\documentclass[11pt,english]{smfart}

\usepackage[T1]{fontenc}
\usepackage[english,francais]{babel}

\usepackage{amssymb,url,xspace,smfthm}

\def\meta#1{$\langle${\it #1}$\rangle$}
\newcommand{\myast}{($\star$)\ }
\makeatletter
    \def\ps@copyright{\ps@empty
    \def\@oddfoot{\hfil\small\copyright 2014, \SmF}}
\makeatother

\newcommand{\SmF}{Soci\'et\'e ma\-th\'e\-ma\-ti\-que de France}
\newcommand{\SMF}{Soci\'et\'e Ma\-th\'e\-Ma\-ti\-que de France}
\newcommand{\BibTeX}{{\scshape Bib}\kern-.08em\TeX}
\newcommand{\T}{\S\kern .15em\relax }
\newcommand{\AMS}{$\mathcal{A}$\kern-.1667em\lower.5ex\hbox
        {$\mathcal{M}$}\kern-.125em$\mathcal{S}$}
\newcommand{\resp}{\emph{resp}.\xspace}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[all]{xy}
\input yoga.sty
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% My added macros
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\tolerance 400
\pretolerance 200


\title{The structure of solvable groups over general fields} 
\date{\today}
\author{Brian Conrad}

\address{Math Dept., Stanford University\\
Stanford, CA 94305, USA}
\email{conrad@math.stanford.edu}
\urladdr{http://smf.emath.fr/}
%\keywords{Monaco, Tax-free. \\  $\hbox{\quad \enskip }$ {\bf   MSC 2000:\!} 14L15, 14L30.}


%\documentclass[12pt,amssymb]{amsart}
%\usepackage{amsmath}
%\usepackage{amssymb}
\usepackage[OT2,T1]{fontenc}
%\DeclareFontFamily{OT1}{rsfs}{}
%\DeclareFontShape{OT1}{rsfs}{n}{it}{<-> rsfs10}{}
%\DeclareMathAlphabet{\mathscr}{OT1}{rsfs}{n}{it}
% \topmargin=0in
%   \oddsidemargin=0in
%   \evensidemargin=0in
%   \textwidth=6.5in
%\textheight=8in
%
%\newcommand{\marginalfootnote}[1]{
%   \footnote{#1}
%   \marginpar{\hfill {\sf\thefootnote}}}
%\newcommand{\edit}[1]{\marginalfootnote{#1}}

%\usepackage[all]{xy}
%\CompileMatrices



\DeclareFontFamily{OT1}{rsfs}{}
\DeclareFontShape{OT1}{rsfs}{n}{it}{<-> rsfs10}{}
\DeclareMathAlphabet{\mathscr}{OT1}{rsfs}{n}{it}

\newcommand{\Add}{{\mathbf{G}}_{\rm{a}}}
\newcommand{\Gm}{{\mathbf{G}}_{\rm{m}}}
\DeclareMathOperator{\wt}{wt}
%\newcommand{\Gm}{{\rm{GL}}_1}
%\DeclareMathOperator{\Hom}{Hom}
%\DeclareMathOperator{\Ext}{Ext}
%\DeclareMathOperator{\Gal}{Gal}
%\DeclareMathOperator{\GL}{GL}
%\DeclareMathOperator{\Lie}{Lie}
%\DeclareMathOperator{\T}{T}
\DeclareMathOperator{\Res}{R}
%\DeclareMathOperator{\Sp}{Sp} 
%\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\X}{X}
%\DeclareMathOperator{\Aut}{Aut}
%\DeclareMathOperator{\Spec}{Spec}
%\DeclareMathOperator{\Proj}{Proj}
%\DeclareMathOperator{\Mod}{mod}
%\DeclareMathOperator{\coker}{coker}
%\DeclareMathOperator{\et}{\acute{e}t}
\newcommand*{\invlim}{\varprojlim}                               % invlimit
\newcommand*{\tensorhat}{\widehat{\otimes}}                      %compl. tensor
\newcommand*{\RRR}{\ensuremath{\mathbf{R}}}                        % reals
\newcommand*{\Z}{\ensuremath{\mathbf{Z}}}                        % integers
\newcommand*{\Q}{\ensuremath{\mathbf{Q}}}                        % rationals
\newcommand*{\F}{\ensuremath{\mathbf{F}}}                        % field
\newcommand*{\C}{\ensuremath{\mathbf{C}}}                        % complex's
\newcommand*{\A}{\ensuremath{\mathbf{A}}}                        % affine/adele
\newcommand*{\N}{\ensuremath{\mathbf{N}}}                        % naturals
\renewcommand*{\P}{\ensuremath{\mathbf{P}}}                        % proj space
\newcommand*{\m}{\mathfrak{m}}                                   % gothic m
\newcommand*{\n}{\mathfrak{n}}                                   % gothic n
\newcommand*{\p}{\mathfrak{p}}                                   % gothic p
\newcommand*{\q}{\mathfrak{q}}                                   % gothic q
\renewcommand*{\r}{\mathfrak{r}}                                 % gothic r
\newcommand*{\barrho}{\overline{\rho}}                           % rho bar
\newcommand*{\Qbar}{\overline{\Q}}                               % alg clos Q
\newcommand*{\Fbar}{\overline{\F}}                               % alg clos F
%\renewcommand*{\calO}{\mathscr{O}}                                  % 'sheaf' O
\DeclareMathOperator{\Det}{det}
%\newcommand*{\calA}{\mathscr{A}}
%\newcommand*{\calE}{\mathscr{E}}                               % `sheaf' E
%\newcommand*{\calF}{\mathscr{F}}                               % 'sheaf' F
%\newcommand*{\calG}{\mathscr{G}}                               % 'sheaf' G
%\newcommand*{\calH}{\mathscr{H}}                               % 'sheaf' H
%\newcommand*{\calI}{\mathscr{I}}                               % 'sheaf' I
%\newcommand*{\calJ}{\mathscr{J}}                               % 'sheaf' J
%\newcommand*{\calL}{\mathscr{L}}                               % `sheaf' L
%\newcommand*{\calS}{\mathscr{S}}                               % `sheaf' S
%\newcommand*{\calHom}{\mathscr{H}\mathit{om}}                  % `sheaf' Hom
\newcommand*{\calExt}{\mathscr{E}\mathit{xt}}                  % `sheaf' Ext
%\renewcommand*{\qedsymbol}{\ensuremath{\blacksquare}}            % end of proof
\newcommand*{\eqdef}{\stackrel{\text{def}}{=}}     % definition
\renewcommand*{\ge}{\geqslant}
\renewcommand*{\le}{\leqslant}





% \swapnumbers
\theoremstyle{plain}
  \newtheorem{theorem}[subsection]{Theorem}
  \newtheorem{proposition}[subsection]{Proposition}
  \newtheorem{lemma}[subsection]{Lemma}
  \newtheorem{corollary}[subsection]{Corollary}


\theoremstyle{definition}
  \newtheorem{definition}[subsection]{Definition}
  \newtheorem{example}[subsection]{Example}
  \newtheorem{remark}[subsection]{Remark}
  \newtheorem{para}[subsubsection]{}
  \newtheorem{notation}{Notation}

\renewcommand{\thepart}{\Roman{part}}



\numberwithin{equation}{section}

\newcounter{bean}

\makeatletter
\newcommand\@dotsep{4.5}
\def\@tocline#1#2#3#4#5#6#7{\relax
  \ifnum #1>\c@tocdepth % then omit
  \else
    \par \addpenalty\@secpenalty\addvspace{#2}%
    \begingroup \hyphenpenalty\@M
    \@ifempty{#4}{%
      \@tempdima\csname r@tocindent\number#1\endcsname\relax
    }{%
      \@tempdima#4\relax
    }%
    \parindent\z@ \leftskip#3\relax
    \advance\leftskip\@tempdima\relax
    \rightskip\@pnumwidth plus1em \parfillskip-\@pnumwidth
    #5\leavevmode\hskip-\@tempdima #6\relax
    \leaders\hbox{$\m@th
      \mkern \@dotsep mu\hbox{.}\mkern \@dotsep mu$}\hfill
    \hbox to\@pnumwidth{\@tocpagenum{#7}}\par
    \nobreak
    \endgroup
  \fi}
\makeatother 

%\usepackage{hyperref}


\usepackage[usenames,dvipsnames]{color}
\usepackage[colorlinks=false,naturalnames]{hyperref}


\begin{document}
\def\smfbyname{}

\begin{abstract}
We explain Tits' structure theory for smooth connected unipotent 
groups over general fields of positive characteristic (especially imperfect fields).
This builds on earlier work of Rosenlicht \cite{rosenlicht} and 
concerns the structure of smooth connected unipotent groups
as well as torus actions on such groups over an arbitrary
ground field of positive characteristic. 
We use it to establish a general structure theorem for solvable smooth connected
affine $k$-groups that replaces (and generalizes) the semi-direct product
structure over perfect $k$. 
\end{abstract}

\begin{altabstract}
Nous expliquons la  th\'eorie de   structure de Tits  des groupes alg\'ebriques  unipotents connexes et lisses
sur un corps  g\'en\'eral de  caract\'eristique positive (en particulier  imparfait).
Ceci s'appuie sur les travaux ant\'erieurs de Rosenlicht \cite{rosenlicht}
concernant la structure des groupes unipotents lisses et connexes
ainsi que des actions de tores sur ces groupes au-dessus d'un corps de base  de caract\'eristique positive.
Nous l'utilisons pour \'etablir un th\'eor\`eme de  structure plus  g\'en\'eral pour  les $k$--groupes affines
r\'esolubles lisses et connexes  qui remplace (et g\'en\'eralise) la structure de produit semi-direct
dans le cas d'un corps parfait  $k$.
\end{altabstract}

\maketitle


\tableofcontents

\section*{Introduction}

Consider a smooth connected solvable group $G$ over a field $k$.
If $k$ is algebraically closed then $G = T \ltimes \mathscr{R}_u(G)$ for any maximal
torus $T$ of $G$ \cite[10.6(4)]{borelag}.
Over more general $k$, an analogous such semi-direct product structure can fail to exist. 

For example, consider an imperfect field $k$ of characteristic $p > 0$
and $a \in k - k^p$, so $k' := k(a^{1/p})$ is a degree-$p$ purely inseparable extension 
of $k$.  Note that $k'_s := k' \otimes_k k_s = k_s(a^{1/p})$ is a separable closure of
$k'$, and ${k'_s}^p \subset k_s$.  The affine Weil restriction $G = {\rm{R}}_{k'/k}(\Gm)$ is an open subscheme of
${\rm{R}}_{k'/k}(\mathbf{A}^1_{k'}) = \mathbf{A}^p_k$, so it is a smooth connected
affine $k$-group of dimension $p > 1$. Loosely speaking, $G$ is ``${k'}^{\times}$ viewed as a $k$-group''.  More precisely,
for $k$-algebras $R$ we have $G(R) = (k' \otimes_k R)^{\times}$
functorially in $R$.  (See Exercise U.4 for a treatment
of Weil restriction in the affine case.)  The commutative $k$-group $G$ contains an evident
1-dimensional torus $T \simeq \Gm$ corresponding to the subgroup $R^{\times} \subset (k' \otimes_k R)^{\times}$,
and $G/T$ is unipotent because $(G/T)(k_s) = (k'_s)^{\times}/(k_s)^{\times}$ is $p$-torsion.
In particular, $T$ is the unique maximal torus of $G$.
Since the group $G(k_s) = {k'_s}^{\times}$ has no nontrivial $p$-torsion, $G$ contains
{\em no} nontrivial unipotent smooth connected $k$-subgroup.   Thus,
$G$ is a commutative counterexample over $k$ to the analogue of the semi-direct product
structure for connected solvable smooth affine groups over $\overline{k}$.

The appearance of imperfect fields in  the preceding counterexample is essential.
To explain this, recall Grothendieck's theorem that over a general field $k$, if $S$ is a maximal $k$-torus in a smooth
affine $k$-group $H$ then
$S_{\overline{k}}$ is maximal in $H_{\overline{k}}$.  (This theorem is an application 
of \cite[XIV, Thm.\,1.1]{sga3} to the smooth
affine $k$-group $Z_H(S)$, since a ``maximal torus'' over $k$ in the sense of
\cite[XII, Def.\,1.3]{sga3} is {\em defined} to be a $k$-torus that is maximal after
scalar extension to $\overline{k}$. For another proof, see \cite[A.1.2]{sga3notes}.)  Thus, by the conjugacy
of maximal tori in $G_{\overline{k}}$, 
$G = T \ltimes U$ for a $k$-torus $T$ and a unipotent smooth
connected normal $k$-subgroup $U \subset G$ if and only if
the subgroup $\mathscr{R}_u(G_{\overline{k}}) \subset G_{\overline{k}}$
is defined over $k$ (i.e., descends to a $k$-subgroup of $G$).  In such cases, 
the semi-direct product structure holds for $G$ over $k$ using
any maximal $k$-torus $T$ of $G$ (and $U$ is unique: it must be a $k$-descent of $\mathscr{R}_u(G_{\overline{k}})$).
If $k$ is perfect then by Galois descent we may always descend $\mathscr{R}_u(G_{\overline{k}})$
to a $k$-subgroup of $G$.  
The main challenge is the case of imperfect $k$.


The results we shall discuss for unipotent groups were presented by Tits in a course at Yale University in 1967,
and lecture notes \cite{titsyale} for that course were circulated but never published.   
Much of the course was concerned with general results on 
linear algebraic groups that are available now in many standard references (such as \cite{borelag},
\cite{humphreys}, and \cite{springer}). 
The original  account (with proofs)
of Tits' structure theory of unipotent groups is his unpublished Yale lecture notes, 
and a summary of the results is given in \cite[Ch.\:V]{oesterle}. 

Our exposition in \S\ref{splitgp}--\S\ref{torsec} is an improvement of \cite[App.\,B]{pred} via simplifications
in some proofs. 
(This simplified treatment of Tits' work will also appear in Appendix B of the second edition of \cite{pred}.)
In some parts we have simply reproduced arguments from Tits' lecture 
notes.  The general solvable case is addressed in \S\ref{solvgps}, where
we include applications to general smooth connected affine $k$-groups.
{\em Throughout the discussion below, $k$ is an arbitrary field with characteristic $p > 0$.} 


{\bf Acknowledgements}. This work was supported by NSF grants DMS-0917686 and DMS-1100784.  I am grateful to
P.\:Gille and L.\:Moret-Bailly for suggestions about the general solvable case, to the referee for several insightful comments,
and to O.\:Gabber and G.\:Prasad for many illuminating discussions related to Tits' work on unipotent groups.


\section{Subgroups of vector groups}\label{splitgp}

The {\em additive group} is denoted $\Add$ 
and the {\em multiplicative group} is denoted $\Gm$, always with the base ring understood from context.


\begin{definition}\label{linstr} A {\em vector group} over a field $k$ 
is a smooth commutative $k$-group  $V$ that admits an isomorphism
to $\Add^n$ for some $n \ge 0$.  The $\Gm$-scaling action arising
from such an isomorphism is a {\em linear structure} on $V$. 
\end{definition}

Observe that the $\Gm$-action on $V$ arising from a linear structure induces the
canonical $k^{\times}$-action on ${\rm{Lie}}(V)$ (e.g., if ${\rm{char}}(k) = p > 0$
then the composition of such a $\Gm$-action on $V$ with the $p$-power map
on $\Gm$ does not arise from a linear structure on $V$ when $V \ne 0$). 

\begin{example} If $W$ is a finite-dimensional $k$-vector space then the {\em associated vector group}
$\underline{W}$ represents the functor $R \rightsquigarrow R \otimes_k W$ on $k$-algebras 
and its formation commutes with any extension of the ground field.  
Explicitly, $\underline{W} = \Spec({\rm{Sym}}(W^{\ast}))$ and it 
has a unique linear structure relative to which the natural identification
of groups $\underline{W}(k_s) \simeq 
W_{k_s}$ carries the linear structure
over to the $k_s^{\times}$-action on $W_{k_s}$ arising from the $k_s$-vector space structure;
call this the {\em canonical} linear structure on $\underline{W}$.  (We can use $k$ instead of $k_s$
in this characterization when $k$ is infinite, as $W(k)$ is Zariski-dense in $\underline{W}$
for infinite $k$.) 
For finite-dimensional $k$-vector spaces $W$ and $W'$, the subset ${\rm{Hom}}_k(W,W') \subset
{\rm{Hom}}_{k{\mbox{-}{\rm{gp}}}}(\underline{W},\underline{W'})$ consists of
precisely the $k$-homomorphisms respecting the canonical linear structures.
\end{example}

When linear
structures are specified on a pair of vector groups, a homomorphism respecting them is called {\em linear}.  
Over a field of characteristic 0 there is a unique linear structure and all homomorphisms are
linear.  Over a field with characteristic $p > 0$ the linear structure is not unique
in dimension larger than 1 (e.g., $a.(x,y) := (ax + (a-a^p)y^p, ay)$ is a linear
structure on $\Add^2$, obtained from the usual one via the non-linear $k$-group automorphism
$(x,y) \mapsto (x+y^p, y)$ of $\Add^2$). 
For a finite-dimensional $k$-vector space $W$, a {\em linear subgroup} of $\underline{W}$
is a smooth closed $k$-subgroup that is stable under the $\Gm$-action. By computing
with $k_s$-points and using Galois descent, it is straightforward to verify that
the linear subgroups of $\underline{W}$ are precisely $\underline{W'}$ for $k$-subspaces $W' \subset W$.

\begin{definition}
A smooth connected solvable $k$-group $G$ is {\em $k$-split} if
it admits a composition series 
$$G = G_0 \supset G_1 \supset \dots \supset G_n = 1$$
consisting of smooth closed $k$-subgroups such that 
$G_{i+1}$ is normal in $G_i$ and the quotient $G_i/G_{i+1}$ is $k$-isomorphic to
$\Add$ or $\Gm$ for all $0 \le i < n$.  (Such $G_i$ must be connected,
so each $G_i$ is also a $k$-split smooth connected solvable $k$-group.)
\end{definition}

 
 In the case of tori this is a widely-used notion, and it satisfies convenient
properties, such as:  (i) every subtorus or quotient torus (over $k$) of a $k$-split $k$-torus is
$k$-split, (ii) every $k$-torus is an almost direct product of its maximal $k$-split subtorus
and its maximal $k$-anisotropic subtorus.  However, in contrast with the case of tori,
it is not true for general smooth connected
solvable $G$ that the $k$-split property is inherited by smooth connected normal $k$-subgroups: 

\begin{example}[Rosenlicht]\label{nonsplit}
Assume $k$ is imperfect and choose $a \in k - k^p$. 
The $k$-group $$\mathbf{U} := \{y^p = x - a x^p\}$$ is a $k$-subgroup of the $k$-split 
$G = \Add^2$ and it becomes isomorphic to $\Add$ over
$k(a^{1/p})$ but there is no non-constant $k$-morphism
$f:\mathbf{A}^1_k \rightarrow \mathbf{U}$, let alone a $k$-group isomorphism $\Add \simeq \mathbf{U}$.
Indeed, the regular compactification $\overline{\mathbf{U}}$ of $\mathbf{U}$ 
has a unique point $\infty_{\mathbf{U}} \in \overline{\mathbf{U}} - \mathbf{U}$, 
and the regular compactification of $\Add$ is $\mathbf{P}^1_k$
via $x \mapsto [x,1]$, so any non-constant map
$f$ extends to a (finite) surjective map
$\mathbf{P}^1_k \rightarrow \overline{\mathbf{U}}$ that must carry
$[1,0]$ to $\infty_{\mathbf{U}}$, an absurdity since $k(\infty_{\mathbf{U}}) = k(a^{1/p}) \ne k$.
\end{example}

Tits introduced an analogue for unipotent $k$-groups of 
the notion of 
anisotropicity for tori over a field. 
This rests on a preliminary understanding of the properties of subgroups of vector groups,
so we take up that study now.  The main case of interest to us will
be imperfect ground fields, due to the fact
that every unipotent smooth connected group over a perfect field is split
(see Exercises U.9(iii)).

\begin{definition}\label{ppolydef} A polynomial $f \in k[x_1,\dots,x_n]$ is a {\em $p$-polynomial} 
if every monomial appearing
in $f$ has the form $c_{ij}x^{p^j}_i$ for some $c_{ij} \in k$; 
that is, $f=\sum f_i(x_i)$ with $f_i(x_i) = \sum_j c_{ij} x_i^{p^j} \in k[x_i]$.
(In particular, $f_i(0) = 0$ for all $i$.  Together with the
identity $f = \sum f_i(x_i)$, this uniquely determines each $f_i$ in terms of $f$. Note that $f(0) = 0$.) 
\end{definition}

\begin{proposition}\label{III.3.3.4}
A polynomial $f\in k[x_1,\ldots,x_n]$ is a $p$-polynomial if and only if  the associated map of 
 $k$-schemes $\Add^n \rightarrow \Add$ is a $k$-homomorphism.  
 \end{proposition}

\begin{proof} This is elementary and is left to the reader.
\end{proof} 


A nonzero polynomial over $k$ is {\em separable} if its zero scheme in affine space is generically $k$-smooth. 


\begin{proposition}\label{fgmap} Let $f \in k[x_1,\dots,x_n]$ be a nonzero polynomial such that $f(0) = 0$.
Then the subscheme $f^{-1}(0) \subset \Add^n$ is a smooth 
$k$-subgroup if and only if $f$ is a separable $p$-polynomial.
\end{proposition}

\begin{proof} The ``if'' direction is clear.  For the converse, we assume 
that $f^{-1}(0)$ is a smooth $k$-subgroup 
and we denote it as $G$.  The smoothness implies that $f$ is separable. To prove that
$f$ is a $p$-polynomial, by Proposition \ref{III.3.3.4}
it suffices to prove that the associated
map of $k$-schemes 
$\Add^n \rightarrow \Add$ is a $k$-homomorphism. Without loss of generality, we may assume
that $k$ is algebraically closed.  

For any $\alpha \in G(k)$, 
$f(x+\alpha)$ and $f(x)$ have the same zero scheme (namely, $G$) inside $\Add^n$.  Thus, 
$f(x+\alpha)=c(\alpha)f(x)$ for a unique $c(\alpha) \in k^{\times}$. 
 Consideration of
a highest-degree monomial term appearing in $f$ implies that 
$c = 1$.
Pick $\beta \in k^n$, so $f(\beta+\alpha)-f(\beta)=0$ for all $\alpha\in G(k)$.  
Thus $f(\beta+x)-f(\beta)$ vanishes on $G$,
so $f(\beta+x)-f(\beta)=g(\beta)f(x)$ for a unique $g(\beta)\in k$.  Consideration of a highest-degree
monomial term in $f$ forces $g(\beta) = 1$.
\end{proof}

\begin{corollary}\label{III.3.3.5}
 Let $G\subset \Add^n$ be a smooth $k$-subgroup of codimension $1$.  
 Then $G$ is the zero scheme of a separable nonzero $p$-polynomial in $k[x_1,\dots,x_n]$. 
 \end{corollary}

\begin{proof}Since $G$ is smooth of codimension 1 in $\Add^n$, it is the  zero scheme of a 
separable nonzero polynomial $f \in k[x_1,\dots,x_n]$.
 By Proposition \ref{fgmap}, $f$ is a $p$-polynomial.
 \end{proof}


\begin{lemma}\label{splitqt}
 If $f:U' \rightarrow U$ is a surjective homomorphism between smooth connected
unipotent $k$-groups and $U'$ is $k$-split then so is $U$.
\end{lemma}

This result is part of \cite[15.4(i)]{borelag}. We give a proof based
on the elementary Proposition \ref{fgmap}.

\begin{proof} Let $\{U'_i\}$ be a descending composition series of $U'$ over $k$ with successive
quotients $U'_i/U'_{i+1}$ isomorphic to $\Add$.  Then the $k$-groups $U_i = f(U'_i)$ are a composition
series for $U$ and $U_i/U_{i+1}$ is a quotient of $U'_i/U'_{i+1} = \Add$.  It
therefore suffices to show that  for any surjective $k$-homomorphism
$q:\Add \rightarrow G$ with $G \ne 1$, necessarily $G \simeq \Add$.  Clearly $q$ is
an isogeny. If $\ker q$ is not \'etale
then $\ker q$ has nontrivial Frobenius kernel.  But
the Frobenius kernel of $\Add$ is $\alpha_p$, so $q$ factors through $\Add/\alpha_p \simeq \Add$.
Hence, by induction  on $\deg q$ we can assume $\ker q$ is \'etale.
By Proposition \ref{fgmap}, the smooth $k$-subgroup $\ker q \subset \Add$ must be the zero scheme
of a 1-variable separable $p$-polynomial $f = \sum c_j t^{p^j}$ (so $c_0 \ne 0$).
But $f:\Add \rightarrow \Add$ is then an isogeny and its kernel $\{f = 0\}$ coincides
with $\ker q$, so $f$ identifies $G = \Add/\ker q$ with $\Add$.
\end{proof}



\begin{definition} If $f=\sum_{i=1}^n f_i(x_i)$ is a $p$-polynomial over $k$ in 
$n$ variables with $f_i(0) = 0$ for all $i$, then the {\em principal part} of $f$ is 
the sum of the leading terms of the $f_i$.
\end{definition}


\begin{lemma}\label{III.3.3.6}
Let $V$ be a vector group of dimension $n \ge 1$ over $k$, and let $f:V\to\Add$ be a $k$-homomorphism.  Then the following are equivalent:
\begin{enumerate}
\item there exists a non-constant $k$-scheme morphism $f':\A^1_k\to V$ such that $f\circ f'=0$;
\item for every $k$-group isomorphism $h:\Add^n\simeq V$, 
the principal part of the $p$-polynomial $f\circ h \in k[x_1,\dots,x_n]$ has a 
nontrivial zero in $k$; 
\item there exists a $k$-group isomorphism $h:\Add^n\simeq V$ such that $f\circ h$ ``only depends on the last $n-1$ coordinates'' $($i.e., $\ker(f\circ h)$ contains the first factor of $\Add^n$$)$.
\end{enumerate}
\end{lemma}

In this lemma, it is not sufficient in (2) to consider just a single choice of $h$.
For example, if $k$ is imperfect and $a \in k - k^p$, then $f := y^p - (x + a x^p)$ has principal
part $y^p - a x^p$ with no zeros on $k^2 - \{0\}$.  Composing $f$ with the $k$-automorphism
$(x,y) \mapsto (x, y + x^p)$ yields the polynomial $y^p + x^{p^2} - (x + ax^p)$ whose principal
part is $y^p + x^{p^2}$, which has zeros on $k^2 - \{0\}$.  

\begin{proof} We will show that $(1)\Rightarrow (2)\Rightarrow (3)\Rightarrow (1)$.  

For $(1)\Rightarrow (2)$, assume that $(1)$ holds and let $\varphi=h^{-1}\circ f'$.  Let 
$\varphi_i:\Add \to \Add$ be the $i$th component of $\varphi$,
and $a_i t^{s_i}$ denote the leading term of $\varphi_i(t)$,
with $s_i = 0$ when $\varphi_i = 0$. For some $i$ we have $s_i > 0$, since some
$\varphi_i$ is non-constant (as $\varphi$ is non-constant, because
of the same for $f'$).  Let $\sum_{i=1}^n c_i x_i^{p^{m_i}}$ be the prinicipal part 
of $f\circ h$, so 
$$0 = f(h(\varphi(t)))=\sum_{i=1}^n c_i a_i^{p^{m_i}} t^{s_i p^{m_i}}+\dots$$ since 
$f\circ h\circ\varphi=f\circ h\circ h^{-1}\circ f'=f\circ f'=0$.  Let $N = \max_i \{s_i p^{m_i}\} > 0$. 
Define $b_i=a_i$ if $s_ip^{m_i}=N$ (so $b_i \ne 0$), and $b_i=0$ if $s_ip^{m_i}<N$.  
Since the coefficient of the term of degree $N$ in $f(h(\varphi(t)))$ must be zero, we have $\sum_{i=1}^n c_i b_i^{p^{m_i}}=0$ with $b_i\in k$ and some $b_i$ is nonzero, so $(2)$ holds. 

To prove $(2)\Rightarrow (3)$, assume $(2)$ holds and let $h:\Add^n\simeq V$ 
be any $k$-group isomorphism.  We may assume $f \ne 0$, so the principal part of $f \circ h$
is nonzero.  The proof will proceed by induction on the sum $d$ of the degrees of nonzero terms of the principal part $\sum_{i=1}^n c_i x_i^{p^{m_i}}$ of 
$f\circ h$.  
If $c_r=0$ for some $r$, we are done by interchanging $x_r$ and $x_1$.  So we may assume that all $c_i$ are nonzero and, upon permuting the coordinates, that $m_1\ge\cdots\ge m_n \ge 0$.   
By (2), there exists $(a_1,\dots, a_n) \in k^n - \{0\}$ such that $\sum_{i=1}^n c_i a_i^{p^{m_i}}=0$.
Let $r \ge 0$ be minimal such that $a_r \ne 0$.  Define
the $k$-group isomorphism $h':\Add^n\simeq \Add^n$ by $h'(y_1,\ldots,y_n)=(x_1,\ldots,x_n)$ with 
$$x_1=y_1,\ldots, x_{r-1}=y_{r-1},$$
$$x_r=a_r y_r,\quad x_{r+1}=y_{r+1}+a_{r+1}y_r^{p^{m_r-m_{r+1}}}, \ldots, x_n=y_n+a_n y_r^{p^{m_r-m_n}}$$
Thus,
$f\circ h\circ h'$ is a $p$-polynomial with principal part
$$\sum_{i\neq r}c_i y_i^{p^{m_i}}+\sum_{i=1}^n c_i a_i^{p^{m_i}} \cdot y_r^{p^{m_r}} = 
\sum_{i \ne r} c_i y_i^{p^{m_i}}$$
since $\sum_{i=1}^nc_i a_i^{p^{m_i}}=0$.  The sum of the degrees of the nonzero terms of the principal part of 
$f\circ h \circ h'$  is strictly smaller than $d$ since $c_r \ne 0$, so the induction hypothesis applies. 

Finally, we assume $(3)$ and prove $(1)$.  Let $h:\Add^n\to V$ be a $k$-isomorphism such that $\ker(f\circ h)$ contains the first factor of $\Add^n$.  Define $\varphi:\Add\to\Add^n$ by $\varphi(t)=(t,0,0,\ldots,0)$.  Finally, let $f'=h\circ\varphi$.  Then $f \circ f' = f\circ h\circ\varphi=0$.\end{proof}

\begin{lemma}\label{III.3.3.7}
If a $p$-polynomial $\sum_{i=1}^n c_i x_i^{p^{m_i}}$ over $k$ has
a zero in $K^n - \{0\}$ for a Galois extension $K/k$
 then it has a zero in $k^n - \{0\}$.
\end{lemma}

\begin{proof}The proof is by induction on $n$.  The terms may be
 ordered so that $m_1\ge m_2\ge \cdots$.  If $n=1$, then since $c_1a_1^{p^{m_1}}=0$ with $a_1\in K^{\times}$ 
we see that $c_1=0$, so $c_1x_1^{p^{m_1}}$ has a zero in $k^{\times}$.  

Now suppose $n > 1$ and 
that  $\sum_{i=1}^n c_i a_i^{p^{m_i}}=0$ with $a_i\in K$ not all zero.  
Let $a = (a_1, \dots, a_n)$. 
If $a_n=0$ then the theorem is true by the induction hypothesis.  If $a_n\neq 0$, we may assume $a_n=1$ by
replacing $a_i$ with $a_i/a_n^{p^{m_n-m_i}}$ for all $i$.  For all 
$\sigma\in\Gal(K/k)$, the point $a - \sigma(a)$ is a zero of  $\sum c_i x_i^{p^{m_i}}$. 
If not all $a_i$ belong to $k$ then $a - \sigma(a) \ne 0$, so
since $a_n-\sigma(a_n)=0$ we may again apply the inductive hypothesis.
\end{proof}

\begin{lemma}\label{III3.3noname}
Let $V$ be a vector group over $k$, $K/k$ a Galois extension,
and $f:V\to\Add$ a $k$-homomorphism.  The equivalent conditions $(1)$, $(2)$, and $(3)$ of Lemma {\rm{\ref{III.3.3.6}}} hold over $K$ if and only if they hold over $k$.
\end{lemma}

\begin{proof}It is clear that if $(1)$ holds over $k$ then  it also holds over $K$.  On the other hand, by 
Lemma \ref{III.3.3.7}, (2) is true over $k$ if it is true over $K$.
\end{proof}

\begin{lemma}\label{embedvec}
Every smooth $p$-torsion commutative affine $k$-group $G$ embeds
as a $k$-subgroup of a vector group over $k$.  Moreover, $G$ admits an \'etale isogeny onto a vector group over $k$, and if $G$ is connected and $k = \overline{k}$
then $G$ is a vector group over $k$.  
\end{lemma}

\begin{proof} 
We first construct the embedding into a vector group over $k$, and then at the end use this to make the \'etale isogeny.  
Consider the canonical $k$-subgroup inclusion
$G \hookrightarrow {\Res}_{k'/k}(G_{k'})$ for any finite extension field $k'/k$.
Since ${\Res}_{k'/k}(\Add) \simeq \Add^{[k':k]}$, it is harmless (for the purpose of finding an embedding
into a vector group over $k$) 
to replace $k$ with a finite extension.   If $G_{\overline{k}}$ embeds as a subgroup of 
$\Add^N$ over $\overline{k}$, the embedding descends to a finite extension $k'/k$ inside
$\overline{k}$.  Hence, for the construction of the embedding into
a vector group we can now assume that $k$ is algebraically closed.

The component group $G/G^0$ is a power of $\Z/p\Z$.  Thus, since $G$ is commutative and
$p$-torsion, the connected-\'etale sequence of $G$ splits.  That is, $G = G^0 \times (\Z/p\Z)^n$
for some $n \ge 0$.  The finite constant $k$-group
$\Z/p\Z$ is a $k$-subgroup of $\Add$, so we can assume that $G$ is connected. 
We shall prove that $G$ is a vector group.
By \cite[10.6(2), 10.9]{borelag}, $G$ has a composition series whose successive quotients
are $\Add$.  By induction on $\dim G$, it suffices to prove that a commutative extension $U$ of
$\Add$ by $\Add$ over $k$ is a split extension if  $p \cdot U = 0$.  

Let $W_2$ be the additive $k$-group  of Witt vectors of length 2, so there is a canonical exact sequence of
$k$-groups
$$0 \rightarrow \Add \rightarrow W_2 \rightarrow \Add \rightarrow 0.$$ 
It is a classical fact (see \cite[Ch.\:VII.9,\:Lemma\:3]{serrecft}) that every commutative 
extension $U$ of $\Add$ by $\Add$ over $k$ is obtained by pullback of
this Witt vector extension along a (unique) $k$-homomorphism
$f:\Add \rightarrow \Add$.  In other words, there is a unique pullback diagram
$$\xymatrix{
0 \ar[r] & \Add \ar[r] \ar@{=}[d] & U \ar[r] \ar[d]^-{f'} & \Add \ar[d]^-{f} \ar[r] & 0 \\
0 \ar[r] & \Add \ar[r] & W_2 \ar[r] & \Add \ar[r] & 0}$$
and we claim that if $U$ is $p$-torsion then $f = 0$ (so the top row is a split sequence).
Clearly $f'(U) \subset W_2[p]$, but the maximal smooth $k$-subgroup
of $W_2[p]$ is the kernel term $\Add$ along the bottom row.  Hence, 
$f'(U)$ is killed by the quotient map along the bottom row, so $f = 0$.

Now return to the setting of a general ground field $k$, and fix a $k$-subgroup inclusion of $G$ into
a vector group $V$, say with codimension $c$.  Choose a linear structure on $V$
(in the sense of Definition \ref{linstr}).  Then 
$W \mapsto {\rm{Lie}}(W)$ is a bijection between the set of linear subgroups of $V$ and the set of linear
subspaces of ${\rm{Lie}}(V)$.   Hence, if we choose $W$ so that ${\rm{Lie}}(W)$ is complementary to
${\rm{Lie}}(G)$ then the natural map $G \rightarrow V/W$ is an isomorphism on Lie algebras, so it is
an \'etale isogeny.  Since $W$ is a linear subgroup of $V$, the quotient $V/W$ is a vector group over $k$. 
\end{proof}



\begin{proposition}\label{III.3.3.8}
Let $V_1, \dots, V_n$ be $k$-groups isomorphic to $\Add$,
and let $V=\prod_{i=1}^n V_i$.  Let $U$ be a smooth $k$-subgroup of $V$ such that
$U_{k_s}$ as a $k_s$-subgroup of $V_{k_s}$
is generated by images of $k_s$-scheme morphisms $\A^1_{k_s} \rightarrow V_{k_s}$ that pass through
$0$.  

There exists a $k$-group automorphism $h:V \simeq V$ such that $h(U)$ is the direct product of some of the $V_i$
inside $V$.  In particular, $U$ is a vector group over $k$ and is a $k$-group direct factor of $V$.
\end{proposition}

\begin{proof}
The proof is by induction on $n$ and is trivial for $n=1$.  Now consider $n>1$.  The case $U = V$ is trivial, so we
can assume $\dim U \le n-1$.  
First assume that $\dim U=n-1 > 0$.  By Corollary \ref{III.3.3.5}, $U$ is the kernel of a $k$-homomorphism $f:V\to\Add$.  By hypothesis, there exists a non-constant $k_s$-scheme morphism
$\A^1_{k_s} \rightarrow U_{k_s}$, so by Lemma \ref{III.3.3.6} (applied over $k_s$)
and Lemma \ref{III3.3noname} there exists a $k$-group 
automorphism $h'$ of $V$ such that $h'(U)\supset V_1$.  But then $h'(U)=V_1\times U'$, where $U'$ denotes the 
projection of $h'(U)$ into $V'=\prod_{i=2}^n V_i$.  Applying the induction hypothesis to $V'$ and $U'$, we are done. 

Suppose now that $\dim U<n-1$, and let $U'$ denote the projection of $U$ into the product $V'$ as defined above.  By the inductive hypothesis, after relabelling $V_2, \dots, V_n$ there exists a $k$-group 
automorphism $h_1:V'\to V'$ such that $h_1(U')=\prod_{i=2}^r V_i$ for some $r<n$.  Setting $$h'={\rm{id}}_{V_1} \times h_1:V\simeq V,$$ we then have 
$h'(U)\subset\prod_{i=1}^r V_i$, and we can again apply induction.  The proof is now complete.
\end{proof}

\begin{corollary}\label{III.3.3.9}
In a smooth $p$-torsion commutative affine $k$-group $G$, every 
smooth $k$-subgroup that is a vector group is a $k$-group direct factor.
\end{corollary}

\begin{proof} This is a consequence of Proposition \ref{III.3.3.8}, provided
that $G$ is a $k$-subgroup of a vector group.  Such an embedding is provided
by Lemma \ref{embedvec}. 
\end{proof}


The following proposition is a useful refinement of Lemma \ref{embedvec}. 

\begin{proposition} \label{III.3.3.1}
Let $k$ be an infinite field of characteristic $p > 0$
and let $U$ be a smooth $p$-torsion commutative affine $k$-group. Then $U$ is 
$k$-isomorphic to a $k$-subgroup of codimension $1$ in a $k$-vector group.
In particular, $U$ is isomorphic $($as a $k$-group$)$ 
to the zero scheme of a separable nonzero $p$-polynomial over $k$.  
\end{proposition}

This proposition is also true for finite $k$ if $U$ is connected since then
$U$ is a vector group;
see Corollary \ref{III.3.3.14}. 

\begin{proof} By Lemma \ref{embedvec}, $U$ can be identified with
a $k$-subgroup of a $k$-vector group $V$. Let $m =\dim V -\dim U$. If $m \le 1$ then we are done
by Corollary \ref{III.3.3.5},
so we assume $m>1$. We will show that $U$ can be embedded in a $k$-vector group $W$ with $\dim W = \dim V -1$,
which will complete the argument via induction on $m$.   The vector group $W$ will arise as a quotient of $V$. 

The $k$-linear subspace $\Lie(U)$ in ${\rm{Lie}}(V)$ has codimension $m$. Fix a choice
of linear structure on $V$ (in the sense of Definition \ref{linstr}).  Since $m \ge 2$, 
the Zariski closure $\Add.U \,(\subset V)$ of the image of the multiplication map
$\Add \times U \rightarrow V$ is a closed subscheme of $V$ with nonzero codimension.
By irreducibility of $V$, the union $\Lie(U) \cup (\Add.U)$ inside $V$ is a proper closed subscheme of $V$.

Since $V(k)$ is Zariski-dense in $V$ (as $k$ is infinite),
there exists $v\in V(k)$ with $$v\notin \Lie(U) \cup (\Add.U).$$ Let $L \subset V$ be the $k$-subgroup
corresponding to the line $kv \subset V(k)$. 
Consider the canonical $k$-homomorphism $\phi: V\rightarrow W := V/L$, and let $\psi = \phi|_{U}$.
We shall prove $\ker \psi = 1$, from which it follows that $\psi$ identifies $U$ with a $k$-subgroup of
$W$.   

It suffices to show that $\Lie(\psi)$ is injective (so $\ker \psi$ is \'etale) and that $\psi|_{U(\overline{k})}$
is injective. The map $\Lie(\psi)$
has kernel $L \cap \Lie(U) =\{0\}$, so it is indeed injective.
If $\psi|_{U(\overline{k})}$ is not injective
then the line $L$ would lie in $\Add.U$ since $\Add.U$ is stable under the $\Add$-multiplication on
$V$.  But the point $v \in L(k)$ does not lie in $(\Add.U)(\overline{k})$, due to how we chose $v$, so
indeed $\psi|_{U(\overline{k})}$ is injective.  
\end{proof}


\section{Wound unipotent groups}\label{wsec}


A smooth connected
unipotent $k$-group $U$ is analogous to an anisotropic torus if 
$U$ does not contain $\Add$ as a $k$-subgroup.  This concrete viewpoint is inconvenient for developing 
a general theory, but eventually we will prove that it gives the right concept.
A more convenient definition to get the theory of such $U$ off the ground requires
going beyond the category of $k$-groups, as follows.  

\begin{definition}\label{IV.4.1}
A smooth connected unipotent $k$-group $U$ is 
{\em $k$-wound} if every map of $k$-schemes
$\A^1_k \rightarrow U$ is a constant map to a point in $U(k)$.
Equivalently, $U(k) = U(k[x])$.
\end{definition}

By considering translation by $k$-points, it is equivalent to say that every
map of pointed $k$-schemes $(\mathbf{A}^1_k, 0) \rightarrow (U,1)$ is constant.

\begin{remark}\label{remtori} An analogous definition for tori using $\mathbf{A}^1 - \{0\}$ recovers
the usual notion of anisotropicity:  if $F$ is any field (possibly of characteristic 0)
and $T$ is an $F$-torus, then the condition $T(F[x,1/x]) = T(F)$ 
(i.e., the constancy of any $F$-scheme map $\Gm \rightarrow T$,
or equivalently the triviality of any map of pointed $F$-schemes $(\Gm,1) \rightarrow (T,1)$)
characterizes $F$-anisotropicity of $T$.   

Indeed, $F$-anisotropicity is
equivalent to the vanishing of 
$\Hom_{F\mbox{-}{\rm{gp}}}(\Gm, T)$, so we just need to check that  in general 
a map of pointed $F$-schemes $(\Gm,1) \rightarrow (T,1)$ is a homomorphism. 
By extending scalars we may assume $F = \overline{F}$, so $T$ is a power
of $\Gm$, and this reduces us to the case $T = \Gm$.  An endomorphism
of the pointed $F$-scheme $(\Gm,1)$ is the ``same'' as an element $u \in F[x,1/x]^{\times}$
satisfying $u(1) = 1$, and such units are precisely $u = x^n$ for $n \in \Z$.
\end{remark}

The main reason that we go beyond the category of $k$-groups in 
Definition \ref{IV.4.1}
is due to the intervention of a non-homomorphic conjugation morphism
$\varphi'$ that arises in the proof of Proposition \ref{IV.4.1.4} below.  The interested reader can easily check
that all appearances of maps from $\mathbf{A}^1$ in \S\ref{splitgp}--\S\ref{wsec}
can be replaced with homomorphisms from $\Add$ without affecting the proofs there.

\begin{remark}\label{remwound} 
The definition of 
``wound'' makes sense in characteristic 0,
where it is only satisfied by $U = 1$
(since a nontrivial smooth connected unipotent group in characteristic 0 contains
$\Add$ as a subgroup over the ground field). Thus, although 
we only work with ground fields of positive characteristic, it is
convenient in practice (for handling some trivialities) to make the convention that ``wound'' means ``trivial''
for smooth connected unipotent groups in characteristic 0. 
\end{remark}

 Whereas anisotropicity for a torus over a field is insensitive
to purely inseparable extension of the ground field but is often lost under a separable algebraic extension
of the ground field, the $k$-wound property 
%will behave
behaves in the opposite manner: we will prove that it is insensitive to
a separable extension on $k$ (such as scalar extension from a global
field to a completion), but it is often lost under a purely inseparable extension on $k$.

\begin{example}\label{woundex}
Assume $k$ is imperfect and choose $a \in k - k^p$.  The $k$-group
$U = \{y^p = x - a x^p\}$ becomes isomorphic to $\Add$ over the  purely inseparable extension $k(a^{1/p})$ but by 
Example \ref{nonsplit} it is $k$-wound.
Observe that the isogeny $y:U \rightarrow \Add$ is \'etale, so
 applying an \'etale isogeny can destroy the wound property.  (Although $y$ is \'etale, its extension to
 a degree-$p$ finite flat covering $\overline{U} \rightarrow \mathbf{P}^1_k$ between regular compactifications
 is not \'etale: explicitly, at the point at infinity the ramification index is 1 but the residue field extension is $k(a^{1/p})/k$.) Hence,
for problems involving wound unipotent groups one must be more attentive to the use of isogenies
than is usually necessary when working with tori.

Note that the wound $k$-group $U$ is a {\em $k$-subgroup} of the $k$-split group $\Add^2$.  
In the opposite direction, there also exist nontrivial $k$-split {\em quotients} of $k$-wound groups
modulo {\em smooth connected} $k$-subgroups.  For instance, in 
\cite[Ch.\:V,\:3.5]{oesterle} there is an example over any imperfect
field $k$ of a 2-dimensional $k$-wound  smooth connected $p$-torsion commutative
affine group $G$ admitting a 1-dimensional (necessarily $k$-wound)
smooth connected $k$-subgroup $G'$ such that $G/G' \simeq \Add$ as $k$-groups.
\end{example}

\begin{example}\label{pwoundex}
Assume $k$ is infinite.  By Corollary \ref{III.3.3.5}, smooth
$p$-torsion commutative affine $k$-groups $G$ are precisely the zero schemes
of separable nonzero $p$-polynomials $f$ over $k$.  Since $G$ is connected if and only if it
is geometrically irreducible (as for any $k$-group scheme of finite type), we see that $G$ is connected
if and only if $f$ is irreducible over $k$, as well as if and only if $f$ is absolutely irreducible over $k$. 
Assume $G$ is connected. 

If the principal part $f_{\rm{prin}}$ of $f$ has no zero on $k^n - \{0\}$ then
by Lemma \ref{III.3.3.6} it follows that $G$ is $k$-wound.   The converse is false, as we
saw following the statement of Lemma \ref{III.3.3.6}.  However, if $f_{\rm{prin}}$ has
a zero on $k^n - \{0\}$ then the calculation in the proof of $(2) \Rightarrow (3)$
in Lemma \ref{III.3.3.6} (taking $h$ to be the identity map of $\Add^n$) shows that
we can find a $p$-polynomial $F \in k[x_1,\dots,x_n]$ having zero scheme $k$-isomorphic to $G$ as a $k$-group
(so $F$ is absolutely irreducible over $k$) with the sum of the degrees of the monomials appearing in
$F_{\rm{prin}}$ strictly less than the corresponding sum for $f_{\rm{prin}}$.  Continuing in this way,
we eventually arrive at a choice of $f$ having zero scheme $G$ (as a $k$-group)
such that $f_{\rm{prin}}$ has no zeros on $k^n - \{0\}$.  In this sense, the zero schemes of
absolutely irreducible $p$-polynomials $f$ over $k$ for which
$f_{\rm{prin}}$ has no nontrivial $k$-rational zero are precisely the $p$-torsion commutative $k$-wound
%groups
smooth connected unipotent $k$-groups 
 (up to $k$-isomorphism). 
\end{example}

\begin{theorem}\label{III.3.3.11}
Every smooth connected $p$-torsion commutative affine $k$-group $U$ is a direct product $U=V\times W$ of a 
vector group $V$ and a smooth connected unipotent $k$-group $W$ such that $W_{k_s}$ is $k_s$-wound.  
In this decomposition, the subgroup $V$ is uniquely determined: $V_{k_s}$
is generated by the images of $k_s$-scheme 
morphisms $\varphi:\A^1_{k_s}\to U_{k_s}$ passing through the identity.
\end{theorem}

\begin{proof} By Galois descent, there is a unique smooth connected
$k$-subgroup $V$ of $U$ such that $V_{k_s}$ is 
generated by the images of $k_s$-scheme morphisms $\varphi:\A^1_{k_s}\to U_{k_s}$ 
that pass through the identity.   By Lemma \ref{embedvec}, 
we can identify $U$ with a $k$-subgroup of a vector group over $k$.
Thus, by Proposition \ref{III.3.3.8}, $V$ is a vector group over $k$ and
(by Corollary \ref{III.3.3.9}) we have $U=V\times W$ as $k$-groups for some $k$-subgroup $W$ of $U$.  
Since $U$ is a smooth connected unipotent $k$-group, so is its direct factor $W$. 
Clearly, $W_{k_s}$ is $k_s$-wound (due to the definition of $V$). 

Now we prove that $V$ in this decomposition is unique.  Consider
any decomposition of $k$-groups 
$U=V'\times W'$, where $V'$ is a vector group over $k$ and $W'$ is a 
smooth connected unipotent $k$-subgroup of $U$ such that
$W'_{k_s}$ is $k_s$-wound.  The image of any $k_s$-scheme morphism $\varphi:\A^1_{k_s}\to U_{k_s}$ 
passing through the identity is contained in $V'_{k_s}$ because
otherwise the composite of $\varphi$ and the canonical projection $U_{k_s} \to W'_{k_s}$ 
would be a non-constant $k_s$-scheme morphism from $\A^1_{k_s}$ to $W'_{k_s}$ (contradicting
that $W'_{k_s}$ is assumed to be $k_s$-wound).  Hence, $V  \subset V'$, so
$V' = V \times V'_1$ with $V'_1$ the image of the vector group $V'$ under the projection
$U \twoheadrightarrow W$.  Since $W_{k_s}$ is $k_s$-wound and $V'$ is a vector group,
$V'_1 = 0$.  That is, $V' = V$. 
\end{proof}

In Theorem \ref{III.3.3.11}, the group $W$ as an abstract $k$-group is unique up to isomorphism,
since it is identified with the quotient $U/V$ modulo the uniquely determined $k$-subgroup $V$.
However, the decomposition of $U$ as $V \times W$ is not unique when $V, W \ne 0$.  That is, there may be more
than one $k$-homomorphic section to $U \rightarrow U/V = W$, or in other words
$\Hom_k(W,V)$ may be nontrivial.  For example, over an imperfect field consider $U = \Add^2 \times \mathbf{U}$
where $\mathbf{U}$ is as in Example \ref{nonsplit}.
Clearly $\Hom_k(\mathbf{U},\Add^2)$ is nontrivial.

\begin{corollary}\label{III.3.3.12}
A smooth connected $p$-torsion commutative affine $k$-group $U$ is $k$-wound if and only if $U_{k_s}$
is $k_s$-wound, and also if and only if there are no nontrivial $k$-homomorphisms
$\Add \rightarrow U$.  The $k$-group $U$ is a vector group over $k$
if and only if $U_{k_s}$ is a vector group over $k_s$.  
\end{corollary}

\begin{proof} This is immediate from Theorem \ref{III.3.3.11}. 
\end{proof}

\begin{corollary}\label{III.3.3.14}
If $k$ is perfect then a smooth connected $p$-torsion commutative affine $k$-group is a vector group.
\end{corollary}

\begin{proof} By Corollary \ref{III.3.3.12}, we may assume that $k$ is algebraically closed.  
This case is part of Lemma \ref{embedvec}.
\end{proof}

To get results on $k$-wound groups beyond the commutative $p$-torsion case, 
we need to study smooth connected $p$-torsion central $k$-subgroups 
in a general smooth connected unipotent $k$-group $U$.    This is taken up in the next section.
%% new sentence below.
We end this section with some examples.

\begin{example}\label{predex} Let $k$ be a field and let
$G$ be a commutative smooth connected affine $k$-group containing
no nontrivial unipotent smooth connected $k$-subgroup.  
The commutativity ensures that there exists a unipotent smooth connected
$k$-subgroup $\mathscr{R}_{u,k}(G)$ in $G$ containing all other such $k$-subgroups,
and by Galois descent $\mathscr{R}_{u,k}(G)_{k_s} = \mathscr{R}_{u,k_s}(G_{k_s})$.  
Assume $\mathscr{R}_{u,k}(G) = 1$.  
(By 
the argument near the start of the Introduction, 
such a $G$ with $\mathscr{R}_u(G_{\overline{k}}) \ne 1$
is ${\rm{R}}_{k'/k}(T')$ for any nontrivial
purely inseparable finite extension $k'/k$ and a nontrivial $k'$-torus $T'$.)  

For 
the maximal $k$-torus $T$ in $G$, consider
the smooth connected commutative unipotent quotient $U = G/T$.  
We claim that $U$ is $k$-wound.  Since $\mathscr{R}_{u,k_s}(G_{k_s}) = \mathscr{R}_{u,k}(G)_{k_s} = 1$,
 we may assume
$k = k_s$, so $T$ is $k$-split.  By definition, we need to prove that any map of $k$-schemes
$f:\mathbf{A}^1_k \rightarrow U$ is constant.

Consider the pullback $G \times_U \mathbf{A}^1_k$.  This is a $T$-torsor
over $\mathbf{A}^1_k$, so it is trivial since $T$ is split and
${\rm{Pic}}(\mathbf{A}^1_k) = 1$.  A choice of splitting defines a $k$-scheme morphism
$\widetilde{f}:\mathbf{A}^1_k \rightarrow G$ over $f$, so it suffices to prove that 
$\widetilde{f}$ is constant.  Using a translation, we may assume $\widetilde{f}(0) = 1$.
We claim that for any smooth connected commutative $k$-group $C$
and any $k$-scheme morphism $h:\mathbf{A}^1_k \rightarrow C$ satisfying
$h(0) = 1$, the smooth connected $k$-subgroup of $C$ generated by the image of $h$ is unipotent.
Applying this to $G$ would then force
$\widetilde{f} = 1$ since $\mathscr{R}_{u,k}(G) = 1$, so we would be done. 

To prove our claim concerning $C$ we may assume 
$k = \overline{k}$,
so $C$ is a direct product of a torus and a unipotent group.  Using projections to factors, it
suffices to treat the case $C = \Gm$. In this case $h$ is a nowhere-vanishing polynomial
in one variable with value 1 at the origin, so $h = 1$.
\end{example}

\begin{example} 
Here is an example (due to Gabber) of a 2-dimensional {\em non-commutative} wound smooth
connected unipotent group $U$ over an arbitrary imperfect field $k$ of characteristic $p > 0$.
Choose $a \in k - k^p$, and consider the smooth connected $k$-subgroups  of $\mathbf{G}_{\rm{a}}^2$
defined by 
$$G = \{x = x^{p^2} + a y^{p^2}\},\,\,\,
C^{\pm} = \{x = \pm (x^p + a y^p)\}.$$
Their closures in $\mathbf{P}^2_k$ are regular with 
a unique point at infinity, and this point is not $k$-rational, so these groups are wound.
We will construct a non-commutative central extension $U$ of $G$ by $C^{-}$,
so $U$ must be a $k$-wound smooth connected unipotent $k$-group.
(The construction will work ``universally'' over the polynomial
ring $\mathbf{F}_p[a]$, yielding the desired $k$-group via base change.)

Define the $k$-morphism $f:G \rightarrow C^{+}$ by $(x,y) \mapsto (x^{p+1},xy^p)$
and consider the symmetric bi-additive 2-coboundary $b = -{\rm{d}}f:G \times G \rightarrow C^{+}$
defined by 
$$b(g,g') = f(g + g') - f(g) - f(g') = (x{x'}^p + x^p x', x {y'}^p + x' y^p)$$
for points $g = (x,y)$ and $g' = (x',y')$ of $G$.  The related map $b^{-}:G \times G \rightarrow C^{-}$ defined by
$$b^{-}((x,y),(x',y')) = (x{x'}^p - x^p x', x {y'}^p - x' y^p)$$
is easily checked to be an alternating bi-additive 2-cocycle, so if $p \ne 2$ then $b^{-}$ is {\em not} symmetric.
Thus, if $p \ne 2$ then the associated $k$-group $U$ with underlying scheme $C^{-} \times G$ and composition law
$$(c,g)(c',g') = (c + c' + b^{-}(g,g'), g + g')$$
is a non-commutative central extension of $G$ by $C^{-}$
(with identity $(0,0)$ and inversion $-(c,g) = (-c,-g)$).

To handle the case $p = 2$, we consider a variant
on this construction.  For any $p$ and $\zeta \in \mathbf{F}_{p^2} - \mathbf{F}_p$ consider
the bi-additive map $b_{\zeta}:G \times G \rightarrow C^{+}$ over $\mathbf{F}_{p^2}[a]$
defined by $b_{\zeta}(g,g') = b(g,\zeta g') = b(\zeta^p g, g')$.  This is easily seen to be a 2-cocycle
that is not symmetric, so it defines a
non-commutative central extension $U_{\zeta}$ of $G$ by $C^{+}$ over $\mathbf{F}_{p^2}[a]$
($U_{\zeta} = C^{+}  \times G$ as $\mathbf{F}_{p^2}[a]$-schemes, equipped with the composition law
$(c,g)(c',g') = (c+c' + b_{\zeta}(g,g'),g+g')$, identity $(0,0)$, and inversion
$-(c,g) = (-c - b_{\zeta}(g,-g),-g)$).  
Taking $p = 2$, so $C^{+} = C^{-}$ and each $\zeta$ is a primitive cube root of unity, we have $\zeta^{-1} = \zeta + 1$
and $b_{\zeta+1} = b_{\zeta} + b = b_{\zeta} - {\rm{d}}f$, so for each $\zeta$ we obtain
an isomorphism of central extensions $U_{\zeta} \simeq U_{\zeta+1}$ 
via $(c,g) \mapsto (c + f(g),g)$.  Letting
$\sigma$ be the nontrivial automorphism of $\mathbf{F}_4$, upon fixing $\zeta$ we have built an
$\mathbf{F}_4[a]$-isomorphism
$[\sigma]:U_{\zeta} \simeq U_{\zeta+1} = \sigma^{\ast}(U_{\zeta})$
corresponding to the automorphism $(c,g) \mapsto (c+f(g),g)$ of $C^{-} \times G$.
%By inspection
By inspection,
 the automorphism $\sigma^{\ast}([\sigma]) \circ [\sigma]$ of $U_{\zeta}$ is the identity 
 %map, so,
 map.   Thus,
$[\sigma]$ defines a descent datum on the central extension $U_{\zeta}$ relative
to the quadratic Galois covering $\Spec(\mathbf{F}_4[a]) \rightarrow \Spec(\mathbf{F}_2[a])$.
The descent is a non-commutative central extension of $G$ by $C^{-}$ over $\mathbf{F}_2[a]$, 
so it yields the desired $k$-group by base change.
\end{example}





\section{The cc$kp$-kernel}

In a smooth connected unipotent $k$-group $U$,
any two smooth connected $p$-torsion central $k$-subgroups generate
a third such subgroup.  Hence, the following definition makes sense.  

\begin{definition}\label{IV.4.1.3}
The maximal smooth connected $p$-torsion central $k$-subgroup of $U$ is the {\em cc$kp$-kernel}.
\end{definition}

Note that if $U \ne 1$ then its cc$kp$-kernel is nontrivial, since the latter 
contains the cc$kp$-kernel of the last nontrivial term of the descending central series of $U$.  
By Galois descent and specialization (as in the proof of \cite[1.1.9(1)]{pred}), 
the formation of the cc$kp$-kernel commutes with any separable extension on $k$.
However, its formation generally does {\em not} commute with purely inseparable
extension on $k$; see Exercise U.10(ii).

\begin{proposition}\label{IV.4.1.4}
Let $U$ be a smooth connected unipotent $k$-group, and let $k'/k$ be a separable
extension.  Let $F$ denote the cc$kp$-kernel of $U$.
Then $U$ is $k$-wound if and only if $U_{k'}$ is $k'$-wound, 
and the quotient $U/F$ is $k$-wound whenever $U$ is $k$-wound.  Also,
the following conditions are equivalent:
\begin{enumerate}
\item $U$ is $k$-wound, 
\item $U$ does not have a central $k$-subgroup $k$-isomorphic to $\Add$, 
\item the cc$kp$-kernel $F$ of $U$ is $k$-wound.
\end{enumerate}
\end{proposition}

This proposition implies that $U$ is $k$-wound if and only if 
$U$ admits no nontrivial $k$-homomorphism from $\Add$.  Such a characterization of the $k$-wound property
is analogous to the characterization of anisotropic tori over a field in terms of homomorphisms from $\Gm$
over the ground field. 

\begin{proof} Obviously $(1)\Rightarrow (2)$.  By Theorem \ref{III.3.3.11}, $(2)$ and $(3)$ are equivalent.  
Also, by specialization (as in the proof
of \cite[1.1.9(1)]{pred}), if $U_K$ is not $K$-wound for
some separable extension $K/k$ then the same holds with $K/k$ taken to be some finite separable
extension.  Thus, to prove the equivalence of (1), (2),  and (3) and the fact that $U_{k'}$ is $k'$-wound
whenever $U$ is $k$-wound, it suffices to show that if $U_{k_s}$ is not $k_s$-wound 
then the cc$kp$-kernel $F$ of $U$ is not $k$-wound.  

Let $\varphi:\A^1_{k_s}\to U_{k_s}$ be a non-constant $k_s$-scheme morphism.  
Composing with a $U(k_s)$-translation if necesary, we may assume $\varphi(0) = 1$. 
We may choose such a $\varphi$ 
so that $\varphi(\A^1_{k_s})$ is central. Indeed, suppose $\varphi(\A^1_{k_s})$ is non-central,
so $U$ is not commutative and there exists $g\in U(k_s)$ 
not centralizing $\varphi(\A^1_{k_s})$.  The $k_s$-scheme morphism $\varphi':\A^1_{k_s}\to 
U_{k_s}$ defined by $\varphi'(x)=g^{-1}\varphi(x)^{-1}g\varphi(x)$ (which is
generally not a homomorphism even when $\varphi$ is a homomorphism) carries $0$ to $1$, so it is
then non-constant, and its image lies in 
derived group $\mathscr{D}(U_{k_s}) = \mathscr{D}(U)_{k_s}$. 
The $k$-subgroup $\mathscr{D}(U)$ has smaller dimension than $U$ and
is nontrivial since
the smooth connected $k$-group $U$ is not commutative.   Hence, 
by iteration with the descending central
series of $U$, 
the required non-constant $\varphi$ with $\varphi(\A^1_{k_s})$ central is eventually obtained.  We may also assume that $\varphi(\A^1_{k_s})$ is $p$-torsion 
by replacing the original $\varphi$ with $p^e \cdot \varphi$
for some $e\ge 0$.  
 
  The 
 nontrivial $k_s$-subgroup generated by 
 $\varphi(\A^1_{k_s})$ lies in the cc$k_sp$-kernel of $U_{k_s}$; i.e., it lies in $F_{k_s}$.  
 Thus $F_{k_s}$ is not $k_s$-wound, so by Corollary \ref{III.3.3.12} the $k$-group $F$ is not $k$-wound. 
 
It remains to show that if $U$ is $k$-wound then $U/F$ is $k$-wound.  For this we may, in view of 
the preceding conclusions, assume that $k=k_s$.  Suppose that $U$ is $k$-wound  and $U/F$ is not $k$-wound.  
Thus, there exists a central $k$-subgroup $A$ of $U/F$ that is $k$-isomorphic to $\Add$. 
Let $\pi$ denote the canonical homomorphism $U\to U/F$.  The $k$-subgroup scheme
 $\pi^{-1}(A)$ in $U$ is an extension of $A$ by $F$, so it is smooth, connected, and unipotent.  
 
 We claim that $\pi^{-1}(A)$ is central in $U$.  If not, 
  we get a non-constant $k$-scheme morphism $\varphi:\A^1_k \to F$ (contradicting that $U$ is $k$-wound) as follows.
 Choose $g\in U(k)$ not centralizing 
 $\pi^{-1}(A)$ (recall $k = k_s$), identify $\Add$ with $A=\pi^{-1}(A)/F$,
  and define $\varphi:\pi^{-1}(A)/F\to F$ by $xF \mapsto gxg^{-1}x^{-1}$.  
  Thus, $\pi^{-1}(A)$ is central in $U$.  Similarly, $\pi^{-1}(A)$ is $p$-torsion 
  because otherwise we would get a non-constant $k$-scheme morphism $\psi:\A^1_k \to F$ via $\psi(xF)=x^p$.  
We have shown that $\pi^{-1}(A)$ lies in the cc$kp$-kernel $F$ of $U$,
so the given inclusion $F \subset \pi^{-1}(A)$ is an equality.
Hence, $A = 1$, which is absurd since $A \simeq \Add$. 
\end{proof}

\begin{corollary}\label{cancckp}
Let $U$ be a $k$-wound smooth connected unipotent $k$-group.
Define the ascending chain of smooth connected normal $k$-subgroups $\{U_i\}_{i \ge 0}$ as follows:
$U_0 = 1$ and $U_{i+1}/U_i$ is the
cc$kp$-kernel of the $k$-wound group $U/U_i$ for all $i \ge 0$.   These subgroups are
stable under $k$-group automorphisms of $U$, their formation commutes
with any separable extension of $k$, and $U_i = U$ for sufficiently large $i$.

Moreover, if $H$ is a smooth $k$-group acting on $U$ then
$H$ carries each $U_i$ into itself.
\end{corollary}

\begin{proof}
Well-posedness of the definition (e.g., that $U/U_1$ is $k$-wound) 
and compatibility with separable extension on $k$ follow from Proposition \ref{IV.4.1.4}.
By dimension considerations, 
$U_i = U$ for sufficiently large $i$ since the cc$kp$-kernel of a nontrivial smooth connected
unipotent $k$-group is nontrivial. 

 Finally, if $H$ is a smooth $k$-group acting on
$U$ then we need to prove that $H$ carries each $U_i$ into itself.
For this we may extend scalars to $k_s$, so $k$ is separably closed.
Then the $H$-stability of $U_i$ is equivalent to the $H(k)$-stability of $U_i$,
and this latter property is a special case of each $U_i$ being stable under all $k$-automorphisms of $U$.  
\end{proof}

As an application of the structure of $k$-wound groups
 we can unify the definitions of ``wound'' for unipotent groups and ``anisotropic'' for
 tori (see Remark \ref{remtori}):
 
 \begin{corollary}\label{unify}
 A unipotent smooth connected $k$-group $U$ is $k$-wound if and only if
 $U(k[x,1/x]) = U(k)$.  More generally, if $h \in k[x]$ is nonzero and separable
 then $U$ is $k$-wound if and only if $U(k[x][1/h]) = U(k)$.
 \end{corollary}
 
 \begin{proof}
 The equality $U(k[x,1/h]) = U(k)$ clearly forces $U$ to be $k$-wound.
 For the converse,  suppose $U$ is $k$-wound, so $U_{k_s}$ is $k_s$-wound
 (Proposition \ref{IV.4.1.4}).  Thus, to prove that
 $U(k[x][1/h]) = U(k)$ we may replace $k$ with $k_s$ (by Galois
 descent). 
 Hence, now $h = c\prod (x - a_i)$ for $c \in k^{\times}$ and pairwise distinct $a_i \in k$. 
 For each $i$,  the $k$-wound property implies
 $U(k(\!(x-a_i)\!)) = U(k[\![x-a_i]\!])$ by \cite[V, \S8]{oesterle} (whose proof
 rests on the existence of a composition series for the $k$-wound $U$ with
successive quotients that are commutative
 $p$-torsion wound hypersurface groups; see Corollary \ref{cancckp},
 Proposition \ref{III.3.3.1}, and Example \ref{pwoundex}).  Writing $h = (x-a_i)q_i$, inside $k(\!(x-a_i)\!)$, we have
 $k[x][1/h] \cap k[\![x-a_i]\!] = k[x][1/q_i]$. Thus,  $U(k[x][1/h]) = \bigcap_i U(k[x][1/q_i]) = U(k[x])$ since
 $\gcd_i (q_i) = 1$, and $U(k[x]) = U(k)$ since $U$ is $k$-wound.
  \end{proof}
 
 \begin{remark}\label{remcompact}
 It is well-known that if $F$ is a 
 %local field 
 non-archimedean local field 
 and $T$ is an $F$-torus 
 then $T(F)$ is compact if and only if $T$ is $F$-anisotropic.  (To prove
compactness of $T(F)$ for $F$-anisotropic $T$, identify ${\rm{X}}(T_{F_s})$ with a quotient of a direct sum of copies
 of the regular representation of ${\rm{Gal}}(F'/F)$ over $\Z$ for a finite Galois
 splitting field $F'/F$ of $T$. This identifies $T$ with an $F$-subgroup of 
${\rm{R}}_{F'/F}(\Gm)^N$ for some $N \ge 1$.  By $F$-anisotropicity, $T$ lies
in $(T^1_{F'/F})^N$, where $T^1_{F'/F}$ is the $F$-torus 
$\ker({\rm{R}}_{F'/F}(\Gm) \rightarrow \Gm)$ of ``norm-1 units''.
Since $T^1_{F'/F}(F) = \calO_{F'}^{\times}$, we are done.) 
 
 There is a similar equivalence
 in the unipotent case, as follows.  We 
 restrict attention to unipotent smooth connected $U$ over a local function field $k$, since
 in characteristic 0 the split condition always holds for unipotent groups and hence
 compactness cannot hold when the unipotent group is nontrivial. 
 Over such $k$, the equivalence of $k$-woundness for $U$
 and compactness for $U(k)$
 is \cite[VI, \S1]{oesterle} (whose proof ultimately reduces to an explicit calculation
 with wound hypersurface groups over $k = \F_q(\!(t)\!)$, using the ``principal part'' criterion at the end of 
 Example \ref{pwoundex}).
  \end{remark}

\begin{remark}\label{unirat}
The separability condition on $h$
in Corollary \ref{unify} cannot be removed. 
For example, if $p = 2$ and $a \in k - k^2$ then the $k$-wound group
$U = \{y^2 = x - ax^2\}$ is a smooth plane conic with $U(k) \ne \emptyset$, so $U$ is $k$-rational.
Explicitly, $U \simeq \Spec k[t,1/(t^2-a)]$ via $t \mapsto (1/(t^2 - a), t/(t^2 - a))$.  
\end{remark}

We will now prove a structure theorem that is analogous to the unique presentation of
a torus over a field as an extension of an anisotropic torus by a split torus. 

\begin{theorem}\label{IV.4.2}
Let $U$ be a unipotent smooth connected 
$k$-group.  There exists a unique 
%smooth connected normal $k$-split 
$k$-split smooth connected normal 
$k$-subgroup $U_{\rm{split}}
\subset U$ such that $U/U_{\rm{split}}$ is $k$-wound.

The subgroup $U_{\rm{split}}$ contains the image of every $k$-homomorphism from a $k$-split smooth connected unipotent $k$-group into $U$.  Also, the kernel of every $k$-homomorphism 
from $U$ into a $k$-wound smooth connected unipotent $k$-group contains $U_{\rm{split}}$,
and the formation of the $k$-subgroup $U_{\rm{split}}$ is compatible
with any separable extension of  $k$. 
\end{theorem}

\begin{proof} The proof is by induction on $\dim U$.  If $U$ is $k$-wound
then $U_{\rm{split}} :=\{1\}$ satisfies the requirements and is unique as such.  
Assume that $U$ is not $k$-wound, and let $A$ be a smooth central $k$-subgroup isomorphic to $\Add$
(Proposition \ref{IV.4.1.4}).  Let $H = U/A$. By induction, there exists a 
smooth connected normal $k$-subgroup $H_{\rm{split}}$ in $H$ with the desired properties in relation to $H$
(in the role of $U$).  Let $U_{\rm{split}}$ be the corresponding subgroup of $U$ containing $A$.  It is $k$-split, and $U/U_{\rm{split}}\simeq H/H_{\rm{split}}$ is $k$-wound.  

Let $U'$ be a smooth connected unipotent $k$-group having a composition series 
$$U'=U'_0\supset U'_1\supset \cdots$$ with 
successive quotients $k$-isomorphic to $\Add$, and let $\varphi:U'\to U$ be a $k$-homomorphism.  There exists a minimal $i$ such that $\varphi(U'_i)\subset U_{\rm{split}}$.  If $i>0$ then there is induced a $k$-homomorphism 
$\Add \simeq U'_{i-1}/U'_i\to U/U_{\rm{split}}$ with nontrivial image. 
This contradicts that $U/U_{\rm{split}}$ is $k$-wound.  Thus, $i=0$; i.e., 
$\varphi(U')\subset U_{\rm{split}}$.  It follows in particular that $U_{\rm{split}}$ is unique. 
Also, for any $k$-homomorphism $\varphi:U\to U''$ into a $k$-wound smooth connected unipotent $k$-group $U''$
we have  $\varphi(U_{\rm{split}})\subset U''_{\rm{split}}=\{1\}$. This says that 
$\ker\varphi$ contains $U_{\rm{split}}$.  

The last assertion of the theorem
 follows from Proposition   \ref{IV.4.1.4}.  Indeed, if
 $k'/k$ is a separable extension and $U' := U_{k'}$ then
 $(U_{\rm{split}})_{k'} \subset U'_{\rm{split}}$ and the $k'$-split
 quotient $U'_{\rm{split}}/(U_{\rm{split}})_{k'}$ is a $k'$-subgroup of the $k'$-group $(U/U_{\rm{split}})_{k'}$
 that is $k'$-wound  (by Proposition \ref{IV.4.1.4}).  This forces $U'_{\rm{split}} = (U_{\rm{split}})_{k'}$.
 \end{proof}
 
 \begin{example}\label{ncsplit}
 An elementary non-commutative example of Theorem \ref{IV.4.2} over
 any imperfect field $k$ of characteristic $p > 0$ is obtained via a central pushout construction, as follows.
 Let $U_3 \subset \GL_3$ be the standard upper triangular unipotent subgroup. Its 
 scheme-theoretic center is the group $Z \simeq \Add$ consisting of points
 $$u(x) = \begin{pmatrix} 1 & 0 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$
   Viewing $U_3$ as a central
 extension of $\Add^2$ by $Z$, let $U$ be the pullback along the inclusion
$y:U' \hookrightarrow \Add^2$ where $U'$ is a 1-dimensional $k$-wound group as in 
 Example \ref{woundex}.  A straightforward calculation shows that $U$ is a non-commutative
2-dimensional smooth connected $k$-subgroup of $U_3$ that is neither split nor wound (since it contains a central $Z = \Add$
 and admits the wound quotient $U'$).  Thus, $Z = U_{\rm{split}}$ and the
 sequence $1 \rightarrow Z \rightarrow U \rightarrow U/Z \rightarrow 1$
 cannot split since $Z$ is the center of $U$ and $U/Z$ is commutative.
 \end{example}
%  
% \begin{example}\label{extits} To illustrate Theorem \ref{IV.4.2} in the commutative case, let $k''/k'/k$ be a tower of purely
% inseparable nontrivial finite extensions of fields of characteristic $p > 0$ with ${k'}^p \subset k$, and 
% $G = {\rm{R}}_{k''/k}(\Gm)/\Gm$.  This is a $k$-wound unipotent smooth connected $k$-group
%(Exercise U.6),  and it contains
%the wound subgroup $H = {\rm{R}}_{k'/k}(\Gm)/\Gm$
%such that $H_{k'}$ is $k'$-split (using the ring structure of $k' \otimes_k k'$, 
%since ${k'}^p \subset k$). 
%By the exactness properties of Weil restriction on smooth groups we have
%$$H/G \simeq {\rm{R}}_{k'/k}({\rm{R}}_{k''/k'}(\Gm)/\Gm),$$
%so $(H/G)_{k'}$ admits the $k'$-wound quotient ${\rm{R}}_{k''/k'}(\Gm)/\Gm$.
%This quotient map is the natural map $q:{\rm{R}}_{k'/k}(U')_{k'} \rightarrow U'$
%defined on $k'$-algebras $A'$ by $U'(k' \otimes_k A') \rightarrow U'(A')$
%with $U' = {\rm{R}}_{k''/k'}(\Gm)/\Gm$.  Such a map $q$ has $k'$-split
%kernel for {\em any} smooth connected $k'$-group $U'$ (why?).
%
%Thus, the commutative smooth connected
%$k'$-group $$\mathscr{U}' := (H/G)_{k'} = ({\rm{R}}_{k''/k}(\Gm)/{\rm{R}}_{k'/k}(\Gm))_{k'}$$
%admits the $k'$-wound group
%${\rm{R}}_{k''/k'}(\Gm)/\Gm \ne 1$ as a quotient modulo a {\em nontrivial} $k'$-subgroup that is
%$k'$-split.  Hence, this $k'$-subgroup must be $\mathscr{U}'_{\rm{split}}$.
%By repeated applications
%of Corollary \ref{III.3.3.9}, since $\mathscr{U}'_{\rm{split}}$ is $p$-torsion and commutative, 
%it is a vector group.  Likewise, if ${k''}^p \subset k'$
%then $\mathscr{U}'$ is $p$-torsion and so
%the vector group $\mathscr{U}'_{\rm{split}}$ splits
%off as a direct factor of $\mathscr{U}'$.  That is, 
%if ${k''}^p \subset k'$ then the maximal $k'$-wound quotient
%of $\mathscr{U}'$ is (non-canonically) a direct factor of $\mathscr{U}'$.
%Hence, the most interesting cases are when $k'' \not\subset {k'}^{1/p}$;
%see Exercise U.11.
%\end{example}
 
 \begin{corollary}\label{mapcrit} A unipotent smooth connected $k$-group $U$ is $k$-split
 if and only if $U \simeq \mathbf{A}^n_k$ as $k$-schemes for some $n \ge 0$.
 It is also equivalent for there to be a dominant $k$-morphism $V = \mathbf{A}^d_k - Z \rightarrow U$ for
 a generically smooth  closed subscheme $Z \subset \mathbf{A}^d_k$.
  \end{corollary}
  
  Before we prove this corollary, we make some observations.  The
  dominance condition on $\mathbf{A}^d_k - Z \rightarrow U$ forces
  $Z \ne \mathbf{A}^d_k$, and by Remark \ref{unirat} we cannot remove the generic smoothness condition on $Z$.  Also,
Corollary \ref{mapcrit} has no analogue for tori, since any torus $T$ over any field $F$ is unirational
(by using an isogeny-splitting of the inclusion of $F$-tori $T \hookrightarrow {\rm{R}}_{F'/F}(T_{F'})$
for a finite separable splitting field $F'/F$ of $T$). Finally, 
the proof of sufficiency below for the second criterion in Corollary \ref{mapcrit} uses Bertini's Theorem
in the affine setting over $k_s$ but the only $Z$ that we actually use in later
applications is a (possibly empty) union of hyperplane slices in distinct coordinate directions,
for which linear algebra works equally well in place of Bertini's Theorem.
 
 \begin{proof}
  First assume that $U$ is $k$-split, and let $n = \dim U$.  We seek to prove that 
 $U \simeq \mathbf{A}^n_k$ as $k$-schemes. The cases $n \le 1$
 are obvious, so we may assume $n > 1$.  Thus, there is a $k$-split smooth connected
 normal $k$-subgroup $U' \subset U$ such that $U/U' \simeq \Add$.
 By induction, $U' \simeq \mathbf{A}^{n-1}_k$ as $k$-schemes.  We claim that the 
 $U'$-torsor $U \rightarrow \Add = \mathbf{A}^1_k$ for the \'etale topology is trivial.  More generally,
 for any {\em affine} $k$-scheme $X$ the cohomology set
 ${\rm{H}}^1(X_{\et},U')$ classifying $U'$-torsors for the \'etale topology on $X$
 is trivial.  Indeed, using a composition series for $U'$ over $k$ reduces
 this to the case of ${\rm{H}}^1(X_{\et},\Add)$, and by \'stale descent theory for
 quasi-coherent sheaves this coincides with ${\rm{H}}^1(X_{\rm{Zar}},\calO) = 0$.
 We conclude that as $k$-schemes, $U \simeq U' \times (U/U') \simeq
 \mathbf{A}^n_k$, as desired.
 
 For the converse, suppose there is a dominant
$k$-morphism $f:V = \mathbf{A}^d_k - Z \rightarrow U$ for a generically smooth closed subscheme
$Z \subset \mathbf{A}^d_k$.   
 To prove that $U$ is $k$-split, we may replace $U$ with the $k$-wound quotient
$U/U_{\rm{split}}$ from Theorem \ref{IV.4.2} to reduce to the case that $U$ is $k$-wound.  In such cases
we seek to prove that $U = 1$, so it suffices to prove that the dominant $f$ is a constant map into $U(k)$. 
 It is harmless to extend scalars to $k_s$, so
$V(k)$ is Zariski-dense in $V$. 
Since $Z$ is generically smooth
and $Z \ne \mathbf{A}^d_k$, by Bertini's Theorem over $k$ there exists a dense open locus
$\Omega$ in the $2(d-1)$-dimensional quasi-projective variety $\mathbf{Gr}_d$ of affine lines in $\mathbf{A}^d_k$
such that the closed subscheme $Z_K \bigcap \ell$ in $\ell$ is 0-dimensional and $K$-smooth
for all $K/k$ and affine lines $\ell$ in $K^d$ corresponding to a point in
$\Omega(K)$.  (If $Z$ is a union of several affine hyperplanes then 
linear algebra gives the same conclusion, without using Bertini's Theorem.) Such a closed subscheme is
$K$-\'etale, so for each affine line $\ell \simeq \mathbf{A}^1_k$ 
corresponding to a point in $\Omega(k)$ the open locus $V \bigcap \ell$ in $\ell$ is the complement
of the zero locus on $\ell$ of a separable polynomial.  Hence,
by Corollary \ref{unify} and the $k$-wound hypothesis on $U$, $f$ has constant restriction to $V \bigcap \ell$
for all $\ell \in \Omega(k)$.   

To prove the constancy of $f$, it suffices to prove the constancy of
$f$ on $V'(k)$ for a dense open $V' \subset V$ (since $k = k_s$). 
The idea is that for a generic pair of distinct points $v$ and $v'$ in $V$,
the line $\ell$ joining them should correspond to a point in $\Omega$ and hence
the constancy of $f$ on $V \bigcap \ell$ forces $f(v) = f(v')$.  To make this idea
rigorous, consider the $2d$-dimensional variety $X = V \times V - \Delta$ of
ordered pairs of distinct points in $V$.  There is an evident morphism
$X \rightarrow \mathbf{Gr}_d$ assigning to any $(v,v') \in X$ the unique
line joining them, and all fibers are 2-dimensional, so for dimension reasons this map is dominant.
Hence, there is a dense open locus $X' \subset X$ that is carried into $\Omega$.
For all $(v,v') \in X'(k)$, the unique line $\ell \subset k^d$ passing through $v$ and $v'$
corresponds to a point in $\Omega(k)$, so $f$ is constant on $V \bigcap \ell$.
In particular, $f(v) = f(v')$.  The projection ${\rm{pr}}_1:X' \rightarrow V$ is dominant, so its image
contains a dense open subset of $V$.  We may choose $v_0 \in V(k)$ in this image, so 
the open subset $V' := X' \bigcap (\{v_0\} \times V)$ in $V$ (via ${\rm{pr}}_2$) is non-empty
and therefore dense. 
Clearly $f(v') = f(v_0)$ for all $v' \in V'(k)$.  
 \end{proof}
     
     \begin{remark}\label{splitrat} The above cohomological proof that $U \simeq \mathbf{A}^n_k$ as $k$-schemes
     for $k$-split unipotent smooth connected $k$-groups $U$ generalizes
     to show that any $k$-split solvable smooth connected affine $k$-group is $k$-isomorphic
     to $\mathbf{A}^{n,m}_k := \mathbf{A}^n_k \times (\mathbf{A}^1_k - \{0\})^m$ for some $n, m \ge 0$.
     (This result is due to Rosenlicht, who gave a non-cohomological proof; see 
     Lemma 2 to Theorem 2 in \cite{rosenlicht}.  A generalization 
     to homogeneous spaces under such groups is \cite[Thm.\,5]{rosenlicht}.)
     To carry out this generalization, first note that a composition series 
     expressing the $k$-split property reduces the problem  to proving
     that for a $k$-split solvable smooth connected $k$-group $G$, every
     $G$-torsor over $\Add$ or $\Gm$ for the \'etale topology is a trivial torsor; i.e., ${\rm{H}}^1((\Add)_{\et},G)$ and
     ${\rm{H}}^1((\Gm)_{\et},G)$ vanish.  
     
     As in the proof of Corollary \ref{mapcrit}, by using a composition series expressing the $k$-split property of
     $G$, the low-degree 6-term exact sequence
     in non-abelian cohomology associated to a short exact sequence
     of smooth affine group schemes reduces the vanishing assertion to the special cases
     $G = \Add$ and $\Gm$.  The case $G = \Add$ was addressed more generally
     in the proof of Corollary \ref{mapcrit}.  The case $G = \Gm$ follows from
     the general equality ${\rm{H}}^1(X_{\et},\Gm) = {\rm{Pic}}(X)$ (via descent theory for line bundles)
     and the PID property for $k[x]$ and $k[x,1/x]$. 
               \end{remark}
     
        
   \begin{corollary}\label{dermap}
 If $G$ is a $k$-split solvable smooth connected affine $k$-group
 then $\mathscr{D}(G)$ is $k$-split.
 \end{corollary}
 
 \begin{proof}
 By the structure theory over $\overline{k}$, $\mathscr{D}(G)$ is unipotent.
 Hence, by Corollary \ref{mapcrit} it suffices
 to construct a dominant $k$-morphism $\mathbf{A}^n_k - Z \rightarrow \mathscr{D}(G)$
 for some $n \ge 1$ and some geometrically reduced closed subscheme $Z \subset \mathbf{A}^n_k$.
 Since the product of several varieties 
 $\mathbf{A}^{n_i}_k - Z_i$ with generically smooth $Z_i$ 
 has the form $\mathbf{A}^{\sum n_i}_k - Z$ for a generically smooth closed subscheme $Z$, and 
 the geometric points of $\mathscr{D}(G)$ can be expressed as a product
 of a universally bounded number of commutators (depending on $G$), by considering such a product
 morphism for a sufficiently large set of commutators we are reduced to constructing a dominant $k$-morphism
 $\mathbf{A}^N_k - Z \rightarrow G$ for some $N \ge 1$ and generically smooth $Z$. 
 By Remark \ref{splitrat} there is a $k$-scheme isomorphism 
 $\mathbf{A}^{n,m}_k \simeq G$, so we are done. 
 \end{proof}
   
           
Let $G$ be a smooth connected affine $k$-group.
 The {\em $k$-unipotent radical} $\mathscr{R}_{u,k}(G)$ is 
 the maximal normal unipotent smooth connected $k$-subgroup of $G$, and the {\em $k$-split
 unipotent radical} $\mathscr{R}_{us,k}(G)$ is the maximal normal $k$-split unipotent smooth
 connected $k$-subgroup of $G$.  
 For any extension field $K/k$ clearly $\mathscr{R}_{u,k}(G)_K \subset \mathscr{R}_{u,K}(G_K)$
 inside $G_K$. This inclusion is an equality when $K/k$ is separable \cite[1.1.9(1)]{pred}, but generally
 not otherwise (e.g., for a nontrivial purely inseparable extension
 $k'/k$ of degree $p = {\rm{char}}(k)$ and $G$ equal to the Weil restriction
 ${\rm{R}}_{k'/k}(\Gm)$ we have $\mathscr{R}_{u,k}(G) = 1$
 but $\mathscr{R}_{u,k'}(G_{k'}) = \Add^{p-1}$;  see \cite[1.1.3, 1.6.3]{pred}).
 
 \begin{corollary}\label{uscor} 
 For 
 any 
 smooth connected affine 
 $k$-group 
  $G$, 
   $\mathscr{R}_{us,k}(G) = \mathscr{R}_{u,k}(G)_{\rm{split}}$.
 In particular, $\mathscr{R}_{u,k}(G)/\mathscr{R}_{us,k}(G)$ is $k$-wound, 
 and the formation of $\mathscr{R}_{us,k}(G)$ commutes with separable extension on $k$.
 \end{corollary}
 
 \begin{proof}
 By Galois descent, $\mathscr{R}_{us,k_s}(G_{k_s})$ descends to
 a smooth connected unipotent normal $k$-subgroup of $G$. 
This
 descent is $k$-split, since the $k$-split property of smooth connected unipotent $k$-groups is
 insensitive to separable extension on $k$ (due to Theorem \ref{IV.4.2}).  Thus,  the descent
 is contained in $\mathscr{R}_{us,k}(G)$, so 
 the inclusion $\mathscr{R}_{us,k}(G)_{k_s} \subset \mathscr{R}_{us,k_s}(G_{k_s})$ is an equality.
 In other words, the formation of $\mathscr{R}_{us,k}(G)$ is compatible with separable algebraic extension on
 $k$.  Hence, to prove the compatibility with general separable extension on $k$ and the agreement with
 the maximal $k$-split smooth connected $k$-subgroup of $\mathscr{R}_{u,k}(G)$, we may
 assume
   $k = k_s$.   
  But 
 $\mathscr{R}_{u,k}(G)_{\rm{split}}$ is a characteristic $k$-subgroup of $G$, so it is normal
 due to the Zariski-density of $G(k)$ in $G$ when $k = k_s$.  This proves
 that $\mathscr{R}_{u,k}(G)_{\rm{split}} \subset
 \mathscr{R}_{us,k}(G)$, so equality holds. 
 
 The compatibility of the formation of $\mathscr{R}_{us,k}(G)$ with respect
 to separable extension on $k$ now follows
 from 
 such a 
 compatibility for 
  the formation of
 $U_{\rm{split}}$ in Theorem \ref{IV.4.2}
   and the formation of $\mathscr{R}_{u,k}(G)$
 \cite[1.1.9(1)]{pred}. 
  \end{proof}
  

\section{Torus actions on unipotent groups}\label{torsec}


Consider the action of a $k$-torus $T$ on a smooth connected
unipotent $k$-group $U$.  This induces a linear representation of $T$ on
$\Lie(U)$, so if $T$ is $k$-split then we get a weight space decomposition of $\Lie(U)$.  
If $U$ is a vector group then it is natural to wonder if this
decomposition of $\Lie(U)$ can be lifted to the group $U$.  When $\dim U > 1$, the $T$-action may not
respect an initial choice of linear structure on $U$ (in the sense of Definition \ref{linstr}) 
since ${\rm{char}}(k) = p > 0$, 
so we first seek a $T$-equivariant linear structure. 

For example, if $U = \Add^2$ with its usual linear structure 
and $T = \Gm$ with the action $t.(x,y) = (tx, (t^p-t) x^p + t y)$ then the $T$-action is not
linear and the action on $\Lie(U) = k^2$ has
the single weight given by the identity character of $T$.  
But note that if we transport the $T$-action by the additive automorphism
$(x,y) \mapsto (x, y + x^p)$ of $U$ then the action becomes $t.(x,y) = (tx, ty)$, which is linear.

Tits proved rather generally that if a $k$-split $T$ acts on $U$ with only nontrivial 
weights on $\Lie(U)$, then there are nontrivial constraints on the possibilities for $U$ as a $k$-group 
and that (after passing to a suitable characteristic composition series
for $U$) the action can always be described in terms of linear representations of $T$. 
To explain his results in this direction, we begin with the following 
result 
that generalizes Lemma \ref{embedvec} by incorporating a torus action.

\begin{proposition}\label{IV.5.2.1}
Let 
$U$ be a smooth $p$-torsion commutative affine $k$-group equipped with an action 
by an affine $k$-group scheme $T$
of finite type. There exists a linear representation of $T$ on a finite dimensional $k$-vector space 
$V$ and a $T$-equivariant
isomorphism of $U$ onto a $k$-subgroup of $V$.
\end{proposition}  

\begin{proof}
Let $\mathbf{Hom}(U,\mathbf{G}_{\rm{a}})$ be the covariant functor assigning
to any $k$-algebra $R$ the $R$-module ${\rm{Hom}}_R(U_R,\mathbf{G}_{\rm{a}})$ of $R$-group morphisms
$\phi:U_R \rightarrow \mathbf{G}_{\rm{a}}$ (with $R$-module
structure defined via the $R$-linear structure on the $R$-group $\mathbf{G}_{\rm{a}}$).
There is a natural $R$-linear injection
$\mathbf{Hom}(U,\mathbf{G}_{\rm{a}})(R) \hookrightarrow
R[U_R] = R \otimes_k k[U]$ defined by $\phi \mapsto \phi^{\ast}(x)$
(where $x$ is the standard coordinate on $\mathbf{G}_{\rm{a}}$), and its
image is the $R$-submodule of ``group-like'' elements: those $f$ satisfying
$m_R^{\ast}(f) = f \otimes 1 + 1 \otimes f$ (where $m:U \times U \rightarrow U$ is
the group law).  This is an $R$-linear condition on $f$ and is functorial in
$R$, so by $k$-flatness the $R$-module
of group-like elements over $R$ is $J_R$ where $J \subset k[U]$ is the $k$-subspace
of group-like elements over $k$.   In particular,
the natural map $R \otimes_k {\rm{Hom}}(U,\mathbf{G}_{\rm{a}}) \rightarrow {\rm{Hom}}_R(U_R,\mathbf{G}_{\rm{a}})$
is an isomorphism.

The (left) $T$-action on $U$ defines a left $T$-action on $\mathbf{Hom}(U,\mathbf{G}_{\rm{a}})$
(via $(t.\phi)(u) = \phi(t^{-1}.u)$) making the
$k$-linear inclusion ${\rm{Hom}}(U,\mathbf{G}_{\rm{a}}) \hookrightarrow k[U]$
a $T$-equivariant map.  Thus, ${\rm{Hom}}(U,\mathbf{G}_{\rm{a}})$ is the directed
union of $T$-stable finite-dimensional $k$-subspaces, due to the same property for
$k[U]$ \cite[1.9--1.10]{borelag}. 
By Lemma \ref{embedvec} there is a $k$-subgroup inclusion
$j:U \hookrightarrow \mathbf{G}_{\rm{a}}^n$ for some $n \geqslant 1$.
Let $W \subset {\rm{Hom}}(U,\mathbf{G}_{\rm{a}})$ be a $T$-stable finite-dimensional $k$-subspace
containing $j^{\ast}(x_1),\dots, j^{\ast}(x_n)$. 
The canonical map $U \rightarrow \underline{W^{\ast}} = {\rm{Spec}}({\rm{Sym}}(W))$ is
a $T$-equivariant closed immersion that is a $k$-homomorphism (since $W$
consists of group-like elements in $k[U]$ that generate $k[U]$ as a $k$-algebra).
\end{proof}


We now apply our work with wound groups to analyze the structure of smooth connected
unipotent $k$-groups
equipped with a sufficiently nontrivial action by a $k$-torus.  

\begin{proposition}\label{IV.5.2.2}
Let $T$, $U$, and $V$ be as in Proposition {\rm{\ref{IV.5.2.1}}}, with $T$ a $k$-torus, and let 
$V_0\times V'$ be the unique $T$-equivariant $k$-linear 
decomposition of $V$ with $V_0 = V^T$ $($so $V'$ 
is the span of the isotypic $k$-subspaces for the nontrivial irreducible representations of $T$ over $k$ 
that occur in $V$$)$. The product map $$\iota: (U\cap {\underline V}_0) \times (U \cap {\underline V}') 
\rightarrow U$$ is an isomorphism and there is a $T$-equivariant
$k$-linear decomposition $V' = V'_1\times V'_2$ of $V'$ and a $T$-equivariant $k$-automorphism 
$\alpha$ of the additive $k$-group $\underline{V}$ such that 
$$\alpha(U) = (\alpha(U)\cap {\underline V}_0)\times {\underline V}'_1.$$
In particular, if $V^T = 0$ then the $k$-group $U$ is a vector group
admitting a $T$-equivariant linear structure.
\end{proposition}

\begin{proof}
Clearly $\underline{V_0} = Z_{\underline{V}}(T)$ as $k$-subgroups of
$\underline{V}$, so $U_0 := U \cap {\underline V}_0$ is $Z_U(T)$.  This is smooth since $U$ is smooth.
We will first prove that $\iota$ is an isomprphism,
so  $U \cap {\underline V}'$ is smooth. 

Since the formation of $V'$ clearly commutes with
scalar extension on $k$,
to establish that $\iota$ is an isomorphism we may assume $k$ is algebraically closed.  
Choose $s \in T(k)$ such that for every weight $\chi$ of $T$ in $V'$, $\chi(s)\ne 1$. Consider the 
$k$-linear map $f:V\to V$ defined by $f(v) = s\cdot v-v$. It is obvious that $f$ maps $V$ onto $V'$ 
with $\ker f = Z_V(s) = V_0$  
and that the restriction of $f$ to $V'$ is a linear automorphism. The image $f(U)$ is a smooth 
$k$-subgroup of $\underline{V}'$,
and it lies in $U$ due to the $T$-stability of $U$ inside $\underline{V}$.  By definition, 
$\underline{V}'$ has a $T$-equivariant composition series whose successive quotients are 
1-dimensional
vector groups with a nontrivial $T$-action.  Hence, 
all $T$-stable $k$-subgroup schemes of ${\underline V}'$ are connected.  In particular, $f(U)$ is connected.

Since $U_0 \cap f(U) = 0$ (as $\underline{V_0} \cap {\underline V}' = 0$), 
under addition $U_0 \times f(U)$ is a $k$-subgroup of $U$.
Thus, $f:U \rightarrow f(U)$ is a map onto a $k$-subgroup of $U$ and the restriction of this map to 
$f(U)$ is therefore an endomorphsm $f(U) \rightarrow f(U)$  with trivial kernel.  But 
$f(U)$ is smooth and connected, so this endomorphism is an automorphism.  
In other words, $f:U \twoheadrightarrow f(U)$ is a projector up to an automorphism of $f(U)$.
Since $U \cap \ker f = U \cap {\underline V}_0 = U_0$, this  shows that the $k$-subgroup
 inclusion $U_0 \times f(U) \hookrightarrow U$ is
an isomorphism, so $f(U) = U \cap {\underline V}'$.  This completes the proof that $\iota$ is an isomorphism.


Let $U' = U\cap {\underline V}'$ and  define $V'_1 = {\rm{Lie}}(U')$. Then $V'_1$ is a $T$-stable $k$-linear subspace of $V'$. Complete reducibility of $k$-linear representations of $T$ provides a $T$-stable $k$-linear complement $V'_2$ of $V'_1$ in $V'$.   Using the decomposition $\underline{V}' = \underline{V'_1} \times \underline{V'_2}$,
the projection $U' \rightarrow \underline{V'_1}$ is 
an isomorphism on Lie algebras, so it is \'etale.  By $T$-equivariance, 
the finite \'etale kernel is $T$-stable and therefore centralized by the connected $T$.
But $Z_{\underline{V}'}(T) = 0$, so this kernel vanishes.  
%In other words,
Hence, 
$U' \rightarrow \underline{V'_1}$ is an isomorphism.   It follows
that the $k$-subgroup $U' \subset \underline{V}' = \underline{V'_1} \times \underline{V'_2}$
is the graph of a $T$-equivariant $k$-homomorphism $g:\underline{V'_1} \rightarrow \underline{V'_2}$. 
The $T$-equivariant $k$-automorphism $\alpha$ of $\underline{V}$ may be taken to be
the automorphism that is the identity on ${\underline V}_0$ and is the inverse of the map $(v_1, v_2) \mapsto 
(v_1, g(v_1) + v_2)$ on $\underline{V'_1} \times \underline{V'_2}$.
\end{proof}


\begin{theorem}\label{IV.5.2.4}
Let $T$ be a $k$-torus and $U$ a smooth $p$-torsion commutative affine $k$-group.  Suppose that there is 
given an action of $T$ on $U$ over $k$.  Then $U=U_0  \times U'$ with $U_0 = Z_U(T)$ and $U'$ a
$T$-stable $k$-subgroup that is a vector group admitting a linear structure  relative to which
$T$ acts linearly.
Moreover, $U'$ is uniquely determined and is functorial in $U$.
\end{theorem}

\begin{proof}
By Propositions \ref{IV.5.2.1} and \ref{IV.5.2.2} we get the existence of $U'$.  To prove
the uniqueness and functoriality of $U'$,  we may assume $k = k_s$. 
Under the decomposition of $U'$ into weight spaces relative to a $T$-equivariant
linear structure on $U'$, all $T$-weights must be nontrivial due to the definition of $U_0$.
Hence, the canonical map $T \times U \rightarrow U$ defined by $(t,u) \mapsto t.u - u$ has
image $U'$.  This proves the uniqueness and functoriality of 
$U'$.
\end{proof}

If $U$ in Theorem \ref{IV.5.2.4} 
is $k$-wound, then it must coincide with $U_0$ and so have trivial $T$-action.
This is a special case of the following general consequence of invariance of the wound property
with respect to separable extension of the ground field (Proposition \ref{IV.4.1.4}):


\begin{corollary}\label{IV.5.3.1} 
Let $T$ be a $k$-torus and $U$ a $k$-wound smooth connected unipotent $k$-group.  
The only $T$-action on $U$  is the trivial one.
\end{corollary}


\begin{proof}  Our aim is to prove that the $k$-subgroup scheme 
$Z_U(T)$ is equal to $U$.  For the $k$-group $G = U \rtimes T$,
we have that the torus centralizer $Z_G(T)$ is equal
to $Z_U(T) \rtimes T$.  But $Z_G(T)$ is smooth and connected, so
the same holds for $Z_U(T)$.  Since $Z_U(T)$ is a scheme-theoretic centralizer,
$\Lie(Z_U(T))$ is the $T$-centralizer in $\Lie(U)$.  Hence, to prove
that $Z_U(T) = U$ it suffices (by smoothness and connectedness
of $U$) to prove that $T$ acts trivially on $\Lie(U)$.  

By Proposition \ref{IV.4.1.4}, we may extend scalars to $k_s$, so $T$ is $k$-split. 
Consider the composition series $\{U_i\}$ from Corollary \ref{cancckp}. 
This is $T$-equivariant, and each $U_{i+1}/U_i$ is $k$-wound,
commutative, and $p$-torsion.  The Lie algebras ${\rm{Lie}}(U_i)$
provide a $T$-equivariant filtration on ${\rm{Lie}}(U)$ whose
successive quotients are the ${\rm{Lie}}(U_{i+1}/U_i)$.
By complete reducibility for the $T$-action on ${\rm{Lie}}(U)$, to
prove triviality of the action it suffices to treat the successive quotients
of a $T$-stable composition series of $k$-subspaces of ${\rm{Lie}}(U)$.
Hence, it suffices to treat each $U_{i+1}/U_i$ separately in place
of $U$, so we may assume that the $k$-wound $U$ is commutative
and $p$-torsion.
Applying the decomposition in Theorem \ref{IV.5.2.4}, we
have $U = Z_U(T) \times U'$ where
$U'$ is a vector group.  Since $U$ is wound, we conclude
that $U' = 1$, so the $T$-action on $U$ is trivial.
\end{proof}

%
%By Proposition \ref{IV.4.1.4}, we may extend scalars to $k_s$, so $T$ is $k$-split.  Assume
%that the $T$-action on ${\rm{Lie}}(U)$ is nontrivial, so the action on ${\rm{Lie}}(U)$
%by some $\Gm$ in $T$ is nontrivial.  We may replace $T$ by this $\Gm$, 
%so there is an action $\lambda:\Gm \times U \rightarrow U$ 
%that induces a nontrivial action on ${\rm{Lie}}(U)$. Composing the action with
%inversion on $\Gm$ allows us to arrange that there is a nonzero weight space
%in ${\rm{Lie}}(U)$ with a positive weight.  To ``exponentiate''
%this weight space to a smooth $k$-subgroup of $U$, we need to
%use the dynamic method from \cite[\S2.1]{pred},
%whose basic features we now summarize
%(with references for further details).
%
%The idea behind the dynamic method is to use the limiting behavior
%as ``$t \rightarrow 0$'' for a $\Gm$-action on a finitely presented affine group $H$
%over a ring $k$ to functorially construct interesting subgroups analogous
%to parabolic subgroups and their unipotent radicals in connected semisimple groups.
%More specifically (for our purposes), if $\mu:\Gm \rightarrow H$ is a 
%$k$-homomorphism then define the subfunctor 
%$\underline{P}_H(\mu) \subset H$ to have as its points in a $k$-algebra $R$
%those $h \in H(R)$ such that the orbit map of $R$-schemes
%$\Gm \rightarrow H_R$ defined by $t \mapsto t.h := \mu(t)h\mu(t)^{-1}$
%extends (necessarily uniquely) to an $R$-morphism $\mathbf{A}^1_R \rightarrow H_R$.
%For $h \in \underline{P}_H(\mu)(R)$, the image of $0$ in $H(R)$ is denoted  $\lim_{t \rightarrow 0} t.h$.
%Define the subfunctor $\underline{U}_H(\mu) \subset \underline{P}_H(\mu)$
%to consist of the points $h$ of $\underline{P}_H(\mu)$ such that
%$\lim_{t \rightarrow 0} t.h = 1$.
%For example, if $H = {\rm{SL}}_2$ and $\mu(t) = {\rm{diag}}(t,1/t)$ over a ring $k$ 
%then $t.(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}) = 
%(\begin{smallmatrix} a & t^2 b \\ t^{-2}c & d \end{smallmatrix})$, so
%$\underline{P}_H(\mu)$ and $\underline{U}_H(\mu)$
%are respectively represented by
%$B = \{(\begin{smallmatrix} a & b \\ 0 & 1/a \end{smallmatrix})\}$
%and $U = \{(\begin{smallmatrix} 1 & x \\ 0 & 1 \end{smallmatrix})\}$.  
%See \cite[Ex.\,2.1.1]{pred} for a generalization to $\GL_n$. 
%
%In general, by \cite[2.1.5]{pred},  $\underline{U}_H(\mu)$
%and $\underline{P}_H(\mu)$ are
%represented by finitely presented closed $k$-subgroups $U_H(\mu) \subset P_H(\mu) \subset H$.
%Moreover, by \cite[2.1.8]{pred}, 
%if $H$ is smooth then: $U_H(\mu)$ and $P_H(\mu)$ are smooth,
%$U_H(\mu)$ has unipotent connected
%fibers, and ${\rm{Lie}}(P_H(\mu))$
%(resp.\,${\rm{Lie}}(U_H(\mu))$) is the span in ${\rm{Lie}}(H)$
%of the weight spaces for the $\Gm$-weights that are $\ge 0$ (resp. $> 0$). 
%This formalism works equally well for 
%finitely presented affine $k$-groups $H'$ equipped with an ``abstract''
%$\Gm$-action, by applying the above considerations
%to the associated semi-direct product $H = H' \rtimes \Gm$
%against the given action (and taking $\mu:\Gm \rightarrow H$ to be the
%natural inclusion, against which the conjugation action recovers
%the given $\Gm$-action on $H'$; see \cite[2.1.11]{pred}). 
%In fact, $P_{H'}(\mu) = P_H(\mu) \cap H'$ and $U_{H'}(\mu) = U_H(\mu)$
%(and $P_H(\mu) = P_{H'}(\mu) \rtimes \Gm$).
%
%Now we apply the dynamic method in our initial setting
%over the field $k$, using the group $U$ equipped with its
%$\Gm$-action $\lambda$.  It is equivalent to use
%$G := U \rtimes \Gm$ equipped with the natural inclusion
%$\Gm \rightarrow G$.  We prefer to write $U_G(\lambda)$
%rather than $U_U(\lambda)$, so we express the conclusions in terms
%of $G$:  the 
%unipotent smooth connected $k$-subgroup $U_G(\lambda) \subset U$ has Lie algebra
%that is the span in ${\rm{Lie}}(U)$ of the positive weight spaces.  We have arranged
%that there exist nonzero positive weight spaces, so ${\rm{Lie}}(U_G(\lambda)) \ne 0$
%and hence $U_G(\lambda) \ne 1$. Pick $u \in U_G(\lambda)(k)$, and consider
%the orbit map $\Gm \rightarrow G$ defined by $t \mapsto t.u$.  By the functorial
%definition of $U_G(\lambda)$, this extends to a $k$-scheme map
%$\mathbf{A}^1_k \rightarrow U_G(\lambda) \subset U$ carrying $0$ to 1.  
%But $U$ is $k$-wound, so this map is constant.  But $1 \in \Gm$ is carried to $u$,
%so $u = 1$.  This shows that $U_G(\lambda)(k) = 1$, a contradiction.
%\end{proof}

%\begin{proposition}\label{B.4.5} Let $S$ be a split torus over
%a field $k$ of characteristic exponent $p \geqslant 1$,
%and let $f:G \rightarrow G'$ be a surjective $S$-equivariant $k$-homomorphism between
%smooth affine $k$-groups equipped with $S$-actions.  Every
%$S$-weight on ${\rm{Lie}}(G')$ is a $p$-power multiple of an $S$-weight on
%${\rm{Lie}}(G)$.
%\end{proposition}
%
%\begin{proof}
%By \cite[9.2, 11.12]{borelag}, 
%the centralizer $G^S$ in $G$ for the $S$-action is smooth and connected, since
%$G^S \times S$ is the centralizer of $S$
%in the smooth connected affine group $G \rtimes S$. 
%A surjective homomorphism $h:H' \rightarrow H$ between smooth connected
%affine $k$-groups induces a surjection between centralizers
%for respective $k$-tori $T' \subset H'$ and $h(T') \subset H$ \cite[Cor.\,2, 11.14]{borelag}.
%Applying this to the surjective homomorphism
%$G \rtimes S \twoheadrightarrow G' \rtimes S$ implies that
%$G^S \rightarrow {G'}^S$ is surjective, so if
%${G'}^S \ne 1$ then $G^S \ne 1$.  Such nontriviality is equivalent
%to $S$ admitting the trivial character as a weight on the Lie algebra,
%due to connectedness and smoothness of these
%$S$-centralizers, so the case of the trivial $S$-weight is settled.
%
%Consider a nontrivial $S$-weight $a \in {\rm{X}}(S)$
%that occurs in ${\rm{Lie}}(G')$.  Let $S_a$ be the codimension-1 subtorus
%$(\ker a)^0_{\rm{red}}$.  We may replace
%$G$ with $G^{S_a}$, $G'$ with ${G'}^{S_a}$, and $S$ with $S/S_a$
%to reduce to the case that $S = \Gm$.  Choosing
%$\lambda \in {\rm{X}}_{\ast}(S)$ such that the composition $a \circ \lambda \in \mathbf{Z}$ is positive, 
%the $a$-weight space in ${\rm{Lie}}(G')$ is supported
%inside ${\rm{Lie}}(U_{G'}(\lambda))$.  By \cite[2.1.9]{pred} (and \cite[2.1.11]{pred}), 
%the natural map $U_G(\lambda) \rightarrow U_{G'}(\lambda)$ is surjective.
%Thus, by \cite[2.1.10]{pred} (and \cite[2.1.11]{pred}) we may rename
%$U_G(\lambda)$ as $G$ and rename $U_{G'}(\lambda)$ as $G'$ to reduce the general problem
%(without reference to $a$) to the case that $G$ is $k$-split unipotent (and nontrivial)
%and all weights in ${\rm{X}}(S) = \mathbf{Z}$ are $> 0$.
%
%Since $G$ is not $k$-wound, by Proposition \ref{IV.4.1.4} it contains
%a nontrivial central vector group $V$.   By Galois descent from $k_s$,
%there is such a $V$ that is $S$-stable, so
%$V' := f(V)$ is an $S$-stable central vector group in $G'$
%(by Theorem \ref{III.3.3.11}).  Any $S$-weight on ${\rm{Lie}}(G')$ occurs
%in either ${\rm{Lie}}(V')$ or ${\rm{Lie}}(G'/V')$, and
%$G/V \rightarrow G'/V'$ is an $S$-equivariant
%surjection with $G/V$ inheriting the $k$-split property from $G$ (Lemma \ref{splitqt}).
%Hence, by induction on $\dim G$ we reduce to the case that
%$G$ and $G'$ are vector groups.    By Theorem \ref{IV.5.2.4} 
%there is a decomposition $G = \prod L_i$ where
%$L_i = \mathbf{G}_{\rm{a}}$ on which $S$ acts
%via $t.x = t^{n_i}x$ for some $n_i > 0$. 
%Running through the same filtration argument again, we may assume
%$G = \mathbf{G}_{\rm{a}}$ with $S$-action
%$t.x = t^n x$ for some $n > 0$. We can also assume $G' \ne 1$, so
%$G' = \mathbf{G}_{\rm{a}}$ with $S$-action $t.x = t^{n'}x$ for some $n' > 0$.
%The map $f:G \twoheadrightarrow G'$ is a nonzero additive polynomial
%in one variable such that $f(t^n x) = t^{n'}f(x)$.   Thus, $f(x) = c x^{p^e}$ for some
%$c \in k^{\times}$ and $e \geqslant 0$ with $n' = p^e n$.
%\end{proof}

\section{Solvable groups}\label{solvgps}

By Theorem \ref{IV.4.2}, if $U$ is a unipotent smooth connected $k$-group then
there is a unique $k$-split smooth connected $k$-subgroup $U_{\rm{split}}$
such that $U/U_{\rm{split}}$ is $k$-wound.  For tori the analogous assertion using
an anisotropic quotient is elementary.
We shall establish a common generalization for solvable smooth
connected affine $k$-groups $G$.
This rests on the following common generalization of
the wound condition in the unipotent case and the anisotropicity condition for tori:

\begin{definition}\label{ansolv}
A solvable smooth connected affine $k$-group $G$ is {\em $k$-wound}
if $G(k[x,1/x]) = G(k)$.
\end{definition}

By Remark \ref{remtori}, if $G$ is a torus then this 
  coincides
with $k$-anisotropicity.  By Corollary \ref{unify}, if $G$ is unipotent
then this coincides with Definition \ref{IV.4.1}.


An obvious but useful reformulation of Definition \ref{ansolv}
is that the specialization homomorphism $G(k[x,1/x]) \rightarrow G(k)$ at $x = 1$
has trivial kernel.  For example, this immediately implies:

\begin{lemma}\label{woundexr} Let $1 \rightarrow G' \rightarrow G \rightarrow G'' \rightarrow 1$
be an exact sequence of solvable smooth connected $k$-groups.
If $G'$ and $G''$ are $k$-wound then so is $G$.
\end{lemma}

The converse of Lemma \ref{woundexr}
fails in the commutative unipotent case, as we noted in Example \ref{woundex}.


\begin{remark}\label{fieldext}
A delicate aspect of Definition \ref{ansolv} is that it is generally poorly behaved with respect
to any nontrivial extension of the ground field.
More specifically, in the unipotent case the separable extensions
 preserve the woundness property and the purely inseparable ones can destroy it, whereas in 
the torus case the purely inseparable extensions preserve
the woundness (i.e., anisotropicity) property and the separable ones can destroy it.
\end{remark}

For a $k$-wound solvable smooth connected affine $k$-group $G$, it is obvious
that any smooth connected $k$-subgroup is $k$-wound 
and that if $G'$ is a $k$-split solvable smooth connected affine $k$-group
then $\Hom_{k\mbox{-}{\rm{gp}}}(G',G) = 1$ (e.g., argue by induction
on $\dim G'$, using a composition series over $k$ whose successive
quotients are $\Add$ or $\Gm$).  In particular, if we drop the $k$-wound hypothesis on $G$
then there is at most one $k$-split smooth connected
normal $k$-subgroup $G_s \subset G$ such that $G/G_s$ is $k$-wound.


Since any quotient
of a $k$-split solvable smooth connected affine $k$-group
is $k$-split, it is elementary that there exists
a unique maximal $k$-split {\em normal} smooth connected
$k$-subgroup $G_{\rm{split}} \subset G$.  By \cite[15.4(i)]{borelag}, 
$G_{\rm{split}}$ is the semi-direct product of
a $k$-split torus against a $k$-split unipotent smooth
connected normal $k$-subgroup of $G_{\rm{split}}$.  (This will be reproved in Theorem \ref{mainsolv} below.) 

The only possibility
for $G_s$ is $G_{\rm{split}}$, so $G_s$ exists if and only if $G/G_{\rm{split}}$
is $k$-wound (in which case $G_{\rm{split}}$ remains maximal in $G$
even without the normality requirement as a $k$-subgroup of $G$). 
The main result of this section is: 

\begin{theorem}\label{mainsolv}  For any  solvable smooth connected affine $k$-group $G$,
the $k$-group $G/G_{\rm{split}}$ is a central extension
 of a $k$-wound unipotent group by a $k$-wound torus
 $($so $G/G_{\rm{split}}$ is $k$-wound$)$.
 In particular, $G$ is $k$-wound if and only if $G_{\rm{split}} = 1$.
 The $k$-group $G_{\rm{split}}$ is the semi-direct product of a maximal
 $k$-split torus against a normal $k$-split unipotent smooth connected $k$-subgroup. 
 
The natural map $G \rightarrow G/G_{\rm{split}}$ is initial among
$k$-homomorphisms from $G$ to $k$-wound
solvable smooth connected affine $k$-groups
and the natural map $G_{\rm{split}} \rightarrow G$
is final among $k$-homomorphisms to $G$ from
$k$-split smooth connected affine $k$-groups.
\end{theorem}

\begin{example}
If $F$ is a perfect field (perhaps of characteristic 0) and $G$ is a solvable smooth connected affine $F$-group then $G = T \ltimes U$ for
an $F$-torus $T$ and an $F$-split unipotent smooth connected $F$-group $U$.
Thus, $G_{\rm{split}} := T_{\rm{split}} \ltimes U$ is an $F$-split normal smooth connected
$F$-subgroup such that $G/G_{\rm{split}} = T/T_{\rm{split}}$
is an $F$-anisotropic $F$-torus.  It follows that Theorem \ref{mainsolv}
is only interesting when $k$ is imperfect.  Likewise, 
Theorem \ref{mainsolv} is only nontrivial when $\mathscr{R}_u(G_{\overline{k}})$
is not defined over $k$ as a $\overline{k}$-subgroup of $G_{\overline{k}}$
(e.g., $G = {\rm{R}}_{k'/k}(\Gm)$ for a nontrivial purely inseparable finite extension $k'/k$). 
\end{example}

\begin{remark}\label{gpcrit}
Although Definition \ref{ansolv} goes beyond the category of $k$-groups
(using $k$-scheme morphisms from $\mathbf{A}^1_k - \{0\}$), it is natural to wonder if
it can be expressed within the category of $k$-groups, as in the case of tori
and unipotent groups.  That is, 
if $G$ is a solvable smooth connected affine $k$-group
and $\Hom_{k\mbox{-}{\rm{gp}}}(\Add,G) = 1$
and $\Hom_{k\mbox{-}{\rm{gp}}}(\Gm,G) = 1$ (equivalently,
$\Hom_{k\mbox{-}{\rm{gp}}}(G',G) = 1$ for all $k$-split
solvable smooth connected affine $k$-groups $G'$)
then is $G$ a $k$-wound group?  This will be immediate
once we prove that $G/G_{\rm{split}}$ is always $k$-wound.
\end{remark}


\begin{lemma}\label{splitum}
Let $U$ be a $k$-split unipotent smooth connected $k$-group,
and $M$ a $($finite type$)$ $k$-group scheme of multiplicative type.
Any exact sequence of affine finite type $k$-groups
$$1 \rightarrow M \rightarrow G \rightarrow U \rightarrow 1$$
is uniquely split: $G = M \times U$ as $k$-groups.
\end{lemma}

\begin{proof}
By the uniqueness claim and Galois descent, we may and do assume
 $k = k_s$.  Hence, $M$ is Cartier dual to a finitely generated $\Z$-module
(so $M$ is a $k$-subgroup of a split $k$-torus).
The uniqueness of the splitting amounts to the assertion that $\Hom_{k\mbox{-}{\rm{gp}}}(U,M) = 1$,
which is obvious (e.g., use an inclusion of $M$ into a $k$-torus).   For the existence,
we first note that $G$ must be a central extension of $U$ by $M$, since
the conjugation action of $G/M = U$ on the commutative normal subgroup $M$
defines a homomorphism from $U$ to the automorphism functor of $M$,
and any such homomorphism is trivial since $U$ is connected and 
$\underline{\rm{Aut}}_{M/k}$  is represented by
a constant $k$-group.   Thus, we aim to prove the triviality of the pointed set 
${\rm{Ex}}_k(U,M)$ of central extensions of $U$ by $M$ (in the category of
affine $k$-group schemes of finite type).  

By using a composition series of $U$ over $k$ with successive quotients isomorphic
to $\Add$, the low-degree $\delta$-functoriality involving $\Hom_{k\mbox{-}{\rm{gp}}}(\cdot,M)$
and ${\rm{Ex}}_k(\cdot,M)$ (or direct bare-hands arguments with
exact sequences and splittings thereof) reduces the problem to the case $U = \Add$.
That is, we seek to prove the vanishing of ${\rm{Ex}}_k(\Add,M)$.
Any central extension $G$ of $\Add$ by $M$ is commutative since
the commutator of $G$ factors through a bi-additive pairing $b:\Add \times \Add \rightarrow M$
that is necessarily trivial since for all $u \in \Add(k) = k$ the map $b(u,\cdot):\Add \rightarrow M$
is a $k$-homomorphism and hence trivial.

Since $M$ is a product of $\Gm$'s and $\mu_n$'s, by low-degree $\delta$-functoriality
considerations in the second variable (rather than the first)
it suffices to separately treat the cases $M = \mu_n$ and $M = \Gm$.
The Kummer sequence $1 \rightarrow \mu_n \rightarrow \Gm \rightarrow \Gm \rightarrow 1$
and the vanishing of $\Hom_{k\mbox{-}{\rm{gp}}}(\Add,\Gm)$ reduce us 
to the special case $M = \Gm$.  That is, we want ${\rm{Ex}}(\Add,\Gm) = 1$.

Consider a central extension $G$ of $\Add$ by $\Gm$, so $G$ is commutative. 
By viewing $G$ as a $\Gm$-torsor over the affine line (for the \'etale topology, and
hence the Zariski topology due to descent theory for line bundles), we see that the quotient
map $\pi:G \rightarrow \Add$ admits a $k$-scheme section $\sigma$.   Using translation by a point in $\Gm(k) =
(\ker \pi)(k)$ we can arrange that $\sigma(0) = e \in G(k)$.
Hence, the resulting identification of $G$ with the pointed $k$-scheme
$(\Gm \times \mathbf{A}^1_k, (1,0))$ carries the group law on $G$
over to a composition law $(c,x) \cdot (c',x') = (cc' f(x,x'),x+x')$
for a symmetric  polynomial $f:\mathbf{A}^2_k \rightarrow \Gm$
satisfying $f(0,0) = 1$.  The only such $f$ is $f = 1$.
\end{proof}

\begin{proof}[Proof of Theorem $\ref{mainsolv}$]
There are no nontrivial $k$-homomorphisms from
a $k$-split solvable smooth connected affine $k$-group
to a $k$-wound solvable smooth connected affine $k$-group,
so the only task is to establish the central
extension structure of $G/G_{\rm{split}}$ and the semi-direct product structure of $G_{\rm{split}}$. 

First consider $H = G_{\rm{split}}$.  The derived group
$\mathscr{D}(H)$ is unipotent (as we may check over $\overline{k}$)
and $k$-split (Corollary \ref{dermap}), and any maximal
$k$-torus of $H$ maps isomorphically onto a maximal
$k$-torus of $H/\mathscr{D}(H)$.  Thus, to prove that $H$ is a semi-direct product
of a maximal $k$-split torus against a normal $k$-split unipotent subgroup $U$
(in which case the $k$-torus $H/U$ is $k$-split, so all maximal $k$-tori in $H$
are $k$-split), we may pass to the $k$-split commutative $C = H/\mathscr{D}(H)$. 
This has a unique maximal $k$-torus $T$ and the quotient $U = C/T$ is $k$-split
unipotent, so by Lemma \ref{splitum} there exists a unique decomposition $C = T \times U$.
Thus, $T$ is a quotient of the $k$-split $C$, so it is $k$-split. 

It remains to understand the structure of $G/G_{\rm{split}}$, which is to say that we can
assume $G_{\rm{split}} = 1$. By Lemma \ref{woundexr} it remains to show that $G$ is a central extension of
a $k$-wound unipotent group by a $k$-wound torus.  
Since $G$ is solvable,  $\mathscr{D}(G)$ is unipotent (as we may check
over $\overline{k}$).  Thus, the formation of $\mathscr{D}(G)_{\rm{split}}$ commutes
with separable extension on $k$ (even though such extension may ruin the hypothesis that
$G_{\rm{split}} = 1$).  By computing with $G(k_s)$-conjugation on $\mathscr{D}(G)_{k_s}$, it
follows that $\mathscr{D}(G)_{\rm{split}}$ is normal in $G$.  But we have arranged that $G_{\rm{split}} = 1$, so
$\mathscr{D}(G)_{\rm{split}} = 1$.  Hence, by the structure theory in the unipotent case, $\mathscr{D}(G)$ is $k$-wound.

Let $T$ be a maximal $k$-torus in $G$.  Since $\mathscr{D}(G)$ is $k$-wound unipotent, 
the conjugation action by $T$ on $\mathscr{D}(G)$ is trivial (Corollary \ref{IV.5.3.1}).
Since $T$ maps isomorphically onto its image $\overline{T}$ in the commutative
$G/\mathscr{D}(G)$ (due to the unipotence of $\mathscr{D}(G)$), 
 the $k$-subgroup $T \times \mathscr{D}(G)$ in $G$ is {\em normal}. 
Thus, the $G(k_s)$-action via conjugation 
on the normal $k_s$-subgroup $T_{k_s} \times \mathscr{D}(G)_{k_s}$ of $G_{k_s}$
preserves the unique maximal $k_s$-torus $T_{k_s}$, so
$T$ is normal in $G$.   The connectedness of $G$ then forces $T$ to be central in $G$.
Since $G_{\rm{split}} = 1$, so $T_{\rm{split}} = 1$,
we see that $T$ is $k$-anisotropic.  The formation of $T$
as the maximal central torus commutes with scalar extension on $k$, even though
such scalar extension may ruin the anisotropicity property of $T$.

The quotient $U = G/T$ now makes sense and is unipotent.  It remains to prove that
$U$ is $k$-wound.   By the structure theory in the unipotent case, it suffices to show that $U_{\rm{split}} = 1$.
The preimage $G'$ of $U_{\rm{split}}$ in $G$ is an extension of $U_{\rm{split}}$ by $T$,
so by Lemma \ref{splitum} there is a unique $k$-group decomposition $G' = U_{\rm{split}} \times T$.
The formation of $G'$ commutes with scalar extension to $k_s$, as does the formation of
$U_{\rm{split}} \subset U$,  so the same holds
for the unique subgroup of $G'$ isomorphically lifting $U_{\rm{split}}$.  That is,
the unique product decomposition of $G'$ commutes with scalar extension to $k_s$, so
consideration of $G(k_s)$-conjugation on $G'_{k_s}$ shows that $U'_{\rm{split}}$ is normal in $G$.
But $G_{\rm{split}} = 1$, so $U'_{\rm{split}} = 1$.
\end{proof}

\begin{corollary}\label{regfield}
Let $G$ be a solvable smooth connected affine $k$-group, and $k'/k$ a regular field extension
$($i.e., separable with $k$ algebraically closed in $k'$$)$.  The natural inclusion
$(G_{\rm{split}})_{k'} \subset (G_{k'})_{\rm{split}}$ is an equality.
\end{corollary}

\begin{proof}
The structure of $G/G_{\rm{split}}$ in Theorem \ref{mainsolv}
reduces the problem to verifying that if $k'/k$ is a regular
extension and $G$ is $k$-wound unipotent (resp.\,a $k$-anisotropic $k$-torus)
then $G_{k'}$ is $k'$-wound unipotent (resp.\,a $k'$-anisotropic $k'$-torus).  
The unipotent case follows from Proposition \ref{IV.4.1.4} since $k'/k$ is separable.
To handle the torus case, by consideration of Galois lattice character groups 
it suffices to prove the surjectivity of 
the restriction map ${\rm{Gal}}(k'_s/k') \rightarrow {\rm{Gal}}(k_s/k)$
relative to an embedding $k_s \rightarrow k'_s$ over $k \rightarrow k'$.
The $k'$-algebra $k' \otimes_k k_s$ is a field contained in $k'_s$ that is moreover Galois over
$k$ with Galois group ${\rm{Gal}}(k_s/k)$ in the evident manner,
so we are done.
\end{proof}

%
%
%Consider the preimage $G' \subset G$ of $U_{\rm{split}} \subset C = G/\mathscr{D}(G)$,
%so $G'$ is an extension of $U_{\rm{split}}$ by $\mathscr{D}(G)$.
%Since $\mathscr{D}(G)$ is unipotent, so is $G'$.  Clearly $G'$ is normalized
%by $T$, so $H' := G' \rtimes T$ is a normal smooth connected $k$-subgroup of $G$
%such that $G/H' = U/U_{\rm{split}}$.  Since $U/U_{\rm{split}}$ is $k$-wound,  $H'$ contains
%$G_{\rm{split}}$.  
%
%The unipotence of $G'$ implies that $(G'_{\rm{split}})_{k_s} = (G'_{k_s})_{\rm{split}}$
%(due to the compatibility with separable extension of the ground field in
%Theorem \ref{IV.4.2}).  The  $k_s$-subgroup $G'_{k_s} \subset H'_{k_s}$ is
%characteristic (i.e., stable under $k_s$-automorphisms) since it is
%the unique maximal smooth connected unipotent $k_s$-subgroup,
%so $(G'_{\rm{split}})_{k_s}$ is also characteristic is $H'_{k_s}$.
%Hence, $(G'_{\rm{split}})_{k_s}$ is stable under
%$G(k_s)$-conjugation on the normal subgroup $H'_{k_s} \subset G_{k_s}$,
%so $G'_{\rm{split}}$ is  normal in $G$.   In particular,
%$G'_{\rm{split}}$ is contained in $G_{\rm{split}}$.
%
%The semi-direct product $G'_{\rm{split}} \rtimes T_{\rm{split}} \subset G$
%is a $k$-split solvable smooth connected $k$-subgroup, and we claim
%that it is normal.  To check this, it suffices to check that $T_{\rm{split}}$ is normal in 
%$G/G'_{\rm{split}}$. The  quotient $G/G'_{\rm{split}}$ is an extension of 
%$G/G' = C/U_{\rm{split}}$ 
%by the wound unipotent $G'/G'_{\rm{split}}$.  Note that $C/U_{\rm{split}}$
%is an extension of the wound $U/U_{\rm{split}}$ by $T$,
%so $N := (G'/G'_{\rm{split}}) \rtimes T$ is normal in $G/G'_{\rm{split}}$.
%By Corollary \ref{IV.5.3.1}, $T$ centralizes
%$G'/G'_{\rm{split}}$, so  $T$ is the unique maximal $k$-torus in $N$.
%Hence, the torus $T$ is normal in the connected $G/G'_{\rm{split}}$,
%so it is central. It follows that $T_{\rm{split}}$ is central
%in $G/G'_{\rm{split}}$, so $G_0 := G'_{\rm{split}} \rtimes T_{\rm{split}}$
%is normal in $G$.  Thus, $G_0 \subset G_{\rm{split}}$.
%
%The quotient $G/G_0$ contains
%the wound (anisotropic) $k$-torus $S = T/T_{\rm{split}}$ as a central subgroup,
%modulo which it becomes the group $G/(G'_{\rm{split}} \rtimes T)$
%that is an extension of $G/H' = U/U_{\rm{split}}$ by $H'/(G'_{\rm{split}} \rtimes T) =
%G'/G'_{\rm{split}}$.  But $U/U_{\rm{split}}$ and $G'/G'_{\rm{split}}$ are wound unipotent $k$-groups,
%so by two applications of Lemma \ref{woundexr} we see that
%$(G/G_0)/S$ is $k$-wound unipotent and 
%$G/G_0$ is $k$-wound.  Thus,
%$G_0 = G_{\rm{split}}$ and we are done. 
%\end{proof}

In Remark \ref{splitrat}, we saw that every $k$-split solvable smooth connected $k$-group
is isomorphic as a $k$-scheme to $\mathbf{A}^{n,m}_k = \mathbf{A}^n_k \times (\mathbf{A}^1_k - \{0\})^m$
for some $n, m \ge 0$.
Here is a converse result for solvable groups in the spirit of the splitting criterion for unipotent groups in Corollary \ref{mapcrit}.

\begin{corollary}\label{anmcrit}
A solvable smooth connected $k$-group $G$ is $k$-split if and only if there
is a dominant $k$-morphism $f:\mathbf{A}^{n,m}_k \rightarrow G$.
\end{corollary}

\begin{proof}
The implication ``$\Rightarrow$'' was shown in Remark \ref{splitrat}, and for the 
converse we will use Theorem \ref{mainsolv}.  Assuming such an $f$ exists,
to prove that $G$ is split we may compose $f$ with the quotient map
$G \rightarrow G/G_{\rm{split}}$ to reduce to the case
that $G$ is $k$-wound, so $G$ is an extension of a $k$-wound unipotent smooth connected $k$-group $U$
by a $k$-anisotropic torus $T$.  Our aim is to prove that $G = 1$.  The composite
map $\mathbf{A}^{n,m}_k \rightarrow U$ is dominant, so $U$ is $k$-split by
Corollary \ref{mapcrit}.  But $U$ is $k$-wound, so $U = 1$.  That is, 
$G = T$ is a $k$-anisotropic torus.  

Since the units in $k[x_1^{\pm 1}, \dots, x_m^{\pm 1}]$ are precisely the monomials $c \prod x_i^{e_i}$
with $c \in k^{\times}$ and $e_i \in \Z$, the same argument as in Remark \ref{remtori} shows that 
any $k$-morphism $\mathbf{A}^{0,m}_k = (\mathbf{A}^1_k - \{0\})^m \rightarrow T$ is a constant map to some  $t \in T(k)$.
Thus, the case $n = 0$ is settled.  The anisotropicity has done its work, as it now suffices
to show that for any $k$-torus $T$ whatsoever and any $k$-morphism
$f:\mathbf{A}^{n,m}_k \rightarrow T$, there is a (unique) factorization
of $f$ through the projection $\mathbf{A}^{n,m}_k \rightarrow \mathbf{A}^{0,m}_k$.
This says that $f^{\ast}:k[T] \rightarrow k[\mathbf{A}^{n,m}_k]$ lands in the Laurent
polynomial subalgebra $k[\mathbf{A}^{0,m}_k]$, for which it is harmless to check
after extending scalars to $k_s$ or even $\overline{k}$. 
Now $T = (\Gm)^N$ for some $N \ge 0$, so we are reduced to the case $T = \Gm$.
Any unit on $\mathbf{A}^{n,m}_k$ is the pullback
of a unit on $\mathbf{A}^{0,m}_k$, so we are done.
\end{proof}


We end our discussion with some applications to general smooth connected
affine $k$-groups $G$.  Our interest is in variants
of the $k$-subgroups $\mathscr{R}_{u,k}(G)$ and $\mathscr{R}_{us,k}(G)$
considered in Corollary \ref{uscor}.  Define
the {\em $k$-radical} $\mathscr{R}_k(G)$ to be the maximal
normal solvable smooth connected $k$-subgroup of $G$,
and the {\em $k$-split radical} $\mathscr{R}_{s,k}(G)$ to be the maximal
normal $k$-split solvable smooth connected $k$-subgroup of $G$.
Obviously $G/\mathscr{R}_k(G)$ has trivial $k$-radical
and $G/\mathscr{R}_{s,k}(G)$ has trivial $k$-split radical.
Beware that $G/\mathscr{R}_k(G)$ may {\em not}  be equal
to its own derived group (in contrast with $G_{\overline{k}}/\mathscr{R}(G_{\overline{k}})$).
Equivalently, there exist $G$ such that $\mathscr{R}_k(G) = 1$ but
$G \ne \mathscr{D}(G)$; see \cite[11.2.1]{pred} for many such $G$
over any imperfect field $k$.

A {\em pseudo-reductive} $k$-group is a smooth
connected affine $k$-group $G$ such that $\mathscr{R}_{u,k}(G) = 1$.
By Galois descent and Theorem \ref{IV.4.2},
$\mathscr{R}_k(G)_{k_s} = \mathscr{R}_{k_s}(G_{k_s})$
and $\mathscr{R}_{us,k}(G)_{k_s} = \mathscr{R}_{us,k_s}(G_{k_s})$. 
There is no analogue of these equalities for $\mathscr{R}_{s,k}$.

\begin{proposition} Let $G$ be a smooth connected affine
$k$-group.  Then $\mathscr{R}_k(G) = 1$ if and only if
$G$ is pseudo-reductive and has no nontrivial central $k$-torus.
\end{proposition}

\begin{proof}
In either direction, $G$ is pseudo-reductive, so we may and do assume 
that $G$ is pseudo-reductive.  Since
pseudo-reductivity is inherited by smooth connected normal
$k$-subgroups (as explained near the beginning of \cite[1.1]{pred}), 
$\mathscr{R}_k(G)$ is solvable
and pseudo-reductive.  But a solvable
pseudo-reductive group is commutative \cite[1.2.3]{pred},
so $\mathscr{R}_k(G)$ is commutative. The unique maximal $k$-torus $S$ in $\mathscr{R}_k(G)$ must
be normal in $G$ and hence central
(due to the connectedness of $G$), and $S \ne 1$ if $\mathscr{R}_k(G) \ne 1$
since $\mathscr{R}_k(G)$ cannot be unipotent when it is nontrivial
(due to the pseudo-reductivity of $G$).  Since any central $k$-torus in $G$
lies in $\mathscr{R}_k(G)$, $S$ is the maximal central $k$-torus in
$G$.  Thus, $\mathscr{R}_k(G) = 1$ if and only if $S = 1$.
\end{proof}

As an application of Corollary \ref{regfield}, we can settle
the following natural question:
clearly $\mathscr{R}_{s,k}(G) \subset \mathscr{R}_k(G)_{\rm{split}}$, but
is this containment is an equality?  It is equivalent to ask
if $\mathscr{R}_k(G)_{\rm{split}}$ is normal in $G$, or 
if the $k$-radical of $G/\mathscr{R}_{s,k}(G)$ is  $k$-wound.
In the proof of the unipotent analogue in Corollary \ref{uscor} it was harmless
to extend scalars to $k_s$, but that technique is not
available in the present circumstances
(and $G(k)$ might fail to be Zariski-dense in $G$).
Nonetheless, we can prove an affirmative answer:

\begin{proposition}\label{rskprop} For a smooth connected affine $k$-group $G$,
$\mathscr{R}_{s,k}(G) = \mathscr{R}_k(G)_{\rm{split}}$.
\end{proposition}

\begin{proof}
This amounts to the assertion that the action map
$$G \times \mathscr{R}_k(G)_{\rm{split}} \rightarrow G$$
defined by $(g,h) \mapsto ghg^{-1}$
factors through $\mathscr{R}_k(G)_{\rm{split}}$.  By Zariski-density
considerations it suffices to check this at the generic point $\eta$ of $G$,
which is to say that for $K = k(G)$
the $K$-group $(\mathscr{R}_k(G)_{\rm{split}})_K$ is carried
into itself under conjugation by the $K$-point $\eta \in G(K)$.
More generally, we claim that the $K$-subgroup $(\mathscr{R}_k(G)_{\rm{split}})_K$
inside $G_K$ is stable under conjugation by the entire group $G(K)$.
Since $(\mathscr{R}_k(G)_{\rm{split}})_K = \mathscr{R}_K(G_K)_{\rm{split}}$
(Corollary \ref{regfield}), it remains to note that 
for any smooth connected solvable group $H$
over a field $F$, the closed $F$-subgroup $H_{\rm{split}}$ is obviously normalized
by $H(F)$.
\end{proof}

\begin{corollary}\label{rsk} For   smooth connected affine $k$-group $G$, 
if $\mathscr{R}_{us,k}(G) = 1$ then $\mathscr{R}_{s,k}(G)$ 
is the maximal central $k$-split torus in $G$. 
In particular, $\mathscr{R}_{s,k}(G) = 1$ if and only if
$\mathscr{R}_{us,k}(G) = 1$ with $G$ containing no nontrivial $k$-split central $k$-torus.
\end{corollary}

\begin{proof}
Consider the $k$-split
solvable smooth connected affine
$k$-group $R := \mathscr{R}_k(G)_{\rm{split}} = \mathscr{R}_{s,k}(G)$.
By the semi-direct product structure of split solvable smooth connected affine groups
as in Theorem \ref{mainsolv}, $R$ is the semi-direct product of a split torus
against a normal split unipotent smooth connected $k$-subgroup $U$
that must be $\mathscr{R}_{us,k}(R) = \mathscr{R}_{u,k}(R)$.
Since the $k_s$-subgroup $U_{k_s} = \mathscr{R}_{u,k_s}(R_{k_s})$
is stable under all $k_s$-automorphisms of $R_{k_s}$,
the normality of $R_{k_s}$ in $G_{k_s}$ implies
that $U_{k_s}$ is normal in $G_{k_s}$, so $U$ is normal in $G$.
Thus, $U \subset \mathscr{R}_{us,k}(G) = 1$, proving that $R$ is a split torus.  But 
the torus $R$ is normal in the connected $k$-group $G$, so $R$ is central in $G$. 
This proves that $R$ is the maximal central $k$-split torus in $G$. 
\end{proof}

\begin{corollary}\label{parequiv} Let $G$ be a smooth connected affine $k$-group.
The following three conditions are equivalent:  
\begin{enumerate}
\item $G/\mathscr{R}_{s,k}(G)$ contains
a nontrivial $k$-split solvable smooth connected $k$-subgroup,
\item $G/\mathscr{R}_k(G)$ contains
 a nontrivial $k$-split solvable smooth connected $k$-subgroup,
\item $G$ contains a proper pseudo-parabolic $k$-subgroup.
\end{enumerate}
In $(1)$ and $(2)$ it is equivalent to contain $\Gm$ as a non-central $k$-subgroup. 
 \end{corollary}
 
 The notion of pseudo-parabolicity is defined in \cite[2.2.1]{pred};
 it coincides with parabolicity in the connected reductive case \cite[2.2.9]{pred}.
 A typical example of a pseudo-parabolic $k$-subgroup that is
 not parabolic is $P := {\rm{R}}_{k'/k}(P') \subset {\rm{R}}_{k'/k}(G') =: G$
 for a nontrivial purely inseparable finite extension $k'/k$ and a proper parabolic $k'$-subgroup
 $P'$ in a connected reductive $k'$-group $G'$.  (Such $P$ are precisely
 the pseudo-parabolic $k$-subgroups of $G$, by \cite[11.4.4]{pred}. The non-parabolicity
 of $P$, which is to say the non-properness
 of $G/P \simeq {\rm{R}}_{k'/k}(G'/P')$,  follows from \cite[A.5.6]{pred} since $\dim G'/P' > 0$.)
 By \cite[2.2.10]{pred}, condition (3) is equivalent to the same
 for the maximal pseudo-reductive quotient $G/\mathscr{R}_{u,k}(G)$,
 and if $G$ is pseudo-reductive then (3) is equivalent to saying
 that $G$ has no non-central $k$-split torus \cite[2.2.3(1)]{pred}.
  
 \begin{proof}
 The kernel
 $\mathscr{R}_k(G)/\mathscr{R}_{s,k}(G) = 
 \ker(G/\mathscr{R}_{s,k}(G) \rightarrow G/\mathscr{R}_k(G))$
  is $k$-wound since $\mathscr{R}_{s,k}(G) = \mathscr{R}_k(G)_{\rm{split}}$
 (Proposition \ref{rskprop}),
 so a nontrivial $k$-homomorphism from $\Add$ or $\Gm$
 to $G/\mathscr{R}_{s,k}(G)$ yields a nontrivial composite
 homomorphism to $G/\mathscr{R}_k(G)$.  Hence, (1) implies (2). 
  
 To prove that (2) implies (3), we may replace
 $G$ with $G/\mathscr{R}_{u,k}(G)$, so $G$ is pseudo-reductive. 
The hypothesis in (2) says that the pseudo-reductive $G/\mathscr{R}_k(G)$ contains
$\Add$ or $\Gm$ as a $k$-subgroup.  By \cite[C.3.8]{pred}, if a pseudo-reductive
$k$-group contains $\Add$ as a $k$-subgroup then it contains
a non-central $\Gm$ as a $k$-subgroup.  Since $\Gm$ as a $k$-subgroup of
$G/\mathscr{R}_k(G)$ cannot be central (as $\mathscr{R}_k(G/\mathscr{R}_k(G)) = 1$), 
it suffices to prove that if
$G/\mathscr{R}_k(G)$ contains a non-central $\Gm$
then so does $G$.  The preimage $H$ in $G$ of such a $\Gm$
is a smooth $k$-subgroup, so a maximal
$k$-torus $T$ in $H$ must map onto this $\Gm$.
Hence, $T$ contains a $k$-subgroup $\Gm$ that is
not in $\mathscr{R}_k(G)$ and thus is not central in $G$.
The existence of a non-central $\Gm$ in the pseudo-reductive $k$-group $G$ is equivalent to (3),
by \cite[2.2.3(2)]{pred}.

Finally, we show that (3) implies (1).  It is harmless to replace
$G$ with $G/\mathscr{R}_{us,k}(G)$, so 
$\mathscr{R}_{us,k}(G) = 1$. 
Thus, $\mathscr{R}_{s,k}(G)$ is the maximal $k$-split central $k$-torus in $G$
(Corollary \ref{rsk}), so $G/\mathscr{R}_{s,k}(G)$ contains no non-trivial normal
$k$-split $k$-tori (as a normal $k$-split $k$-torus in $G/\mathscr{R}_{s,k}(G)$
has preimage in $G$ that is a $k$-split normal $k$-torus, and such a normal
torus must be central due to connectedness of $G$, contradicting the maximality of
$\mathscr{R}_k(G)$). 
From the definition of pseudo-parabolicity,  (3) implies that $G$ contains
a non-central $\Gm$.  Its image in $G/\mathscr{R}_{s,k}(G)$ is a
non-central $k$-subgroup isomorphic to $\Gm$.
 \end{proof}

\begin{proposition}\label{parbij}
Let $G$ be a smooth connected affine $k$-group.  The solvable smooth connected
normal $k$-subgroup $R := \mathscr{R}_k(G)/\mathscr{R}_{u,k}(G)$ 
in the maximal pseudo-reductive quotient $G' := G/\mathscr{R}_{u,k}(G)$ is a central $k$-subgroup, 
and if $N$ is a normal closed  $k$-subgroup scheme of
$\mathscr{R}_k(G)$ then the formation of images and preimages under
$G \rightarrow G/N$ defines a bijection between
the sets of pseudo-parabolic $k$-subgroups of $G$ and $G/N$.
\end{proposition}

\begin{proof}
Clearly $R = \mathscr{R}_k(G')$, so to prove the centrality of $R$ in $G'$ we can replace
$G$ with $G'$ to reduce to the case when $G$ is pseudo-reductive.  
Thus, by \cite[Lemma 1.2.1]{pred}, to prove the triviality of the smooth connected commutator $(R, G)$
it suffices to prove that $(R,G)_{\overline{k}} \subset \mathscr{R}_u(G_{\overline{k}})$.
In other words, we claim that $R_{\overline{k}}$ has central image in
the connected reductive group $H := G_{\overline{k}}/\mathscr{R}_u(G_{\overline{k}})$.
But this image is a solvable smooth connected normal subgroup of $H$, so 
it is a central torus in $H$ due to the reductivity of $H$. 

The centrality of $R$ in $G'$ implies that it lies in every pseudo-parabolic $k$-subgroup of
$G'$ (as pseudo-parabolic subgroups always contain the scheme-theoretic center).
For any 1-parameter $k$-subgroup
$\lambda:\Gm \rightarrow G'/R$  there exists $n \ge 1$ such that
$\lambda^n$ lifts to a 1-parameter $k$-subgroup $\Gm \rightarrow G'$ (since
split tori lift lift up to isogeny through any {\em smooth}
surjective $k$-homomorphism between smooth connected affine $k$-groups),
so it follows formally from
\cite[2.1.7, 2.1.9]{pred} and the role of 1-parameter subgroups in the definition of pseudo-parabolicity that 
the formation of images and preimages under the quotient map $G' \rightarrow G'/R$
defines a bijective correspondence between the sets of parabolic $k$-subgroups of 
$G'$ and $G'/R$.  But the formation of images and preimages under
$G \rightarrow G'$ likewise defines a bijection between the sets of pseudo-parabolic 
$k$-subgroups of $G$ and $G'$ (see \cite[Prop.\,2.2.10]{pred}), so we
conclude that the same holds for the formation of images and preimages under
$G \rightarrow G/\mathscr{R}_k(G) = G'/R$.
The analogous such bijectivity for the map $G \rightarrow G/N$ is
therefore reduced to verifying that the evident containment $\mathscr{R}_k(G)/N \subset \mathscr{R}_k(G/N)$
inside $G/N$ is an equality.   This equality holds because the normal smooth connected $k$-subgroup
$$\mathscr{R}_k(G/N)/(\mathscr{R}_k(G)/N) \subset (G/N)/(\mathscr{R}_k(G)/N) = G/\mathscr{R}_k(G)$$
is solvable and hence trivial (due to the definition of $\mathscr{R}_k(G)$).
\end{proof}

\medskip\medskip

\centerline{\sc Exercises on unipotent groups}

\medskip

\noindent
U.1.  Let $k$ be a field and $U_n$ the standard strictly upper-triangular unipotent $k$-subgroup of
${\rm{GL}}_n$.  

(i)  Prove that no nontrivial $k$-group scheme is isomorphic to a closed $k$-subgroup of
both 
$\Add$ and ${\mathbf{G}}_m$.  (If ${\rm{char}}(k) = p > 0$, the key is to prove that $\mu_p$ is not a $k$-subgroup of
$\Add$.)
Deduce that the scheme-theoretic intersection $T \cap U_n$ is trivial
 for any $k$-torus $T$ in ${\rm{GL}}_n$.
 
 (ii) Using the Lie--Kolchin theorem, prove that $T \cap U = 1$ for any $k$-torus $T$ and unipotent 
 smooth connected $k$-group $U$
 in a linear algebraic $k$-group $G$.

\medskip\noindent
U.2. Let $k$ be a nonzero commutative ring, and let $E = 
{\rm{End}}_{k\mbox{-}{\rm{gp}}}(\Add)$, an associative ring containing
$k$ via $c.x = cx$ ($c \in k$).

(i) If $k$ is a $\Q$-algebra, prove $k = E$.  Deduce that 
the functor $\underline{{\rm{Aut}}}_{\Q\mbox{-}{\rm{gp}}}(\Add^n)$ 
on $\Q$-schemes is represented by $\GL_n$.

(ii) If $x \mapsto x^n$ lies in $E$ with $n > 1$, prove that $n$ is a power of a prime
$p$ and $k$ is a $\Z/p\Z$-algebra.


(iii) If $k$ is an $\F_p$-algebra prove that $E = \{\sum c_j t^{p^j}\,|\, c_j \in k\}$.   Now see Exercise U.7(iv).


%\medskip\noindent
%U.3.  Use the method of proof of \cite[Prop.\,4.10]{borelag}
%to prove that if $k$ is a field and a smooth unipotent affine $k$-group $G$ is equipped with a left
%action on a quasi-affine $k$-scheme $V$ of finite type then for any $v \in V(k)$
%the smooth locally closed image of the orbit map $G \rightarrow V$ defined by $g \mapsto gv$ is
%actually closed in $V$.  

%(Hint: to begin, let $k[V]$ denote the $k$-algebra of global functions on $V$
%and prove that $R \otimes_k k[V]$ is the $R$-algebra of global functions on $V_R$ for any $k$-algebra $R$.
%Use this to construct a functorial $k$-linear representation of $G$ on $k[V]$ respecting the $k$-algebra structure.
%Borel's $K$ should be replaced with $k$ after passing to the case
%$k = \overline{k}$.  Note that it is not necessary to assume Borel's $F$ is non-empty; the argument directly proves $J$ meets
%$k^{\times}$, so $J = (1)$ and hence $F$ is empty.)

\medskip\noindent
U.3.  Let $G$ be a group scheme of finite type over a field $k$.

(i) Prove that $(G_{\overline{k}})_{\rm{red}}$ is a closed $\overline{k}$-subgroup of
$G_{\overline{k}}$, and prove it is {\em smooth}. 
 Deduce that the identity component $G^0$ is {\em geometrically irreducible}.  
 
 (ii) Over any imperfect field $k$, one can make a non-reduced $k$-group $G$ such that
 $G_{\rm{red}}$ is {\em not} a $k$-subgroup.  (See \cite[Ex.\,A.8.3(i)]{pred} and \cite[VI$_{\rm{A}}$, 1.3.2(2)]{sga3}.)
 Where does an attempted proof to the contrary get stuck?

(iii) Assume $k$ is imperfect, ${\rm{char}}(k) = p > 0$. For 
$a \in k - k^p$, define a natural $k$-group structure on 
the hyper surface 
$$G = \{x_0^p + a x_1^p + \dots + a^{p-1} x_{p-1}^p = 1\} \subset \mathbf{A}_k^p$$
(hint: consider the kernel of ${\rm{N}}_{k(a^{1/p})/k}:
{\rm{R}}_{k(a^{1/p})/k}(\Gm) \rightarrow \Gm$, using Weil restriction
as in Exercise U.4). Show that $G$ is reduced but $G_{\overline{k}}$ is non-reduced,
so the reduced $k$-group $G$ is {\em not} smooth.

(iv) Prove that the condition $t^n = 1$ defines a finite closed $k$-subgroup
$\mu_n \subset {\mathbf{G}}_m$, and show its preimage $G$ under $\Det:\GL_N \rightarrow {\mathbf{G}}_m$
is a $k$-subgroup of $\GL_N$.  Prove $G^0 = {\rm{SL}}_N$, and for $k = \Q$ and $n = 5$
prove that $G - G^0$ is connected but over $\overline{k}$ has 4 connected components. 



\medskip\noindent
U.4. This exercise develops the important concept of {\em Weil restriction of scalars} in the affine
case (see \cite[7.6]{neron} and \cite[A.5, A.7]{pred} for further information).  It is an analogue
of viewing a complex manifold as a real manifold with twice the dimension (and ``complex points''
become ``real points'').  Let $k$ be a ring, $k'$ a $k$-algebra that is
finite and locally free as a $k$-module, and $X'$ an affine $k'$-scheme of finite
type.  Consider the functor ${\rm{R}}_{k'/k}(X'):A \rightsquigarrow X'(k' \otimes_k A)$ on $k$-algebras. 

(i) By considering $X' = \mathbf{A}^n_{k'}$ and then any $X'$ via a closed
immersion into an affine space over $k'$, prove that this functor is represented by an affine
$k$-scheme of finite type, again denoted ${\rm{R}}_{k'/k}(X')$.  Prove its formation
naturally commutes with products in $X'$, and compute
${\rm{R}}_{k'/k}(\Gm)$ inside ${\rm{R}}_{k'/k}(\mathbf{A}^1_{k'})$.  What if $k' = 0$?  
What if $k' = \prod k'_i$ for $k$-algebras $k'_i$?

(ii) Prove ${\rm{R}}_{k'/k}(\Spec k') = \Spec k$, and explain why ${\rm{R}}_{k'/k}(X')$ is naturally
a $k$-group when $X'$ is a $k'$-group.  In case $k$ is a complete valued field
and $k'$ is an extension field equipped with the extended valuation, prove
that the $k$-analytic topology on ${\rm{R}}_{k'/k}(X')(k)$  coincides
with the $k'$-analytic topology on $X'(k')$.

(iii) Use the infinitesimal
criterion to prove that ${\rm{R}}_{k'/k}(X')$ is $k$-smooth when $X'$ is $k'$-smooth.  
(Can you see it directly from the construction?) 
Warning: if $k'/k$ is a finite extension of fields that is not separable then
${\rm{R}}_{k'/k}(X')$ can be empty (resp.\:disconnected) even when $X'$ is non-empty (resp.\:geometrically
integral)!

(iv) For a $k$-algebra $K$, prove that ${\rm{R}}_{k'/k}(X')_K \simeq
{\rm{R}}_{K'/K}(X'_{K'})$ for $K' = k' \otimes_k K$.  
 If $k'/k$ is a separable extension of fields, prove 
${\rm{R}}_{k'/k}(X')_{k_s} \simeq \prod_{\sigma} \sigma^{\ast}(X')$ with $\sigma$ varying through
${\rm{Hom}}_k(k', k_s)$.   Transfer the natural
${\rm{Gal}}(k_s/k)$-action on the left over to the right and describe it. This recovers Weil's 
definition of Weil restriction of scalars (in the affine case), from which we see
that if $k'/k$ is a finite separable extension of fields 
 then the functor ${\rm{R}}_{k'/k}$ preserves the following conditions on smooth affine
groups:  connected (see Exercise U.6(ii)), torus, unipotent, reductive, semisimple.

(v) Assume ${\rm{char}}(k) = p > 0$ and let $k'/k$ be a non-trivial
purely inseparable finite extension of fields. 
For $n > 0$, compute ${\rm{R}}_{k'/k}(\GL_n)(\overline{k})$ 
and rigorously prove that $G := {\rm{R}}_{k'/k}(\GL_n)$ is not reductive
by relating the geometric unipotent radical to the filtration on
$k' \otimes_k \overline{k}$ by powers of its nilpotent maximal ideal. 
Use the universal property of Weil restriction to show that $G$
contains no nontrivial unipotent smooth connected normal $k$-subgroup!
(See \cite[1.1.3, 1.6.1]{pred} for more surprises with non-\'etale Weil restriction.)

\medskip\noindent
U.5.  Let $U$ be a unipotent smooth connected commutative group scheme
over a field $k$, 
and assume $U$ is $p$-torsion if ${\rm{char}}(k) = p > 0$.

(i) If ${\rm{char}}(k) > 0$ and $U$ is $k$-split, use Corollary \ref{III.3.3.9} to prove that 
$U$ is a vector group.  

(ii) Assume ${\rm{char}}(k) = 0$
(so all $k$-group schemes of finite type are smooth, by
Cartier's theorem). Prove that any short exact sequence $0 \rightarrow
\Add \rightarrow G \rightarrow \Add \rightarrow 0$ with commutative $G$ 
is split.  Deduce that $U \simeq \Add^N$, and prove that any action on
$U$ by a $k$-split torus $T$ respects this linear structure.  Also prove that
every unipotent $k$-group is {\em connected} and $k$-split. 

(iii) Prove that any commutative extension of $\Add$ by $\Gm$ is uniquely split
over $k$. (Hint: first make a scheme splitting using that ${\rm{Pic}}(\Add) = 1$.) 

%\medskip\noindent
%U.6. For a field $k$ with ${\rm{char}}(k)=p>0$, let $k'/k$ be a purely inseparable finite extension.
%
%(i) Prove  that $U := {\rm{R}}_{k'/k}(\Gm)/\Gm$ is smooth and connected 
%of dimension $[k':k]-1$, and is $[k':k]$-torsion.  Deduce it is unipotent.   
%
%(ii) Prove that ${\rm{R}}_{k'/k}(\Gm)(k_s)[p] = 1$.
%Deduce that ${\rm{R}}_{k'/k}(\Gm)$ contains no nontrivial unipotent
%smooth connected $k$-subgroup and that
%$U$ in (i) does not contain $\Add$ as a $k$-subgroup!  
%(Hint: use Exercise U.5(iii).)  For a salvage, see \cite[B.1.10]{pred}:
%a $p$-torsion smooth connected commutative affine group over
%any field of characteristic $p > 0$ admits an \'etale isogeny onto a vector group.
%
%(iii) Generalize (i) and (ii) with $\Gm$ replaced by any nontrivial $k'$-torus $T'$.
%(Hint: use the equality ${\rm{Gal}}(k_s/k) = {\rm{Gal}}(k'_s/k')$ to descend
%$T'$ to a $k$-torus, and extend scalars to $k_s$ to reduce problems to the
%case of $\Gm$).

\medskip\noindent
U.6.  (i) Prove that if a connected scheme $X$ of finite type over a field $k$ has a $k$-rational point,
then $X_{k'} = X \otimes_k k'$ is connected for every finite extension $k'/k$
(hint: $X_{k'} \rightarrow X$ is open and closed; look at a fiber over
$X(k)$).  Deduce that $X_{k'}$ is connected
for {\em every} extension $k'/k$ (i.e., $X$ is {\em geometrically connected} over $k$).

(ii) Prove that if $X$ and $Y$ are geometrically connected of finite type over $k$, so
is $X \times Y$; give a counterexample over $k = \Q$ if ``geometrically'' is removed.   
Deduce that if $G$ is a $k$-group then the identity component $G^0$
is a $k$-subgroup whose formation commutes with any extension on $k$. 

\medskip\noindent
U.7.   Let $X$ be a scheme over a field $k$, and $x \in X(k)$.
Recall that ${\rm{Tan}}_x(X)$ is identified as a set with the fiber of
$X(k[\epsilon]) \rightarrow X(k)$ over $x$. 
Let $k[\epsilon,\epsilon'] = k[t,t']/(t,t')^2$, so this is 3-dimensional 
over $k$ with basis $\{1, \epsilon, \epsilon'\}$.

(i)  For $c \in k$, consider the $k$-algebra endomorphism of $k[\epsilon]$
defined by $\epsilon \mapsto c \epsilon$.  Show that the resulting endomorphism of
$X(k[\epsilon])$ over $X(k)$ restricts to scalar multiplication by $c$ on 
${\rm{Tan}}_x(X)$. 

(ii) Using the two natural quotient maps $k[\epsilon, \epsilon'] \twoheadrightarrow k[\epsilon]$
via killing $\epsilon$ or $\epsilon'$, define a natural map 
$$X(k[\epsilon, \epsilon']) \rightarrow
X(k[\epsilon]) \times_{X(k)} X(k[\epsilon])$$
and prove it is bijective.  Using the natural quotient map $k[\epsilon, \epsilon'] \twoheadrightarrow
k[\epsilon]$ defined by $\epsilon, \epsilon' \mapsto \epsilon$, show that the resulting map
$$X(k[\epsilon]) \times_{X(k)} X(k[\epsilon]) \stackrel{\simeq}{\leftarrow} 
X(k[\epsilon, \epsilon']) \rightarrow X(k[\epsilon])$$
induces addition on ${\rm{Tan}}_x(X)$: 
the {\em functor of points} of $X$ encodes the $k$-linear structure on ${\rm{Tan}}_x(X)$!

(iii) For a $k$-group $(G,e)$, relate addition on ${\rm{Tan}}_e(G)$ to the group 
law as follows:  for $m:G \times G \rightarrow G$, show that
$${\rm{Tan}}_e(G) \times {\rm{Tan}}_e(G) =
{\rm{Tan}}_{(e,e)}(G \times G) \stackrel{{\rm{d}}m(e,e)}{\rightarrow} {\rm{Tan}}_e(G)$$ 
is addition. 

(iv) For a representable functor $F \simeq \Hom_k(\cdot,X)$ on the category of $k$-schemes, 
if the natural map $\varinjlim F(A_i) \rightarrow F(\varinjlim A_i)$ is
bijective for any directed system of $k$-algebras $\{A_i\}$
then $X$ is locally of finite type over $k$.
Prove this if $X$ is affine, and for the general case see \cite[IV$_3$, 8.14.2]{ega}. 
In particular, for such $F$ the fibers of $F(k[\epsilon]) \rightarrow F(k)$
are {\em  finite-dimensional} over $k$ when equipped
with their natural $k$-linear structure as given in functorial terms in (i) and (ii). 
Using
this, deduce from Exercise U.2(iii) that $\underline{{\rm{End}}}_{k\mbox{-}{\rm{gp}}}(\Add)$
and $\underline{\Aut}_{k\mbox{-}{\rm{gp}}}(\Add^n)$ (with $n > 0$) are {\em not} 
representable when ${\rm{char}}(k) = p > 0$.

\medskip\noindent
U.8.  Let $U$ be a unipotent smooth connected affine group over
a field $k$ with ${\rm{char}}(k) = p > 0$.  Let
$k'/k$ be a purely inseparable finite extension such that $U_{k'}$ is $k'$-split.

(i) Choose $n \ge 1$ such that ${k'}^{p^n} \subset k$.
Prove that the $n$-fold base change $U^{(p^n)}$ through
the $p^n$-power endomorphism of $k$ is $k$-split.

(ii) For any $m \ge 1$ and any $k$-group scheme $G$,
prove that the $m$-fold relative Frobenius morphism $F_{G/k,m}:G \rightarrow G^{(p^m)}$ over $k$
is a $k$-homomorphism, and that it 
is a finite flat surjection when $G$ is also smooth (see \cite[Exer.\:1.6.8]{sga3notes}). Use 
$F_{U/k,n}$ to prove
that if $U \ne 1$ then $\Hom_{k\mbox{-}{\rm{gp}}}(U,\Add) \ne 1$.

\medskip\noindent
U.9.  Let $U$ be a unipotent smooth connected
group over a field $k$ with any characteristic. Recall from \cite[11.5(2), 10.5]{borelag} that
there exists a closed immersion of $k$-groups $j:U \hookrightarrow U_N$ for some $N > 0$.
Assume $k$ is {\em perfect}.

(i) If $G$ is a $k$-group scheme of finite
type, prove that $G_{\rm{red}}$ is a smooth $k$-subgroup of $G$.  (Note by Exercise U.3
that perfectness must be used to prove that $G_{\rm{red}}$ is a $k$-subgroup scheme.)

(ii) If $\dim U = 1$, construct a finite Galois extension $k'/k$ such that
$U_{k'} \simeq \Add$.  (Hint: by perfectness, $\overline{k}$ is exhausted by subfields that are finite Galois
over $k$).   Using the vanishing of ${\rm{H}}^1({\rm{Gal}}(k'/k),{k'}^{\times})$ to build
a $k$-isomorphism $U \simeq \Add$.

(iii) Using a ``$\Add$ composition series'' for $U_N$ and induction on $\dim U$ (and (i)),
construct a composition series
for $U$ consisting of smooth connected $k$-subgroups such that the successive quotients
are 1-dimensional (and unipotent).  Deduce with the help of (ii) that $U$ is $k$-split. 

\medskip\noindent
U.10.  Let $k$ be an imperfect field of characteristic $p > 0$.
Let $k''/k$ be a purely inseparable finite extension
such that  ${k''}^{p^2} \subset k$ and $k' := k'' \cap k^{1/p} \ne k''$. Let $U = {\rm{R}}_{k''/k}(\Gm)/\Gm$.

(i)  For any smooth connected affine $k'$-group $G'$, prove that the natural map ${\rm{R}}_{k'/k}(G')_{k'} \rightarrow G'$
defined functorially on $k'$-algebras by $G'(k' \otimes_k A') \rightarrow G'(A')$ is a smooth surjection
with $k'$-split unipotent smooth connected kernel.  Describe
$(U_{k'})_{\rm{split}}$ and $U_{k'}/(U_{k'})_{\rm{split}}$, and show each is $p$-torsion and nontrivial.
Deduce that $U_{k'} \rightarrow U_{k'}/(U_{k'})_{\rm{split}}$ has no $k'$-homomorphic section,
so $U_{k'}$ is {\em not} a direct product of split and wound $k'$-groups.

(ii) Show that $(U_{k'})_{\rm{split}}$ is the cc$k'p$-kernel of $U_{k'}$ whereas
${\rm{R}}_{k'/k}(\Gm)/\Gm$ is the cc$kp$-kernel of $U$. (Hint: compute on $k'_s$-points and $k_s$-points respectively.)
Why does this illustrate the failure of the formation of the cc$kp$-kernel to commute with
non-separable extension on $k$, and why is the non-smoothness of the $p$-torsion
a necessary condition for any such example?

(iii) Does there exist a unipotent smooth connected $k$-group that is not an extension
of a $k$-split group by a $k$-wound group, perhaps even a commutative example?


\begin{thebibliography}{ram}

%\bibitem[Bo1]{borelaff} A.\,Borel, {\em On affine algebraic homogeneous spaces},
%Arch. Math. {\bf 45}(1985), 74--78. 

\bibitem[Bo]{borelag} A.\,Borel, {\em Linear algebraic groups}  
(2nd ed.) Springer-Verlag, New York, 1991.


%\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[BoTi1]{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]{lie3} N.\,Bourbaki, {\em Lie groups and Lie algebras} (Ch.\:1--3), Springer-Verlag, New York, 1989. 
%
%\bibitem[Bou2]{bourbaki} N.\,Bourbaki,
%{\em Lie groups and Lie algebras} (Ch.\:4--6), Springer-Verlag, New York, 2002.
%
%\bibitem[Bri]{brion} M. Brion, {\em Anti-affine algebraic groups}, Journal of Algebra {\bf 321}(2009),
%934--952.
%
%\bibitem[BrTi]{BTI} F.\,Bruhat, J.\,Tits, {\em Groupes r\'eductifs sur un corps local I}, Publ.\,Math.\,IHES\,{\bf 41}(1972), 5--251.
% 
%
%\bibitem[Chev]{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[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[Con1]{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[Con2]{conrad} B.\,Conrad, {\em Finiteness theorems for algebraic groups over function fields}, 
%in preparation (2010).

\bibitem[CGP]{pred} B.\,Conrad, O.\,Gabber, G.\,Prasad, {\em Pseudo-reductive groups},
Cambridge University Press, Cambridge, 2010.

\bibitem[C]{sga3notes} B.\,Conrad, {\em Reductive group schemes}, these Proceedings. 

%
%
%\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[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 $1$}, 
%Paris, 1961. 
%
%\bibitem[Hum1]{humlie} J.\,Humphreys, {\em Introduction to Lie algebras and representation theory}, 
%Springer--Verlag, New York, 1972. 

\bibitem[Hum]{humphreys} J.\,Humphreys, {\em Linear algebraic groups} (2nd ed.), 
Springer--Verlag, New York, 1987.

%\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[Mat]{crt} H.\,Matsumura, {\em Commutative ring theory}, Cambridge Univ.\,Press, 1990.
%
%\bibitem[Mum]{mumford} D.\,Mumford, {\em Abelian varieties}, Oxford Univ.\,Press, 1970.
%

\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[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[PY2]{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[Ros]{rosenlicht} M.\,Rosenlicht, {\em Questions of rationality for solvable
groups over non-perfect fields}, Ann. Mat. Pura Appl. (4) {\bf 61} (1963), 97--120.

%
%
%\bibitem[Ri]{ri} R.\,W.\,Richardson, {\em Affine coset spaces of reductive algebraic groups},
%Bull.\, London Math.\, Soc.\,{\bf 9}(1977), 38--41. 
%
%\bibitem[Ru]{russell} P.\,Russell, {\em Forms of the affine line and its additive group},
%Pacific Journal 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[Ser]{serrecft} J-P.\, Serre {\em Algebraic groups and class fields}, Springer-Verlag, New York, 1988. 

%
%\bibitem[Ser2]{serrelie} J-P.\,Serre, {\em Lie groups and Lie algebras}, LNM 1500, Springer-Verlag, New York, 1992. 
%
%\bibitem[Ser3]{serre} J-P.\,Serre, {\em Galois cohomology}, Springer-Verlag, New York, 1997.

\bibitem[Spr]{springer} T.\,A.\,Springer, {\em Linear algebraic groups} (2nd ed.), Birkh\"auser,
New York, 1998.  
%
%\bibitem[Ti0]{tits66} J.\,Tits, ``Classification of algebraic semisimple groups''
%in {\em Algebraic groups and discontinuous groups}, Proc.\,Symp.\,Pure Math.,vol.\,9, AMS, 1966.


\bibitem[Ti]{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.
%


\end{thebibliography}



\end{document}