of ${\rm{R}}_{Z_q/S}(\Gm)$ whose square lies in $\Gm$
(so it is a commutative extension of $\mu_2$ by $\Gm$). 

For $n = 2$, the containment
${\rm{R}}_{Z_q/S}(\Gm) \subset {\rm{GSpin}}(q)$ of smooth $S$-groups
 is an equality for fibral
connectedness and dimension reasons. When $n = 2m \ge 4$ the $S$-group
${\rm{SO}}(q)$ is semisimple (with geometric
fibers of type ${\rm{D}}_m$, where ${\rm{D}}_2 := {\rm{A}}_1 \times {\rm{A}}_1$),
so the derived group  (in the sense of
Theorem \ref{dergp})
\index{ aaaspinq@${\rm{Spin}}(q)$}
\index{spin group|(}
$${\rm{Spin}}(q) := \mathscr{D}({\rm{GSpin}}(q))$$
is a central extension of ${\rm{SO}}(q)$ by an $S$-subgroup
$\mu \subset \Gm$ that must be finite fppf (Proposition \ref{isogfiber}). 
Since we saw that ${\rm{GSpin}}(q)$ is an extension of ${\rm{SO}}(q)$
by $\Gm$, it follows that ${\rm{GSpin}}(q) = \Gm \times^{\mu} {\rm{Spin}}(q)$.

\begin{lemma}\label{mu2ker} For even $n \ge 4$, the intersection
$\mu = \Gm \bigcap {\rm{Spin}}(q)$ is equal to $\mu_2 \subset \Gm$.
The central extension ${\rm{Spin}}(q)$ of ${\rm{SO}}(q)$ by $\mu_2$ is the simply
connected central cover, and $Z_{{\rm{Spin}}(q)}$ 
is a form of $\mu_4$ if $n \equiv 2 \bmod 4$ and is a form of $\mu_2 \times \mu_2$ if $4|n$.
\end{lemma}

\begin{proof} Consider 
the {\em main anti-involution} $\alpha$ of ${\rm{C}}(V,q)$ induced by
\index{Clifford algebra!main anti-involution}
the anti-involution of the tensor algebra of $V$ via
$v_1 \otimes \dots \otimes v_j \mapsto v_j \otimes \dots \otimes v_1$.
Note that $\alpha$ restricts to the identity on $V$ and the identity on the center $\calO_S$.
(The effect of $\alpha$ on the rank-2 \'etale central $\calO_S$-algebra $Z_q$ in ${\rm{C}}_0(V,q)$
is the identity when $4|n$ and is the unique fiberwise nontrivial algebra automorphism
when $n \equiv 2 \bmod 4$.  We will not need this, but to prove it we just need to consider $q = q_n$ over $\Z$.
The decomposition $Z_{q_n} = \Z z \times \Z (1-z)$ in Remark \ref{evencenter}
for a fiberwise nontrivial idempotent $z$ described over $\Z[1/2]$ in terms of an explicit element $w$ shows 
that $\alpha(z) = 1-z$ when $n \equiv 2 \bmod 4$
and $\alpha(z) = z$ when $4|n$ because $\alpha(w) = (-1)^{n/2} w$. This argument holds for $n=2$ as well.)

For $u \in {\rm{GPin}}(q)$, the operator $h = \pi_q(u) \in {\rm{O}}(q)$
as in (\ref{piq}) 
satisfies $uv = \varepsilon_u h(v)u$ in ${\rm{C}}(V,q)$ for all $v \in V$,
where $\varepsilon_u = (-1)^{\deg_q(u)}$ as in (\ref{piq}). 
Applying $\alpha$ gives $-v \alpha(u) = \varepsilon_u \alpha(u) (-h(v))$.
Hence, $\alpha(u)uv = \varepsilon_u \alpha(u) h(v)u = v \alpha(u)u$,
so the point $\alpha(u)u$ in ${\rm{GPin}}(q)$ centralizes
$V$ in ${\rm{C}}(V,q)$ and thus is central in ${\rm{C}}(V,q)$.
In other words, $\alpha(u)u \in \Gm$ (so $\alpha(u)u = u \alpha(u)$, as
$u^{-1}\alpha(u)u = \alpha(u)uu^{-1} = \alpha(u)$).   Since
$\alpha$ is an anti-automorphism, so 
$$\alpha(uu')uu' = \alpha(u')\alpha(u)uu' = (\alpha(u)u)(\alpha(u')u'),
$$
the map $u \mapsto \alpha(u)u$ is an $S$-homomorphism
\begin{equation}\label{evennorm}
\nu_q: {\rm{GPin}}(q) \rightarrow \Gm
\end{equation} 
(called the {\em Clifford norm})
\index{ aaanuq@$\nu_q$}
\index{Clifford norm}
whose restriction to the central $\Gm$ is $t \mapsto t^2$.
The Clifford norm must kill the semisimple ${\rm{Spin}}(q)$, so $\mu := \Gm \bigcap {\rm{Spin}}(q) \subset \mu_2$.
We claim that this inclusion between finite fppf $S$-groups is an isomorphism.
It suffices to check on fibers. 

For $S = \Spec k$ with a field $k$ we must rule out the
possibility $\Gm \cap {\rm{Spin}}(q) = 1$.  If this happens then
$\pi_q:{\rm{Spin}}(q) \rightarrow {\rm{SO}}(q)$ is an isomorphism.  
This map is equivariant for the natural actions by ${\rm{O}}(q) = {\rm{GPin}}(q)/\Gm$ on each side,
so if $\pi_q$ were an isomorphism then ${\rm{Spin}}(q)$ would contain
a $\mu_2$ that is centralized by ${\rm{GPin}}(q)$ 
(lifting $Z_{{\rm{O}}(q)} = \mu_2 \subset {\rm{SO}}(q)$) and is not contained in the central $\Gm \subset {\rm{C}}(V,q)$.
But ${\rm{GPin}}(q)$ generates the $k$-algebra ${\rm{C}}(V,q)$ 
(since ${\rm{GPin}}(q)$ contains the Zariski-dense open
$U = \{q \ne 0\} \subset V$) and this algebra has center
$k$, so $\Gm = Z_{{\rm{GPin}}(q)}$. Hence, no such $\mu_2$ subgroup can exist.

Returning to the relative setting, 
we conclude that for even $n = 2m \ge 4$, the semisimple $S$-group ${\rm{Spin}}(q)$ is 
a central extension of ${\rm{SO}}(q)$ by $\mu_2$. Since 
$\#Z_{{\rm{Spin}}(q)} = 2 \cdot \#Z_{{\rm{SO}}(q)} = 4$ and the fundamental group for
the root system ${\rm{D}}_m$ ($m \ge 2$) has order 4, ${\rm{Spin}}(q)$ must be simply connected.
Thus, ${\rm{Spin}}(q)$ is the simply connected central cover of ${\rm{SO}}(q)$.
The structure of  $Z_{{\rm{Spin}}(q)}$ can be read off from the fact that it is a form of
the Cartier dual of the quotient $P/Q$ of the weight lattice modulo the root lattice for
the root system ${\rm{D}}_m$ ($m = n/2  \ge 2$). 
\end{proof}
\index{spin group|)}
 
\begin{remark}\label{normp}
For odd $n$, the definitions
of $\alpha$ and $\nu_q$ as in the preceding proof carry over but
$Z_q$ is 
a rank-2 finite flat $\calO_S$-algebra not contained
in the even part.  This leads to some complications when $n$ is odd, as we discuss soon.

In \cite[Def.\,1.8, (3.7)]{abs} and \cite[IV, 6.1]{knus} the Clifford norm $\nu_q$
is defined by replacing
$\alpha$ with the unique anti-involution $\alpha^{-}$ extending negation on $V$.
The signless $\alpha$ and associated Clifford norm that we use agree with 
\cite[II.3.5]{chevquad},  \cite[App.\,I]{frohlich}, \cite[\S55]{omeara}, and \cite[Ch.\:9, \S3]{scharlau}, and also implicitly with
\cite{serrewitt} (see Remark \ref{switt}).
Both sign conventions for the definition of the Clifford norm 
yield the same restriction to the even part of the Clifford algebra.
\end{remark} 
%
%We conclude that for even $n = 2m \ge 4$, the semisimple $S$-group ${\rm{Spin}}(q)$ is 
%a central extension of ${\rm{SO}}(q)$ by $\mu_2$. Since 
%$\#Z_{{\rm{Spin}}(q)} = 2 \cdot \#Z_{{\rm{SO}}(q)} = 4$ and the fundamental group for
%${\rm{D}}_m$ ($m \ge 2$) has order 4, ${\rm{Spin}}(q)$ must be simply connected.
%Thus, ${\rm{Spin}}(q)$ is the simply connected central cover of ${\rm{SO}}(q)$.
%(By construction, the central pushout of ${\rm{Spin}}(q)$ along $\mu_2 \hookrightarrow \Gm$ is
%the relative identity component ${\rm{GSpin}}(q)$
%of ${\rm{GPin}}(q)$  inside  ${\rm{C}}(V,q)^{\times}$.)

By construction, the central pushout of ${\rm{Spin}}(q)$ along $\mu_2 \hookrightarrow \Gm$ is
the relative identity component ${\rm{GSpin}}(q)$
of ${\rm{GPin}}(q)$  inside  ${\rm{C}}(V,q)^{\times}$.
Since
${\rm{GSpin}}(q) = \Gm \times^{\mu_2} {\rm{Spin}}(q)$, we see
that ${\rm{Spin}}(q)$ is the kernel of the Clifford norm $\nu_q:{\rm{GSpin}}(q) \rightarrow \Gm$.
This description of ${\rm{Spin}}(q)$ via the Clifford norm on ${\rm{GSpin}}(q)$ makes sense even when $n = 2$ (whereas
the ``derived group'' definition of the spin group for even $n \ge 4$ is not suitable, due to the commutativity of
${\rm{GSpin}}(q) = {\rm{R}}_{Z_q/S}(\Gm)$ when $n = 2$), so we use it as the {\em definition}
of ${\rm{Spin}}(q)$ when $n = 2$;  direct calculation with $q_2$ shows that
\index{spin group!Clifford norm}
if $n = 2$ then the Clifford norm $\nu_q:{\rm{GSpin}}(q) = {\rm{R}}_{Z_q/S}(\Gm) \rightarrow \Gm$ arises from the $\calO_S$-algebra
norm on $\calO_{Z_q}$. Thus, if $n = 2$ then ${\rm{Spin}}(q)$ is the group of norm-1 units in the rank-2 torus
${\rm{R}}_{Z_q/S}(\Gm)$ and the resulting map ${\rm{Spin}}(q) \rightarrow {\rm{SO}}(q)$
is a degree-2 isogeny with kernel $\mu_2 \subset \Gm$, so Lemma \ref{mu2ker} holds for $n = 2$.
Likewise, by inspection we have ${\rm{GSpin}}(q) = \Gm \times^{\mu_2} {\rm{Spin}}(q)$
when $n = 2$ as well. 

\medskip

Now we turn to spin groups and other related groups for {\em odd} $n = 2m + 1 \ge 1$. 
As in the case of even $n$, we abuse notation by letting ${\rm{C}}(V,q)^{\times}$
denote the ``$S$-group scheme of units of ${\rm{C}}(V,q)$''.
Recall from the discussion following Definition \ref{gpineven} that for even $n$, all points
of ${\rm{GPin}}(q)$ are locally homogeneous.  For odd $n$ the same definition
``makes sense'' but will not be used as the definition of ${\rm{GPin}}(q)$ in such cases
because for such a definition (applied to odd $n$) local homogeneity generally fails, as we
can already see with the $\Z/2\Z$-graded center.  The definition of 
the ${\rm{GPin}}(q)$ for odd $n$ will require  the insertion of a local homogeneity condition
that is automatically satisfied for even $n$. 

The Zariski-closed subgroup scheme ${\rm{C}}(V,q)^{\times}_{\rm{lh}}$ 
\index{ aaacvqlh@${\rm{C}}(V,q)_{\rm{lh}}$}
of locally homogeneous units in ${\rm{C}}(V,q)^{\times}$
meets the unit group $Z_q^{\times} = {\rm{R}}_{Z_q/S}(\Gm)$ in
the group $(Z_q^{\times})_{\rm{lh}}$ of locally homogeneous units of $Z_q$. The $S$-group
$(Z_q^{\times})_{\rm{lh}}$ is 
\index{ aaazqlh@$(Z_q^{\times})_{\rm{lh}}$}
an extension of $(\Z/2\Z)_S$ by $\Gm$, consisting of sections that locally lie in 
either $\Gm$ or the $\Gm$-torsor of local generators of the degree-1 line of $Z_q$. 
Thus, the isomorphism $Z_q \otimes_{\calO_S} {\rm{C}}_0(V,q) = {\rm{C}}(V,q)$
from Proposition \ref{cliffodd} shows that
${\rm{C}}(V,q)_{\rm{lh}}^{\times}$ is an extension of $(\Z/2\Z)_S$ by ${\rm{C}}_0(V,q)^{\times}$
for the Zariski topology. 
In particular, ${\rm{C}}(V,q)^{\times}_{\rm{lh}}$ is smooth with relative identity component
${\rm{C}}_0(V,q)^{\times}$ that is a form of ${\rm{GL}}_{2^m}$ ($n = 2m+1$).
Note that $Z_q^{\times}$ is a rank-2 torus over $S[1/2]$ but it has non-reductive fibers
at points in characteristic 2. Local homogeneity for $(Z_q^{\times})_{\rm{lh}}$ eliminates
the intervention of such non-reductivity in what follows.

\begin{definition}\label{gpindef} For odd $n \ge 1$, the {\em naive Clifford group} over $S$ is 
the closed subgroup 
\index{ aaagpinqp@${\rm{GPin}}'(q)$}
\index{ aaagpinq@${\rm{GPin}}(q)$}
\index{Clifford group}
$${\rm{GPin}}'(q) =  \{u \in {\rm{C}}(V,q)^{\times}\,|\,uVu^{-1} = V\}$$  
of ${\rm{C}}(V,q)^{\times}$, and the {\em Clifford group} over $S$ is 
$${\rm{GPin}}(q) = {\rm{GPin}}'(q) \bigcap {\rm{C}}(V,q)^{\times}_{\rm{lh}} = {\rm{GPin}}'(q) \bigcap
{\rm{C}}(V,q)_{\rm{lh}}.$$
\end{definition}

\begin{remark} Note that for even $n$, Definition \ref{gpindef} makes  sense and we have seen 
in the discussion after Definition \ref{gpineven} that the two resulting groups coincide.
For odd $n$ they are fiberwise 
distinct (see Lemma 
\ref{unitchev}). It is ${\rm{GPin}}'(q)$ rather than ${\rm{GPin}}(q)$ that is called the
{\em Clifford group} in \cite[II, \S3]{chevquad}.  For odd $n$ both groups will yield the same 
``Spin'' and ``GSpin'' groups, but the associated ``Pin groups'' 
(see \S\ref{pinsubsec}) will only agree when $n \equiv 1 \bmod 4$ 
and $S$ is a $\Z[1/2]$-scheme (see Remark \ref{samepin}).  
In \cite[II, \S3]{chevquad}, the case of odd $n$ is ruled out
of consideration in characteristic 2 essentially by definition and Pin groups are not considered.
\end{remark}

By the same calculation as with even $n$ before (\ref{negref}), for 
odd $n$ and any point $u$ of ${\rm{GPin}}'(q)$ 
the induced linear automorphism $v \mapsto uvu^{-1}$ of $V$
lies in ${\rm{O}}(q)$, and moreover $Z_q^{\times} \subset
\ker({\rm{GPin}}'(q) \rightarrow {\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q))$.

\begin{lemma}\label{unitchev} For odd $n$, 
the inclusion $Z_q^{\times} \times^{(Z_q^{\times})_{\rm{lh}}} {\rm{GPin}}(q) \subset
{\rm{GPin}}'(q)$ is an equality.  In particular,
${\rm{GPin}}'(q)/{\rm{GPin}}(q)$ is smooth with geometric fiber $\Gm$ away
from characteristic $2$ and $\Add$ in characteristic $2$.
\end{lemma}

\begin{proof}
Let $u'$ be a point of ${\rm{GPin}}'(q)$ valued in some $S$-scheme $S'$,
so we may rename $S'$ as $S$.  Consider the global decomposition
$u'_{+} + u'_{-}$ of $u'$ as a sum of even and odd parts
in ${\rm{C}}(V,q)$.  We claim that Zariski-locally on
$S$, at least one of $u'_{+}$ or $u'_{-}$ is a unit in
${\rm{C}}(V,q)$.  To prove this we may and do assume
$S = \Spec k$ for an algebraically closed field $k$.  
By the classical  theory over fields (see \cite[II.3.2]{chevquad}), 
$u' = z u$ where $u$ is an element of ${\rm{GPin}}(q)$ and
$z \in Z_q$.  It follows that $z \in Z_{q}^{\times}$.
Since ${\rm{C}}(V,q) = Z_q \otimes {\rm{C}}_0(V,q)$, we can scale
$z$ and $u$ by reciprocal elements of $Z^1_{q} - \{0\}$ if necessary 
to arrange that $u$ has degree 0.  In particular,
$u'_{\pm} = (z_{\pm}) u$, where $z_{\pm}$ are the homogeneous
components of $z$.  But a nonzero homogeneous element of 
$Z_q$ is a {\em unit}, and $u$ lies in ${\rm{GPin}}(q)$, so 
one of $u_{\pm}$ is a unit in ${\rm{C}}(V,q)$.

Over the original $S$, locally scale 
$u'$ by a generator of $Z_q^1$ if necessary so that
$u'_{+}$ is a unit.  For every local section $v$ of $V$,
the local section $\widetilde{v} := u'v{u'}^{-1} $ of $V$ satisfies
$u'_{+}v + u'_{-}v = u'v = \widetilde{v}u' = \widetilde{v}u'_{+} + \widetilde{v}u'_{-}$.
Comparing odd-degree terms on both sides gives that $u'_{+} \in {\rm{GPin}}(q)$,
so we may assume $u'_{+} = 1$.  Writing $u = 1 + x$ with $x$ of degree 1,
the relation $uv = \widetilde{v}u$ for $v \in V$ and $\widetilde{v} := uvu^{-1} \in V$
says $v + xv = \widetilde{v} + \widetilde{v}x$.  Comparing
odd-degree parts implies $\widetilde{v} = v$, so $u$ centralizes
$V$ and hence is central in ${\rm{C}}(V,q)$ (i.e., $u \in Z_q^{\times}$). 
\end{proof}

Fix {\em odd} $n \ge 1$.  The locally constant degree $\deg_q:{\rm{GPin}}(q) \rightarrow (\Z/2\Z)_S$
has restriction to $(Z_q^{\times})_{\rm{lh}}$ that kills the degree-0 part $\mathbf{G}_{\rm{m}}$
and induces the unique isomorphism $(Z_q^{\times})_{\rm{lh}}/\mathbf{G}_{\rm{m}} \simeq (\Z/2\Z)_S$.
Hence, the {\em even Clifford group}  $${\rm{GSpin}}(q) := \ker(\deg_q) \subset {\rm{GPin}}(q)$$
\index{Clifford group!even}
\index{ aaagspinq@${\rm{GSpin}}(q)$}
is  open and closed, with ${\rm{GPin}}(q)/{\rm{GSpin}}(q) = (\Z/2\Z)_S$.
Since $$(Z_q^{\times})_{\rm{lh}} \bigcap {\rm{GSpin}}(q) = \mathbf{G}_{\rm{m}},$$ we have
\begin{equation}\label{zgs}
(Z_q^{\times})_{\rm{lh}} \times^{\mathbf{G}_{\rm{m}}} {\rm{GSpin}}(q) = {\rm{GPin}}(q).
\end{equation}
In particular,
$${\rm{GPin}}(q)/{\rm{GSpin}}(q) = (Z_q^{\times})_{\rm{lh}}/\Gm = (\Z/2\Z)_S$$
for odd $n$, just as we saw for even $n$ in (\ref{spinqt}). 

For odd $n$, the calculation (\ref{negref}) carries over without change and so leads us to define 
$\pi_q:{\rm{GPin}}(q) \rightarrow {\rm{GL}}(V)$
by $\pi_q(u)(v) = (-1)^{{\rm{deg}}_q(u)}uvu^{-1}$ as for even $n$
(so for $u \in V \subset {\rm{C}}_1(V,q)$ such that $q(u)$ is a unit, the automorphism $\pi_q(u):V \simeq V$ is the reflection
through $u$ relative to $q$). 
This is the twist of the conjugation action by
the $\mu_2$-valued character given by exponentiating $\deg_q$.
Since $\mu_2$-scaling has no effect on $q$, 
we see that $\pi_q$ is valued in ${\rm{O}}(q)$.
Also, since ${\rm{GSpin}}(q) \subset {\rm{C}}_0(V,q)$, the restriction of $\pi_q$ to ${\rm{GSpin}}(q)$
is the action on $V$ through conjugation inside ${\rm{C}}(V,q)$.

\begin{proposition}\label{gpinprop} Fix an odd $n \ge 1$.
\begin{enumerate}
%\item The homomorphism 
%$\pi_q:{\rm{GSpin}}(q) \rightarrow {\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$
%factors through ${\rm{SO}}(q)$ and defines a diagram of $S$-groups
% The representation ${\rm{GPin}}'(q) \rightarrow {\rm{GL}}(V)$
%defined by the conjugation action $u.v = uvu^{-1}$ on $V$ inside
%${\rm{C}}(V,q)$ is valued in ${\rm{SO}}(q)$, and defines a diagram of $S$-groups 
%$$1 \rightarrow Z_q^{\times} \rightarrow {\rm{GPin}}'(q) \rightarrow {\rm{SO}}(q) \rightarrow 1$$
%that is short exact for the Zariski topology on the category of $S$-schemes.
%In particular, ${\rm{GPin}}'(q)$ is $S$-smooth with connected fibers that are
%reductive over $S[1/2]$ and non-reductive in characteristic $2$.
\item The homomorphism 
$\pi_q:{\rm{GSpin}}(q) \rightarrow {\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$
factors through ${\rm{SO}}(q)$ and defines a diagram of $S$-groups
$$1 \rightarrow \mathbf{G}_{\rm{m}}  \rightarrow {\rm{GSpin}}(q) \rightarrow {\rm{SO}}(q) \rightarrow 1$$
that is short exact for the Zariski topology on the category of $S$-schemes.  In particular,
${\rm{GSpin}}(q)$ is $S$-smooth with
connected reductive fibers. 
\item The natural map 
$$(\Z/2\Z)_S \times {\rm{SO}}(q) = ((Z_q^{\times})_{\rm{lh}}/\mathbf{G}_{\rm{m}}) \times
({\rm{GSpin}}(q)/\mathbf{G}_{\rm{m}}) \rightarrow {\rm{GPin}}(q)/\mathbf{G}_{\rm{m}}$$
is an isomorphism $($so ${\rm{GPin}}(q)$ is $S$-smooth$)$. Composing its inverse with 
the canonical homomorphism $(\Z/2\Z)_S \rightarrow \mu_2$
recovers the homomorphism ${\rm{GPin}}(q)/\mathbf{G}_{\rm{m}} \rightarrow \mu_2 \times {\rm{SO}}(q) = 
{\rm{O}}(q)$ induced by $\pi_q$.  In particular, over $S[1/2]$
the $S$-group ${\rm{GPin}}(q)$ is an extension of ${\rm{O}}(q)$ by
$\mathbf{G}_{\rm{m}}$.
\end{enumerate}
\end{proposition}

Keep in mind that the representation $\pi_q$ of ${\rm{GPin}}(q)$ on $V$ is the twist of the conjugation action
by the quadratic character $u \mapsto (-1)^{{\rm{deg}}_q(u)}$. 

\begin{proof}
Since ${\rm{GPin}}(q)$ is defined in terms of ${\rm{GPin}}'(q)$, it will be convenient to formulate 
an analogue for (1) for ${\rm{GPin}}'(q)$ from which we shall deduce (1): 
\begin{quote} ($\ast$)\,\,\,The representation ${\rm{GPin}}'(q) \rightarrow {\rm{GL}}(V)$
defined by the conjugation action $u.v = uvu^{-1}$ on $V$ inside
${\rm{C}}(V,q)$ is valued in ${\rm{SO}}(q)$, and defines a diagram 
$$1 \rightarrow Z_q^{\times} \rightarrow {\rm{GPin}}'(q) \rightarrow {\rm{SO}}(q) \rightarrow 1$$
that is short exact for the Zariski topology on the category of $S$-schemes.
(In particular, ${\rm{GPin}}'(q)$ is $S$-smooth with connected fibers that are
reductive over $S[1/2]$ and non-reductive in characteristic $2$.)
\end{quote}

\medskip

Rather than directly show that ${\rm{GPin}}'(q)$ is carried
into ${\rm{SO}}(q)$, we will first show that all points $g$ of ${\rm{SO}}(q)$ Zariski-locally lift to ${\rm{GPin}}'(q)$
under the conjugation action on $V$ (without the quadratic twist). 
Since ${\rm{SO}}(q) \subset {\rm{O}}(q)$, by the functoriality of Clifford algebras
there exists a unique algebra automorphism $[g]$ of ${\rm{C}}(V,q)$ extending $g$ on $V$.
Consider the induced automorphism of $Z_q$. 
We claim that this is trivial, due to the condition that $g$ lies in ${\rm{SO}}(q)$.
The construction $g \mapsto [g]|_{Z_q}$ defines 
a homomorphism of $S$-groups ${\rm{SO}}(q) \rightarrow \underline{\rm{Aut}}_{Z_q/S}$
and we are claiming that this is trivial.  By Lemma \ref{stdq} it suffices
to consider $q = q_n$ over $S = \Spec \Z$.  Since ${\rm{SO}}_n$ is $\Z$-flat, it suffices
to treat  the problem over $\Z[1/2]$, or even over $\Q$.  We have 
reduced to the case when $Z_q$ is \'etale over $S$, so its automorphism scheme is
$(\Z/2\Z)_S$.  Since ${\rm{SO}}(q) \rightarrow S$ is smooth with connected fibers, 
$\Hom_{S\mbox{-}{\rm{gp}}}({\rm{SO}}(q),(\Z/2\Z)_S) = 1$. 

Now returning to the original base $S$,  since $[g]$ is an automorphism of the $Z_q$-algebra
${\rm{C}}(V,q)$ that becomes a matrix algebra over $Z_q$ fppf-locally on $S$, 
by the relative Skolem--Noether theorem (as in the proof of Proposition \ref{disconnok}) 
we may Zariski-localize on the base so that $[g]$ is inner.  That is,
there exists a unit $u$ of ${\rm{C}}(V,q)$ such that 
$[g](x) =  uxu^{-1}$ for all points $x$ of ${\rm{C}}(V,q)$.
Clearly $u$ is a point of ${\rm{GPin}}'(q)$ that lifts $g$.

Since we have proved that all points of ${\rm{SO}}(q)$ Zariski-locally lift
into ${\rm{GPin}}'(q)$, and ${\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$,
to show that ${\rm{GPin}}'(q)$ is carried into ${\rm{SO}}(q)$
it suffices to check that if $\zeta \in \mu_2(S')$ for an $S$-scheme $S'$ and it arises
from $u \in {\rm{GPin}}'(q)(S')$ then $\zeta = 1$.  We may and do rename $S'$ as $S$. The automorphism
$v \mapsto \zeta v$ of $V$ uniquely extends to an algebra automorphism $f$ of ${\rm{C}}(V,q)$
that is the identity on ${\rm{C}}_0(V,q)$ and is multiplication by $\zeta$ on ${\rm{C}}_1(V,q)$.
By hypothesis, the inner automorphism $x \mapsto uxu^{-1}$ of ${\rm{C}}(V,q)$
agrees with $f$ on $V$, so it agrees with $f$ (since $V$ generates ${\rm{C}}(V,q)$ as an algebra).
In particular, $f$ is trivial on the $\Z/2\Z$-graded central subalgebra $Z_q$.  But the odd part $Z_q^1$
is an invertible sheaf on which $f$ acts as multiplication by $\zeta$, so $\zeta = 1$. 
This completes the proof of ($\ast$).

The representation $\pi_q$ of ${\rm{GSpin}}(q) = {\rm{GPin}}'(q) \bigcap {\rm{C}}_0(V,q)^{\times}$ on $V$
is via conjugation
inside ${\rm{C}}(V,q)$ (the quadratic twist is trivial in even degree), 
so since $Z_q^{\times}$ has degree-0 part $\mathbf{G}_{\rm{m}}$ we can deduce (1)
from ($\ast$) once we show that the quotient map ${\rm{GPin}}'(q) \rightarrow {\rm{SO}}(q)$
in ($\ast$) carries ${\rm{GSpin}}(q)$ onto ${\rm{SO}}(q)$ Zariski-locally on $S$.
By Lemma \ref{unitchev}, ${\rm{GPin}}(q)$ is carried onto
${\rm{SO}}(q)$ under the Zariski-quotient map in ($\ast$).  Since scaling by a Zariski-local generator
of $Z_q^1$ swaps ${\rm{GSpin}}(q)$ with the fiber of ${\rm{GPin}}(q) \rightarrow (\Z/2\Z)_S$ over 1,
it follows via (\ref{zgs}) that ${\rm{GSpin}}(q)$ is also carried onto ${\rm{SO}}(q)$ as required in (1).

The initial isomorphism assertion in (2) is now clear, and for the rest it remains to analyze the $S$-homomorphism
$$(Z_{q}^{\times})_{\rm{lh}} \stackrel{\pi_q}{\rightarrow} {\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q).$$
The kernel contains $\mathbf{G}_{\rm{m}}$, so it induces an $S$-homomorphism
$$(\Z/2\Z)_S = (Z_{q}^{\times})_{\rm{lh}}/\mathbf{G}_{\rm{m}} \rightarrow \mu_2 \times {\rm{SO}}(q).$$
Our problem is to show that the second component of this map is trivial 
and that the first component is the canonical $S$-homomorphism $(\Z/2\Z)_S \rightarrow \mu_2$.
In other words, we have to show that a local generator of $Z_q^1$ is carried to
$-1 \in {\rm{O}}(q)(S)$ under $\pi_q$.  But this is obvious, since $\pi_q$ on ${\rm{GPin}}(q)$ is defined as the quadratic
twist of conjugation inside ${\rm{C}}(V,q)$ against $u \mapsto (-1)^{\deg_q(u)}$.
\end{proof}

For odd $n$, Proposition \ref{gpinprop} shows that 
${\rm{GSpin}}(q)$ is the relative identity component of the smooth $S$-group ${\rm{GPin}}(q)$ 
(which we took as the {\em definition} of ${\rm{GSpin}}(q)$ for even $n$), and our definition
${\rm{GSpin}}(q) = {\rm{C}}_0(V,q) \bigcap {\rm{GPin}}(q)$ for odd $n$ was
earlier seen to be valid for even $n$.  Hence, our descriptions of
the relationships between ${\rm{GPin}}(q)$ and ${\rm{GSpin}}(q)$
for even and odd $n$ are consistent. 

The fibral connectedness of the naive Clifford group ${\rm{GPin}}'(q)$
for odd $n$ forces us to use the grading if we wish to define
${\rm{GSpin}}'(q)$ for any $n \ge 1$: it is ${\rm{C}}_0(V,q) \bigcap {\rm{GPin}}'(q)$.
But this is  ${\rm{GSpin}}(q)$, so it provides nothing new.

\begin{corollary}\label{gpinp} Assume $n \ge 1$ is odd. The subgroup ${\rm{GSpin}}(q) \subset {\rm{GPin}}(q)$
is a central extension of ${\rm{SO}}(q)$ by $\Gm$, and 
$${\rm{GPin}}(q) = (Z_q^{\times})_{\rm{lh}} \times^{\Gm} {\rm{GSpin}}(q),\,\,\,
{\rm{GPin}}'(q) = Z_q^{\times} \times^{\Gm} {\rm{GSpin}}(q).$$
\end{corollary}

This result is a relative version of \cite[II.3.2]{chevquad}. 

\begin{proof}
Since $Z_q^{\times}$ has degree-0 part $\Gm$ and 
local bases of $Z_q^1$ are units, Lemma \ref{unitchev}
and Proposition \ref{gpinprop} yield the assertions immediately.
\end{proof}

In the remainder of \S\ref{spinsec} we are primarily interested in odd $n$
(e.g., to construct the simply connected central cover of ${\rm{SO}}(q)$
for odd $n \ge 3$), so for ease of notation 
through the end of \S\ref{spinsec} we denote by $Z_q$ the center of the {\em entire}
algebra ${\rm{C}}(V,q)$ regardless of the parity of $n$.
For even $n$ this is {\em not} the center of ${\rm{C}}_0(V,q)$ (in contrast with Proposition \ref{cliffstr}),
but since our main focus is on odd $n$ this will not create confusion.

For any $n \ge 1$, the {\em main anti-involution} $\alpha$ of ${\rm{C}}(V,q)$ is
\index{Clifford algebra!main anti-involution}
defined as in the proof of Lemma \ref{mu2ker}: it is the unique anti-involution
that restricts to the identity on $V$, which is to say that it
carries $v_1 \cdots v_r$ to $v_r \cdots v_1$.  For
any $u \in {\rm{GPin}}'(q)$, the same calculation
as in the proof of Lemma \ref{mu2ker} (but setting
$\varepsilon_u = 1$ and $h(v) = uvu^{-1}$) shows that $\alpha(u)u \in Z_q^{\times}$
and that the resulting $S$-morphism 
\index{ aaanuqp@$\nu'_q$}
$$\nu'_q:{\rm{GPin}}'(q) \rightarrow Z_q^{\times}$$
given by $u \mapsto \alpha(u)u = u \alpha(u)$
is a homomorphism of $S$-groups; it is called the {\em unrestricted Clifford norm}.   
When $n$ is even, so $Z_q = \calO_S$, $\nu'_q$ restricts to the squaring map on $Z_q^{\times}$.
When $n$ is odd, we have: 

\begin{lemma}\label{cnorm} For odd $n \ge 1$, $\alpha:Z_q \rightarrow Z_q$ is
the identity map when $n \equiv 1 \bmod 4$ and is the canonical algebra
involution $z \mapsto {\rm{Tr}}_{Z_q/\calO_S}(z) - z$ when $n \equiv 3 \bmod 4$.
In particular, $\nu'_q$ is the squaring map on
$Z_q^{\times}$ when $n \equiv 1 \bmod 4$ and it is the restriction of the algebra norm $Z_q \rightarrow \calO_S$
when $n \equiv 3 \bmod 4$. 
\end{lemma}

\begin{proof}
By working fppf-locally on $S$, we may assume $q = q_n$ for $n = 2m+1$ with $m \ge 0$.  Hence, if
$\{e_0,\dots,e_{2m}\}$ denotes the standard basis, then 
we saw in the proof of Proposition \ref{cliffodd} that 
$Z_q = \calO_S \oplus \calO_S z_0 = \calO_S[t]/(t^2 - 1)$, where
$$z_0 = e_0 \prod_{i=1}^m (1 - 2e_{2i-1} e_{2i})$$
and $e_{2i-1} e_{2i} + e_{2i} e_{2i-1} = 1$.
The main anti-involution $\alpha$ carries $e_j$ to $e_j$,
so $\alpha(1 - 2e_{2i-1} e_{2i}) = 1 - 2e_{2i}e_{2i-1} = 1 - 2(1 - e_{2i-1}e_{2i}) = -(1 - 2e_{2i-1} e_{2i})$.  Hence,
$\alpha(z_0) = (-1)^{m}z_0$, so
$\alpha$ is the identity on $Z_q$ for even $m$ and is the 
canonical involution $x \mapsto {\rm{Tr}}_{Z_q/\calO_S}(x) - x$
for odd $m$. 
\end{proof}

The analogue of Lemma \ref{cnorm} for even $n$ was seen early in the proof
of Lemma \ref{mu2ker}:  for even $n \ge 2$, the effect of $\alpha$ on the rank-2 finite \'etale center of ${\rm{C}}_0(V,q)$ is
the identity when $4|n$ and is the unique fiberwise nontrivial automorphism when $n \equiv 2 \bmod 4$.


We are primarily interested
in the restriction $\nu_q$ of $\nu'_q$ to the subgroup ${\rm{GPin}}(q)$ of locally homogeneous points
$u$ for any $n \ge 1$.
For such $u$, $\nu_q(u)$ is the product of the locally homogeneous points $u$ and $\alpha(u)$
of the same degree, so $\nu_q: u \mapsto u \alpha(u)$ is valued in
the degree-0 part $\mathbf{G}_{\rm{m}}$ of $Z_q^{\times}$.
This is the {\em Clifford norm}
\begin{equation}\label{anynorm}
\nu_q:{\rm{GPin}}(q) \rightarrow \mathbf{G}_{\rm{m}}
\end{equation}
for any $n \ge 1$.
\index{Clifford norm}
\index{ aaanuq@$\nu_q$}

\begin{remark}\label{nuqcomp}
As in Remark \ref{normp} with even $n$, for any $n \ge 1$ there is
an anti-involution $\alpha^{-}$ of ${\rm{C}}(V,q)_{\rm{lh}}$
defined similarly to $\alpha$ except with a sign twist on the odd part.
This yields a variant $\nu^{-}_q$ of (\ref{anynorm}) that agrees
\index{ aaanuqminus@$\nu^{-}_q$}
with $\nu_q$ on the even part ${\rm{GSpin}}(q)$.
Recall from (\ref{spinqt}) for even $n$ and from Proposition \ref{gpinprop}(2) for odd $n$ that
${\rm{GPin}}(q)/{\rm{GSpin}}(q)$ is identified with $(\Z/2\Z)_S$ via
the restriction of $\deg_q: {\rm{C}}(V,q)_{{\rm{lh}}} \rightarrow (\Z/2\Z)_S$.
It is the map $\nu_q^{-}$ that is used in \cite[(3.7)]{abs} and \cite[IV, \S6.1]{knus},
whereas $\nu_q$ is used in \cite[App.\,I]{frohlich}

We claim that $\nu_q^{-}$ is obtained from $\nu_q$ via multiplication against
$(-1)^{\deg_q}$.  To prove this claim we may work
Zariski-locally on $S$ so that there exists $v_0 \in V(S)$
satisfying $q(v_0) \in \calO(S)^{\times}$.   The description of
${\rm{GPin}}(q)/{\rm{GSpin}}(q)$ implies that
${\rm{GPin}}(q)$ is generated (for the \'etale or fppf topolgies)
by ${\rm{GSpin}}(q)$ and a single such $v_0$.  But $\nu_q^{-}$ and $\nu_q$ agree on
${\rm{GSpin}}(q)$ since $\alpha$ and $\alpha^{-}$ agree on ${\rm{C}}_0(V,q)^{\times}$,
so comparing $\nu_q^{-}$ and $\nu_q$ up to the desired quadratic twist amounts to comparing
their values on $v_0$.  By definition,
$\nu_q(v_0) = v_0^2 = q(v_0)$ whereas 
$\nu_q^{-}(v_0) = -q(v_0)$. 
Their ratio is $-1 = (-1)^{\deg_q(v_0)}$.
\end{remark}

Now assume $n = 2m+1 \ge 3$ (so $m \ge 1$).
The group ${\rm{SO}}(q)$ is semisimple
of type ${\rm{B}}_m$, so the derived group 
\index{spin group|(}
$${\rm{Spin}}(q) := \mathscr{D}({\rm{GSpin}}(q))$$
is a semisimple $S$-group that is a central extension of
${\rm{SO}}(q)$ by a finite fppf subgroup of $\Gm$ (Proposition \ref{isogfiber}).  
The group ${\rm{SO}}(q)$ is adjoint since
$n$ is odd, and calculations with ${\rm{B}}_m$ 
show that the simply connected central cover of ${\rm{SO}}(q)$ has degree 2.
We can now
adapt arguments in the proof of Lemma \ref{mu2ker} 
to show that  the central isogeny
$\pi_q:{\rm{Spin}}(q) \rightarrow {\rm{SO}}(q)$ has kernel
equal to $\mu_2 \subset \Gm$, so this is the simply connected central cover:

\begin{proposition} Assume $n \ge 3$ is odd. The map $\pi_q$ identifies 
${\rm{Spin}}(q)$ with the simply connected central cover of ${\rm{SO}}(q)$.
\end{proposition}

\begin{proof}
By Lemma \ref{stdq}, it suffices to treat the case $q = q_n$ over $\Z$. The finite multiplicative
type kernel $\ker \pi_q$ is flat and so has constant
degree that must be 1 or 2, so  we 
just have to rule out the possibility of degree 1.  Equivalently, it suffices to prove that over a
field $k$ of characteristic $\ne 2$ (or even just algebraically closed of characteristic 0), 
 the {\em central} subgroup $\mu_2 \subset \Gm$ in ${\rm{GSpin}}(q)$ is contained in the derived group ${\rm{Spin}}(q)$.

The first step is to reduce to the case $n = 3$. 
We have $n = 2m+1 \ge 3$ and $q = q_n$.   Let $V_3 \subset V = k^n$ be the span of
$\{e_0, e_1, e_2\}$ (so $q|_{V_3} = q_3$) and let $V'$ be the span of $\{e_3,\dots,e_{2m}\}$
(so $V' = 0$ if $n = 3$).  
The inclusion $(V_3,q_3) \hookrightarrow (V,q)$ induces 
an injective homomorphism of $\Z/2\Z$-graded algebras $j:{\rm{C}}(q_3) \rightarrow {\rm{C}}(V,q)$.
Since $V_3$ is orthogonal to $V'$, the {\em even} subalgebra ${\rm{C}}_0(q_3)$ centralizes
${\rm{C}}(V',q|_{V'}) \subset {\rm{C}}(V,q)$ because
for $i \in \{0, 1, 2\}$ and $i' > 2$ we have
$e_i e_{i'} = -e_{i'} e_i$ (treating $i=0$ separately from $i = 1, 2$, using that 
$q(e_0 + e_{i'}) = 1$ and $q(e_0) = 1$). 
Hence, a unit in ${\rm{C}}_0(q_3)^{\times}$
whose conjugation action on ${\rm{C}}_0(q_3)$ preserves $V_3$ is carried
into the group ${\rm{GSpin}}(q)$ of even units in ${\rm{C}}(V,q)$ whose conjugation
action on ${\rm{C}}(V,q)$ preserves $V$.  In other words, $j$ carries
${\rm{GSpin}}(q_3)$ into ${\rm{GSpin}}(q)$, so  on derived groups it
carries ${\rm{Spin}}(q_3)$ into ${\rm{Spin}}(q)$.   But $j$ carries $\Gm$ to $\Gm$ via the identity map,
so  to prove that $\mu_2 \subset {\rm{Spin}}(q)$ it suffices to treat the case of $q_3$.  
That is, we may and do now assume $n = 3$.

If ${\rm{Spin}}(q)$ does not contain the central $\mu_2$ then the natural map
${\rm{Spin}}(q) \rightarrow {\rm{SO}}(q)$ is an isomorphism. The inverse
would provide a section to ${\rm{GSpin}}(q) \rightarrow {\rm{SO}}(q)$
that identifies ${\rm{GSpin}}(q)$ with $\Gm \times {\rm{SO}}(q)$.
In particular,  the kernel of the Clifford norm $\nu_q:{\rm{GSpin}}(q) \rightarrow \Gm$
would equal $\mu_2 \times {\rm{SO}}(q)$, which is disconnected (since ${\rm{char}}(k) \ne 2$).
Hence, it suffices to prove that the kernel of $\nu_q$ on ${\rm{GSpin}}(q)$ is irreducible.

For the standard basis $\{e_0, e_1, e_2\}$ of $V_3$,
let $e = e_0 e_1, e' =  e_0 e_2, e'' = e_1 e_2$, so $\{1, e, e', e''\}$ is a $k$-basis of ${\rm{C}}_0(q_3)$.
We claim that the inclusion ${\rm{GSpin}}(q_3) \subset {\rm{C}}_0(q_3)^{\times}$
between $k$-groups is an equality.  Since ${\rm{GSpin}}(q_3)$
is smooth and connected of dimension 4 (a central extension of
${\rm{SO}}_3$ by $\mathbf{G}_{\rm{m}}$), it suffices
to show the same for ${\rm{C}}_0(q_3)^{\times}$.
For any finite-dimensional associative algebra $A$ over a field $F$, the associated $F$-group
$\underline{A}^{\times}$ of units is 
defined by the non-vanishing on the affine space $\underline{A}$ of the determinant of the left multiplication
action of $A$ on itself. In particular, $\underline{A}^{\times}$ is smooth
and connected of dimension equal to $\dim_F A$.  Hence,
${\rm{C}}_0(q_3)^{\times}$ is smooth and connected of dimension 4
as desired.

Computing as in Remark \ref{evencenter}, the algebra structure on ${\rm{C}}_0(q_3)$ is determined by the relations
\begin{eqnarray*}
e^2 = {e'}^2 = 0, {e''}^2 = e'', ee' = -e'', e'e = e'' - 1, \\ e'e'' = e', e''e' = 0, ee'' = 0, e''e = e,
\end{eqnarray*} 
and the restriction to ${\rm{C}}_0(q_3)$ of the main anti-involution
$\alpha$ of ${\rm{C}}(q_3)$ is given by 
$$u = t + ae + be' + ce'' \mapsto (t+c) - ae - be' - ce'' =: u^{\ast}.$$
Using the above relations, we compute that $\nu(u) := u^{\ast}u = ab + t^2 + ct$. 
By inspection, $\nu -  1$ is an irreducible polynomial in $k[t,a,b,c]$ and hence
its non-unit restriction over the open subset
${\rm{GSpin}}(q_3)$ in the affine 4-space ${\rm{C}}_0(q_3)$
(which has zero locus ${\rm{Spin}}(q_3)$) has irreducible zero locus. 
 \end{proof}

Fix an odd $n \ge 1$.  If $n \ge 3$ then $\mu_2 \subset {\rm{Spin}}(q)$
and this must exhaust the order-2 center, so 
${\rm{GSpin}}(q) = {\rm{Spin}}(q) \times^{\mu_2} \Gm$
(by Proposition \ref{gpinprop}(1)) and ${\rm{Spin}}(q) = \ker(\nu_q|_{{\rm{GSpin}}(q)})$.
Hence, Corollary \ref{gpinp} gives 
$${\rm{GPin}}(q) = {\rm{Spin}}(q) \times^{\mu_2} (Z_q^{\times})_{\rm{lh}},\,\,\,
{\rm{GPin}}'(q) = {\rm{Spin}}(q) \times^{\mu_2} Z_q^{\times}.$$
These hold if $n = 1$ by {\em defining} ${\rm{Spin}}(q) := \mu_2$ inside
${\rm{GSpin}}(q) = \Gm$ if $n = 1$.  
\index{spin group|)}

\begin{remark}\label{buildspin} Using notation as in the setting of Remark \ref{halfspin}
(which is applicable whenever $(V,q)$ is split), we obtain vector bundle representations
of the even Clifford group ${\rm{GSpin}}(q) \subset {\rm{C}}_0(V,q)$ on which the central $\Gm$
acts as ordinary scaling:
\index{spin \& half-spin representations}
if $n$ is even then use the vector bundles $A_{+}$ and $A_{-}$, and 
if $n$ is odd then use the vector bundle $A$ given by the exterior algebra of $W$.
The kernel ${\rm{Spin}}(q) $ of the Clifford norm
${\rm{GSpin}}(q) \rightarrow \Gm$ is a central extension of ${\rm{O}}(q)$ by $\mu_2$.
Thus, the central $\mu_2$ acts by ordinary scaling (hence fiberwise nontrivially)
for the restriction to ${\rm{Spin}}(q)$ of the action of ${\rm{C}}_0(V,q)^{\times}$
on $A_{\pm}$ when $n$ is even and on $A$ when $n$ is odd. 
The actions on $A_{+}$ and $A_{-}$ for even $n$ are the {\em half-spin} representations
of ${\rm{Spin}}(q)$, and the action on $A$ for odd $n$ is the {\em spin} representation
of ${\rm{Spin}}(q)$; these do {\em not} factor through ${\rm{SO}}(q)$. 
\end{remark}


\subsection{Pin groups and spinor norm}\label{pinsubsec}


For any $n \ge 1$, the {\em Pin group} ${\rm{Pin}}(q)$ is the kernel of 
\index{pin group}
\index{ aaapinq@${\rm{Pin}}(q)$}
the Clifford norm $\nu_q: {\rm{GPin}}(q) \rightarrow \Gm$
as in (\ref{anynorm}).
For odd $n \ge 1$, 
the {\em naive Pin group} ${\rm{Pin}}'(q)$ is the kernel of the unrestricted Clifford norm
$\nu'_q:{\rm{GPin}}'(q) \rightarrow Z_q^{\times}$
(so ${\rm{Pin}}(q) \subset {\rm{Pin}}'(q)$).  


\begin{remark} For the modified Clifford norm $\nu_q^{-}$
as in Remark \ref{nuqcomp}, its kernel is an $S$-subgroup ${\rm{Pin}}^{-}(q)$ of ${\rm{GPin}}(q)$
\index{ aaapinqminus@${\rm{Pin}}^{-}(q)$}
distinct from ${\rm{Pin}}(q)$ in general.  For example, if $S$ is over $\Z[1/2]$
and $v \in V(S)$ satisfies $q(v) = 1$
then $v \in {\rm{Pin}}(q)(S)$ but $v \not\in {\rm{Pin}}^{-}(q)(S)$ (since $-q(v) = -1 \ne 1$).
Since $\nu_q$ and $\nu_q^{-}$ agree on the $S$-group ${\rm{C}}_0(V,q)^{\times}$, we have
${\rm{Pin}}^{-}(q) \cap {\rm{C}}_0(V,q) = {\rm{Spin}}(q)$.
In \cite[Thm.\,3.11]{abs}  and \cite[IV, \S6.2]{knus} it is ${\rm{Pin}}^{-}(q)$ that is called
the ``Pin group'' attached to $(V,q)$ whereas in \cite[App.\,I]{frohlich}
the group ${\rm{Pin}}(q)$ is used (denoted  there as $\widetilde{{\rm{Pin}}}(q)$ to avoid
conflict with the notation in \cite{abs}).
\end{remark}


Suppose $n$ is even, so (as we saw early in \S\ref{spinsec}) ${\rm{GSpin}}(q)$ is an open and closed
subgroup of ${\rm{GPin}}(q)$.  The kernel of the Clifford norm on ${\rm{GSpin}}(q)$ is
the smooth $S$-group ${\rm{Spin}}(q)$ (as we saw below Remark \ref{normp}), so
${\rm{Spin}}(q)$ is an open and closed subgroup of ${\rm{Pin}}(q)$.  But
${\rm{GPin}}(q)$ maps onto ${\rm{O}}(q)$ with
kernel $\Gm$, and $\nu_q$ carries this kernel 
onto $\Gm$ via $t \mapsto t^2$, so the kernel ${\rm{Pin}}(q)$ of $\nu_q$
\index{pin group!relation with spin group|(}
\index{spin group!relation with pin group|(}
fits into an fppf central extension 
\begin{equation}\label{pino}
1 \rightarrow \mu_2 \rightarrow {\rm{Pin}}(q) \rightarrow {\rm{O}}(q) \rightarrow 1.
\end{equation}
The subgroup ${\rm{Spin}}(q) \subset {\rm{Pin}}(q)$ is compatibly a central extension of
${\rm{SO}}(q)$ by $\mu_2$ (by Lemma \ref{mu2ker} for even $n \ge 4$, 
and direct arguments with the definition of ${\rm{Spin}}(q)$ when $n = 2$),
so ${\rm{Pin}}(q)/{\rm{Spin}}(q) = {\rm{O}}(q)/{\rm{SO}}(q) = (\Z/2\Z)_S$ when $n$ is even.
Hence, ${\rm{Pin}}(q)$ is a smooth $S$-affine $S$-group for even $n$.

For odd $n$ the analogous assertions
are more subtle to prove, due to complications in characteristic 2. 
Suppose $n \ge 1$ is odd.
By Lemma \ref{cnorm},
if $n \equiv 1 \bmod 4$ then 
\begin{equation}\label{pin3}
{\rm{Pin}}(q) = {\rm{Spin}}(q) \times^{\mu_2} (Z_q^{\times})_{\rm{lh}}[2],\,\,\,\,\,
{\rm{Pin}}'(q) = {\rm{Spin}}(q) \times^{\mu_2} Z_q^{\times}[2],
\end{equation}
(writing $A[2]$ to denote the 2-torsion in $A$) whereas if $n \equiv 3 \bmod 4$  then
$${\rm{Pin}}(q) = {\rm{Spin}}(q) \times^{\mu_2} \ker({\rm{N}}_{Z_q/S}|_{(Z_q^{\times})_{\rm{lh}}}),\,\,\,\,\,
{\rm{Pin}}'(q) = {\rm{Spin}}(q) \times^{\mu_2} \ker({\rm{N}}_{Z_q/S}|_{Z_q^{\times}}).
$$
Hence, if $n \equiv 1 \bmod 4$ then $${\rm{Pin}}(q)/{\rm{Spin}}(q) = (Z_q^{\times})_{\rm{lh}}[2]/\mu_2,\,\,\,
{\rm{Pin}}'(q)/{\rm{Spin}}(q) = Z_q^{\times}[2]/\mu_2,$$
whereas if $n \equiv 3 \bmod 4$ then 
$${\rm{Pin}}(q)/{\rm{Spin}}(q) = \ker({\rm{N}}_{Z_q/S}|_{(Z_q^{\times})_{\rm{lh}}})/\mu_2, \,\,\,
{\rm{Pin}}'(q)/{\rm{Spin}}(q) = \ker({\rm{N}}_{Z_q/S}|_{Z_q^{\times}})/\mu_2.$$
The norm map ${\rm{N}}_{Z_q/S}:Z_q^{\times} \rightarrow \mathbf{G}_{\rm{m}}$
is an fppf surjection whose kernel is a torus over $S[1/2]$
and has geometric fiber $\mathbf{G}_{\rm{a}}$ in characteristic 2.
Hence, if $n \equiv 3 \bmod 4$ then ${\rm{Pin}}'(q)$ is smooth.
We will see later that if $n \equiv 1  \bmod 4$ then ${\rm{Pin}}'(q_n)$ is not even $\Z$-flat
(due to dimension-jumping of the fiber at $\Spec \F_2$).

\begin{proposition}\label{pinso}
Fix an odd $n \ge 1$.  The quotient ${\rm{Pin}}(q)/{\rm{Spin}}(q)$ is identified with $(\Z/2\Z)_S$
$($so ${\rm{Pin}}(q)$ is smooth$)$
and there is a natural central extension
$$1 \rightarrow \mu_2 \rightarrow {\rm{Pin}}(q) \rightarrow (\Z/2\Z)_S \times {\rm{SO}}(q) \rightarrow 1$$
whose restriction over $S[1/2]$ coincides with the restriction 
to ${\rm{Pin}}(q)|_{S[1/2]}$ of $\pi_q:{\rm{GPin}}(q) \rightarrow {\rm{O}}(q) =
\mu_2 \times {\rm{SO}}(q)$
via the unique $S[1/2]$-isomorphism $\mu_2 \simeq (\Z/2\Z)_{S[1/2]}$
\end{proposition}

%The quotient ${\rm{Pin}}'(q)/{\rm{Spin}}(q)$ has geometric fiber $\mathbf{G}_{\rm{a}}$
%in characteristic $2$, and its restriction over $S[1/2]$ is $(\Z/2\Z)_S$
%when $n \equiv 3 \bmod 4$ and is a rank-$1$ torus when $n \equiv 1 \bmod 4$.
%In particular, for odd $n$, ${\rm{Pin}}(q) = {\rm{Pin}}'(q)$ if and only if $n \equiv 3 \bmod 4$
%and $S$ is a $\Z[1/2]$-scheme.


Together with our preceding considerations for even $n$, we conclude that 
${\rm{Pin}}(q)/{\rm{Spin}}(q) = (\Z/2\Z)_S$ for {\em all} $n$
(proved in \cite[IV, (6.4.1)]{knus} for ${\rm{Pin}}^{-}(q)$)
and $\pi_q:{\rm{Pin}}(q) \rightarrow {\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$
is not an fppf quotient map on fibers in characteristic 2 when $n$ is odd.

\begin{proof}
First assume $n \equiv 1 \bmod 4$.  The exact sequence
$$1 \rightarrow \Gm \rightarrow (Z_q^{\times})_{\rm{lh}} \rightarrow (\Z/2\Z)_S \rightarrow 1$$
for the fppf topology (for any  odd $n$) identifies $(Z_q^{\times})_{\rm{lh}}[2]/\mu_2$ with $(\Z/2\Z)_S$, so
if $n \equiv 1 \bmod 4$ then
${\rm{Pin}}(q)/{\rm{Spin}}(q) = (\Z/2\Z)_S$ by (\ref{pin3}). 
%Likewise, 
%${\rm{Pin}}'(q)/{\rm{Spin}}(q) = Z_q^{\times}[2]/\mu_2 = (Z_q^{\times}/\Gm)[2]$,
%and over $S[1/2]$ this is uniquely isomorphic to the constant group $(\Z/2\Z)$ but its
%geometric fibers in characteristic 2 are $\Ga$.

Assume $n \equiv 3 \bmod 4$.
Under the identification of $(Z_{q_n}^{\times})_{\rm{lh}}$ with 
$\mathbf{G}_{\rm{m}} \times (\Z/2\Z)_S$ as in the proof
of Lemma \ref{cnorm}, 
${\rm{N}}_{Z_{q_n}/S}:(Z_{q_n}^{\times})_{\rm{lh}} \rightarrow \mathbf{G}_{\rm{m}}$
is identified with the product of the squaring map on $\mathbf{G}_{\rm{m}}$
and the canonical map $(\Z/2\Z)_S \rightarrow \mu_2 \hookrightarrow \mathbf{G}_{\rm{m}}$,
so $\ker({\rm{N}}_{Z_{q_n}/S}|_{(Z_{q_n}^{\times})_{\rm{lh}}})$ is identified with the
subgroup of $\Gm \times (\Z/2\Z)_S$ consisting of $(\zeta,c)$
such that $\zeta^2 = (-1)^c$.  Hence, the finite subgroup scheme
$${\rm{Pin}}(q)/{\rm{Spin}}(q) \subset
(\mu_4/\mu_2) \times (\Z/2\Z)_S \simeq \mu_2 \times (\Z/2\Z)_S$$
is (uniquely) isomorphic to $(\Z/2\Z)_S$ since $\mu_4/\mu_2 \simeq \mu_2$ via $\zeta \mapsto \zeta^2$.  
%In contrast, 
%the quotient 
%${\rm{Pin}}'(q)/{\rm{Spin}}(q) = \ker({\rm{N}}_{Z_q/S}|_{Z_q^{\times}})/\mu_2$ is a rank-1 torus over $S[1/2]$
%and has geometric fiber $\mathbf{G}_{\rm{a}}$ in characteristic 2 since
%$Z_q$ is \'etale of rank 2 over $S[1/2]$ and has geometric fiber
%algebra $k[x]/(x^2 - 1)$ in characteristic 2. 
\end{proof}
\index{pin group!relation with spin group|)}
\index{spin group!relation with pin group|)}


\begin{remark}\label{samepin} The analogous result for ${\rm{Pin}}'(q)$ in place of ${\rm{Pin}}(q)$ is that 
${\rm{Pin}}'(q)/{\rm{Spin}}(q)$ has geometric fiber $\mathbf{G}_{\rm{a}}$
in characteristic $2$ and restriction over $S[1/2]$ that is $(\Z/2\Z)_S$
if $n \equiv 1 \bmod 4$ but is  a rank-$1$ torus if $n \equiv 3 \bmod 4$.
In particular, for odd $n$, ${\rm{Pin}}(q) = {\rm{Pin}}'(q)$ if and only if $n \equiv 1 \bmod 4$
and $S$ is a $\Z[1/2]$-scheme.

To prove these assertions, 
for $n \equiv 1 \bmod 4$ the same exact sequence argument as in the preceding proof
shows that ${\rm{Pin}}'(q)/{\rm{Spin}}(q) = Z_q^{\times}[2]/\mu_2 = (Z_q^{\times}/\Gm)[2]$.
Over $S[1/2]$ this is uniquely isomorphic to the constant group $(\Z/2\Z)$ but its
geometric fibers in characteristic 2 are $\Add$.
For $n \equiv 3 \bmod 4$, the quotient 
${\rm{Pin}}'(q)/{\rm{Spin}}(q) = \ker({\rm{N}}_{Z_q/S}|_{Z_q^{\times}})/\mu_2$ is a rank-1 torus over $S[1/2]$
and has geometric fiber $\mathbf{G}_{\rm{a}}$ in characteristic 2 since
$Z_q$ is \'etale of rank 2 over $S[1/2]$ and has geometric fiber
algebra $k[x]/(x^2 - 1)$ in characteristic 2. 

The jumping of fiber dimension
implies that for $n \equiv 1 \bmod 4$ the naive Pin group ${\rm{Pin}}'_n := {\rm{Pin}}'(q_n)$ over $\Z$ is not flat 
at $\Spec \F_2$ (whereas we have seen that ${\rm{Pin}}'_n$ is
smooth when $n \equiv 3 \bmod 4$).
\end{remark}

Pin groups exhibit some subtleties under unit-scaling of $q$, as follows. 
For $c \in \calO(S)^{\times}$, ${\rm{C}}(V,cq)$ is not easily related to ${\rm{C}}(V,q)$ when 
$c$ is a non-square but the equality
${\rm{O}}(cq) = {\rm{O}}(q)$ inside ${\rm{GL}}(V)$ implies
${\rm{SO}}(cq) = {\rm{SO}}(q)$. Over the latter equality there is a unique isomorphism
${\rm{Spin}}(c q) \simeq {\rm{Spin}}(q)$ as simply connected central extensions by $\mu_2$.
However, if $S$ is local and we assume $n$ is even in case of residue characteristic 2
(so ${\rm{Pin}}(q)$ is a central extension of ${\rm{O}}(q) = {\rm{O}}(cq)$ by $\mu_2$) then 
${\rm{Pin}}(cq)$ and ${\rm{Pin}}(q)$ are never $S$-isomorphic as central extensions
when $c$ is not a square in $\calO(S)^{\times}$.  This rests on
the {\em spinor norm}, as we explain in Example \ref{spcalc}.

Likewise, for $S$ over $\Z[1/2]$, the groups ${\rm{Pin}}^{-}(q)$ and ${\rm{Pin}}(c q)$ are {\em never} 
isomorphic as central extensions of 
${\rm{O}}(q)={\rm{O}}( cq)$ by $\mu_2$ (even though their relative identity components are uniquely isomorphic
over ${\rm{SO}}(q) = {\rm{SO}}(cq)$ and for $c=-1$ they meet $V$ in the same locus $\{q=-1\}$ 
of non-isotropic vectors inside the respective Clifford algebras ${\rm{C}}(V,q)$
and ${\rm{C}}(V,-q)$). It suffices to check this on geometric
fibers, so consider $S = {\rm{Spec}}(k)$
for a field $k$ with ${\rm{char}}(k) \ne 2$. The reflection $r_v \in {\rm{O}}(q)(k)$
through non-isotropic $v$ has preimages 
in ${\rm{Pin}}^{-}(q)(\overline{k})$ and ${\rm{Pin}}(cq)(\overline{k})$ respectively identified with 
$\{\pm v/\sqrt{-q(v)}\}$ and $\{\pm v/\sqrt{cq(v)}\}$ in the Clifford algebras
${\rm{C}}(V,q)$ and ${\rm{C}}(V,cq)$. The squares of the elements in these preimages are equal to $-1$
and $1$  in $\mu_2(k)$ respectively, so elements of the preimages have respective orders 4 and 2. 
For example, if $\dim V = 1$ then the $k$-groups ${\rm{Pin}}^{-}(q)$ and
${\rm{Pin}}(cq)$ are each finite \'etale of order 4, but the first of these is cyclic 
whereas the second is 2-torsion.

We will see below that over any field $k$, with
$n$ even if ${\rm{char}}(k) = 2$, the central extensions ${\rm{Pin}}^{-}(q)$
and ${\rm{Pin}}(-q)$ of ${\rm{O}}(q) = {\rm{O}}(-q)$ by $\mu_2$ yield the {\em same}
connecting homomorphism ${\rm{O}}(q)(k) \rightarrow {\rm{H}}^1(k,\mu_2)$. However, the preceding
shows that these central extensions are {\em not} $k$-isomorphic when ${\rm{char}}(k) \ne 2$.

\begin{remark}\label{switt} Suppose $(V,q) = (k^n, \sum x_j^2)$ over a field $k$
with ${\rm{char}}(k) \ne 2$.  Assume
$q(k^{\times}) \subset  (k^{\times})^2$, as when $k = \overline{k}$ or $k = \mathbf{R}$ with positive-definite $q$, 
so ${\rm{Pin}}(q)(k) \rightarrow {\rm{O}}(q)(k)$ is surjective (since 
${\rm{O}}(q)(k)$ is generated by reflections $r_v$ in non-isotropic vectors $v \in V$ \cite[I.5.1]{chevquad},
and $v/\sqrt{q(v)}$ is a lift of $r_v$ with $\sqrt{q(v)} \subset k^{\times}$).
Under the permutation representation
\index{pin group!central extension of symmetric group}
\index{ aaasn@$\mathfrak{S}_n$}
$\mathfrak{S}_n \rightarrow {\rm{O}}(q)(k)$ of the symmetric group, the central extension ${\rm{Pin}}(q)(k)$
of ${\rm{O}}(q)(k)$ by $\mu_2(k)$ pulls back to a central extension $E_n$ of $\mathfrak{S}_n$ 
by $\{1,-1\}$.  Elements of $E_n$ lying over a transposition visibly have
order 2, and if $n \ge 4$ then elements of $E_n$ lying over a product of two transpositions with disjoint
support have order 4, so $E_n$ is the central extension 
denoted as $\widetilde{\mathfrak{S}}_n$ in \cite[\S1.5]{serrewitt}.

By contrast, the surjective ${\rm{Pin}}^{-}(-q)(k) \rightarrow {\rm{O}}(-q)(k) = 
{\rm{O}}(q)(k)$ pulls back to a central extension of
$\mathfrak{S}_n$ by $\{1,-1\}$ in which elements lying over any transposition have order 4.
The quadratic form $-q$ on $\mathbf{R}^n$
is used in \cite[\S2]{abs} because ${\rm{Spin}}({\mathbf{R}}^n, -q)$ is the anisotropic
$\mathbf{R}$-form of ${\rm{Spin}}_n$, so \cite{abs} uses the Pin$^{-}$ construction resting on
the signed Clifford norm  because ${\rm{Pin}}^{-}(-q)(\mathbf{R}) \rightarrow {\rm{O}}(-q)(\mathbf{R})$ is surjective
whereas ${\rm{Pin}}(-q)(\mathbf{R}) \rightarrow {\rm{O}}(-q)(\mathbf{R})$ is not surjective (due to obstructions
provided by the spinor norm, as explained below).
\end{remark}

As a prelude to defining the spinor norm,
observe that for $(V,q)$ with rank $n \ge 1$ over a scheme $S$, if 
$v_0 \in V(S)$ satisfies $q(v_0) = 1$
then $\nu_q(v_0) = v_0^2 = 1$.  
%(The role of the sign is due to the intervention of signs in the definition of
%the anti-involution $\alpha$ that is used to define the Clifford norm $\nu_q$.)
Hence,
$v_0$ belongs to the group  ${\rm{Pin}}(q)(S)$ of $S$-points of the kernel of the Clifford norm $\nu_q$ on
${\rm{GPin}}(q)$,
and it lies over the reflection $r_{v_0} \in {\rm{O}}(q)(S)$.

More generally, assuming $S$ is a $\Z[1/2]$-scheme when $n$ is odd (but arbitrary when $n$ is even),
the  structure on ${\rm{Pin}}(q)$ as a {\em central} extension of
${\rm{O}}(q)$ by $\mu_2$ yields a connecting homomorphism
(called the {\em spinor norm})
\index{ aaaspq@${\rm{sp}}_q$}
\index{spinor norm}
\index{orthogonal group!spinor norm}
$${\rm{sp}}_q: {\rm{O}}(q)(S) \rightarrow {\rm{H}}^1(S,\mu_2)$$
that carries the reflection $r_v$ through $v \in q^{-1}(\Gm) \cap V(S)$
to the class of the $\mu_2$-torsor of square
roots of $q(v)$ since over the fppf cover $S' \rightarrow S$
defined by $t^2 = q(v)$ we have $q(v/t) = 1$ and $r_{v/t} = r_v$.  Via the equality 
${\rm{O}}(cq) = {\rm{O}}(q)$ inside ${\rm{GL}}(V)$ for $c \in \calO(S)^{\times}$, observe that 
${\rm{sp}}_{cq}(r_v) =  [c] \cdot {\rm{sp}}_q(r_v)$ where $[c]$ is the image of $c$ under
$\calO(S)^{\times}/(\calO(S)^{\times})^2 \hookrightarrow {\rm{H}}^1(S,\mu_2)$. 


If $S = {\rm{Spec}}(k)$ for a field $k$, with $n$ even when ${\rm{char}}(k) = 2$, then 
${\rm{O}}(q)(k)$ is generated by such reflections except if
$k = \mathbf{F}_2$ with $\dim V = 4$ \cite[I.5.1]{chevquad}. Hence,
the condition ${\rm{sp}}_q(r_v) = q(v)$ determines
${\rm{sp}}_q$ (as $k^{\times}/(k^{\times})^2 = 1$ when $k = \mathbf{F}_2$),
and ${\rm{sp}}_{cq} = [c]^{\pi_0} \cdot {\rm{sp}}_q$ for $c \in k^{\times}$
where $\pi_0: {\rm{O}}(q) \twoheadrightarrow {\rm{O}}(q)/{\rm{SO}}(q) = \Z/2\Z$
is projection to the component group. 

\begin{example}\label{spcalc}  Suppose $S$ is local, 
with $n$ even in case of residue characteristic 2. Consider
$c \in \calO(S)^{\times}$ such that ${\rm{Pin}}(cq) \simeq {\rm{Pin}}(q)$
as central extensions of ${\rm{O}}(cq) = {\rm{O}}(q)$ by $\mu_2$.  Pick
residually non-isotropic
$v \in V$, so $q(v) \in \calO(S)^{\times}$.
It follows that ${\rm{sp}}_{cq}(r_v) = {\rm{sp}}_q(r_v)$.  But
${\rm{sp}}_{cq}(r_v) = [c] \cdot {\rm{sp}}_q(r_v)$, so
$c$ must be a square on $S$.
Hence, if there exists such an isomorphism of Pin groups
as central extensions then $c$ is a square on $S$ (and the converse is obvious).
\end{example}

\begin{example}
For fiberwise non-degenerate quadratic spaces $(V,q)$ and $(V',q')$ over $S$, 
the Clifford algebra
of the orthogonal sum $(V \oplus V', q \perp q')$ is naturally isomorphic to the super-graded
tensor product of ${\rm{C}}(V,q)$ and ${\rm{C}}(V',q')$ (as we noted
in the proof of Proposition \ref{cliffodd}).  It follows
that if $g \in {\rm{O}}(q)(S)$ and $g' \in {\rm{O}}(q')(S)$ then 
\begin{equation}\label{nuprod}
{\rm{sp}}_{q \perp q'}(g \oplus g') = {\rm{sp}}_q(g) {\rm{sp}}_{q'}(g')
\end{equation}
since $g \oplus g' = (g \oplus 1)\circ (1 \oplus g')$ and the inclusion
${\rm{C}}(V,q) \hookrightarrow {\rm{C}}(V \oplus V', q \perp q')$ carries
${\rm{Pin}}(q)$ into ${\rm{Pin}}(q \perp q')$ over
the inclusion ${\rm{O}}(q) \hookrightarrow {\rm{O}}(q \perp q')$.
\end{example}

For $(V,q)$ over a field $k$ 
with ${\rm{char}}(k) \ne 2$, there is a useful formula due to Zassenhaus \cite[\S2, Cor.]{zass}
for spinor norms that does not require
expressing $g \in {\rm{O}}(q)(k)$ in terms of reflections.
Below we use the structure of Clifford algebras to establish
such a formula. 
See Remark \ref{hahnrem} for a discussion of the case
${\rm{char}}(k) = 2$. 

Before we state and prove Zassenhaus' result,
it is convenient to recall some elementary properties
of generalized eigenspaces for orthogonal
transformations.  Consider $g \in {\rm{O}}(q)(k)$, so the $g$-action on $V$ is an automorphism that leaves invariant 
the associated symmetric bilinear form $B_q(v,v') = q(v+v') - q(v) - q(v')$
(which is non-degenerate, as ${\rm{char}}(k) \ne 2$).  
For any $\lambda \in \overline{k}^{\times}$, the generalized $\lambda$-eigenspace $V_{\overline{k}}(\lambda)$ of $g$ on
$V_{\overline{k}}$ is orthogonal to $V_{\overline{k}}(\mu)$ except possibly when $\lambda \mu = 1$.
Indeed, suppose $\lambda^{-1} \ne \mu$ and choose large $n$ so that 
the operator $(g - \lambda)^n$ on $V_{\overline{k}}(\lambda)$ vanishes.
Note that $g - \lambda^{-1}$ is invertible on $V_{\overline{k}}(\mu)$, so any $v' \in V_{\overline{k}}(\mu)$
can be written as $v' = (g - \lambda^{-1})^n(v'')$ for some $v'' \in V_{\overline{k}}(\mu)$ and hence
\begin{eqnarray*}
B_q(v,v') = B_q(v, (g - \lambda^{-1})^n(v'')) &=& B_q((g^{-1} - \lambda^{-1})^n(v),v'') \\ &=&
B_q((\lambda g)^{-n}(\lambda - g)^n(v), v'') \\ &=& 0.
\end{eqnarray*}
Since $V_{\overline{k}}$ is the direct sum of generalized eigenspaces for $g$, 
the non-degeneracy of $B_q$ on $V$ implies that
$V_{\overline{k}}(1/\lambda)$ and 
$V_{\overline{k}}(\lambda)$ are in perfect duality under $B_q$ for all $\lambda$
(even if $\lambda = \pm 1$).

Letting $V_0 \subset V$ be the generalized $-1$-eigenspace for $g$ and 
$V' \subset V$ be its $g$-stable $B_q$-orthogonal,
we conclude that $V_0 \oplus V' \rightarrow V$ is an isometry with $q$ non-degenerate on each of $V_0$
and $V'$.  Define $q_0 = q|_{V_0}$, $q' = q|_{V'}$,
$g_0 = g|_{V_0}$, and $g' = g|_{V'}$.
 The compatibility (\ref{nuprod}) implies 
$${\rm{sp}}_q(g) = {\rm{sp}}_{q_0}(g_0) {\rm{sp}}_{q'}(g').$$
But the effect of $-g$ on $V_0$ is visibly {\em unipotent}, and the $k^{\times}/(k^{\times})^2$-valued 
spinor norm kills unipotent elements
of ${\rm{O}}(q)(k)$ because ${\rm{char}}(k) \ne 2$, so ${\rm{sp}}_{q_0}(g_0) = {\rm{sp}}_{q_0}(-1)$.
Moreover, the element $g' \in {\rm{O}}(q')(k)$ does not have $-1$ as an eigenvalue,
so the eigenvalues of $g'$ aside from 1 occur in reciprocal pairs
with generalized eigenspaces for $\lambda, 1/\lambda \in \overline{k}^{\times} - \{1, -1\}$
having the same dimension.  Hence, $\Det(g') = 1$, so $g' \in {\rm{SO}}(q')$.
Note that $(V',q',g') = (V,q,g)$ when $-1$ is not an eigenvalue of $g$
(recovering that every $g \in {\rm{O}}(q) - {\rm{SO}}(q)$ has
$-1$ as an eigenvalue).

\begin{theorem}[Zassenhaus]\label{main}
Via the standard representation
${\rm{O}}(q) \hookrightarrow {\rm{GL}}(V)$ and the preceding notation, 
$${\rm{sp}}_q(g) = {\rm{disc}}(q_0) \cdot \Det((1+g')/2) \bmod (k^{\times})^2.$$
In particular, if $\Det(1+g) \ne 0$ then ${\rm{sp}}_q(g) = \Det((1+g)/2) \bmod (k^{\times})^2$.
\end{theorem}
\index{spinor norm!formula}
\index{Zassenhaus' formula}

%The sign in the discriminant factor is due to the sign in the definition of the anti-involution $\alpha$
%that intervenes in the definition of the Clifford norm.  Zassenhaus preferred to work with $\alpha^+$, corresponding
%to the modified Pin group ${\rm{Pin}}^+(q)$
%whose associated connecting homomorphism ${\rm{O}}(q)(k) \rightarrow k^{\times}/(k^{\times})^2$
%carried the reflection $r_v$ to the square class of $q(v)$ for 
%any non-isotropic $v \in V$, so the discriminant factor in his version of the formula is ${\rm{disc}}(q_0)$.

In view of the preceding calculations, the proof of Theorem \ref{main} reduces to separately
treating $(V',q',g')$ and $(V_0,q_0,-1)$.  By renaming
each of $(V',q')$ and $(V_0,q_0)$ as $(V,q)$, 
this amounts to the general identities ${\rm{sp}}_q(-1) = {\rm{disc}}(q)$ and
${\rm{sp}}_q(g) = \Det((1+g)/2) \bmod (k^{\times})^2$ for any
$g \in {\rm{SO}}(q)$ that does not have $-1$ as an eigenvalue.

\begin{lemma} For non-degenerate $(V,q)$ over a field $k$ with ${\rm{char}}(k) \ne 2$,
${\rm{sp}}_q(-1) = {\rm{disc}}(q)$.
\end{lemma}

\begin{proof}  Since ${\rm{char}}(k) \ne 2$, we can diagonalize $q$; i.e., 
$(V,q)$ is an orthogonal sum of 1-dimensional non-degenerate quadratic spaces. Verifying
${\rm{sp}}_q(-1) = {\rm{disc}}(q)$ therefore reduces to
showing that ${\rm{sp}}_{ax^2}(-1) = a \bmod (k^{\times})^2$.
The $\Z/2\Z$-graded $k$-algebra $${\rm{C}}(k,ax^2) = k[t]/(t^2-a) = k \oplus kt$$ is commutative 
and on the algebraic group of homogeneous units the 
Clifford norm is the squaring map (since $\alpha|_{Z_q}$ is the identity map
when $n \equiv 1 \bmod 4$), on which the kernel ${\rm{Pin}}(ax^2)$
is the functor of points $\{c + c' t\}$ where $2cc' = 0$ and $c^2 + a {c'}^2 = 1$.
The fiber of $\pi_{ax^2}: {\rm{Pin}}(ax^2) \rightarrow {\rm{O}}(ax^2) = \mu_2$ over $-1$
is the functor of points $c' t$ satisfying ${c'}^2 = a^{-1}$, so
${\rm{sp}}_{ax^2}(-1)$ is the $\mu_2$-torsor over $k$ classified by $a^{-1}$; i.e.,
${\rm{sp}}_{ax^2}(-1) = a^{-1} (k^{\times})^2 = a (k^{\times})^2$. 
\end{proof}

Now consider $g \in {\rm{SO}}(q)$ with $-1$ not an eigenvalue of $g$, so
$\Det(1+g) \ne 0$.  We claim that
${\rm{sp}}_q(g)$ is represented by
$\Det((1+g)/2)$.  The case $n=1$ is trivial (as then ${\rm{SO}}(q) = 1$), 
so we assume for the rest of the proof of Theorem \ref{main} that $n \ge 2$.
The cases of even and odd $n$ will be treated by similar arguments with different details
due to how the  structure of ${\rm{C}}_0(V,q)$ 
depends on the parity of $n$.

{\bf Step 1};  Suppose $n = 2m$ is even (with $m \ge 1$). 
By Proposition \ref{cliffstr} the $k$-algebra ${\rm{C}}_0(V,q)$ is
central simple of rank $2^{n-2}$ over a degree-2 finite \'etale
$k$-algebra $k'$ (which is the center of ${\rm{C}}_0(V,q)$), and the 
algebraic group of units ${\rm{C}}_0(V,q)^{\times}$ contains ${\rm{Spin}}(q)$.  If $(V,q)$ is split
then this $k$-algebra is a product of two copies of ${\rm{Mat}}_{2^{m-1}}(k)$, and the resulting
two $2^{m-1}$-dimensional representations of ${\rm{Spin}}(q)$ are the
\index{spin \& half-spin representations}
{\em half-spin representations} from Remark \ref{buildspin}.  For $m = 1$ -- i.e., $n=2$ -- and split $(V,q)$,
the group ${\rm{Spin}}(q)$ is a 1-dimesional split torus and these two 1-dimensional representations
are the two faithful 1-dimensional representations of such a torus.

Assume instead that $n = 2m+1$ is odd ($m \ge 1$). By Proposition \ref{cliffodd}
the $k$-algebra ${\rm{C}}_0(V,q)$ is central 
simple of rank $2^{n-1}$, and its unit group contains ${\rm{Spin}}(q)$. If moreover $(V,q)$ is split then 
this is a matrix algebra ${\rm{Mat}}_{2^{m}}(k)$, and the resulting
$2^{m}$-dimensional representation of ${\rm{Spin}}(q)$ is the {\em spin representation}
from Remark \ref{buildspin}.

{\bf Step 2}. We shall now reduce to the case when ${\rm{C}}_0(V,q)$ is a matrix algebra over its center
(though $(V,q)$ might not be split).
The reason for interest in this case is that if ${\rm{C}}_0(V,q)$ is a matrix algebra over its center then 
we obtain a $k$-descent $\rho$ of the direct sum of the half-spin representations of ${\rm{Spin}}(q)_{k_s}$ 
for even $n$ (using a choice of
$k$-basis of the center $k'$ of ${\rm{C}}_0(V,q)$) and a $k$-descent of the
spin representation of ${\rm{Spin}}(q)_{k_s}$ for odd $n$. 

We may certainly assume $k$ is finitely generated over its prime field, so there exists 
a finitely generated $\Z[1/2]$-subalgebra 
$R \subset k$  with fraction field $k$ such 
that: $(V,q)$ arises from a fiberwise non-degenerate quadratic space $(\mathscr{V},\mathscr{Q})$
of rank $n$ over $R$, $g \in  {\rm{SO}}(\mathscr{Q})(R)$, and $\Det(1+g) \in R^{\times}$.
Consider the spinor norm $${\rm{sp}}_{\mathscr{Q}}: {\rm{SO}}(\mathscr{Q})(R) \rightarrow {\rm{H}}^1(R, \mu_2).$$
By replacing $R$ with $R[1/r]$ for a suitable nonzero $r \in R$
we may arrange that the image of ${\rm{sp}}_{\mathscr{Q}}(g)$
in ${\rm{H}}^1(R, {\rm{GL}}_1) = {\rm{Pic}}(R)$ is trivial, so 
${\rm{sp}}_{\mathscr{Q}}(g)$ lies in the subgroup $R^{\times}/(R^{\times})^2 \subset {\rm{H}}^1(R,\mu_2)$.
To control this square class, at least after further Zariski-localization on $R$,
we now reduce to working over finite fields.

If ${\rm{char}}(k) > 0$ then we may use generic smoothness over the perfect
field $\mathbf{F}_p$ to find a nonzero $r' \in R$
such that $R[1/r']$ is $\mathbf{F}_p$-smooth. 
 Likewise, if ${\rm{char}}(k) = 0$
then generic smoothness over $\Q$ provides a nonzero $r' \in R$ such that
$R[1/r']$ is $\Z[1/N]$-smooth for a sufficiently divisible integer $N > 0$.
Hence, in all cases we can replace $R$ with a suitable $R[1/r']$
to arrange that $R$ is normal (i.e., integrally closed).
It suffices to show that ${\rm{sp}}_{\mathscr{Q}}(g) = \Det(1+g) \bmod (R^{\times})^2$
(as then localizing at the generic point will conclude the proof).

Functoriality of Kummer theory with respect to base change implies that
for any closed point $\xi \in {\rm{Spec}}(R)$ and its finite residue field $\kappa = \kappa(\xi)$, 
the specialization $g(\xi) \in
{\rm{SO}}(\mathscr{Q})(\kappa)$ has spinor norm in $\kappa^{\times}/(\kappa^{\times})^2$ that 
is equal to the $\xi$-specialization of ${\rm{sp}}_{\mathscr{Q}}(g)$.  
We claim that a unit $r$ in $R$ is a square in $R$ if and only if its $\xi$-specialization
in $\kappa(\xi)^{\times}$ is a square for all $\xi$, in which case taking $r$ to be
a representative of $\Det(1+g)^{-1} {\rm{sp}}_{\mathscr{Q}}(g)$ would reduce
our task to the case of finite fields.  More generally:

\begin{lemma} Let $X$ be a connected normal $\Z$-scheme of finite type. 
If a finite \'etale cover $X' \rightarrow X$ has split fibers
over all closed points then it is a split covering.
In particular, if $X$ is a $\Z[1/2]$-scheme and an element  $u \in \calO(X)^{\times}$
has square image in the residue field $\kappa(x)$ at every closed point $x$
then $u$ is a square in $\calO(X)$.  
\end{lemma}

\begin{proof}  The assertion for square roots of units follows from the rest by using the 
cover $X' \rightarrow X$ defined by $t^2 = u$.  In general, let $d = \dim(X)$ and 
let $m$ be the degree of $X'$ over $X$.
Consider the zeta functions $\zeta_X(s)$ and $\zeta_{X'}(s)$; these are
absolutely and uniformly convergent products on ${\rm{Re}}(s) \ge d+\varepsilon$
for any $\varepsilon > 0$ \cite[\S1.3]{serrez}. The zeta function of any
$\Z$-scheme $Y$ of finite type with dimension $d$ has a meromorphic
continuation to the half-plane ${\rm{Re}}(s) > d-1/2$ with pole at $s=d$
equal to the number of  $d$-dimensional irreducible components \cite[\S1.4, Cor.\,1]{serrez}.

The normal noetherian 
$X$ is connected and hence irreducible, so $\zeta_X(s)$ has a simple pole at $s=d$.
The hypotheses imply
that $\zeta_{X'} = \zeta_X^m$ where $m = [X':X]$, 
so $X'$ has an $m$th-order pole at $s=d$ and hence
has $m$ irreducible components of dimension $d$.  But these are also the connected components of $X'$ 
(as $X'$ is finite \'etale over the normal noetherian $X$), so by degree considerations each
connected component of $X'$ maps isomorphically onto $X$.
\end{proof}

\begin{remark} The meromorphicity assertion
used above is not proved in \cite{serrez}, though it can be deduced from the Lang--Weil estimate
for geometrically irreducible schemes over finite fields \cite[\S2, Cor.\,2]{lw} (or from Weil's
Riemann Hypothesis for curves) via a fibration technique.
Here is a short proof via Deligne's mixedness
bounds \cite[Cor.\,3.3.4]{weilii}.

Via stratification and dimension induction, we may assume $Y$ is separated, irreducible, and reduced.
If the function field $F$ of $Y$ has characteristic $p > 0$
and $\kappa$ is the algebraic
closure of $\F_p$ in $F$ then $Y$ is geometrically irreducible over $\kappa$.
Thus,  the top-degree
cohomology ${\rm{H}}^{2d}_c(Y_{\overline{\kappa}}, \mathbf{Q}_{\ell})$ is $\Q_{\ell}(-d)$.
The Grothendieck--Lefschetz formula for $\zeta_Y(s)$ as a rational function in $q^{-s}$
($q = \#\kappa$) therefore has a factor of $1 - q^{d-s}$ in the denominator arising from the action of geometric
Frobenius on the top-degree
cohomology, and all other cohomological contributions in the denominator are non-vanishing for ${\rm{Re}}(s) > d-1/2$
due to Deligne's bounds.  This settles the case ${\rm{char}}(F) > 0$.

Assume ${\rm{char}}(F) = 0$, so $Y_{\Q}$ is geometrically irreducible of dimension $d-1$ over
the algebraic closure $K$ of $\Q$ in $F$.  Replacing $Y$ with a dense open subscheme
brings us to the case that $Y$ is an $\calO_K$-scheme with geometrically irreducible fiber
$Y_v$ of dimension $d-1$
over the closed points $v$ in a dense open subset $U \subset  \Spec(\calO_K)$ \cite[IV$_3$, 9.7.7(i)]{ega}. 
Since $\zeta_Y(s) = \prod_{v \in U} \zeta_{Y_v}(s)$ with $Y_v$ geometrically irreducible
of dimension $d-1$ over $\kappa(v)$, the analysis just given
in positive characteristic exhibits $\zeta_Y(s)$ as the product of $\zeta_U(s-(d-1))$ and a holomorphic
function in the half-plane ${\rm{Re}}(s) > 1 + (2(d-1)-1)/2 = d-1/2$.  By meromorphicity of
$\zeta_{U}$ on $\C$ with a simple pole at $s=1$, we are done.
\end{remark}
%
%This is immediate from 
%the generalized Chebotarev Density Theorem for the \'etale fundamental group of $X$
%(density of the union of Frobenius conjugacy classes at closed points). 
%
%The only literature
%reference we know for a proof of the generalized Chebotarev theorem is
%\cite[Ch.\:VI, \S2.1]{faltings}, but that proof rests on Deligne's deep mixedness results \cite[Cor.\,3.3.4]{weilii},
%so we give a proof below resting on the more elementary Lang--Weil estimate
%for geometrically irreducible schemes over finite fields \cite[\S2, Cor.\,2]{lw},
%itself an immediate consequence of \cite[Cor.\,3.3.4]{weilii}.
%(Although \cite{faltings} addresses $X$ whose function field $F$ has
%characteristic 0, it adapts to ${\rm{char}}(F) = p > 0$: the case $d := \dim X = 0$ is trivial 
%and if $d > 0$ then  a member of a separating transcendence basis of $F$
%over $\F_p$ expresses a dense open in $X$ as smooth with fibers of pure dimension $d-1$ over a dense open
%in $\mathbf{A}^1_{\F_p}$, so $\Spec(\Z)$ can be replaced by $\mathbf{A}^1_{\F_p}$ in the argument.)
%
%First assume that the function field $F$ of $X$ has positive characteristic $p$,
%and define $\kappa$ to be the algebraic closure of
%$\mathbf{F}_p$ in $F$, so $X$ is geometrically connected (and hence geometrically
%irreducible) over $\kappa$.
%Assuming $X'$ is connected, we seek a contradiction.
%Let $\kappa'$ be the algebraic closure of $\kappa$
%in the function field of $X'$,
%so $[\kappa':\kappa] \le 2$.  If $\kappa''/\kappa$ is a finite extension with odd degree
%then $\kappa' \otimes_{\kappa} \kappa''$ is a field,
%so $X' \otimes_{\kappa} {\kappa''}$ is connected (as $X'$ is geometrically connected over $\kappa'$). 
%Hence, geometric connectedness
%of $X'$ over $\kappa$ is reduced to that of $X' \otimes_{\kappa} \kappa''$ over $\kappa''$ for one such $\kappa''$.
%
%By the Lang--Weil estimate applied to $X$ we have $X(\kappa'') \ne \emptyset$ for all such $\kappa''$
%with sufficiently large size, and by the residual hypothesis on $u$
%the fiber of $X' \rightarrow X$ over any $x \in X(\kappa'')$ contains a $\kappa''$-point $x'$.
%Thus, the connected normal $\kappa''$-scheme $X' \otimes_{\kappa} {\kappa''}$ cannot have constant field larger than
%$\kappa''$, so $\kappa' = \kappa$ and hence $X'$ is geometrically connected over $\kappa$.
%
%For any finite extension $\kappa''$ of $\kappa$,
%the fiber of $X' \rightarrow X$ over any $\kappa''$-point is non-empty due to the 
%hypothesis on $u$, so all such fibers have 2 distinct points.
%It follows that $\#X'(\kappa'') = 2 \#X(\kappa'')$.
%But if $d$ is the common dimension of $X$ and $X'$
%then Lang--Weil for $X'$ and $X$ implies $\#X'(\kappa'') \sim |\kappa''|^d \sim \#X(\kappa'')$
%as $|\kappa''| \rightarrow \infty$, a contradiction.  Thus, $X'$ is disconnected. 
%
%Suppose instead that the function field $F$ of $X$ has characteristic 0, so 
%$X_{\Q}$ is geometrically connected over the number field $K$ given by the algebraic
%closure of $\Q$ in $F$.  Once again we assume
%$X'$ is connected and seek a contradiction.  By normality, $X$ is an $\calO_K$-scheme.
%The generic fiber of $X \rightarrow \Spec(\calO_K)$ is 
%geometrically irreducible, so the same holds for all fibers over 
%a dense open locus $Y \subset \Spec(\calO_K)$ \cite[IV$_3$, 9.7.7(i)]{ega}. 
%Let $K'/K$ be the algebraic closure of $K$ in the function field of $X'_{\Q}$,
%so $X'_{\Q}$ is geometrically irreducible over
%$K'$.  Let $Y'$ be the normalization of $Y$ in
%$K'/K$, so the natural map $X' \rightarrow Y'$
%over $X \rightarrow Y$ has geometrically irreducible fibers $X'_{y'}$ for $y'$ in a dense open subset
%$U' \subset Y'$. 
%Since $[K':K] \le 2$, we may choose a closed point $y \in Y$ over which 
%the fiber in $Y'$ consists of a single reduced point $y'$, with $y' \in U'$. 
%The settled case of positive characteristic
%applies to the double cover $X'_{y} \rightarrow X_y$, so
%$X'_y$ is disconnected.  But $X'_{y} = X'_{y'}$ due to the choice of $y$,
%and $X'_{y'}$ is connected since $y' \in U'$, a contradiction.
%\end{proof}

{\bf Step 3}. Now we may assume $k$ is finite, so ${\rm{C}}_0(V,q)$ is a matrix algebra over its center.
Let $\rho: {\rm{Spin}}(q) \rightarrow {\rm{GL}}(W)$ be the associated $k$-descent
of the spin representation for odd $n \ge 3$ and of the direct sum of the two half-spin representations
for even $n \ge 2$.  There is a general identity (brought to my attention by
Z.\,Yun) on the {\em entirety} of ${\rm{Spin}}(q)$
without any intervention of squaring ambiguity or finiteness hypotheses on $k$: 

\begin{lemma}\label{yun} Let $(V,q)$ be a non-degenerate quadratic space
of dimension $n \ge 2$ over an arbitrary field $k$ with ${\rm{char}}(k) \ne 2$.
For any $\widetilde{h} \in {\rm{Spin}}(q)$ with image $h \in {\rm{SO}}(q)$,
$$(1/2)\Det(1+ h) = {\rm{Tr}}(\rho(\widetilde{h}))^2 \mbox{ for odd } n,\,\,\,
\Det(1+h) = {\rm{Tr}}(\rho(\widetilde{h}))^2 \mbox{ for even } n.$$
\end{lemma}

Granting this, we shall prove
${\rm{sp}}_q(g) = \Det((1+g)/2) \bmod (k^{\times})^2$ for any $g \in {\rm{SO}}(q)(k)$ without $-1$
as an eigenvalue, where $k$ is finite (as above). By finiteness
of $k$, it is equivalent to prove ${\rm{sp}}_q(g) = 1$ if and only if
$\Det((1+g)/2)$ is a square in $k^{\times}$.   If
${\rm{sp}}_q(g) = 1$ then $g$ lifts to some $\widetilde{g} \in {\rm{Spin}}(q)(k)$
and Lemma \ref{yun} gives that $\Det((1+g)/2)$ is a square in $k$.
Conversely, if $\Det((1+g)/2)$ is a square in $k$ then we want to show
that the degree-2 finite \'etale $g$-fiber in ${\rm{Spin}}(q)$ splits. 

Suppose not, so for the quadratic extension $k'/k$
there is a point $\widetilde{g} \in {\rm{Spin}}(q)(k')$ over $g$
and its $k'/k$-conjugate is $\widetilde{g}z$ for the nontrivial central element $z$ in ${\rm{Spin}}(q)(k)$.
But ${\rm{Tr}}(\widetilde{g})^2/\Det((1+g)/2) \in (k^{\times})^2$ by Lemma \ref{yun}, and by hypothesis
$\Det((1+g)/2) \in (k^{\times})^2$, so ${\rm{Tr}}(\rho(\widetilde{g})) \in k^{\times}$.  Galois-equivariance
of $\rho$ then implies ${\rm{Tr}}(\rho(\widetilde{g}z)) = {\rm{Tr}}(\rho(\widetilde{g}))$.
By construction of the spin representation for odd
$n$ and both half-spin representations for even $n$,
 the central involution $z$ satisfies $\rho(z) = -1$ (as we may check over $k_s$) 
and hence 
$${\rm{Tr}}(\rho(\widetilde{g})) = {\rm{Tr}}(\rho(\widetilde{g}z)) = -{\rm{Tr}}(\rho(\widetilde{g})),$$
forcing ${\rm{Tr}}(\rho(\widetilde{g})) = 0$.  But the square of this trace is $\Det(1+g) \ne 0$ by Lemma \ref{yun},
so we have a contradiction and hence $g$ lifts into ${\rm{Spin}}(q)(k)$ as desired.

{\bf Step 4}. It remains to prove
Lemma \ref{yun}. We may assume $k$ is {\em algebraically closed}, so $(V,q)$ is split.  We treat the case of
even and odd $n$ separately, based on how the structure of
the Clifford algebra depends on the parity of $n$.

First consider odd $n$, so the center $Z_q$ of ${\rm{C}}(V,q)$
is finite \'etale of degree 2 over $k$ (i.e., $Z_q = k \times k$ as $k$-algebras) and the natural map
$Z_q \otimes_k {\rm{C}}_0(V,q) \rightarrow {\rm{C}}(V,q)$ is an isomorphism.
The conjugation action on the $\Z/2\Z$-graded algebra ${\rm{C}}(V,q)$
by the subgroup ${\rm{Spin}}(q) \subset {\rm{C}}_0(V,q)^{\times}$ is trivial on
$Z_q$, so ${\rm{C}}(V,q)$ as a representation of ${\rm{Spin}}(q)$ is
a direct sum of two copies of ${\rm{C}}_0(V,q) = {\rm{End}}_k(\rho) = \rho \otimes \rho^{\ast}$
where $\rho$ is the spin representation.

Consider the conjugation action on ${\rm{C}}(V,q)$ by any $\widetilde{h} \in {\rm{Spin}}(q)$.
We compute the trace of this action in two different ways.  On the one hand,
this is $2 \chi_{\rho \otimes \rho^{\ast}}(\widetilde{h}) = 2 \chi_{\rho}(\widetilde{h}) \chi_{\rho^{\ast}}(\widetilde{h})$,
yet $\chi_{\rho^{\ast}} = \chi_{\rho}$ in characteristic 0 (by highest-weight theory)
and hence in general by specialization from characteristic 0 (consider
the Clifford algebra of the standard split quadratic space of rank $n$ over $\Z$),
so the trace of the action is $2 \chi_{\rho}(\widetilde{h})^2$.

On the other hand, by definition of
${\rm{Spin}}(q)$ inside the Clifford algebra, 
this action preserves $V$ with resulting representation on $V$  that is the composition of
the standard quotient map ${\rm{Spin}}(q) \rightarrow {\rm{SO}}(q)$ and the inclusion
${\rm{SO}}(q) \hookrightarrow {\rm{GL}}(V)$, so this action preserves
the {\em filtration} of ${\rm{C}}(V,q)$ defined by degree of (possibly mixed) tensors.  The trace of the 
$\widetilde{h}$-conjugation on ${\rm{C}}(V,q)$ is therefore the same as that of its effect on the associated
graded space for this filtration, which is the exterior algebra $\wedge^{\bullet}(V)$.
In other words, the trace of $\widetilde{h}$-conjugation is the trace of
the action of $h \in {\rm{SO}}(q)$ on $\wedge^{\bullet}(V)$, which is $\Det(1+h)$.


Now consider even $n$.  Exactly as for odd $n$, the trace of $\widetilde{h}$-conjugation
on ${\rm{C}}(V,q)$ is $\Det(1+h)$.  On the other hand, for a Lagrangian (i.e., maximal isotropic) subspace
$W \subset V$ of dimension $n/2$ and the associated graded components
$A_{+} = \oplus \wedge^{2j}(W)$ and $A_{-} = \oplus \wedge^{2j+1}(W)$
of the exterior algebra of $W$, there is an isomorphism of algebras
$${\rm{C}}(V,q) \simeq {\rm{End}}(A) =
({\rm{End}}(A_{+}) \times {\rm{End}}(A_{-})) \oplus
({\rm{Hom}}(A_{+},A_{-}) \oplus {\rm{Hom}}(A_{-},A_{+}))$$
(depending on a choice of Lagrangian complement of $W$ in $V$) 
in which ${\rm{C}}_0(V,q)$ is identified with the subalgebra 
${\rm{End}}(A_{+}) \times {\rm{End}}(A_{-})$ of linear endomorphisms of $A$ that respect
its $\Z/2\Z$-grading.  This identifies $A_{+}$ and $A_{-}$ as underlying spaces
of the two half-spin representations $\rho_{\pm}$ of ${\rm{Spin}}(q) \subset {\rm{C}}_0(V,q)^{\times}$.

We conclude that ${\rm{C}}(V,q)$  as a representation space for ${\rm{Spin}}(q)$ via
conjugation is isomorphic to
$$(\rho_{+} \otimes \rho_{+}^{\ast}) \oplus (\rho_{-} \otimes \rho_{-}^{\ast}) \oplus
(\rho_{-} \otimes \rho_{+}^{\ast}) \oplus (\rho_{+} \otimes \rho_{-}^{\ast}).$$
Since the half-spin representations each have a self-dual character (by specialization
from characteristic 0, as in the treatment of odd $n$), it follows that the
character of this representation is therefore equal to
$$\chi_{\rho_{+}}^2 + \chi_{\rho_{-}}^2 + 2 \chi_{\rho_{+}} \chi_{\rho_{-}} = \chi_{\rho_{+} \oplus \rho_{-}}^2.$$
But $\rho_{+} \oplus \rho_{-} = \rho$ by definition, so we are done.

\index{spinor norm!characteristic 2|(}
\begin{remark}\label{hahnrem}
Consider a non-degenerate $(V,q)$ over a field $k$
with characteristic 2.  If $n = \dim V$ is even then ${\rm{Pin}}(q)$ is a central extension
of ${\rm{O}}(q)$ by $\mu_2$ yielding a spinor norm ${\rm{sp}}_q: {\rm{O}}(q)(k)
\rightarrow k^{\times}/(k^{\times})^2$ characterized by the property
${\rm{sp}}_q(r_v) = q(v) \bmod (k^{\times})^2$ for non-isotropic $v \in V$.
A formula for ${\rm{sp}}_q(g)$ in the spirit of Zassenhaus' theorem is given in \cite[Cor.\,2.7]{hahn},
building on earlier work of Wall \cite{wall1}, \cite{wall2}.
However, this formula involves an ``anisotropic'' factor that is difficult to use in theoretical arguments.
(Note that \cite{hahn} uses ${\rm{sp}}^{-}_q$, as shown by
the Example after \cite[Thm.\,1.4]{hahn}, but this sign aspect is invisible
in characteristic 2.)  

If $n$ is odd then the disconnected
${\rm{Pin}}(q)$ is {\em not} a central extension of the connected ${\rm{O}}(q) = \mu_2 \times {\rm{SO}}(q)$ by
$\mu_2$.  However, ${\rm{O}}(q)(k) = {\rm{SO}}(q)(k)$ in such cases, so the central
extension ${\rm{Spin}}(q)$ of ${\rm{SO}}(q)$ by $\mu_2$ (for the fppf topology) defines a connecting homomorphism
$${\rm{O}}(q)(k) = {\rm{SO}}(q)(k) \rightarrow {\rm{H}}^1(k, \mu_2) = k^{\times}/(k^{\times})^2$$
that we call the ``spinor norm'' ${\rm{sp}}_q$; it encodes the obstruction to
lifting $g \in {\rm{O}}(q)(k) = {\rm{SO}}(q)(k)$ to ${\rm{Spin}}(q)(k)$.
This homomorphism is determined by its values on reflections $r_v$ in non-isotropic $v$, and we may
restrict attention to $v \not\in V^{\perp}$  since $r_v = 1$ when $v \in V^{\perp} - \{0\}$. For $v \not\in V^{\perp}$
we claim that ${\rm{sp}}_q(r_v) = q(v) \bmod (k^{\times})^2$, exactly as for even $n$.
Indeed, since $v \not\in V^{\perp}$ there exists $w \in V$ such that $B_q(v,w) \ne 0$,
and $w$ is linearly independent from $v$ since $B_q(v,v) = 0$ (as ${\rm{char}}(k)=2$). 
Thus, for the span $P$ of $v$ and $w$ we see that $(P,q|_P)$ is a quadratic space containing $v$
on which $B_q$ has discriminant $B_q(v,w)^2 \ne 0$, so $P$ is non-degenerate and $V = P \oplus P^{\perp}$.
Consequently, ${\rm{sp}}_q(r_v)$ coincides with the analogue for $P$. But $\dim P$ is even, so
the claim follows.

The restriction $q|_{V^{\perp}}$ to the defect line
has the form $cx^2$ for $c \in k^{\times}$ well-defined up to $(k^{\times})^2$-multiple,
so for any nonzero $v \in V^{\perp}$ we have $q(v) \in c (k^{\times})^2$ yet $r_v = 1$.
Hence, if $V^{\perp}$ does not contain a unit vector
(i.e., if $c$ is not a square) there is {\em no} well-defined homomorphism 
${\rm{O}}(q)(k) \rightarrow k^{\times}/(k^{\times})^2$ carrying $r_v$ to the class of $q(v)$
for {\em all} non-isotropic $v$, but ${\rm{sp}}_q$
achieves this for all such $v \not\in V^{\perp}$.   
The composition of this spinor norm with the quotient map
$$k^{\times}/(k^{\times})^2 \rightarrow k^{\times}/\langle q(V^{\perp} - \{0\}) \rangle =
k^{\times}/\langle c, (k^{\times})^2 \rangle$$
carries $r_v$ to the residue class of $q(v)$ for {\em all} non-isotropic $v \in V$,
and it is this composite map that is called the ``spinor norm'' in \cite{hahn}, where a formula in the spirit of Zassenhaus'
theorem is given in \cite[Cor.\,2.7]{hahn} (involving a non-explicit ``anisotropic'' factor as for even $n$).
\end{remark}
\index{spinor norm!characteristic 2|)}


\subsection{Accidental isomorphisms}\label{acc}

The study of (special) orthogonal groups provides many
{\em accidental isomorphisms} between low-dimensional members of 
distinct ``infinite families'' of algebraic groups.  This is analogous to 
 isomorphisms between small members of distinct ``infinite families''
of finite groups. Using the hyperplane 
$H = \{\sum x_i = 0\} \subset \mathbf{A}^n$ over
$\Z$, quadratic form $q = \sum x_i^2$ on $H$ over $\Z[1/2]$,
and line $L = \{x_1 = \dots = x_n\} \subset H$ over $\F_p$ with $p|n$, 
the natural action on $\mathbf{A}^n$ by the symmetric group $\mathfrak{S}_n$ yields 
\index{ aaasn@$\mathfrak{S}_n$}
isomorphisms:
\begin{itemize}
\item $\mathfrak{S}_3 \simeq {\rm{SL}}_2(\mathbf{F}_2)$ (use $H \subset \mathbf{A}^3$), 
\item $\mathfrak{S}_4 \simeq {\rm{PGL}}_2(\mathbf{F}_3)$ (identify
${\mathbf{P}}^1_{\F_3}$ with the smooth conic $\{q = 0\}$ in
$\mathbf{P}(H^{\ast})$ for $H \subset \mathbf{A}^4_{\F_3}$),
\item $\mathfrak{S}_5 \simeq {\rm{PGL}}_2(\mathbf{F}_5)$
(identify $\mathbf{P}^1_{\F_5}$ with the smooth conic $\{\overline{q} = 0\}$ in  the plane
$\mathbf{P}((H/L)^{\ast})$, where
$H \subset \mathbf{A}^5_{\F_5}$ and
$L$ is the defect line for $q$ on $H$),
\item $\mathfrak{A}_6 \simeq {\rm{SL}}_2(\F_9)/\langle -1 \rangle$ 
(for $H \subset \mathbf{A}^6_{\F_3}$ and the defect line $L$ of $q$ on $H$, 
$(H/L, \overline{q}) \simeq x^2+y^2+z^2-t^2$ and the $\mathfrak{S}_6$-action on the projective quadric
$\{\overline{q} = 0\} \subset \mathbf{P}((H/L)^{\ast})$ defines 
a map $\mathfrak{S}_6 \rightarrow {\rm{O}}(\overline{q})(\F_3)/\langle -1 \rangle \subset
({\rm{O}}(\overline{q})/\mu_2)(\F_3)$
that must be injective and 
carry the simple $\mathfrak{A}_6$ into the normal subgroup
${\rm{Spin}}(\overline{q})(\F_3)/Z_{{\rm{Spin}}(\overline{q})}(\F_3)$ that 
is identified with ${\rm{SL}}_2(\F_9)/\langle -1 \rangle$ in Remark \ref{so4rem}),
\item $\mathfrak{S}_6 \simeq {\rm{Sp}}_4(\F_2)$ (use $(H/L, \psi)$
where $H \subset \mathbf{A}^6_{\F_2}$, $L$ 
is the defect line for $Q = \sum_{i < j} x_i x_j$ on $H$, and $\psi$ is induced
by the symplectic form $B_Q(x,y) = \sum_{i \ne j} x_i y_j$), 
\item $\mathfrak{S}_8 \simeq {\rm{O}}_6(\F_2)$ (use the non-degenerate quadratic space $(H/L,\overline{Q})$
where $H  \subset \mathbf{A}^8_{\F_2}$, $Q = \sum_{i < j} x_i x_j$,
and $L$ is the defect line for $Q|_H$). 
\end{itemize}

When accidental isomorphisms among algebraic groups 
are applied to rational points over {\em finite} fields one
obtains some of the accidental isomorphisms among small finite simple groups.
This is seen in the discussion of $\mathfrak{A}_6$ above.  As another illustration, 
in Example \ref{so6} we will see that ${\rm{SL}}_4/\mu_2 \simeq {\rm{SO}}_6$
as $\Z$-groups, so by passing to $\F_2$-points and using 
the vanishing of ${\rm{H}}^1(\F_2,\mu_2)$ (fppf Kummer theory) we see 
that ${\rm{SL}}_4(\F_2) = {\rm{SO}}_6(\F_2)$, an index-2 subgroup of 
${\rm{O}}_6(\F_2) = \mathfrak{S}_8$. The only such subgroup
is $\mathfrak{A}_8$, so ${\rm{SL}}_4(\F_2) = \mathfrak{A}_8$. 

The case of $\mathfrak{A}_6$ above can be pushed a bit further.  
First, we note that $\mathfrak{S}_6 \not\simeq {\rm{PGL}}_2(\F_9)$,
despite the isomorphism between their index-2 perfect commutator subgroups,
since $\mathfrak{S}_6$ has no element of order 8.  More interesting is 
that we can interpret the nontrivial outer automorphism of $\mathfrak{S}_6$.
(For $n > 2$, ${\rm{Out}}(\mathfrak{S}_n)$ is trivial for $n \ne 6$ and
has order 2 for $n = 6$.)  This rests on an isomorphism
associated to any non-degenerate quadratic space $(W,Q)$ of
rank 4 over a finite field $k$:  by Example \ref{so4} below, for some degree-2 \'etale $k$-algebra $k'$ we have 
$${\rm{O}}(Q)/\mu_2 \simeq \Aut_{{\rm{R}}_{k'/k}(\mathbf{P}^1_{k'})/k} =
{\rm{R}}_{k'/k}({\rm{PGL}}_2) \rtimes \langle -1 \rangle$$
where the nontrivial $k$-automorphism of $k'$ defines the action of
$\langle -1 \rangle$. 
On identity components this gives ${\rm{SO}}(Q)/\mu_2 \simeq {\rm{R}}_{k'/k}({\rm{PGL}}_2)$,
so if $Q$ is not $k$-split (hence ${\rm{SO}}(Q)/\mu_2$ is not $k$-split, by Proposition \ref{projsim})
then $k'$ is a field.
%${\rm{H}}^1(k,{\rm{O}}(Q)/\mu_2)$ has order 2.  Since the natural map 
%map ${\rm{H}}^1(k,{\rm,{O}}(Q)) \rightarrow {\rm{H}}^1(k,{\rm{O}}(Q)/\mu_2)$
%is surjective (as ${\rm{H}}^2(k,\mu_2) = {\rm{Br}}(k)[2] = 1$), it is bijective. 
%Thus, we deduce the exactness of
%$$1 \rightarrow {\rm{O}}(Q)(k)/\mu_2(k) \rightarrow ({\rm{O}}(Q)/\mu_2)(k)
%\rightarrow {\rm{H}}^1(k,\mu_2) \rightarrow 1,$$
%so if ${\rm{char}}(k) \ne 2$ then ${\rm{O}}(Q)(k)/\mu_2(k)$ has index 2 in
%$({\rm{O}}(Q)/\mu_2)(k) = {\rm{PGL}}_2(k') \rtimes \langle -1 \rangle$.
We conclude that $\mathfrak{S}_6$ is a subgroup of ${\rm{PGL}}_2(\F_9) \rtimes \langle -1 \rangle$
making $\mathfrak{A}_6$ the index-4 image of ${\rm{SL}}_2(\F_9)$, so
$\mathfrak{S}_6$ is an index-2 subgroup of ${\rm{PGL}}_2(\F_9) \rtimes \langle -1 \rangle$
(the unique one distinct from ${\rm{PGL}}_2(\F_9)$ that does
not contain the Galois involution). This index-2 inclusion
defines the nontrivial outer automorphism of $\mathfrak{S}_6$.

Just as isomorphisms among small finite groups are due to the limited possibilities
for finite groups of small size, accidental isomorphisms between low-dimensional semisimple
groups are due to a limitation in the possibilities for a ``small'' case of the root datum
that governs the (geometric) isomorphism
class of a connected semisimple  group.  We now work out the accidental isomorphisms 
for ${\rm{SO}}_n$ with $2 \le n \le 6$; for $3 \le n \le 6$ these correspond to equalities between
distinct classical infinite families of root systems, as we will explain.
(See \cite[IV, \S15]{bookinv} for further discussion over fields,
using algebras with involution, and \cite[V]{knus} for further discussion over rings.)
In what follows, $(V,q)$ is a quadratic space (so $q$ is $\mathscr{O}_S$-valued and non-degenerate);
some aspects go through with little change for line bundle-valued $q$, especially when
the odd part of the Clifford algebra does not intervene (see \cite[\S5]{auel} for $n = 2, 4, 6$). 

\begin{example}\label{so2}
Suppose $n = 2$. In this case, Lemma \ref{stdq}
and calculations with $q_2 = xy$ show 
that ${\rm{SO}}(q)$ is a rank-1 torus
and $E_q := \{q = 0\} = {\rm{Proj}}({\rm{Sym}}(V^{\ast})/(q)) \subset \mathbf{P}(V^{\ast})$
is a degree-2
finite \'etale $S$-scheme. 
By Proposition \ref{rzq}(1) (and its proof), the coordinate ring of $E_q$ over $\calO_S$ 
naturally coincides with the quadratic \'etale commutative
$\calO_S$-algebra $Z_q = {\rm{C}}_0(V,q)$ and 
there is a natural isomorphism between ${\rm{GSO}}(q) = \Gm \times^{\mu_2} {\rm{SO}}(q)$ and the  
Weil restriction torus ${\rm{R}}_{E_q/S}(\Gm)$.  This isomorphism respects the natural inclusion of $\Gm$ into each
torus 
and identifies ${\rm{SO}}(q)$ with the norm-1 subtorus of ${\rm{R}}_{E_q/S}(\Gm)$.  
The discussion preceding Lemma \ref{mu2ker} identifies the $S$-group 
${\rm{GSpin}}(q) = \Gm \times^{\mu_2} {\rm{Spin}}(q)$ 
with the torus ${\rm{R}}_{Z_q/S}(\Gm)$ (essentially by definition), 
respecting the natural inclusion of $\Gm$ into each and (as we saw in the discussion following
Remark \ref{normp}) identifying ${\rm{Spin}}(q)$ with the norm-1 subtorus of ${\rm{R}}_{Z_q/S}(\Gm)$.  
In this way we obtain a natural isomorphism between the tori ${\rm{SO}}(q)$
and ${\rm{Spin}}(q)$ via their identifications with the norm-1 subtori in the respective
tori ${\rm{R}}_{E_q/S}(\Gm)$ and ${\rm{R}}_{Z_q/S}(\Gm)$ that we have seen are naturally isomorphic.
(These isomorphisms are also established in \cite[V, 2.5.2]{knus}.) 
\index{spin group!binary case}

This natural identification of ${\rm{SO}}(q)$
and its double cover ${\rm{Spin}}(q)$ is a special case of the general fact that for any $d \ge 1$ and degree-$d$ isogeny
$f:T' \rightarrow T$ between rank-1 tori (such as the degree-2 isogeny ${\rm{Spin}}(q) \rightarrow {\rm{SO}}(q)$)
there is a unique isomorphism $T' \simeq T$ that identifies $f$ with $t \mapsto t^d$ on $T$.
To see this agreement of isomorphisms, we have to show that the composition of the isomorphism
${\rm{SO}}(q) \simeq {\rm{Spin}}(q)$ between norm-1 subtori of
${\rm{R}}_{E_q/S}(\Gm)$ and ${\rm{R}}_{Z_q/S}(\Gm)$ 
with the natural isogeny ${\rm{Spin}}(q) \rightarrow {\rm{SO}}(q)$ is the squaring endomorphism
of ${\rm{SO}}(q)$.  It suffices to check this comparison of tori on geometric fibers
over $S$, so we may assume $S = \Spec k$ for an algebraically closed field $k$
with $V = k e_1 \oplus k e_2$ and $q = xy$.  For
$a \in k^{\times}$, the element ${\rm{diag}}(a,1/a) \in {\rm{SO}}_2(k)$
is associated to the unit 
$$u(a) = a e_1 e_2 + (1/a)(1 - e_1 e_2) = 1/a + (a - 1/a) e_1 e_2 \in  {\rm{Spin}}(q_2)(k) \subset Z_{q_2}^{\times}$$
due to the computations in the proof of Proposition \ref{rzq}(1). 
Since $e_1 e_2 e_1 = e_1$ and $e_2 e_1 e_2 = e_2$ inside
${\rm{C}}(k^2, q_2)$ and $u(a)^{-1} = u(1/a)$, the image of $u(a)$ in ${\rm{GL}}_2(k)$ under the natural isogeny
${\rm{Spin}}(q_2) \rightarrow {\rm{SO}}_2$ is the linear automorphism of $k^2$ 
defined by
$$e_1 \mapsto u(a) e_1 u(a)^{-1} = a^2 e_1,\,\,\,
e_2 \mapsto u(a) e_2 u(a)^{-1} = (1/a)^2 e_2.$$
Thus, we obtain the point ${\rm{diag}}(a,1/a)^2 \in {\rm{SO}}_2(k)$ as desired.

Via the action of
${\rm{PGL}}(V)$ on $\mathbf{P}(V^{\ast})$ we can interpret the torus 
${\rm{SO}}(q)/\mu_2 = {\rm{GSO}}(q)/\Gm$ in an interesting way, as follows.   Consideration of the relative homogenous
coordinate ring shows that the ${\rm{PGL}}(V)$-stabilizer
$G_q$ of $E_q$ is ${\rm{GO}}(q)/\Gm$.  Thus, we get a homomorphism
$$\alpha_q:{\rm{GO}}(q) \twoheadrightarrow
{\rm{GO}}(q)/\Gm = G_q \rightarrow \underline{\rm{Aut}}_S(E_q) = (\Z/2\Z)_S.$$
We claim that $\alpha_q$ is the enhanced Dickson invariant ${\rm{GD}}_q$ from (\ref{Gdickson}) and Remark \ref{bettergso}
(for $n=2$),
so ${\rm{GSO}}(q)/\Gm$ is the subgroup of points of $G_q$ whose action on $E_q$ is trivial.
The map $\alpha_q$ must kill the fiberwise connected open and closed subgroup ${\rm{GSO}}(q)$,
so it factors through the quotient ${\rm{GO}}(q)/{\rm{GSO}}(q)$ that ${\rm{GD}}_q$ identifies with
$(\Z/2\Z)_S$.  Hence, $\alpha_q$ is the composition of ${\rm{GD}}_q$ 
with a uniquely determined endomorphism of $(\Z/2\Z)_S$,
and we have to show that this endomorphism is the identity map.  The identity map is the unique
automorphism of $(\Z/2\Z)_S$, so our problem is reduced to the geometric fibers,
where it amounts to the assertion that the $G_q$-action on $E_q$ is nontrivial over an algebraically closed field $k$.
This nontriviality is easily verified by using a point of ${\rm{O}}_2(k)$ not in ${\rm{SO}}_2(k)$
(e.g., the point that swaps the standard basis vectors).
\end{example}

\index{exceptional isomorphisms|(}
\index{accidential isomorphisms|(}
\begin{example}\label{so3}
Suppose $n = 3$.  This  corresponds to 
 the equality ${\rm{A}}_1 = {\rm{B}}_1$ for adjoint groups. 
Consider the zero scheme
$C_q = \{q = 0\} \subset \mathbf{P}(V^{\ast})$.  
By the definition of non-degeneracy, this is a smooth conic
in a $\mathbf{P}^2$-bundle, 
so it is a $\mathbf{P}^1$-bundle.  The automorphism scheme
$\Aut_{C_q/S}$ is therefore a form of
$\Aut_{\mathbf{P}^1_S/S} = {\rm{PGL}}_2$.
Under the left action of ${\rm{PGL}}(V)$ on $\mathbf{P}(V^{\ast})$,
the stabilizer $G_q$ of $C_q$ is ${\rm{GO}}(q)/\Gm = {\rm{SO}}(q)$
(see Lemma \ref{pgoq} with $n=3$),
and we claim that the action map
$$\alpha_q:{\rm{SO}}(q) \rightarrow \Aut_{C_q/S}$$
to a form of ${\rm{PGL}}_2$ is an isomorphism.  
(For $q=q_3$ this is a map ${\rm{SO}}_3 \rightarrow {\rm{PGL}}_2$.)

The source and target of $\alpha_q$ are smooth
with fibers that are connected of the same dimension, and it is easy
to check that $\alpha_q$ is  injective on geometric
points (since a smooth plane conic
over an algebraically closed field contains many triples of points
in general position). Thus, $\alpha_q$ is 
a purely inseparable isogeny on fibers over each $s \in S$, so by Proposition \ref{isogfiber}
(and Definition \ref{ceniso}) it is 
finite flat as an $S$-morphism. To prove that $\alpha_q$ has constant degree 1, we may first pass to 
$q = q_3$ over $\Z$ (by Lemma \ref{stdq}) and then
check over $\Q$.  A purely inseparable isogeny in characteristic 0 is an isomorphism,
so we are done.

In the special case $q = q_3$, an isomorphism in the opposite direction can be 
described by the following alternative procedure.
Consider the linear ``conjugation'' action of 
${\rm{PGL}}_2 = {\rm{GL}}_2/\Gm$ on the rank-3 affine space 
$\mathfrak{sl}_2$.   This action preserves the non-degenerate  
quadratic form $Q$ on $\mathfrak{sl}_2$ given by the determinant.
Explicitly, $Q(\begin{smallmatrix} x & y \\ z & -x \end{smallmatrix})
= -(x^2 + yz)$ is, up to sign, the quadratic form $q_3$.
Preservation of $q_3$ is the same as that of $-q_3$, so the sign does not affect the orthogonal group. 
Thus, we obtain a homomorphism
${\rm{PGL}}_2 \rightarrow {\rm{O}}_3 = \mu_2 \times {\rm{SO}}_3$ 
over $\Z$ with trivial kernel. By computing over $\Q$, 
the map to the $\mu_2$-factor must be trivial.  Thus, 
the map ${\rm{PGL}}_2 \rightarrow {\rm{O}}_3$ factors through 
${\rm{SO}}_3$. 
Since ${\rm{PGL}}_2$ is smooth and fiberwise connected of dimension 3, it follows that 
the monic map ${\rm{PGL}}_2 \rightarrow {\rm{SO}}_3$ is an isomorphism on fibers and hence
is an isomorphism (Lemma \ref{flatclosed}).  We leave it to the interested
reader to relate this isomorphism to $\alpha_{q_3}^{-1}$ (relative to a suitable
isomorphism $C_{q_3} \simeq \mathbf{P}^1$). 
\end{example}

\begin{example}\label{so4}
Suppose $n = 4$.  This corresponds to the equality of root systems
${\rm{D}}_2 = {\rm{A}}_1 \times {\rm{A}}_1$.
More specifically, in this case ${\rm{SO}}(q)$ is not ``absolutely simple''; i.e., 
on geometric fibers it 
contains nontrivial smooth connected proper normal subgroups.
(This is the only $n \ge 3$ for which 
that happens.)  In more concrete terms, we claim that
\begin{equation}\label{slso}
({\rm{SL}}_2 \times {\rm{SL}}_2)/M \simeq {\rm{SO}}_4
\end{equation}
with $M = \mu_2$ diagonally embedded in the evident central manner.  
As in Example \ref{so3}, we will also show that 
there is a geometrically-defined isomorphism in the opposite direction
from ${\rm{SO}}(q)/\mu_2$ onto an adjoint semisimple group
of type ${\rm{A}}_1 \times {\rm{A}}_1$ for {\em any} $q$ (not just $q_4$). 

First we explain the concrete isomorphism (\ref{slso}) for $q = q_4$, since the relative geometry
underlying the isomorphism in the opposite
direction for general $q$ is more complicated than in Example \ref{so3}. 
Apply a 
sign to the third standard coordinate to convert $q_4$ into 
$Q = x_1 x_2 - x_3 x_4$, which we recognize as the determinant of a $2 \times 2$ matrix.
The group ${\rm{SL}}_2$ acts on the rank-4 space of
such matrices in two evident commutating ways, via $(g,g').x = gx{g'}^{-1}$, 
and these actions preserve the determinant by the definition of ${\rm{SL}}_2$.
This defines a homomorphism
${\rm{SL}}_2 \times {\rm{SL}}_2 \rightarrow {\rm{SO}}'(Q) \simeq {\rm{SO}}'_4$
whose kernel is easily seen to be $M$. 
This map visibly lands in
${\rm{SO}}_4$ since ${\rm{SL}}_2$ is fiberwise connected
and ${\rm{O}}_4/{\rm{SO}}_4 = \Z/2\Z$.  Hence, we obtain a monomorphism
$({\rm{SL}}_2 \times {\rm{SL}}_2)/M \rightarrow {\rm{SO}}_4$
that must be an isomorphism on fibers (as both sides have smooth
connected fibers of the same dimension), and thus is an isomorphism.

For general $q$ with $n = 4$, we shall build an isomorphism
$\varphi:{\rm{SO}}(q)/\mu_2 \simeq {\rm{R}}_{S'/S}(G')$ for a canonically
associated degree-2 finite \'etale
cover $S' \rightarrow S$ and $S'$-form $G'$ of ${\rm{PGL}}_2$.
(For $q = q_4$ we will have
$S' = S \coprod S$ and $G' = {\rm{PGL}}_2$ over $S'$, 
yielding a canonical  isomorphism
$${\rm{SO}}_4/\mu_2 \simeq {\rm{PGL}}_2 \times {\rm{PGL}}_2$$
that the interested reader can relate to the isomorphism
of adjoint groups induced by (\ref{slso}).) 
There are two methods to construct $(S' \rightarrow S, G', \varphi)$:
an algebraic method via Clifford algebras and a geometric
method via automorphism schemes.   The algebraic method is simpler, so we
explain that one first (see \cite[V, 4.4]{knus} for a related discussion). 

By Proposition \ref{cliffstr} (and its proof), 
the even part ${\rm{C}}_0(V,q)$ of the Clifford algebra ${\rm{C}}(V,q)$
is a quaternion algebra (i.e., rank-4 Azumaya algebra) over a rank-2 finite \'etale $\calO_S$-algebra $Z_q$.
Denote by ${\rm{C}}_0(V,q)^{\times}$ the associated unit group over $Z_q$. This
is an inner form of ${\rm{GL}}_2$ over $Z_q$, and 
${\rm{GSpin}}(q)$ is a reductive closed subgroup of ${\rm{R}}_{Z_q/S}({\rm{C}}_0(V,q)^{\times})$.
The derived group ${\rm{Spin}}(q)$ has fibers of 
dimension 6, yet the reductive ${\rm{R}}_{Z_q/S}({\rm{C}}_0(V,q)^{\times})$ also has
derived group with fibers of dimension 6, so the containment 
$${\rm{Spin}}(q) \subset \mathscr{D}({\rm{R}}_{Z_q/S}({\rm{C}}_0(V,q)^{\times}))$$
is an equality.  Passing to adjoint quotients, we get
$${\rm{SO}}(q)/\mu_2 = {\rm{Spin}}(q)/Z_{{\rm{Spin}}(q)} \simeq
{\rm{R}}_{Z_q/S}({\rm{C}}_0(V,q)^{\times}/\Gm).$$
Thus, we take $S' = Z_q$ and $G' = {\rm{C}}_0(V,q)^{\times}/\mathbf{G}_{\rm{m}}$. 

The geometric method rests on the zero scheme $\Sigma_q = \{q = 0\} \subset
\mathbf{P}(V^{\ast})$, a smooth proper $S$-scheme with geometric 
fibers over $S$ given by a ruled quadric $\mathbf{P}^1 \times \mathbf{P}^1$ in $\mathbf{P}^3$.
The ${\rm{PGL}}(V)$-stabilizer of $\Sigma_q$ is ${\rm{GO}}(q)/\Gm = {\rm{O}}(q)/\mu_2$, so 
we get an action map 
$$f_q:{\rm{O}}(q)/\mu_2 \rightarrow \Aut_{\Sigma_q/S}.$$
We claim that for any smooth proper
map $X \rightarrow S$ whose geometric fibers are
$\mathbf{P}^1 \times \mathbf{P}^1$ (e.g., $\Sigma_q$), there is a unique triple
$(S' \rightarrow S, C', \varphi)$ consisting of a degree-2 finite \'etale
cover $S' \rightarrow S$, a $\mathbf{P}^1$-bundle $C' \rightarrow S'$,
and an $S$-isomorphism $\varphi:X \simeq {\rm{R}}_{S'/S}(C')$.
Here, ``unique'' means that if $(S'', C'', \psi)$ is second such triple
then there is a unique pair consisting of an $S$-isomorphism $\alpha:S' \simeq S''$
and an isomorphism $C' \simeq C''$ over $\alpha$ such that
the induced $S$-isomorphism ${\rm{R}}_{S'/S}(C') \simeq
{\rm{R}}_{S''/S}(C'')$ coincides with $\psi \circ \varphi^{-1}$.

To prove the existence
and uniqueness of $(S',C',\varphi)$, by limit arguments and the uniqueness assertion 
we may assume $S = \Spec A$ for a noetherian local ring $A$.
Uniqueness allows us to work fppf-locally, so we may increase
the residue field $k$ by a finite amount to make the special fiber
isomorphic to $\mathbf{P}^1_k \times \mathbf{P}^1_k$. 
Since ${\rm{H}}^1(\mathbf{P}^1_k \times \mathbf{P}^1_k, \calO) = 0$, 
by standard cohomological and deformation theory arguments
 the Isom-functor $\underline{\rm{Isom}}(X,\mathbf{P}^1 \times \mathbf{P}^1)$
is formally smooth. Hence, by formal GAGA this functor
has an $\widehat{A}$-point.  By the uniqueness assertion and
fpqc descent, it suffices to solve existence and uniqueness over $\widehat{A}$, so 
we may assume $X = \mathbf{P}^1 \times \mathbf{P}^1$.
It now suffices to prove that the $\Z$-homomorphism
$$({\rm{PGL}}_2 \times {\rm{PGL}}_2) \rtimes (\Z/2\Z) \rightarrow \Aut_{\mathbf{P}^1
\times \mathbf{P}^1}$$
is an isomorphism, and this is part of Exercise \ref{1.1}(iv).

Since $(S',C',\varphi)$ has been built for $X  = \Sigma_q$, we have an $S$-homomorphism
\begin{equation}\label{oqaut}
{\rm{O}}(q)/\mu_2 \rightarrow {\Aut}_{{\rm{R}}_{S'/S}(C')/S}.
\end{equation}
This automorphism scheme is an \'etale form of
$({\rm{PGL}}_2 \times {\rm{PGL}}_2) \rtimes (\Z/2\Z)_S$, so it
is a smooth $S$-affine $S$-group and the natural map
${\rm{R}}_{S'/S}(\Aut_{C'/S'}) \rightarrow
\Aut_{{\rm{R}}_{S'/S}(C')/S}$
is an open and closed immersion onto the open relative identity component.
Thus, we get an induced $S$-homomorphism
\begin{equation}\label{somu2}
{\rm{SO}}(q)/\mu_2 \rightarrow {\rm{R}}_{S'/S}(\Aut_{C'/S'})
\end{equation}
between adjoint semisimple $S$-groups with the same constant fiber dimension.
Arguing similarly to the treatment
of $\alpha_q$ in Example \ref{so3} (by passing to $q_4$ over $\Z$),
it follows that (\ref{oqaut})
is an isomorphism.  

To link the algebraic and geometric
methods, we claim that the composite $S$-group isomorphism
$${\rm{R}}_{S'/S}(\Aut_{C'/S'}) \simeq 
{\rm{SO}}(q)/\mu_2 \simeq {\rm{R}}_{Z_q/S}({\rm{C}}_0(V,q)^{\times}/\Gm)$$
arises from a unique pair 
$(\beta_q,h_q)$ consisting of an $S$-isomorphism
$\beta_q:S' \simeq Z_q$ and isomorphism $h_q:\Aut_{C'/S'} \simeq 
{\rm{C}}_0(V,q)^{\times}/\Gm$ over $\beta_q$. 
Since ${\rm{PGL}}_2 \times {\rm{PGL}}_2$ is the open relative identity component in its own automorphism scheme
and ${\rm{SO}}(q)$ has connected fibers, 
the claim follows from a descent argument similar to the general construction  of
$(S' \rightarrow S,C',\varphi)$ above. 
Thus,  $S' \rightarrow S$ corresponds to the center $Z_q$ of ${\rm{C}}_0(V,q)$, 
and the map $\delta:{\rm{H}}^1(Z_q, {\rm{PGL}}_2) \rightarrow {\rm{Br}}(Z_q)[2]$ carries the class
of the $\mathbf{P}^1$-bundle (more precisely,
the associated ${\rm{PGL}}_2$-torsor) $C'$ over $S' = Z_q$ to the class of the quaternion algebra ${\rm{C}}_0(V,q)$
over $Z_q$.
However, $\ker \delta$ can be nontrivial (e.g., when $S$ is Dedekind
and ${\rm{Pic}}(Z_q) \ne 1$), and working with cohomology only keeps track of
torsors and Azumaya algebras up to an equivalence. Hence, it is better 
to express the link between geometry
and algebra via the canonical $\beta_q$ and $h_q$.
\end{example}

\begin{remark}\label{so4rem}
Example \ref{so4} can be made concrete for 
$S = \Spec k$ with a finite field $k$, as follows. Since central simple
algebras over finite fields are matrix algebras, and
$\mathbf{P}^1$-bundles over finite fields are trivial, we get 
an isomorphism ${\rm{SO}}(q)/\mu_2 
\simeq {\rm{R}}_{k'/k}({\rm{PGL}}_2)$ 
for some quadratic \'etale $k$-algebra $k'$.  
Assume that $q$ is non-split, so ${\rm{SO}}(q)$ is not $k$-split (by Proposition \ref{projsim}) and hence 
$k'$ must be a field.  The isomorphism
${\rm{SO}}(q)/\mu_2 \simeq {\rm{R}}_{k'/k}({\rm{PGL}}_2)$
lifts uniquely to an isomorphism ${\rm{Spin}}(q) \simeq {\rm{R}}_{k'/k}({\rm{SL}}_2)$
between the simply connected central covers, so 
for non-split $q$ of rank 4 over a finite field $k$, 
$${\rm{Spin}}(q)(k)/Z_{{\rm{Spin}}(q)}(k) \simeq
{\rm{R}}_{k'/k}({\rm{SL}}_2)(k)/{\rm{R}}_{k'/k}(\mu_2)(k) = 
{\rm{SL}}_2(k')/\mu_2(k')$$
with $k'$ a quadratic extension field. 
\end{remark}

\begin{example}\label{so5}
Suppose $n = 5$.  This 
corresponds to the isomorphism
${\rm{B}}_2 = {\rm{C}}_2$ for adjoint groups, which says that
${\rm{SO}}_5$ is  isomorphic to the quotient of
${\rm{Sp}}_4$ by its center $\mu_2$. 
We shall work over a base scheme $S$ that is $\Z_{(2)}$-flat
at all points of residue characteristic 2. This includes the base scheme $\Spec \Z$, so 
we will get an isomorphism ${\rm{Sp}}_4 \simeq {\rm{Spin}}_5$ over $\Z$, hence over
any scheme (including $\mathbf{F}_2$-schemes) by base change. 

To that end, consider a rank-4 symplectic space $(V,\omega_0)$ over such an $S$, 
with $\omega_0 \in \wedge^2(V)^{\ast} = \wedge^2(V^{\ast})$ the given
symplectic form.  The rank-6 vector bundle
$\wedge^2(V)$ contains the rank-5 subbundle $W$ of sections
killed by $\omega_0$, and on $\wedge^2(V)$ there exists a natural
non-degenerate quadratic form $q$ valued in $L = \Det(V)$ defined
%by $q(\omega, \eta) = \omega \wedge \eta$.  
by $\omega \mapsto \omega \wedge \omega$ that is actually valued in the subsheaf $2L \subset L$.  
Thus, by the $\Z_{(2)}$-flatness hypothesis on $S$ we can define the $L$-valued quadratic
form $q(\omega) = (1/2)(\omega \wedge \omega)$ that is readily checked to be fiberwise
non-degenerate.  The action of
${\rm{SL}}(V)$ clearly preserves $q$, the restriction
$q|_W$ is non-degenerate (by calculation), and ${\rm{Sp}}(\omega_0)$
preserves $W$ (due to the definition of $W$).  
Thus, the ${\rm{Sp}}(\omega_0)$-action on $W$ defines a homomorphism
${\rm{Sp}}(\omega_0) \rightarrow {\rm{O}}(q|_W)$
that obviously kills the center $\mu_2$.
We claim that the resulting homomorphism $f:{\rm{Sp}}(\omega_0)/\mu_2 \rightarrow
{\rm{O}}(q|_W)$ is an isomorphism onto ${\rm{SO}}(q|_W)$.
For this purpose we may work Zariski-locally on $S$ so that the symplectic space
$(V, \omega_0)$ is standard.
Now our problem is a base change from $\Z$, so
we may assume $S = \Spec \Z$.

By using connectedness of symplectic groups over $\Q$ we see that
$f$ factors through ${\rm{SO}}(q|_W)$, so we obtain 
a map $h:{\rm{Sp}}_4/\mu_2 \rightarrow {\rm{SO}}_5$ over $\Z$
that we need to prove is an isomorphism.  
A computation shows 
that $\ker h$ has trivial intersection with the ``diagonal'' maximal torus, so $\ker h$ is quasi-finite
(as it is normal in a reductive group) 
and hence $h$ is surjective for fibral dimension reasons. 
 This forces $h$ to be finite flat (Proposition \ref{isogfiber}),
 so $h$ has locally constant fiber degree that we claim is 1.
It suffices to compute the fiber degree over $\Q$.  Isogenies between connected
semisimple groups of adjoint type are isomorphisms in characteristic 0
(false in characteristic $>0$, via Frobenius).
\end{example}

\begin{example}\label{so6}
Suppose $n = 6$.  This case corresponds
to the equality ${\rm{D}}_3 = {\rm{A}}_3$ for groups that are
neither adjoint nor simply connected: we claim that the $\Z$-group ${\rm{SO}}_6$ is the quotient of
${\rm{SL}}_4$ by the subgroup $\mu_2$ in the central $\mu_4$ (and hence likewise over
any scheme by base change). 
To construct the isomorphism, we work in the setup of Example \ref{so5} (over $\Spec \Z$) and again use 
the natural action of ${\rm{SL}}(V)$ on the rank-6 bundle $\wedge^2(V)$
equipped with the non-degenerate quadratic form $q(\omega) = (1/2)(\omega \wedge \omega)$
valued in the line bundle $\Det V$. The
homomorphism ${\rm{SL}}(V) \rightarrow {\rm{O}}(q)$ defined in this way clearly kills the central
$\mu_2$, and it factors 
through ${\rm{SO}}(q)$ (as ${\rm{O}}(q)/{\rm{SO}}(q) = \Z/2\Z$). 

To prove that the resulting map $h:{\rm{SL}}(V)/\mu_2 \rightarrow {\rm{SO}}(q)$
is a $\Z$-isomorphism, as in Example \ref{so5} we reduce to the isomorphism
problem over $\Q$.  Isogenies between smooth connected
groups in characteristic 0 are always central, and $\mu_4/\mu_2$ is not killed
by $h$, so we are done.
\end{example}
\index{exceptional isomorphisms|)}
\index{accidential isomorphisms|)}



%
%\begin{example}\label{weylgp}  The proof of Proposition \ref{findcenter} helps us to work out
%an important calculation over a field $k$: we will compute $N_G(T)(k)/Z_G(T)(k)$
%for $G = {\rm{SO}}'_n$ and $T$ the ``diagonal'' $k$-torus with $n \ge 3$. 
%(The case $n = 2$ is of no interest since $G = \Gm$ is commutative in such cases.) 
%The case ${\rm{char}}(k) = 2$ with even $n \ge 4$ will turn out to be anomalous; 
%it will underlie our proof that ${\rm{SO}}'_n$ is {\em disconnected} in such cases.

% In the preceding proof we saw
%$Z_{{\rm{SO}}'_n}(T) = T$. The problem is to compute $N_{{\rm{SO}}'_n}(T)(k)/T(k)$. 
%To proceed in a characteristic-free way, we will first compute
%$N_{{\rm{O}}_n}(T)(k)/T(k)$, and then we will figure out which subgroup comes
%from ${\rm{SO}}'_n$.   This latter step is where characteristic 2 will exhibit different behavior
%(since ${\rm{SO}}'_n = {\rm{O}}_n$ when $n$ is even and ${\rm{char}}(k) = 2$). 

%Let $m = \dim T$, so $n = 2m$ or $n = 2m+1$.  A natural copy of
%$S_m \ltimes (\Z/2\Z)^m$ inside of $N_{{\rm{O}}_n}(T)$ having the natural faithful action on the
% character group
%${\rm{X}}(T) = {\rm{Hom}}(T,{\rm{GL}}_1) \simeq \Z^m$ (via permutation and negation of coordinates)
%is constructed as follows. 
%Let the standard basis be denoted $\{e_1, e'_1, \dots, e_m, e'_m\}$ when $n = 2m$
%and $\{e_0, e_1, e'_1, \dots, e_m, e'_m\}$ when $n = 2m+1$, so $q = q_n$ where 
%$$q_{2m} = x_1 x'_1 + \dots + x_m x'_m,\,\,\,\,\,q_{2m+1} = x_0^2 + x_1 x'_1 + \dots + x_m x'_m.$$
%There is a natural homomorphism $S_m \rightarrow {\rm{O}}_{n}(k)$ 
%via permutation of the $m$ ordered pairs $(e_i, e'_i)$. There
%is a natural homomorphism $(\Z/2\Z)^m \rightarrow
%{\rm{O}}_{n}(k)$ whose definition depends on the parity of
%$n$:  if $n$ is odd then the $i$th factor $\Z/2\Z$ swaps $e_i$ and $e'_i$ and negates $e_0$ while 
%leaving all other basis vectors invariant, and if $n$ is even then the action is the same
%except there is no mention of $e_0$.  The image of $S_m$ is readily checked
%to lie in ${\rm{SO}}'_n$, and the same holds for
%$(\Z/2\Z)^m$ when $n$ is odd (due to the sign on $e_0$). In contrast, if $n$ is even then 
%the image of a point $(a_1,\dots,a_m) \in (\Z/2\Z)^m$
%has determinant $(-1)^{\sum a_j}$ (i.e., determinant 1 
%when the number of nonzero $a_j$ is even, and determinant $-1$ when the number is odd).
%In this way we have defined a homomorphism 
%$S_m \ltimes (\Z/2\Z)^m \rightarrow N_{{\rm{O}}_n}(T)(k)$ which yields the action on 
%${\rm{X}}(T) = \Z^m$ in the expected faithful manner as promised, and if
%$n$ is odd then the image is even contained in $N_{{\rm{SO}}'_n}(T)$.

%Recall from the proof of Proposition \ref{findcenter} that
%$Z_{{\rm{O}}_n}(T) = T$ when $n$ is even, whereas
%$Z_{{\rm{O}}_n}(T) = \mu_2 \times T$ when $n$ is odd.  More uniformly,
%$Z_{{\rm{O}}_n}(T) = D \bigcap {\rm{O}}_n$ where $D$ is the diagonal torus of ${\rm{GL}}_n$.
%The natural map of $k$-groups 
%\begin{equation}\label{snm}
%(S_m \ltimes (\Z/2\Z)^m) \ltimes Z_{{\rm{O}}_n}(T) \rightarrow N_{{\rm{O}}_n}(T)
%\end{equation}
%is a closed immersion, since $D$ clearly has trivial intersection with the finite
%constant subgroup $S_m \ltimes (\Z/2\Z)^m$ (due to the {\em faithfulness} of
%the action on $T$, which commutes with $D$).  Clearly the $S_m$-action preserves
%the ``hyperplane''
%$$H_m = \{(a_1,\dots,a_m) \in (\Z/2\Z)^m\,|\,\sum a_j = 0\},$$
%so $S_m \ltimes H_m$ is a subgroup in $S_m \ltimes (\Z/2\Z)^m$, visibly of index 2.
%If  ${\rm{char}}(k) \ne 2$ and $n$ is even then this subgroup is the intersection of
%$S_m \ltimes (\Z/2\Z)^m$ with ${\rm{SO}}_n$. 

%We now show that (\ref{snm}) is bijective on $k$-valued points.
%Consider any element $g \in {\rm{O}}_n(k)$ that normalizes $T$.
%The standard basis lines in $V = k^n$ have an intrinsic characterization in terms of
%$T$: they are the only line stable under the $T$-action on $V$, with pairwise
%distinct eigencharacters $T \rightarrow {\rm{GL}}_1$.    Thus, $g$ must permute these
%lines. 

% If $n$ is odd then $g$ must carry $k e_0$ into itself since this is the unique line
%with trivial $T$-action, and in general if $k e_i$ is carried into
%$\{k e_j, k e'_j\}$ then $k e'_i$ is also carried to the other member of
%this unordered pair.  To see this latter claim, which is to say that
%$g$ permutes the unordered pairs $\{k e_i, k e'_i\}$, we simply observe
%that the eigencharacters $\chi_i, \chi'_i:T \rightrightarrows \Gm$
%giving the $T$-action on the respective lines $k e_i$ and $k e'_i$
%are the maps $t \mapsto t_i, t_i^{-1}$, so these share the same kernel and
%all other eigencharacters have a different kernel.  But this permutation among
%the unordered pairs $\{e_i, e'_i\}$, together with swaps within each unordered
%pair (coupled with a sign on $e_0$ when $n$ is odd) gives {\em precisely}
%the action on $S_m \ltimes (\Z/2\Z)^m$.   Hence, translating $g$ by an element of
%this finite group brings us to the case when $g$ {\em preserves} each basis line. 
%But then $g$ is a diagonal matrix. That is,
%$g \in D \bigcap {\rm{O}}_n = Z_{{\rm{O}}_n}(T)$, as desired.

%Recall that $T = Z_{{\rm{SO}}'_n}(T) =
%{\rm{SO}}'_n \bigcap Z_{{\rm{O}}_n}(T)$. Thus, from our determination of $N_{{\rm{O}}_n}(T)(k)$,
%$Z_{{\rm{O}}_n}(T)$, and $(S_m \ltimes (\Z/2\Z)^m) \bigcap {\rm{SO}}'_n(k)$, we can now
%compute $N_{{\rm{SO}}'_n}(k)/T(k)$. 
%We treat the case of even and odd $n$ separately.  If $n = 2m+1$ then we have shown that 
%${\rm{O}}_n = {\rm{SO}}'_n \times \mu_2$ in all characteristics, 
%and $S_m \ltimes (\Z/2\Z)^m \subset {\rm{SO}}'_n$.  Hence, 
%$$N_{{\rm{SO}}'_{2m+1}}(T)(k) = {\rm{SO}}'_{2m+1}(k) \bigcap {\rm{N}}_{{\rm{O}}_{2m+1}}(T)(k) =
%(S_m \ltimes (\Z/2\Z)^m) \ltimes T(k).$$
%Suppose instead that $n = 2m$.  Then $Z_{{\rm{O}}_n}(T) = T$, so
%$$N_{{\rm{SO}}'_{2m}}(T)(k) = {\rm{SO}}'_{2m}(k) \bigcap N_{{\rm{O}}_{2m}}(T)(k) =
%((S_m \ltimes (\Z/2\Z)^m) \bigcap {\rm{SO}}'_{2m}(k)) \ltimes T(k).$$
%This intersection $(S_m \ltimes (\Z/2\Z)^m) \bigcap {\rm{SO}}_{2m}(k)$ has already been
%computed:  it is $S_m \ltimes H_m$ if ${\rm{char}}(k) \ne 2$ and it is
%$S_m \ltimes (\Z/2\Z)^m$ if ${\rm{char}}(k) = 2$. 
%\end{example}
%
%What about the functorial center of ${\rm{O}}(q)$ over a field
%$k$, say assuming again that $\dim V \ge 3$?  
%In the non-smooth case
%(i.e., the defect-1 case in characteristic 2), 
%we actually have shown that ${\rm{O}}(q) = {\rm{SO}}'(q) \times \mu_2$.
%So in these cases there is not much to say.  Likewise, if
%${\rm{char}}(k) = 2$ and $\dim V$ is even then we have proved
%that ${\rm{O}}(q) = {\rm{SO}}'(q)$, so again there's nothing new to be said.
%If ${\rm{char}}(k) \ne 2$ and $\dim V$ is odd then 
%the scalar $-1$ in ${\rm{GL}}(V)$ provides a splitting
%${\rm{O}}(q) = {\rm{SO}}'(q) \times \mu_2$ exactly
%as in characteristic 2.   
%
%For the remaining case that 
%$\dim V$ is even and ${\rm{char}}(k) \ne 2$, we claim that
%${\rm{O}}(q)$ has the same center as ${\rm{SO}}'(q)$.  It suffices to check this
%after extending scalars to $\overline{k}$, and then by using the standard form we
%can easily find a $\mu_2$ subgroup carried isomorphically onto
%$\mu_2$ by the determinant homomorphism.  Since
%$\Det$ on ${\rm{O}}(q)$ is $\mu_2$-valued whenever $B_q$ is non-degenerate
%(such as when ${\rm{char}}(k) \ne 2$), we thereby get a semi-direct product decomposition
%\begin{equation}\label{semio}
%{\rm{O}}(q) = \mu_2 \ltimes {\rm{SO}}'(q).
%\end{equation}
%But now $\mu_2 = \Z/2\Z$ as $k$-groups, and its conjugation action on
%${\rm{SO}}(q)$ is visibly nontrivial.   Thus, it follows easily (check!)
%that $Z_{{\rm{O}}(q)} = Z_{{\rm{SO}}'(q)}$.  
%
%\begin{remark}\label{disconnrem} One consequence of this semi-direct product description
%(\ref{semio}) is that
%if $k$ is a field
%with ${\rm{char}}(k) \ne 2$ then ${\rm{O}}(q) - {\rm{SO}}(q)$ is non-empty and geometrically connected,
%so ${\rm{O}}(q)$ has two connected components, each geometrically connected.
%Beware that we do {\em not} claim to be able to find a copy of the component group
%$\Z/2\Z$ 
%inside ${\rm{O}}(q)$ over $k$ in general as we did in the Bureau of Standards case.
%(The most subtle case is when the projective hypersurface $(q = 0)$ has no $k$-rational points,
%in which case it is a bit hard to dig holes into ${\rm{O}}(q)$ to see what is going on
%over $k$.)
%\end{remark}
%
%The disconnectedness over a field $k$ away from characteristic 2 in
%Remark \ref{disconnrem} is a contrast with what we found for connectedness in characteristic 2:
%${\rm{O}}(q)_{\rm{red}} = {\rm{SO}}'(q)$ in all such cases, so ${\rm{O}}(q)$ is connected
%when ${\rm{char}}(k) = 2$ and $n \ge 3$ with $n$ odd.  
%In contrast, when $n \ge 2$ with $n$ even, 
%we have shown that ${\rm{O}}(q) = {\rm{SO}}'(q)$ and that this has geometric
%component group of order 2.

\newpage

\section{Proof of Existence Theorem over $\C$}\label{csec}

In this appendix, we prove the Existence Theorem
over $\C$. The main difficulty 
is the construction of connected semisimple groups
whose semisimple root datum is simply connected and has
irreducible underlying root system.
(See Definition \ref{sssc} and Lemma \ref{existlemma}.) 
Over any algebraically closed field $k$ of
characteristic 0, algebraic methods can easily settle the case of
semisimple reduced root data that are adjoint (see
Proposition \ref{adjointexist}).  From that case, one can
bootstrap to simply connected
semisimple reduced root data via an 
existence theorem for {\em projective} representations (modeled on the Borel--Weil
construction for semisimple complex Lie groups). This method is explained
in \cite[15.3, 23.1]{bible}, and is cited in \cite[XXV, 1.4]{sga3}.
In this appendix, we present an alternative analytic argument 
via covering space methods over the 
ground field $\C$, linking up the analytic viewpoint
with algebraic techniques. 
The references we shall use  are \cite{brtd} for the ``algebraicity'' of 
compact Lie groups
and \cite{hoch} for general facts related to complex Lie groups
and maximal compact subgroups of Lie groups with a finite component group.
An elegant summary (without proofs)
of much of this analytic background is given in \cite[VIII]{serreclie}.

But first, we sketch an algebraic proof based on enveloping algebras; this
can be extracted from \cite{hochcover} and was pointed out 
by P.\,Polo. 
Let $\mathfrak{g}$ be a semisimple Lie algebra over $k$ having the desired
root system (this exists, by \cite[18.4(a)]{humlie}). Consider the universal
enveloping algebra $U(\mathfrak{g})$, an associative $k$-algebra
whose representation theory coincides with that of $\mathfrak{g}$.
As $J$ varies through the 2-sided ideals of finite codimension in
$U(\mathfrak{g})$, the dual spaces $(U(\mathfrak{g})/J)^{\ast}$
exhaust the $k$-algebra generated by the ``matrix coefficients'' 
(on $U(\mathfrak{g})$) of the finite-dimensional representations
of $\mathfrak{g}$.  This 
equips $H(\mathfrak{g}) := \varinjlim (U(\mathfrak{g})/J)^{\ast}$
with a natural commutative $k$-algebra structure and a compatible 
Hopf algebra structure that defines an affine $k$-group scheme $G(\mathfrak{g}) = \Spec H(\mathfrak{g})$
(see \cite[\S3]{hochcover}).
Every dominant integral weight for $\mathfrak{g}$ is a $\Z_{\ge 0}$-linear
combination of the finitely many fundamental weights (dual basis to the coroots), so by
highest-weight theory for $\mathfrak{g}$ the $k$-algebra 
$H(\mathfrak{g})$ is finitely generated (see the end of \cite[\S5]{hochlie}).
The linear dual of $U(\mathfrak{g})$ 
is a $k$-algebra that contains
$H(\mathfrak{g})$ as a subalgebra, so $H(\mathfrak{g})$ is a domain since
$U(\mathfrak{g})^{\ast}$ is a formal power series ring (see \cite[\S2]{hochlie}).
Hence, $G(\mathfrak{g})$ is connected and smooth. 

By \cite[3.1]{hochcover} (or \cite[6.1]{hochcover}), 
$\mathfrak{g}$ is identified with ${\rm{Lie}}(G(\mathfrak{g}))$
(so $G(\mathfrak{g})$ is semisimple; see Exercise \ref{rad}).
Every finite-dimensional $\mathfrak{g}$-representation
is naturally an $H(\mathfrak{g})$-comodule, which is to say
a $G(\mathfrak{g})$-representation, and this is inverse to applying
the Lie functor to a $G(\mathfrak{g})$-representation.
Since every representation $\mathfrak{g} \rightarrow \mathfrak{gl}(V)$
has just been ``integrated'' to a representation $G(\mathfrak{g}) \rightarrow \GL(V)$, an argument 
with highest-weight theory for $\mathfrak{g}$ (as in the proof
of ``$\Rightarrow$'' in Proposition \ref{scgt}) shows $G(\mathfrak{g})$
has a simply connected root datum.

\subsection{Preliminary steps}\label{ssirr}

By the self-contained (and rather formal) Lemma \ref{existlemma}, 
 to prove the Existence Theorem over $\C$ it suffices to consider semisimple root data 
$R$ that are simply connected.  We will approach such $R$
through preliminary consideration of the adjoint case, beginning with
a basic existence result over 
any algebraically closed field $k$ with ${\rm{char}}(k) = 0$.

\begin{proposition}\label{adjointexist} Let 
$R = (X,\Phi,X^{\vee},\Phi^{\vee})$ be an adjoint semisimple reduced root datum.
There exists a connected semisimple $k$-group $(\mathscr{G},\mathscr{T})$
such that $R(\mathscr{G},\mathscr{T}) \simeq R$.
\end{proposition}
\index{Existence Theorem!adjoint case (char.\,0)}

\begin{proof}
By \cite[18.4(a)]{humlie}, there 
exists a semisimple Lie algebra $\mathfrak{g}$ over $k$ and a Cartan subalgebra
$\mathfrak{t}$ such that the root system $\Phi(\mathfrak{g},\mathfrak{t})$ 
(see \cite[8.5, 15.3]{humlie}) coincides
with $(X_{\Q},\Phi)$.  By Ado's theorem \cite[I, \S7.3, Thm.\,2]{lie1}, 
there exists an injective map of Lie algebras
$\mathfrak{g} \hookrightarrow \mathfrak{gl}_n$ over $k$.
Since $\mathfrak{g}$ is its own derived subalgebra (due to semisimplicity)
and ${\rm{char}}(k) = 0$, it is an ``algebraic''
subalgebra of $\mathfrak{gl}_n$ in the sense that there exists a (unique) connected linear
algebraic $k$-subgroup $G \subset \GL_n$ satisfying 
${\rm{Lie}}(G) = \mathfrak{g}$ inside $\mathfrak{gl}_n$
\cite[7.9]{borelag}. 
The $k$-group $G$ is necessarily semisimple, since the Lie algebra
${\rm{Lie}}(\mathscr{R}(G))$ of the radical
is a solvable ideal in the semisimple $\mathfrak{g}$
and hence vanishes (so $\mathscr{R}(G) = 1$). 

Let $T \subset G$ be a maximal torus, so the abelian subalgebra ${\rm{Lie}}(T) \subset \mathfrak{g}$
is its own Lie-theoretic centralizer (as  $T = Z_G(T)$ and ${\rm{char}}(k) = 0$).  Hence,
${\rm{Lie}}(T)$ is a Cartan subalgebra of the semisimple Lie algebra $\mathfrak{g}$ (i.e., it is maximal
among commutative subalgebras whose elements have semisimple adjoint action on $\mathfrak{g}$).  All Cartan
subalgebras of $\mathfrak{g}$ are in the same ${\rm{Aut}}(\mathfrak{g})$-orbit
\cite[16.2]{humlie}, so we get an isomorphism of root systems 
$$({\rm{X}}(T)_{\Q}, \Phi(G,T)) = \Phi(\mathfrak{g}, {\rm{Lie}}(T)) \simeq \Phi(\mathfrak{g},\mathfrak{t}) =
(X_{\Q},\Phi).$$
This identifies the Weyl groups $W(\Phi(G,T))$ and $W(\Phi)$
since the root system determines the Weyl group 
(without reference to a Euclidean structure).
The roots determine the coroots since the 
reflection $s_a:v \mapsto v - \langle v, a^{\vee} \rangle a$ in $a$ is uniquely
determined by preservation of the set of roots, 
so the isomorphism ${\rm{X}}(T)_{\Q} \simeq X_{\Q}$
identifies $\Phi(G,T)^{\vee}$ with $\Phi^{\vee}$ as well. 

Since ${\rm{X}}(T/Z_G) = \Z \Phi(G,T)$ by Corollary \ref{computez}(1), and
$\Z \Phi = X$ by our hypothesis that $R$ is adjoint, the equality $\Phi(G,T) = \Phi$
via ${\rm{X}}(T)_{\Q} \simeq X_{\Q}$ forces the compatible identification
${\rm{X}}(T/Z_G) \simeq X$.  This yields an isomorphism of root data
$R(G/Z_G,T/Z_G)  \simeq (X, \Phi, X^{\vee}, \Phi^{\vee}) = R$.
Set $(\mathscr{G}, \mathscr{T}) = (G/Z_G, T/Z_G)$. 
\end{proof}

Fix a semisimple  reduced root datum $R = (X,\Phi,X^{\vee},\Phi^{\vee})$ that is simply connected.
Let $R^{\rm{ad}} = (\Z \Phi,\Phi, (\Z \Phi)^{\ast}, \Phi^{\vee})$ denote the associated adjoint semisimple
root datum.  Let $k$ be an algebraically closed field of characteristic 0.
By Proposition \ref{adjointexist},
there exists a connected semisimple $k$-group $(G,T)$ such that
$R(G,T) \simeq R^{\rm{ad}}$.  We seek a connected semisimple $k$-group
$G'$ and an isogeny $h:G' \rightarrow G$
such that for the maximal torus $T' = h^{-1}(T)$ in $G'$
there exists a (necessarily unique) isomorphism $R(G',T') \simeq R$
over the isomorphism $R(G,T) \simeq R^{\rm{ad}}$
via the central isogenies $R(G',T') \rightarrow R(G,T)$
and $R \rightarrow R^{\rm{ad}}$. 
Over $\C$ we can
build covers via topology, so now we  set $k = \C$ and pass to the analytic theory.

\subsection{Compact and complex Lie groups}\label{ccgp}

For a connected semisimple $\C$-group $G$, 
the group $G(\C)$ is connected for the analytic topology
(since for an open cell $\Omega \subset G$ around 1, $\Omega(\C)$ is visibly connected
and $g\Omega(\C)$ meets $\Omega(\C)$ for any $g \in G(\C)$,
due to the irreducibility of $G$).  Thus, $G(\C)$ is a connected Lie group.
To build isogenous covers of $G$, we will use maximal compact
subgroups of $G(\C)$ and topological facts from the theory of Lie groups.
%Now we recall some basic facts concerning complex Lie
%groups and compact subgroups of general Lie groups.
%As a preliminary step, we record a basic finiteness
%property over $\R$ (though it is not needed in what follows):
%
%\begin{lemma}\label{finitepi0} If $G$ is a linear algebraic $\R$-group then $\pi_0(G(\R))$ is finite.
%\end{lemma}
%
%\begin{proof}
%Since $G^0(\R)$ is normal in $G(\R)$ with quotient contained in the finite group
%$(G/G^0)(\R)$, we may assume $G$ is connected.
%The unipotent radical $U = \mathscr{R}_u(G)$ has a composition series
%over $\R$ whose successive quotients are isomorphic to $\Ga$
%(since $\R$ is perfect, or via $\log_U$), so the vanishing
%of ${\rm{H}}^1(\R, \Ga)$ implies that ${\rm{H}}^1(\R,U) = 1$.   Hence,
%the open immersion $G(\R)/U(\R) \rightarrow (G/U)(\R)$ is an equality, so
%we can assume that $G$ is reductive.  
%
%Letting $Z$ denote the center of $G$, it is an elementary exercise
%to check that ${\rm{H}}^1(\R,Z)$ is finite (as for any $\R$-group of multiplicative type in the role of $Z$),
%so the open normal 
%subgroup $G(\R)/Z(\R) \subset (G/Z)(\R)$ has finite index. We may therefore replace $G$ with $G/Z$
%to reduce to the case that $G$ is semisimple with $Z = 1$, so
%${\rm{Ad}}_G:G \rightarrow \GL(\mathfrak{g})$ is a closed immersion.
%
%Since $G(\R)$ is the fixed point locus in the adjoint image of $G(\C)$
%under the involution on $\GL(\mathfrak{g}_{\C})$ arising
%from the complex conjugation on $\mathfrak{g}_{\C}$, 
%a general criterion (see \cite[XV, \S2]{hoch}) for adjoint images in
%the presence of a complex conjugation on a semisimple Lie
%algebra over $\C$ implies that $G(\R) = K \times V$ under
%multiplication, where $K$ is a compact Lie subgroup of $G(\R)$ and $V$ is a vector group 
%that is closed in $G(\R)$.  Hence, $\pi_0(G(\R)) = \pi_0(K)$ is finite.
%\end{proof}
%
%\begin{remark} By using the structure theory for the group of 
%rational points of a connected reductive group over an arbitrary field (such as $\R$), one
%can do much better: if $G$ is a connected linear algebraic $\R$-group
%and $S$ is a maximal $\R$-split torus in $G$ then $\pi_0(S(\R)) \rightarrow \pi_0(G(\R))$
%is surjective, so $\pi_0(G(\R))$ is an elementary abelian 2-group. 
%See \cite[14.4]{boreltits}.
%\end{remark}

\begin{proposition}\label{gagagp}
The functor $G \rightsquigarrow G(\C)$ from connected reductive $\C$-groups
to complex Lie groups is fully faithful.  More generally, if
$G$ is a connected reductive $\C$-group and $G'$ is a 
linear algebraic $\C$-group then every analytic homomorphism
$f:G(\C) \rightarrow G'(\C)$ arises from a unique $\C$-homomorphism $G \rightarrow G'$.
\end{proposition}
\index{reductive group!and complex Lie groups|(}

The essential image of the functor in Proposition \ref{gagagp}  is identified
in Example \ref{cx}. 


\begin{proof}
First we treat semisimple $G$, and then we 
allow a central torus.   Assume $G$ is semisimple, so 
the Lie algebra $\mathfrak{g}$ is semisimple (Exercise \ref{rad}).
The graph $\Gamma_f$ is a connected closed complex Lie subgroup
of $G(\C) \times G'(\C)$, so its Lie algebra  is a subalgebra
of $\mathfrak{g} \bigoplus \mathfrak{g}'$ that projects isomorphically
onto the semisimple $\mathfrak{g}$ and hence is its
own derived subalgebra. But in characteristic 0,
a Lie subalgebra $\mathfrak{h}$ of the Lie algebra of a linear algebraic
group arises from a connected linear algebraic subgroup
provided that $\mathfrak{h}$ is its own derived subalgebra \cite[7.9]{borelag}.
We thereby get a connected linear algebraic subgroup
$H \subset G \times G'$ such that ${\rm{Lie}}(H) = {\rm{Lie}}(\Gamma_f)$
inside $\mathfrak{g} \bigoplus \mathfrak{g}'$, and $H$ is semisimple since
${\rm{Lie}}(H) \simeq \mathfrak{g}$ is semisimple.   
The connected closed Lie subgroups $H(\C)$ and $\Gamma_f$ in $G(\C) \times G'(\C)$
must therefore be equal.    Thus,
the algebraic projection $H \rightarrow G$ induces a holomorphic
isomorphism on $\C$-points, so it is \'etale and bijective on $\C$-points, hence an isomorphism
(by Zariski's Main Theorem).
It follows that the $\C$-subgroup $H$ in $G \times G'$ is the graph
of a $\C$-homomorphism $G \rightarrow G'$; this clearly 
analytifies to $f$.

Now consider the general case, so $G = (Z \times \mathscr{D}(G))/\mu$ for the
maximal central torus $Z$, connected 
semisimple derived group $\mathscr{D}(G)$, and a finite central subgroup $\mu
\subset Z \times \mathscr{D}(G)$.  The algebraic
theory of quotients by finite subgroups over $\C$ analytifies to the analogous theory on the analytic side,
so it suffices to treat the cases of $T$ and $\mathscr{D}(G)$.  We have already settled $\mathscr{D}(G)$,
so we may now assume $G = T$, and more specifically $G = \Gm$.   Our problem is
to prove the algebraicity of any holomorphic homomorphism
$\C^{\times} \rightarrow G'(\C)$.  Using a faithful representation $G' \hookrightarrow {\rm{GL}}_n$,
we are reduced to checking the algebraicity of any holomorphic action of
$\C^{\times}$ on a finite-dimensional $\C$-vector space $V$.  The action of each finite subgroup
$\C^{\times}[n] = \mu_n$ diagonalizes, and the $\C^{\times}$-action preserves
each $\mu_n$-isotypic subspace of $V$.  Thus, by induction on $\dim V$
we can assume that {\em every} $\mu_n$ acts through
a character $\chi_n:\mu_n \rightarrow \C^{\times}$.
This says that the given holomorphic map $\rho:\C^{\times} \rightarrow {\rm{GL}}(V)$
carries each $\mu_n$ into the central $Z = \C^{\times}$, so $\rho$ does as well
(since the complex-analytic subgroup  $\rho^{-1}(Z) \subset \C^{\times}$ contains
all $\mu_n$, so it is infinite and thus exhausts the connected 1-dimensional $\C^{\times}$).
In other words, we are reduced to showing that 
a holomorphic homomorphism $\chi:\C^{\times} \rightarrow \C^{\times}$
must be $\chi(z) = z^m$ for some $m \in \Z$.  
Since $\C^{\times} = \C/2\pi i \Z$ with $\C$ simply connected, this clearly holds.
\end{proof}

\begin{remark}
The analogue of Proposition \ref{gagagp} over $\RRR$ fails, even for semisimple
groups whose set of $\RRR$-points is connected.  For example, the natural
isogeny ${\rm{SL}}_{2n+1} \rightarrow
{\rm{PGL}}_{2n+1}$ of degree $2n+1$ induces an isomorphism on $\RRR$-points,
the inverse of which is non-algebraic.
\end{remark}

\begin{corollary}\label{corss}
The finite-dimensional
holomorphic representations of $G(\C)$ are completely reducible
for any connected reductive $\C$-group $G$.
\end{corollary}

\begin{proof}
By Proposition \ref{gagagp}, it suffices to
prove the analogous result for the finite-dimensional
algebraic representations of $G$.  As in the proof
of Proposition \ref{gagagp}, we immediately reduce
to the separate cases of tori and semisimple $G$.
The case of tori is clear (weight spaces),
so we can assume that $G$ is semisimple.
Let $\rho:G \rightarrow {\rm{GL}}(V)$ be a finite-dimensional
representation, and let $\mathfrak{g} \rightarrow {\rm{End}}(V)$
be the associated Lie algebra representation.
Since we are in characteristic 0 and
$G$ is connected, a linear subspace of $V$ is $G$-stable if and only if it is
$\mathfrak{g}$-stable.  Hence, it is sufficient to prove
complete reducibility of the representation theory of
$\mathfrak{g}$.  This in turn follows from the fact that
$\mathfrak{g}$ is semisimple (by Exercise \ref{rad}).
\end{proof}
\index{reductive group!complex Lie groups|)}


The mechanism by which we will keep track of algebraicity when working on the analytic
side is to use maximal compact subgroups of complex Lie groups
and the ``algebraicity'' of the theory of compact Lie groups:

\index{reductive group!compact Lie groups|(}
\index{compact Lie groups|(}
\begin{theorem}[Chevalley--Tannaka]\label{anisotropic}
The functor $H \rightsquigarrow H(\RRR)$ is an equivalence
from the category of $\RRR$-anisotropic 
reductive $\RRR$-groups whose connected components
contain $\RRR$-points to the category of
compact Lie groups.

The $\RRR$-group $H$ is connected if and only if
$H(\RRR)$ is connected,
and $H$ is semisimple if and only if the Lie group $H(\RRR)$ has semisimple Lie algebra.
\end{theorem}

\begin{proof}
The semisimplicity criterion for $H$ follows from 
Exercise \ref{rad}
since ${\rm{Lie}}(H(\RRR)) = {\rm{Lie}}(H)$ (which is semisimple over
$\RRR$ if and only if the complex Lie algebra ${\rm{Lie}}(H)_{\C} = {\rm{Lie}}(H_{\C})$ is
semisimple).  It is not at all obvious that $H(\RRR)$ is compact
when $H$ is $\RRR$-anisotropic, nor that $H(\RRR)$ is connected
when $H$ is moreover connected.
We begin by proving the compactness of $H(\RRR)$ via a variant of the 
method used by G.\,Prasad in his elementary proof of the analogous
result over non-archimedean local fields in \cite{prasad}.

Fix a faithful 
finite-dimensional representation $\rho:H \hookrightarrow \GL(V)$  over $\RRR$ (i.e., $\ker \rho = 1$).
By Exercise \ref{1.6}, $V_{\C}$ is completely reducible as an $H_{\C}$-representation. 
 Let $\bigoplus W_j$ be a decomposition of $V_{\C}$ into a direct
sum of irreducible subrepresentations of $H(\C)$.
The group $H^0(\RRR)$ is Zariski-dense in $H^0$ by the unirationality of connected reductive
groups over fields \cite[18.2(ii)]{borelag}.  By hypothesis every connected
component of $H$ contains an $\RRR$-point
and hence is an $H(\RRR)$-translate of $H^0$, so $H(\RRR)$ is Zariski-dense in $H$.
Thus, $H(\RRR)$ is Zariski-dense in $H_{\C}$, so 
each $W_j$ is irreducible as a representation of $H(\RRR)$.
If $H(\RRR)$ has compact image in each
$\GL(W_j)$ then $H(\RRR)$ is compact.   
The image $H_j$ of $H$ in ${\rm{R}}_{\C/\RRR}(\GL(W_j))$
is $\RRR$-anisotropic, and $H_j(\RRR)$ must be Zariski-dense
in $H_j$ since even the image of $H(\RRR)$ in $H_j$ is Zariski-dense.
We may therefore replace $H$ with each $H_j$ to reduce to the case
that $H$ is a closed subgroup of ${\rm{R}}_{\C/\RRR}(\GL(W))$
with $W$ irreducible as a $\C$-linear representation of $H(\RRR)$.

For $h \in H(\RRR)$, its eigenvalues $\lambda$ on $W$
satisfy $|\lambda| = 1$.  Indeed, by Jordan decomposition in
$H(\RRR)$ we may assume $h$ is semisimple, and it is harmless
to replace $h$ with $h^n$ for $n > 0$.  Thus, we can assume
that $h$ lies in the identity component of the 
commutative semisimple Zariski closure of $h^{\Z}$.  That identity component is 
an $\RRR$-torus $T$.  All $\RRR$-tori in $H$
are $\RRR$-anisotropic and hence have a compact group of $\RRR$-points,
so the eigenvalues of the integral powers of $h$ are bounded in
$\C$.  This forces the eigenvalues to lie on the unit circle.

Since $H(\RRR)$ acts irreducibly on $W$, its
image in $\GL(W)$ must generate the $\C$-algebra ${\rm{End}}(W)$ (Burnside),
so $H(\RRR)$ spans ${\rm{End}}(W)$ over $\C$. 
Choose $\{h_j\}$ in $H(\RRR)$ that is a $\C$-basis of
${\rm{End}}(W)$, and consider the dual basis $\{L_j\}$
under the perfect $\C$-bilinear trace pairing $(T,T') = {\rm{Tr}}(T \circ T')$
on ${\rm{End}}(W)$.  The eigenvalue condition
forces $|{\rm{Tr}}(hh_j)| \le n := \dim W$ for all $h \in H(\RRR)$, 
so (exactly as in the proof of finite generation of integer rings
of number fields) the coefficients of $\rho(h)$ relative to $\{L_j\}$
are bounded in $\C$ independently of $h$.  Hence, the closed set $\rho(H(\RRR))$ 
in ${\rm{GL}}(W)$ is bounded in
${\rm{End}}(W)$. It is also closed
in ${\rm{End}}(W)$ because $|\Det(\rho(h))| = 1$
for all $h \in H(\RRR)$ (due to the eigenvalues of $\rho(h)$
lying on the unit circle), so $\rho(H(\RRR))$ is compact.

%
%
%\begin{theorem}\label{maxcompact}
%Let $\mathscr{H}$ be a Lie group with finite component group.  Every compact subgroup of 
%$\mathscr{H}$
%is contained in a maximal one, and all maximal compact subgroups $K$ of
%$\mathscr{H}$ are conjugate to each other.  Moreover, for any $K$ 
%there are closed vector subgroups $V_1, \dots, V_n \subset \mathscr{H}$
%such that the multiplication map
%$K \times V_1 \times \dots \times V_n \rightarrow \mathscr{H}$ is a $C^{\infty}$ isomorphism.
%In particular, $K$ is a deformation retract of $\mathscr{H}$, so
%$\pi_0(K) = \pi_0(\mathscr{H})$ and $\pi_1(K,1) = \pi_1(\mathscr{H},1)$.
%\end{theorem}
%
%\begin{proof}
%This is \cite[XV, 3.1]{hoch}.  (We only need the
%case of connected $\mathscr{H}$.
%The connected case does not seem to formally imply the general case.)
%\end{proof}

%Thus, if $H(\R)^0$ is non-compact then
%there exists a closed subgroup $V$ that is a nonzero vector group.
%For $h \in V - \{0\}$, the subgroup $\langle h \rangle$ is
%infinite cyclic, so the identity component of its
%Zariski closure in $H$ is a nontrivial commutative connected
%$\R$-subgroup $Z$.  The group
%$Z(\R)$ meets $V$ in a closed
%subgroup that contains the unbounded $\langle h \rangle$ in $V$,
%so $Z(\R)$ is non-compact.
%As for any connected commutative $\R$-group,
%$Z = T \times U$ for an $\R$-torus $T$ and $U = \mathscr{R}_u(Z)$.
%By the $\R$-anisotropicity of $H$, necessarily $T(\R)$ is compact.
%Thus, $U \ne 1$.    In other words, assuming $H(\R)^0$ is non-compact,
%we have constructed a nontrivial smooth connected
%unipotent $\R$-subgroup of $H$, and hence $\Ga$ as an $\R$-subgroup of $H$.
%
%To reach a contradiction, we pass to the Lie algebra of this $\Ga$,
%which provides a nonzero nilpotent element $X_0$ in
%the semisimple Lie algebra $\mathfrak{h} = {\rm{Lie}}(H)$ over $\R$.
%By the Jacobson--Morosov theorem \cite[VIII, \S11.2, Prop.\,2]{lie7},
%there exists an injection of Lie algebras
%$\mathfrak{sl}_2(\R) \hookrightarrow \mathfrak{h}$
%carrying $(\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix})$ to $X_0$.
%A graph argument similar to the one in the proof of Proposition \ref{gagagp}
%yields a connected closed $\R$-subgroup $H' \subset H$
%with Lie algebra $\mathfrak{h}' = \mathfrak{sl}_2(\R)$ inside $\mathfrak{h}$.
%By Lemma \ref{sl2r} below, 
%$H'/Z_{H'} = {\rm{PGL}}_2$ as an $\R$-group.
%Hence, $H'/Z_{H'}$ contains an $\R$-torus $T = \Gm$, so likewise for $H'$
%(via identity component of the preimage of $T$ in $H'$).  
%This contradicts the $\R$-anisotropicity of $H$ and so 
%completes the proof of compactness of $H(\R)^0$ (and thus of $H(\R)$). 
%
%\begin{lemma}\label{sl2r}
%Let $G$ be a connected semisimple group over a field $k$
%with ${\rm{char}}(k) = 0$ and Lie algebra $\mathfrak{g} \simeq \mathfrak{sl}_2$.
%Then $G/Z_G \simeq {\rm{PGL}}_2$.
%\end{lemma}
%
%\begin{proof}
%The adjoint representation defines a closed immersion
%$G/Z_G \hookrightarrow \Aut(\mathfrak{g}) = \Aut(\mathfrak{sl}_2)$. 
%It is classical that the natural map
%${\rm{PGL}}_2 \rightarrow \Aut(\mathfrak{sl}_2)$ is an isomorphism of $k$-groups.
%Since $\dim G/Z_G = 3$, we are done.
%\end{proof}
%
%We will prove this following an argument of Borel from \cite[24.6c(ii)]{borelag}.
%
%\begin{lemma} The $\R$-group $H$ is generated by $\R$-tori.
%\end{lemma}
%
%\begin{proof} Let $N$ be the connected smooth closed $\R$-subgroup generated
%by the $\R$-tori.  This is normal in $H$, since $H(\R)$ is Zariski-dense in $H$ (as for any smooth connected
%$\R$-scheme with an $\R$-point).  Hence, $H/N$ is a connected reductive
%$\R$-group which contains no nontrivial $\R$-tori.  This forces $N = H$ by
%applying the (much simpler) characteristic-0 case of
%Grothendieck's theorem on tori in connected reductive groups over fields
%(Appendix \ref{torusapp}).  But here is an alternative simple proof that $N = H$ by using
%arguments specific to $\R$. 
%
%By Exercise \ref{gross1}(iv),
%the locus of regular semisimple elements in $(H/N)(\C)$ is Zariski-dense and open.
%By Galois descent, the corresponding open
%subset of $(H/N)_{\C}$ descends to a Zariski-dense open subset of $H/N$.
%Zariski-density of $(H/N)(\R)$ in $H/N$ then implies that 
%there exists a regular semisimple $g \in (H/N)(\R)$, and by deforming $g$ over $\R$ 
%we can ensure (by countability considerations) that $g$ does not have finite order.  Thus,
%the identity component of the Zariski closure of $\langle g \rangle$ in $H/N$
%has positive dimension, so it is a nontrivial $\R$-torus in $H/N$, a contradiction.
%We conclude that $H = N$.
%\end{proof}
%
%We now obtain finitely many $\R$-tori
%$T_1, \dots, T_n$ of $H$ such that the product map 
%$\prod T_j \rightarrow H$ is surjective. 
%By \cite[7.6]{borelag}, the conjugates
%of the ${\rm{Lie}}(T_j)_{\C}$'s under $H(\C)$ span ${\rm{Lie}}(H)_{\C}$, so 
% the algebraicity of ${\rm{Ad}}_H$ over $\R$ and the Zariski-density of $H(\R)$ in $H$
% ensure that the $H(\R)$-conjugates of the ${\rm{Lie}}(T_j)$'s span
% ${\rm{Lie}}(H)$ over $\R$.  
% 
% Thus, at the cost of enlarging the collection of $\R$-tori, we can arrange
% that the ${\rm{Lie}}(T_j)$'s span ${\rm{Lie}}(H)$.
% Each $T_j(\R)$ is compact (a power of $S^1$), 
%and the multiplication map $\prod T_j \rightarrow H$
%is a submersion at the identity, so there is a Zariski-open neighborhood $V$ 
%of $1$ in $H$ such that $V(\R)$ is contained in the compact image of $\prod T_j(\R)$.
%This compact image must then contain the closure of $V(\R)$ in $H(\R)$,
%which in turn contains $H(\R)^0$ (since $V$ is Zariski-open in $H$ around 1).
%Hence, $H(\R)^0$ is compact, so $H(\R)$ is compact. . 

To establish the desired equivalence of categories between
anisotropic reductive $\RRR$-groups and compact Lie groups (including
the connectedness aspect), we shall now proceed
in reverse by constructing a functor from  compact Lie groups
to reductive $\RRR$-groups.   
This rests on the {\em real representation algebra} $R(K)$ of a compact
Lie group $K$: the $\RRR$-algebra generated inside
\index{real representation algebra}
\index{ aaark@$R(K)$, $R(K, \C)$}
the $\RRR$-algebra of continuous $\RRR$-valued functions on $K$ by the
matrix entries of the finite-dimensional continuous representations of
$K$ over $\RRR$.  Let $R(K,\C)$ denote the analogue defined using 
continuous linear representations over $\C$.  The operation
of scalar extension of $\RRR$-linear representations to $\C$-linear
representations defines an $\RRR$-algebra
injection $R(K) \rightarrow R(K,\C)$.  There is a natural involution $\iota$ of
$R(K,\C)$ over complex conjugation on $\C$ (using 
scalar extension through complex conjugation on
$\C$-linear representations of $K$), and this acts trivially on $R(K)$. 
Galois descent gives that $R(K,\C) = \C \otimes_{\RRR} R(K,\C)^{\iota=1}$,
so the natural map of $\C$-algebras
\begin{equation}\label{rgcisom}
\C \otimes_{\RRR} R(K) \rightarrow R(K,\C)
\end{equation}
is injective.  But every $\C$-linear representation
can be viewed as an $\RRR$-linear representation, so
(via a choice of $i = \sqrt{-1} \in \C$) if $f = u + iv \in R(K, \C)$
with $\RRR$-valued $u$ and $v$ then $u, v \in R(K)$.  In other words,
(\ref{rgcisom}) is an isomorphism.

There exists a faithful representation $\rho:K \hookrightarrow \GL_n(\RRR)$
(see \cite[III, 4.1]{brtd}, which rests on the Peter--Weyl theorem).
Upon choosing such a $\rho$, we get an $\RRR$-algebra map
$\RRR[x_{ij}][1/\Det] \rightarrow R(K)$. 
The representation algebra is generated by the matrix
functions from any single faithful
representation \cite[III, 1.4(iii), 2.7(i)]{brtd}, so this map is surjective. In particular, 
$R(K)$ is finitely generated over $\RRR$, and likewise $R(K,\C)$ is finitely generated over $\C$.
There is a natural Hopf algebra structure on $R(K)$ that is
contravariant in $K$ (see \cite[III, \S7]{brtd}),
and by Tannaka duality \cite[III, 7.15]{brtd} the natural map
$K \rightarrow (\Spec R(K))(\RRR)$ defined via evaluation
is an isomorphism of Lie groups.  
The linear algebraic $\RRR$-group $K^{\rm{alg}} := \Spec R(K)$ may be disconnected,
but we can detect its connected components using $\RRR$-points:
\index{ aaakalg@$K^{\rm{alg}}$}

\begin{lemma}\label{kprop} The locus $K = K^{\rm{alg}}(\RRR)$ is Zariski-dense in $K^{\rm{alg}}$, 
and $(K^{\rm{alg}})^0$ is $\RRR$-anisotropic reductive.
In particular, each connected component of $K^{\rm{alg}}$ contains an $\RRR$-point and hence
is geometrically connected over $\RRR$.
\end{lemma}

\begin{proof}
  By definition $R(K)$ is a subalgebra of
the $\RRR$-algebra of continuous $\RRR$-valued functions on $K$, so
every element of the finite type $\RRR$-algebra $R(K)$ is uniquely
determined by its restriction to the set $K$ of $\RRR$-points.
This exactly encodes the Zariski-density property in the affine algebraic
$\RRR$-scheme $K^{\rm{alg}}$.

To prove reductivity, assume to the contrary.  Any nontrivial connected
unipotent group over a perfect field contains the additive group as a closed
subgroup, so $K^{\rm{alg}}$ contains 
$\Add$ as a closed $\RRR$-subgroup.  Hence, 
$\RRR = \Add(\RRR)$ is a closed subgroup
of the group $K^{\rm{alg}}(\RRR) = K$ that is compact, a contradiction.  The same reasoning
shows that the connected reductive group $(K^{\rm{alg}})^0$ is $\RRR$-anisotropic.
\end{proof}

Every continuous homomorphism $f:K \rightarrow K'$
between compact Lie groups  yields 
a map of Hopf algebras $R(K') \rightarrow R(K)$ by composing
$K'$-representations with $f$.  The corresponding
$\RRR$-group homomorphism $f^{\rm{alg}}:K^{\rm{alg}} \rightarrow 
{K'}^{\rm{alg}}$ recovers $f$ on $\RRR$-points,
and it is uniquely determined by this condition
since $K$ is Zariski-dense in $K^{\rm{alg}}$.  
In particular, for {\em any} $\RRR$-group map $\varphi:K^{\rm{alg}} \rightarrow {K'}^{\rm{alg}}$,
necessarily $\varphi = f^{\rm{alg}}$
where $f$ is the restriction of $\varphi$ to $\RRR$-points. 
Also, if $K$ is a closed subgroup of $K'$
then $R(K') \rightarrow R(K)$ is surjective \cite[III, 4.3]{brtd}, so 
$f^{\rm{alg}}$ is a closed immersion when $f$ is injective.
%Likewise, if $f$ is surjective then $R(K') \rightarrow R(K)$ is obviously injective
%and hence is faithfully flat (as for any injective map between Hopf algebras over a field).
%In other words, $f^{\rm{alg}}_{\R}$ is faithfully flat when $f$ is surjective.

\begin{lemma} $K^0 = (K^{\rm{alg}})^0(\RRR)$ and 
$(K^{\rm{alg}})^0 = (K^0)^{\rm{alg}}$. 
\end{lemma}

\begin{proof}
Let $j:K^0 \rightarrow K$ be the natural map, 
so $j^{\rm{alg}}$ is a closed immersion since $j$ is injective.  
The dimensions of $(K^0)^{\rm{alg}}$ and $K^{\rm{alg}}$ coincide
because the Lie algebra of a
linear algebraic $\RRR$-group can be naturally computed using the Lie group of $\RRR$-points.
Thus, $j^{\rm{alg}}$ is an open and closed immersion, so 
the identity component of $K^{\rm{alg}}$ is the same as that of $(K^0)^{\rm{alg}}$.  

We claim that $(K^0)^{\rm{alg}}$ is the identity component of
$K^{\rm{alg}}$; i.e., $(K^0)^{\rm{alg}}$ is connected.  Since $(K^{\rm{alg}})^0(\RRR)$ is an 
open and closed subgroup of $K^{\rm{alg}}(\RRR) = K$, it contains $K^0$.
But we just saw that $(K^{\rm{alg}})^0 \subset (K^0)^{\rm{alg}}$, so 
$$K^0  \subset (K^{\rm{alg}})^0(\RRR) \subset (K^0)^{\rm{alg}}(\RRR) = K^0,$$
forcing the containment $(K^{\rm{alg}})^0 \subset (K^0)^{\rm{alg}}$ to be an equality
on $\RRR$-points and hence an equality of $\RRR$-groups (due to the Zariski-density of
$\RRR$-points in $K^{\rm{alg}}$ and $(K^0)^{\rm{alg}}$). 
\end{proof}

To summarize, we have defined a fully faithful functor $K  \rightsquigarrow K^{\rm{alg}}$
from the category of compact Lie groups into the category of 
anisotropic reductive $\RRR$-groups all of whose connected
components contain $\RRR$-points, and this functor is compatible
with identity components (i.e., $K$ is connected if and only if $K^{\rm{alg}}$ is connected).

It remains to show that if 
$G$ is an anisotropic reductive $\RRR$-group whose
connected components have $\RRR$-points then for 
the compact Lie group $K := G(\RRR)$
(which is Zariski-dense in $G$) there exists an $\RRR$-group isomorphism
$G \simeq K^{\rm{alg}}$ extending the equality of $\RRR$-points.
(Such an isomorphism is unique if it exists, due to the Zariski-density of $\RRR$-points
in these $\RRR$-groups, and likewise it is necessarily functorial in $G$.)

Since $\mathscr{O}(G)$ is exhausted by finite-dimensional
$G$-stable $\RRR$-subspaces \cite[I, \S1.9]{borelag}, we get a restriction
map from $\mathscr{O}(G)$ to the real representation algebra $R(K)$.
This map is injective, due to the Zariski-density of $K$ in $G$. 
To complete the proof of Theorem \ref{anisotropic}, we just have to check that
the map $\mathscr{O}(G) \hookrightarrow R(K) = \mathscr{O}(K^{\rm{alg}})$
of $\RRR$-algebras is an isomorphism of Hopf algebras. 

The corresponding $\RRR$-scheme map $K^{\rm{alg}} \rightarrow G$ induces
a group isomorphism between $\RRR$-points, so it is an $\RRR$-group map
(as $K$ is Zariski-dense in $K^{\rm{alg}}$).
In other words, $\mathscr{O}(G) \hookrightarrow R(K)$ is a map of Hopf algebras.
To check surjectivity, we note that a choice of faithful representation
$G \hookrightarrow \GL_n$ over $\RRR$ defines a faithful representation
$K \hookrightarrow \GL_n(\RRR)$.  The induced pullback map
$\mathscr{O}(\GL_n) \rightarrow R(K)$ is surjective, as we noted in 
the discussion preceding Lemma \ref{kprop}, so this
forces the map $\mathscr{O}(G) \hookrightarrow R(K)$ to be surjective.
\end{proof}





%
%It remains to show that $G(\R)$ is connected.
%Let $K$ denote the connected Lie group $G(\R)^0$ that we know is compact.
%The natural map $f: K \rightarrow G(\C)$ 
%factors uniquely through a holomorphic map $f':K^{\rm{an}} \rightarrow G(\C)$:
%$$\xymatrix{
%K \ar[r]^-{j} \ar[dr]_{f} & K^{\rm{an}} \ar[d]^-{f'} \\
%& G(\C)}$$
%where $j$ is the natural inclusion.
%Both maps ${\rm{Lie}}(j)$ and ${\rm{Lie}}(f)$  induce
%$\C$-linearized Lie algebra maps that are isomorphisms 
%(see \cite[III, 8.4]{brtd} for $j$), so 
%${\rm{Lie}}(f')$ is an isomorphism.  Hence, $f'$ is surjective (as $G(\C)$ is connected).

%We shall prove that $f'$ is an isomorphism, and use this to deduce
%the connectedness of $G(\R)$.  The isomorphism property of $f'$ amounts
%to the triviality of its kernel, which we will prove with the help of an auxiliary 
%faithful linear representation of $G$ over $\R$.  Fix such a representation, so it 
%restricts to a closed embedding of $K$ into some $\GL_n(\R)$. 
%We claim that the induced composite map $\mathscr{O}(\GL_n) \rightarrow R(K)
%\rightarrow \mathscr{O}(K^{\rm{alg}})$ is surjective.
%It is harmless to apply a conjugation, so we can assume $K$ is contained in 
%the standard compact orthogonal group ${\rm{O}}_{q_n}(\R)$
%for $q_n = \sum x_j^2$.  Since an injective homomorphism $K_1 \rightarrow K_2$
%induces a closed immersion $K_1^{\rm{alg}} \rightarrow K_2^{\rm{alg}}$,
%we get a closed immersion $K^{\rm{alg}} \rightarrow {\rm{O}}_{q_n}(\R)^{\rm{alg}}$.
%By Example \ref{onex} we have ${\rm{O}}_{q_n} = {\rm{O}}_{q_n}(\R)^{\rm{allg}}$,
%so the inclusion $\mathscr{O}({\rm{O}}_{q_n}(\R)^{\rm{alg}})
%\hookrightarrow \mathscr{O}({\rm{O}}_{q_n})$ is an equality.  Hence, 
%the surjectivity of $\mathscr{O}(\GL_n) \rightarrow \mathscr{O}(K^{\rm{alg}})$ is 
%reduced to the evident fact that ${\rm{O}}_{q_n} \rightarrow \GL_n$ is a closed immersion.

%Applying scalar extension by $\R \rightarrow \C$ to the surjection 
%$\mathscr{O}(\GL_n) \rightarrow \mathscr{O}(K^{\rm{alg}})$
%gives that the coordinate ring of 
%$\GL_n$ over $\C$ maps onto $\mathscr{O}((K^{\rm{alg}})_{\C}) = R(K,\C)$
%(Lemma \ref{cpt}).  That is, under the injection $K \hookrightarrow \GL_n(\C)$
%that factors through $f'$, 
%the algebraic functions on $\GL_n(\C)$ pull back to
%$\C$-algebra generators of $R(K,\C)$.  
%Since $K^{\rm{an}} = (\Spec R(K,\C))(\C)$, we conclude that $f'$ is injective
%and therefore is an isomorphism.

%The connectedness of $K$ ensures that $K^{\rm{an}} = K^{\rm{alg}}(\C)$,
%so by Proposition \ref{gagagp} the map
%$f'$ arises from a $\C$-group isomorphism $K^{\rm{alg}}_{\C} \rightarrow G_{\C}$.
%On Lie algebras this map is the scalar extension to $\C$ of the evident
%equality ${\rm{Lie}}(K^{\rm{alg}}) = {\rm{Lie}}(G)$, so
%the isomorphism $K^{\rm{alg}}(\C) \rightarrow G(\C)$
%is equivariant for the action of complex conjugation on the underlying {\em connected}
%real Lie groups (as we can check equivariance on Lie algebras).
%Thus, by Galois descent we obtain an isomorphism of $\R$-groups $K^{\rm{alg}} \simeq G$,
%so $G(\R) = K^{\rm{alg}}(\R) = K$. 
%\end{proof}


 Recall
that for any connected Lie group $H$, 
the universal covering space $q:\widetilde{H} \rightarrow H$
(equipped with a marked point over the identity of $H$)
has a unique compatible structure of real Lie group,
and the discrete closed normal subgroup $\ker q = \pi_1(H,1)$ is central (as 
in any connected Lie group).  This recovers
the fact that $\pi_1(H,1)$ is abelian, so $\ker q = {\rm{H}}_1(H,\Z)$.
In the compact case this universal covering
has finite degree:

\begin{proposition}[Weyl]\label{kcover} If $K$ is a connected compact Lie group and
${\rm{Lie}}(K)$ is semisimple then the universal cover $\widetilde{K} \rightarrow K$
is a finite-degree covering.  In particular, $\widetilde{K}$ is compact.
\end{proposition}

The hypothesis on the Lie algebra cannot be dropped; consider $K = S^1$.

\begin{proof}
The kernel of the covering map is the group ${\rm{H}}_1(K,\Z)$
that is finitely generated (as $K$ is a compact manifold), 
so this group is finite if and only if its finite-order quotients have bounded cardinality.
In other words, we seek an upper bound on the degree of
isogenies $f:K' \rightarrow K$ from connected compact Lie groups $K'$ onto the given group $K$.
By Theorem \ref{anisotropic}, $f$ ``algebraizes''
to an isogeny of connected semisimple $\RRR$-groups
${K'}^{\rm{alg}} \rightarrow K^{\rm{alg}}$.  By extending
scalars to $\C$, we are reduced to checking that 
for any connected semisimple $\C$-group $G$ there exists an upper bound on the degree
of all connected isogenous covers $f:G' \rightarrow G$.

Such a bound is obtained from the root datum of $G$, as follows.  Let $T \subset G$ be a maximal torus
and $T' = f^{-1}(T)$ the associated maximal torus in $G'$, so 
the centrality of $\ker f$ forces $\ker f = \ker(T' \rightarrow T)$.
Hence, to bound the degree of $f$ it is equivalent to bound the index
of the inclusion of lattices ${\rm{X}}(T) \rightarrow {\rm{X}}(T')$.
The centrality of $\ker f$ implies that the isomorphism ${\rm{X}}(T)_{\Q} \simeq {\rm{X}}(T')_{\Q}$
identifies $\Phi(G,T)$ with $\Phi(G',T')$ and  that the dual isomorphism
identifies $\Phi(G,T)^{\vee}$ with $\Phi(G',T')^{\vee}$.
(This is seen most concretely by using that $f$ even identifies
$\mathfrak{g}'$ with $\mathfrak{g}$, as we are in characteristic 0.)  Thus, by
(\ref{xphi}) we have
$$\Z \Phi(G,T) \subset {\rm{X}}(T) \subset {\rm{X}}(T') \subset (\Z \Phi(G,T)^{\vee})^{\ast}.$$
Hence, an upper bound on $\deg f$ is the index of $\Z \Phi(G,T)$
in $(\Z \Phi(G,T)^{\vee})^{\ast}$ (i.e., the absolute determinant of the
non-degenerate pairing between $\Z\Phi(G,T)$ and $\Z \Phi(G,T)^{\vee}$).
\end{proof}

The following hard result concerning maximal compact subgroups is fundamental: 


\begin{theorem}\label{maxcompact}
Let $\mathscr{H}$ be a Lie group with finite component group.  Every compact subgroup of 
$\mathscr{H}$
is contained in a maximal one, and all maximal compact subgroups $K$ of
$\mathscr{H}$ are conjugate to each other.  Moreover, for any $K$ 
there are closed vector subgroups $V_1, \dots, V_n \subset \mathscr{H}$
such that the multiplication map
$K \times V_1 \times \dots \times V_n \rightarrow \mathscr{H}$ is a $C^{\infty}$ isomorphism.
In particular, $K$ is a deformation retract of $\mathscr{H}$, so
$\pi_0(K) = \pi_0(\mathscr{H})$ and $\pi_1(K,1) = \pi_1(\mathscr{H},1)$.
\end{theorem}
\index{maximal compact subgroup}

\begin{proof}
This is \cite[XV, 3.1]{hoch}.  (We only need the
case of connected $\mathscr{H}$ in the proof of the Existence Theorem over $\C$.
An application in the disconnected case is given in Example \ref{morecx}.
The connected case does not seem to formally imply the general case.)
\end{proof}

As an elementary illustration of Theorem \ref{maxcompact} in the disconnected case, consider 
$\mathscr{H} = {\rm{GL}}(W)$
for a finite-dimensional nonzero $\RRR$-vector space $W$.
We can take $K = {\rm{O}}(q)$ for a positive-definitive quadratic form $q$ on $W$.
Let $V$ be the vector space of endomorphisms of $W$ that are self-adjoint with
respect to the symmetric auto-duality $W \simeq W^{\ast}$ arising from $B_q$.
Exponentiation identifies $V$ with the closed subgroup $P \subset \mathscr{H}$ of self-adjoint automorphisms of $W$
whose eigenvalues (all in $\RRR^{\times}$, by the spectral theorem) are positive.
The isomorphism $K \times V \simeq \mathscr{H}$ is defined by $(k, T) \mapsto k e^T$,
recovering the classical ``polar decomposition'' when $W = \RRR^n$ and $q = \sum x_i^2$.
\index{reductive group!compact Lie groups|)}
\index{compact Lie groups|)}

\subsection{Complexification}\label{cxtion}

To go further, we need to link up compact Lie groups
and complex Lie groups.  For any Lie group $H$,
the {\em complexification} of $H$ is an initial object
\index{complexification}
among $C^{\infty}$ homomorphisms from $H$ to complex Lie groups;
i.e., it is a $C^{\infty}$ homomorphism
$j_H:H \rightarrow H_{\C}$ to a complex Lie group
\index{ aaajh@$j_H$}
such that any $C^{\infty}$ homomorphism $f:H \rightarrow \mathscr{H}$
to a complex Lie group has the form $F \circ j_H$ for a unique holomorphic
homomorphism $F:H_{\C} \rightarrow \mathscr{H}$.
We emphasize that this definition is intrinsic to the $C^{\infty}$ and holomorphic
theories, having {\em no reliance on the algebraic theory}.  In particular,
if $H = G(\RRR)$ for a linear algebraic $\RRR$-group $G$
then it is not at all clear if the canonical map
$H \rightarrow G(\C)$ is a complexification, 
nor is it clear if $\ker j_H = 1$ (or if $\dim \ker j_H = 0$).

Here is a basic example of complexifications (whose verification is easy): if we denote by $\exp_{\RRR}(\mathfrak{h})$
the unique connected and simply connected real Lie group $H$ such that 
${\rm{Lie}}(H)$ is equal to a given real Lie algebra $\mathfrak{h}$, and similarly for complex Lie algebras,
then $\exp_{\RRR}(\mathfrak{h}) \rightarrow \exp_{\C}(\mathfrak{h}_{\C})$
is a complexification.  In the commutative case, this says that if $V$ is a real vector space
then the inclusion $V \hookrightarrow \C \otimes_{\RRR} V$ is a complexification.

If a Lie group $H$ admits a complexification $H \rightarrow H_{\C}$ 
and $\Lambda$ is a discrete central subgroup of $H$
whose image $\Lambda'$ in $H_{\C}$ is discrete
then $H/\Lambda \rightarrow H_{\C}/\Lambda'$ is a complexification.
(For example, the inclusion 
$S^1 = \RRR/\Z \hookrightarrow \C/\Z \simeq \C/2\pi i \Z \simeq \C^{\times}$ given by 
$\theta \mapsto \exp(2\pi i \theta)$, where $i^2 = -1$, is a complexification.)
A formal argument using the settled simply connected
case (and a pushout construction when $H$ is disconnected) shows that 
a complexification exists for any $H$ \cite[III, 6.10, Prop.\,20]{lie1} (also
see \cite[XVII, \S5]{hoch}).  The construction gives
that $H_{\C}$ is connected
when $H$ is connected (though this is also clear by applying the universal property
of $H_{\C}$ to $H \rightarrow (H_{\C})^0 \hookrightarrow H_{\C}$)
and that the $\C$-linearization $\C \otimes_{\RRR} {\rm{Lie}}(H) \rightarrow
{\rm{Lie}}(H_{\C})$ is surjective, but in general $\ker j_H$
can have positive dimension, as the following
example from \cite[Ch.\,1, \S4.1, Thm.\,7.2]{vinberg} shows.

\begin{example} A maximal compact subgroup of $\SL_2(\RRR)$ is 
${\rm{SO}}_2(\RRR) = S^1$, so the universal cover $G$ of ${\rm{SL}}_2(\RRR)$ fits into an exact sequence
$$1 \rightarrow \Z \rightarrow G \rightarrow {\rm{SL}}_2(\RRR) \rightarrow 1$$
and the induced natural map $G \rightarrow {\rm{SL}}_2(\C)$ is the complexification of $G$
(since ${\rm{SL}}_2(\C)$ is simply connected; i.e.,
${\rm{SL}}_2(\C) = \exp_{\C}(\mathfrak{sl}_2(\C))$).  
Consider the central pushout $H$ of $G$ along an injection $j:\Z \hookrightarrow S^1$,
so $H = (S^1 \times G)/\Z$.  Then 
$H_{\C}$ is the quotient of $\C^{\times} \times {\rm{SL}}_2(\C)$ modulo the unique minimal
complex Lie subgroup containing the central subgroup $j(\Z)$ in the first factor.   In other words,
$H_{\C} = {\rm{SL}}_2(\C)$, so $\ker(H \rightarrow H_{\C}) = S^1$.
\end{example}

Fortunately, in the compact case the pathological situation $\dim \ker j_H > 0$ does not arise:

\begin{proposition}\label{kcx}
If $K$ is a compact Lie group then
$K = K^{\rm{alg}}(\RRR) \rightarrow K^{\rm{alg}}(\C) =: K^{\rm{an}}$ is a complexification
and $K$ is a maximal compact subgroup of $K^{\rm{an}}$.
In particular, the complexification $j_K:K \rightarrow K_{\C}$ is a closed embedding 
and the $\C$-linearization of ${\rm{Lie}}(j_K)$ is an isomorphism $($so ${\rm{Lie}}(K_{\C})$
is semisimple when $K$ has finite center$)$.
\end{proposition}
\index{ aaakan@$K^{\rm{an}}$}
\index{compact Lie group!and complexification|(}

See Example \ref{morecx} for a discussion of examples
and counterexamples related to complexification beyond the compact case.
The main difficulty in the proof of Proposition \ref{kcx} is that we are demanding
the universal property for maps into arbitrary complex Lie groups; the weaker version
of Proposition \ref{kcx} that only asserts the universal property
relative to maps to complex Lie groups arising from linear algebraic groups over $\C$
(which is entirely sufficient for our needs) is much easier to prove (see \cite[III, 8.6]{brtd}). 

\begin{proof}
Every connected component of $K^{\rm{an}}$ meets $K = K^{\rm{alg}}(\RRR)$ (as
$(K^{\rm{an}})^0 = (K^{\rm{alg}})^0(\C)$ and the
connected components of $K^{\rm{alg}}$ 
have $\RRR$-points), and $(K^{\rm{alg}})^0 = (K^0)^{\rm{alg}}$.
Granting the case with connected $K$, we deduce the general case as follows.
Let $f:K \rightarrow \mathscr{H}$ be a Lie group homomorphism to a complex-analytic Lie group,
so the restriction $f^0:K^0 \rightarrow \mathscr{H}$ uniquely extends to a 
holomorphic homomorphism $F:K^{\rm{alg}}(\C)^0 = (K^0)^{\rm{alg}}(\C) \rightarrow \mathscr{H}$.
We just need to check that $F$ uniquely extends to a holomorphic
homomorphism $K^{\rm{alg}}(\C) \rightarrow \mathscr{H}$ satisfying $F|_K = f$. 
Such an $F$ is unique if it exists since $K \cdot K^{\rm{alg}}(\C)^0 = K^{\rm{alg}}(\C)$ (as
$K$ meets every connected component of $K^{\rm{alg}}(\C)$, since all
connected components of $K^{\rm{alg}}$ have an $\RRR$-point and so are
geometrically connected), and the  existence of $F$ is a 
group-theoretic problem:  we just have to check that for $k \in K$, $k$-conjugation on
$K^{\rm{alg}}(\C)^0$ is compatible via $F$ with $f(k)$-conjugation on $\mathscr{H}$.
This is a comparison of two holomorphic homomorphisms $K^{\rm{alg}}(\C)^0 \rightrightarrows \mathscr{H}$,
so by the assumed universal property in the connected case it suffices to check equality
of their restrictions to $K^0$.  But $F|_{K^0} = f^0$ by design. 

Now we may assume $K$ is connected, so $K^{\rm{alg}}$ is connected.  
Let $K'=\mathscr{D}(K^{\rm{alg}})(\RRR)$, and let
$T = \mathscr{Z}(\RRR)$ for the maximal central $\RRR$-torus $\mathscr{Z}$ in $K^{\rm{alg}}$, so
$K'$ and $T$ are connected and 
$K' \times T \rightarrow K$ is {\em surjective} (due to connectedness of $K$)
with finite central kernel $\mu$.  Clearly $\mu^{\rm{alg}}$ is a central constant finite $\RRR$-subgroup of
${K'}^{\rm{alg}} \times T^{\rm{alg}}$, and upon passing to $\RRR$-points the natural map
${K'}^{\rm{alg}}(\RRR) \times T^{\rm{alg}}(\RRR) \rightarrow K^{\rm{alg}}(\RRR)$
with kernel $\mu^{\rm{alg}}(\RRR)$ is exactly the surjection
$K' \times T \rightarrow K$ with kernel $\mu$.

The finite complex Lie group associated
to the finite abelian group $\mu$ is a complexification of $\mu$, and a complexification 
of $K = (K' \times T)/\mu$ is given by $(K'_{\C} \times T_{\C})/\mu_{\C}$
where $\mu_{\C}$ is the image of $\mu$ in $K'_{\C} \times T_{\C}$.
Provided that $T_{\C} = T^{\rm{alg}}(\C)$ and 
$K'_{\C} = {K'}^{\rm{alg}}(\C)$  via the natural maps, it follows that $\mu_{\C} = \mu^{\rm{alg}}(\C)$
and $K_{\C} = K^{\rm{alg}}(\C)$ with the desired canonical map
from $K = (K' \times T)/\mu$.   Hence, it 
suffices to treat $K'$ and $T$ rather than $K$.
The case of $T = (S^1)^r$ is elementary, so now we may
assume that $K^{\rm{alg}}$ is semisimple. Equivalently,
${\rm{Lie}}(K)$ is semisimple.

Let $j_K:K \rightarrow K_{\C}$ be the complexification.
The natural map $j:K \rightarrow K^{\rm{an}}$ uniquely factors as $f \circ j_K$ for a holomorphic map $f:K_{\C} \rightarrow K^{\rm{an}}$
between connected complex Lie groups, and $f$ is an isomorphism on Lie algebras
since $\C \otimes_{\RRR} {\rm{Lie}}(K) \rightarrow {\rm{Lie}}(K^{\rm{an}})$
is an isomorphism and the map 
$\C \otimes_{\RRR} {\rm{Lie}}(K) \rightarrow {\rm{Lie}}(K_{\C})$ via $j_K$ is surjective.
Hence, the $\C$-linearization of
${\rm{Lie}}(j_K)$ is an isomorphism and $f$ is a quotient map modulo a discrete subgroup.
Thus, the real Lie group $K_{\C}$ has dimension $2 \dim K$ and to show that $f$ is an isomorphism it 
suffices to show that $f$ is injective.  Note
that $j_K$ is injective (and so is a closed embedding)
since we have factored it through the injective map $K \rightarrow K^{\rm{an}}$.
%By \cite[XVII, 5.1]{hoch}, the compact subgroup
%$K$ in $K_{\C}$ is maximal.  
%Also, we have shown that the real dimension of $K_{\C}$ is equal to $2 \dim K$ in general.

By Proposition \ref{kcover}, the universal covering $\pi:\widetilde{K} \rightarrow K$ 
has finite degree.  Let $q:\widetilde{K}^{\rm{alg}} \rightarrow K^{\rm{alg}}$
be the corresponding map between $\RRR$-anisotropic connected
reductive $\RRR$-groups, and let $\mu = \ker q$ be its finite central kernel, so 
$K^{\rm{an}} = \widetilde{K}^{\rm{an}}/\mu(\C)$.
The induced map $\pi_{\C}:\widetilde{K}_{\C} \rightarrow K_{\C}$ is 
an isomorphism on Lie algebras, so it is surjective between the connected
complexifications and has discrete central kernel.
If we can handle the simply connected case
(i.e., if $\widetilde{K}_{\C} \rightarrow \widetilde{K}^{\rm{an}}$ is
an isomorphism) then $\pi_{\C}$ has degree dividing $\deg \pi^{\rm{an}} = \#\mu(\C)$
with equality if and only if $K_{\C} \rightarrow K^{\rm{an}}$ is an isomorphism.

Thus, to reduce to the case that $K$ is simply connected we just need
to show that $\ker \pi_{\C}$ has size at least $\#\mu(\C)$.  Let
$T \subset K$ be a maximal (compact) torus, so its preimage
$\widetilde{T} = \pi^{-1}(T)$ in $\widetilde{K}$ is also a maximal torus
(due to the conjugacy and self-centralizing properties of maximal tori
in connected compact Lie groups), and clearly
$\ker \pi = \ker(\widetilde{T} \rightarrow T)$.    Since the case of tori
is settled and we are granting the simply connected case as also being settled,
the natural map of complexifications $\widetilde{T}_{\C} \rightarrow \widetilde{K}_{\C}$
is identified with $\widetilde{T}^{\rm{an}} \rightarrow \widetilde{K}^{\rm{an}}$ and hence
is {\em injective}, so 
$\ker \pi_{\C}$ has size at least that of $\ker(\widetilde{T}_{\C} \rightarrow T_{\C})$.
But $\widetilde{T}_{\C} \rightarrow T_{\C}$
is the map on $\C$-points arising from the map $\widetilde{T}^{\rm{alg}} \rightarrow T^{\rm{alg}}$
induced by $q:\widetilde{K}^{\rm{alg}} \rightarrow K^{\rm{alg}}$ between maximal $\RRR$-tori, 
so this map between maximal $\RRR$-tori has kernel $\mu$ as well and hence $\mu(\C)$ is identified
with the kernel of $\widetilde{T}_{\C} \rightarrow T_{\C}$. Now we can assume $K$ is simply connected.

By the construction of complexification
for simply connected Lie groups, to show
$f:K_{\C} \rightarrow K^{\rm{an}}$ is an isomorphism it suffices to show that $K^{\rm{an}}$
is simply connected if $K$.  The group $K^{\rm{alg}}(\RRR) = K$ is a compact
subgroup of $K^{\rm{an}}$, and any connected
Lie group has a deformation retract onto its maximal compact subgroups
(and so has their homotopy type) by Theorem \ref{maxcompact}.

It now suffices to prove that if $K$ is {\em any} connected compact Lie group
(not necessarily simply connected) then it is maximal as
a compact subgroup of $K^{\rm{an}}$.
Since the maximal compact subgroups of
$K^{\rm{an}}$ are connected, 
it suffices to show that the common dimension of
the maximal compact subgroups of $K^{\rm{an}}$
is $\dim K$.

To control the possibilities for the maximal compact
subgroups of $K^{\rm{an}}$,
we consider its finite-dimensional holomorphic representations. 
The group $K^{\rm{an}}$ has a faithful finite-dimensional
holomorphic representation (using a faithful representation of
the $\RRR$-group $K^{\rm{alg}}$) and the 
finite-dimensional holomorphic representations of
$K^{\rm{an}}$ are completely reducible
(Corollary \ref{corss}).  Thus, the connected complex Lie group $K^{\rm{an}}$
is ``reductive'' in the sense of \cite[XVII, \S5]{hoch}.
It then follows from \cite[XVII, 5.3]{hoch} that
$K^{\rm{an}}$ is the complexification of its maximal compact
subgroups.  That is, if $\mathscr{K}$ is a maximal compact
subgroup of $K^{\rm{an}}$ then
the map $\mathscr{K}_{\C} \rightarrow K^{\rm{an}}$
induced by the inclusion $\mathscr{K} \hookrightarrow K^{\rm{an}}$
is an isomorphism.    But we showed earlier in the proof that the complexification of a compact Lie group
has twice the real dimension in general, so applying that to 
$\mathscr{K}$  and using the equality $\mathscr{K}_{\C} = K^{\rm{an}} = K^{\rm{alg}}(\C)$
gives that $\dim \mathscr{K} = (1/2) \dim_{\RRR} K^{\rm{alg}}(\C) = \dim K^{\rm{alg}} = \dim K$.
\end{proof}

\begin{example}\label{cx}
By Proposition \ref{kcx}, every 
compact Lie group 
is a maximal compact subgroup of its complexification, whose component group is finite.
Turning this around, let 
$K$ be the maximal compact subgroup of a complex Lie group
$\mathscr{G}$ with finite component group, and
assume ${\rm{Lie}}(\mathscr{G})$ is semisimple (over $\RRR$ or $\C$;
these are equivalent).  The inclusion $j:K \hookrightarrow \mathscr{G}$
factors through a unique holomorphic homomorphism $K_{\C} \rightarrow \mathscr{G}$,
and we claim that this latter map is an isomorphism.  
\index{maximal compact subgroup!complexification}

In other words, we claim that 
every complex Lie group with semisimple Lie algebra and finite component group 
is the complexification of its maximal compact
subgroups. (In particular, 
using Proposition \ref{gagagp}, $G \rightsquigarrow G(\C)$ is an equivalence
from connected semisimple $\C$-groups to connected complex Lie groups
with semisimple Lie algebra. The analogous equivalence for connected
reductive $\C$-groups is onto the category of connected complex Lie groups
$\mathscr{H}$ such that ${\rm{Lie}}(\mathscr{H})$ is reductive and
$Z_{\mathscr{H}}^0 \simeq (\C^{\times})^r$ for some $r \ge 0$.) 

To prove that $\mathscr{G}$ is the complexification of $K$, we
use the criterion in \cite[XVII, 5.3]{hoch}: it suffices to prove
that $\mathscr{G}$ has a faithful finite-dimensional
representation (i.e., an injective holomorphic homomorphism $\mathscr{G}
\rightarrow {\rm{GL}}(V)$ for a finite-dimensional $\C$-vector space $V$)
and that the holomorphic finite-dimensional representations of
$\mathscr{G}$ are completely reducible.  Complete reducibility
is inherited from semisimplicity of ${\rm{Lie}}(\mathscr{G})$
(and finiteness of $\pi_0(\mathscr{G})$), 
as in the proof of Corollary \ref{corss}.  The existence
of a faithful finite-dimensional representation of $\mathscr{G}$ lies
deeper (and its analogue in the $\RRR$-theory is false; e.g., 
the universal cover of ${\rm{SL}}_2(\RRR)$). We only need
the case when $\mathscr{G}$ is the Lie group of $\C$-points of a connected semisimple $\C$-group,
for which the existence of a faithful finite-dimensional representation is obvious. 
The general case is \cite[XVII, 3.2]{hoch}.
\end{example}
\index{compact Lie group!and complexification|)}


\subsection{Construction of covers and applications}\label{concover}

Returning to the root datum $R$ and the connected semisimple $\C$-group $G$ of adjoint type at the end of 
\S\ref{ssirr}, we now use \S\ref{ccgp}--\S\ref{cxtion} 
to build an isogeny $G' \rightarrow G$ from a connected
semisimple $\C$-group $G'$ such that the semisimple root datum of
$G'$ is simply connected.  Let $H$ be the connected Lie
group $G(\C)$ whose Lie algebra is semisimple, and 
let $K$ be a maximal compact subgroup of $H$, so $K^{\rm{alg}}$ is an $\RRR$-anisotropic connected
reductive $\RRR$-group.  By Proposition \ref{kcx}
and Example \ref{cx}, $K^{\rm{alg}}(\C) \simeq H = G(\C)$.  
It follows that $K^{\rm{alg}}(\C)$ has finite center, so the connected reductive
$\RRR$-group $K^{\rm{alg}}$ is semisimple.   By Proposition \ref{gagagp}, 
the analytic isomorphism $K^{\rm{alg}}(\C) \simeq G(\C)$
arises from a $\C$-group isomorphism $K^{\rm{alg}}_{\C} \simeq G$; i.e.,
$K^{\rm{alg}}$ is an $\RRR$-descent of $G$ (the ``compact form'' of $G$). 

Since $K^{\rm{alg}}$ is an 
$\RRR$-descent of $G$, we now convert our problem for 
isogenous covers of $G$ over $\C$ into an analogous problem for isogenous covers of 
$K^{\rm{alg}}$ over $\RRR$.   Inspired by Theorem \ref{anisotropic},
we will  make constructions in the category of compact connected Lie groups,
and then pass back to the algebraic theory over $\RRR$ (and finally extend scalars to $\C$).
To this end, we use the connected compact universal covering
space $\widetilde{K} \rightarrow K$ (see Proposition \ref{kcover}).  

Passing to the corresponding $\RRR$-anisotropic connected semisimple $\RRR$-groups,
we get an isogeny $\widetilde{K}^{\rm{alg}} \rightarrow K^{\rm{alg}}$.
Extending scalars to $\C$ defines  an isogeny 
$$f:G' := \widetilde{K}^{\rm{alg}}_{\C} \rightarrow K^{\rm{alg}}_{\C} = G$$
between connected semisimple $\C$-groups. 
Let $T' = f^{-1}(T)$ be the maximal torus of $G'$ corresponding
to a choice of maximal torus $T$ of $G$, so we get an 
isogeny of root data 
$$R(G',T') \rightarrow R(G,T) = R^{\rm{ad}}.$$
We will prove that $R(G',T')$ is simply connected
(i.e., the coroots span the cocharacter group
of $T'$), so $R(G',T') = R$ compatibly with the initial identification
$R(G,T) = R^{\rm{ad}}$.  
This will complete the proof of the Existence Theorem over $\C$.

By Proposition \ref{kcx},  the Lie group $G'(\C) = \widetilde{K}^{\rm{alg}}(\C)$ contains
$\widetilde{K} = \widetilde{K}^{\rm{alg}}(\RRR)$ as a maximal compact subgroup. Thus,  $G'(\C)$ inherits the simply
connectedness property from $\widetilde{K}$ (Theorem \ref{maxcompact}).
Our problem is therefore reduced to verifying a relationship between
combinatorial and topological notions of being ``simply connected'':  

\begin{proposition}\label{scgt}
If $(G',T')$ is a connected semisimple $\C$-group then $G'(\C)$
is topologically simply connected  if and only if 
$R(G',T')$ is simply connected in the sense of root data.
\end{proposition}
\index{simply connected!topological versus combinatorial}

\begin{proof}
We will use an ``existence theorem'' (for highest-weight representations)
in the representation theory of semisimple Lie algebras over $\C$
and exponentiation of Lie algebra representations
to Lie group representations in the topologically simply connected case.

Note that the $\Q$-vector space $(\Z {\Phi'}^{\vee})_{\Q} = {\rm{X}}_{\ast}(T')_{\Q}$
has complexification that is naturally identified with $\mathfrak{t}' = {\rm{Lie}}(T')$
(by identifying $\lambda \in \Hom(\Gm, T')$ with 
${\rm{d}}\lambda := {\rm{Lie}}(\lambda)(z \partial_z) \in \mathfrak{t}'$). 
The set  ${\Phi'}^{\vee}$ of coroots spans ${\rm{X}}_{\ast}(T')$  over $\Z$ 
provided that every $\Z$-linear form
$\ell:\Z {\Phi'}^{\vee} \rightarrow \Z$ is
$\Z$-valued on ${\rm{X}}_{\ast}(T')$ (i.e., arises from ${\rm{X}}(T')$) when
$\ell$ is viewed as a $\Q$-linear form on ${\rm{X}}_{\ast}(T')_{\Q}$,
or equivalently when $\ell$ is viewed as a $\C$-linear form on ${\rm{X}}_{\ast}(T')_{\C} =
{\mathfrak{t}'}$.  This integrality property of $\ell$ is invariant
with respect to the action of $W_{G'}(T')$, so to check whether or not it holds 
for a particular $(G',T')$ there is no loss of generality in 
picking a positive system of roots ${\Phi'}^{+}$ and imposing the additional requirement on 
the element $\ell \in {\rm{X}}(T)_{\Q}$ that it lies in the associated Weyl chamber of ${\rm{X}}(T)_{\RRR}$
(i.e., $\langle \ell, a^{\vee} \rangle \ge 0$ for all $a \in {\Phi'}^{+}$). 

Let ${\rm{d}}\ell = {\rm{Lie}}(\ell): \mathfrak{t}' \rightarrow {\rm{Lie}}(\Gm) =  \C$
be the 
linear form associated to $\ell \in (\Z{\Phi'}^{\vee})^{\ast}$ satisfying
$\langle \ell, a^{\vee} \rangle \ge 0$ for all $a \in {\Phi'}^{+}$.  We have 
$\langle {\rm{d}}\ell, {\rm{d}}(a^{\vee}) \rangle =
\langle \ell, a^{\vee} \rangle \in \Z_{\ge 0}$ for all $a \in \Phi' = \Phi(\mathfrak{g}', \mathfrak{t}')$.
Thus, by the existence theorem for highest-weight representations
of semisimple Lie algebras over $\C$ (Theorem \ref{highwtlie}),
there exists a (unique) irreducible representation
\begin{equation}\label{lierep}
\mathfrak{g}' \rightarrow \mathfrak{gl}(V) = {\rm{Lie}}(\GL(V))
\end{equation}
having ${\rm{d}}\ell$ as its highest weight (relative to the
Cartan subalgebra $\mathfrak{t}'$ and  ${\Phi'}^+ \subset \Phi(\mathfrak{g}', \mathfrak{t}')$). 

Now assume that $G'(\C)$ is simply connected, 
so every finite-dimensional representation of $\mathfrak{g}'$ over $\C$ 
exponentiates to a holomorphic representation of $G'(\C)$ on the same
vector space.  By Proposition \ref{gagagp}, any finite-dimensional holomorphic 
representation of $G'(\C)$ 
arises from a unique algebraic representation of $G'$ on the same
vector space, due to the semisimplicity of $G'$.
Thus, (\ref{lierep}) arises from 
a $\C$-group representation $\rho:G' \rightarrow \GL(V)$, and 
the highest weight vector $v \in V$ for $(\mathfrak{g}', \mathfrak{t}')$ 
(which is unique up to $\C^{\times}$-scaling) 
is a $T'$-eigenvector since 
it is a $\mathfrak{t}'$-eigenvector.
The corresponding weight homomorphism
$w:T' \rightarrow \Gm$ via the $T'$-action on $\C v$ induces ${\rm{d}}\ell$ on Lie algebras,
so $w = \ell$ in ${\rm{X}}(T')_{\C}$. In particular, $\ell \in {\rm{X}}(T')$. That is,
${\rm{X}}(T') = (\Z {\Phi'}^{\vee})^{\ast}$, or equivalently ${\rm{X}}_{\ast}(T') = \Z {\Phi'}^{\vee}$,
so $R(G',T')$ is simply connected.

For the converse, assume that $R(G',T')$ is simply connected.
Let $K$ be a maximal compact subgroup of $G'(\C)$, so
$K^{\rm{alg}}(\C) \simeq G'(\C)$ by Proposition \ref{kcx} and Example \ref{cx}.
Hence, $K^{\rm{alg}}$ is an $\RRR$-descent of $G'$, by Proposition \ref{gagagp}.
Consider the finite-degree 
universal cover $f:\widetilde{K} \rightarrow K$.
By Propositions \ref{maxcompact} and Proposition \ref{kcx}, 
$\widetilde{K}^{\rm{alg}}(\C)$ is simply connected in the topological sense.
Since $f$ ``algebraizes'' to an isogeny
$f^{\rm{alg}}:\widetilde{K}^{\rm{alg}} \rightarrow K^{\rm{alg}}$ between
connected semisimple $\RRR$-groups, $\widetilde{K}^{\rm{alg}}(\C)$ is the universal cover
of $K^{\rm{alg}}(\C) = G'(\C)$. (This covering map is the
complexification $f_{\C}$ of $f$, and it arises from $f^{\rm{alg}}$ on $\C$-points.) 
The description of the effect of central isogenies at
the level of root data (see Example \ref{rootisog}) shows that any isogeny
onto $G'$ from a connected semisimple $\C$-group is an isomorphism
because $R(G',T')$ is simply connected.
Thus, the $\C$-homomorphism $(f^{\rm{alg}})_{\C}$ is an isomorphism, so
its analytification $f_{\C}$ is an isomorphism and therefore 
$G'(\C)$ is topologically simply connected.
\end{proof}
%
%\begin{corollary}
%The functor $G \rightsquigarrow G(\C)$ is an equivalence from 
%connected semisimple $\C$-groups to 
%connected complex Lie groups with semisimple Lie algebra.
%\end{corollary}

%This result fails over $\R$, even when the group of $\R$-points is connected; e.g., ${\rm{SL}}_2$.

\begin{example}\label{morecx} As an application of our
work with complex Lie groups and the Existence Theorem over $\C$,
we now relate the Lie group notion of complexification 
to the algebraic notion of scalar extension from $\RRR$ to $\C$
for Lie groups arising from semisimple $\RRR$-groups, going beyond the 
$\RRR$-anisotropic case. This is not used elsewhere in these notes.
\index{complexification!scalar extension}

Let $H$ be a connected 
semisimple $\RRR$-group (so $H_{\C}$ denotes
the associated semisimple $\C$-group, not to be
confused with the complexification $H(\RRR)_{\C}$
of the Lie group $H(\RRR)$).  Even when $H(\RRR)$ is connected, it can happen that
the natural map $j:H(\RRR) \rightarrow H(\C)$ is
not the complexification. For example, the isomorphism ${\rm{SL}}_3(\RRR) \simeq {\rm{PGL}}_3(\RRR)$
provides a homomorphism ${\rm{PGL}}_3(\RRR) \rightarrow {\rm{SL}}_3(\C)$
to a degree-3 connected cover of ${\rm{PGL}}_3(\C)$.
 This  problem disappears if the root datum
for $H_{\C}$ is simply connected, as we now explain.
%
%We first recall the notion of {\em Cartan involution} $\Theta_K$ associated
%to a maximal compact subgroup $K$ of a Lie group $H$
%with finite component group and semisimple Lie algebra $\mathfrak{h}$.
%This is an involution whose fixed-point locus is $K$,
%and it is uniquely determined by these conditions.
%(For example, if $H = {\rm{SL}}_n(\R)$ and $K = {\rm{SO}}_{q_n}(\R)$
%for $q_n = \sum x_j^2$, then $\Theta_K(h) = (h^{\top})^{-1}$.
%As another example, if $H = G(\C)$ for an anisotropic 
%connected semisimple $\R$-group $G$
%then for $K = G(\R)$ we have $\Theta_K(h) = \overline{h}$.) 

Consider a homomorphism
$f:H(\RRR) \rightarrow \mathscr{G}$ to a complex Lie group
$\mathscr{G}$. The map ${\rm{Lie}}(f)$ is a homomorphism
from $\mathfrak{h} = {\rm{Lie}}(H)$ into the underlying real Lie algebra
of ${\rm{Lie}}(\mathscr{G})$, so it linearizes to
a map of complex Lie algebras
$\widetilde{f}:\mathfrak{h}_{\C} \rightarrow {\rm{Lie}}(\mathscr{G})$.
But $\mathfrak{h}_{\C} = {\rm{Lie}}(H)_{\C} = {\rm{Lie}}(H_{\C}) = 
{\rm{Lie}}(H(\C))$,
and $H(\C)$ is simply connected in the topological sense
by Proposition \ref{scgt} (whose proof relied on our 
proof of the Existence Theorem over $\C$).  Hence, 
$\widetilde{f}$ exponentiates to a holomorphic homomorphism
$F:H(\C) \rightarrow \mathscr{G}$.  The maps 
$F \circ j, f:H(\RRR) \rightrightarrows \mathscr{G}$ agree
on Lie algebras by construction of $F$, so they coincide 
on $H(\RRR)^0$.  Likewise, by Lie algebra 
considerations, 
$F$ is uniquely determined on  $H(\C)$ by the equality $F \circ j = f$ on $H(\RRR)^0$.

It remains to prove that $H(\RRR)$ is connected when $H_{\C}$
has a simply connected root datum.
The connectedness
of $H(\RRR)$ in such cases is a deep result of E.\,Cartan, originally proved
by Riemannian geometry (going through the theory of
compact groups, for which the main connectedness ingredient is proved in 
\cite[VII, Thm.\,8.2]{helgason}). 

Here is a sketch of a proof of Cartan's connectedness
theorem via an algebraic connectedness result of Steinberg. 
\index{Steinberg!connectedness theorem}
The real Lie group underlying $H(\C)$ is connected
with  an involution $\theta$ (complex conjugation) 
having fixed-point locus $H(\RRR)$. 
Note that $H(\RRR)$ has {\em finite} component group, as follows
either from a general result of Whitney on $\RRR$-points of affine algebraic varieties
(see \cite[App.\,A]{milnor}, which rests on \cite[\S1, Lemma]{andreotti})
or from a result of Matsumoto for $\RRR$-groups (see \cite[14.4, 14.5]{boreltits}).
Thus, Theorem \ref{maxcompact} is applicable to $H(\RRR)$ (so $H(\RRR)$
admits a good theory of maximal compact subgroups).

By a result of Mostow \cite[\S6]{mostow}, 
there is a $\theta$-stable maximal compact subgroup $K'$ of $H(\C)$
such that $K := K' \bigcap H(\RRR)$ is a maximal compact
subgroup of $H(\RRR)$.  
(Mostow's proof gives  that 
$K' \mapsto K' \bigcap H(\RRR)$ is a bijection from the set of
$\theta$-stable maximal compact subgroups of $H(\C)$ to the set
of maximal compact subgroups of $H(\RRR)$, with $H$ any connected
reductive $\RRR$-group.)
Fix such a $K'$, so $K'$ is connected (as $H(\C)$ is connected) 
and $\pi_0(H(\RRR)) = \pi_0(K)$ for the maximal compact
subgroup $K = K' \bigcap H(\RRR) = {K'}^{\theta}$ in
$H(\RRR)$ (applying Theorem \ref{maxcompact} to $H(\RRR)$).  Connectedness of $H(\RRR)$ is reduced to
connectedness of
$K$, so it suffices to show 
that the fixed-point locus of {\em any} involution $\theta$ of $K'$ is connected. 
Note that the $\C$-group $({K'}^{\rm{alg}})_{\C} = H_{\C}$ has a simply connected root datum.

 The involution $\theta$ 
 of the connected compact Lie group $K'$ arises from an involution $\theta^{\rm{alg}}$
 of 
${K'}^{\rm{alg}}$, so $({K'}^{\rm{alg}})^{\theta^{\rm{alg}}}$ is a closed $\RRR$-subgroup of 
${{K'}}^{\rm{alg}}$ whose group of $\RRR$-points is ${K'}^{\theta}$.
The fixed-point subgroup for an involution of a connected reductive
$\RRR$-group has reductive identity component (see \cite[2.2]{pryu},
or \cite[2.4]{pryu} for an algebraic proof),
so $({K'}^{\rm{alg}})^{\theta^{\rm{alg}}}$
has $\RRR$-anisotropic reductive identity component. 
(This identity component may not be semisimple;
e.g., for $K' = {\rm{SU}}(2)$ we have $K = S^1$.) 
Thus, by Theorem \ref{anisotropic}, if $({K'}^{\rm{alg}})^{\theta^{\rm{alg}}}$
is connected for the Zariski topology then its group $K$ of
$\RRR$-points is connected for the analytic topology.

We are reduced to an algebraic assertion over $\RRR$: for 
any $\RRR$-anisotropic connected semisimple $\RRR$-group $G$
such that the semisimple $\C$-group $G_{\C}$ has a simply connected
root datum (e.g, $G = {K'}^{\rm{alg}}$)
and any involution $\iota$ of $G$ (e.g., $\theta^{\rm{alg}}$),  the linear algebraic $\RRR$-subgroup
$G^{\iota=1}$ of $G$ is connected for the Zariski topology. 
We can reformulate this assertion over any field $k$:
if $G$ is a connected semisimple $k$-group
such that the root datum for $G_{\overline{k}}$ is simply connected
and if $\theta$ is an automorphism of $G$ that is semisimple (i.e.,
the induced automorphism of the coordinate ring is semisimple,
such as an involution when ${\rm{char}}(k) \ne 2$),
then the closed fixed-point scheme $G^{\theta}$ is connected for the Zariski topology.
It suffices to consider
algebraically closed $k$ and to work with $k$-valued points, in which case
the connectedness assertion is a theorem of Steinberg \cite[8.1]{steinberg}
(which was partially motivated by 
the desire for an algebraic version of Cartan's connectedness theorem).
\end{example}

\end{appendix}


\newpage

%\printindex

\newpage

\begin{thebibliography}{rammmm}

\bibitem[AV]{vogan} J.\,Adams, D.\,Vogan, {\em Contragredients}, {\tt{http://arxiv.org/abs/1201.0496}}, preprint, 2012.

\bibitem[AF]{andreotti} A.\,Andreotti, T.\,Frankel, {\em The Lefschetz theorem on hyperplane sections},
Annals of Math.\,{\bf 69} no.\,3 (1959), 713--717.

\bibitem[Arf]{arf} C.\,Arf, {\em Untersuchungen \"{u}ber quadratische Formen in  K\"{o}rpern 
der Charakteristik $2$}, (Teil 1) Crelle {\bf 183} (1941), 148--167.  

\bibitem[Ar69a]{Artapprox} M.\,Artin, {\em Algebraic approximation of
structures over complete local rings}, Publ.\:Math.\:IHES {\bf 36} (1969), 23--58.

\bibitem[Ar69b]{alg} M.\,Artin,
``Algebraization of formal moduli: I''
in {\em Global analysis $($papers in honor of K. Kodaira$)$},
Univ. of Tokyo Press, Tokyo (1969), 21--71.

\bibitem[Ar74]{versal} M.\,Artin, {\em Versal deformations and algebraic stacks}, 
Inv.\, Math.\, {\bf 27} (1974),  165--189.

\bibitem[ABS]{abs} M.\,Atiyah, R.\,Bott, A.\,Shapiro, {\em Clifford modules}, 
Topology {\bf 3} (1964), 3--38.

\bibitem[Au]{auel} A.\,Auel, {\em Clifford invariants of line bundle-valued quadratic forms},
MPIM preprint series 2011-33 (2011). 

\bibitem[Bar]{barsotti} I.\,Barsotti, {\em Structure theorem for group varieties}, 
Ann.\,Mat.\,Pura Appl.\,(4) {\bf 38} (1955), 77--119.

%\bibitem[Bo1]{borelsem} A.\,Borel, ``Properties and linear representations of
%Chevalley groups'' in {\em Seminar on algebraic groups and related finite groups},
%Lecture Notes in Math {\bf 131}, Springer--Verlag, New York, 1970.

\bibitem[BK]{bk} W.\,Bichsel, M-A.\,Knus, ``Quadratic forms with values in line bundles''
in {\em Recent developments in real algebraic geometry and quadratic forms},
Comtemp.\:Math., {\bf 155} AMS, Providence, 1994. 

\bibitem[Bo69]{borelarith} A.\,Borel, {\em Introduction aux groupes arithm\'etiques}, Hermann, Paris, 1969.

\bibitem[Bo85]{borelaff} A.\,Borel, {\em On affine algebraic homogeneous spaces},
Arch. Math. {\bf 45} (1985), 74--78. 

\bibitem[Bo91]{borelag} A.\,Borel, {\em Linear algebraic groups}  
(2nd ed.) Springer-Verlag, New York, 1991.

%\bibitem[Bo4]{essays} A.\,Borel, {\em Essays in the history of Lie groups  and algebraic groups}, 
%American Math.\,Soc., Providence, 2001.

%
%\bibitem[BoSe]{bs} A.\,Borel, J-P.\,Serre, {\em Th\'eor\`emes de finitude en cohomologie galoisienne},
%Comm.\,Math.\,Helv. {\bf 39}(1964), 111--164.

%\bibitem[BoSp]{bsp} A.\,Borel, T.\,Springer, {\em Rationality properties of linear
%algebraic groups {\rm{II}}}, Tohoku Math. Journal (2) {\bf 20}(1968), 443--497. 

\bibitem[BoTi]{boreltits} A.\,Borel, J.\,Tits, {\em Groupes r\'eductifs}, Publ.\,Math.\,IHES {\bf 27}(1965),  
55--151.

%
%\bibitem[BoTi2]{bt2} A.\,Borel, J.\,Tits, {\em Th\'eor\`emes de structure et de
%conjugasion pour les groupes alg\'ebriques lin\'eaires}, C.\,R.\, Acad.\, Sci.\, Paris 
%{\bf 287}(1978),  55--57.

\bibitem[BLR]{neron} S.\,Bosch, W.\,L\"utkebohmert,
M.\,Raynaud, {\em N\'eron models}, Springer-Verlag, New York, 1990. 

%\bibitem[Bou1]{balg} N.\,Bourbaki, {\em Algebra} (Ch.\:9), Hermann, Paris, 1959. 

\bibitem[Bou1]{lie1} N.\,Bourbaki, {\em Lie groups and Lie algebras} (Ch.\:1--3), Springer-Verlag, Berlin, 1989. 

\bibitem[Bou2]{lie4} N.\,Bourbaki,
{\em Lie groups and Lie algebras} (Ch.\:4--6), Springer--Verlag, Berlin, 2002.

\bibitem[Bou3]{lie7} N.\,Bourbaki,
{\em Lie groups and Lie algebras} (Ch.\:7--9), Springer--Verlag, Berlin, 2005. 

%
%\bibitem[Bri]{brion} M. Brion, {\em Anti-affine algebraic groups}, Journal of Algebra {\bf 321}(2009),
%934--952.

\bibitem[BSU]{brion} M.\,Brion, P.\,Samuel, V.\,Uma, {\em Lectures on the structure of algebraic
groups and geometric applications}, 
{\tt{http://www-fourier.ujf-grenoble.fr/$\sim$mbrion/chennai.pdf}}, 
preprint, 2011.

\bibitem[BtD]{brtd} T.\,Br\"ocker, T.\,tom Dieck, {\em Representations of compact Lie groups},
GTM 98, Springer--Verlag, New York, 1985.


%
%\bibitem[BrTi]{brt2} F.\,Bruhat, J.\,Tits, {\em Groupes r\'eductifs sur un corps local {\rm{II}}}, 
%Publ.\,Math.\,IHES\,{\bf 60} (1983), 197--376.


\bibitem[CP]{map} C-Y.\,Chang, M.\,Papanikolas, {\em Algebraic independence of periods
and logarithms of Drinfeld modules}, Journal of the AMS {\bf 25} no.\:1 (2012), 123--150.

\bibitem[Chev60]{chevold} C.\,Chevalley, {\em Une d\'emonstration d'un th\'eor\`eme sur les
groupes alg\'ebriques}, J.\,Math\'ematiques Pures et Appliq\'ees, {\bf 39} (1960),
307--317.

\bibitem[Chev61]{chevbour} C.\,Chevalley, {\em Certain sch\'emas de groupes semi-simples},
Sem.\,Bourbaki {\bf 219}, 1960/61, 219--234.

\bibitem[Chev97]{chevquad} C.\,Chevalley, {\em The algebraic theory of spinors},
Springer--Verlag, 1997. 

\bibitem[BIBLE]{bible} C.\,Chevalley, {\em Classification des groupes alg\'ebriques semi-simples}
(with Cartier, Grothendieck, Lazard),
Collected Works, volume 3, Springer--Verlag, 2005.

%\bibitem[Chow]{chow} W-L.\, Chow, ``On the projective embedding of homogeneous varieties'' in
%{\em Algebraic geometry and topology.  A symposium in honor of S. Lefschetz}, 122--128, Princeton
%Univ. Press, Princeton, 1957. 

\bibitem[Con02]{chevc} B.\,Conrad, {\em A modern proof of Chevalley's theorem on algebraic groups},
Journal of the Ramanujan Math.\,Society, {\bf 17} (2002), 1--18.


\bibitem[Con11]{conrad} B.\,Conrad, {\em Finiteness theorems for algebraic groups over function fields}, 
Compositio Math. {\bf 148} (2012), 555--639. 

\bibitem[Con14]{gross} B.\,Conrad, {\em Non-split reductive groups over $\Z$}, these
Proceedings. 

\bibitem[CGP]{pred} B.\,Conrad, O.\,Gabber, G.\,Prasad, {\em Pseudo-reductive groups},
Cambridge University Press, 2010.

\bibitem[D]{weilii} P.\,Deligne, {\em La conjecture de Weil II}, Publ.\,IHES {\bf 52} (1980), 137--252.

\bibitem[SGA7]{sga7} P.\,Deligne, N.\,Katz, {\em Groupes de monodromie en g\'eom\'etrie alg\'ebrique} II, 
LNM 340, Springer--Verlag, New York, 1973. 

\bibitem[DR]{dr} P.\,Deligne, M.\,Rapoport, {\em Les sch\'emas de modules des courbes elliptiques}
in Modular Functions of One Variable II, Springer LNM {\bf 349} (1973), 143--316.

\bibitem[DG]{dg} M.\,Demazure, P.\,Gabriel, {\em Groupes alg\'ebriques}, Masson, Paris, 1970.


\bibitem[SGA3]{sga3} M.\,Demazure, A.\,Grothendieck, {\em Sch\'emas en groupes}
I, II, III, Lecture Notes in Math {\bf 151, 152, 153}, Springer-Verlag, New York (1970).

%\bibitem[Dem]{demazure} M.\, Demazure, {\em Sch\'emas en groupes r\'eductifs},
%Bull. S.M.F. {\bf 93} (1965), 369--413.

\bibitem[DF]{farb} R.K.\,Dennis, B.\,Farb, {\em Non-commutative algebra}, GTM 145, 
Springer--Verlag, New York, 1993.

\bibitem[Di]{dickson} L.E.\,Dickson, {\em Linear groups with an exposition of the Galois
field theory}, Dover, 1958. 

\bibitem[Dieu]{classique} J.\,Dieudonn\'e,
{\em On the automorphisms of the classical groups}
(with a supplement by L-K. Hua), AMS Memoirs {\bf 2}, 1951.

\bibitem[EKM]{ekm} R.\,Elman, N.\,Karpenko, A.\,Merkurjev, {\em The algebraic and geometric theory of
quadratic forms}, AMS Colloq. Publ. {\bf 56}, 2008.

%\bibitem[FW]{faltings} G.\,Faltings, G.\,W\"ustholz, {\em Rational Points} (3rd ed.), Freidr.\,Vieweg \& Sohn,
%Wiesbaden, 1992.

\bibitem[Fr]{frohlich} A.\,Fr\"ohlich, {\em Orthogonal representations of Galois groups, Steifel--Whitney classes,
and Hasse--Witt invariants}, J.\,Reine Angew.\,Math.\, {\bf 360} (1985) 84--123.

\bibitem[FH]{fultonharris} W.\,Fulton, J.\,Harris, {\em Representation theory: a first course},
GTM 129, Springer--Verlag, New York, 1991.

\bibitem[GGP]{gross2} W-T.\,Gan, B.\,Gross, D.\,Prasad,  J-L.\,Waldspurger, 
{\em Sur les conjectures de Gross et Prasad}. I.,
Ast\'erisque {\bf 346} (2012).

\bibitem[GY]{ganyu} W-T.\,Gan, J-K.\,Yu, {\em Group schemes and local densities},
Duke Mathematical Journal, {\bf 105} (3), 2000, 497--524.

\bibitem[G]{gillesga3} P.\,Gille, {\em Sur la classification des sch\'emas en groupes semi-simples}, these
Proceedings.

\bibitem[GM]{gille} P.\,Gille, L.\,Moret-Bailly, ``Actions alg\'ebriques de groupes arithm\'etiques'' in 
{\em Torsors, \'etale homotopy theory, and applications to rational points}
(Edinburgh, 2011), edited by V.\,Batyrev and A.\,Skorobogatov, 
LMS Lecture Notes {\bf 405} (2013), 231--249.

%
%\bibitem[Gre]{green} M. Greenberg, {\em Rational points in henselian discrete valuation rings},
%Publ.\,Math.\,IHES {\bf 31}(1966), 59--64.



\bibitem[EGA]{ega} A.\,Grothendieck, {\em El\'ements de G\'eom\'etrie Alg\'ebrique},
Publ.\,Math.\,IHES {\bf 4, 8, 11, 17, 20, 24, 28, 32}, 1960--7.  

\bibitem[SGA1]{sga1} A.\,Grothendieck, {\em S\'eminaire  de g\'eom\'etrie alg\'ebrique $1$}, 
LNM {\bf 224}, Springer-Verlag, Berlin, 1971. 

\bibitem[SGA4]{sga4} A.\,Grothendieck, {\em S\'eminaire de g\'eom\'etrie alg\'ebrique $4$} (vol.\,2),
LNM {\bf 220}, Springer-Verlag, Berlin, 1972.

\bibitem[Gr68]{brauer3} A.\,Grothendieck, ``Le groupe de Brauer III: exemples et complements''
in {\em Dix Expos\'es sur la Cohomologie des Sch\'emas}, North-Holland, Amsterdam, 1968.

\bibitem[Ha]{hahn} A.\,Hahn, {\em Unipotent elements and the spinor norms of Wall and Zassenhaus},
Archiv der Mathematik {\bf 32} (1979), no.\,2, 114--122.

\bibitem[He]{helgason} S.\,Helgason, {\em Differential geometry, Lie groups,
and symmetric spaces}, Graduate Studies in Mathematics {\bf 34}, AMS, Providence, 2001.

\bibitem[Ho59]{hochlie} G.\,Hochschild, {\em Algebraic Lie algebras and representative
functions}, Illinois Journal of Math. {\bf 3} (1959), 499--523.

\bibitem[Ho65]{hoch} G.\,Hochschild, {\em The structure of Lie groups},  Holden--Day, San Francisco, 1965. 

\bibitem[Ho70]{hochcover} G.\,Hochschild, {\em Algebraic groups and Hopf Algebras}, 
Illinois Journal of Math. {\bf 34} (1970), 52--65.

\bibitem[Hum72]{humlie} J.\,Humphreys, {\em Introduction to Lie algebras and representation theory}, 
Springer--Verlag, New York, 1972. 

\bibitem[Hum87]{humphreys} J.\,Humphreys, {\em Linear algebraic groups} (2nd ed.), 
Springer--Verlag, New York, 1987.

\bibitem[Hum98]{humams} J.\,Humphreys, {\em Modular representations of
simple Lie algebras}, Bulletin of the AMS {\bf 35} no.\,2 (1998), 105--122.
%

\bibitem[Jan]{jantzen} J.\,Jantzen, {\em Representations of algebraic groups}, 
second edition, Mathematical Surveys and Monographs {\bf 107}, AMS, 2003.

%\bibitem[KMT]{414} T.\,Kambayashi, M.\,Miyanishi, M.\,Takeuchi, {\em Unipotent algebraic groups},
%Lecture Notes in Math {\bf 414}, Springer-Verlag, New York (1974). 

%\bibitem[Kem]{Kempf} G.\,Kempf, {\em Instability in invariant theory}, Ann.\,Math.\,{\bf 108}(1978), 299-316.

\bibitem[Kne]{kneser} M.\,Kneser, {\em Composition of binary quadratic forms}, 
Journal of Number Theory {\bf 15} (1982), 406--413.

\bibitem[Knus]{knus} M-A.\,Knus, {\em Quadratic and hermitian forms over rings}, 
Grundlehren der mathematischen Wissenschaften {\bf 294}, Springer--Verlag, New York, 1991.

\bibitem[KO]{ko} M-A.\,Knus, M.\,Ojanguren, {\em The Clifford algebra and a metabolic space},
Arch.\:Math.\:(Basel) {\bf 56} no.\,5 (1991),  440--445.

\bibitem[KMRT]{bookinv} M-A.\,Knus, A.\,Merkurjev, M.\,Rost,  J-P.\,Tignol, {\em The book of involutions},
AMS Colloq. Publ. {\bf 44}, Providence, 1998.

\bibitem[Knut]{knutson} D. Knutson,
{\em Algebraic spaces}, Lecture Notes in Math. {\bf 203},
Springer-Verlag, New York, 1971.

\bibitem[LW]{lw} S.\,Lang, A.\,Weil,  {\em Number of points of varieties in finite fields}, Amer. J. Math.
{\bf 76} (1954), 819--827.


\bibitem[LMB]{lmb} G. Laumon, L. Moret-Bailly, {\em Champs alg\'ebriques}, 
Ergebnisse der Mathematik {\bf 39}, Springer-Verlag, Berlin, 2000. 


\bibitem[Mat]{crt} H.\,Matsumura, {\em Commutative ring theory}, Cambridge Univ.\,Press, 1990.

\bibitem[Mil68]{milnor} J.\,Milnor, {\em Singular points of complex hypersurfaces},
Annals of Math Studies {\bf 61}, Princeton University Press, Princeton, 1968.

\bibitem[Mi80]{milne} J.\,Milne, {\em \'Etale cohomology}, Princeton University Press, Princeton, 1980.

\bibitem[Mi13]{milner} J.\,Milne, {\em A proof of the Barsotti--Chevalley theorem on algebraic groups}, 
{\tt{http://arxiv.org/abs/1311.6060}} preprint, 2013.

%\bibitem[Moch]{mot} S.\,Mochizuki, {\em Foundations of $p$-adic Teichm\"uller theory}, 
%AMS/IP Advance Studies in Mathematics {\bf 11},  Providence, 1999.

\bibitem[Mos]{mostow} G.\,Mostow, ``Some new decomposition theorems for semi-simple groups'' 
in {\em Lie groups and Lie algebras}, 
AMS Memoirs {\bf 14}, 1955, 31--54.

\bibitem[Mum]{mumford} D.\,Mumford, {\em Abelian varieties}, Oxford Univ.\,Press, 1970.

\bibitem[Mur]{murre} J.\,Murre, {\em Representation of unramified functors.  Applications.}
S\'eminaire Bourbaki, Exp.\,294, 1965, 243--261.


\bibitem[Oes]{oesterle} J.\,Oesterl\'e, {\em Groupes de type multiplicatif et sous-tores des sch\'emas en groupes}, 
these Proceedings.

\bibitem[O]{omeara} O.T.\,O'Meara, {\em Introduction to quadratic forms} (3rd ed.), Springer--Verlag, Berlin, 1973.

%
%\bibitem[Oes]{oesterle}
%J.\,Oesterl\'e, {\em Nombres de Tamagawa et groupes unipotents
%en caract\'eristique $p$},
%Inv.\:Math.\,{\bf 78}(1984), 13--88.


\bibitem[OV]{vinberg} A.L.\,Onishchik, E.B.\,Vinberg, {\em Lie groups and Lie algebras {\rm{III}}: structure
of Lie groups and Lie algebras}, Encyclopedia of Math. {\bf 41}, Springer--Verlag, New York, 1994.

\bibitem[PS]{ps} R.\,Parimala, R.\,Sridharan, ``Reduced norms and Brauer--Severi schemes'' in 
{\em Recent developments in real algebraic geometry and quadratic forms},
Comtemp.\:Math., {\bf 155} AMS, Providence, 1994. 


\bibitem[PS]{petrov} V.\,Petrov, A.\,Stavrova, {\em The Tits indices over semilocal rings},
Transform. Groups {\bf 16} no.\,1 (2011), 193--217.

%\bibitem[Pink]{pink}
%R.\,Pink, {\em On Weil restriction of reductive groups
%and a theorem of Prasad}, Math.\,Z.\,{\bf 248}(2004), 449--457. 

%\bibitem[PY1]{pryu} G.\,Prasad, J-K.\,Yu, {\em On finite group actions on reductive groups and buildings},
%Inv.\:Math.\,{\bf 147}(2002), 545--560.

\bibitem[Pr]{prasad} G.\,Prasad, {\em Elementary proof of a theorem of
Bruhat--Tits--Rousseau and of a theorem of Tits}, Bull.\,Soc.\, Math.\,France {\bf 110} (1982), 197--202.

\bibitem[PY02]{pryu} G.\,Prasad, J-K.\,Yu, {\em On finite group actions on reductive groups and buildings},
Inv.\:Math.\,{\bf 147} (2002), 545--560.


\bibitem[PY06]{py} G.\,Prasad, J-K.\,Yu, {\em On quasi-reductive group schemes},
Journal of Algebraic Geometry {\bf 15} (2006),  507--549.
%
%\bibitem[RG]{rg} M.\,Raynaud, L.\, Gruson, {\em Crit\`eres de platitude et de projectivit\'e.
%Techniques de ``platification'' d'un module}, Inv.\, Math.\, {\bf 13} (1971), 1--89. 

\bibitem[Ra]{raynaud} M.\,Raynaud,
{\em Faisceaux amples sur les sch\'emas en groupes et les espaces homog\`enes},
LNM {\bf 119}, Springer--Verlag, 1970. 

\bibitem[Ri]{ri} R.\,W.\,Richardson, {\em Affine coset spaces of reductive algebraic groups},
Bull.\, London Math.\,Soc.\, {\bf 9} (1977), 38--41. 

\bibitem[Ro]{rosenlicht} M.\,Rosenlicht, {\em Some basic theorems on algebraic groups},
American Journal of Mathematics {\bf 78} (1956), 401--443.

\bibitem[Ru]{russell} P.\,Russell, {\em Forms of the affine line and its additive group},
Pacific J.\,of Math.\,{\bf 32} (1970), 527--539. 
%
%\bibitem[SS]{ss} C.\,Sancho de Salas, F.\,Sancho de Salas, {\em Principal bundles, quasi-abelian
%varieties, and structure of algebraic groups}, Journal of Algebra {\bf 322}(2009), 2751--2772.

\bibitem[Sch] {scharlau}W.\,Scharlu, {\em Quadratic and hermitian forms}, 
Grundlehren der mathematischen Wissenschaften {\bf 270}, Springer-Verlag, Berlin, 1985.

\bibitem[Sel]{seligman} G.\,Seligman, {\em Modular Lie algebras}, Springer--Verlag, New York, 1967.

\bibitem[Ser63]{serrez} J-P.\,Serre, ``Zeta and $L$-functions'' in {\em Arithmetical algebraic geometry},
Harper \& Row, New York, 1963.

%\bibitem[Ser1]{serrecft} J-P.\, Serre {\em Algebraic groups and class fields}, Springer-Verlag, New York, 1988. 

\bibitem[Ser79]{serrelocal} J-P.\,Serre, {\em Local fields}, Springer-Verlag, New York, 1979.

\bibitem[Ser84]{serrewitt} J-P. Serre, {\em L'invariant de Witt de la forme ${\rm{Tr}}(x^2)$},
Comment.\,Math.\,Helv.\, {\bf 59} (1984), 651--676.

\bibitem[Ser92]{serrelie} J-P.\,Serre, {\em Lie groups and Lie algebras}, LNM 1500, Springer-Verlag, New York, 1992. 

\bibitem[Ser97]{serre} J-P.\,Serre, {\em Galois cohomology}, Springer-Verlag, New York, 1997.

\bibitem[Ser01]{serreclie} J-P.\,Serre, {\em Complex semisimple Lie algebras}, 
Springer--Verlag, Berlin, 2001.

\bibitem[Spr]{springer} T.\,A.\,Springer, {\em Linear algebraic groups} (2nd ed.), Birkh\"auser,
New York, 1998.  

\bibitem[St67]{steinyale} R.\,Steinberg, {\em Lectures on Chevalley groups}, Yale Math Dept., 1967.

\bibitem[St68]{steinberg} R.\,Steinberg, {\em Endomorphisms of linear algebraic groups}, 
AMS Memoirs {\bf 80}, Providence, 1968.

\bibitem[Ti66a]{tits66} J.\,Tits, ``Classification of algebraic semisimple groups''
in {\em Algebraic groups and discontinuous groups}, Proc.\,Symp.\,Pure Math.,vol.\,9, AMS, 1966.

\bibitem[Ti66b]{tits66b} J.\,Tits, {\em Normalisateurs de tores. {\rm{I}}.
Groupes de Coxeter \'etendus}, J.\,of Algebra {\bf 4} (1966), 96--116.

\bibitem[Ti66c]{tits66c} J.\,Tits, {\em Sur les constantes de structure et le th\'eor\`eme d'existence
des alg\`ebres de Lie semisimples}, Publ.\,Math.\,IHES {\bf 31} (1966), 21--58.

%\bibitem[Ti1]{titsyale} J.\,Tits, {\em Lectures on algebraic groups}, Yale Univ., New Haven, 1967.

%\bibitem[Ti2]{titscf12} J.\,Tits, {\em Th\'eorie des groupes}, Annuaire du Coll\`ege de France, 1991--92.


%\bibitem[Ti3]{titscf23} J.\,Tits, {\em Th\'eorie des groupes}, Annuaire du Coll\`ege de France, 1992--93.

\bibitem[Wat]{water} W.\,Waterhouse, {\em Affine group schemes}, GTM 66, Springer--Verlag, Berlin, 1979.

\bibitem[Wa59]{wall1} G.E.\,Wall, {\em The structure of a unitary factor group},
Publ.\,Math.\,IHES {\bf 1} (1959), 7--23.

\bibitem[Wa63]{wall2} G.E.\,Wall, {\em On the conjugacy classes in the unitary, symplectic, and orthogonal
groups}, J.\,Australian Math.\,Soc.\, {\bf 3} (1963), 1--62.

\bibitem[Za]{zass} H.\,Zassenhaus, {\em On the spinor norm}, Archiv der Mathematik {\bf 13} (1962), 434--451.

\end{thebibliography}

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\printindex

\end{document}

