\documentclass[a4paper,10pt,twoside]{article}
%\marginparwidth=15pt
\oddsidemargin=19pt
\evensidemargin=19pt
\textwidth=411pt
\textheight=620pt
% PACKAGES
\usepackage[latin1]{inputenc}
% RAVI's FONT.
%\usepackage{mathpazo}
%\usepackage{euler}
\newcommand{\marginalfootnote}[1]{
\footnote{#1}
\marginpar{\hfill {\sf\thefootnote}}}
\newcommand{\edit}[1]{\marginalfootnote{#1}}
\DeclareFontFamily{OT1}{rsfs}{}
\DeclareFontShape{OT1}{rsfs}{n}{it}{<-> rsfs10}{}
\usepackage[OT2, T1]{fontenc}
\usepackage[UKenglish]{babel}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{fancyhdr}
\usepackage{color}
\usepackage{makeidx}
\usepackage{amsmath}
\usepackage[all,cmtip]{xy}
\usepackage{blindtext}
\usepackage{parskip}
\usepackage{hyperref}
\usepackage[notlof,notlot]{tocbibind}
\usepackage{mathrsfs}
\usepackage{tikz}
%%%<
\usepackage{verbatim}
\usepackage{verbatim}
%\usepackage[active,tightpage]{preview}
%\PreviewEnvironment{tikzpicture}
%\setlength\PreviewBorder{5pt}%
%%%>
\usepackage{fancyhdr}
\pagestyle{fancy}
%\addtocounter{section}{24}
%\setlength{\parindent}{0pt}
%\setlength{\headheight}{15.2pt}
%\renewcommand{\sectionmark}[1]{\markboth{#1}{}}
\fancyhf{} % sets both header and footer to nothing
\renewcommand{\headrulewidth}{0pt}
\lhead{}
\fancyhead[OC]{\footnotesize\textsc{\'etale cohomology of algebraic number fields}}
\fancyhead[CE]{\footnotesize\textsc{brian conrad \ \ \ alessandro maria masullo}}
\fancyhead[LE,OR]{{\footnotesize\thepage}}
%\lfoot{}
%\cfoot{}
%\rfoot{}
\makeindex
\definecolor{red}{rgb}{1,0,0}
\definecolor{lblue}{rgb}{0.3,0.4,1}
\definecolor{orange}{rgb}{1,0.6,0.2}
\definecolor{green}{rgb}{0.1,0.7,0.1}
% MEANINGS
\newtheoremstyle{thm}% name of the style to be used
{3pt}% measure of space to leave above the theorem. E.g.: 3pt
{3pt}% measure of space to leave below the theorem. E.g.: 3pt
{\em}% name of font to use in the body of the theorem
{0pt}% measure of space to indent
{\bfseries}% name of head font
{}% punctuation between head and body
{5pt}% space after theorem head
{}% Manually specify head
\newtheoremstyle{rem}% name of the style to be used
{3pt}% measure of space to leave above the theorem. E.g.: 3pt
{3pt}% measure of space to leave below the theorem. E.g.: 3pt
{}% name of font to use in the body of the theorem
{0pt}% measure of space to indent
{\bfseries}% name of head font
{}% punctuation between head and body
{5pt}% space after theorem head
{}% Manually specify head
% MACROS
\theoremstyle{thm}
%\newtheorem*{them}{Theorem}
\newtheorem{thm}{Theorem}[subsection]
\newtheorem{prop}[thm]{Proposition}%[subsection]
%\newtheorem*{propn}{Proposition}
\newtheorem{cor}[thm]{Corollary}%[subsection]
\newtheorem{lemma}[thm]{Lemma}%[subsection]
\theoremstyle{rem}
\newtheorem{eg}[thm]{Example}%[subsection]
\newtheorem{defin}[thm]{Definition}%[subsection]
\newtheorem{rem}[thm]{Remark}%[subsection]
% ADDITIONAL SYMBOLS
\newcommand\ab{{\rm ab}}
\newcommand\ad{{\rm ad}}
\newcommand\aff{{\rm aff}}
\newcommand\alg{{\rm alg}}
\newcommand\ant{{\rm ant}}
\newcommand\codim{{\rm codim}}
\newcommand\diag{{\rm diag}}
\newcommand\et{{\rm et}}
\newcommand\gr{{\rm gr}}
\newcommand\gp{{\rm gp}}
\newcommand\coker{{\rm coker}}
\newcommand\id{{\rm id}}
\newcommand\lin{{\rm lin}}
\renewcommand\log{{\rm log}}
\newcommand\op{{\rm op}}
\newcommand\ord{{\rm ord}}
\newcommand\red{{\rm red}}
\newcommand\reg{{\rm reg}}
\newcommand\rk{{\rm rk}}
\newcommand\Aff{{\rm Aff}}
\newcommand\Aut{{\rm Aut}}
\newcommand\Bl{{\rm Bl}}
\newcommand\End{{\rm End}}
\newcommand\GL{{\rm GL}}
\newcommand\Ga{{\mathbb G}_a}
\newcommand\Gm{{\mathbb G}_m}
\newcommand\Gal{{\rm Gal}}
\newcommand\Gr{{\rm Gr}}
\newcommand\Hilb{{\rm Hilb}}
\newcommand\Quot{{\rm Quot}}
\newcommand\fX{{\mathfrak{X}}}
\newcommand\fY{{\mathfrak{Y}}}
\newcommand\ccF{{\rm \mathcal{F}}}
\newcommand\ccD{{\rm \mathcal{D}}}
\newcommand\ccO{{\rm \mathcal{O}}}
\newcommand\Spf{\mathrm{Spf}}
\newcommand\cSpec{\underline{\mathrm{Spec}}}
\newcommand\disc{{\mathrm{disc}}}
\newcommand\Frob{{\mathrm{Frob}}}
\newcommand\rR{{\mathrm{R}}}
\newcommand\Tate{{\mathrm{Tate}}}
\newcommand\fG{{\mathfrak{G}}}
\newcommand\fH{{\mathfrak{H}}}
\newcommand\Ann{{\rm Ann}}
\newcommand\Spv{{\rm Spv}}
\newcommand\RZ{{\rm RZ}}
\newcommand\Cont{{\rm Cont}}
\newcommand\Hom{{\rm Hom}}
\newcommand\Ind{{\rm Ind}}
\newcommand\Irr{{\rm Irr}}
\newcommand\Lie{{\rm Lie}}
\newcommand\M{{\rm M}}
\newcommand\Mor{{\rm Mor}}
\newcommand\varpic{{\rm Pic}}
\newcommand\Proj{{\rm Proj}}
\newcommand\PGL{{\rm PGL}}
\newcommand\PSL{{\rm PSL}}
\newcommand\SL{{\rm SL}}
\newcommand\Spec{{\rm Spec}}
\newcommand\Spa{{\rm Spa}}
\newcommand\Sp{{\rm Sp}}
\newcommand\Supp{{\rm Supp}}
\newcommand\Sym{{\rm Sym}}
\newcommand\Univ{{\rm Univ}}
\newcommand\Ass{{\rm Ass}}
\newcommand\Frac{{\rm Frac}}
\newcommand\Tor{{\rm Tor}}
\newcommand\cTor{{\mathcal Tor}}
\newcommand\cHom{{\underline{\rm Hom}}}
\newcommand\RSchemes{{\bfseries aff} $R$-{\bfseries Sch}}
\newcommand\Sch{{\mathbf{Sch}}}
\newcommand\Top{{\mathbf{Top}}}
\newcommand\Cov{{\mathrm{Cov}}}
\newcommand\Sets{\mathbf{Set}}
\newcommand\Func{\mathbf{Func}}
\newcommand\Pref{{\bfseries Presh}_X}
\newcommand\Sheaf{{\bfseries She}_X)}
\newcommand\Shv{\mathbf{Shv}}
\newcommand\PShv{\mathbf{PShv}}
\newcommand\Open{{\mathbf{Op}}_X}
\newcommand\Ab{{\mathbf{Ab}}}
\newcommand\Grps{\mathbf{Grp}}
\newcommand\RMod{\mathbf{Mod}_R}
\newcommand\RAlg{\mathbf{Alg}_R}
\newcommand\Ob{\textup{Ob}}
\newcommand\bA{{\mathbb A}}
\newcommand\bC{{\mathbf C}}
\newcommand\bF{{\mathbf F}}
\newcommand\bG{{\mathbb G}}
\newcommand\bI{{\mathbf I}}
\newcommand\bN{{\mathbf N}}
\newcommand\bP{{\mathbb P}}
\newcommand\bQ{{\mathbf Q}}
\newcommand\bR{{\mathbf R}}
\newcommand\bS{{\mathbb S}}
\newcommand\bV{{\mathbf V}}
\newcommand\bX{{\mathbf X}}
\newcommand\bZ{{\mathbf Z}}
\newcommand\cA{{\mathscr A}}
\newcommand\cB{{\mathscr B}}
\newcommand\cC{{\mathscr C}}
\newcommand\cD{{\mathscr D}}
\newcommand\cE{{\mathscr E}}
\newcommand\cF{{\mathscr F}}
\newcommand\cG{{\mathscr G}}
\newcommand\cH{{\mathscr H}}
\newcommand\cI{{\mathscr I}}
\newcommand\cJ{{\mathscr J}}
\newcommand\cL{{\mathscr L}}
\newcommand\cM{{\mathscr M}}
\newcommand\cN{{\mathscr N}}
\newcommand\cO{{\mathscr O}}
\newcommand\cQ{{\mathscr Q}}
\newcommand\cR{{\mathscr R}}
\newcommand\cS{{\mathscr S}}
\newcommand\cT{{\mathscr T}}
\newcommand\cV{{\mathscr V}}
\newcommand\cW{{\mathscr W}}
\newcommand\cX{{\mathscr X}}
\newcommand\cY{{\mathscr Y}}
\newcommand\cZ{{\mathscr Z}}
\newcommand\cProj{{\mathcal Proj}}
\newcommand\fg{{\mathfrak g}}
\newcommand\fh{{\mathfrak h}}
\newcommand\ft{{\mathfrak t}}
\newcommand\fD{{\mathfrak D}}
\newcommand\fS{{\mathfrak S}}
\newcommand\fm{{\mathfrak m}}
\newcommand\fn{{\mathfrak n}}
\newcommand\fp{{\mathfrak p}}
\newcommand\fq{{\mathfrak q}}
\newcommand\ccU{{\mathfrak U}}
\newcommand\cU{{\mathscr U}}
\newcommand\bJ{{\mathbf J}}
\newcommand\bH{{\mathbf H}}
\newcommand\fA{{\mathfrak A}}
\newcommand\fB{{\mathfrak B}}
\newcommand\fC{{\mathfrak C}}
\newcommand\SR{{\fS(R_{\mathrm{fpqc}})}} % descent 1
\newcommand\tg{\tilde{g}}
\renewcommand\th{\tilde{h}}
\newcommand\tlambda{\tilde{\lambda}}
\newcommand\tp{\tilde{p}}
\newcommand\tq{\tilde{q}}
\newcommand\tG{\tilde{G}}
\newcommand\tM{\tilde{M}}
\newcommand\tV{\tilde{V}}
\newcommand\tX{\tilde{X}}
\newcommand\tY{\tilde{Y}}
\newcommand\tZ{\tilde{Z}}
\newcommand{\fAb}{{\mathfrak{Ab}}}
\newcommand\tcZ{\tilde{\mathcal Z}}
\newcommand\Pic{{\rm Pic}}
\newcommand\ualpha{\underline{\alpha}}
\newcommand\ulambda{\underline{\lambda}}
\newcommand\BT{\mathbf{BT}}
\newcommand\fotimes{{\widehat{\otimes}}}
\newcommand\Inf{{\mathrm{Inf}}}
\newcommand\uomega{{\underline{\omega}}}
\newcommand\Ext{{\mathrm{Ext}}}
\newcommand\rH{{\rm H}}
\newcommand\rV{{\rm V}}
\newcommand\rT{{\rm T}}
\newcommand\fa{{\mathfrak a}}
\newcommand\rad{{\mathrm{rad}}}
\newcommand\bD{{\mathbf{D}}}
\newcommand\ccG{{\mathcal{G}}}
\newcommand\ob{{\rm ob}}
\DeclareSymbolFont{cyrletters}{OT2}{wncyr}{m}{n}
\DeclareMathSymbol{\Sha}{\mathalpha}{cyrletters}{"58}
\newcommand\bmu{{\mathbb \mu}}
\newcommand\Tr{{\rm Tr}}
\newcommand\dd{{\rm d}}
\newcommand\bL{{\mathbf{L}}}
\newcommand\Exal{{\mathbf{Exal}}}
\newcommand\rD{{\mathrm{D}}}
\newcommand\Der{{\mathrm{Der}}}
\newcommand\Mod{{\mathbf{Mod}}}
%\renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}}
%\bFancyhf{} % delete current setting for header and footer
%\bFancyhead[LE,RO]{\thepage}
%\bFancyhead[LO]{\bfseries\rightmark}
%\bFancyhead[RE]{\bfseries\leftmark}
%\renewcommand{\headrulewidth}{0.5pt}
%\renewcommand{\bFootrulewidth}{0pt}
%\addtolength{\headheight}{0.5pt} % make space for the rule
%\bFancypagestyle{plain}{%
% \bFancyhead{} % get rid of headers on plain pages
% \renewcommand{\headrulewidth}{0pt} % and the line
%}
\definecolor{mikadoyellow}{rgb}{1.0, 0.77, 0.05}
\definecolor{mahogany}{rgb}{0.80, 0.20, 0.0}
\definecolor{palemagenta}{rgb}{1.0, 0.48, 1.1}
\definecolor{palered-violet}{rgb}{0.90, 0.41, 0.61}
\hypersetup{
colorlinks = true,
linktoc = page,
citecolor = blue,
linkcolor = palered-violet,
}
\usepackage{tocloft}
\usepackage{titlesec}
\titleformat{\section}[block]{\scshape\filcenter}{\thesection}{1em}{}
\titleformat{\subsection}[block]{\scshape\filcenter}{\thesubsection}{1em}{}
\renewcommand{\cfttoctitlefont}{\hfill\scshape}
\renewcommand{\cftaftertoctitle}{\hfill\hfill}
\renewcommand\cftsecfont{\normalfont}
\renewcommand\cftsecpagefont{\normalfont}
\setlength{\cftbeforesecskip}{0pt}
%\renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}}
%\fancyhf{} % delete current setting for header and footer
%\fancyhead[LE,RO]{\thepage}
%\fancyhead[LO]{\bfseries\rightmark}
%\fancyhead[RE]{\bfseries\leftmark}
%\renewcommand{\headrulewidth}{0.5pt}
%\renewcommand{\footrulewidth}{0pt}
%\addtolength{\headheight}{0.5pt} % make space for the rule
%\fancypagestyle{plain}{%
% \fancyhead{} % get rid of headers on plain pages
% \renewcommand{\headrulewidth}{0pt} % and the line
%}
\title{\textbf{\Large \textsc{\'etale cohomology of algebraic number fields}}}
\author{{\ \ \footnotesize BRIAN CONRAD \ \ \ \ ALESSANDRO MARIA MASULLO}}
\date{}
\begin{document}
\maketitle
{
\abstract{We use \'etale cohomology to obtain a conceptual proof of global Tate duality by means of Artin--Verdier duality, elaborating on Zink's
method of appending ``real points'' to $\Spec(\ccO_K)$ to extend Mazur's proof in the totally imaginary case.\\
It is organized in six sections. \\
After some introductory review in \S\ref{intro},
including statements of Tate's main theorems, in \S\ref{secet} we discuss the foundations of the \'etale topology on algebraic number fields and its modification to allow consideration of real places. In \S\ref{cohomsec} we deal with the cohomology theory arising from what we call extended small \'etale sites, introduced in
\S\ref{secet}. \\
In \S\ref{secfinite} we prove one of the main theorems (including a finiteness aspect) via a series of explicit computations and reductions.\\
In \S\ref{avsec} we prove the basics of Artin--Verdier duality. Finally, in \S\ref{tatesec} we use Artin-Verdier duality to recover Tate's theorems, including the 9-term long exact sequence for finite ramification as a local cohomology sequence.}
}
%\setcounter{tocdepth}{1}
\tableofcontents
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{intro}
\subsection{Some history}
Given a field $k$ and a separable closure $\overline{k}$ of $k$, we denote by $G_k$ the Galois group $\Gal(\overline{k}/k)$. We consider the category
$\Mod_{G_k}$ of discrete
$G_k$-modules (called ``Galois modules over $k$'', or simply ``Galois modules'' when $k$ is understood).
A very interesting $G_k$-module is $\overline{k}^{\times}$, which plays an essential role in $\Mod_{G_k}$ when $k$ is a non-archimedean local field.
Our main interest will be the case when $k$ is a number field, but first we discuss some more basic cases that feed into the study of the number field case.
Suppose first $k$ is a non-archimedean local field. Let $M$ be a finite Galois module over $k$, and define
$$M^{\vee} := \Hom_{{\rm{Ab}}}(M,\overline{k}^{\times}) = \Hom_{{\rm{Ab}}}(M, \mu_{\infty})$$
where $\mu_{\infty}$ is the Galois module of roots of unity in $\overline{k}^{\times}$;
$M^{\vee}$ is a finite Galois module in the evident manner. Tate local duality says that cup-product induces a perfect pairing
of {\em finite} abelian groups
$$\rH^i(G_k, M)\times \rH^{2-i}(G_k, M^{\vee}) \to\rH^2(G_k,\overline{k}^{\times}) \xrightarrow{\simeq} \bQ/\bZ$$
for $0 \le i \le 2$ and all finite $M$ of order prime to the characteristic of $k$. Here, the isomorphism $\rH^2(G_k,\overline{k}^{\times})\simeq\bQ/\bZ$ comes from local class field theory. Tate also proved $$\chi(M) := \frac{h^0(M)h^2(M)}{h^1(M)} = \frac{1}{|\#M|}$$ for the normalized $|\cdot|$ on $k$
(with $h^i(M) := \#{\rm{H}}^i(G_K, M)$).
These results are proved in \S5.2 of Chapter II of Serre's book {\em Galois cohomology} using local class field theory
and general results on the cohomology of profinite groups; the same methods are also used there to show
that for such $M$ the cohomology $\rH^i(G_k, M)$ vanishes for $i > 2$ (so $G_k$ has cohomological dimension 2 relative
to torsion coefficients with torsion-orders prime to ${\rm{char}}(k)$). There is also a duality result when $p = {\rm{char}}(k) > 0$
and $p|\#M$ but then it involves Cartier duality and fppf cohomology, so we do not discuss that here. The analogue
for archimedean local fields is elementary but will be important in some later considerations:
\begin{rem}\label{realtate}
The field $k = \bR$ admits a duality using Tate cohomology: for a finite $G_k$-module $M$, the cup product
$$\rH^i_T(G_k, M) \times \rH^{2-i}_T(G_k, M^{\vee}) \rightarrow \rH^2_T(G_k, \overline{k}^{\times}) =
\rH^2(G_k, \overline{k}^{\times}) = {\rm{Br}}(k) = \bZ/2\bZ$$
is a perfect pairing of finite groups (with $\rH^{\bullet}_T(G, \cdot)$ denoting Tate cohomology for a finite group $G$). To prove this,
note that since the Tate cohomologies are 2-torsion, $\Hom(\cdot, \bZ/2\bZ)$ is exact
on long exact cohomology sequences in such Tate cohomologies. We may thereby reduce to the case
when $M$ is a simple $G_k$-module, so it is killed by some prime $\ell$.
If $\ell$ is odd then the Tate cohomologies vanish and there is nothing to do.
Hence, we can assume $M$ is 2-torsion. Since $G_k$ has order 2, it follows that $M$ has a composition series
of $G_k$-submodules whose successive quotients are $\bZ/2\bZ$. In this way we may
assume $M = \bZ/2\bZ$.
Let $\omega\in\rH^2(G_k, \bZ)$ be the unique generator (an element of order 2). By the theory of Tate cohomology for cyclic groups,
cup product with $\omega$ induces and isomorphism
${\rH}_T^p(G_k, N)\to{\rH}_T^{p+2}(G_k, N)$ for every $G_k$-module $N$.
Using the $\delta$-functoriality of Tate cohomology cup products, the pairing
$${\rH}_T^{n}(G_k, M)\times{\rH}_T^m(G_k, M^{\vee})\to\rH^{n+m}(G_k, \overline{K}^{\times})$$
depend on $n$ and $m$ only through their parities. Hence, it suffices
to prove that
$$\rH^{-n}_T(G_k, \bZ/2\bZ) \times \rH^n_T(G_k, \bZ/2\bZ) \rightarrow \rH^0_T(G_k, \bZ/2\bZ) = \rH^0_T(G_k, \overline{K}^{\times})$$
(final equality via inclusion of $\mu_2$ into $\overline{K}^{\times}$!)
is perfect for $n=0,1$. This is a trivial verification.
\end{rem}
\begin{rem}
If $k$ is a finite field, so $G_k = \widehat{\bZ}$ has cohomological dimension 1 (and strict cohomological dimension 2),
the ``dualizing module'' for $G_k$ is $\bQ/\bZ$ (playing a role analogous to that of $\mu_{\infty}$
for local fields above). Hence, we consider the Pontryagin dual Galois module $M^D := \Hom(M,\bQ/\bZ)$ and we get that for all finite $M$
(no condition on $\gcd(\#M, {\rm{char}}(k))$!) there is a perfect pairing
of finite abelian groups
$\rH^i(G_k, M)\times\rH^{1-i}(G_k, M)\to {\rm{H}}^1(G_k, \bQ/\bZ) = \bQ/\bZ$
induced by cup-product. One also has $\chi(M) := h^0(M)/h^1(M) = 1$ from the classical theory of Herbrand quotients for
the cohomology of (pro-)cyclic groups.
%
%\begin{rem}\label{finiteext}
%A refinement of the duality theorem for finite fields is that for any finite discrete $G_k$-module $M$,
%there is a natural isomorphism ${\rm{Ext}}^i_{G_k}(M, \bQ/\bZ) \simeq {\rm{H}}^i(G_k, M^D)$ (so in particular
%${\rm{Ext}}^i_{G_k}(M, \bQ/\bZ) = 0$ for $i \ge 2$). To see this, the key point is to show that the identification
%${\rm{Hom}}(M, N)^{G_k} \simeq {\rm{Hom}}_{G_k}(M, N)$ for discrete $G_k$-modules $N$
%(note that ${\rm{Hom}}(M, N)$ is discrete as a $G_k$-module due to the finiteness of $M$) promotes
%to a spectral sequence
%$${\rm{H}}^i(G_k, {\rm{Ext}}^j(M, N)) \Rightarrow {\rm{Ext}}^{i+j}_{G_k}(M, N);$$
%once that is built, setting $N$ to be the injective abelian group $\bQ/\bZ$ makes the ${\rm{Ext}}^j$'s vanish for $j > 0$,
%so this collapses to the desired isomorphism.
%
%Making the spectral sequence entails showing that ${\rm{Hom}}(M, I)$ is $G_k$-acyclic for any injective
%abelian group $I$. We may reduce to the case that $M$ is killed by a prime $\ell$, and then ${\rm{Hom}}(M, I) =
%{\rm{Hom}}(M, I[\ell])$ with $I[\ell]$ clearly injective in the category of discrete $\mathbf{F}_{\ell}[G_k]$-modules.
%We can compute the $G_k$-cohomology of an $\ell$-torsion object by working with derived functors
%on that category, so the original acyclic problem is reduced to the case of $\ell$-torsion objects.
%But then ${\rm{Hom}}(M, I)$ is even an injective discrete $G_k$-module, since
%$${\rm{Hom}}_{G_k}(\cdot, {\rm{Hom}}(M, I)) = {\rm{Hom}}_{G_k}(M \otimes_{\mathbf{F}_{\ell}}(\cdot), I)$$
%is a composition of exact functors and hence is exact.
\end{rem}
\medskip
In the case $k$ is a global field, there is a duality principle, called global Tate duality and first announced by Tate
at the 1962 ICM \cite{Tate}. He never published his proofs, which were later documented in \cite{Milne} and \cite{Hab}.
Based entirely on techniques in Galois cohomology, the proofs (and even the some aspects of the statements) were sufficiently complicated that
they did not provide a great conceptual
insight into ``why'' they held (in contrast with the duality above for local fields, for which the statements are more concrete and calculations
with $\mu_n$ have explicit meaning that leads to the general case).
As first worked out by Mazur in the totally imaginary case in \cite{Mazur},
the formalism of Artin--Verdier duality provides an elegant approach to Tate's main global results; e.g., the famous ``9-term exact sequence''
(see Theorem \ref{9term}) which seems so mysterious in the traditional
formulation (as it was originally built in a somewhat ad hoc manner as a device to record a few
other dualities built by prior means)
is obtained directly as a long exact ``local cohomology'' sequence.
In these notes we use or extend arguments from \cite{Mazur}
to recover Tate's results on global duality for algebraic number fields via \'etale cohomology
on spectra of rings of $S$-integers
augmented with some ``real places at infinity'', expanding on ideas of Zink introduced in the Appendix of \cite{Hab}.
This provides topological insight into Tate's results that make ${\rm{Spec}}(\ccO_{K,S})$ for a number field $K$
(and finite set $S$ of places of $K$ containing the archimedean places) behave as if it is a 3-manifold.
The analogy with 3-manifolds
can be seen directly in the function field case: if $X$ is a smooth proper and geometrically
connected curve over a finite field $\mathbf{F}$ of characteristic $p$ then for a dense open $j:U \hookrightarrow X$ the structure map
$f:U \rightarrow {\rm{Spec}}(\mathbf{F})$ is viewed as a ``family'' of open Riemann surfaces fibered over a loop, ``hence'' of dimension 3
for cohomological purposes with torsion coefficients away from $p$. More specifically, we have
the Leray spectral sequence ${\rm{H}}^n(\mathbf{F}, \rR^mf_{!}(\cdot)) \Rightarrow \rH^{n+m}_c(U, \cdot)$
with $\mathbf{F}$ of cohomological dimension 1 and $f_{!}$ of cohomological dimension $2$
on torsion objects.
\subsection{Summary of results}
First, we set some notation to be fixed for the entire discussion.
We let $K$ be a number field, and $\ccO_K$ its ring of integers. We set $X := \Spec(\ccO_K)$.
Let $S$ be a finite set of places of $K$ that contains
the archimedean places, and denote by $\{v_1, \dots, v_r\}$ the set of real places of $K$ (which may be empty).
As usual, denote by $K_S$ the maximal algebraic extension of $K$ which is unramified outside $S$, and by $G_S$ its Galois group over $K$.
Let $U:= {\rm{Spec}}(\ccO_{K,S}) \subset X$. Since finite extensions of $K$ inside $K_S$ ``correspond''
(via normalization) to connected finite \'etale covers of $U$, we have $G_S = \pi_1(U, \overline{\eta})$ where $\overline{\eta}: {\rm{Spec}}(\overline{K}) \rightarrow U$
is a geometric generic point (over $K_S$).
One of the results we will prove takes care of global cohomology beyond degree 2:
\begin{thm}\label{Tate1} Let $M$ be a finite $G_S$-module whose order is an $S$-unit.
For $i \ge 3$ there is a canonical isomorphism:
$$\rH^i(G_S, M) \to \bigoplus_{j=1}^r\rH^i(D_{v_j}, M)$$
where $D_v$ is a decomposition group at a real place $v$.
\end{thm}
This will emerge as a piece of a local cohomology sequence (the flanking terms
vanishing due to the ``3-manifold'' property of $\Spec(\ccO_{K,S})$); see Theorem
\ref{newtate}. Before going further, we state and prove a basic finiteness result:
\begin{prop}\label{hfinite} If $M$ is finite and $\#M$ an $S$-unit then ${\rm{H}}^i(G_S, M)$ is finite for all $i$.
\end{prop}
Finiteness also holds for
cases with $\#M$ not an $S$-unit, and such $M$ are certainly of much importance in Galois deformation theory (so their
avoidance in Proposition \ref{hfinite} does not mean that such cases of are no interest). Proposition \ref{hfinite} also follows from
a general finiteness result of Artin--Verdier (part of Theorem \ref{dualT}) via a local cohomology argument and a comparison
isomorphism in Appendix \ref{contet} between continuous cohomology of the fundamental group $G_S$ of $\Spec(\ccO_{K,S})$ and \'etale cohomology on
${\rm{Spec}}(\ccO_{K,S})$ with locally constant coefficients; the essential arithmetic input from class field theory remains
the same under both approaches.
\begin{proof} Theorem \ref{Tate1} settles the case $i \ge 3$, so the remaining issue is $i = 1, 2$. If $K'/K$ is a finite Galois splitting field for $M$
(so $K' \subset K_S$) and $S'$ is the set of places of $K'$ over $S$ then the Hochschild--Serre spectral sequence
$${\rm{H}}^i(K'/K, {\rm{H}}^j(G_{S'}, M)) \Rightarrow {\rm{H}}^{i+j}(G_S,M)$$
reduces the finiteness assertion to the case when $M$ has trivial $G_S$-action, so we can assume $M = \bZ/n\bZ$ for some $n >0$
that is an $S$-unit. In a similar manner we can replace $K$ with $K(\zeta_n)$ so that $M = \mu_n$.
Note that $\ccO_{K_S, S}^{\times}$ is $n$-divisible since $n$ is an $S$-unit.
The extension $K_S/K$ is the directed union
of finite-degree Galois subextensions $F/K$, for all of which $\Spec(\ccO_{F,S})$ is a connected finite \'etale Galois cover
of $\Spec(\ccO_{K,S})$. Hence, \'etale descent for line bundles identifies
${\rm{H}}^1(F/K, \ccO_{F,S}^{\times})$ with $\ker({\rm{Pic}}(\ccO_{K,S}) \rightarrow {\rm{Pic}}(\ccO_{F,S}))$.
Passing to the limit over such $F/K$, we get
$${\rm{H}}^1(G_S, \ccO_{K_S, S}^{\times}) = \ker({\rm{Pic}}(\ccO_{K,S}) \rightarrow {\rm{Pic}}(\ccO_{K_S, S})).$$
But every ideal class for $\ccO_{K,S}$ becomes trivial over the ring of $S$-integers of the Hilbert class field of $K$
by the Principal Ideal Theorem, so the above kernel coincides with ${\rm{Pic}}(\ccO_{K,S})$. Hence, the exact sequence
$$1 \rightarrow M \rightarrow \ccO_{K_S, S}^{\times} \stackrel{t^n}{\rightarrow} \ccO_{K_S, S}^{\times} \rightarrow 1$$
of discrete $G_S$-modules provides an exact sequence
$$1 \rightarrow \ccO_{K,S}^{\times}/(\ccO_{K,S}^{\times})^n \rightarrow {\rm{H}}^1(G_S, M) \rightarrow {\rm{Pic}}(\ccO_{K,S})[n] \rightarrow 0.$$
The $S$-unit theorem and finiteness for class groups thereby settle the case $i=1$ for this $M$.
Likewise, by Hilbert 90 the natural inflation map
${\rm{H}}^2(G_S, M) \rightarrow {\rm{H}}^2(K, \mu_n) = {\rm{Br}}(K)[n]$ is injective,
so to prove finiteness it suffices to check that for $v \not\in S$ the image of this injection has
vanishing local restriction into ${\rm{Br}}(K_v)$. This is reduced to the vanishing of
${\rm{H}}^2(K_v^{\rm{un}}/K_v, \mu_n)$, that in turn is immediate
since ${\rm{Gal}}(K_v^{\rm{un}}/K_v) = \widehat{\bZ}$ has cohomological dimension 1.
(Note that in ${\rm{Br}}(K_v) = {\rm{H}}^2(K_v^{\rm{un}}/K_v, (K_v^{\rm{un}})^{\times})$ the $n$-torsion
has {\em nothing to do} with ${\rm{H}}^2(K_v^{\rm{un}}/K_v, \mu_n) = 1$ for $n > 1$ because the $n$-power endomorphism of
$(K_v^{\rm{un}})^{\times}$ is not surjective! That is, there is no $n$-power Kummer sequence at
the level of $K_v^{\rm{un}}/K_v$ when $n > 1$.)
\end{proof}
For each $v \in S$ we denote by $\rH^p(K_v, M)$ the group $\rH^p(D_v, M)$ for finite places $v$ in $S$
and the Tate cohomology group ${\rH}^p_T(D_v,M)$ for real places $v$.
Tate established the following 9-term exact sequence that we will recover as a local cohomology sequence in \'etale cohomology:
\begin{thm}\label{9term} Let $M$ be a finite $G_S$-module such that $\#M$ is an $S$-unit. Then there exists a natural exact sequence:
\begin{displaymath}
\xymatrix{0\ar[r] & \rH^0(G_S, M)\ar[r] & \bigoplus_{v\in S}\rH^0(K_v, M)\ar[r] & \rH^2(G_S, M^{\vee})^D\ar[r]^-{f_M} & \rH^1(G_S,M)\\
\ar[r] & \bigoplus_{v \in S} \rH^1(K_v, M)\ar[r] & \rH^1(G_S, M^{\vee})^D\ar[r]^-{f'_M} & \rH^2(G_S, M)\ar[r] & \bigoplus_{v \in S}\rH^2(K_v, M)\\
\ar[r] & \rH^0(G_S, M^{\vee})^D\ar[r] & 0 & & }
\end{displaymath}
where $M^{\vee} := \Hom(M, \overline{K}^{\times})$ is the dual Galois module, $(\cdot )^D$ denotes the $\bQ/\bZ$-dual of a finite abelian group,
and all maps aside from $f_M$ and $f'_M$ are the evident ones.
\end{thm}
This exact sequence encodes some deep duality theorems. To see this, define
$$\Sha^i_S(K,M) := \ker({\rm{H}}^i(G_S, M) \rightarrow \prod_{v \in S} {\rm{H}}^i(K_v, M))$$
(with the ``modified'' definition for real $v$ with Tate cohomology as noted above).
A ``justification'' for this notation is that if $A$ is an abelian variety over $K$ with good reduction outside $S$
then the analogous definition $\Sha^1_S(K, A) \subset {\rm{H}}^1(G_S, A(K_S))$ actually coincides
with $\Sha(A)$ inside ${\rm{H}}^1(K, A)$ (because
${\rm{H}}^1(K_v^{\rm{un}}/K_v, A(K_v^{\rm{un}})) = 0$ for good $v$,
proved using N\'eronian properties of abelian schemes and Lang's theorem on torsors
for {\em connected} group varieties over finite fields; this local triviality at unramified places has no counterpart
for finite coefficient modules $M$!)
The 9-term exact sequence provides an exact sequence
\begin{equation}\label{sha2}
0\to \Sha_1^2(K,M)^D\to \rH^2(G_S,M)\to\bigoplus_{v\in S}\rH^2(K_v, M)
\end{equation}
and this expresses exactly a perfect
{\em duality} pairing
$$\Sha^2_S(K, M) \times \Sha^1_S(K, M^{\vee}) \rightarrow \bQ/\bZ$$
that will arise for us as an instance of ``Poincar\'e duality'' on the ``3-manifold''
$\Spec(\ccO_{K,S})$.
As a consequence of the above theorems, we can define a useful
notion of Euler-Poincar\'e characteristic for any $M$ as above, via an alternating product of the sizes $\#\rH^p(G_S, M)$.
There is a complication: if $K$ has a real place
and $M$ has even size then the cohomology groups will be nonzero for an infinite number of values of $p$ in general
(see Theorem \ref{Tate1}). We ignore this and simply define
$$\chi(G_S, M) :=\frac{\#\rH^0(G_S, M)\cdot \#\rH^2(G_S, M)}{\#\rH^1(G_S, M)}$$
in general (with $\#M$ an $S$-unit; one can drop this condition on $\#M$ but then more ideas are needed to establish useful results building
on Tate's).
By adapting the technique with Grothendieck groups used to prove his local Euler characteristic formula
(which is proved in \S5.7 of Chapter II of Serre's book on Galois cohomology),
Tate proved the following formula (again, with $\#M$ an $S$-unit):
$$\chi(G_S, M ) = \prod_{v\ \textup{arch}} \frac{\#\rH^0(G_v, M)}{|\# M|_v},$$
where $|\cdot|_v$ is the normalized absolute value (which for complex $v$ is understood to be the square of the usual complex absolute value,
as needed in the product formula); see \cite[\S5, Ch.\,I]{Milne} for the details on this proof (which is not proved in a new way via the \'etale cohomology method).
\begin{eg} Consider $M=\bZ/n\bZ$ with $n > 0$ and $S$ the set of infinite primes together with those dividing $n$,
so $\ccO_{K,S} = \ccO_K[1/n]$. Clearly $\rH^0(G_v, M) = M^{G_v} = \bZ/n\bZ$, and $|\# M|_v$ is equal to
$n$ or $n^2$ depending on whether $v$ is real or complex respectively. The formula yields
$$\chi(G_S, \bZ/n\bZ) = n^{-s}$$
with $s$ the number of complex places.
\end{eg}
\begin{eg}
To illustrate the power of Tate's global Euler characteristic formula when combined with the 9-term exact sequence,
we compute a ``formula'' for $\#{\rm{H}}^1(G_S, M)$ in general (remembering that $\#M$ is required to be an $S$-unit!).
We have $\#\Sha_S^2(K,M) = \#\Sha_S^1(K,M^{\vee})$ since these two finite groups are
Pontryagin dual to each other, and $\#\rH^2(K_v, M) = \# \rH^0(K_v, M^{\vee})$ for all $v$
(by Tate local duality for finite $v$, and some care for real $v$). Combining this with the Euler characteristic formula and (\ref{sha2}), we get
$$\#\rH^1(G_S, M) =
\#\Sha_S^1(K, M^{\vee})\cdot\frac{\#\rH^0(G_S, M)}{\#\rH^0(G_S, M^{\vee})}\cdot\prod_{v\in S}\#\rH^0(G_S, M^{\vee})\cdot\prod_{v\ \textup{arch}}\frac{|\#M|_v}{\#\rH^0(G_S, M)}.$$
This formula has a mystery term, namely $\Sha^1_S(K, M^{\vee})$.
The merit of this formula is that the groups $\Sha^i_S(K, \cdot)$ are more robust than ${\rm{H}}^i(G_S, \cdot)$ for some purposes
since they admit a good duality theory (which the ${\rm{H}}^i(G_S, \cdot)$'s do not); it is in this sense
that the above formula for $\#{\rm{H}}^1(G_S, M)$ can be useful. (Wiles established a important generalization
comparing sizes of such ${\rm{H}}^1$'s for $M$ and $M^{\vee}$ when it is {\em not} assumed that $\#M$ is an $S$-unit.)
\end{eg}
We have seen that the 9-term exact sequence encodes some duality results among
subgroups $\Sha^i_S(K, \cdot)$ of $G_S$-cohomologies. In Tate's original work, he first directly established
the duality between $\Sha^1_S(K, M^{\vee})$ and $\Sha^2_S(K, M)$ and then {\em defined} the 9-term
exact sequence in terms of that global duality and his local duality theorems.
In the \'etale cohomology approach,
we will begin with the Artin--Verdier duality theorem stated below that is of entirely different nature,
relating cohomology and Ext's (much as in Grothendieck--Serre coherent duality and Verdier's topological duality),
applicable to rather general constructible coefficient sheaves.
From that foundation, we will obtain the 9-term exact sequence as an exact ``local cohomology'' sequence
and {\em deduce} the duality between $\Sha^1_S(K, M^{\vee})$ and $\Sha^2_S(K, M)$ from that 9-term sequence
(so the logic is {\em opposite} to Tate's original approach).
For totally imaginary $K$, Mazur wrote up an account \cite{Mazur} with all of the essential arithmetic ideas.
But there are many interesting number fields that are not totally imaginary,
the most basic being $\bQ$ itself. In an appendix to \cite{Hab}, Zink introduced the idea of including extra ``points''
for real places to extend Mazur's aproach so that one recovers the entirety of Tate's results for all number fields. We have provided
some additional explanations and define an appropriate
$\delta$-functorial notion of $\rH^{\bullet}_c(U, \cdot)$ for dense open
$U \subset \Spec(\ccO_K)$ (which is not addressed in Zink's exposition)
to clarify some aspects. This appears in the statement of:
\begin{thm}{\textup{(Artin-Verdier)}}\label{dualT} Let $U \subset X = \Spec(\ccO_K)$ be a dense open subscheme. There are natural isomorphisms
$\rH^3_c(U, \mathbf{G}_{m,U}) \simeq \rH^3_c(X, \mathbf{G}_{m,X}) \simeq \bQ/\bZ$
and for constructible $F$ on $U$ there is a $\delta$-functorial perfect pairing
$$\rH^i_c(U, F) \times \textup{Ext}_U^{3-i}(F,\mathbf{G}_{m,U}) \to {\rH}^3_c(U, \mathbf{G}_{m,U}) = \bQ/\bZ,$$
of finite abelian groups for all $i \in \bZ$.
\end{thm}
Beware that $\rH^i_c(U, F)$ doesn't generally vanish for $i < 0$ when there are real places
(but by definition the Ext's vanish in negative degrees);
see Remark \ref{negdegree}.
Also, as is noted in \cite{Mazur}, there is {\em no} hypothesis concerning stalks of $F$ having order that is a unit on $U$!
Theorem \ref{dualT} is the basis upon which everything else will depend in the approach of
these notes (and it will be easily reduced to the case $U=X$
once the definitions are in place). Incorporating the real places
in a systematic way will require quite a bit of preparatory work. But as with any cohomological machine,
once the foundations are laid we can reap many rewards from some key calculations.
%
%The main link between this Theorem and Tate's results is
%the comparison between $G_S$-cohomology
%and \'etale cohomology on
% ${\rm{Spec}}(\ccO_{K,S})$ in Appendix \ref{contet}.
{\bf Notation}. For a scheme $X$, denote by $X_{\rm \acute{e}t}$ the small \'etale site of $X$; i.e., the category $\mathbf{{\acute{E}t}}_X$
consists of \'etale morphisms $U\to X$ together with $X$-morphisms as arrows, endowed with the \'etale topology for covers of such $U$.
We denote by $\Ab_X$ the category of abelian sheaves on $X_{\et}$. If $X$ is of finite type over $\bZ$, we denote by $X^0$
the set of closed points of $X$.
To avoid overload of notation, with any scheme $X$:
\centerline{\em from now on, by ``abelian sheaf on $X$'' we shall means an abelian sheaf on $X_{\et}$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\'Etale topology on algebraic number fields}\label{secet}
This section is organized as follows. We briefly review some definitions and basic facts from the theory of abelian sheaves on the small \'etale site on a scheme,
for which the main case of interest is the scheme $\Spec(\ccO_K)$ and schemes \'etale over it.
We also build an extension of the category of abelian \'etale sheaves on $\Spec(\ccO_K)$ so that its objects have ``stalks'' at real places of $K$.
Finally, we introduce and study an appropriate notion of ``finite morphism'' in this extended setting, especially the exactness
of an associated pushforward functor; that wil be a valuable tool to reduce some problems to the case of
totally imaginary $K$ (which don't involve the intervention of real places, so ordinary \'etale cohomology of schemes does the job).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Abelian sheaves represented by constant $X$-schemes}
Given an abelian group $G$, we set $\underline{G}_X$ to be the functor
$$\Sch\to\Grps$$
represented by the $X$-group $G_X := \coprod_{g \in G} X$; this assigns to any $X$-scheme $Y$
the group of locally constant functions $|Y| \rightarrow G$.
This is a sheaf for the \'etale topology on $X$
because of the openness of \'etale maps (check!), or in fancier terms because {\em every} representable functor on the category of $X$-schemes
is a sheaf for the Zariski and fpqc topologies (one of the key theorems of descent theory).
\begin{defin} Every abelian sheaf on $X$ which is represented by an abelian $X$-group of the form $G_X$, for some abelian group $G$, is called {\em constant}.
\end{defin}
We also recall the following:
\begin{defin} An abelian sheaf $F$ on $X$ is called {\em locally constant} if there exists an \'etale covering
$$\{f_i : U_i\to X\}_{i\in I}$$
such that each $F|_{U_i} := f_i^*(F)$ is constant as an abelian sheaf on $U_i$.
\end{defin}
\begin{eg} \label{continuousmod} Let $k$ be a field, and $G$ its Galois group. For $X = \Spec(k)$, $\Ab_X$ is equivalent to the category of discrete $G$-modules. By Galois theory, a sheaf is locally constant if an open subgroup of $G$ acts trivially on the corresponding discrete $G$-module.
For example, if $n \ge 1$ is not divisible by ${\rm{char}}(k)$ then $\mu_n$ is locally constant and correspondingly $\mu_n(\overline{k})$
has trivial action by the open subgroup corresponding to the finite extension $k(\zeta_n)$; in contrast, ${\bf{G}}_m$ is not locally
constant in general (aside from quirky cases such as separably closed $k$, or $k = \bR$).
\end{eg}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Artin's decomposition lemma and applications}
An important source of motivation for how to systematically add ``real points'' to $\Spec(\ccO_K)$
is obtained from a result of Artin that describes how to reassemble
\'etale sheaves on a scheme from \'etale sheaves on constituents of a stratification of the underlying
topological space.
To explain this, let $X$ be a scheme (we only need the case of Dedekind $X$), $j:U \hookrightarrow X$ an open subscheme,
and $Z = X - U$ its closed complement (with any closed subscheme structure we wish, such as the reduced structure;
it doesn't matter since killing nilpotents has no effect on the \'etale site and hence no effect on \'etale sheaf theory).
Let $i:Z \hookrightarrow X$ be the canonical closed immersion.
Let $F$ be an abelian sheaf on $X$, and define $F_U = j^{\ast}(F)$ and $F_Z = i^{\ast}(F)$. Using the natural map $F \rightarrow j_{\ast}(F_U)$,
applying $i^{\ast}$ gives a natural map
$$\varphi _F: F_Z \rightarrow i^{\ast} j_{\ast}(F_U).$$
We now denote by $\fC_{X,U}$ the category whose objects are triples $(F', F'', \varphi)$
consisting of an abelian sheaf $F'$ on $Z$, an abelian sheaf $F''$ on $U$, and a morphism
$\varphi:F' \to i^{\ast}j_{\ast}(F'')$. We define a {\em morphism} between two such triples
in the evident manner (involving a commutative square). This yields a ``decomposition'' functor
$$\textup{dec}: F\mapsto (F_Z, F_U, \varphi_F)$$
from ${\mathbf{Ab}}_X$ to the category $\fC_{X,U}$.
Artin proved the crucial {\em decomposition lemma}, which is so important that we record it as a Theorem:
\begin{thm}{\scshape(Artin, \cite[\S 3.2.5]{Art})} \label{dec} The functor $\textup{dec}$ is an equivalence of categories.
\end{thm}
The key to the proof is to answer the puzzle: how to recover $F$ from $F_Z$ and $F_U$? It will be built as an appropriate fiber product.
\begin{proof}
Using the deep theorem that formation of strict henselization of a local ring commutes
with passage to quotients by ideals, for an abelian sheaf $F'$ on $Z$ we naturally have
$$(i_{\ast}(F'))_{\overline{x}} = \left\{\begin{array}{cc}
0 & \textup{if}\ x\notin i(Z)\\
F'_{\overline{x_0}} & \textup{if}\ x = i(x_0)
\end{array}\right.
$$
For an abelian sheaf $F''$ on $U$ we have the more trivial identification $(j_{\ast}(F''))_{\overline{x}} = F''_{\overline{x_0}}$ if $x = j(x_0)$,
but the stalk at points of $Z$ is rather less easy to write down (it is the group of global sections of the pullback of $F''$
over the pullback of $U$ along ${\rm{Spec}}(\ccO_{X, \overline{x}}^{\rm{sh}}) \rightarrow X$).
We now use these two basic calculations.
Given some $F$ on $X_{\et}$, we claim that the commutative square
\begin{displaymath}
\xymatrix{ F \ar[d]\ar[r] & j_*(F_U)\ar[d]\\
i_*(F_Z)\ar[r] & i_*i^*j_*(F_U)}
\end{displaymath}
is Cartesian. In this diagram the vertical maps express the natural transformation ${\rm{id}} \rightarrow i_{\ast} i^{\ast}$
(applied to $F$ and to $j_{\ast}(F_U)$), and commutativity just expresses naturality with respect
to $F \rightarrow j_{\ast}(F_U)$.
It suffices to check the Cartesian property on stalks at geometric points $\overline{x}$ over $x \in X$.
The situation at stalks over $U$ is trivial because the bottom of the square has vanishing stalks at such points and the top collapses
to an isomorphism on such stalks. If instead $x \in Z$ then applying $\overline{x}$-stalks to
the natural transformation ${\rm{id}} \rightarrow i_{\ast}i^{\ast}$ always yields an isomorphism
between {\em functors} on abelian sheaves on $X_{\et}$. The Cartesian property is then a tautology as well.
Inspired by this construction, for
a triple $(F', F'', \varphi)$ in $\fC_{X,U}$ we define
$$F:= i_{*}(F')\times_{i_{\ast}(\varphi), i_{\ast}i^{\ast}j_{\ast}(F'')}j_{\ast}(F'').$$
We need to show this gives an inverse to the ``decomposition'' functor.
The formation of such a fiber product sheaf commutes with any pullback (left-exactness of pullback), so
the restriction $F_U$ over $U$ is naturally identified with $F''$
(more precisely: the natural projection $F \rightarrow j_{\ast}(F'')$ restricts over $U$ to an isomorphism $F_U \simeq F''$).
Likewise,
$$F_Z = F' \times_{\varphi, i^{\ast}i_{\ast}i^{\ast}(j_{\ast}(F''))} i^{\ast}j_{\ast}(F''),$$
where the second component of the fiber product is given by $i^{\ast}$ applied to the natural transformation
$h_i:{\rm{id}} \rightarrow i_{\ast}i^{\ast}$ on abelian sheaves on $X$. But by consideration of stalks
we see that $i^{\ast}(h_i)$ is always an {\em isomorphism}, so the fiber-product description of $F_Z$ collapses
to make the projection $F_Z \rightarrow F'$ an isomorphism.
The upshot is that under the fiber product definition of $F$, the first projection restricts to an isomorphism over $Z$ and the
second restricts to one over $U$. In particular, the natural map
$$F \rightarrow i_{\ast}(F_Z) \times j_{\ast}(F_U)$$
is injective, and more specifically recovers exactly the {\em definition} of $F$ as a fiber product inside a direct product!
By its definition, $F$ is thereby identified with the fiber product along the two maps
$$i_{\ast}(F_Z) \stackrel{i^{\ast}(\varphi)}{\longrightarrow} i_{\ast}i^{\ast}j_{\ast}(F_U) \stackrel{\rm{can}}{\longleftarrow}
j_{\ast}(F_U)$$
where $\varphi:F_Z = F' \rightarrow i^{\ast}j_{\ast}(F'') = i^{\ast}j_{\ast}(F_U)$ is the initially given map, so we have to show that $\varphi = \varphi_F$.
It is equivalent to prove equality of such maps over $Z$ after applying $i_{\ast}$,
and it is harmless to check equality after composing with the {\em surjection} $F \twoheadrightarrow i_{\ast}(F_Z)$.
This becomes a problem of comparing two composite maps $F \rightrightarrows i_{\ast}i^{\ast}j_{\ast}(F_U)$.
Ah, but by {\em definition} of $F$ as a fiber product, the composite map resting on $\varphi$ is equal to the composition of
the {\em canonical} second projection $F \rightarrow j_{\ast}(F_U)$ with the effect of ${\rm{id}} \rightarrow i_{\ast}i^{\ast}$ applied
to $j_{\ast}(F_U)$. However, this latter composition is {\em exactly} what arises in the initial considerations we made
when realizing every abelian sheaf $H$ on $X$ as a fiber square using $\varphi_H$. Thus, indeed
$\varphi = \varphi_F$, so we are done!
\end{proof}
\begin{eg}\label{ext0}
If $F''$ is an abelian sheaf on $U$ then $j_{!}(F'')$ corresponds to $(0, F'', \varphi = 0)$. By contrast,
$j_{\ast}(F'')$ corresponds to the triple $(j_{\ast}(F'')|_Z, F'', \varphi)$ where $\varphi$ is the identity map on $j_{\ast}(F'')|_Z$.
\end{eg}
Before specializing the decomposition lemma to the global setting of interest, we first discuss an instructive local example.
\begin{eg}\label{Galoisoverdvr}
Let $R$ be a complete (or henselian)
discrete valuation ring with fraction field $L$, and consider $X := \Spec(R)$. Let $G := \Gal(L^{\rm sep}/L)$,
with $I$ the inertia subgroup. Let $k$ be the residue field of $R$ and $G_k := \Gal(k^{\rm sep}/k)$. Then we claim that the decomposition lemma provides
an equivalence of $\mathbf{Ab}_{X_{\rm\acute{e}t}}$ onto
the category of triples $(M, N, \varphi)$ with $M$ a discrete $G_k$-module, $N$ a discrete $G$-module, and $\varphi : M\to N^I$ a
$G_k$-module homomorphism.
Setting $Z := \Spec(k)\to X$ and $U := \Spec(L)\to X$, clearly ${\mathbf{Ab}}_Z$ is the category of discrete $G_k$-modules
and ${\mathbf{Ab}}_U$ is the category of discrete $G$-modules. The only issue with content is to show that the functor $i^{\ast}j_{\ast}$
from discrete $G$-modules to discrete $G_k$-modules is precisely the formations of $I$-invariants.
Using the link between stalks at a geometric point and global sections of the pullback to a strictly henselian local ring,
the discrete $G_k$-module associated to $i^{\ast}(j_{\ast}(N))$ is precisely the $G_k$-module of global sections of the
pullback of $j_{\ast}(N)$ along $\pi: X^{\rm{sh}} := {\rm{Spec}}(R^{\rm{sh}}) \rightarrow {\rm{Spec}}(R) = X$ (where $G_k$ acts through its identification with
${\rm{Aut}}(R^{\rm{sh}}/R) = G/I$!). But expressing $R^{\rm{sh}}$ as a direct limit of local finite \'etale extensions
$R'$ inside $L^{\rm sep}$ and using the compatibility of the formation of global sections with passage to limits of qcqs schemes
(with affine transition maps) shows that
$$\Gamma(X^{\rm{sh}}, \pi^{\ast}(j_{\ast}(N))) = \varinjlim \Gamma({\rm{Spec}}(R'), j_{\ast}(N)) = \varinjlim \Gamma(L', N) = \varinjlim N^{{\rm{Gal}}(L^{\rm sep}/L')} =
N^I,$$
the final equality because $N$ is a discrete $G$-module.
\end{eg}
\medskip
Now we finally come to the situation of interest: {\em for the remainder of these notes}
we write $X = \Spec(\ccO_K)$ for a number field $K$.
Let $\eta: \Spec(K)\to X$ be the generic point, and likewise $$\overline{\eta}: \Spec(\overline{K})\to X$$
for a fixed separable closure $\overline{K}$ of $K$. We set $G := \Gal(\overline{K}/K)$, and for any closed point $$x :\Spec(\kappa) \to X,$$ we set
$G_x := \Gal(\overline{\kappa}/\kappa)$ when an algebraic closure $\overline{\kappa}$ of $\kappa$ is specified.
For each closed point $x$, we fix a decomposition subgroup $D_x\hookrightarrow G$, and recall $D_x$ is unique up to conjugation inside $G$.
The choice of $D_x$ amounts to a choice of place on $\overline{K}$ extending the $x$-adic place on $K$, and relative to this
choice we get an inclusion of $\overline{K}$ into an algebraic closure of the completion $K_x$ at $x$,
thereby identifying $D_x$ with $\Gal(\overline{K_x}/K_x)$. We denote by $I_x$ the inertia subgroup of $D_x$.
This data identifies the residue field of $\overline{K_x}$ (or equivalently of $\overline{K}$ at the chosen place over $x$)
with an algebraic closure of the residue field $\kappa$ at $x$, so it defines $\overline{\kappa}$ and identifies $G_x$ with $D_x/I_x$.
By Example \ref{continuousmod}, for any abelian sheaf $F$ on $X$ we identify
$F_{\eta} := \eta^*(F)$ with a discrete $G$-module, each $F_x := x^*(F)$ with a discrete $G_x$-module,
and the pullback of $F$ over $\Spec(\cO_{X,x})$ corresponds to the triple $(F_x, F_{\eta}, \varphi_x)$ where
$$\varphi_x : F_x\to F_{\eta}^{I_x}$$
is called the {\em specialization map} at $x$. The decomposition lemma (Theorem \ref{dec}) now becomes:
\begin{lemma}\label{decomposition} Let $S = \{x_1,\dots, x_n\}$ be a finite set of closed points of $X$, and $U := X-S$. Then for any abelian sheaf $F$ on $X$, the functor $\textup{dec} : \Ab_X \to \fC_{X,U}$ takes the form:
$$\textup{dec}(F) = \left(\bigoplus_{k=1}^n (x_k)_*F_{x_k}, F_U, \bigoplus_{k=1}^n \varphi_{x_k}\right)$$
for the specialization maps $\varphi_{x_k} : F_{x_k}\to F_{\eta}^{I_{x_k}}$ where $F_{\eta} := \eta^{\ast}(F_U)$.
\end{lemma}
This lemma says that $F$ is determined by: the skyscraper sheaves $F_{x_k}$ ($1 \le k \le n$),
the sheaf $F_U$ on $U$, and the specialization maps from each $F_{x_k}$ into the $I_{x_k}$-invariants
of $\eta^{\ast}(F_U) = F_{\eta}$. The case when every $\varphi_{x_k}$ vanishes corresponds to
$$F = j_{!}(F_U) \oplus \bigoplus_{k=1}^n (x_k)_{\ast}(F_{x_k}).$$
Recall that in general a {\em constructible} abelian \'etale sheaf on a noetherian scheme is one that becomes
locally constant with finite geometric fibers on the constituents of a finite stratification by locally closed sets
(and these are {\em characterized} as the noetherian objects in the category of abelian \'etale sheaves).
Thus, it is a simple exercise to check that an abelian sheaf $F$ on $X$ is {\em constructible} if
it is locally constant with finite fibers on some dense open $U \subset X$
and has stalks $F_x$ that are finite $G_x$-modules for all $x \in X = U$.
If $F$ is locally constant and constructible (equivalently, locally constant with finite stalks)
we say $F$ is {\em lcc}.
\begin{eg} \label{lcc} A constructible abelian sheaf $F$ is locally constant on an \'etale neighborhood of a closed point $x \in X$
if and only if two conditions hold: $I_x$ acts trivially on $F_{\eta}$
and $\varphi_x:F_x \rightarrow F_{\eta}^{I_x} = F_{\eta}$ is an isomorphism.
Hence, a constructible abelian etale sheaf $F$ is characterized in terms of the associated
data $F_{\eta}$, $F_x$ for closed $x \in X$, and specialization maps $\varphi_x:F_x \rightarrow F_{\eta}^{I_x}$
by the conditions: $F_{\eta}$ is finite (hence by $G$-discreteness it is unramified at all but finitely many $x$!),
every $F_x$ is finite, and $\varphi_x$ is an isomorphism for all but finitely many $x$.
Turning this around, if we are given such data
$$(F_{\eta}, \{F_x\}_{x \in X^0}, \{\varphi_x: F_x \rightarrow F_{\eta}^{I_x}\}_{x \in X^0})$$
satisfying these finiteness and isomorphism conditions
then we claim that it arises from a canonically determined constructible $F$ on $X$.
Indeed, the finite $F_{\eta}$ comes from a locally constant sheaf $F_U$ on some $U_{\et}$
governed by the finitely many ramified places for $F_{\eta}$ as a finite discrete $G$-module, so then by removing from $U$ any closed
$u \in U$ for which $\varphi_u$ isn't an isomorphism, we
we equivalently package this data in terms of $(F_U, \{F_x\}_{x \in X-U}, \{\varphi_x\}_{x \in X - U})$
to which the decomposition lemma really is applicable.
\end{eg}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Adjoining the real places to $X = \Spec(\ccO_K)$}
We define $\overline{X}$ to be the union of $X$ and the set $X_{\infty} := \{v_1,\dots, v_r\}$ of all real places of $K$, and we endow $\overline{X}$ with the topology whose closed sets are finite subsets of $\overline{X} - \{\eta\}$ (so $X$ is an open subspace of $\overline{X}$).
{\bf Notation}. Let $\overline{v}$ be a fixed extension of
a real place $v$ to a place on a fixed algebraic closure $\overline{K}$ of $K$. Denote by $D_{v}$ the decomposition group of $\overline{v}$,
so uniquely $D_{v}\simeq \bZ/2\bZ$ since $v$ is real.
We define $I_{v} := D_{v}$ due to the convention in algebraic number theory
that a real place is ``unramified'' when it is totally split (such as in the definition of Hilbert class fields), so we define $G_{v} := 1$.
In particular, {\em real points should be treated as if they are geometric points}!
For any dense open subscheme $U \subset X$, we denote by $\overline{U}$ the union of $U$ and $X_{\infty}$; this is an open subspace of $\overline{X}$
(i.e., it is the result of applying to $U = \Spec(\ccO_{K,S})$ the same formalism we applied to $X$ when defining $\overline{X}$).
We also use the notation $\widetilde{U}$ for a typical open subset of $\overline{X}$
(which may contains several or no real points). Given
a choice of $\widetilde{U}$, we denote by $\widetilde{U}_{\infty} = \{\widetilde{v_1},\dots,\widetilde{v_s}\}$
its overlap with $X_{\infty}$, and define $U := \widetilde{U}\cap X$
(so if $\widetilde{U}$ is {\em non-empty} then $\overline{U}$ makes sense and contains $\widetilde{U}$).
\medskip
Next, inspired by Artin's decomposition lemma,
we shall now define notions of: ``abelian \'etale sheaf'' on non-empty open
subsets $\widetilde{U}$ of $\overline{X}$, ``stalk'' at real places (simply an abelian group, since we view real points as geometric points),
and constructibility for such ``sheaves'':
\begin{defin} \label{abelian tilde} An {\em abelian sheaf} $\widetilde{F}$ on a non-empty open subset $\widetilde{U} \subset \overline{X}$ is a tuple
$$\widetilde{F} := (\{\widetilde{F}_{\widetilde{v}}\}, F, \{\varphi_{\widetilde{v}}\})$$
where $\widetilde{v}$ varies through $\widetilde{U}_{\infty}$,
the $\widetilde{F}_{\widetilde{v}}$ are abelian groups, $F$ is an abelian sheaf on $U := \widetilde{U} \cap X$, and $\varphi_{\widetilde{v}}: \widetilde{F}_{\widetilde{v}}
\rightarrow F_{\eta}^{I_{\widetilde{v}}}$ is a group homomorphism. We call $\widetilde{F}_{\widetilde{v}}$ the {\em stalk} of $\widetilde{F}$ at $\widetilde{v}$, and we also denote by $\widetilde{F}_x$ the stalk $F_x$ at any $x\in U$. The ``sheaf'' $\widetilde{F}$ is called {\em constructible} if
$F$ is constructible and each $\widetilde{F}_{\widetilde{v}}$ is finite.
\end{defin}
\begin{rem}
The fixed field $\overline{K}^{I_v}$ is the algebraic closure $K_v^{\rm{alg}}$ of $K$ inside its completion $K_v \simeq \mathbf{R}$ (isomorphism
unique!). In particular, $K_v^{\rm{alg}}$ is {\em intrinsic} to $K$ (given $v$),
in contrast with $\overline{K}$. Since $F_{\eta}^{I_v}$ is identified with the group $F(K_v^{\rm{alg}})$ of global sections of the pullback of
$F$ over $\Spec(K_v^{\rm{alg}})$, we see that it is intrinsic to $(K, v)$ in a way that
$F_{\eta}^{I_x} = F(\overline{K}^{I_x}) = F({\rm{Frac}}(\cO_{X,x}^{\rm{sh}}))$ is not for $x \in U^0$ (a geometric point over $x$
is needed to define the strict henselization $\cO_{X,x}^{\rm{sh}}$).
We can likewise identify $F_{\eta}^{I_v}$ with the group $F(K_v)$ of global sections of the pullback of $F$
over $\Spec(K_v)$, saving a bit of notation.
Consequently, we will sometimes prefer to speak in terms of $F(K_v)$ or $F(K_v^{\rm{alg}})$ rather than $F_{\eta}^{I_v}$
to emphasize that it doesn't depend on base points. However, the notation $F_{\eta}^{I_v}$
is closer to that at closed points $x$ as in the decomposition lemma (with $D_x \ne I_x$ for $x \in U^0$!).
\end{rem}
The abelian category of such ``sheaves'' is denoted $\fAb_{\widetilde{U}}$. The reason that we do not (yet)
denote it as $\mathbf{Ab}_{\widetilde{U}}$ is that later we will define an {\em \'etale site} for
$\widetilde{U}$, so the category of abelian sheaves on that site will deserve the name $\mathbf{Ab}_{\widetilde{U}}$;
in Theorem \ref{abshvs tilde} we will show that this naturally coincides with $\fAb_{\widetilde{U}}$ (after which we shall pass
between the two notions interchangeably). At this moment,
objects in $\fAb_{\widetilde{U}}$ are not yet genuinely realized as sheaves on an actual site, so
the usual Grothendieck cohomological formalism is {\em not yet applicable} to them.
In the special case that $\widetilde{U}$ contains no real points,
$\fAb_{\widetilde{U}}$ trivially recovers ${\mathbf{Ab}}_U$.
Our aim is, for now, to gain information on $\widetilde{F}$ from knowledge of information of $F \in \Ab_U$ and the real stalks. In effect,
we shall establish a special case of the decomposition lemma in our new framework (but later we will have to do work to
establish a general version of the decomposition lemma in our new setting, appropriately interpreted as sheaves on a site).
In the setting of \'etale sheaf theory on schemes,
the functors of pushforward and pullback are related by an adjunction,
as are extenson by zero $j_{!}$ and pullback $j^{\ast}$ relative to an open immersion $j$, and
pushforward $i_{\ast}$ and extraordinary pullback $i^{!}$ relative to a closed immersion $i$
(with $\Hom(i_{\ast}(F'), \widetilde{F}) = \Hom(F', i^{!}(\widetilde{F}))$; i.e., $i^{!}$ is a right adjoint to $i_{\ast}$).
\begin{rem}\label{suppx}
For usual \'etale sheaf theory and the inclusion $i$ of a {\em geometric} closed point, a right-adjoint $i^{!}$
to $i_{\ast}$ is given by the functor ``global sections vanishing away from the point''.
If instead $i$ is the inclusion of a general closed point and its open complement has associated inclusion $j$
then $\ker(i^{\ast} \rightarrow i^{\ast} j_{\ast}j^{\ast}) = i^{\ast}(\ker({\rm{id}} \rightarrow j_{\ast}j^{\ast}))$ gives
such a right-adjoint $i^{!}$ to $i_{\ast}$.
\end{rem}
Motivated by how these functors
are described in terms of the decomposition lemma (e.g., see Example \ref{ext0} for extension by zero and pushforward from an open set),
for the open embedding of topological spaces $j : U\to \widetilde{U}$
and the closed embedding of topological spaces
$i_{\widetilde{v}}: \{\widetilde{v}\}\to \widetilde{U}$ for a real point $\widetilde{v}$ of $\widetilde{U}$
we {\em define} functors
\begin{displaymath}
\xymatrix{\Ab_U\ar @< 12.5pt> [r]^{j_!}
\ar @<-12.5pt> [r]^{j_*} & {\fAb}_{\widetilde{U}}\ar[l]_{j^*} \ar @< 12.5pt> [r]^{(i_{\widetilde{v}})^{!}}
\ar @<-12.5pt> [r]^{(i_{\widetilde{v}})^{*}} & {\mathbf{Ab}}\ar[l]_{(i_{\widetilde{v}})_{*}}}
\end{displaymath}
as follows (with our notation conventions as used above):
\begin{flalign*}
j_!(F) :=&\ (0,\dots, 0, F, 0\to F_{\eta}^{I_{\widetilde{v_1}}}, \dots, 0\to F_{\eta}^{I_{\widetilde{v_s}}})\\
j^*(\widetilde{F}) :=&\ F\\
j_*(F) :=&\ (F_{\eta}^{I_{\widetilde{v_1}}}, \dots, F_{\eta}^{I_{\widetilde{v_s}}}, F, \textup{id}_{F_{\eta}^{I_{\widetilde{v_1}}}}, \dots, \textup{id}_{F_{\eta}^{I_{\widetilde{v_s}}}})\\
i^*_{\widetilde{v_k}}(\widetilde{F}) :=&\ \widetilde{F}_{\widetilde{v_k}}\\
i_{\widetilde{v_k}*}(A_k) :=&\ (0,\dots, 0, A_k,\dots, 0, 0\to 0, \dots, A_k \to 0, \dots, 0\to 0)\\
i_{\widetilde{v_k}}^!(\widetilde{F}) :=&\ \ker(\varphi_{\widetilde{v_k}}).
\end{flalign*}
It is easy to check that the first three satisfy the expected adjunctions, as do the $i_{\widetilde{v}\ast}$ and $i_{\widetilde{v}}^{\ast}$, and that
$i^{!}_{\widetilde{v}}$ is right adjoint $i_{\widetilde{v}\ast}$. Hence, these notations are all ``justified''.
\begin{rem}
It is {\em irrelevant} to try to make sense of such $j$ and $i_{\widetilde{v}}$ as ``morphisms'' of geometric objects;
all that actually matters are the 6 associated functors as defined above. Speaking in terms of $j$ and $i_{\widetilde{v}}$
is mainly for ease of exposition when we want to argue similarly to what is done with actual open immersions
and closed immersions between schemes.
\end{rem}
\begin{eg}\label{jadj}
A moment's reflection yields a canonical map $\widetilde{F} \rightarrow j_{\ast}j^{\ast}(\widetilde{F})$
that realizes the adjunction between $j_{\ast}$ and $j^{\ast}$.
\end{eg}
\begin{rem}\label{goodphi}
We can reformulate the map $\varphi_{\widetilde{v}}$ appearing in the definition of
$i_{\widetilde{v}}^{!}$ as follows: for $F := j^{\ast}(\widetilde{F})$ clearly
$$i_{\widetilde{v}}^*j_*j^*(\widetilde{F}) = i_{\widetilde{v}}^*j_*(F) = i_{\widetilde{v}}^*(F_{\eta}^{I_{\widetilde{v_1}}}, \dots, F_{\eta}^{I_{\widetilde{v_s}}}, F, \textup{id}_{F_{\eta}^{I_{\widetilde{v_1}}}}, \dots, \textup{id}_{F_{\eta}^{I_{\widetilde{v_s}}}}) = F_{\eta}^{I_{\widetilde{v}}},$$
so $i_{\widetilde{v}}^!(\widetilde{F})$ is identified
with the kernel of a map $\varphi_{\widetilde{v}} : i_{\widetilde{v}}^*(\widetilde{F}) \to i_{\widetilde{v}}^*j_*(F)$,
and we easily identify this map as $i_{\widetilde{v}}^{\ast}$ applied to the {\em canonical}
map $\widetilde{F} \rightarrow j_{\ast}j^{\ast}(\widetilde{F})$ as in Example \ref{jadj}.
This calculation will ensure
that once we have realized $\fAb_{\widetilde{U}}$ as a genuine sheaf category for some site,
the map $\varphi_{\widetilde{v}}$ are built exactly as in the decomposition lemma on $X$.
\end{rem}
\begin{rem} \label{injectives} The functors $j_!, j^*, i^*$ and $i_*$ are exact, and therefore
their respective right adjoints $j^*, j_*, i_*, i^!$ preserve injectives. It is also clear from the termwise construction of kernels
that a map in $\fAb_{\widetilde{U}}$ has vanishing kernel if the same holds
after applying $j^{\ast}$ and $i_{\widetilde{v}}^{\ast}$ for all $\widetilde{v} \in \widetilde{U}_{\infty}$.
If we let $J$ be an injective object in $\Ab_U$ and $J_{\widetilde{v_k}}$ be an injective abelian group for $k=1,\dots, s$, then
$$j_*(J) \oplus\bigoplus i_{\widetilde{v_k}*}(J_{\widetilde{v_k}})$$
is injective in $\fAb_{\widetilde{U}}$, so it is clear that $\fAb_{\widetilde{U}}$
has enough injectives (and more specifically of this special form).
Of course, the existence of enough injectives will also follow from general sheaf theory on sites
once we define an appropriate site whose category of abelian sheaves is $\fAb_{\widetilde{U}}$;
however, for some later theoretical calculations it will be useful to know that ``enough'' injectives
can be found in this explicit form.
\end{rem}
\begin{rem}\label{stalks}
The collection of functors $\{i_x^{\ast}\}_{x \in \widetilde{U}}$ has the familiar properties
one wants for a collection of ``enough stalk functors'' on $\fAb_{\widetilde{U}}$ (with $i_x^{\ast}$ valued
in $\mathbf{Ab}$ for real $x$, and valued in the category of abelian sheaves on $\Spec(\kappa(x))$ for $x \in U$):
each such functor is exact, we can detect equality for a pair of maps between abelian sheaves by checking equality
after applying every $i_x^{\ast}$, and a complex in $\fAb_{\widetilde{U}}$ is exact if and only if it is
exact after applying every $i_x^{\ast}$. These assertions are all obvious by inspection how
$i_x^{\ast}$ is defined for real $x$ and how the formation of kernels
and cokernels and images is expressed in the language of triples that {\em define} objects of
$\fAb_{\widetilde{U}}$.
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Completed small \'etale sites}
We now wish to show that the category $\fAb_{\widetilde{U}}$ of abelian sheaves on $\widetilde{U}$ is actually equivalent to the category of abelian sheaves
for a suitable Grothendieck topology. This will {\em provide us with a cohomology theory} on
$\widetilde{U}$.
One could try to avoid this and impose the later explicit formula for the global sections functor given in Example \ref{gsec}
out of thin air as a definition, and then define cohomology as its derived functors (since we have seen
directly that $\fAb_{\widetilde{U}}$ has enough injectives). That would be a mistake: avoiding the completed site introduced below
would entail a lack the insight and techniques that come for free
once we can view $\fAb_{\widetilde{U}}$ as sheaves on an actual site.
We first note that the set $\{\widetilde{v_1},\dots,\widetilde{v_s}\} = \widetilde{U}_{\infty}$ can be regarded as a subset of
$$U(\bR) = {\rm{Hom}}(\Spec(\bR), U).$$
In Definition \ref{abelian tilde} we rigged the notion of abelian sheaf on $\widetilde{U}$ as if $\widetilde{U}$ were ``stratified''
by the pair $(U, \widetilde{U}_{\infty})$. Therefore, let us introduce the category $\widetilde{\Sch}_X$, whose objects are pairs $(Y, Y_{\infty})$
for an \'etale $X$-scheme $Y$ and
a subset $Y_{\infty} \subset Y(\bR)$, and morphisms $(Y, Y_{\infty})\to (Y', Y'_{\infty})$ are morphisms of
$X$-schemes $Y\to Y'$ sending $Y_{\infty}$ into $Y'_{\infty}$
(at the level of $\bR$-points).
\begin{rem}
We are forcing into the definition of
$\widetilde{\Sch}_X$ that there is no ``residue field extension'' at real points (as the latter are meant to behave
like geometric points; we want to treat $\mathbf{C}$ as ``ramified'' over $\mathbf{R}$, and more algebraically
$\overline{K}$ as ``ramified'' over $K_v^{\rm{alg}}$ for real $v$).
\end{rem}
\begin{rem} The category
$\widetilde{\Sch}_X$ has
fibered products: for $(Y, Y_{\infty})\to (T, T_{\infty}) \leftarrow (Y', Y'_{\infty})$ two morphisms in $\widetilde{\Sch}_X$,
by writing $\widetilde{Y} := (Y, Y_{\infty})$ and likewise for $\widetilde{T}$ and $\widetilde{Y'}$ we have:
$$\widetilde{Y}\times_{\widetilde{T}}\widetilde{Y'} = (Y\times_TY', Y_{\infty}\cup Y'_{\infty}).$$
\end{rem}
By definition, the category $\mathbf{\acute{E}t}_{\widetilde{U}}$ has as its objects the morphisms
$\widetilde{Y} \rightarrow \widetilde{U}$ in $\widetilde{\Sch}_X$ as its objects, and the morphisms in this category are defined
over $\widetilde{U}$ in the evident manner. A family of morphisms $\{\widetilde{f_i} : \widetilde{Y_i}\to \widetilde{Y}\}$ is a
{\em covering} in $\mathbf{\acute{E}t}_{\widetilde{U}}$ precisely when $\{f_i : Y_i\to Y\}$ is a covering in $\mathbf{\acute{E}t}_U$ and moreover
$$\bigcup_{i\in I}f_i(\widetilde{Y}_{i\infty}) = \widetilde{Y}_{\infty}.$$
These coverings define the {\em \'etale topology} on $\widetilde{U}$, so in this way we have defined the small \'etale site $\widetilde{U}_{\rm\acute{e}t}$.
We denote by $\Ab_{\widetilde{U}}$ the category of abelian sheaves on $\widetilde{U}_{\rm \acute{e}t}$.
The main theorem of this section is:
\begin{thm} \label{abshvs tilde} The categories $\fAb_{\widetilde{U}}$ and $\Ab_{\widetilde{U}}$ are equivalent.
\end{thm}
We first need to set some notation. First, for a real point $\widetilde{v}$ of $\widetilde{U}$ let $T_{\widetilde{v}}$ be a copy of the category of sets,
equipped with the topology generated by jointly surjective families of morphisms. Let
$$j_{\widetilde{U}} : \mathbf{\acute{E}t}_{\widetilde{U}}\to \mathbf{\acute{E}t}_U$$
be the forgetful functor, and
$$i_{\widetilde{v}}: \mathbf{\acute{E}t}_{\widetilde{U}}\to T_{\widetilde{v}}$$
the functor associating to $\widetilde{Y}$ the inverse image of $\widetilde{v}$ under the map
$$\widetilde{Y}_{\infty}\to \widetilde{U}_{\infty}.$$
Manifestly, the category $\Ab_{\widetilde{v}}$ of abelian sheaves on $T_{\widetilde{v}}$ is the category
of abelian groups. From the functors $j := j_{\widetilde{U}}$ and $i_{\widetilde{v}}$ we obtain functors:
\begin{displaymath}
\xymatrix{\Ab_U
\ar @<8.5pt> [r]^{j_*} & \Ab_{\widetilde{U}}\ar @<8.5pt> [l]_{j^*} \ar @< 8.5pt> [r]^{(i_{\widetilde{v}})^*}
& \Ab_{\widetilde{v}}\ar @<8.5pt> [l]_{(i_{\widetilde{v}})_*}}
\end{displaymath}
where $j_{\ast}$ is defined by composing a sheaf with the functor $j$, likewise for $(i_{\widetilde{v}})_{\ast}$,
and $j^{\ast}$ and $i_{\widetilde{v}}^{\ast}$ are their respective left adjoints.
Explicitly, $j^{\ast}$ is restriction of a sheaf
to the full subcategory $\mathbf{\acute{E}t}_U \subset \mathbf{\acute{E}t}_{\widetilde{U}}$,
and $(i_{\widetilde{v}})^{\ast}(\widetilde{F})$ is the sheaf on $T_{\widetilde{v}}$ whose value on the singleton is the direct limit of the diagram of abelian groups
$\widetilde{F}(\widetilde{Y})$ for all $\widetilde{Y} \rightarrow \widetilde{U}$
with non-empty $\widetilde{v}$-fiber (and with maps in the diagram arising from morphisms among such $\widetilde{Y}$'s);
the same limit is attained by limiting attention to those $\widetilde{Y}$'s with a singleton over $\widetilde{v}$.
\begin{proof}\!(of Theorem \ref{abshvs tilde})
We want to define a functor $\tau : \Ab_{\widetilde{U}}\to\fAb_{\widetilde{U}}$ yielding an equivalence of categories.
Given $M$ in $\Ab_{\widetilde{U}}$, we shall cook up a triple
$$(\{\widetilde{F}_{\widetilde{v}}\}, F, \{\varphi_{\widetilde{v}}\})$$
with group homomorphisms $\varphi_{\widetilde{v}} : \widetilde{F}_{\widetilde{v}}\to F_{\eta}^{I_{\widetilde{v}}} = F(K_{\widetilde{v}})$ for all $v\in \widetilde{U}_{\infty}$.
We denote by $j: U\to \widetilde{U}$ and $i_x : x\to \widetilde{U}$ the natural maps (where $x$ is a closed point, possibly ``real''), so we get the triple
$$(\{i_{\widetilde{v}}^*(M)\}, j^*(M), \{\varphi_{\widetilde{v}}\})$$
with $\varphi_{\widetilde{v}}: i_{\widetilde{v}}^*(M)\to i_{\widetilde{v}}^*j_*j^*(M)$ the
canonical ``specialization'' maps. Clearly $i_{\widetilde{v}}^*(M)$ is an abelian group for all $\widetilde{v}$, $j^*(M)$ an abelian sheaf on $U$, and $\varphi_{\widetilde{v}}$ is a group homomorphism.
We now check that $\varphi_{\widetilde{v}}$ has target of the required form. Clearly
$$i_{\widetilde{v}}^*j_*j^*(M)(\{\widetilde{v}\}) = \varinjlim_{(W,w)} j^*(M)(W)$$
with objects $W$ in $\mathbf{\acute{E}t}_{\widetilde{U}}$ and $w\in W(\bR)$ mapping to $\widetilde{v}$; the limit can be interpreted as evaluating $j^*(M)$ at a maximal extension of $K$ {\em unramified} at $\widetilde{v}$. Since $\overline{K}^{I_{\widetilde{v}}}$ is such extension, we get
$$i_{\widetilde{v}}^*j_*j^*(M)(\{\widetilde{v}\}) = (j^*M)_{\eta}^{I_{\widetilde{v}}}$$
as desired. We can now define
$$\tau(M) := (\{i_{\widetilde{v}}^*(M)\}, j^*(M), \{\varphi_{\widetilde{v}} : i_{\widetilde{v}}^*(M)\to (j^*M)_{\eta}^{I_{\widetilde{v}}}\})$$
and it is clear that $\tau$ is functorial in $M$ with values in the category $\fAb_{\widetilde{U}}$.
We seek a quasi-inverse. For ease of notation, suppose $\widetilde{U}$ contains
one real point $\widetilde{v}$. (The general case will go similarly, just with more notation.) Let $\widetilde{F}$ be the object of $\fAb_{\widetilde{U}}$ given by
a triple $$(\{\widetilde{F}_{\widetilde{v}}\}, F, \{\varphi_{\widetilde{v}}\}).$$
In the spirit of the proof of the decomposition lemma, define
$$M := i_{*}(\widetilde{F}_{\widetilde{v}}) \times_{i_*(\varphi_{\widetilde{v}}),i_*i^*j_*(F)} j_*(F)$$
where we ease the notation by writing $i := i_{\widetilde{v}}$ and $j := j_U$. Heuristically,
$\widetilde{F}_{\widetilde{v}}$ is viewed as an abelian sheaf on the punctual site $\{v\}$ and the inclusion $i$ ``yields'' the functor $i_* : \Ab\to \Ab_{\widetilde{U}}$.
We would like to argue exactly the same way as in the proof of Theorem \ref{dec}, and to do so we need stalk functors for every $x$ point in $\widetilde{U}$, with the usual desired properties. For $x \in U$ and an abelian sheaf $\widetilde{F}$ on $\widetilde{U}$ we define
$$\widetilde{F}_x := (j^*\widetilde{F})_x;$$
this inherits the usual exactness properties from the classical theory of stalks for \'etale sheaves.
Consider $x$ a real point of $\widetilde{U}$, so a cofinal system of \'etale neighbourhoods of $x$ is
of the form: $(W, w)$, with $w \in W(\bR)$ mapping to $x$ and $W\to U$ \'etale. Maps between such pairs
are defined in the evident manner, and by definition
$$\widetilde{F}_x := \varinjlim_{(W,w)\to \widetilde{U}} \widetilde{F}(W,w),$$
for which exactness is immediate. By design of the pullback functors $i_x^*$, it is readily checked that $\widetilde{F}_x = i_x^*(\widetilde{F})$
in accordance with \'etale sheaf theory on schemes.
Now the argument in the proof of Theorem \ref{dec} applies.
\end{proof}
%By \cite[Chapter III \S 2.4]{Art}, the main content for proving Theorem \ref{abshvs tilde} is the following lemma:
%\begin{lemma} \label{equiv} The following assertions hold true:
%\begin{enumerate}
%\renewcommand{\labelenumi}{\textup{(\arabic{enumi})}}
%\item $i_{\widetilde{v_k}}^*$ and $j^*$ are exact.
%\item $i_{\widetilde{v_k}}^*$, resp. $j^*$, are left-adjoint to $i_{\widetilde{v_k}*}$, resp, $j_*$.
%\item $i_{\widetilde{v_k}*}$ and $j_*$ are fully faithful.
%\item An abelian sheaf $\widetilde{F}$ in $\Ab_{\widetilde{U}}$ is of the form
%$$\bigoplus_{k=1}^si_{\widetilde{v_k}*}(M_{\widetilde{v_k}})$$
%for abelian sheaves $M_{\widetilde{v_k}}$ in $\Ab_{\widetilde{v_k}}$ for all $k$, if and only if $j^*(\widetilde{F}) = 0$.
%\item The functor $i_{\widetilde{v_k}}^*j_*(-)$ is given by $F\mapsto F_{\eta}^{I_{\widetilde{v_k}}}$.
%\end{enumerate}
%\end{lemma}
%\begin{proof}
%The first two assertion are easy (the first is part of the standard formalism for constructing pullback functors from suitable morphisms of sites).
%We prove $(3)$. For $F_1, F_2$ in $\mathbf{Ab}_U$, clearly
%$$\Hom(j_{\ast}(F_1), j_{\ast}(F_2)) = \Hom(j^{\ast}j_{\ast}(F_1), F_2).$$
%It therefore suffices (for full faithfulness of $j_{\ast}$) to show that the natural map $j^{\ast}(j_{\ast}(F)) \rightarrow F$ is an isomorphism for all $F$
%in $\mathbf{Ab}_U$. But this is clear from the explicit descriptions of $j^{\ast}$ and $j_{\ast}$ (and of
%the resulting adjunction morphism $j^{\ast}j_{\ast} \rightarrow {\rm{id}}$).
%The same game works for $(i_{\widetilde{v}_k})_{\ast}$ since the adjunction
%$(i_{\widetilde{v}_k})^{\ast} (i_{\widetilde{v}_k})_{\ast} \rightarrow {\rm{id}}$ is an isomorphism
%(again, seen from the explicit descriptions of these functors).
%
%$$\Hom(j_*F(\widetilde{W}), j_*G(\widetilde{W})) = \Hom(j^*j_*F({W}), G(W)).$$
%Moreover, if $\widetilde{W}_{\infty}$ is empty, then $j^*\widetilde{F}(W) = \widetilde{F}(\widetilde{W})$. Choose $\widetilde{F} = j_*(F)$ and we conclude:
%$$\Hom(j_*F(\widetilde{W}), j_*G(\widetilde{W})) = \Hom(j^*j_*F({W}), G(W)) = \Hom(j_*F(\widetilde{W}), G(W)) = \Hom(F(W), G(W)),$$
%whence $\Hom(j_*F,j_*G) = \Hom(F,G)$, and $j_*(-)$ is fully faithful.
%Thus, $(3)$ is proved.\\
%For the proof of $(4)$, it is elementary from the definitions that any $i_{\widetilde{v}}$-pushforward is killed by $j^{\ast}$.
%For the converse, consider $\widetilde{F}$ such that
%$j^*(\widetilde{F})=0$.
%Let $\widetilde{U_k}$ be the object $(U,\widetilde{v_k})$, and set $M_{\widetilde{v_k}} := \widetilde{F}(\widetilde{U_k})$. Let $$(W, w\in W(\bR))\to (U,\widetilde{v_k})$$
%be an object of $\mathbf{\acute{E}t}_{\widetilde{U}}$, and $\{V_j\to U\}_{j\in J}$ a covering in $\mathbf{\acute{E}t}_U$. Then $(V_j,\varnothing)\to (U,\widetilde{v_k})$ together with $(W,w)\to (U,\widetilde{v_k})$ is a covering and from the sheaf axioms we deduce $\widetilde{F}(W,w)\to\widetilde{F}(U,\widetilde{v_k}) = M_{\widetilde{v_k}}$ is an isomorphism. We can cover any $\widetilde{W}$ by objects of the form $(W,w)$, and then we deduce $\widetilde{F}$ is of the form $\bigoplus i_{\widetilde{v_k}*}(M_{\widetilde{v_k}})$. The converse is clear, and $(4)$ is proved.\\
%Finally, to achieve $(5)$ we note that any {\em sheaf} on $T_{\widetilde{v_k}}$ is uniquely determined by its value at $\{\widetilde{v_k}\}$. By definition, we have
%$$i_{\widetilde{v_k}}^*j_*F(\{\widetilde{v_k}\}) = \varinjlim_{(W,w)} F(W),$$
%with $(W,w)$ objects in $\mathbf{\acute{E}t}_{\widetilde{U}}$ and $w\in W(\bR)$ mapped to $\widetilde{v_k}$. The limit can be interpreted as evaluating $F$ at a maximal extension of $K$ {\em unramified} at $\widetilde{v_k}$. Since $\overline{K}^{I_{\widetilde{v_k}}}$ is such extension, we get:
%$$i_{\widetilde{v_k}}^*j_*F(\{\widetilde{v_k}\}) = F_{\eta}^{I_{\widetilde{v_k}}}$$
%as desired.
%\end{proof}
Next, we record the Decomposition Lemma adapted to the study of
sheaves on the \'etale site of $\widetilde{U}$. (This goes beyond the version for schemes, but
is easily deduced from that due to what has been shown above, including Remark \ref{goodphi}). %and Lemma \ref{equiv}(5).)
\begin{lemma}\label{gen decomposition} Let $j: \widetilde{U}'\to \widetilde{U}$ be a non-empty open subset, and $S = \{v_1,\dots, v_n\}$
the complement of $\widetilde{U}'$ in $\widetilde{U}$. For any abelian sheaf $\widetilde{F}$ on $\widetilde{U}$, the assignment:
$$\textup{dec}(\widetilde{F}) := (\widetilde{F}_{{v_1}},\dots, \widetilde{F}_{{v_n}}, j^*(\widetilde{F}), \varphi_{v_1}: \widetilde{F}_{v_1}\to \widetilde{F}_{\eta}^{I_{v_1}},\dots, \varphi_{v_n}: \widetilde{F}_{v_n}\to \widetilde{F}_{\eta}^{I_{v_n}})$$
with $\widetilde{F}_{v_k}$ being the discrete $G_{v_k}$-module $i_{v_k}^*(\widetilde{F})$ for each $k$, and $$\varphi_{v_k} : \widetilde{F}_{v_k} = i_{v_k}^*(\widetilde{F}) \to i_{v_k}^*\eta_*\eta^*(\widetilde{F}) = \widetilde{F}_{\eta}^{I_{v_k}}$$ the natural $G_{v_k}$-module homomorphism for all $k$, is a functor which yields an equivalence between the category $\Ab_{\widetilde{U}}$ and the category of tuples:
$$(M_1,\dots, M_n, F', \varphi_1,\dots, \varphi_n)$$
with $M_k$ a discrete $G_{v_k}$-module for each $k$, $F'$ an abelian sheaf on $\widetilde{U}'$, and $\varphi_k : M_k \to {F'_{\eta}}^{I_{v_k}}$ a $G_{v_k}$-module homomorphism for all $k$.
\end{lemma}
To appreciate the power of Theorem \ref{abshvs tilde}, we discuss an example.
\begin{eg}\label{gsec} The constant
sheaf $\bZ_{\widetilde{U}}$ on the site $\widetilde{U}$ corresponds to $(\{\bZ\}_{\widetilde{v}}, \bZ_U, \{{\rm{id}}_{\bZ}\}_{\widetilde{v}}) \in
\fAb_{\widetilde{U}}$.
Since $\widetilde{F}(\widetilde{U}) = \Hom(\bZ_{\widetilde{U}}, \widetilde{F})$, we immediately obtain the formula
$$\widetilde{F}(\widetilde{U}) = F(U) \times_{F_{\eta}} \widetilde{F}_{\widetilde{v_1}}\times_{F_{\eta}}\cdots\times_{F_{\eta}} \widetilde{F}_{\widetilde{v_s}},$$
where the fibre product is taken with respect to the natural map $F(U) \to F_{\eta}$ and the specialization maps $\varphi_{\widetilde{v_k}} : \widetilde{F}_{\widetilde{v_k}}
\to F_{\eta}$. The right side thereby {\em defines} the ``global sections'' functor $\Gamma(\widetilde{U}, \cdot)$ on the category of triples $\fAb_{\widetilde{U}}$.
\end{eg}
\subsection{Push/pull functors, finite morphisms, and traceable sheaves}
We finish this preliminary discussion by introducing pushforward and pullback functors beyond
the setting open (and closed) immersions; they will be especially useful in the case of {\em finite} morphisms
(appropriately defined!).
Fix a pair of number fields $K'$ and $K$ and the associated schemes
$X' = \Spec(\ccO_{K'})$ and $X = \Spec(\ccO_K)$.
We have the corresponding ``spaces'' $\overline{X}$ and $\overline{X}'$ and associated \'etale sheaf theories.
[It would be more natural to permit $K$ and $K'$ to be finite \'etale $\bQ$-algebras, allowing disconnectedness of the spaces,
as disconnectedness arises when forming fiber products. However, this will not be necessary for our needs and it overloads
the notation too much to keep track of the connected components and associated abundance of generic points. Hence,
we impose connectedness throughout.]
For non-empty open subsets $\widetilde{U}' \subset \overline{X}'$
and $\widetilde{U} \subset \overline{X}$ with respective ``finite parts'' $U' \subset X'$ and $U \subset X$ (complement
of the respective real points in each), a {\em morphism}
$\widetilde{f}: \widetilde{U}' \rightarrow \widetilde{U}$ is defined to be a morphism
$f: U' \rightarrow U$ such that on $\mathbf{R}$-points $f$ carries $\widetilde{U}'$ into $\widetilde{U}$.
Composition of such morphisms is defined in the evident manner.
\begin{defin}\label{infpushpull} Let $\widetilde{f}: \widetilde{U}' \rightarrow \widetilde{U}$ be a morphism. The {\em pullback} $\widetilde{f}^{\ast}: \fAb_{\widetilde{U}} \rightarrow
\fAb_{\widetilde{U}'}$ is
$$\widetilde{f}^{\ast}(\{\widetilde{F}_{\widetilde{v}}\}, F, \{\varphi_{\widetilde{v}}\}) = (\{\widetilde{F'}_{\widetilde{v'}}\}, F', \{\varphi'_{\widetilde{v'}}\})$$
where $F' = f^{\ast}(F)$, $\widetilde{F'}_{\widetilde{v'}} = \widetilde{F}_{\widetilde{f}(\widetilde{v'})}$, and $\varphi'_{\widetilde{v'}} =
\varphi_{\widetilde{f}(\widetilde{v'})}$. The {\em pushforward}
$\widetilde{f}_{\ast}: \fAb_{\widetilde{U}'} \rightarrow \fAb_{\widetilde{U}}$ is
$$\widetilde{f}_{\ast}(\{\widetilde{F'}_{\widetilde{v'}}\}, F', \{\varphi'_{\widetilde{v'}}\}) =
(\{\widetilde{F}_{\widetilde{v}}\}, F, \{\varphi_{\widetilde{v}}\})$$
where $F = f_{\ast}(F')$,
$$\widetilde{F}_{\widetilde{v}} = (\bigoplus_{\widetilde{v'} \in \widetilde{f}^{-1}(\widetilde{v})} \widetilde{F'}_{\widetilde{v'}}) \times_{
\bigoplus_{\widetilde{v'} \in \widetilde{f}^{-1}(\widetilde{v})} F'(K'_{v'})} (f_{\ast} F')(K_v) =
(\bigoplus_{\widetilde{v'} \in \widetilde{f}^{-1}(\widetilde{v})} \widetilde{F'}_{\widetilde{v'}}) \times_{F'(\prod_{v' \mapsto v} K'_{v'})}
F(K_v)$$
(using the maps $\varphi'_{\widetilde{v'}}$ for the first component of the fiber product and the natural projection from
$(f_{\ast}F')(K_v) = F'(K' \otimes_K K_v) = F(K_v)$ onto its ``real factors'' for the second component),
and for $\widetilde{v} \in \widetilde{U}$ the map
$\varphi_{\widetilde{v}}: \widetilde{F}_{\widetilde{v}} \rightarrow F_{\eta}^{I_v} = F(K_v)$ is the second
projection from the fiber product.
\end{defin}
\begin{rem}\label{pushstalk}
If $w'$ varies through the {\em non-real} places of $K'$ over a real place $v$ of $K$
then $$(\widetilde{f}_{\ast}\widetilde{F'})_{\widetilde{v}} = \bigoplus_{\widetilde{v'} \mapsto \widetilde{v}} \widetilde{F'}_{\widetilde{v'}} \times
\bigoplus_{w'} F'(K'_{w'}).$$
The appearance of such places $w'$ in general will underlie the adjointness between $\widetilde{f}_{\ast}$ and $\widetilde{f}^{\ast}$! (Think
about the case when $K'$ has {\em no} real places to see that complex places of $K'$ over $v$ must be taken into account
to possibly have adjoint functors over $v$.)
\end{rem}
\begin{eg}\label{pushformula}
For open $\widetilde{V} \subset \widetilde{U}$ there is an evident notion of {\em preimage}
$\widetilde{f}^{-1}(\widetilde{V}) \subset \widetilde{U}'$, and building on Example \ref{gsec} it is very instructive (and a good exercise!) to build a
natural isomorphism $$(\widetilde{f}_{\ast}(\widetilde{F'}))(\widetilde{V}) = \widetilde{F'}(\widetilde{f}^{-1}(\widetilde{V})).$$
This is rather interesting: the appearance of complex stalks in the definition of $\widetilde{f}_{\ast}$ is wiped out
at the level of global sections as on the right side; this apparent paradox
is resolved by the role of fiber products over the generic stalk in the description of
the global-sections functors.
This formula for $(\widetilde{f}_{\ast}(\widetilde{F'}))(\widetilde{V})$ with varying open $\widetilde{V} \subset \widetilde{U}$
cannot be used to {\em define} $\widetilde{f}_{\ast}$ (why not?).
\end{eg}
For a second morphism $\widetilde{F'}: \widetilde{U}'' \rightarrow \widetilde{U}'$
involving a third number field $K''$, we define the isomorphism
$\widetilde{f}_{\ast} \circ \widetilde{F'}_{\ast} \simeq (\widetilde{f} \circ \widetilde{F'})_{\ast}$ in the evident manner,
and this satisfies an obvious associativity condition.
\begin{lemma} The functor $\widetilde{f}^{\ast}$ is exact and is a left adjoint to $\widetilde{f}_{\ast}$.
\end{lemma}
\begin{proof}
It is elementary to check that exactness of $\widetilde{f}^{\ast}$ by computing
on stalks (see Remark \ref{stalks}!), and the left-adjoint property is a straightforward
diagram-chase (underlying the reason that we incorporate the stalks of $F'$ at {\em all} points of $\Spec(K' \otimes_K K_v)$ in the definition of
$\widetilde{f}_{\ast}(\widetilde{F'})_{\widetilde{v}}$!)
\end{proof}
\begin{eg}\label{kerR}
The $\widetilde{v}$-stalk of the kernel of $\widetilde{F} \rightarrow \widetilde{f}_{\ast}\widetilde{f}^{\ast}(\widetilde{F})$
vanishes if $v$ lifts to a real place of $K'$ and otherwise it is $\ker \varphi_{\widetilde{v}} = i_{\widetilde{v}\ast} i_{\widetilde{v}}^{!}(\widetilde{F})$
(since the stalk of
$f^{\ast}(F)$ at any complex place over $v$ computes the geometric points of the generic fiber $F_{\eta}$).
That the kernel can have a nonzero stalk at a real place that does not lift to a real place is
geometrically reasonable.
\end{eg}
\begin{defin}\label{finiteR} Let $\widetilde{f}: \widetilde{U}' \rightarrow \widetilde{U}$ be a morphism, with $K \rightarrow K'$
the associated injection arising from the dominant map $f:U' \rightarrow U$. We say that $\widetilde{f}$ is {\em finite} if $f$ is finite
and $\widetilde{U}'_{\infty}$ coincides with the preimage of $\widetilde{U}_{\infty}$ under the map
$X'(\mathbf{R}) \rightarrow X(\mathbf{R})$ induced by $K'/K$.
\end{defin}
Put in other words, if $K'/K$ is a given finite extension and $\widetilde{U}$ is a dense open subset of $\overline{X}$
then there is an evident notion of ``preimage'' of $\widetilde{U}$ in $\overline{X}'$ and finiteness says exactly
that $\widetilde{U}'$ is that preimage (informally, it says that $\widetilde{U}'$ is the ``normalization'' of $\widetilde{U}$
in $\overline{X}'$). One subtlety is that if $K$ has real places but $K'$ is totally imaginary then
$\widetilde{f}^{\ast}(\widetilde{F})$ loses all information about the stalks of $\widetilde{F}$ at real points of $\widetilde{U}$, so
the natural map $\widetilde{F} \rightarrow \widetilde{f}_{\ast} \widetilde{f}^{\ast}(\widetilde{F})$ is the zero map
on stalks at real points of $\widetilde{U}$ that do not lift to real points of $\widetilde{U}'$.
The purpose of the preceding abstract nonsense is so that we can obtain:
\begin{prop}\label{pushR} If $\widetilde{f}: \widetilde{U}' \rightarrow \widetilde{U}$ is finite then $\widetilde{f}_{\ast}$ is exact
and $$\ker(\widetilde{F} \rightarrow \widetilde{f}_{\ast} \widetilde{f}^{\ast}(\widetilde{F})) =
\bigoplus_{v \in \Sigma} i_{\widetilde{v}\ast}i_{\widetilde{v}}^{!}(\widetilde{F})$$ where $\Sigma$
is the set of real places of $\widetilde{U}$ that do not lift to a real place of $\widetilde{U}'$.
In particular, if $U \subset X$ and $U' \subset X'$ are dense opens with $f:U' \rightarrow U$
a finite map and $\overline{f}: \overline{U}' \rightarrow \overline{U}$ the associated finite morphism
then $$\ker(\overline{F} \rightarrow \overline{f}_{\ast} \overline{f}^{\ast}\overline{F}) =
\bigoplus_{v \in U(\mathbf{R}) - f(U'(\mathbf{R}))} i_{\widetilde{v}\ast}i_{\widetilde{v}}^{!}(\overline{F}).$$
\end{prop}
\begin{proof}
This is an elementary computation on stalks (see Remark \ref{stalks}), requiring just a bit of attention to the role
of complex stalks in the definition of $\widetilde{f}_{\ast}$.
The determination of the kernel is a matter of unwinding definitions.
\end{proof}
\begin{eg}\label{finiteinj}
If $\widetilde{F}_{\widetilde{v}} = 0$ for all real points $\widetilde{v} \in \widetilde{U}$
then $\widetilde{F} \rightarrow \widetilde{f}_{\ast} \widetilde{f}^{\ast} \widetilde{F}$ has trivial kernel.
But beware that $\widetilde{f}_{\ast} \widetilde{f}^{\ast}\widetilde{F}$ generally has nonzero stalks
at such $\widetilde{v}$ (if $K'$ has complex places over $v$).
\end{eg}
Finally, we consider $f:U' \rightarrow U$ that is a connected finite \'etale cover and ask when the usual
trace map ${\rm{tr}}: f_{\ast}f^{\ast} \rightarrow {\rm{id}}$ extends to a ``trace map''
$\overline{f}_{\ast}\overline{f}^{\ast} \rightarrow {\rm{id}}$. Over $U$ we know what to do, but
how should we make a definition for stalks at a real point $\widetilde{v}$? If $\Sigma$ is the set of real places on $K'$ over
$v$ and $\Sigma'$ is the set of non-real places over such a $v$ then on $\widetilde{v}$-stalks we seek a suitable map
$$\prod_{v' \in \Sigma} \widetilde{F}_{\widetilde{v'}} \times \prod_{w' \in \Sigma'} F_{\eta}^{I_{w'}} \rightarrow \widetilde{F}_{\widetilde{v}}.$$
There is no apparent such
map when $\Sigma'$ is non-empty except if $\varphi_{\widetilde{v}}: \widetilde{F}_{\widetilde{v}} \rightarrow F_{\eta}^{I_v} = F(K_v)$
is an isomorphism, in which case
$(\overline{f}_{\ast}\overline{f}^{\ast}\overline{F})_{\widetilde{v}} = F(K' \otimes_K K_v)$
and hence we may {\em define} the $\widetilde{v}$-stalk map
$${\rm{tr}}_{\widetilde{v}}: F(K' \otimes_K K_v) \rightarrow F(K_v)$$
to be the map on $K_v$-points by the usual trace map $f_{\ast} f^{\ast}(F) \rightarrow F$.
This motivates:
\begin{defin}\label{trdef} A sheaf $\overline{F}$ on $\overline{U}$ is {\em traceable} if $\varphi_{\widetilde{v}}: \overline{F}_{\widetilde{v}} \rightarrow F_{\eta}^{I_v} = F(K_v)$
is an isomorphism for all real points $v$ of $U$. For such $\overline{F}$ that are locally constant and constructible on $U$,
and connected finite \'etale cover $f:U' \rightarrow U$,
the {\em trace map}
$${\rm{tr}}_{f,\overline{F}}: \overline{f}_{\ast} \overline{f}^{\ast}(\overline{F}) \rightarrow \overline{F}$$
is defined to be the usual trace $f_{\ast}f^{\ast}F \rightarrow F$ over $U$ and
its effect on $K_v$-points at each real place $v$ (so the necessary diagrams commute, hence ${\rm{tr}}_{f, \overline{F}}$ is a morphism of sheaves on $\overline{U}$).
\end{defin}
\begin{rem}\label{trrem}
It is clear that $\overline{F} \rightsquigarrow \overline{F}|_U$ is an equivalence between the categories
of traceable sheaves on $\overline{U}$ and abelian sheaves on $U$. The inverse functor
is $j_{\ast}$ where $j:U \rightarrow \overline{U}$ is the natural ``map'';
explicitly, $j_{\ast}(F) = (\{F(K_v)\}, F, \{\rm{id}\})$. Note that $\overline{F}$ is locally constant and constructible
if and only if $F$ is.
For such $\overline{F}$ that
are locally constant and constructible and any connected finite \'etale cover $f:U' \rightarrow U$, the composition
$\overline{F} \rightarrow \overline{F}$ of
${\rm{tr}}_{f, \overline{F}}$ with the natural map
$\overline{F} \rightarrow \overline{f}_{\ast}\overline{f}^{\ast}\overline{F}$ coincides
with multiplication by the degree of $f$, as we may verify by computing over $U$
since $\overline{F}$ is lcc (or can do directly on stalks separately at real points
and usual points of $U$). This will be {\em very useful} to reduce some later cohomological problems
with $K$ to the case of a totally imaginary extension $K'/K$ that is easier to work with (e.g., no real points).
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cohomology theory}\label{cohomsec}
We now develop what is necessary for \'etale cohomology over $\widetilde{U}$ with real points
$\widetilde{v_1}, \dots, \widetilde{v_s}$.
\subsection{Two basic $\delta$-functors}\label{twodelta}
Since $\Gamma(\widetilde{U}, \cdot)$ is left-exact, we can make the habitual definition:
\begin{defin} \label{cohomology} For any abelian sheaf $\widetilde{F}$ in $\Ab_{\widetilde{U}}$, we set:
$$\rH^p(\widetilde{U}, \widetilde{F}) := \rR^p\Gamma(\widetilde{U}, \widetilde{F}),\ p\ge 0.$$
\end{defin}
For $x \in \widetilde{U}_{\infty}$, recall from Remark \ref{goodphi} that the functor
$$\widetilde{F} \rightsquigarrow \ker(\varphi_x: \widetilde{F}_x \rightarrow i_x^{\ast} j_{\ast} j^{\ast}(\widetilde{F}))$$
has the right-adjunction property to deserve the notation $i_x^!$.
More importantly, in view of the explicit formula describing $\Gamma(\widetilde{U}, \cdot)$ in Example \ref{gsec},
we see that if $\widetilde{V} = \widetilde{U} - \{x\}$ then the functor
$$\Gamma_x(\widetilde{U}, \cdot) := \ker(\Gamma(\widetilde{U},\cdot) \rightarrow \Gamma(\widetilde{V}, \cdot))$$
of ``global sections with supports in $x$'' coincides with $i_x^{!}$.
Thus, the functor
$$\rH^{\bullet}_x(\widetilde{U}, \cdot) = \rR^{\bullet}\Gamma_x(\widetilde{U}, \cdot)$$
of {\em local cohomology with supports in $x \in \widetilde{U}_{\infty}$} coincides with $\rR^{\bullet}i_{x}^!$.
For a usual closed point $x \in U$, the functor $\Gamma_x(\widetilde{U}, \cdot)$ of ``global sections with supports in $x$''
is the composition of $\Gamma(x, \cdot)$ with $i_x^{!}$. The modified \'etale site
treats the real points as if they are geometric points, but the usual closed points of $U$ are cohomologically {\em nontrivial}
(so the output of $i_x^{!}$ must be viewed as a sheaf or Galois module, not just an abelian group, when $x \not\in \widetilde{U}_{\infty}$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The local cohomology sequence}\label{localcohom}
Let $j:\widetilde{V}\hookrightarrow \widetilde{U}$ be a non-empty open subset, and call $S$ its complement.
For any abelian sheaf $\widetilde{F}$ on $\widetilde{U}$, we set:
$$\Gamma_S(\widetilde{U},\widetilde{F}):=\ker(\widetilde{F}(\widetilde{U})\to\widetilde{F}(\widetilde{V})).$$
It is easy to see that the functor
$$\Gamma_S(\widetilde{U}, \cdot):\Ab_{\widetilde{U}}\to\Ab$$
from the category of abelian sheaves on $\widetilde{U}$ to the category of abelian groups is a left-exact functor. We denote by $\rH^p_S(\widetilde{U},\cdot)$ the right-derived functors $\rR^p\Gamma_S(\widetilde{U},\cdot)$.
For $\widetilde{I}$ an injective sheaf on $\widetilde{U}$, we have that the sequence:
$$0\to\Gamma_S(\widetilde{U}, \widetilde{I})\to\Gamma(\widetilde{U}, \widetilde{I})\to \Gamma(\widetilde{V},\widetilde{I})$$
is also right-exact. To see this, we claim it is enough to consider sheaves of the form described in Remark \ref{injectives}. Indeed, if $\widetilde{I}$ is contained into another abelian sheaf $\widetilde{J}$, then it is a direct summand of this latter, by injectivity, and it suffices to treat $\widetilde{J}$. But by Remark \ref{injectives}, every abelian sheaf on $\widetilde{U}$ can be embedded into one of the injectives described there as an abelian subsheaf, and we conclude. The right-exactness of the above sequence for this explicit class of injectives is clear, because it boils down to the analogous claim for injectives on $U_{\rm \acute{e}t}$.
We now let $\widetilde{F}\to\widetilde{I^{\bullet}}$ be an injective resolution, and take cohomology of the short-exact sequence of complexes:
$$0\to\Gamma_S(\widetilde{U}, \widetilde{I}^{\bullet})\to\Gamma(\widetilde{U}, \widetilde{I}^{\bullet})\to \Gamma(\widetilde{V},\widetilde{I}^{\bullet})\to 0$$
It yields the following {\em local cohomology sequence}
$$\cdots\to \bigoplus_{x\in S}\rH^p_x(\widetilde{U},\widetilde{F})\to \rH^p(\widetilde{U},\widetilde{F})\to \rH^p(\widetilde{V},\widetilde{F}\vert_{\widetilde{V}})\to \cdots$$
%Another way to build this, as early in SGA2, is by applying $\Ext^{\bullet}_{\Ab_{\widetilde{U}}}(-,\widetilde{F})$ to the exact sequence
%$$0\to j_!j^*(\bZ)\to \bZ\to i_*i^*(\bZ)\to 0$$
%and showing that $\rH^p(\widetilde{U}, j_!(\widetilde{F})) = 0$ for all $p\ge 1$ and abelian sheaves $\widetilde{F}$ on $\widetilde{V}$. To do this, as before one needs to check that $j_!(\widetilde{I})$ is acyclic for injective $\widetilde{I}$, which is done using the fact that $i^*$ preserves injectives, and hence
%$$0\to j_!(\widetilde{I})\to j_*(\widetilde{I}) \to i_*i^*j_*(\widetilde{I})\to 0$$
%is an injective resolution.
%
%THIS WAS CORRECT FOR COHOMOLOGY OVER A DVR!!
\begin{rem}\label{filtered colims}
A consequence of the local cohomology sequence is that \'etale cohomology of $\widetilde{U}$ commutes with filtered colimits. Indeed, it reduces this to the case of usual \'etale cohomology of $U$, and cohomology of $I_v$, which are both known. This will be used without comment!
\end{rem}
\begin{rem}\label{localsign}
It is natural to wonder if the connecting map $\rH^p(\widetilde{V}, \widetilde{F}|_{\widetilde{V}}) \rightarrow
\rH^{p+1}_S(\widetilde{U}, \widetilde{F})$ is $\delta$-functorial in the sense that for a short exact sequence
$$0 \rightarrow \widetilde{F}' \rightarrow \widetilde{F} \rightarrow \widetilde{F}'' \rightarrow 0$$
of abelian sheaves on $\widetilde{V}$, the diagram of connecting maps
$$\xymatrix{
\rH^p(\widetilde{V}, \widetilde{F}''|_{\widetilde{V}}) \ar[r] \ar[d] & \rH^{p+1}_S(\widetilde{U}, \widetilde{F}'') \ar[d] \\
\rH^{p+1}(\widetilde{V}, \widetilde{F}'|_{\widetilde{V}}) \ar[r] & \rH^{p+2}_S(\widetilde{U}, \widetilde{F}')}$$
commutes. This is not an entirely idle question: it will arise later when we need
to convert an abstract isomorphism into {\em concrete terms} to recover one of Tate's theorems.
The diagram actually does not commute, but it comes close: it {\em anti-commutes}. This sign issue is a well-known phenomenon
for the analogous situation in usual topological and \'etale cohomologies, and by going back to the definitions of the maps
it expresses a basic fact in homological algebra: given a commuting diagram of short exact sequences of complexes
$$\xymatrix{
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d] &\\
0 \ar[r] & A' \ar[r] \ar[d] & A \ar[r] \ar[d] & A'' \ar[d] \ar[r] & 0\\
0 \ar[r] & B' \ar[d] \ar[r] & B \ar[r] \ar[d] & B'' \ar[d] \ar[r] & 0\\
0 \ar[r] & C' \ar[d] \ar[r] & C \ar[r] \ar[d] & C'' \ar[d] \ar[r] & 0\\
& 0 & 0 & 0 &}$$
in an abelian category,
the resulting diagram of connecting maps from the snake lemma
$$\xymatrix{
\rH^n(C'') \ar[d] \ar[r] & \rH^{n+1}(A'') \ar[d] \\
\rH^{n+1}(C') \ar[r] & \rH^{n+2}(A) }$$
anti-commutes; see Exercise 10.2.6 in Weibel's book on homological algebra for a broader context on the significance of this fact.
\end{rem}
\medskip
Setting $\widetilde{V} := \widetilde{U}-\widetilde{U}_{\infty}$, we have $$\rH^p(\widetilde{V},\widetilde{F}\vert_{\widetilde{V}}) = \rH^p(U,F).$$
It follows that problems for $\rH^p(\widetilde{U},\widetilde{F})$ (such as finiteness properties)
can sometimes be reduce to separate consideration of usual \'etale cohomology groups and of
local cohomology with support at real points.
We deal with this latter case in the following:
\begin{lemma}\label{support} For $\widetilde{v} \in \widetilde{U}_{\infty}$, we have canonical isomorphisms:
$$\rH^0_{\widetilde{v}}(\widetilde{U},\widetilde{F}) = \ker(\varphi_{\widetilde{v}}),\ \ \ \rH^1_{\widetilde{v}}(\widetilde{U},\widetilde{F}) = \coker(\varphi_{\widetilde{v}})$$
$$\rH^p_{\widetilde{v}}(\widetilde{U},\widetilde{F}) = \rH^{p-1}(I_{\widetilde{v}},\widetilde{F}_{\eta}),\ \ \ p\ge 2.$$
In this latter isomorphism, the right side is group cohomology.
\end{lemma}
\begin{proof}
We only need to check that the right sides form a $\delta$-functor such that
$$\rH^0_{\widetilde{v}}(\widetilde{U},\widetilde{F}) = \ker(\varphi_{\widetilde{v}})$$
and the ``erasability'' property
$$\coker(\varphi_{\widetilde{v}}) = \rH^{p-1}(I_{\widetilde{v}}, \widetilde{F}_{\eta}) = 0,\ \ p\ge 2$$
holds for all injective sheaves $\widetilde{F}$.
In \S\ref{twodelta} we reviewed the reason that $\rH^0_{\widetilde{v}}(\widetilde{U}, \widetilde{F}) = \ker(\varphi_{\widetilde{v}})$.
Likewise, if $\widetilde{F}$ is injective then to show that $\coker(\varphi_{\widetilde{v}}) = 0$ it suffices to show that
$\widetilde{F} \rightarrow j_{\ast}j^{\ast}(\widetilde{F})$ is an epimorphism. If $\widetilde{F}$ is embedded in another
abelian sheaf then it splits off as a direct summand, and so this epimorphism problem is reduced to the ``cofinal'' system of
injectives as built in Remark \ref{injectives}. For those specific injectives the epimorphism property is obvious.
Now consider a short-exact sequence of abelian sheaves on $\widetilde{U}$:
$$0\to\widetilde{F'}\to\widetilde{F}\to\widetilde{F''}\to 0$$
Using the long-exact sequence for group cohomology, we find a commutative diagram with exact lines:
\begin{displaymath}
\xymatrix{0\ar[r] & \widetilde{F'}_{\widetilde{v}}\ar[d]_{\varphi'_{\widetilde{v}}}\ar[r] & \widetilde{F}_{\widetilde{v}}\ar[r]\ar[d]_{\varphi_{\widetilde{v}}} & \widetilde{F''}_{\widetilde{v}}\ar[r]\ar[d]_{\varphi''_{\widetilde{v}}} & 0 & \\
0\ar[r] & \rH^0(I_{\widetilde{v}}, \widetilde{F'}_{\eta})\ar[r] & \rH^0(I_{\widetilde{v}}, \widetilde{F}_{\eta})\ar[r] & \rH^0(I_{\widetilde{v}}, \widetilde{F''}_{\eta})\ar[r] & \rH^1(I_{\widetilde{v}}, \widetilde{F'}_{\eta})\ar[r] & \cdots}
\end{displaymath}
Taking the ker-coker sequence on each column, we deduce that the right sides in the assertions of the Lemma indeed form a $\delta$-functor.
Using Remark \ref{injectives} once again, and noting that tautologically $i_{\widetilde{v}}^*(M)_{\eta} = 0$
for any abelian group $M$, we are reduced to showing that any injective sheaf $F$ on $U$ satisfies
$$\rH^p(I_{\widetilde{v}}, F_{\eta}) = 0$$
for all $p\ge 1$. Here we are computing cohomology on the field $\overline{K}^{I_{\widetilde{v}}}$ of real algebraic numbers over $K$.
This field is the directed limit of Zariski-localized rings of integers of number fields inside $\overline{K}^{I_{\widetilde{v}}}$.
This corresponds via Spec with the inverse system of connected \'etale affine
$U$-schemes $Y$ whose function field is equipped with an embedding into $\overline{K}^{I_{\widetilde{v}}}$.
Since \'etale cohomology of qcqs schemes commutes with inverse limits having affine transition maps,
$$\rH^p(I_{\widetilde{v}}, F_{\eta}) = \varinjlim_{Y}\rH^p(Y, F|_Y).$$
Pullback along \'etale maps $Y \rightarrow U$ preserves injectives, so the colimit on the right side vanishes for all $p\ge 1$.
\end{proof}
As an immediate corollary of the local cohomology sequence and the fact that $\rH^p(I_{\widetilde{v}}, M) = 0$ for $p\ge 1$
when $M$ is an $I_{\widetilde{v}}$-module on which 2 acts invertibly (e.g., a torsion group with trivial 2-primary part), we get:
\begin{cor}\label{HSerre} If $2$ acts invertibly on $F_{\eta}$ then
$$\rH^p(\widetilde{U}, \widetilde{F} ) = \rH^p(U,F)$$
for all $p\ge 2$.
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cohomology of $\mathbf{G}_{m,\overline{U}}$}\label{cohomology multiplicative}
Let $U \subset X$ be a dense open subscheme.
We shall define the multiplicative group $\mathbf{G}_{m, \overline{U}}$ over $\overline{U}$ and compute its cohomology.
Recall that $\overline{U}$ contains all real points. In \S\ref{first} we will motivate a new canonical procedure
to extend abelian sheaves $F$ on $U$ to abelian sheaves $\widehat{F}$ on $\overline{U}$
(not extension-by-zero from $U$, nor pushforward from $U$; it will be right-exact and generally not left-exact!).
The definition given directly below will be an instance of that more general formalism applied
to the sheaf $F = \mathbf{G}_{m,U}$ (see Example \ref{Gm modified});
here we give the direct definition without any broader context and see how to work with it.
For each $v \in X_{\infty} = \overline{U}_{\infty}$, denote by $K_v^{\rm alg}$ the algebraic closure of $K$ inside its completion $K_v$ at the real place $v$.
We write $K_v^{\rm alg, +}$ for its positive part, positivity being determined via the unique isomorphism $K_v\simeq\bR$
(or by virture of $K_v^{\rm{alg}}$ being a real closed field, so it has a unique order structure as a field).
Clearly $K_v^{\rm alg, +}$ is a multiplicative subgroup of $K_v^{\rm alg, \times}$, and we denote by $\varphi_v$ this inclusion of groups.
%Recall that the fixed valuations $\overline{v_k}$ define embeddings $\overline{v_k}: \overline{K}\hookrightarrow\bC$ for all $k= 1,\dots, r$. We denote by $\overline{K}_+^{I_{v_k}}$ the set of those elements of $\overline{K}$ which are mapped into $\bR_{>0}$ by $\overline{v_k}$. Recall that $I_{v_k}$ is the decomposition subgroup of $\overline{v_k}$, for each $k$. Clearly we have: $\overline{K}_+^{I_{v_k}}\hookrightarrow \overline{K}^{I_{v_k}}$, which yelds a group homomorphism, for all $k=1,\dots, r$:
%$$\varphi_{\overline{v_k}}: \overline{K}_+^{\times, I_{v_k}}\hookrightarrow \overline{K}^{\times, I_{v_k}}.$$
\begin{defin} \label{Gm} The sheaf $\mathbf{G}_{m,\overline{U}} \in \fAb_{\overline{U}} = \mathbf{Ab}_{\overline{U}}$ is defined to be the triple
$$((K_v^{\rm alg, +})_{v\in X_{\infty}}, \mathbf{G}_{m, U}, (\varphi_v)_{v\in X_{\infty}}).$$
\end{defin}
We want to calculate the cohomology of $\mathbf{G}_{m,\overline{U}}$ over $\overline{U}$, by adapting
Artin's method for usual curves. To that end, we need some modifications of constructions that arise
in Artin's method.
Given the embedding of the generic point $\eta_U : \Spec(K)\to U$ and the embedding $j_{U} : U\to \overline{U}$, we have functors
$$\eta_{U*} : \Ab_{\eta}\to\Ab_U,\ \ \ j_{{U}*} : \Ab_U\to \Ab_{\overline{U}}.$$
Denote by $\mu_*$ their composition. From the definitions we get at once
$$\mu_*(\mathbf{G}_{m,\eta}) = ((K_v^{\rm alg})_{v\in X_{\infty}}, \eta_{U*}\mathbf{G}_{m, \eta},
(K_v^{\rm alg}\xrightarrow{\textup{id}} K_v^{\rm alg})_{v\in X_{\infty}})$$
and an evident exact sequence of abelian shaves on $\overline{U}$
$$0\to\mathbf{G}_{m,\overline{U}}\to \mu_*(\mathbf{G}_{m,\eta}) \to \bigoplus_{x\in U^0}i_{x*}(\bZ) \oplus \bigoplus_{k=1}^r i_{\widetilde{v_k}*}(\bZ/2\bZ)\to 0$$
in which the first map corresponds to the identification of $\mathbf{G}_{m,\eta}$ with $\mu^{\ast}(\mathbf{G}_{m, \overline{U}})$.
If we compute the cohomology of the latter two terms in this exact sequence then we can hope to compute the cohomology of $\mathbf{G}_{m, \overline{U}}$.
This will be achieved by using a lot of class field theory.
For $p > 0$ we have $\rR^p\eta_{U*}(\mathbf{G}_{m,\eta})=0$ by applying Proposition \ref{comp1} after base change
to the henselization (not completion!) of $\ccO_K$ at closed points of $U$. We deduce, as in the proof of Lemma \ref{support}:
$$\rR^pj_{U*}(\eta_{U\ast}\mathbf{G}_{m,\eta}) = \bigoplus_{k=1}^r i_{\widetilde{v_k}*}\rH^p(I_{v_k}, \mathbf{G}_{m,\eta}),\ \ p\ge 1.$$
Using Grothendieck's composition of functors spectral sequence, we thereby get:
$$\rR^p\mu_*(\mathbf{G}_{m,\eta}) \simeq\bigoplus_{k=1}^r i_{\widetilde{v_k}*}\rH^p(I_{v_k}, \mathbf{G}_{m,\eta}),\ \ p\ge 1.$$
Now we use the Leray spectral sequence for the functor $\mu_*$:
$$E_2^{p,q} = \rH^p(\overline{U}, \rR^q\mu_*(\mathbf{G}_{m,\eta}) ) \Rightarrow\rH^{p+q}(\eta,\mathbf{G}_{m,\eta}).$$
Since $\rR^q\mu_*(\mathbf{G}_{m,\eta})$ is concentrated in the real places for $q \ge 1$,
we deduce $E_2^{p,q} = 0$ for all $p,q\ge 1$. Therefore, we have the following exact sequence:
\begin{displaymath}
\xymatrix{ 0 \ar[r] & \rH^1(\overline{U},\mu_*(\mathbf{G}_{m,\eta})) \ar[r] & \rH^1(\eta,\mathbf{G}_{m,\eta})\ar[r] & \bigoplus_{k=1}^r\rH^1(I_{v_k}, \overline{K}^{\times})\\
\ar[r] & \rH^2(\overline{U},\mu_*(\mathbf{G}_{m,\eta})) \ar[r] & \rH^2(\eta,\mathbf{G}_{m,\eta})\ar[r] & \bigoplus_{k=1}^r\rH^2(I_{v_k}, \overline{K}^{\times})\\
\ar[r] & \rH^3(\overline{U}, \mu_*(\mathbf{G}_{m,\eta})) \ar[r] & \rH^3(\eta,\mathbf{G}_{m,\eta}). & }
\end{displaymath}
By Hilbert's Theorem 90, we have
$$\rH^1(\eta,\mathbf{G}_{m,\eta}) = 0 = \rH^1(I_{v_k}, \overline{K}^{\times})$$
and by class field theory we get an exact sequence:
\begin{equation}\label{h2seq}
0\to\rH^2(\eta,\mathbf{G}_{m,\eta}) \rightarrow \bigoplus_v\textup{Br}(K_v) \xrightarrow{\textup{sum}} \bQ/\bZ\to 0
\end{equation}
where the direct sum runs over all places $v$ of $K$. The sequence from before becomes:
\begin{displaymath}
\xymatrix{ 0 \ar[r] & \rH^1(\overline{U},\mu_*(\mathbf{G}_{m,\eta})) \ar[r] & 0 & \\
0\ar[r] & \rH^2(\overline{U},\mu_*(\mathbf{G}_{m,\eta})) \ar[r] & \rH^2(\eta,\mathbf{G}_{m,\eta})\ar[r] & \bigoplus_{k=1}^r\textup{Br}(K_{v_k})\\
\ar[r] & \rH^3(\overline{U}, \mu_*(\mathbf{G}_{m,\eta})) \ar[r] & \rH^3(\eta,\mathbf{G}_{m,\eta}). & }
\end{displaymath}
From (\ref{h2seq}) we get an exact equence:
$$0\to\rH^2(\overline{U},\mu_*(\mathbf{G}_{m,\eta})) \xrightarrow{\textup{inv}}\bigoplus_{x\in X^0}\textup{Br}(K_x)\xrightarrow{\textup{sum}} \bQ/\bZ\to 0$$
and that
$$\rH^2(\eta,\mathbf{G}_{m,\eta})\to \bigoplus_{k=1}^r\textup{Br}(K_{v_k})$$
is surjective. But $\rH^3(\eta, \mathbf{G}_{m,\eta}) = 0$ by class field theory, so we obtain exact sequences:
\begin{displaymath}
\xymatrix{ 0 \ar[r] & \rH^1(\overline{U},\mu_*(\mathbf{G}_{m,\eta})) \ar[r] & 0& & \\
0\ar[r] & \rH^2(\overline{U},\mu_*(\mathbf{G}_{m,\eta})) \ar[r] & \rH^2(\eta,\mathbf{G}_{m,\eta})\ar[r] & \bigoplus_{k=1}^r\textup{Br}(K_{v_k})\ar[r] & 0\\
0\ar[r] & \rH^3(\overline{U}, \mu_*(\mathbf{G}_{m,\eta})) \ar[r] & 0 & & }
\end{displaymath}
We summarize what we have shown so far in the following:
\begin{prop} \label{first piece} We have
$$\rH^0(\overline{U}, \mu_*(\mathbf{G}_{m,\eta})) = K^{\times},\ \ \ \rH^1(\overline{U}, \mu_*(\mathbf{G}_{m,\eta})) = 0$$
the exact sequence:
$$0\to\rH^2(\overline{U}, \mu_*(\mathbf{G}_{m,\eta})) \to\bigoplus_{x\in X^0}\bQ/\bZ\to\bQ/\bZ\to 0$$
and
$$\rH^3(\overline{U},\mu_*(\mathbf{G}_{m,\eta})) = 0.$$
\end{prop}
We also have
$$\rH^p\left(\overline{U}, \bigoplus_{x\in U^0} i_{x*}(\bZ)\oplus\bigoplus_{k=1}^ri_{\widetilde{v_k}*}(\bZ/2\bZ) \right) = 0$$
if $p\neq 0,2$, and it is equal to $\bigoplus_{x\in U^0}\bQ/\bZ$ if $p=2$.
Indeed, by Remark \ref{filtered colims},
$$\rH^p\left(\overline{U}, \bigoplus_{x\in U^0} i_{x*}(\bZ)\oplus\bigoplus_{k=1}^ri_{\widetilde{v_k}*}(\bZ/2\bZ) \right) = \bigoplus_{x\in U^0}\rH^p(\overline{U}, i_{x*}(\bZ)) \oplus \bigoplus_{k=1}^r\rH^p(\overline{U}, i_{\widetilde{v_k}*}(\bZ/2\bZ)),$$
and exactness of the pushforwards from the (finite and real) closed points identifies
the cohomologies on the right side with derived functors of the composition of $\Gamma(\overline{U}, \cdot)$ with the pushforward.
At real points such composite functors are the identity, so their higher derived functors vanish. At finite closed
points such composite functors are global sections for \'etale sheaves on the spectrum of a finite field,
so the higher derived functors are Galois cohomology of $\widehat{\mathbf{Z}}$.
This gives that $\rH^p(\overline{U}, i_{x*}(\bZ))$
is $\bQ/\bZ$ for $p=2$ and vanishes for all other positive $p$, and that $\rH^p(\overline{U}, i_{\widetilde{v_k}*}(\bZ/2\bZ)) = 0$ for all $p > 0$.
The case $p=0$ is trivial.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Putting everything together}\ \\
Given the exact sequence:
$$0\to\mathbf{G}_{m,\overline{U}}\to \mu_*(\mathbf{G}_{m,\eta}) \to \bigoplus_{x\in U^0}i_{x*}(\bZ) \oplus \bigoplus_{k=1}^r i_{\widetilde{v_k}*}(\bZ/2\bZ)\to 0$$
we apply $\rH^0(\overline{U}, \cdot)$ to get an exact sequence
$$0\to\rH^0(\overline{U}, \mathbf{G}_{m,\overline{U}})\to K^{\times} \stackrel{f}{\to} \bigoplus_{x\in U^0}\bZ \oplus \bigoplus_{k=1}^r \bZ/2\bZ\to \rH^1(\overline{U}, \mathbf{G}_{m,\eta})\to 0$$
where $$f: K^{\times} \to \bigoplus_{x\in U^0}\bZ \oplus \bigoplus_{k=1}^r \bZ/2\bZ$$
has $x$-component given by the normalized valuation for each
$x \in U^0$ and has $v_k$-component given by the sign relative to $v_k$.
Thus, $\rH^0(\overline{U}, \mathbf{G}_{m,\overline{U}})$ is the group $\ccO_{K,X-U}^{\times,+}$ of totally
positive elements of $K$ which are units over $U$; for $U = X$ we shall denote this as $\ccO_K^{\times, +}$.\\
By classical weak approximation, the composition of $f$
with projection onto the direct sum of the real components is surjective with kernel $K_+^{\times}$ consisting of the totally positive elements of $K$. Thus,
$$\rH^1(\overline{U},\mathbf{G}_{m,\overline{U}}) = \bigoplus_{x\in U^0} \bZ/\pi(K_+^{\times})$$
where $\pi: K^{\times}_+ \rightarrow \bigoplus_{x \in U^0} \bZ$ is the evident map.
Hence, $\rH^1(\overline{U}, \mathbf{G}_{m, \overline{U}})$
is a refinement of the Picard group of $U$ via a positivity condition, so we denote it as $\Pic^+(U)$.
Combining the preceding calculations with the commutative diagram
$$\xymatrix{
\rH^2(\overline{U}, \mu_{\ast}\mathbf{G}_{m,\eta}) \ar[r] \ar[d] & \rH^2(\overline{U}, i_{x\ast}\bZ) \ar@{=}[r] & \rH^2(x, \bZ) \\
\rH^2(\eta, \mathbf{G}_{m,\eta}) \ar[r] & {\rm{Br}}(K_x) & {\rm{H}}^2(K_x^{\rm{un}}/K_x, \mathbf{G}_m) \ar[l]^-{\simeq} \ar[u]_-{\simeq}}$$
(in which the two maps emanating from the lower-right corner are shown to be isomorphisms in local class field theory), we obtain:
\begin{prop} \label{HGm} For $U\neq X$, we have
$$\rH^0(\overline{U},\mathbf{G}_{m,\overline{U}}) = \ccO_{K, X-U}^{\times, +},\,\,\, \rH^1(\overline{U}, \mathbf{G}_{m,\overline{U}}) = \Pic^+(U),$$
an exact sequence
$$0\to \rH^2(\overline{U},\mathbf{G}_{m,\overline{U}}) \to \bigoplus_{x\in (X-U)^0}\bQ/\bZ \to\bQ/\bZ\to 0,$$
and $\rH^3(\overline{U},\mathbf{G}_{m,\overline{U}}) = 0$.
For $U=X$, we have:
$$\rH^0(\overline{X},\mathbf{G}_{m,\overline{X}}) = \ccO_{K}^{\times,+},\,\,\,\rH^1(\overline{X},\mathbf{G}_{m,\overline{X}}) = \Pic^+(X),\,\,\,
\rH^2(\overline{X}, \mathbf{G}_{m,\overline{X}}) = 0,\,\,\,
\rH^3(\overline{X},\mathbf{G}_{m,\overline{X}}) = \bQ/\bZ.$$
\end{prop}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Finiteness and vanishing}\label{secfinite}
\subsection{Main goal}
The purpose of this section is to show the following:
\begin{thm}\label{finitezero} Let $U$ be an non-empty
open subscheme of $X$, and $\overline{F}$ a constructible sheaf on $\overline{U}$. Then the groups $\rH^p(\overline{U},\overline{F})$ are finite abelian groups, and $\rH^p(\overline{U},\overline{F}) = 0$ for all $p\ge 4$.
\end{thm}
First consider when $K$ is totally imaginary or the stalk $F_{\eta}$ has odd order.
In these cases, the assertions amount to the same for usual \'etale cohomology on
$U$: this is obvious if $K$ is totally imaginary (as then there are no real places),
and in the second case it is Corollary \ref{HSerre}.
Upon decomposing the constructible $F$ into its $\ell$-primary parts for
varying primes $\ell$ (only finitely many of which have nontrivial contribution to $F$), the
general vanishing for each $\ell$-primary part beyond degree 3 in the totally imaginary case is part of \cite[Exp.\,X, \S 6.1]{SGA},
the proof of which amounts to many applications of results in Serre's book on Galois cohomology and also applies
to ordinary \'etale cohomology of $\ell$-primary parts for odd $\ell$ {\em without} a totally imaginary hypothesis
(as will be useful shortly).
Suppose instead that $F_{\eta}$ has odd order but $K$ may have real places.
In that case the local cohomology sequence attached to the dense open $U \subset \overline{U}$ and its complement
consisting of the real points has vanishing local cohomologies at the real points in degrees $\ge 3$
by Lemma \ref{support}. Hence, the restriction map $\rH^p(\overline{U}, \overline{F}) \rightarrow \rH^p(U,F)$
is an isomorphism for $p \ge 3$ in such cases, so the vanishing in degree $\ge 4$ reduces to the case of
ordinary \'etale cohomology that we have noted is established in \cite{SGA} without a totally imaginary hypothesis.
Next, consider the finiteness assertion if either $K$ is totally imaginary or $F_{\eta}$ has odd order.
In both cases the finiteness concerns ordinary \'etale cohomology, so for
those cases the finiteness assertion is a consequence of:
\begin{prop}\label{finiteH} For an arbitrary number field $K$
and dense open subscheme $U \subset \Spec(\ccO_K)$, the \'etale cohomology groups $\rH^p(U, F)$ are finite for all constructible sheaves $F$ on $U$.
\end{prop}
Before we prove this result, we require a technical lemma that justifies the idea of local cohomology at a point
only depending on a ``small open set'' around the point:
\begin{lemma}\label{henselization}
Let $F$ be a sheaf on $X_{\rm \acute{e}t}$ for a scheme $X$. Let $x \in X$ be a closed point, and
denote by $X^{\rm h}_x$ the henselization of $X$ at $x$. For the natural map $f_x^{\rm h}: X_x^{\rm h}\to X$,
there is a natural isomorphism
$$\rH^p_x(X, F) = \rH^p_x(X_x^{\rm h}, (f_x^{\rm h})^*F),\ \ p\ge 0.$$
\end{lemma}
\begin{proof} It is a general fact (used all the time when working with \'etale cohomology) that if $\{Y_i\}$ is an inverse system of qcqs schemes with
affine transition maps and $\{F_i\}$ is a compatible system of \'etale sheaves on them
(i.e., for $i' \ge i$, the pullback of $F_i$ along $Y_{i'} \rightarrow Y_i$ is identified with $F_{i'}$
in a ``transitivie'' manner) then for the inverse limit $Y$ of the $Y_i$'s (i.e., over any affine open $U$ in a fixed $Y_{i_0}$
we form Spec of the direct limit of the corresponding coordinate rings from the affine open preimages in each $Y_i$ of
$U$) and the common pullback $F$ of the $F_i$'s to $Y$,
the natural map $\varinjlim \rH^p(Y_i, F_i) \rightarrow \rH^p(Y, F)$ is an isomorphism for all $p \ge 0$.
Applying this to cohomology over $X$ and $X - Z$ for closed $Z \subset X$, the local cohomology sequence
shows that $\rH^p_Z(X, \cdot)$ satisfies the same limiting behavior. In particular, for $X_x := \Spec(\cO_{X,x})$, to show that
$\rH^p_x(X, F) \rightarrow \rH^p_x(X_x, F|_{X_x})$ is an isomorphism it suffices to show
that $\rH^p_x(X, F) \rightarrow \rH^p_x(U, F|_U)$ is an isomorphism for every open $j:U \subset X$ containing $X$.
But this is obvious since $\Gamma_x(X,\cdot) = \Gamma_x(U, (\cdot)|_U)$
and $j^{\ast}$ preserves injectivity (as its left adjoint $j_{!}$ is exact).
Since $X_x^{\rm h}$ is the inverse limit of the spectra of the residually trivial local-\'etale extensions of $\cO_{X,x}$,
each of which is Zariski-localizing at a $k(x)$-point over $x$ on an \'etale $\cO_{X,x}$-scheme,
it remains suffices to check that $\rH^p_x$ is unaffected by pullback to a residually trivial pointed \'etale neighborhood of $(X,x)$.
That is, if $\pi:X' \rightarrow X$ is \'etale and $x' \in \pi^{-1}(x)$ is a $k(x)$-point of $X'$ over $x$
then we claim that the natural map
$$\rH^p_x(X, \cdot) \rightarrow \rH^p_{x'}(X', \pi^{\ast}(\cdot)).$$
The functor $\pi^{\ast}$ has left adjoint $\pi_{!}$ that is exact, the same reasoning as in the proof of Zariski-local invariance
reduces the task to checking that $\Gamma_x(X, F) \rightarrow \Gamma_{x'}(X', F)$ is an isomorphism
for any \'etale abelian sheaf $F$ on $X$.
The settled passage to the local case allows us to now assume $X$ is local, so $X' \rightarrow X$ is surjective.
We may also replace $X'$ with an open around $x'$ so that $x'$ is the only point over $x$.
In particular, $X' - \{x'\}$ maps onto $X - \{x\}$.
Hence, $F(X) \rightarrow F(X')$ is injective and $F(X) \cap \Gamma_{x'}(X', F) = \Gamma_x(X, F)$.
For any $\sigma' \in \Gamma_{x'}(X', F) \subset F(X')$, to show that
$\sigma'$ comes from $\Gamma_x(X, F)$ it is therefore equivalent to show that $\sigma'$ comes from the subgroup $F(X) \subset F(X')$.
In other words, the two pullbacks of $\sigma'$ in $F(X' \times_X X')$ coincide. Both pullbacks vanish over the part
of $X' \times_X X'$ lying over $X - \{x\}$ since $X' - \{x'\}$ is the preimage of $X - \{x\}$ in $X'$ (check!).
Consequently, our problem is concentrated in the stalks at points of $X' \times_X X'$ over $x$. But
we have arranged that $x'$ is the unique point of $X'$ over $x$ and that $k(x') = k(x)$,
so the $x$-fiber of $X' \times_X X'$ is a single $k(x)$-point too. Since this schematic point (as a morphism from $\Spec(k(x))$ over $X$) is {\em invariant}
under the $X$-automorphism of $X' \times_X X'$ that swaps the two factors (thereby interchanging the two projections),
the desired equality of pullbacks follows.
\end{proof}
We turn to the proof of Proposition \ref{finiteH}.
Let's first see that for a given $F$ we may shrink $U$ as much as
we wish. Let $j:V \hookrightarrow U$ be a non-empty open subscheme with finite complement $S$, so the local cohomology sequence
$$\dots \rightarrow \rH^p_S(U, F) \rightarrow \rH^p(U,F) \rightarrow \rH^p(V, F|_V) \rightarrow \dots$$
allows us to replace $U$ with $V$ provided that the local cohomologies are finite.
Since $\Gamma_S = \bigoplus_{x \in S} \Gamma_x(U, \cdot)$, for the finiteness of local cohomology
we may use Lemma \ref{henselization} to reduce to showing the finiteness of
$\rH^p_{\mathfrak{m}}(\Spec(A), F)$ for a henselian discrete valuation ring $A$ with maximal ideal $\mathfrak{m}$
and a constructible abelian \'etale sheaf $F$ on $\Spec(A)$. This in turn reduces to finiteness of the
\'etale cohomology of constructible sheaves on $\Spec(A)$ and on its open generic point $\Spec(K)$.
Since $A$ is henselian, the Galois theory of $K$ coincides with that of its completion
(due to Krasner's Lemma), so finiteness for the Galois cohomology of $K$
on finite Galois modules is reduced to the known case of local fields of characteristic 0.
The henselian property also ensures that cohomology on $\Spec(A)$ coincides
with that at its closed point (for the stalk of a sheaf) %, which is discussed in the following Lemma \ref{henselclosed}. Hence,
and hence the elementary finiteness properties
of the Galois cohomology of finite fields completes the proof of Proposition \ref{finiteH}.
%\begin{lemma} \label{henselclosed}
%\end{lemma}
\begin{rem}\label{local0}
Since finite fields have cohomological dimension 1 (hence strict cohomological dimension $\le 2$) and non-archimedean local fields of characteristic 0 have
cohomological dimension 2 (hence strict cohomological dimension $\le 3$), the preceding considerations
relating local cohomology to Galois cohomology shows that for any $x \in U^0$ the functor $\rH^p_x(U, \cdot)$ vanishes for $p \ge 4$ on constructible sheaves (hence on all torsion sheaves, as they are always a limit of constructible subsheaves)
and for $p \ge 5$ on arbitrary abelian sheaves.
\end{rem}
Now we may shrink $U$ so that $F$ is locally constant on $U$.
For such $F$ the finiteness in degree 0 is clear. Let $f:U' \rightarrow U$ is a connected finite \'etale cover
and define $F' = f^{\ast}(F)$. In the exact sequence
$$0 \rightarrow F \rightarrow f_{\ast}(F') \rightarrow H \rightarrow 0$$
all terms are locally constant. Since $\rH^p(U, f_{\ast}(\cdot)) = \rH^p(U', \cdot)$,
we can induct upwards on $p$
to reduce to replacing $(U, F)$ with $(U', F')$. In this way we can arrange that $F$ is constant, and then
even $F = \bZ/n\bZ$ for some $n > 0$. Further use of the local cohomology sequence allows us
to shrink $U$ to make $n$ a unit on $U$, and then to replace $U$ with a connected finite \'etale cover
so that $\mu_n = \bZ/n\bZ$ on $U$ and $K$ is totally imaginary. The Kummer sequence on $U = \overline{U}$
and Proposition \ref{HGm} now do the job in view
of standard finiteness properties of unit groups and class groups for
rings of $S$-integers.
We have established Theorem \ref{finitezero} when either $K$ is totally imaginary or $F_{\eta}$ has odd order.
In fact, we can now establish the finiteness assertion in general (but will have to work harder
for the vanishing beyond degree 3 in general). Indeed,
by Proposition \ref{finiteH} (which had no ``totally imaginary'' hypothesis) the groups $\rH^p(U,F)$ are finite for all $p$, so to prove
that $\rH^p(\overline{U}, \overline{F})$ is finite for all $p$ we may use
the local cohomology sequence to reduce to proving the finiteness of
$\rH^p_{\widetilde{v}}(\overline{U}, \overline{F})$ for all $p$. But $\overline{F}$ is constructible, so
the finiteness of these local cohomologies at the real points is an immediate consequence of Lemma \ref{support}.
It remains to prove the vanishing assertion for $p \ge 4$ when we do not assume either
that $K$ is totally imaginary or $F_{\eta}$ has odd size. For this we need some preliminary results.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Preliminaries and reductions}
%We begin with a definition:
%\begin{defin}\label{galoiscovbar} Let $\overline{f} : \overline{Y}\to\overline{U}$ be an object of $\mathbf{\acute{E}t}_{\overline{U}}$. Assume a finite group $G$ acts on $\overline{f}$. We say $\overline{f}$ is an {\em unramified Galois covering} with group $G$ if the following conditions are satisfied:
%\begin{enumerate}
%\renewcommand{\labelenumi}{\textup{(\arabic{enumi})}}
%\item The underlying map of schemes $f: Y\to U$ with its canonical $G$-action is an unramified Galois covering.
%\item $G$ acts on $\overline{Y}_{\infty}$ with no fixed points.
%\item The orbit space $\overline{Y}_{\infty}/G$ is mapped bijectively onto $\overline{U}_{\infty}$.
%\end{enumerate}
%This is the same notion as that of principal homogeneous $G$-space over $\overline{U}_{\rm\acute{e}t}$.
%\end{defin}
We need one more lemma before we can complete the proof of Theorem \ref{finitezero}.
\begin{lemma}\label{effaceable} Let $\overline{F}$ be a constructible sheaf on $\overline{U}$.
For fixed $p > 0$ there exists an injection $\overline{F}\to\overline{F}'$ of $\overline{F}$ into a constructible sheaf $\overline{F}'$ such that the induced map
$$\rH^p(\overline{U},\overline{F})\to\rH^p(\overline{U},\overline{F}')$$
vanishes.
\end{lemma}
\begin{proof} By Remark \ref{filtered colims}, $\rH^p(\overline{U}, \cdot)$ commutes with direct limits. Also recall that $\overline{F}$ is a subsheaf of an injective
$J$ that is a direct sum of $j_{U*}(I)$ and $i_{\widetilde{v_k}*}(I_k)$ for an injective $I$ on $U_{\et}$ and injective
abelian groups $I_k$. More specifically, we may pick an integer $m > 0$ killing $\overline{F}$
and take such $I$ and $I_k$'s to be injective as $\bZ/m\bZ$-module sheaves. But
{\em every} torsion abelian sheaf on a noetherian scheme is a direct limit of its constructible subsheaves,
and the analogue for torsion abelian groups (as a direct limit of finite subgroups) is clear.
Hence, since $j_{U*}$ and each $i_{\widetilde{v_k}*}$ commute with direct limits (clearly from their definitions),
and the constructible $\overline{F}$ satisfies the ascending chain condition for directed systems
of (necessarily constructible!) subsheaves, we can find a sufficiently large constructible subsheaf
$\overline{F}' \subset J$ containing $\overline{F}$ such that the vanishing of the image of the {\em finite}
group $\rH^p(\overline{U}, \overline{F})$ in $\rH^p(\overline{U}, J)$ already occurs in $\rH^p(\overline{U}, F')$.
\end{proof}
We are ready to complete the proof of Theorem \ref{finitezero}, whose statement the reader is encouraged to review.
\begin{proof}[Proof of Theorem \ref{finitezero}] Recall that it remains to prove the vanishing assertion for $p \ge 4$.
Consider a dense open subscheme $V\to U$ and the local cohomology sequence:
$$\bigoplus_{x\in (U-V)^0} \rH^p_x(\overline{U},\overline{F}) \to\rH^p(\overline{U},\overline{F}) \to\rH^p(\overline{V},\overline{F}) \to\bigoplus_{x\in (U-V)^0} \rH^{p+1}_x(\overline{U},\overline{F}).$$
Note that none of the points $x$ here are real; they are all ordinary closed points. Hence,
$\rH^p_x(\overline{U},\overline{F}) = 0$ for all $p\ge 4$ by Remark \ref{local0}.
Therefore, we can replace $U$ by any dense open subscheme $V$.
By Lemma \ref{effaceable}, there exists an injection $\alpha: \overline{F}\to\overline{F}'$ into a constructible sheaf $\overline{F}'$ with the property that $\rH^p(\overline{U},\overline{F})\to\rH^p(\overline{U},\overline{F}') $ is the zero map. Call $\overline{F}''$ the cokernel of $\alpha$. From the long exact sequence we obtain a surjection
$$\rH^{p-1}(\overline{U},\overline{F}'')\twoheadrightarrow\rH^p(\overline{U},\overline{F}).$$
By induction on $p \ge 4$ we thereby reduce to the case $p=4$.
Running the preceding argument for $\overline{F}$ with $p=4$,
we can shrink $U$ so that: $U \ne X$, $F''$ is locally constant on $U$ and the order of $F''_{\eta}$ is a unit on $U$. It suffices to show that
in this case $\rH^3(\overline{U},\overline{F}'')=0$. This is achieved in the following result (applied to $\overline{F}''$).
\end{proof}
\begin{prop} \label{base} Let $\overline{F}$ be a constructible sheaf on $\overline{U}$ such that $F$ is locally constant on $U$
with order that is a unit on $U$. Then $\rH^3(\overline{U},\overline{F}) = 0$.
\end{prop}
Let us first reduce to the case that $\overline{F}$ is traceable (in the sense of Definition \ref{trdef}).
For $F = \overline{F}|_U$, consider the traceable sheaf $j_{\ast}(F)$ as defined in Remark \ref{trrem}.
There is an evident map $\overline{F} \rightarrow j_{\ast}(F)$ whose kernel and cokernel are supported at
the real points of $U$; in fact, by Lemma \ref{support} we have an exact sequence
$$0 \rightarrow \bigoplus_{v \in X_{\infty}} i_{\widetilde{v}\ast}i_{\widetilde{v}}^{!}(\overline{F}) \rightarrow
\overline{F} \rightarrow j_{\ast}(F) \rightarrow \bigoplus_{v \in X_{\infty}} i_{\widetilde{v}\ast} \rR^1 i_{\widetilde{v}}^{!}(\overline{F}) \rightarrow 0.$$
The cohomology functors $\rH^m(\overline{U}, \cdot)$ kill all pushforwards from real points $v$ for $m > 0$
because $\Gamma(\overline{U}, i_{\widetilde{v}\ast}(\cdot))$ is the identity functor. It follows that the natural map
$$\rH^p(\overline{U}, \overline{F}) \rightarrow \rH^p(\overline{U}, j_{\ast}(F))$$
is an isomorphism for all $p > 0$. Hence, for our purposes we may replace $\overline{F}$ with $j_{\ast}(F)$
to reduce to the case that $\overline{F}$ is traceable.
To go further, we need a lemma that is an application of Sylow subgroups:
\begin{lemma} \label{Gal covering}
Let $U$ be a connected normal noetherian scheme. Let $\ell$ be a prime number and
$F$ a locally constant sheaf of finite-dimensional $\bF_\ell$-vector
spaces on $U_{\et}$. There exists a connected finite \'etale cover
$f : V \to U$ of degree prime to $\ell$ such that $f^{*}(F)$
has a finite filtration whose successive quotients are
$\underline{\bZ/\ell\bZ}_V$.
\end{lemma}
\begin{proof}
Since $U$ is connected and $F$ is locally constant and constructible, it has
constant fiber-rank $r$. We may therefore choose a Galois connected finite
\'etale cover $f:V \rightarrow U$ such that $f^{*}(F)$ is isomorphic to
$\underline{\bZ/\ell \bZ}^{\oplus r}$.
Let $G$ be the Galois group of this cover; this is the Galois group of the field extension
at the generic points. Observe that the action of $G$ on $V$ lifts to
an action of $G$ on
$f^{*}(F) \simeq \underline{\bZ/\ell \bZ}^{\oplus r}$.
Looking at the stalk in the generic point gives a representation
$\rho : G \to \GL_r(\bF_\ell)$.
Let $H \subset G$ be an
$\ell$-Sylow subgroup. We claim that $V/H \to U$ works.
The action of $\pi_1(V/H)$ (based at its generic point) associated to the pullback of
$F$ to $V/H$ corresponds to $\rho|_H$, but
$H$ is a finite $\ell$-group and hence
$\rho\vert_H$ is a successive extension of copies of
$\bZ/(\ell)$ with trivial $H$-action. Since
$V/H \to U$ has degree $[G:H]$ that is prime to $\ell$, it does the job.
\end{proof}
Continuing with the proof of Proposition \ref{base},
since $K(\zeta_{\ell})$ is unramified over $K$ away from $\ell$ and has degree at most $\ell - 1 < \ell$
we can find a connected finite \'etale cover $f:U' \rightarrow U$ of degree prime to $\ell$
over which $\mu_{\ell}$ becomes constant. Using Lemma \ref{Gal covering},
we can moreover arrange that $f^{\ast}(F)$ has a filtration whose successive quotients are
copies of the constant sheaf $\bZ/(\ell) = \mu_{\ell}$. Since $\overline{F}$ is now traceable, so
it coincides with $j_{\ast}(F)$, we have the composite map
$$\overline{F} \rightarrow \overline{f}_{\ast}\overline{f}^{\ast}(\overline{F}) \rightarrow \overline{F}$$
that is multiplication by $\deg(f)$ (see Remark \ref{trrem}).
This degree acts
invertibly on the $\ell$-torsion $\overline{F}$, so likewise
$\rH^p(\overline{U}, \overline{F})$ is thereby realized as a direct summand of
$\rH^p(\overline{U}', \overline{f}^{\ast}(\overline{F}))$.
It therefore suffices to prove that the latter vanishes for $p=3$.
This permits us to replace $(\overline{U}, \overline{F})$ with $(\overline{U}', \overline{f}^{\ast}(\overline{F}))$;
note that this step preserves the ``traceable'' condition (as it is clear by inspection that the functors
$\overline{f'}^{\ast}$ and $F' \rightsquigarrow \widehat{F'}$ naturally commute for any
connected finite \'etale cover $f':V \rightarrow U$). The filtration provided
by Lemma \ref{Gal covering} thereby reduces us to the case $\overline{F} = j_{\ast}(\mu_{\ell})$.
%We first handle the case $\ell=2$. In this case, $\overline{f} : \overline{V}\to\overline{U}$ is also an unramified Galois covering as in Definition \ref{galoiscovbar}, because each one of the $I_{v_k}$ has order $2$ and hence cannot be contained in $G$.
%Therefore, we can apply Lemma \ref{HS seq} and get the sequence:
%This sequence degenerates because the order of $G$ is prime to $\ell$. Therefore,
%$$\rH^q(\overline{V},\overline{f}^*(\overline{F})) ^G = \rH^q(\overline{U},\overline{F}),$$
%and we reduce to showing that $\rH^3(\overline{V},\overline{f}^*(\overline{F})) = 0$. As remarked above, there is a short-exact sequence of sheaves on $Y_{\rm\acute{e}t}$:
%$$0\to\bZ/2\bZ\to f^*(F)\to F'\to 0$$
%and by induction we reduce to the case $F=\bZ/2\bZ$.
If $\ell$ is odd then there are no real places in $K$ (as we have arranged that $K$ contains a primitive $\ell$th root of unity),
so in such cases $\overline{U} = U$ and $j_{\ast}(\mu_{\ell}) = \mu_{\ell}$. But if $\ell = 2$ then $K$ might have real places, and so
then $j_{\ast}(\mu_{\ell})$ does not naturally lie inside $\mathbf{G}_{m, \overline{U}}$
(there are problems at any real place since $-1 < 0$). Thus, rather than work with $j_{\ast}(\mu_{\ell})$,
it is better to work with $j_{!}(\mu_{\ell})$.
Indeed, for odd $\ell$ this is the same as $\mu_{\ell} = j_{\ast}(\mu_{\ell})$
and so for any $\ell$ the natural map $j_{!}(\mu_{\ell}) \rightarrow j_{\ast}(\mu_{\ell})$
has kernel and cokernel that are pushforwards from real points.
Hence, these two sheaves have the same higher cohomology over $\overline{U}$,
whence it is equivalent to prove the vanishing of $\rH^3(\overline{U}, j_{!}(\mu_{\ell}))$.
The reason it is advantageous to consider $j_{!}(\mu_{\ell})$ for all $\ell$ is that it fits into an exact ``Kummer sequence''
\begin{equation}\label{newkummer}
0\to j_!(\bZ/\ell\bZ) \to\mathbf{G}_{m,\overline{U}} \to\mathbf{G}_{m,\overline{U}}\to 0
\end{equation}
over $\overline{U}$. Now we may conclude via Proposition \ref{HGm} as follows.
We arranged by preliminary shrinking that $U \ne X$, so
$\rH^3(\overline{U}, \mathbf{G}_{m, \overline{U}}) = 0$. Thus,
$$\rH^3(\overline{U}, j_{!}(\mu_{\ell})) = \rH^2(\overline{U}, \mathbf{G}_{m,\overline{U}})/(\ell).$$
Thus, we just need to show that $\rH^2(\overline{U}, \mathbf{G}_{m, \overline{U}})$ is $\ell$-divisible.
But we have an exact sequence
$$0 \rightarrow \rH^2(\overline{U}, \mathbf{G}_{m, \overline{U}}) \rightarrow \bigoplus_{x \in (X - U)^0} \bQ/\bZ \rightarrow \bQ/\bZ \rightarrow 0,$$
so the snake lemma applied to the $\ell$-power endomorphism of this short exact sequence identifies the cokernel of $\ell$
on $\rH^2(\overline{U}, \mathbf{G}_{m, \overline{U}})$ with the cokernel of
$$\bigoplus_{x \in (X - U)^0} (\bQ/\bZ)[\ell] \rightarrow (\bQ/\bZ)[\ell].$$
This latter map is surjective, so we are done. This completes the proof
of Proposition \ref{base}.
%
%. In fact, multiplication by $\ell$ is surjective on $\bQ/\bZ$, because this latter is $\ell$-divisible. We have the following commutative diagram with exact rows:
%\begin{displaymath}
%\xymatrix{ & & \bigoplus_{x\in (X-U)^0}\frac{1}{\ell}\bZ/\bZ\ar[r]^{\rm\ \ \ \ \ \ sum}\ar[d] & \frac{1}{\ell}\bZ/\bZ\ar[d] & \\
%0\ar[r] & \rH^2(\overline{U},\mathbf{G}_{m,\overline{U}}) \ar[d]^{\delta}\ar[r] & \bigoplus_{x\in (X-U)^0}\bQ/\bZ \ar[d]^{\times\ell}\ar[r]^{\rm \ \ \ \ \ \ \ sum} & \bQ/\bZ\ar[r] \ar[d]^{\times\ell} & 0\\
%0\ar[r] & \rH^2(\overline{U},\mathbf{G}_{m,\overline{U}}) \ar[r]\ar[d] & \bigoplus_{x\in (X-U)^0}\bQ/\bZ \ar[r]^{\rm \ \ \ \ \ \ \ sum}\ar[d] & \bQ/\bZ\ar[r]\ar[d] & 0\\
% & \coker(\delta) & 0 & 0 & }
%\end{displaymath}
%so by the snake lemma, the following is an exact sequence of abelian groups:
%$$\cdots\to\bigoplus_{x\in (X-U)^0}\frac{1}{\ell}\bZ/\bZ\xrightarrow{\rm sum} \frac{1}{\ell}\bZ/\bZ\to \coker(\delta)\to 0$$
%Since the sum map is surjective, $\delta$ is surjective, and since we have the exact sequence:
%$$\rH^2(\overline{U},\mathbf{G}_{m, \overline{U}})\xrightarrow{\delta}\rH^2(\overline{U},\mathbf{G}_{m,\overline{U}})\to\rH^3(\overline{U},j_!(\bZ/\ell\bZ))\to 0$$
%we conclude
%$$\rH^3(\overline{U}, j_!(\bZ/\ell\bZ)) = 0,$$
%as desired.
%
%In the case $\ell\neq 2$ the proof stays the same with the slight modification that one has to start from the Hochshild-Serre sequence for $f: Y\to U$ making use of Corollary \ref{HSerre}.
%\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Global Galois cohomology beyond degree $2$}
We are ready to show Theorem \ref{Tate1}, which we restate here for convenience of the reader:
\begin{thm}\label{newtate} Let $S$ be a finite set of places of $K$ that contains the archimedean places,
and let $\{v_1, \dots, v_r\}$ be the real places $($with corresponding decomposition groups $D_{v_k} = I_{v_k} \subset G_S$$)$.
Let $M$ be a finite discrete $G_S$-module whose order is an $S$-unit. For $p \ge 3$ the natural restriction map
$$\rH^p(G_S, M) \to \bigoplus_{k=1}^r\rH^p(D_{v_k}, M)$$
is an isomorphism.
\end{thm}
\begin{proof}
Let $U = \Spec(\ccO_{K,S})$, so $M = F_{\eta}$ for a locally constant constructible sheaf $F$ on $U$
whose fiber-rank is a unit on $U$. Hence, for the traceable sheaf $j_{\ast}(F)$ we have
$\rH^p(\overline{U}, j_{\ast}(F)) = 0$ for all $p \ge 3$ by Proposition \ref{base} (for $p=3$)
and Theorem \ref{finitezero} (for $p \ge 4$). The local cohomology sequence arising from the inclusion
$U \hookrightarrow \overline{U}$ thereby takes the form
$$\dots \rightarrow \bigoplus_{k=1}^r \rH^p_{\widetilde{v_k}}(\overline{U}, j_{\ast}(F)) \rightarrow
\rH^p(\overline{U}, j_{\ast}(F)) \rightarrow \rH^p(U, F) \rightarrow \dots.$$
The vanishing of $\rH^p(\overline{U}, j_{\ast}(F))$ for $p \ge 3$ therefore implies that the connecting map
$$\rH^p(U, F) \rightarrow \bigoplus_{k=1}^r \rH^{p+1}_{\widetilde{v_k}}(\overline{U}, j_{\ast}(F))$$
is an isomorphism for all $p \ge 3$.
Lemma \ref{support} identifies the local cohomology in degree $p+1$ at a real point $\widetilde{v}$
with the $I_v$-cohomology in degree $p$ for the generic fiber $(j_{\ast}(F))_{\eta} = F_{\eta} = M$.
Putting it all together, we have natural isomorphisms
$$\rH^p(U, F) \simeq \bigoplus_{k=1}^r \rH^p(I_v, M)$$
for all $p \ge 3$. An application of Theorem \ref{galet} (whose proof uses
Theorem \ref{finitezero} and Proposition \ref{base}, but not the present result we are aiming to prove!)
then identifies $\rH^p(U, F)$ with $\rH^p(G_S, M)$, so we get isomorphisms
\begin{equation}\label{fpmmap}
f^p_M: \rH^p(G_S, M) \to \bigoplus_{k=1}^r\rH^p(D_{v_k}, M)
\end{equation}
for $p \ge 3$.
It remains to check that $f^p_M$ is the natural restriction map for $p \ge 3$.
This is verified in Proposition \ref{fpm} as special case of a more general compatibility argument.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Artin-Verdier duality for algebraic number fields}\label{avsec}
\subsection{A first look}\label{first}
Let us recall that our aim is to show Tate's global duality theorem, together with the 9-term exact sequence for finite ramification, by means of the
Duality Theorem (Theorem \ref{dualT}).
Such exact sequence is intended to be an instance for $\Spec(\ccO_{K,S}) \subset \overline{X}$ of
the local cohomology sequence which for dense open $\widetilde{V}$ inside $\widetilde{U}$ with complement denoted $S$ takes the form:
$$\cdots\to \bigoplus_{x\in S}\rH^p_x(\widetilde{U},\widetilde{F})\to \rH^p(\widetilde{U},\widetilde{F})\to \rH^p(\widetilde{V},\widetilde{F}\vert_{\widetilde{V}})\to \cdots$$
As a first step, since we will be given only a sheaf $F$ on $\Spec(\ccO_{K,S})$ (such as arising from a finite discrete $G_S$-module $M$)
yet we want to exploit local cohomology relative to the inclusion of $U$ into $\overline{X}$ (accounting for all the places in $S$, including
the real places!), we need an appropriate functorial extension of abelian sheaves on $U$ to abelian sheaves
on $\overline{X}$. Extending to $X$ will be achieved via extension-by-zero, but
to extend from $X$ to $\overline{X}$ involves an entirely different idea.
More generally, for any dense open subscheme $U$ of $X$ and abelian sheaf $F$ on $U$ we wish to naturally extend
to it to an abelian sheaf $\widehat{F}$ on $\overline{U}$ in a manner that interacts
well with local cohomology (and will then mainly need for $U=X$, though considering
more general $U$ is illuminating and useful too). We will also need the functors $\rH^p(\overline{X}, \widehat{F})$
to assemble into a $\delta$-functor of $F$; this will be a bit trick since it turns out that the appropriate functor
$\widehat{F}$ will {\em not} be exact in $F$ (due to complications in stalks at real points) but merely right-exact.
In particular, calling $j$ the ``open embedding'' of $U$ into $\overline{U}$,
neither of the left-exact functors $j_{!}$ nor $j_{\ast}$ will do the job we need.
For motivation, assume we have some such functor $F \rightsquigarrow \widehat{F}$,
so for a real point $v$ we have an exact sequence
\begin{equation}\label{excision}
0\to i_v^!(\widehat{F}) \to \widehat{F}_v \xrightarrow{\varphi_v} i_v^*j_*j^*(\widehat{F})\to\rR^1i_v^!(\widehat{F})\to 0
\end{equation}
(we used $i_v^*$ is exact and that $i_v^*i_{v*} \to\textup{id}$ is an isomorphism).
Note that the third term $i_v^{\ast}j_{\ast}j^{\ast}(\widehat{F}) = i_v^{\ast}j_{\ast}(F) = (j_{\ast}(F))_v$ is equal to $F(K_v) = \rH^0(I_v, F_{\eta})$
by the definition of $j_{\ast}$.
By Lemma \ref{support}, we also
have $$i_v^!(\widehat{F}) = \rH^0_v(\overline{U}, \widehat{F}),\,\,\,\rR^1i_v^!(\widehat{F}) = \rH^1_v(\overline{U}, \widehat{F}).$$
What should these be {\em as functors of $F$}?
To guess correctly, we will use {\em Tate cohomology}
(as developed in \cite[Chapter IV]{CF}).
Since Lemma \ref{support} yields canonical isomorphisms $$\rH^p_v(\overline{X}, \widehat{F}) \simeq \rH^{p-1}(I_v, F_{\eta}) = {\rH}_T^{p-1}(I_v, F_{\eta})$$
for $p\ge 2$ ($(\cdot)_T$ denoting Tate cohomology), the $\delta$-functor $F \rightsquigarrow \rH^{\bullet}_v(\overline{X}, \widehat{F})$
we hope to have in degrees $\ge 0$ would extend
{\em two degrees to the left} the $\delta$-functor of Tate cohomology of $F_{\eta}$
in degrees $\ge 1$. Hence, whatever $\widehat{F}$ is to be (as a functor of $F$),
it is natural to want $$\rH^p_v(\overline{U}, \widehat{F}) \simeq{\rH}_T^{p-1}(I_v, F_{\eta})$$ for $p=0,1$ since Tate cohomology
in degrees $\ge -1$ provides
the only erasable
$\delta$-functorial extension to degrees $\ge -1$ of Tate cohomology in degrees $\ge 1$. In particular, (\ref{excision}) would then take the form of an exact
sequence
$$0 \rightarrow {\rH}_T^{-1}(I_v, F_{\eta})\rightarrow \widehat{F}_v \stackrel{\varphi_v}{\rightarrow} \rH^0(I_v, F_{\eta}) \rightarrow {\rH}_T^0(I_v, F_{\eta}) \rightarrow 0$$
for some unknown maps relating the terms in the sequence.
But $\rH_T^p$ is {\em defined} for $p = 0, 1$ via a tautological exact sequence
$$0 \rightarrow {\rH}_T^{-1}(I_v, F_{\eta}) \rightarrow \rH_0(I_v, F_{\eta}) \stackrel{{\rm{N}}_v}{\rightarrow} \rH^0(I_v, F_{\eta}) \rightarrow \rH_T^0(I_v, F_v)
\rightarrow 0$$
where ${\rm{N}}_v$ is the norm map for the $I_v$-module $F_{\eta}$. Hence, this motivates us to {\em define}
$\widehat{F} \in \fAb_{\overline{U}}$ via real stalks $\widehat{F}_v := \rH_0(I_v, F_{\eta})$ with $\varphi_v := {\rm{N}}_v$ for all $v$.
In other words, we now make:
\begin{defin}\label{modified} The {\em modified sheaf} $\widehat{F}$ on $\overline{U}$ is given by:
$$(\rH_0(I_{v_1}, F_{\eta}),\dots,\rH_0(I_{v_r}, F_{\eta}), F, \rH_0(I_{v_1}, F_{\eta})\xrightarrow{\textup{N}_v}\rH^0(I_{v_1}, F_{\eta}), \dots, \rH_0(I_{v_r}, F_{\eta})\xrightarrow{N} \rH^0(I_{v_r}, F_{\eta})).$$
\end{defin}
\begin{eg}\label{Gm modified} Since the norm map $\overline{K} \rightarrow K_v^{\rm{alg}}$ has image
$K_v^{{\rm{alg}},+}$ (or upon completion, the norm map $\mathbf{C}^{\times} \rightarrow \bR^{\times}$ has
image $\bR^{\times}_{>0}$), by computing $\rH_0(I_v, \mathbf{G}_{m, \eta})$ and the norm maps for all real places $v$ one readily sees that
$$\widehat{\mathbf{G}_{m, U}} = \mathbf{G}_{m,\overline{U}}.$$
\end{eg}
The first basic feature of $\widehat{F}$ as a functor of $F$ is that it is not left-exact. Instead, we have:
\begin{lemma} The functor $\fAb_U \rightarrow \fAb_{\overline{U}}$
$($or $\Ab_U\to\Ab_{\overline{U}}$$)$ given by $F\mapsto \widehat{F}$ is right-exact and preserves injectives;
explicitly, for $j: U \hookrightarrow \overline{U}$ naturally $\widehat{F} \simeq j_{\ast}(F)$ for injective $F$.
\end{lemma}
\begin{proof} Right exactness can be checked over $U$ and on the real stalks. Over $U$ we have
$\widehat{F}|_U = F$, and the real stalks are group homologies in degree 0, which are right-exact.
Now assuming that $F$ is injective, so $j_*(F)$ is injective as an abelian sheaf on $\overline{U}$ (as its left adjoint $j^{\ast}$ is exact),
we will show that the natural map $\widehat{F} \rightarrow j_{\ast}(F)$ adjoint to the isomorphism $j^{\ast}(\widehat{F}) \simeq F$ is itself an isomorphism
(so $\widehat{F}$ is indeed then injective). In view of how $j_{\ast}$ is defined, the content is that the maps $\varphi_v$
associated to $\widehat{F}$ are all isomorphisms, or in other words that the kernel and cokernel of each $\varphi_v$ vanishes.
This is precisely the assertion that the Tate cohomology groups
${\rH}_T^p(I_{v_k}, F_{\eta})$ vanish for $p = 0, 1$ for any injective $F$ on $U$.
The injectivity of $F$ and $j_{\ast}(F)$ implies
that the cohomologies
$$\rH^p(\overline{U}, j_{\ast}(F)),\,\,\,\rH^p(U, F)$$
vanish for all $p > 0$, so the local cohomology sequence relating them gives that $\rH^p_v(\overline{U}, j_{\ast}(F)) = 0$ for $p \ge 2$ and real points $v$.
Hence, $\rH^p(I_v, F_{\eta}) = 0$ for all $p\ge 1$ by Lemma \ref{support}. Since the cohomology of $I_{v}$ is doubly periodic, it follows that all
${\rH}_T^p(I_{v}, F_{\eta})$ are zero.
\end{proof}
\subsection{Modified \'etale cohomology}\label{modet}
As with Tate cohomology as a refinement of cohomology of finite groups,
we are going to define a modified $\delta$-functorial cohomology theory $\widehat{\rH}^{\bullet}(U, \cdot)$ that coincides with
$\rH^{\bullet}(\overline{U}, \widehat{(\cdot)})$ in degrees $\ge 0$ but extends to negative degrees accounting
for the fact that $\rH^0(\overline{U}, \widehat{F})$ is generally {\em not} left-exact in $F$.
One subtlety is that since $\widehat{F}$ is not left-exact in $F$ (the real stalks can fail to be left-exact),
some argument is needed to explain how $\rH^{\bullet}(\overline{U}, \widehat{(\cdot)})$ is at all a $\delta$-functor
in degrees $\ge 0$!
Consider a short-exact sequence of abelian sheaves on $U$,
$$0\to F'\to F\to F''\to 0,$$
and apply the modification functor $\widehat{(\cdot)}$. By means of the homology long-exact sequence for $I_{v}$
with real points $v$ and the equality of Tate cohomology in degrees $\le -2$ with group homology in degrees $> 0$ up to a
re-indexing with a sign, we obtain the following long-exact sequence extending on the left:
\begin{displaymath}
\xymatrix{\cdots \ar[r] & \bigoplus_{k=1}^ri_{v_k*}\rH^{-2}(I_{v_k}, F_{\eta}) \ar[r] & \bigoplus_{k=1}^ri_{v_k*}\rH^{-2}(I_{v_k}, F_{\eta}'') & & \\
\ar[r] & \widehat{F'}\ar[r] & \widehat{F}\ar[r] & \widehat{F''}\ar[r] & 0}
\end{displaymath}
We let $M = \ker(\widehat{F}\to\widehat{F''})$, so there is an induced map $\widehat{F'}\twoheadrightarrow M$.
We call $N$ the kernel of this latter map, so also $N = \ker(\widehat{F'} \rightarrow \widehat{F})$. Hence,
$N = \bigoplus_v i_{v\ast}(N_{v})$ for
$$N_v := \ker({\rm{H}}_0(I_v, F'_{\eta}) \rightarrow {\rm{H}}_0(I_v, F_{\eta})) = \coker(\rH^{-2}_T(I_v, F_{\eta}) \rightarrow \rH^{-2}_T(I_v, F''_{\eta})).$$
Hence, $\rH^p(\overline{U}, N) = 0$ for $p \ge 1$ since $\Gamma(\overline{U}, i_{v \ast}(\cdot))$ is the identity functor for real $v$.
We deduce the following exact sequences:
$$0\to \rH^0(\overline{U}, N)\to \rH^0(\overline{U},\widehat{F'})\to \rH^0(\overline{U}, M)\to 0$$
$$0\to \rH^0(\overline{U}, M)\to \rH^0(\overline{U},\widehat{F})\to \rH^0(\overline{U}, \widehat{F''})\to \rH^1(\overline{U}, M)\to\cdots$$
together with the canonical isomorphism:
$$\rH^p(\overline{U}, \widehat{F'}) \simeq\rH^p(\overline{U}, M),\ \ p\ge 1.$$
Hence, we get a long exact sequence
$$0 \rightarrow \rH^0(\overline{U}, N) \rightarrow \rH^0(\overline{U}, \widehat{F'}) \rightarrow \rH^0(\overline{U}, \widehat{F}) \rightarrow
\rH^0(\overline{U}, \widehat{F''}) \rightarrow \rH^1(\overline{U}, \widehat{F'}) \rightarrow \rH^1(\overline{U}, \widehat{F}) \rightarrow \dots$$
The description of $N$ in terms of $N_v$'s also gives an exact sequence
$$\bigoplus_{k=1}^r{\rH}_T^{-2}(I_{v_k}, F_{\eta}) \to \bigoplus_{k=1}^r{\rH}_T^{-2}(I_{v_k}, F_{\eta}'') \to \rH^0(\overline{U}, N)\to 0$$
This motivates the following:
\begin{defin}\label{modifiedcohom} For all $p\in\bZ$, the {\em modifield \'etale cohomology groups} are:
$$\widehat{\rH}^p(U,F) := \left\{\begin{array}{cc}
\rH^p(\overline{U}, \widehat{F}), & p\ge 0\\
\bigoplus_{k=1}^r{\rH}^{p-1}_T(I_{v_k}, F_{\eta}), & p<0.
\end{array}\right.$$
\end{defin}
The preceding considerations make the modified \'etale cohomology groups into a $\delta$-functor: to
each the short-exact sequence of abelian sheaves
$$0\to F'\to F\to F''\to 0$$
on $U$ we have naturally associated a long exact sequence
$$\cdots\to\widehat{\rH}^p(U, F')\to \widehat{\rH}^p(U, F)\to \widehat{\rH}^p(U, F'')\to \widehat{\rH}^{p+1}(U, F')\to \cdots$$
with $p \in \bZ$. Voila, $\widehat{\rH}^{\bullet}(U, \cdot)$ has been made into
a $\delta$-functor on $\mathbf{Ab}_U$ despite the failure of $\widehat{F}$ to be exact in $F$.
\subsection{Modified cohomology with support}
Using Lemma \ref{support} and the definition of $\widehat{(\cdot)}$, for all real $v$ and $p \ge 0$
\begin{equation}\label{htate}
\rH^p_v(\overline{U},\widehat{F}) = {\rH}_T^{p-1}(I_v, F_{\eta}).
\end{equation}
This fails for $p = 0, 1$ if instead of $\widehat{F}$ we use a general abelian sheaf $\overline{F} = (\{\overline{F}_v\}, F, \{\varphi_v\})$
on $\overline{U}$ (rather than one of the form $\widehat{F}$). Indeed, for such $\overline{F}$ we have
$$\rH^0_v(\overline{U}, \overline{F}) = \ker(\varphi_v: \overline{F}_v \rightarrow F_{\eta}^{I_v}),\,\,\,
\rH^1_v(\overline{U}, \overline{F}) = \coker(\varphi_v).$$
Hence, if $\overline{F} = j_{!}(F)$ for the inclusion $j$ of $U$ into
$\overline{U}$ then $\overline{F}_v = 0$ for all real $v$, so $\rH^0_v(\overline{U}, \overline{F}) = 0$, but
generally ${\rH}_T^0(I_v, F_{\eta}) \ne 0$. Hence, motivated by (\ref{htate}) as a functor of $F$, we are led to make:
\begin{defin}\label{localcoho} For all $p\in\bZ$ and real $v$, the {\em modified local cohomology groups} on $U$ are
$$\rH_v^p(U,F) := \left\{\begin{array}{cc}
\rH^p_v(\overline{U}, \widehat{F}), & p\ge 0\\
{\rH}_T^{p-1}(I_{v}, F_{\eta}), & p<0
\end{array}\right.$$
\end{defin}
We make the convention that the functors $\rH^p_x(U, \cdot)$ for $x \in U^0$ and $\rH^p(U, \cdot)$ vanish for $p < 0$
since they are left-exact for $p=0$ (whereas for real $v$ the functor $\rH^0_v(U, \cdot) := \rH^0_v(\overline{U}, \widehat{(\cdot)})$
is {\em not} left-exact, so its non-vanishing extension to negative degrees is appropriate).
Using that convention, for any dense open subset $V \subset U$ we get a long exact ``local cohomology'' sequence
\begin{equation}\label{modlocal}
\to\cdots \bigoplus_{s\in\overline{U} - V} \rH^p_s(U,F)\to\widehat{\rH}^p(U,F)\to\rH^p(V,F\vert_V)\to\bigoplus_{s\in\overline{U}-V}\rH^{p+1}_s(U,F)\to\cdots
\end{equation}
with $p \in \bZ$. This is just a reformulation of the local cohomology sequence for $\widehat{F}$ on $\overline{U}$ relative to $V$
for $p \ge 0$ and tautological identifications for $p < 0$ via the considerations in \S\ref{modet}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Compactly supported cohomology and duality preparations}\label{prepsec}
We let $j$ be the open immersion of an open subscheme $U$ into $X$, and $F$ be an abelian sheaf on $U$.
We make use of the notation developed so far with no further mention.
\begin{defin} \label{Hc} For $p \in \bZ$, the {\em compactly supported cohomology groups} are
$$\rH^p_c(U, F) := \widehat{\rH}^p(X, j_!(F))$$
for abelian sheaves $F$ on $U$.
\end{defin}
First of all, we observe that $\rH_c^{\bullet}(U, \cdot)$ is a covariant $\delta$-functor because $\widehat{\rH}^{\bullet}(X, \cdot)$ is
such a $\delta$-functor and $j_!$ is exact.
For $U = X$ we have $j_! = \textup{id}$, by definition for every abelian sheaf $F$ on $X$ and every integer $p$ we have
$$\rH^p_c(X,F) = \widehat{\rH}^p(X,F).$$
Likewise, if $h:V \hookrightarrow U$ is an inclusion of dense open subsets of $X$ then for any sheaf $F$ on $U$ there are natural maps
\begin{equation}\label{excise}
\rH^p_c(V, F|_V) \rightarrow \rH^p_c(U,F)
\end{equation}
defined by applying $\rH^p_c(U, \cdot)$ to the map $h_{!}(F|_V) \rightarrow F$.
Here is an important example for arithmetic duality:
\begin{eg}\label{cGm}
Let $j:U \hookrightarrow X$ be a dense open subscheme. We claim that naturally
\begin{equation}\label{trisom}
\rH^3_c(U,\mathbf{G}_{m, U}) \simeq \rH^3_c(X, \mathbf{G}_{m,X}) \simeq \bQ/\bZ.
\end{equation}
For the second isomorphism, we note that by definition $\rH^3_c(X, \mathbf{G}_{m,X}) = \rH^3(\overline{X}, \mathbf{G}_{m,\overline{X}})$
by Example \ref{Gm modified}, so it is identified with $\bQ/\bZ$ by
Proposition \ref{HGm}.
The first map in (\ref{trisom}) is the map (\ref{excise}) for the inclusion $U \hookrightarrow X$
and the sheaf $\mathbf{G}_{m,X}$ on $X$; explicitly, it is obtained by applying $\rH^3(\overline{X}, \cdot)$ to the inclusion
\begin{equation}\label{jux}
(j_{!}(\mathbf{G}_{m,U}))^{\wedge} \hookrightarrow \mathbf{G}_{m, \overline{X}}
\end{equation}
over $\overline{X}$ whose cokernel is supported on $X - U$ (no real points!) with stalk at $x \in X - U$ given by the Galois module
$(\cO_{X,x}^{\rm{sh}})^{\times}$ for $G_x = {\rm{Gal}}(\overline{\kappa(x)}/\kappa(x)) \simeq \widehat{\bZ}$.
Since $G_x$ has cohomological dimension 1 and hence strict cohomological dimension $\le 2$, the map obtained from applying
$\rH^3(\overline{X}, \cdot)$ to (\ref{jux})
is surjective with kernel that is a quotient of $\bigoplus_{x \in X - U} \rH^2(\kappa(x), (\cO_{X,x}^{\rm{sh}})^{\times})$.
Now it suffices to show that if $A$ is a henselian (not necessarily complete!) excellent discrete valuation ring with finite residue field $k$
then ${\rm{H}}^2(k, (A^{\rm{sh}})^{\times}) = 1$. This is the limit of the finite-layer cohomology groups
$\rH^2(k'/k, {A'}^{\times})$ for finite extensions $k'/k$ and the corresponding local finite \'etale extension $A'$ of $A$.
But the Tate cohomology of cyclic groups is doubly periodic, so this degree-2 cohomology coincides with the degree-0 Tate cohomology
that is the cokernel of the norm map on local unit groups ${A'}^{\times} \rightarrow A^{\times}$. Hence, it suffices to show
that such norm maps are surjective.
If $A$ were complete then such surjectivity is clear by a successive approximation argument. In general
surjectivity is a consequence of the complete case via the Artin approximation theorem, which applies
to the existence of solutions to system of polynomial equations over any henselian excellent discrete valuation ring.
(A similar use of Artin approximation occurs in the proof of Proposition \ref{comp1}.)
\end{eg}
%
%\begin{eg} Let $\ell$ be a fixed prime number, and $K$ a number field containing $\mu_{\ell}$. Let $S$ be the set of infinite places together with the finite places dividing $\ell$, and call $U := \Spec(\ccO_{K,S})$. Then the abelian sheaf $\mu_{\ell, U}$ is isomorphic to $\underline{\bZ/\ell\bZ}_U$ via a choice of primitive
%$\ell$th root of unity over $U$.
%
%
%We call $j$ the immersion of $U$ into $\overline{X}$, $j'$ the inclusion of $\overline{U}$ into $\overline{X}$, and $j''$ the inclusion of $U$ into $\overline{U}$
%(so $j = j' \circ j''$).
%There is an evident ``extension by zero'' functor $j'_{!}$ left-adjoint to restriction to $\overline{U}$ obtained via ``gluing'' over $U = X \cap \overline{U}$
%with the usual extension-by-zero from $U$ to $X$ of a restriction to $U$; this is given in the language of triples for $\fAb_{\overline{U}} \rightarrow
%\fAb_{\overline{X}}$ by applying usual extension-by-zero to the middle term of a triple, so it is exact as expected.
%The composition of $j'_{!}$ with the extension-by-zero $j''_{!}: \mathbf{Ab}_U \rightarrow \mathbf{Ab}_{\overline{U}}$
%is identified with the analogous functor $j_{!}$ in an evident manner.
%
%Apply this functor $j'_!: \mathbf{Ab}_{\overline{U}} \rightarrow \mathbf{Ab}_{\overline{X}}$ to the Kummer sequence
%$$0\to j''_{!}(\underline{\bZ/\ell\bZ}) \to\mathbf{G}_{m,\overline{U}}\to\mathbf{G}_{m,\overline{U}}\to 0$$
%(see (\ref{newkummer})) to obtain an exact sequence
%$$0\to j_!(\underline{\bZ/\ell\bZ}) \to j_!'(\mathbf{G}_{m,\overline{U}})\to j_!'(\mathbf{G}_{m,\overline{U}})\to 0$$
%But $j'_{!}(\mathbf{G}_{m, \overline{U}})$ is identified with applying $\widehat{(\cdot)}$ to
%the extension by zero of $\mathbf{G}_{m,U}$ to $X$, so applying ${\rH}^{\bullet}(\overline{X}, \cdot)$
%gives an exact sequence
%$$\cdots\to\rH^2_c(U,\mathbf{G}_m)\to\rH^3(\overline{X},j_!(\underline{\bZ/\ell\bZ})) \to \rH^3_c(U,\mathbf{G}_m)
%\stackrel{\ell}{\to}\rH^3_c(U,\mathbf{G}_m)\cdots$$
%
%By Proposition \ref{HGm} we know $\rH^2_c(U, \mathbf{G}_m) = 0$, so since $\rH^3_c(U, \mathbf{G}_m) = \bQ/\bZ$ we get
%$${\rH}^3(\overline{X},j_!(\underline{\bZ/\ell\bZ})) = (1/\ell)\bZ/\bZ.$$
%The cokernel $F_0$ of the map $j_!(\underline{\bZ/\ell\bZ})\to \underline{\bZ/\ell\bZ}_{\overline{X}}$ is concentrated at the real places. Therefore, the kernel of
%$$\rH^3(\overline{X}, j_!(\underline{\bZ/\ell\bZ}))\to\rH^3_c(X,\underline{\bZ/\ell\bZ})$$
%is
%$$\rH^2(\overline{X}, F_0) = \bigoplus_{v\ \textup{real}} \rH^2_v(\overline{X}, F_{0v}) = \bigoplus_{v\ \textup{real}}\rH^1_T(I_v, F_{0\eta}) = 0$$
%as $F_{0\eta}=0$.
%
%It follows $\rH^3_c(X,\underline{\bZ/\ell\bZ})$ is nonzero, as $\frac{1}{\ell}\bZ/\bZ$ injects into it.
%Instead, by the proof of Proposition \ref{base}, we have:
%$$\rH^3_c(U, \underline{\bZ/\ell\bZ}) = 0,$$
%which shows how $\rH^p_c(U,\underline{\bZ/\ell\bZ})$ depends on $U$.
%
%
%We call $j$ the immersion of $U$ into $\overline{X}$, and apply $j_!$ to the Kummer sequence:
%$$0\to\underline{\bZ/\ell\bZ}\to\mathbf{G}_{m,U}\to\mathbf{G}_{m,U}\to 0$$
%and apply $\widehat{\rH}^{\bullet}(X,-)$ to obtain and exact sequence
%$$\cdots\to\rH^2_c(U,\mathbf{G}_m)\to\rH^3_c(U,\underline{\bZ/\ell\bZ}) \to \rH^3_c(U,\mathbf{G}_m)[\ell] = (1/\ell)\bZ/\bZ.$$
%By Proposition \ref{HGm}, we deduce $\rH^2_c(U, \mathbf{G}_m) = 0$, and hence we obtain
%$${\rH}^3_c(U, \underline{\bZ/\ell\bZ}) = (1/\ell)\bZ/\bZ.$$
%Since $\widehat{\rH}^3(X,j_!(\underline{\bZ/\ell\bZ}) \hookrightarrow\widehat{\rH}^3(X,\underline{\bZ/\ell\bZ})$, this latter is nonzero, and equals $\rH^3_c(X,\underline{\bZ/\ell\bZ})$.
%Instead, by the proof of Proposition \ref{base}, we have:
%$$\rH^3_c(U, \underline{\bZ/\ell\bZ}) = 0,$$
%which shows how $\rH^p_c(U,\underline{\bZ/\ell\bZ})$ depends on $U$.
%\end{eg}
%
%\medskip
%
%For $F$ on $U$ and the inclusion $j:U \rightarrow X$, the local cohomology sequence (\ref{modlocal}) for $j_!(F)$ relative to
%the inclusion of $U$ into $X$ takes the form:
%$$\cdots \to\bigoplus_{s\in \overline{X}-U} \rH^p_s(X, j_!(F))\to \widehat{\rH}^p(X, j_!(F))\to\rH^p(U, F)\to \bigoplus_{s \in \overline{X}-U}
%\rH^{p+1}_s(X, j_!(F))\to\cdots$$
%for $p \in \bZ$. We claim that for all integers $p$ and points $s \in \overline{X} - U$, naturally $$\rH^p_s(X, j_!(F)) = \rH^p_s(U,F).$$
%For real points $s$ the local cohomology at $s$ of $j_!(F)$ is read off the generic fiber which
%coincides with the generic fiber of $F$, so the local cohomology sequence actually reads:
%$$\cdots \to\bigoplus_{s\notin U} \rH^p_s(U, F)\to {\rH}_c^p(U, F)\to\rH^p(U, F)\to \bigoplus_{s\notin U} \rH^{p+1}_s(U, F)\to\cdots$$
%for $p \in \bZ$.
%
\begin{rem}\label{tropen}
Since the trace $\textup{Tr}_U : \rH^3_c(U,\mathbf{G}_m)\to\bQ/\bZ$ is {\em defined} to be a composition
$$\rH^3_c(U,\mathbf{G}_m)\to\rH^3_c(X,\mathbf{G}_m)\xrightarrow{\textup{Tr}}\bQ/\bZ$$
(with ${\rm{Tr}}$ an isomorphism),
for dense open $h:V \hookrightarrow U$ we wish to express a compatibility between ${\rm{Tr}}_U$ and ${\rm{Tr}}_V$.
Calling $j_U$ the inclusion of $U$ into $X$, and $j_V$ the inclusion of $V$ into $X$,
by definition
$$\rH^3_c(U,\mathbf{G}_m) = \rH^3(\overline{X}, (j_{U!}\mathbf{G}_m)^{\wedge}),\,\,\,
\rH^3_c(V,\mathbf{G}_m) = \rH^3(\overline{X}, (j_{V!}\mathbf{G}_m)^{\wedge}).
$$
The identification $j_{V!} = j_{U!}\circ h_!$ implies that the inclusion
$(j_{V!}\mathbf{G}_{m, V})^{\wedge}\hookrightarrow\mathbf{G}_{m,\overline{X}}$ factors through
$(j_{U!}\mathbf{G}_{m, U})^{\wedge}\hookrightarrow\mathbf{G}_{m,\overline{X}}$, so we obtain a natural map
$$f:\rH^3_c(V, \mathbf{G}_{m,V}) \rightarrow \rH^3_c(U, \mathbf{G}_{m,U})$$
by applying $\widehat{\rH}^3(\overline{X},\cdot)$ to
the inclusion $(j_{V!}\mathbf{G}_{m, V})^{\wedge}\hookrightarrow
(j_{U!}\mathbf{G}_{m, U})^{\wedge}$.
By functoriality of $\rH^3(\overline{X}, \cdot)$,
the diagram
\begin{displaymath}
\xymatrix{\rH^3_c(V,\mathbf{G}_m)\ar[d]_-{f} \ar[r] & \rH^3_c(X,\mathbf{G}_m) \ar@{=}[d] \\
\rH^3_c(U,\mathbf{G}_m)\ar[r] & \rH^3_c(X,\mathbf{G}_m)}
\end{displaymath}
commutes. It follows that $f$ is an isomorphism and ${\rm{Tr}}_V = {\rm{Tr}}_U \circ f$.
\end{rem}
\medskip
We have established most of the necessary terminology and
preliminary results to begin to prove Theorem \ref{dualT}, but we require one more definition in
order for the entire statement to make sense. Let us first recall the statement of the result
to be proved:
\begin{thm}{\textup{(Artin-Verdier)}} Let $F$ be a constructible abelian \'etale sheaf on a dense open $U \subset X$. The Yoneda pairing
$${\rH}_c^p(U, F)\times \textup{Ext}_U^{3-p}(F,\mathbf{G}_{m,U}) \to{\rH}_c^3(U, \mathbf{G}_{m,U}) = \bQ/\bZ$$
is a perfect pairing of finite abelian groups for all $p \in \bZ$.
\end{thm}
\begin{rem}\label{negdegree}
In this statement, the Ext-group is defined to be 0 when $p > 3$ (so part of the assertion is that $\rH^p_c(U, F) = 0$ if $p > 3$).
Also, $\rH^p_c(U, F)$ doesn't generally vanish for $p < 0$ when there are real places since
$\widehat{\rH}^p(\overline{X}, \cdot)$ doesn't generally vanish for $p < 0$ when there are real places: see
Definition \ref{modifiedcohom} (and Definition \ref{Hc}). It may seem totally ridiculous to comtemplate
$p < 0$, but if there are real places then $\rH^0_c$ is {\em not} left-exact and so to push through the method
of proof (which is an induction from the left) it is actually essential to incorporate negative degrees
in the presence of real places (in which case $\rH^p_c$ for $p < 0$ is
Tate cohomology for decomposition groups at real places;
these can be handled quite concretely, at the cost of some actual work to ensure that the connecting
map from degree $-1$ to degree 0 interacts well with the constructions we'll make!).
\end{rem}
\medskip
The gap which remains before we can begin the proof is to explain what the ``Yoneda pairing'' is; the technical problem
is that the definition of $\rH^{\bullet}_c(U, \cdot)$ involves the intervention of the non-exact
functor $\widehat{(\cdot)}$. The role of such a non-exact functor made it a bit surprising that $\rH^{\bullet}_c(U, \cdot)$
admits a $\delta$-functor structure, but $\widehat{(\cdot)}$ applied to complexes of sheaves does not preserve quasi-isomorphisms
and hence doesn't make sense at the level of derived categories. Consequently, the standard method
of defining Yoneda pairings via the identification of ${\rm{Ext}}^n(F, G)$ with the set of derived-category homomorphisms $F \rightarrow G[n]$ (where
$G[n]$ is the one-term complex $G$ supported in degree $-n$)
cannot be used. We shall proceed in another way.
Let $j:U \hookrightarrow X$ be the open immersion. By definition
$\rH^p_c(U, F) \simeq \rH^p_c(X, j_{!}(F))$ for any abelian sheaf $F$
(even $\delta$-functorial!), $\rH^3_c(U, \mathbf{G}_{m,U}) \simeq \rH^3_c(X, \mathbf{G}_{m,X})$
(see Example \ref{cGm}), and
$${\rm{Ext}}_X^i(j_{!}(F), F') \simeq {\rm{Ext}}^i_U(F, j^{\ast}(F'))$$
induced by restriction along $j$ for any $F$ on $U$ and any $F'$ on $X$
($\delta$-functorial in $F$).
Setting $F' = \mathbf{G}_{m,X}$, we see that to define the pairing over $U$ for $F$ in a manner
that is $\delta$-functorial it suffices to treat the situation over $X$
and then apply that to $j_{!}(F)$ (as $j_{!}$ is exact). Hence,
for the rest of the proof we may focus on $X$.
If we let $\overline{j}:X \hookrightarrow \overline{X}$ be the inclusion
then for abelian sheaves $F$ and $F'$ on $X$ we have
$${\rm{Ext}}^m_X(F, F') = {\rm{Ext}}^m_X(\overline{j}^{\ast}(\widehat{F}), F') =
{\rm{Ext}}^m_{\overline{X}}(\widehat{F}, \overline{j}_{\ast}(F'))$$
due to the adjunction between restriction $\overline{j}^{\ast}$ and pushforward $\overline{j}_{\ast}$. Hence,
the usual derived-category Yoneda method gives a natural pairing
$$\rH^n(\overline{X}, \widehat{F}) \times {\rm{Ext}}^m_{\overline{X}}(\widehat{F}, \overline{j}_{\ast}(F')) \rightarrow
\rH^{n+m}(\overline{X}, \overline{j}_{\ast}(F'))$$
for $n \ge 0$. The natural isomorphism $\overline{j}^{\ast}(\widehat{F'}) \simeq F'$ is adjoint to a natural homomorphism
$\widehat{F'} \rightarrow \overline{j}_{\ast}(F')$ whose kernel and cokernel are supported at real points. Thus,
applying $\rH^p(\overline{X}, \cdot)$ to this gives an isomorphism for $p > 0$, so with $p=3$ we have a natural isomorphism
$$\rH^3_c(X, F') \simeq \rH^3(\overline{X}, \overline{j}_{\ast}(F')).$$
Using the inverse of this gives a pairing
$$\rH^p(\overline{X}, \widehat{F}) \times {\rm{Ext}}^{3-p}_{\overline{X}}(\widehat{F}, \overline{j}_{\ast}(F')) \rightarrow
\rH^3(\overline{X}, \overline{j}_{\ast}(F')) = \rH^3_c(X, F')$$
for $p \ge 0$.
Setting $F' = \mathbf{G}_{m,X}$, we get a natural pairing
$$\rH^p_c(X, F) \times {\rm{Ext}}^{3-p}_X(F, \mathbf{G}_{m,X}) \rightarrow \rH^3(X, \mathbf{G}_{m,X}) = \bQ/\bZ$$
for $p \ge 0$.
That is the Yoneda pairing {\em by definition} for $p \ge 0$, and applying it to the extension-by-zero to $X$ of an abelian sheaf on $U$
defines such a pairing over any dense open $U$ in $X$ (for any abelian sheaf $F$ on $U$).
For $p < 0$, by definition we seek a pairing
$$(\bigoplus_v \rH^{p-1}_T(I_v, F_{\eta})) \times {\rm{Ext}}^{3-p}_X(F, \mathbf{G}_{m,X}) \rightarrow \bQ/\bZ.$$
To define this, we need to analyze the Ext's in degree $\ge 4$. This require a lemma:
\begin{lemma}\label{global sections} For every constructible sheaf $F$ on a dense open $U \subset X$, the edge map
$$\rH^i(U,\cHom_U(F,\mathbf{G}_m))\to\Ext_U^i(F,\mathbf{G}_m)$$
in the local-to-global spectral sequence
$\rH^m(U, \underline{\rm{Ext}}^n_U(F, \mathbf{G}_m)) \Rightarrow \Ext^{m+n}_U(F, \mathbf{G}_m)$ is an isomorphism
if either $i \ge 4$ or if $F$ is locally constant with order that is a unit on $U$.
\end{lemma}
\begin{proof}
%We claim that in general $\cHom_X(F, \cdot)$ carries injectives to $\Gamma(X, \cdot)$-acyclics,
%so there is a Leray spectral sequence
%$$\rH^p(X, \mathscr{E}xt^q_X(F, \cdot)) \Rightarrow \Ext^{p+q}_X(F, \cdot).$$
%This then provides an edge map
%
%Provided that $\mathscr{E}xt^q_X(F, \mathbf{G}_m) = 0$ for all $q > 0$, the edge map
%from $(p, 0)$ would then
%
First consider the case of general $F$ and $i \ge 4$.
Let $j:V \hookrightarrow U$ be a dense open such that $F|_V$ is locally constant with order
that is a unit on $V$. By Remark \ref{local0}, the local cohomology functor $\rH^i_x(U, \cdot)$ vanishes
on torsion abelian sheaves for $i \ge 4$. Hence, the restriction map
$\rH^i(U, F') \rightarrow \rH^i(V, F'|_V)$
is an isomorphism for all $i \ge 4$ and all torsion abelian sheaves $F'$.
We shall apply that isomorphism with $F'$ equal to any of the $\underline{\rm{Ext}}$-sheaves in the local-to-global Ext spectral sequence.
The formation of that spectral sequence is pullback-functorial with respect to working locally on $U$, so it follows that the formation of the edge
map is not only compatible with restriction from $U$ to $V$ but moreover,
those restriction maps between edge terms as well as between the global abutments are {\em isomorphisms} in all degrees $\ge 4$. Hence, we may
assume $F$ is locally constant with order that is a unit on $U$.
%
%We first assume $F$ is concentrated at a closed point $x\in X$.
%In such cases $\cHom(F,\mathbf{G}_m)$ vanishes, so we want the right side to vanish for $p \ge 4$.
%Let $i_x:x\to U$ be the inclusion,
%so $F = i_{x*}(M)$ for some finite discrete $G_x$-module $M$. Since $i_x^{!}$
%has exact left adjoint $i_{x*}$ (so it carries injectives to injectives), we get a spectral sequence:
%$$\Ext_{x}^p(M,\rR^qi_x^!(\mathbf{G}_m)) \Rightarrow\Ext^p_U(F,\mathbf{G}_m).$$
%Moreover, we have $$\rR^1i_x^!(\mathbf{G}_m) = \bZ\ \ \textup{and}\ \ \rR^qi_x^!(\mathbf{G}_m) = 0,\ q\neq 1$$
%by Proposition \ref{comp2}, so the spectral sequence collapses to isomorphisms
%$$\Ext_{x}^{p-1}(M,\bZ) = \Ext_U^p(F,\mathbf{G}_m).$$
%Hence, we want to prove $\Ext^i_{x}(M, \bZ) = 0$ for any $i \ge 3$
%and finite discrete $G_x$-module $M$.
%
%The exact sequence of $G_x$-modules (with trivial $G_x$-action)
%$$0 \rightarrow \bZ \rightarrow \bQ \rightarrow \bQ/\bZ \rightarrow 0$$
%induces an isomorphism $\Ext^i_{x}(M, \bZ) \simeq \Ext^{i-1}_x(M, \bQ/\bZ)$ for $i \ge 1$ since $\Ext^j_x(M, \bQ) = 0$
%for all $j \ge 0$ (as $M$ is killed by $\#M$ yet that multiplication on $\bQ$ is an automorphism).
%But $\Ext^j_x(M, \bQ/\bZ) = 0$ for all $j \ge 2$,
%as was proved in Remark \ref{finiteext}, so the punctual case is settled.
%
%Now let $F$ be a general constructible sheaf on $U$. We choose a dense open set $j:V\to U$ such that $F\vert_V$ is locally constant with
%order a unit on $U$. Consider the short exact sequence
%$$0 \rightarrow j_!(F\vert_V) \to F \rightarrow F' \rightarrow 0$$
%where $F'$ is punctual. We do {\em not} claim that the edge map in question is $\delta$-functorial
%(indeed, it would make no sense to claim that, since a-priori $\cHom(F, \mathbf{G}_m)$ is not exact in $F$),
%but the settled punctual case gives that both sides of the isomorphism of interest vanish at $F'$ in degrees $\ge 4$
It suffices to show that for those special class of $F$'s, the higher Ext-sheaves
$\underline{\rm{Ext}}_U^i(F, \mathbf{G}_m)$ vanish for all $i > 0$.
This assertion is of \'etale-local nature on $U$, so by passing to a suitable connected finite \'etale
cover we may assume $F = \bZ/n\bZ$ for some $n > 0$ that is a unit on $U$. Then the exact sequence of constant sheaves
$$0 \rightarrow \bZ \rightarrow \bZ \rightarrow \bZ/n\bZ \rightarrow 0$$
reduces the task to showing two facts:
\begin{itemize}
\item[(i)] $\underline{\rm{Ext}}_U^i(\bZ, \mathbf{G}_m) = 0$ for $i > 0$,
\item[(ii)] multiplication by $n$ on $\underline{\rm{Ext}}^0(\bZ, \mathbf{G}_m) = \mathbf{G}_m$ is a surjection of \'etale sheaves.
\end{itemize}
Assertion (ii) is clear since $n$ is a unit on $U$, and in general
$\underline{\rm{Ext}}^i_U(\bZ, \mathbf{G}_m)$ is the sheafification of
$$V \rightsquigarrow \Ext^i_V(\bZ, \mathbf{G}_m) = \rH^i(V, \mathbf{G}_m).$$
This sheafification clearly vanishes when $i > 0$ (as for cohomology of any sheaf), settling (i).
\end{proof}
For $i \ge 4$ and any {\em constructible} $F$ on $U$, Lemma \ref{global sections} allows us to define a composite map
$$\Ext^i_U(F,\mathbf{G}_m) = \rH^i(U, \cHom(F, \mathbf{G}_m)) \rightarrow
\rH^i(\eta, \cHom(F_{\eta}, \mathbf{G}_{m, \eta})) = \rH^i(K, F_{\eta}^{\vee})$$
via localization to the generic point,
where $F_{\eta}^{\vee} = \Hom(F_{\eta}, \mathbf{G}_{m,\eta})$ is the dual Galois module.
(The formation of $\cHom(F, \cdot)$ commutes with pullback to $\eta$ because $F$ is {\em constructible}; in general
the formation of $\cHom$-sheaves does {\em not} commute with non-\'etale localization.) Pulling back further
to cohomology at the real places $v_1, \dots, v_r$ of $K$ thereby provides a canonical map
$$\Ext_U^i(F,\mathbf{G}_{m}) \to \bigoplus_{k=1}^r\rH^i(I_{v_k},F_{\eta}^{\vee}).$$
\begin{cor} \label{euler char disguise} The preceding canonical map is an isomorphism for all $i \ge 4$.
\end{cor}
\begin{proof}
In the proof of Lemma \ref{global sections}, we saw that since $i \ge 4$ we can shrink $U$ without affecting
the Ext's in degree $\ge 4$. Hence, we may assume $F$ is locally constant with order that is a unit on $U$.
Our task is equivalent to showing that the collective restriction map
$$\rH^i(U, F^{\vee}) \rightarrow \bigoplus_v \rH^i(I_v, F_{\eta}^{\vee})$$
is an isomorphism for $i \ge 4$, with $F^{\vee} = \cHom(F, \mathbf{G}_m)$ the Cartier dual sheaf
(also lcc, and naturally $F \simeq F^{\vee\vee}$).
Writing $U = \Spec(\ccO_{K,S})$ (where $S$ contains the archimedean places of $K$)
and focusing on the finite discrete $G_S$-module corresponding to $F_{\eta}^{\vee}$,
this was established in the proof of Theorem \ref{newtate}.
\end{proof}
\begin{rem}\label{finiteeuler}
For later purposes we need to address a compatibility with finite pushforward.
Consider a finite extension $K'/K$ and the finite normalization $f:U' \rightarrow U$ of $U$ in $K'/K$.
Let $\eta' \in U'$ be the generic point. For a constructible abelian sheaf $F'$ on $U'_{\et}$,
so $F'$ is lcc over the preimage of a dense open in $U$,
naturally $(f_{\ast}F')_{\eta}^{\vee} \simeq f_{\ast}((F'_{\eta'})^{\vee})$. We claim that the isomorphism in Corollary
\ref{euler char disguise} is compatible with $f_{\ast}$ in the sense that for $i \ge 4$ the diagram
$$\xymatrix{
\Ext^i_{U'}(F', \mathbf{G}_{m,U'}) \ar[r]^-{\simeq} \ar[d] & \bigoplus_v \bigoplus_{w|v} \rH^i(I_w, (F'_{\eta'})^{\vee}) \ar[d] \\
\Ext^i_U(f_{\ast}(F'), \mathbf{G}_{m,U}) \ar[r]_-{\simeq} & \bigoplus_v
\rH^i(I_v, (f_{\ast}F')^{\vee}_{\eta})
}$$
commutes, where the left side rests on the exactness of $f_{\ast}$ and the norm map $f_{\ast}\mathbf{G}_{m,U'} \rightarrow \mathbf{G}_{m,U}$.
To prove this commutativity, we can use the compatibility of the {\em isomorphisms} in Lemma \ref{global sections}
and Corollary \ref{euler char disguise} with respect to Zariski localization to reduce to the case when $F'$ is lcc on $U'$
and $f$ is \'etale. Hence, the natural composite map
$$f_{\ast}\cHom_{U'}(F', \mathbf{G}_{m,U'}) \rightarrow \cHom_U(f_{\ast}F', f_{\ast}\mathbf{G}_{m,U'}) \rightarrow
\cHom_U(f_{\ast}F, \mathbf{G}_{m,U})$$
is an {\em isomorphism} (here we use both that $F'$ is lcc and that $f$ is finite \'etale).
Since $f_{\ast}$ is exact and preserves injectives (so it carries Cartan--Eilenberg resolutions to Cartan--Eilenberg resolutions), it follows
that for $i \ge 0$ the edge maps in Lemma \ref{global sections} for $F'$ on $U'$ and for
$f_{\ast}F'$ on $U$ are compatible in the sense that the
diagram
$$\xymatrix{
\rH^i(U',\cHom_{U'}(F',\mathbf{G}_{m,U'})) \ar[r] \ar[d] & \Ext_{U'}^i(F', \mathbf{G}_{m,U'}) \ar[d]\\
\rH^i(U, \cHom_U(f_{\ast}F', \mathbf{G}_{m,U})) \ar[r] & \Ext_U^i(f_{\ast}F', \mathbf{G}_{m,U})
}$$
commutes (as we check by chopping the diagram into two
smaller squares via an intermediate row involving $f_{\ast}\mathbf{G}_{m,U'}$, in terms of which the bottom square
commutes by functoriality of the edge map over $U$ relative the norm map $f_{\ast}\mathbf{G}_{m,U'} \rightarrow \mathbf{G}_{m,U}$).
\end{rem}
\medskip
Returning to the task of defining the Yoneda pairing for negative $p$ with {\em constructible} $F$, by Corollary \ref{euler char disguise}
it suffices
to define pairings
\begin{equation}\label{tatepair}
\rH^{p-1}_T(I_v, F_{\eta}) \times \rH^{3-p}(I_v, F_{\eta}^{\vee}) \rightarrow (1/2)\bZ/\bZ \subset \bQ/\bZ
\end{equation}
for $p < 0$. But Tate cohomology coincides with group cohomology in positive degrees, so
the usual cup product for Tate cohomology gives a pairing valued in
$\rH^2(I_v, \mathbf{G}_m) = \rH^0_T(I_v, \mathbf{G}_m)$, and this latter group is the cokernel of
the norm map $\overline{K}^{\times} \rightarrow (K^{\rm{alg}}_v)^{\times}$ (with $K^{\rm{alg}}_v$ the real closed
field given by the algebraic closure of $K$ in $K_v = \mathbf{R}$). The image of the norm is
the subgroup of positive elements in the real closed field $K_v^{\rm{alg}}$, so its cokernel has order 2.
That defines the Yoneda pairing for negative degrees.
In order to do nontrivial things with the Yoneda pairing as just defined, we need:
\begin{prop}\label{yoneda} The pairing $\rH^p_c(U, F) \times {\rm{Ext}}^{3-p}_U(F, \mathbf{G}_{m,U}) \rightarrow \bQ/\bZ$
for constructible abelian $F$ on $U_{\et}$ is $\delta$-functorial for $p \in \bZ$.
\end{prop}
\begin{proof} Since extension by zero from $U$ to $X$ is exact, by design the problem reduces to the case $U = X$.
Consider a short exact sequence of abelian sheaves on $X$
$$0 \rightarrow F' \stackrel{f}{\rightarrow} F \stackrel{f'}{\rightarrow} F'' \rightarrow 0$$
and apply the right-exact $\widehat{(\cdot)}$ to get a pair of exact sequences
$$0 \rightarrow N \rightarrow \widehat{F'} \rightarrow M \rightarrow 0,\,\,\,
0 \rightarrow M \rightarrow \widehat{F} \rightarrow \widehat{F''} \rightarrow 0$$
with $M := \ker \widehat{f'}$ and $N := \ker \widehat{f}$, with $N$ supported at the real points.
The $\delta$-functor structure of $\rH^{\bullet}_c(X, \cdot)$ was defined in degrees $\ge 0$ to be the maps
$$\delta^p: \rH^p(\overline{X}, \widehat{F''}) \rightarrow \rH^{p+1}(\overline{X}, M) \simeq \rH^{p+1}(\overline{X}, \widehat{F'})$$
for $p \ge 0$.
Since $M \rightarrow \widehat{F}$ restricts to an isomorphism over $X$, applying ${\rm{Ext}}^{\bullet}_{\overline{X}}(\cdot,
\overline{j}_{\ast}(G))$ to that map gives an isomorphism for any $G$ on $X$ (such as $G = \mathbf{G}_{m,X}$). Hence,
the $\delta$-functoriality of the usual Yoneda pairing does the job for connecting maps in non-negative degrees.
Up to here, the constructibility of $F$ has not been used.
Now let's consider the connecting maps in degrees $\le 0$; i.e., from degree $p$ to degree $p+1$
for $p < 0$. Recall that the pairing in negative degrees has only been defined
for constructible $F$. For $p < -1$ this expresses the basic fact that cup product in Tate cohomology is $\delta$-functorial
(in all degrees). Hence, it remains to analyze the connecting map from degree $-1$ to degree 0. This is proved
in Proposition \ref{dyoneda}.
\end{proof}
%
%By means of Example \ref{cGm}, we are led to believe that the Yoneda pairing:
%$$\rH_c^p(U, F)\times\Ext_U^{3-p}(F,\mathbf{G}_{m,U})\to\rH^3_c(U,\mathbf{G}_m)\simeq\bQ/\bZ$$
%is perfect for every $U\hookrightarrow X$.
%
%Note that, calling $j$ the open immersion of $U$ into $X$, we get $j^*(\mathbf{G}_{m,X}) = \mathbf{G}_{m,U}$, and we have:
%$$\rR\underline{\Hom}_{X_{\rm\acute{e}t}}(j_!(F), \mathbf{G}_{m,X}) = \rR\underline{\Hom}_{U_{\rm\acute{e}t}}(F, j^*(\mathbf{G}_{m,X})) = \rR\underline{\Hom}_{U_{\rm\acute{e}t}}(F, \mathbf{G}_{m,U})$$
%Since if $F$ is constructible, then so is $j_!(F)$. We apply Theorem \ref{dualT}, and by means of Example \ref{cGm}, perfectness of the above pairing follows.
%
%
%
%We give some more general context with a brief:
%\begin{rem} Verdier duality in topology says roughly that for a map $f:X\to Y$ between sufficiently nice spaces,
%the total derived functor $\rR f_! : \bD^+(X)\to\bD^+(Y)$ of the ``direct image with compact supports'' functor $f_{!}$ has a right adjoint $f^!$
%yielding the canonical bijection $$\rR\Hom(\rR f_! (F), G) = \rR\Hom(F, f^!(G))$$ for all objects $F\in\bD^+(X)$, $G\in\bD^+(Y)$.
%Unpackaging $\rR\Hom$ in terms of Ext's on homologies, and choosing nice $F$, one gets pairings between Ext's and cohomology
%much as in Artin--Verdier duality stated above (up to the modification account for real places).
%This inspires many duality principles in geometry, including Poincar\'e duality for manifolds,
%Grothendieck duality for coherent cohomology, and Artin--Verdier duality in \'etale cohomology.
%\end{rem}
%
%
The proof of Artin--Verdier duality will involve pulling down results
from a finite extension of $K$ that is totally imaginary (over which the theorem
has been proved in \cite[Thm.\,2.4]{Mazur}). In addition to the $\delta$-functoriality already established, we also require
compatibility with pushforward along finite maps. More specifically,
let $K'/K$ be a finite extension, $U \subset \Spec(\ccO_K)$ a dense open subset,
and $U' \subset \Spec(\ccO_{K'})$ the preimage of $U$.
We get a finite morphism $f: U' \to U$ and will use the following preliminary result:
\begin{lemma} \label{cohomfinite} For any abelian sheaf $F'$ on $U'$ and $p \ge 0$ we have canonical isomorphisms
$$\widehat{\rH}^p(U', F') \simeq \widehat{\rH}^p(U, f_*(F')).$$
\end{lemma}
This isomorphism in nonnegative degrees is also $\delta$-functorial by construction, but we won't need that.
Such isomorphisms also exist for $p < 0$ (and $\delta$-functoriality holds in all integral degrees). We discuss these further aspects
in \S\ref{paircomp}, where we establish a result that is crucial
in subsequent arguments: the isomorphism is compatible with the Yoneda pairing in Proposition \ref{yoneda}
(under a constructibility hypothesis when $p < 0$).
\begin{proof} We need to compare
$\rH^p(\overline{U}', \widehat{F'})$ and $\rH^p(\overline{U}, \widehat{f_{\ast}(F')})$.
Using the map $\overline{f}: \overline{U}' \rightarrow \overline{U}$ that is ``finite''
in the sense of Definition \ref{finiteR}, the functor $\overline{f}_{\ast}$
as in Definition \ref{infpushpull} is exact by Proposition \ref{pushR} and it preserves injectivity
(since its left adjoint $\overline{f}^{\ast}$ is exact), so
$$\rH^p(\overline{U}', \cdot) \simeq \rH^p(\overline{U}, \overline{f}_{\ast}(\cdot)).$$
Hence, it suffices to construct a natural isomorphism
$$\overline{f}_{\ast}(\widehat{F'}) \simeq (f_{\ast}(F'))^{\wedge}.$$
Over $U$ this is tautological, so we just need to compare stalks at real points
$v$ of $K$ along with the associated maps $\varphi_v$. With the help of Remark \ref{pushstalk}, this amounts to
elementary bookkeeping with real and complex places
(such as factor fields of $K' \otimes_K \overline{K_v} = (K' \otimes_K K_v) \otimes_{K_v} \overline{K_v}$
for real places $v$ of $K$) that we leave to the reader.
\end{proof}
%We are ready to prove the main Theorem of this handout, which is Theorem \ref{dualT}. By Example \ref{Gm modified}, we have: $$\mathbf{G}_{m,\overline{U}} = \widehat{\mathbf{G}}_{m, U}.$$ By Proposition \ref{HGm} we also get
%$$\widehat{\rH}^3(X,\mathbf{G}_{m,X}) = \bQ/\bZ.$$
%Let us agree that $\Ext^p_{(-)}(-,-)$ is zero for negative $p$. We restate Theorem \ref{dualT}.
%\begin{thm} Let $F$ be a constructible sheaf on $X$. The Yoneda pairing:
%$$\widehat{\rH}^p(X, F)\times \textup{Ext}_X^{3-p}(F,\mathbf{G}_{m,X}) \to\widehat{\rH}^3(X, \mathbf{G}_{m,X}) = \bQ/\bZ, \ \ p\in\bZ$$
%is perfect.
%\end{thm}
%
%Note that if $K$ is totally imaginary, then this recovers \cite[Theorem 2.4]{Mazur}.
%To set this up, rather generally for
%any abelian sheaves $F$ and $F'$ on $S_{\et}$ for a scheme $S$ and any injective resolution $F' \rightarrow I^{\bullet}$,
%the complex $\cHom(F, I^{\bullet})$ in non-negative degrees has kernel $\cHom(F, F')$ in degree 0.
\subsection{Proof of Artin--Verdier duality}
We are ready to prove Theorem \ref{dualT}.
By definition, for the case $p < 0$ this amounts to the statement that if $v$ is a real place and $M$ is a finite $I_v$-module
with dual $M^{\vee} = \Hom(M, \overline{K}^{\times})$ then
$${\rH}^{2-p}_T(I_{v}, M)\times{\rH}_T^p(I_{v}, M^{\vee})\to\rH^2(I_v, \overline{K}^{\times}) = \rH^2(I_v, \mu_2) = (1/2)\bZ/\bZ$$
is a perfect pairing. This is exactly Remark \ref{realtate}.
Now consider the case $p\ge 0$.
This was settled by Mazur for totally imaginary $K$ in \cite{Mazur} using a massive amount of class field theory. (In such cases
$\rH^0_c(X, \cdot) = \rH^0(X, \cdot)$ is left exact!) We will establish the general case by induction on $p \ge 0$,
using the settled cases of $p < 0$
and of totally imaginary $K$ (for all $p \in \bZ$)
to get the induction started. That induction is the following lemma, which concludes the proof.
\begin{lemma}\label{induction} Assume that for some $p_0 \ge 0$
the pairing
is perfect for any $K$ and any constructible sheaf $F$ with any $p 0$).
\begin{thm}\label{finite ramification} Let $M$ be a finite discrete
$G_S$-module whose order is a unit on $U$. Then there exists an exact sequence:
\begin{displaymath}
\xymatrix{0\ar[r] & \rH^0(G_S, M)\ar[r] & \bigoplus_{s\in S}\rH^0(K_s, M)\ar[r] & \rH^2(G_S, M^{\vee})^D\ar[r] & \rH^1(G_S,M)\\
\ar[r] & \bigoplus_{s\in S} \rH^1(K_s, M)\ar[r] & \rH^1(G_S, M^{\vee})^D\ar[r] & \rH^2(G_S, M)\ar[r] & \bigoplus_{s\in S}\rH^2(K_s, M)\\
\ar[r] & \rH^0(G_S, M^{\vee})^D\ar[r] & 0 & & }
\end{displaymath}
naturally in $M$, where $M^{\vee} := \Hom(M, \mathbf{G}_{m,\eta})$ is the Cartier dual Galois module
and we denote by $(\cdot)^D$ the Pontryagin dual.
\end{thm}
\begin{proof}
Since $G_S$ is the fundamental group of $U$, $M$ defines a locally constant sheaf $F_M$ on $U$. We denote by $j: U\to X$ the inclusion.
By Lemma \ref{global sections}, we have canonical isomorphisms:
$$\Ext_X^p(j_!(F_M),\mathbf{G}_{m,X})\simeq\Ext_U^p(F_M,\mathbf{G}_{m,U}) \simeq\rH^p(U,\cHom(F_M,\mathbf{G}_{m,U})).$$
We denote the dual sheaf $\cHom(F_M,\mathbf{G}_{m,U})$ by $F^{\vee}_M$. Since the order of $F_M$ is
a unit on $U$, $F_M^{\vee}$ is locally constant on $U$; its generic fiber is $M^{\vee}$. By Theorem \ref{comparison} applied to $F_M$, we get an isomorphism:
\begin{equation}\label{exth}
\Ext_X^p(j_!(F_M),\mathbf{G}_{m,X})\simeq\rH^p(G_S, M^{\vee}).
\end{equation}
The local cohomology sequence of \S\ref{localcohom} becomes
\begin{equation}\label{9term}
\cdots\to\bigoplus_{s\in S}\rH^p_s(X,j_!(F_M))\to\widehat{\rH}^p(X,j_!(F_M))\to\rH^p(U,F_M)\to\bigoplus_{s\in S}\rH^{p+1}_s(X,j_!(F_M)).
\end{equation}
This sequence will be the desired 9-term long-exact sequence for finite ramification upon unraveling
its terms and some maps (e.g., to ensure that $\Sha^1_S(K, \cdot)$ and $\Sha^2_S(K, \cdot)$
and the perfect duality between them can be extracted from the long exact sequence).
By Theorem \ref{comparison} we have
$$\rH^p(U,F_M) = \rH^p(G_S,M).$$
Moreover, by Artin--Verdier duality (!) and dualizing (\ref{exth}) we get:
$$\widehat{\rH}^p(X,j_!(F_M))\simeq\Ext_X^{3-p}(j_!(F_M),\mathbf{G}_{m,X})^D\simeq\rH^{3-p}(G_S, M^{\vee})^D.$$
We are left to interpret $\rH^p_s(X,j_!(F_M))$ and some of the maps in the long exact sequence (\ref{9term}).
Suppose $s$ is a real point. From Definition \ref{localcoho}, we have:
$$\rH^p_s(X, j_!(F_M)) = {\rH}_T^{p-1}(I_s, M) = \rH^{p-1}(K_s, M)$$
(where the final equality is our {\em definition} of $\rH^{\bullet}(K_s, \cdot)$ for real $s$, coinciding
with usual Galois cohomology in positive degrees).
Let us now assume $s\in S$ is a finite point, and denote by $f_s : X_s^{\rm h}\to X$ the henselization of $X$ at $s$. By Lemma \ref{henselization} we have:
$$\rH^p_s(X, j_!(F_M)) = \rH^p_s(X_s^{\rm h}, f^*_sj_!(F_M)).$$
The latter group can be computed from the local cohomology sequence for a henselian discrete valuation ring $A$ (such
as that of $X$ at $s$), giving an isomorphism
$$\rH^p_s(X,j_!(F_M)) \simeq\rH^{p-1}(D_s, M) = \rH^{p-1}(K_s, M)$$
because of the {\em vanishing} of the the flanking terms $\rH^p(\Spec(A), j'_{!}(G))$ for the open immersion $j': \Spec(L) \hookrightarrow \Spec(A)$
of the generic point and an \'etale sheaf $G$ on the generic point (since the degree-$p$
cohomology of a sheaf on $\Spec(A)$ coincides with the degree-$p$ cohomology of the fiber sheaf at the closed point due to $A$
being henselian, so it vanishes when that fiber sheaf is 0).
Next, we describe some of the maps that emerge via these identifications:
\begin{lemma}\label{maps} For $M$ as in Theorem $\ref{finite ramification}$ and $p \ge 0$, the composite isomorphism
$$\rH^p(G_S, M) \simeq \rH^p(U, F_M) \rightarrow \bigoplus_{s \in S} \rH^{p+1}_s(X, j_{!}(F_M)) \simeq \bigoplus_{s \in S} \rH^p(K_s, M)$$
is the restriction map and moreover the map
$\bigoplus_{s \in S} \rH^1(K_s, M) \rightarrow \rH^1(U, M^{\vee})^D$
defined via identification with the outer terms in
$$\bigoplus_{s \in S} \rH^2_s(X, j_{!}(F_M)) \rightarrow \widehat{\rH}^2(X, j_{!}(F_M)) \simeq {\rm{Ext}}^1_X(j_{!}(F_M), \mathbf{G}_{m,X})^D
\simeq \rH^1(G_S, M^{\vee})^D$$
is Pontryagin dual to the map $\rH^1(G_S, M^{\vee}) \rightarrow \bigoplus_{s \in S} \rH^1(K_s, M)^D$
defined by Tate local duality $\rH^1(K_s, M) \times \rH^1(K_s, M^{\vee}) \rightarrow \bQ/\bZ$ for all $s \in S$ $($including real $s$!$)$.
\end{lemma}
These descriptions ensure that the 9-term exact sequence we will build yields perfect pairings between
$\Sha^1_S(K, M^{\vee})$ and $\Sha^2_S(K, M)$ (entirely opposite to Tate's approach that directly
constructed such perfect pairings and used that to {\em define} parts of the 9-term exact sequence).
The degree-1 identification is not entirely a tautology, since (for non-real $s$) the degree-1 local duality pairing changes by
a sign if we swap the factors in the pairing!
\begin{proof} Consider the $s$-component of the first assertion. If $s$ is real then is reduced to
the description for $p \ge 0$ of connecting maps from global to local cohomology at a real point
for arbitrary abelian sheaves on $X$
as in the proof of Theorem \ref{newtate}. The case of finite $s$ is handled by
a simpler version of the same limit trick as in the case of real points, namely we use compatibility with
residually trivial pointed \'etale neighborhoods of $s$ to reduce
to the analogous identification when $X$ is replaced with $\Spec(\cO_{X,s}^{\rm{h}})$, in which case
our task is a tautology in view of how $\rH^{p+1}_s(X, j_{!}(F_M))$ is identified with $\rH^p(K_s, M)$ via
a local cohomology sequence on $\Spec(\cO_{X,s}^{\rm{h}})$.
The link to Tate local duality in degree 1 lies deeper, and we address it in
Proposition \ref{tatedual1}.
\end{proof}
It remains to check exactness at the endpoints of the 9-term sequence; i.e., that (\ref{9term}) is injective from
$\rH^0(U, F_M)$ and is surjective onto $\widehat{\rH}^3(X, j_{!}(F_M))$. By Lemma \ref{maps},
the injectivity expresses the obvious fact that the natural map $\rH^0(G_S, M) \rightarrow \bigoplus_{s \in S} \rH^0(K_s,M)$
is injective: if $S$ contains a finite point $s_0$ then even the projection to the $s_0$-factor is injective, and if
$S$ consists only of the real places then $U = X$ yet the order of $M$ is a unit on $U$, forcing $M=0$.
By Lemma \ref{maps} the surjectivity is the assertion that the natural restriction map
$$\rH^3(G_S, M)\to\bigoplus_{s\in S}\rH^3(K_s, M)$$
is injective, and even its projection to the direct sum over
the real points is injective (in fact, an isomorphism!) due to Theorem \ref{newtate}. This completes the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\section{Relating Galois cohomology and \'etale cohomology}\label{contet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Continuous cohomology vs \'etale cohomology}\label{contet}
\subsection{A global comparison}
We begin by proving Theorem \ref{comparison}, which we restate here for convenience.
\begin{thm}\label{galet} Let $F$ be a locally constant constructible sheaf on $U$ whose fiber-rank is a unit on $U$. There is a natural isomorphism
of $\delta$-functors
$$\rH^p(G_S, F_{\eta}) = \rH^p(U_{\et},F)$$
for $p\ge 0$, and it is compatible with cup products.
\end{thm}
\begin{proof}
We consider $U_{\et}$, which is equivalent to the category of finite \'etale $K$-algebras unramified outside $S$. Therefore, $\rH^{\bullet}(U_{\rm f\acute{e}t},\cdot)$ is $G_S$-cohomology on discrete $G_S$-modules. We consider the functor
$$U_{\rm f\acute{e}t}\to U_{\et}$$
assigning to each finite \'etale $U$-scheme the same scheme viewed as an \'etale $U$-scheme. This yields a morphism of categories of sheaves of sets
$$\mu = (\mu_{\ast}, \mu^{\ast}): {\rm{Shv}}(U_{\et}) \rightarrow {\rm{Shv}}(U_{\rm f\acute{e}t})$$
for which $\mu_*$ sends an \'etale sheaf of sets on $U_{\rm\acute{e}t}$ its restriction on $U_{\rm f\acute{e}t}$
and the corresponding pullback $\mu^*$ assigns to each sheaf of sets $F$ on $U_{\rm f\acute{e}t}$ the sheafification of the
presheaf $V\mapsto F(V')$ where $V'$ is the unique maximal finite \'etale $U$-scheme dominated by $V$.
(If $V = \emptyset$ then $V' = \emptyset$.)
Since $\mu^*$ is exact, implying that $\mu_*$ preserves injectives, the equality
$$\Gamma(U_{\rm\acute{e}t}, \cdot) = \Gamma(U_{\rm f\acute{e}t}, \cdot)\circ\mu_*$$
yields a Grothendieck spectral sequence
$$\rH^p(U_{\rm f\acute{e}t}, \rR^q\mu_*(A)) \Rightarrow \rH^{p+q}(U_{\rm\acute{e}t}, A)$$
for any \'etale abelian sheaf $A$ on $U$.
We shall prove that the edge map
$$\rH^p(G_S, \mu_{\ast}(F)_{\eta}) = \rH^p(U_{\rm f\acute{e}t}, \mu_{\ast}(F)) \rightarrow \rH^p(U_{\et}, F)$$
is an isomorphism when $F$ is lcc with order a $U$-unit (and then at the end we will address why this is a map of $\delta$-functors
compatible with cup products).
It suffices to prove that for every such $F$, $\mu_*(F)_{\eta} = F_{\eta}$ and $\rR^q\mu_*(F)$ vanishes for all $q>0$.
The equality $\mu_{\ast}(F)_{\eta} = F_{\eta}$ is immediate from the definition of $\mu_{\ast}$ since
the lcc hypothesis on $F$ implies that $F_{\eta}$ as a ${\rm{Gal}}(K_s/K)$-module is a $G_S$-module.
(In general, if $F$ is an arbitrary abelian sheaf on $U_{\et}$ then $\mu_{\ast}(F)_{\eta} =
F_{\eta}^{{\rm{Gal}}(\overline{K}/K_S)}$.)
Higher direct images are always computed as ``sheafified cohomology'', so upon renaming a connected finite \'etale cover of $U$ as
$U$ it suffices to show that
$$\varinjlim \rH^p(U'_{\rm f\acute{e}t}, F ) =0$$
for all $p>0$, the colimit running over all connected finite \'etale covers $U'\to U$ dominated by the chosen $K_S$.
Up to replacing again $U$ with a suitable connected finite \'etale cover, we reduce to the case $F = \bZ/n\bZ$ with $n$ invertible on $U$ such that the function field $K$ of $U$ contains the $n$-th roots of unity and is totally imaginary. For any connected finite \'etale
$U' \rightarrow U$ we will use the Kummer sequence
$$0\to\bZ/n\bZ\to\mathbf{G}_{m, U'}\xrightarrow{\times n}\mathbf{G}_{m,U'}\to 0$$
that yields the exact sequence:
$$
\xymatrix{0\ar[r] & \rH^0(U,\bZ/n\bZ) \ar[r] & \ccO_{K,S}^{\times}\ar[r]^{t^n} & \ccO_{K,S}^{\times}\ar[r] & \\
\rH^1(U,\bZ/n\bZ) \ar[r] & \Pic(U)\ar[r]^{\times n} & \Pic(U) \ar[r] & \rH^2(U,\bZ/n\bZ) \ar[r] & \\
\rH^2(U,\mathbf{G}_{m, U})\ar[r]^{\times n} & \rH^2(U,\mathbf{G}_{m, U})\ar[r] & \rH^3(U,\bZ/n\bZ) & &}
$$
Since $K$ is totally
imaginary, so $\overline{U} = U$, Theorem \ref{finitezero}
and Proposition \ref{base} imply $\rH^p(U,\bZ/n\bZ) = 0$ for all $p\ge 3$.
To show $\varinjlim \rH^p(U',\bZ/n\bZ) = 0$ for $p=1,2$ as $U'$ varies
through the connected finite \'etale covers of $U$ dominated by $K_S$, first note that
$\varinjlim \ccO_{U'}^{\times} = \ccO_{K_S,S}^{\times}$ is $n$-divisible (as $n$ is an $S$-unit),
and $\varinjlim {\rm{Pic}}(U') = 1$ due to the Principal Ideal Theorem (as we saw in the proof
of Proposition \ref{hfinite}); this gives the vanishing for $p=1$.
To settle $p=2$ it suffices to prove
$$\varinjlim_{U'}\rH^2(U'_{\rm\acute{e}t},\mathbf{G}_{m,U'})[n] = 0,$$
By Grothendieck's work on Brauer groups in the case of
Dedekind schemes
(applying the end of \cite[Ch.\,13]{Milne} to a connected Dedekind scheme), $\rH^2(U'_{\rm\acute{e}t},\mathbf{G}_{m,U'})$ is the group of Brauer classes of the function field $K'$ which are locally trivial at all places corresponding to closed points.
Hence, it suffices to show that every class in $\textup{Br}(K')[n]$ locally trivial outside $S$ is split by some finite extension of $K'$ unramified outside $S$.
Let $L/K'$ be an extension which is unramified along $U'$. Let $w$ be a place of $K'$ over a place $u\in S$. By local class field theory we get a commutative diagram:
\begin{displaymath}
\xymatrix{\textup{Br}(K'_u)\ar[d]\ar[r]^{\textup{inv}} & \frac{1}{n}\bZ/\bZ\ar[d]^{[L_{w} : K'_u]}\\
\textup{Br}(L_{w})\ar[r]_{\textup{inv}} & \frac{1}{n}\bZ/\bZ}
\end{displaymath}
It is sufficient to construct an extension $L/K'$ such that it is unramified along $U'$ and all local degrees $[L_w : K'_u]$ are divisible by $n$ for all $u\in S$ and $w|u$. If we let $H'$ be the Hilbert class field of $K'$ then all maximal ideals $q\in S$ become principal in $H'$ by the Principal Ideal Theorem ,
say generated by some $f_q$. The extension
$$L := H'(\{f_q^{1/n}, q\in S\})$$
does the job. Indeed, passing to $L$ makes all classes in $\textup{Br}(K')[n]$ locally trivial at all places (he archimedean ones being complex), hence globally trivial. This completes the proof of the desired natural isomorphism.
It remains to show that the edge-map isomorphisms $f^p_F:\rH^p(G_S, F_{\eta}) \simeq \rH^p(U_{\et}, F)$ for lcc $F$
constitute a map of $\delta$-functors compatible with cup products.
The compatibility with $\delta$-functoriality is a special case of a general property of edge maps along the bottom in the Grothendieck spectral sequence:
if $G:C \rightarrow C'$ and $G':C' \rightarrow C''$ are left-exact functors between abelian categories
with enough injectives and $G$ carries injectives to $G'$-acyclics then we claim that the edge map
$$\rR^p(G') \circ G \rightarrow \rR^p(G' \circ G)$$
respects the $\delta$-functor structure on both sides when applied to a short exact sequence $0 \rightarrow F_1 \rightarrow F_2 \rightarrow F_3 \rightarrow 0$
in $C$ {\em for which} applying $G$ preserves short-exactness (we apply this
with $G = \mu_{\ast}$, $G' = \Gamma(U_{\rm f\acute{e}t}, \cdot)$, and the $F_j$ an abelian sheaf on $U_{\et}$ arising from
a discrete $G_S$-module $M_j$). This is an elementary diagram-chasing exercise in reviewing how the Grothendieck
spectral sequence is constructed by means of Cartan--Eilenberg resolutions, so it is left to the interested reader.
For {\em any} discrete $G_S$-module $M$ and the associated abelian sheaf $F_M$ on $U_{\et}$, we have
$\mu_{\ast}(F_M)_{\eta} = M$ as discrete modules over ${\rm{Gal}}(\overline{K}/K)$ (acting on $M$ through its quotient $G_S$). Hence, we have edge maps
$$f^p_M: \rH^p(G_S, M) = \rH^p(U_{\rm f\acute{e}t}, \mu_{\ast}(F_M)) \rightarrow \rH^p(U_{\et}, F_M)$$
for any such $M$. (This edge map is {\em not} claimed to be an isomorphism for all such $M$, especially without torsion hypotheses.)
The $\delta$-functoriality argument just given applies to all such $M$; i.e., there no need to assume $M$ is finite with order a unit on $U$.
But in that generality the class of $M$'s under consideration is far richer than just finite discrete $G_S$-modules
with order a unit on $U$. In particular, $\rH^{\bullet}(G_S, \cdot)$ is a {\em derived functor} on this huge class of such $M$'s,
so we can apply Grothendieck's theorem on the universality of erasable $\delta$-functors to
conclude that the collection of edge maps $\{f^p_M\}_p$ is the {\em unique} such map of $\delta$-functors.
Hence, to establish compatibility with cup products it suffices to produce {\em some} map of
$\delta$-functors $\rH^{\bullet}(G_S, M) \rightarrow \rH^{\bullet}(U_{\et}, F_M)$ in discrete $G_S$-modules $M$
such that it is compatible with cup products. (The point is that for $M$ of finite order that is a unit on $U$
such an alternative map must recover the isomorphisms that are the focus of interest in the statement of Theorem \ref{galet},
so these isomorphisms would be compatible with cup products as desired.)
The key observation is that if we view $M$ as a sheaf on $U_{\rm f\acute{e}t}$ then via the natural isomorphism $M \simeq \mu_{\ast}(F_M)$
we get an adjoint map $h_M: \mu^{\ast}(M) \rightarrow F_M$ and the latter is {\em also} an isomorphism. To check this latter isomorphism property we may use
compatibility with direct limits in $M$ to reduce to the case when $M$ is finitely generated as a discrete $G_S$-module,
so $M$ is split by a connected finite \'etale cover of $U$. The formation of the map $h_M$ is compatible with pullback to a connected
finite \'etale cover of $U$, so we may reduce to the case when the $G_S$-action on $M$ is trivial, so $M$ on $U_{\rm f\acute{e}t}$ is a constant sheaf
and hence $\mu^{\ast}(M)$ is the ``same'' constant sheaf on $U_{\et}$ because if $V \rightarrow U$ is \'etale with $V$ connected
(and non-empty) then the maximal finite \'etale $U$-scheme $V'$ through which $V \rightarrow U$ factors is also connected.
Since $F_M$ is also the ``same'' constant sheaf, in this way we see that $h_M$ is identified with the identity map
on that constant sheaf and so it is an isomorphism.
Now our aim is to give an ``alternative'' construction of a map of $\delta$-functors
$\rH^{\bullet}(U_{\rm f\acute{e}t}, \cdot) \rightarrow \rH^p(U_{\et}, \mu^{\ast}(\cdot))$ which is compatible with cup products.
But for any map of topoi (such as $(\mu_{\ast}, \mu^{\ast})$!) there is an associated $\delta$-functorial pullback map
in abelian cohomology, and it is {\em always} compatible with cup products (as cup products
may always be computed in terms of composition of maps in derived categories, as an instance of Ext-pairings).
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Computations}\label{computations}
%Before performing a few needed computations, we review a general construction due to Artin which is called the {\em mapping cylinder}, and specialize it to some examples. We are given two abelian categories $\fA$ and $\fB$, and a left exact functor $u: \fA\to\fB$. Artin constructs a new category $\fC := \fC_{u}$ which is called the mapping cylinder of $u$. The objects of $\fC$ are triples $(M,N,\varphi)$ where $M$ is an object of $\fA$, $N$ an object of $\fB$ and $\varphi: N\to uM$ a morphism in $\fB$. Morphisms in $\fC$ are given by pairs $(f,g) : (M,N)\to (M',N')$ yielding the commutative diagram:
%\begin{displaymath}
%\xymatrix{N\ar[d]_{g}\ar[r]^{\varphi} & uM\ar[d]^{uf}\\
% N'\ar[r]_{\varphi'} & uM'}
%\end{displaymath}
%
%Using left-exactness of $u$ one can check $\fC$ is again an abelian category, and that it has enough injectives as soon as $\fA$ and $\fB$ do.
%
%\begin{eg}\label{dvr}
%As an example, let us consider a complete discrete valuation ring $R$ with fraction field $K$ and finite residue field $k$. Call $G_K$ and $G_k$ the Galois groups of $K$ and $k$, respectively. Call $I$ the inertia group. We call $\pi$ the reduction map $G_K\twoheadrightarrow G_k$ and $\alpha : G_K/I\xrightarrow{\simeq} G_k$ the canonical isomorphism. We let $u := \alpha^*\pi_*(-)$ be our left exact functor between the appropriate categories of Galois modules:
%$$u: \Mod_{G_K}\to \Mod_{G_k}$$
%which, explicitly, is given by sending a $G_K$-module $M$ to $M^I$. We can define the category of Galois modules over $R$ to be the mapping cylinder of $u$. Therefore, the assignment of a Galois $R$-module is that of a diagram $\varphi: N\to M$ where $N$ is a $G_k$-module, $M$ is a $G_K$-module and $\varphi$ is an abelian group homomorphism sending $N$ into $M^I$ and that is a $G_k$-module homomorphism when regarded as a map $N\to M^I$.
%\end{eg}
%
%There are various functors relating the categories $\fA$, $\fB$ and $\fC$. They are:
%
%\paragraph{Pullbacks}\ \\
%
%$$i^*:\fC\to\fB\ \ \textup{and}\ \ j^*:\fC\to\fA$$
%given by
%$$i^*(M,N,\varphi) = N\ \ \textup{and}\ \ j^*(M,N,\varphi) = M.$$
%
%\paragraph{Direct images}\ \\
%
%$$i_* : \fB\to\fC\ \ \textup{and}\ \ j_*: \fA\to\fC$$
%given by
%$$i_*(N) = (0, N, 0)\ \ \textup{and}\ \ j_*(M) = (M, uM, \textup{id}).$$
%
%\paragraph{Extension by zero over a closed point}\ \\
%
%$$j_! : \fA\to\fC$$
%given by
%$$j_!(M) = (M,0,0).$$
%
%\paragraph{Sections with support on a closed point}\ \\
%
%$$i^!: \fC\to \fB$$
%given by
%$$i^!(M,N,\varphi) = \ker(\varphi).$$
%
%We can recollect all these functors into the following diagram:
%
%\begin{displaymath}
%\xymatrix{\fA\ar @< 12.5pt> [r]^{j_!}
% \ar @<-12.5pt> [r]^{j_*} & \fC\ar[l]_{j^*} \ar @< 12.5pt> [r]^{i^*}
% \ar @<-12.5pt> [r]^{i^!} & \fB\ar[l]_{i_*}}
%\end{displaymath}
%
%Suppose $X$ is a scheme, $j: U\to X$ an open immersion and $i : S :=X-U\to X$ the corresponding closed immersion. Let $\fA$ be the category of abelian sheaves on $U_{\rm\acute{e}t}$ and $\fB$ the category of abelian sheaves on $S_{\rm\acute{e}t}$. The above functors satisfy some properties which we review below:
%\begin{enumerate}
%\renewcommand{\labelenumi}{\textup{(\arabic{enumi})}}
%\item We have the following adjunctions: $(i^*, i_*)$, $(i_*, i^!)$, $(j_!, j^*)$, $(j^*, j_*)$.
%
%For example, let us show that for $C$ an object of $\fC$ and $B$ an object of $\fB$, we get:
%$$\Hom(i_*B,C) = \Hom(B, i^!C).$$
%
%Let $C= (C_1,C_2,\varphi)$. Giving a morphism $i_*B\to C$ is equivalent to giving maps $0\to C_1, B\to C_2$, together with the commutative diagram:
%\begin{displaymath}
%\xymatrix{B\ar[d]\ar[r]^{0} & 0\ar[d]^{0}\\
% C_2\ar[r]_{\varphi} & uC_1}
%\end{displaymath}
%
%which is equivalent to assigning a map $B\to C_2$ wth the property that its composition with $\varphi$ factors through the zero map, that is, a map $B\to\ker(\varphi)$, by the universal property of kernels. This proves the claim.
%
%\item The functors $i^*$, $i_*$, $j^*$ and $j_!$ are exact, while $j_*$ and $i^!$ are left-exact.
%\item The functors $j^*$, $j_*$, $i^!$, $i_*$ map injectives to injectives.
%\item The compositions $i^*j_!$, $i^!j_!$, $i^!j_*$, $j^*i_*$ are zero.
%\item The functors $i_*$ and $j_*$ are fully faithful.
%\end{enumerate}
%
%\paragraph{Back to Galois modules}\ \\
%Let us step back to our Example \ref{dvr}. We
\subsection{Local computations}
Now let $A$ be a discrete valuation ring with fraction field $K$ and residue field $k$.
For the open immersion $j: \Spec(K) \rightarrow \Spec(A)$ and its closed complement $i:\Spec(k) \rightarrow \Spec(A)$ we have
the usual exact sequence
%Consider the Galois $R$-module $\mathbf{G}_{m,R}$ given by the map $U_{\rm unr}\to \overline{K}^{\times}$, $U_{\rm unr}$ being the unit group of the maximal unramified extension $K_{\rm unr}$ of $K$, regarded as a $G_k$-module. We have an exact sequence in the category of Galois $R$-modules:
%\begin{displaymath}
%\xymatrix{0\ar[d]\ar[r] & U_{\rm unr}\ar[d]\ar[r] & K_{\rm unr}^{\times}\ar[d]\ar[r]^v & \bZ\ar[r]\ar[d] & 0\ar[d]\\
% 0\ar[r] & \overline{K}^{\times}\ar[r] & \overline{K}^{\times}\ar[r] & 0\ar[r] & 0}
%\end{displaymath}
%which can be rewritten the following way:
$$(*)\ \ 0 \to \mathbf{G}_{m,A}\to j_*(\mathbf{G}_{m,K})\to i_*(\bZ)\to 0$$
as in Artin's calculation of cohomology on smooth curves over a field.
\begin{prop}\label{comp1}
Assume $A$ is henselian, its residue field $k$ is perfect, and $\widehat{K}$ is separable over $K$. Then
$$\rR^pj_*(\mathbf{G}_{m,K}) = 0,\ \ p>0.$$
\end{prop}
The separability of $\widehat{K}$ over $K$ is automatic in characteristic 0 and in general says that $A$ is excellent.
\begin{proof}
The stalk at the generic point vanishes, ultimately because $j_{\ast}$ carries injectives to injectives
and $j^{\ast}j_{\ast} \simeq {\rm{id}}$ (good exercise). Thus, we need to compute the stalk at a geometric
closed point. As for higher direct images in general, this is the direct limit of
the cohomologies of the $K$-fibers of local-etale neighborhoods of the closed point,
which is to say ${\rm{H}}^p(K^{\rm{sh}}, \mathbf{G}_m)$ where $K^{\rm{sh}}$ is the fraction field
of $A^{\rm{sh}}$, or equivalently the maximal unramified extension of $K$. We may rename
$A^{\rm{sh}}$ as $A$ to reduce to the case that $k$ is algebraically closed.
Hilbert 90 gives vanishing for $p = 1$, and for $p=2$ the assertion is that ${\rm{Br}}(K) = 1$.
Granting the vanishing of the Brauer group of such $K$ in general, the same then holds for all of its finite separable extensions too.
Consequently, by the criterion in \S3.1 of Chapter II of Serre's {\em Galois Cohomology},
the field $K$ has cohomological dimension $\le 1$ (i.e., its Galois cohomology
vanishing beyond degree 1 on torsion discrete Galois modules). But then it is strict
cohomological dimension $\le 2$ (i.e., vanishing Galois cohomology beyond degree 2 on {\em all} discrete Galois modules), by
\S3.2 in Chapter I of Serre's book. Hence, we would get the desired vanishing for all $p > 2$.
It remains to prove that ${\rm{Br}}(K) = 1$ when $k$ is algebraically closed.
If $A$ is complete then this is Example (b) in \S7 of Chapter X of Serre's ``Local Fields''.
To settle the general case, first note that since $A$ is henselian, by Krasner's Lemma
completion defines an equivalence between the finite separable extensions of $K$ and $\widehat{K}$.
Hence, the Galois groups of $K$ and $\widetilde{K}$ coincide. In particular, for any $n > 0$
not divisible by ${\rm{char}}(K)$ the natural map
$${\rm{H}}^2(K, \mu_n) \rightarrow {\rm{H}}^2(\widehat{K}, \mu_n)$$
is an isomorphism. But this is the map between $n$-torsion in the Brauer groups, so it follows that the map
$${\rm{Br}}(K) \rightarrow {\rm{Br}}(\widehat{K})$$
is an equality if ${\rm{char}}(K)=0$ and away from $p$-primary parts if ${\rm{char}}(K) = p > 0$.
To settle the case $p = {\rm{char}}(K) > 0$, we will use Artin approximation (which applies to systems
of polynomial equations over the henselian local $A$ since it is excellent); one can avoid Artin approximation by
instead appealing to the deep result in Lang's thesis that such fields $K$ are C$_1$-fields.
More specifically, one of the several sufficient conditions for the Brauer groups of all finite separable extensions of $K$ to vanish
is that for every finite separable extension $K'/K$ and finite Galois extension $L/K'$ the norm map
$L^{\times} \rightarrow {K'}^{\times}$ is surjective. It is harmless to rename such $K'$ as $K$
and we want to show that the norm map $L^{\times} \rightarrow K^{\times}$ is surjective.
The completion $\widehat{L}$ coincides with $\widehat{K} \otimes_K L$, and
the settled result in the complete case ensures that the norm map
$\widehat{L}^{\times} \rightarrow \widehat{K}^{\times}$ is surjective.
In particular, for every $c \in K^{\times}$ the norm equation ${\rm{N}}_{L/K}(x) = c$
viewed as a polynomial equation over $K$ (via a $K$-basis of $L$)
has a solution in $\widehat{K} = K \otimes_A \widehat{A}$. This norm equation is easily rewritten
as a polynomial equation over $A$ which has a solution in $\widehat{A} - \{0\}$,
so by Artin approximation there exists a nearby solution in $A$.
%
%
%We have, in general:
%$$\rR^pj_*(\mathbf{G}_{m,K}) = (\rR^pu(\mathbf{G}_{m,K}), 0,0).$$
%Moreover, we have:
%$$\rR^pu(\mathbf{G}_{m,K}) = \rH^p(I, \overline{K}^{\times}),$$
%which is zero by local class field theory for all $p\ge 1$, as desired.
\end{proof}
Finally, we recall that there is a right adjoint $i^{!}$ to $i_{\ast}$ (i.e., $\Hom(i_{\ast}(M), F) = \Hom(M, i^{!}(F))$
for a discrete $G_k$-module $F$ and \'etale sheaf $F$ on $\Spec(A)$). In the language of Artin's decomposition lemma
applied to $\Spec(A)$, it is easily checked that
$$i^{!}(M, N, \varphi:M \rightarrow N^I) = \ker \varphi$$
has the correct adjunction property. Since
$j_{\ast}(N) = (N^I, N, {\rm{id}}_{N^I})$ it follows
that $i^{!} \circ j_{\ast} = 0$. Likewise, since
$i_{\ast}(M) = (M, 0, 0)$, clearly $i^{!} i_{\ast} = {\rm{id}}$. Using this, we shall prove:
\begin{prop}\label{comp2}
We have:
$$\rR^pi^!(\mathbf{G}_{m,A}) = \left\{\begin{array}{cc}
0 & \textup{if}\ p\neq 1\\
\bZ & \textup{if}\ p=1
\end{array}\right.$$
\end{prop}
\begin{proof}
To prove this we use the exact sequence $(*)$, and compute the outer terms. We claim:
$$(\rR^pi^!)j_*(\mathbf{G}_{m,K}) = 0,\ \ p\ge 0.$$
Indeed, $j_*$ preserves injectives, and hence by Grothendieck's composition of functors sequence:
$$(\rR^pi^!)(\rR^qj_*)(\mathbf{G}_{m,K}) \Rightarrow \rR^{p+q}(i^!j_*)(\mathbf{G}_{m,K}).$$
Next we use the fact that $i^!j_* = 0$, and deduce the claim from Proposition \ref{comp1}.
Now it suffices to show
$$(\rR^pi^!)i_*(\bZ) =\left\{\begin{array}{cc}
\bZ & \textup{if}\ p=0\\
0 & \textup{if}\ p\neq 0.
\end{array}\right.$$
Indeed, $i_*$ is exact and preserves injectives, and then we have:
$$(\rR^pi^!)i_* = \rR^p(i^!i_*) = \rR^p(\textup{id}).$$
This gives the result.
\end{proof}
\section{Homological compatibilities}
To avoid disrupting the exposition with long compatibility checks that may not be so interesting, we
place some of those verifications in this appendix.
\subsection{Global to local restriction maps}
\begin{prop}\label{fpm} The isomorphism $f^p_M$ in $(\ref{fpmmap})$ for $p \ge 3$ is the natural restriction map.
\end{prop}
\begin{proof} To determine $f^p_M$, we consider it as a special case of a much more general problem,
going far beyond isomorphisms and locally constant sheaves.
In Lemma \ref{support} we
computed the $\delta$-functor $\rH^{\bullet}_{\widetilde{v}}(\overline{U}, G)$ in a general abelian sheaf $G$
on $\overline{U}$. The local cohomology sequence
for $U \hookrightarrow \overline{U}$ provides connecting maps
$$\rH^p(U, G|_U) \rightarrow \bigoplus_{\widetilde{v}} \rH^{p+1}_{\widetilde{v}}(\overline{U}, G)$$
for $p \ge 0$ that are $\delta$-functorial by Remark \ref{localsign} (where
the sign problem doesn't arise at real points since
Tate cohomology at real points is 2-torsion and hence insensitive to signs!).
In view of how $f^p_M$ was built for $p \ge 3$, it therefore
suffices to show that for all such $G$ and all $p \ge 1$ the composite map
$$\rH^p(U, G|_U) \rightarrow \bigoplus_{\widetilde{v}} \rH^{p+1}_{\widetilde{v}}(\overline{U}, G) \simeq \bigoplus_{\widetilde{v}} \rH^p(I_v, G_{\eta})$$
is the natural restriction map.
By standard dimension-shifting arguments (using crucially that we allow {\em arbitrary} $G$ so that we may avail
ourselves of the existence of enough injectives in the category of all abelian sheaves on $\widetilde{U}$),
the $\delta$-functoriality in Remark \ref{localsign} (again, no sign problem with Tate cohomology
at real places) and the $\delta$-functoriality in Lemma \ref{support}
reduce this task to the case $p=1$. We need to push one step further into degree 0, as follows.
In Lemma \ref{support} we identified the $\delta$-functor $\rH^{\bullet}_{\widetilde{v}}(\overline{U}, G)$
in degree 1 with the quotient
$$\coker(\varphi_{\widetilde{v}}: G_{\widetilde{v}} \rightarrow \rH^0(I_v, G_{\eta}))$$
of $\rH^0(I_v, G_{\eta})$. Moreover, by {\em construction}, the $\delta$-functor structure on the
concrete side of the statement of Lemma \ref{support}
is induced in degree 0 by the usual connecting map in group cohomology.
Hence, running the same argument once again reduces our task to checking for
{\em arbitrary} abelian sheaves $G$ on $\overline{U}$ that the composite map
$$f_G:\rH^0(U,G|_U) \rightarrow \bigoplus_{\widetilde{v}} \rH^1_{\widetilde{v}}(\overline{U}, G) \simeq
\bigoplus_{\widetilde{v}} \coker(\varphi_{\widetilde{v}}: G_{\widetilde{v}} \rightarrow \rH^0(I_v, G_{\eta}))$$
is equal to the composition of the natural restriction map
$\rH^0(U, G|_U) \rightarrow \bigoplus_{\widetilde{v}} \rH^0(I_v, G_{\eta})$ followed by the natural quotient map.
We may and do focus on describing
the composition $f_{G,v}$ of $f_G$ with projection to the $v$-component for a {\em fixed} choice of real place $v$ of $K$.
It is easy to check from the construction that $f_{G,v}$ is compatible with pullback to a connected \'etale $U$-scheme $U'$ arising
from a number field $K'/K$ inside $K_v^{\rm{alg}}$ (equipped with the induced replace place
over $v$), and the concrete map we want to prove coincides with $f_{G,v}$ is also
compatible with such pullback. Hence, we can pass to the limit over such $U'$ to reduce
to the analogous compatibility question when $U$ is replaced with $\Spec(K_v^{\rm{alg}})$,
we use the real place $v'$ on
the real closed field $K_v^{\rm{alg}}$ arising from its tautological inclusion into the completion $K_v = \mathbf{R}$,
and $\rH^{\bullet}_{\widetilde{v}}(\overline{U}, \cdot)$ is replaced with the derived functor of
the analogue of $\rH^0_{\widetilde{v}}(\overline{U}, \cdot)$ for $\Spec(K_v^{\rm{alg}})$ equipped with $v'$.
Note that $I_v$ is naturally identified with the Galois group of $K_v^{\rm{alg}}$, so
the role of $G_{\eta}^{I_v}$ collapses to the global sections over $K_v^{\rm{alg}}$!
Our new setup is as follows. We have a (real closed) field $E$, the category $\mathbf{Ab}_Y$ of abelian \'etale sheaves on $Y = \Spec(E)$,
and an ``extended'' category $\mathbf{Ab}_{\overline{Y}}$ consisting of triples $G = (G_0, G_{\eta}, \varphi)$ where $G_{\eta} \in \mathbf{Ab}_Y$,
$G_0$ is an abelian group,
and $\varphi:G_0 \rightarrow G_{\eta}(E)$ is a homomorphism into the global sections. By the analogue of Example \ref{gsec}, the functor
$\Gamma(\overline{Y},\cdot)$ assigns to $G$ the fiber product $G_{\eta}(E) \times_{G_{\eta}(\overline{E}), \varphi} G_0 = G_0$
and the restriction functor $\Gamma(\overline{Y}, G) \rightarrow \Gamma(Y, G)$ is $\varphi:G_0 \rightarrow G_{\eta}(E)$. Hence,
the role of $\rH^0_{\widetilde{v}}(U,\cdot)$ is played by the functor $\rH^0_{\ast}: G \rightsquigarrow \ker \varphi$.
Our task is to prove that via the isomorphism
$\rH^1_{\ast}(G) \simeq \coker(\varphi)$ provided by (the proof of) the analogue of Lemma \ref{support} for the present setting, the natural
connecting map $G_{\eta}(E) = \Gamma(Y, G) \rightarrow \rH^1_{\ast}(G) \simeq \coker(\varphi)$ is {\em exactly} the canonical quotient map.
Consider the initial short exact sequence $0 \rightarrow G \rightarrow I \rightarrow N \rightarrow 0$ arising from an injective resolution of $G$.
This yields a commutative diagram with exact rows and columns as indicated:
$$\xymatrix{
& & 0 & &\\
0 \ar[r] & \ker \varphi_N \ar[r] & N_0 \ar[u] \ar[r]^-{\varphi_N} & N_{\eta}(F) & \\
0 \ar[r] & \ker \varphi_I \ar[u]_-{f} \ar[r] & I_0 \ar[u] \ar[r]^-{\varphi_I} & I_{\eta}(F) \ar[r] \ar[u] & 0\\
0 \ar[r] & \ker \varphi_G \ar[r] \ar[u] & G_0 \ar[r]^-{\varphi_G} \ar[u] & G_{\eta}(F) \ar[u] & \\
& 0 \ar[u] & 0 \ar[u] & 0 \ar[u] & }$$
Applying the snake lemma to the diagram given by the right two columns provides a connecting isomorphism
$\coker f \simeq \coker \varphi_G$ that is exactly the isomorphism $\rH^1_{\ast}(G) \simeq \coker \varphi_G$
provided by the proof of Lemma \ref{support}. Applying the snake lemma to the diagram given by the upper two rows
provides a connecting isomorphism $\coker \varphi_G \simeq \coker f$ whose composition with
the natural quotient map $G_{\eta}(F) \rightarrow \coker \varphi_G$ is the definition of the connecting map
$G_{\eta}(F) \rightarrow \rH^1_{\ast}(G)$ in the ``local cohomology'' exact sequence.
Our task is now reduced to showing that the connecting isomorphisms that we have just constructed
in opposite directions between $\coker \varphi_G$ and $\coker f$ (one using the right two columns, the other using the upper two rows)
are inverse to each other. As an assertion in homological algebra for any such commutative diagram with the indicated exactness
in the rows and columns, this is a trivial diagram chase (traversing in opposite directions along
the same zig-zag staircase path between $G_{\eta}(F)$ and
$\ker \varphi_N$ passing through $I_{\eta}(F)$, $I_0$, and $N_0$).
\end{proof}
\subsection{Properties of some pairings}\label{paircomp}
For a dense open $U \subset X = \Spec(\ccO_K)$ and an abelian sheaf $F$ on $U_{\et}$,
in \S\ref{prepsec} we defined general pairings
\begin{equation}\label{genpair}
{\rH}_c^p(U, F)\times \textup{Ext}_U^{3-p}(F,\mathbf{G}_{m,U}) \to \bQ/\bZ
\end{equation}
naturally in $F$ for any $p \in \bZ$, assuming $F$ is constructible when $p < 0$.
This relied on Ext-pairings on $\overline{X}$ for $p \ge 0$ (using
the extension-by-zero of $F$ to $X$) and an entirely different procedure with
Tate cohomology when $p < 0$ (using Corollary \ref{euler char disguise}
and (\ref{tatepair})). In this section we establish some properties of (\ref{genpair}) that
underlie how it is analyzed in proofs (such as for Artin--Verdier duality).
The proof of Proposition \ref{yoneda} addresses the $\delta$-functoriality of (\ref{genpair}) except for the case
that links up the two aspects of how the pairing is defined (i.e., relating $p \ge 0$ and $p < 0$), which we now address:
\begin{prop}\label{dyoneda} The pairing {\rm{(\ref{genpair})}} is $\delta$-functorial from degree $-1$ to degree $0$
for constructible abelian $F$.
\end{prop}
\begin{proof}\footnote{To be filled in.}
\end{proof}
Now we turn to refinements of the isomorphism in Lemma \ref{cohomfinite}: defining it in all degrees $p \in \bZ$ (not just $p \ge 0$),
establishing its $\delta$-functoriality, and especially proving that it is compatible with pairings as in (\ref{genpair}) on $U$ and $U'$
via the composite map
$${\rm{Ext}}^i_{U'}(F', \mathbf{G}_{m,U'}) \rightarrow {\rm{Ext}}^i_U(f_{\ast}(F'), f_{\ast}(\mathbf{G}_{m,U'}))
\rightarrow {\rm{Ext}}^i_U(f_{\ast}(F'), \mathbf{G}_{m,U})$$
(first step using exactness of $f_{\ast}$, and second step using the evident ``norm'' map); this composite map
is an isomorphism for constructible $F'$ by \cite[Thm.\,2.7]{Mazur} (see Remark \ref{normpf}) but that is not
relevant to the compatibility assertion. This compatibility
with finite pushforward is essential for bootstrapping the proof of Artin--Verdier duality from the totally imaginary case.
The definition of (\ref{genpair}) rests on the global trace isomorphism
${\rm{Tr}}_U: \rH^3_c(U, \mathbf{G}_{m,U}) \simeq \bQ/\bZ$ whose compatibility with shrinking $U$ was addressed in
Remark \ref{tropen}. A further compatibility we shall require involves comparing local and global trace maps,
in the sense of a commutative diagram
\begin{equation}\label{localglobal}
\xymatrix{\textup{Br}(K_v)\ar[d]\ar[r]^{\ \textup{inv}_v} & \bQ/\bZ\ar@{=}[d]\\
\rH^3_c(U,\mathbf{G}_m)\ar[r]_-{\simeq}^-{{\rm{Tr}}_U} & \bQ/\bZ}
\end{equation}
for all points $v \in \overline{X} - U$. This will be largely a matter of review of definitions (such as of
${\rm{Tr}}_U$ in terms of ${\rm{Tr}}_X$), and is especially useful for finite $v$ (i.e., $v \in X - U$)
since in such cases ${\rm{inv}}_v$ is an isomorphism (thereby making the left side of (\ref{localglobal}) an isomorphism).
We first define the map along the left side of this diagram
before we prove that it commutes. It is convenient to initially work with a general abelian sheaf
$F$ on $U$ rather than specifically with $\mathbf{G}_{m,U}$.
Letting $j:U \hookrightarrow X$ be the open immersion,
the local cohomology sequence (\ref{modlocal}) for $j_{!}F$ relative to the dense open inclusion of $U$ into $X$ has the form
\begin{equation}\label{localU}
\cdots\to\bigoplus_{v\in \overline{X} - U}\rH^p_v(X, j_{!}(F))\to \rH^p_c(U, F)\to \rH^p(U,F)\to\cdots
\end{equation}
where the local term at real $v$ coincides with $\rH^p_v(U, F)$ as in Definition \ref{localcoho} (using (\ref{htate})).
For any $v \in \overline{X} - U$, the $v$-term in this local cohomology sequence
is $\rH^{p-1}(K_v, F_{\eta})$ by arguments treating real points
and points of $X-U$ separately; this is shown the proof of Theorem \ref{finite ramification} in a special case,
and for the convenience of the reader we now reproduce the argument in general.
If $v$ is real then by Definition \ref{localcoho} we have
$$\rH^p_v(X, j_!(F)) = {\rH}_T^{p-1}(I_s, F_{\eta}) = \rH^{p-1}(K_s, F_{\eta})$$
(where the final equality is our {\em definition} of the notation $\rH^{\bullet}(K_s, \cdot)$ for real $s$, coinciding
with usual Galois cohomology in positive degrees).
For $x \in X - U$ and $f_x : X_x^{\rm h}\to X$ the henselization of $X$ at $x$, by Lemma \ref{henselization} we have
$$\rH^p_x(X, j_!(F)) = \rH^p_x(X_x^{\rm h}, f^*_xj_!(F)).$$
The latter group can be computed from the local cohomology sequence for a henselian discrete valuation ring $A$ (such
as that of $X$ at $x$), giving an isomorphism
$$\rH^p_x(X,j_!(F)) \simeq\rH^{p-1}(D_x, F_{\eta}) = \rH^{p-1}(K_x, F_{\eta})$$
because of the {\em vanishing} of the the flanking terms $\rH^p(\Spec(A), j'_{!}(G))$ for the open immersion $j': \Spec(L) \hookrightarrow \Spec(A)$
of the generic point and an \'etale sheaf $G$ on the generic point (since the degree-$p$
cohomology of a sheaf on $\Spec(A)$ coincides with the degree-$p$ cohomology of the fiber sheaf at the closed point due to $A$
being henselian, so it vanishes when that fiber sheaf is 0).
The left vertical arrow in (\ref{localglobal}) is {\em defined} to be the ``$v$-summand'' of the resulting map
$$\bigoplus_{v \in \overline{X} - U} {\rm{Br}}(K_v) =
\bigoplus_{v\in \overline{X} - U}\rH^2_v(X, j_{!}(\mathbf{G}_{m,U})) \to \rH^3_c(U,\mathbf{G}_m).$$
Via the definition of ${\rm{Tr}}_U$ in terms of ${\rm{Tr}}_X$
and the computation of ${\rH}^3(\overline{X}, \mathbf{G}_{m, \overline{X}})$ performed in \S\ref{cohomology multiplicative},
the invariant map $\textup{inv}_v : \textup{Br}(K_v)\to\bQ/\bZ$ makes (\ref{localglobal}) commute.
This local-global compatibility will have a very useful consequence for the global trace relative to finite
pushforward, as we shall see in Example \ref{normex}.
Now let $K'/K$ be a finite extension, and $U'$ the $K'$-normalization of $U$, with $\nu: U'\to U$ the
resulting finite surjection. For any abelian sheaf $A$ on $U'_{\rm\acute{e}t}$ and $p \in \bZ$ we aim to construct an isomorphism
$$\omega_{\nu,A, p}: \rH^p_c(U, \nu_*A)\to\rH^p_c(U', A)$$
that is $\delta$-functorial in $A$.
For $X' := \Spec(\ccO_{K'})$, the open immersion $j':U'\to X'$, and the normalization map
$\nu_X: X' \rightarrow X$, the following diagram is cartesian:
\begin{displaymath}
\xymatrix{U'\ar[d]_{\nu}\ar[r]^{j'} & X'\ar[d]^{\nu_X}\\
U\ar[r]_{j} & X}
\end{displaymath}
Since the natural map $j_!\nu_* \rightarrow \nu_{X\ast}j_!'$ is an isomorphism (as can be checked on stalks) and $\nu_X$
is finite, we obtain a composition of natural maps
%\begin{flalign*}
$$\rH^p_c(U, \nu_*A ) := \widehat{\rH}^p(X, j_!\nu_*A) = \widehat{\rH}^p(X, \nu_{X*} j'_!A)
\simeq \widehat{\rH}^p(X', j'_!A) =: \rH^p_c(U', A)$$
(using Lemma \ref{cohomfinite} for the third identification).
For $p\ge 0$, we define $\omega_{\nu, A, p}$ to be this composite isomorphism.
Now consider $p< 0$, so by definition ${\rH}^p_c(U, \nu_*A) = \bigoplus_{v \in X_{\infty}}\rH^{p-1}_T(I_v, (\nu_*A)_{\eta})$.
Denoting by $\eta' \in U'$ the generic point, we have
${\rH}^p_c(U', A) = \bigoplus_{w \in X'_{\infty}}\rH^{p-1}_T(I_w, A_{\eta'})$.
Tate cohomology in degree $< -1$ is positive-degree group homology, and for any real place $v$ the
\'etale sheaf $(\nu_{\ast} A)_{\eta}$ viewed as an $I_v$-module is the pushforward of $A$ through
$$\coprod_{v'|v} \Spec(K'_{v'}) = \Spec(K' \otimes_K K_v) \rightarrow \Spec(K_v).$$ The contributions from complex places over $v$ are
induced $I_v$-modules, so their higher group homology vanishes. The contributions to this pushforward from real places $w$ of $K'$
over $v$ contribute $A$ viewed as an $I_w$-module. Thus, for $p < 0$ we have natural isomorphisms
$$\omega_{\nu,A, p}: \rH^p_c(U, A) := \rH^{p-1}_T(I_v, (\nu_{\ast} A)_{\eta}) = \bigoplus_{w \in X'_{\infty}} \rH^{p-1}_T(I_w, A_{\eta'}) =:
\rH^p_c(U', A)$$
(dropping the vanishing contribution from complex places of $K'$ over real places of $K$!).
\begin{lemma}\label{domega} The isomorphisms $\omega_{\nu, A, p}$ for $p \in \bZ$ are $\delta$-functorial in $A$.
\end{lemma}
\begin{proof}
We need to check compatibility with connecting maps $\rH^p_c \rightarrow \rH^{p+1}_c$
arising from every short exact sequence of abelian sheaves
$$0\to A'\to A\to A''\to 0$$
on $U'$ and its short exact $\nu$-pushforward on $U$. The case $p < -1$ is a trivial assertion
in higher group homology. For $p \ge 0$,
since $j_!\nu_*\to \nu_{X*}j_!'$ is an isomorphism of exact functors and the isomorphism
$\rH^{\bullet}(\overline{X}, \overline{\nu_X}_{\ast}(\cdot)) \simeq \rH^{\bullet}(\overline{X}', \cdot)$
(as in the proof of Lemma \ref{cohomfinite}) is visibly $\delta$-functorial, a review of how $\widehat{\rH}^{\bullet}(X, \cdot)$
and $\widehat{\rH}^{\bullet}(X', \cdot)$ are
made into $\delta$-functors in \S\ref{modet} gives the result.
It remains to consider $p=-1$. By definition of $\rH^p_c(U, \cdot)$ and $\rH^p_c(U', \cdot)$
we reduce to the case $U = X$ (so $U' = X'$). Let
$$0\to A'\to A\to A''\to 0$$
be a short exact sequence of abelian sheaves on $X'$, so we get a diagram of long exact sequences
\begin{displaymath}
\xymatrix{ \cdots\ar[r] & \bigoplus_{v}\rH_T^{-2}(I_v, (\nu_*A'')_{\eta})\ar[d]^{\omega_{\nu, A'', -1}}\ar[r] & \widehat{\rH}^0(X, \nu_*A')\ar[r]^{\alpha}\ar[d]^{\omega_{\nu, A', 0}} & \widehat{\rH}^0(X, \nu_*A)\ar[r]\ar[d]^{\omega_{\nu, A, 1}} & \cdots\\
\cdots\ar[r] & \bigoplus_{v}\bigoplus_{w\mid v}\rH_T^{-2}(I_w, A''_{\eta'})\ar[r] & \widehat{\rH}^0(X', A')\ar[r]_{\beta} & \widehat{\rH}^0(X', A)\ar[r] & \cdots}
\end{displaymath}
where $v$ and $w$ vary through real places.
Defining $M := \ker(\alpha)$ and $M' := \ker(\beta)$, it suffices to check commutativity of
the resulting diagram
\begin{displaymath}
\xymatrix{ \bigoplus_{v}\rH_T^{-2}(I_v, (\nu_*A'')_{\eta})\ar[d]_{\omega_{\nu, A'', -1}}\ar[r] & M \ar[d]^{\omega_{\nu, A',0}}\\
\bigoplus_{v}\bigoplus_{w\mid v}\rH_T^{-2}(I_w, A''_{\eta'})\ar[r] & M'}
\end{displaymath}
As we saw in \S\ref{modet} when making $\widehat{\rH}^{\bullet}(X, \cdot)$ into a $\delta$-functor,
by design of $\widehat{(\cdot)}$ we have
$$M \subset \bigoplus_{v} \rH^{-1}_T(I_v, (\nu_*A')_{\eta}),\,\,\, M' \subset \bigoplus_{v}\bigoplus_{w\mid v} \rH^{-1}_T(I_w, A'_{\eta'})$$
intertwining $\omega_{\nu, A', 0}$ with the naturally induced map between $\rH^{-1}_T$'s (via the description of
$(\nu_{\ast} A')_{\eta}$ as a Galois module for $K_v$ using pushforward through
$\Spec(K' \otimes_K K_v) \rightarrow \Spec(K_v)$ and the vanishing of $\rH^{\bullet}_T(I_v, \cdot)$ on induced $I_v$-modules
arising from complex places of $K'$ over $v$).
Thus, it suffices to check the commutativity of the diagram
\begin{displaymath}
\xymatrix{\bigoplus_{v}\rH_T^{-2}(I_v, (\nu_*A'')_{\eta})\ar[d] \ar[r] & \bigoplus_{v}\rH^{-1}_T(I_v, (\nu_*A')_{\eta}) \ar[d]\\
\bigoplus_{v}\bigoplus_{w\mid v}\rH_T^{-2}(I_w, A''_{\eta'})\ar[r] & \bigoplus_{v}\bigoplus_{w\mid v}\rH^{-1}_T(I_w, A'_{\eta'})}
\end{displaymath}
where the maps are the obvious ones. Hence, for each real place $w$ of $K'$
and its induced real place $v$ of $K$ it remains to note that the diagram of group homologies
\begin{displaymath}
\xymatrix{\rH_1(I_v, (\nu_*A'')_{\eta})\ar[d] \ar[r] & \rH_0(I_v, (\nu_*A')_{\eta}) \ar[d]\\
\rH_1(I_w, A''_{\eta'})\ar[r] & \rH_0(I_w, A'_{\eta'})}
\end{displaymath}
(using evident vertical maps, via the isomorphism $I_w \simeq I_v$) obviously commutes.
\end{proof}
\begin{eg}\label{normex}
The natural ``norm'' map $\nu_*\mathbf{G}_{m,U'}\to\mathbf{G}_{m,U}$ yields a map
$$\textup{N}_{U'/U}: \rH^3_c(U',\mathbf{G}_{m,U'})\to\rH^3_c(U,\mathbf{G}_{m,U})$$
given by composing the inverse of $\omega_{\nu, \mathbf{G}_{m,U'}, 3}:
\rH^3_c(U,\nu_*\mathbf{G}_{m,U'})\xrightarrow{\simeq}\rH^3_c(U',\mathbf{G}_{m,U'})$ with the map
$\rH^3_c(U,\nu_*\mathbf{G}_{m,U'})\to\rH^3_c(U,\mathbf{G}_{m,U})$ induced by $\rH^3_c(U, \cdot)$-functoriality.
It will be important that this is compatible with global trace maps:
$${\rm{Tr}}_U \circ {\rm{N}}_{U'/U} = {\rm{Tr}}_{U'}.$$
To prove that identity, we shall use the local-global compatibility in (\ref{localglobal}) to reduce the problem to a fact
in local class field theory. For dense open $V \subset U$ and its preimage $V' \subset U'$, it is easy to see from the definitions of the
isomorphisms
\begin{equation}\label{exc3}
\rH^3_c(V, \mathbf{G}_{m,V}) \simeq \rH^3_c(U, \mathbf{G}_{m,U}), \,\,\,
\rH^3_c(V', \mathbf{G}_{m,V'}) \simeq \rH^3_c(U', \mathbf{G}_{m,U'})
\end{equation}
that they intertwine ${\rm{N}}_{U'/U}$ and ${\rm{N}}_{V'/V}$. The
isomorphisms (\ref{exc3}) also respectively intertwine global traces by Remark \ref{tropen}, so to prove the desired compatibility of
${\rm{N}}_{U'/U}$ with global traces it is equivalent to do the same for ${\rm{N}}_{V'/V}$.
Hence, we may assume that $U \ne X$, so there exists a point $x \in X - U$.
Rather generally, for any $v \in X - U$ (which exists!) and $w \in X' - U'$ over $v$, we claim that the diagram
\begin{equation}\label{normBr}
\xymatrix{\textup{Br}(K'_w)\ar[d]_{\textup{Norm}}\ar[r] & \rH^3_c(U', \mathbf{G}_{m,U'})\ar[d]^{\textup{N}_{U'/U}}\\
\textup{Br}(K_v)\ar[r] & \rH^3_c(U, \mathbf{G}_{m,U})}
\end{equation}
commutes, where the left vertical map is induced by the usual norm map between local Brauer groups (defined similarly
to ${\rm{N}}_{U'/U}$, using Shapiro's Lemma) and
the horizontal maps are the ones defined in (\ref{localglobal}). Once this is proved,
since we have {\em isomorphisms} along the horizontal sides of (\ref{normBr})
(due to the commutativity of (\ref{localglobal})) it will follow that the compatibility of ${\rm{N}}_{U'/U}$ with global traces
is reduced to the compatibility of norm maps between Brauer groups with local invariant maps
in local class field theory. This local compatibility is a well-known fact:
see Prop.\,1(ii) in \S2 of Chapter XI in Serre's book ``Local Fields'' (applicable
since the norm map between Brauer groups is an instance of ``corestriction'' in Galois cohomology).
To check commutativity of (\ref{normBr}), if we go back to how the horizontal maps in (\ref{normBr})
are {\em defined} via local cohomology sequences and we use that
$$X' \times_X X_v^{\rm{h}} \simeq \coprod_{w|v} {X'}_w^{\rm{h}}$$
then we are reduced to proving the commutativity of
\begin{displaymath}
\xymatrix{\rH^2({K'}^{\rm{h}}_w, \mathbf{G}_m)\ar[d]\ar[r]^-{\simeq} & \rH^3_w(\Spec({\ccO}_{X',w}^{\rm h}), j_{w!}(\mathbf{G}_{m}))
\ar[d]^{\textup{N}}\\
\rH^2(K_v^{\rm{h}}, \mathbf{G}_m)\ar[r]_-{\simeq} & \rH^3_v(\Spec({\ccO}_{X,v}^{\rm h}), j_{v!}(\mathbf{G}_{m}))}
\end{displaymath}
where $j_v: \Spec(K_v^{\rm{h}}) \hookrightarrow \Spec(\ccO_{X,v}^{\rm{h}})$ is the open generic point
and likewise for $j_w$. This final commutativity is an immediate consequence of the compatibility of the local cohomology sequence with
respect to {\em finite} pushforward and the functoriality of the local cohomology sequence in the sheaf
(such as with respect to the norm map $\nu_{v\ast}(j_{w!}\mathbf{G}_m) \rightarrow j_{v!}\mathbf{G}_m$
defined by applying $j_{v!}$ to the ``$w$-component'' of the
$K_v^{\rm{h}}$-pullback of the norm $\nu_{\ast}\mathbf{G}_{m,U'} \rightarrow \mathbf{G}_{m,U}$).
\end{eg}
The following refinement of Lemma \ref{domega} is crucial in the proof of Artin--Verdier duality beyond the totally complex case:
\begin{prop}\label{Yoneda comp} For any abelian sheaf $F'$ on $U'$, the natural $\delta$-functorial isomorphisms
$$\omega_{\nu, F', p}: \rH^p_c(U, \nu_*(F')) \xrightarrow{\simeq}\rH^p_c(U', F')$$
are compatible with the Yoneda pairing {\rm{(\ref{genpair})}} and its analogue on $U'$ for all $p \in \bZ$,
assuming $F'$ is constructible when $p < 0$.
\end{prop}
The Yoneda pairing for $p < 0$ is only defined for constructible $F'$.
\begin{proof}
In Example \ref{normex} we defined a natural ``norm'' map
$$\textup{N}_{U'/U}: \rH^3_c(U',\mathbf{G}_{m,U'})\to\rH^3_c(U,\mathbf{G}_{m,U})$$
and proved that it is compatible with global traces. To define what it means to say that
$\omega_{\nu, F, p}$ is compatible with the Yoneda pairing, we shall use ${\rm{N}}_{U'/U}$ and
the natural composite map
$$\Ext^q_{U'}(F',\mathbf{G}_{m,U'})\to \Ext^q_U(\nu_{\ast}F', \nu_{\ast}\mathbf{G}_{m,U'}) \to \Ext^q_U(\nu_*F', \mathbf{G}_{m,U})$$
(with $q = 3-p$) that we also call the ``norm'' and denote as $\textup{N}$.
We claim commutativity (in the evident sense) of the diagram
\begin{displaymath}
\xymatrix{\rH^p_c(U', F') & \times & \Ext^{3-p}_{U'}(F',\mathbf{G}_{m,U'})\ar[d]_{\textup{N}} \ar[r] & \rH^3_c(U',\mathbf{G}_{m,U'})\ar[d]^{\textup{N}}\ar[r]^-{\textup{Tr}}_-{\simeq} & \bQ/\bZ\ar@{=}[d]\\
\rH^p_c(U, \nu_*F')\ar[u]_-{\simeq}^{\omega_{\nu, F', p}} & \times & \Ext^{3-p}_{U}(\nu_*F',\mathbf{G}_{m,U})\ar[r] & \rH^3_c(U,\mathbf{G}_{m,U})\ar[r]^{\ \ \ \textup{Tr}}_{\ \ \ \simeq} & \bQ/\bZ}
\end{displaymath}
where the abelian sheaf $F'$ is assumed to be constructible when $p < 0$.
Commutativity of the right square allows us to reduce our considerations to checking commutativity of
\begin{displaymath}
\xymatrix{\rH^p_c(U', F') & \times & \Ext^{3-p}_{U'}(F',\mathbf{G}_{m,U'})\ar[d]_{\textup{N}} \ar[r] & \rH^3_c(U',\mathbf{G}_{m,U'})\ar[d]^{\textup{N}}\\
\rH^p_c(U, \nu_*F')\ar[u]_{\ \ \ \simeq}^{\omega_{\nu, F', p}} & \times & \Ext^{3-p}_{U}(\nu_*F',\mathbf{G}_{m,U})\ar[r] & \rH^3_c(U,\mathbf{G}_{m,U})}
\end{displaymath}
where $F'$ is any abelian sheaf on $U'$, with $F'$ constructible when $p < 0$.
For $p\ge 4$ the Ext's vanish and there is nothing to do.
Next, consider $0 \le p \le 3$. In view of how the Yoneda pairing and $\rH^p_c(U, \cdot)$ (for $p \ge 0$) are {\em defined} by using
an extension by zero to $X'$, the problem for $F'$ is equivalent to the same for
its extension by zero to $X'$. Hence, we may assume $U=X$.
The Yoneda pairing over $X$ is initially valued in $\rH^3(\overline{X}, \overline{j}_{\ast}(\mathbf{G}_{m,X}))$
(with $\overline{j}:X \rightarrow \overline{X}$
the canonical ``morphism''), and is then made to take values in $\rH^3_c(X, \mathbf{G}_{m,X}) =
\rH^3(\overline{X}, \mathbf{G}_{m, \overline{X}})$ via inverting the {\em isomorphism}
obtained by applying $\rH^3(\overline{X}, \cdot)$ to the map
$\mathbf{G}_{m, \overline{X}} \rightarrow \overline{j}_{\ast}(\mathbf{G}_{m,X})$
adjoint to the natural isomorphism $\overline{j}^{\ast}\mathbf{G}_{m, \overline{X}} \simeq \mathbf{G}_{m,X}$.
Using $\overline{j}':X' \hookrightarrow \overline{X}'$, we get a natural ``norm'' homomorphism
$$\overline{\rm{N}}:\overline{\nu}_{\ast}(\overline{j'}_{\ast}\mathbf{G}_{m,X'}) =
\overline{j}_{\ast}(\nu_{\ast}\mathbf{G}_{m,X'}) \rightarrow \overline{j}_{\ast}
\mathbf{G}_{m,X}$$
over $\overline{X}$, and applying $\rH^3(\overline{X}, \cdot)$ defines the left side of the diagram
$$
\xymatrix{\rH^3(\overline{X'}, \overline{j'}_*\mathbf{G}_{m,X'}) \ar[d]^{\textup{N}} & \rH^3_c(X', \mathbf{G}_{m,X'}) \ar[l]_-{\simeq}
\ar[d]^{\textup{N}} \\
\rH^3(\overline{X}, \overline{j}_*\mathbf{G}_{m,X}) & \rH^3_c(X,\mathbf{G}_{m,X}) \ar[l]^-{\simeq}
}
$$
that is easily verified to commute. Thus, we are reduced to proving the commutativity of the diagram
\begin{displaymath}
\xymatrix{\Ext^p_{\overline{X'}}(\underline{\bZ}, \widehat{F'}) & \times & \Ext^{3-p}_{\overline{X'}}(\widehat{F'},\overline{j'}_*\mathbf{G}_{m,X'}) \ar[d]_{\textup{N}} \ar[r] & \Ext^3_{\overline{X'}}(\underline{\bZ},\overline{j'}_*\mathbf{G}_{m,X'})\ar[d]^{\textup{N}}\\
\Ext^p_{\overline{X}}(\underline{\bZ}, \overline{\nu}_*\widehat{F'})\ar[u]_{\ \ \ \simeq}^{\omega_{\nu, F',p}} & \times & \Ext^{3-p}_{\overline{X}}(\overline{\nu}_*\widehat{F'},\overline{j}_*\mathbf{G}_{m,X})\ar[r] &
\Ext^3_{\overline{X}}(\underline{\bZ},\overline{j}_*\mathbf{G}_{m,X})}
\end{displaymath}
whose vertical maps are defined via composition in the derived categories of abelian sheaves
on $\overline{X'}$ and $\overline{X}$. The inverse to $\omega_{\nu, F', p}$ is the map that applies
the exact (injective-preserving) functor $\overline{\nu}_{\ast}$ to a morphism $\underline{\bZ} \rightarrow F'[p]$
in the derived category over $\overline{X}'$, and the two other maps are computed by applying
$\overline{\nu}_{\ast}$ and then composing with (a shift of) $\overline{\rm{N}}$, so the desired commutativity is just a special case of the
functoriality of $\overline{\nu}_{\ast}$ between the derived categories.
%
%The rightmost square commutes by functoriality and the fact that the following diagram is cartesian:
%\begin{displaymath}
%\xymatrix{X'\ar[r]^{\overline{j'}}\ar[d]_{\nu} & \overline{X'}\ar[d]^{\overline{\nu}}\\
% X\ar[r]^{\overline{j}} & \overline{X}}
%\end{displaymath}
%We have $\overline{\nu}_*\overline{j'}_* = \overline{j}_*\nu_*$ (as can be checked on stalks).
%
%Commutativity of the left part of the diagram is seen, to sum up before checking it explicitly, by means of the description of the $\Ext$ groups as groups of homotopy classes of maps, the corresponding description of the pairing, by exactness of $\overline{\nu}_* : \Ab_{\overline{X'}_{\rm\acute{e}t}}\to \Ab_{\overline{X}_{\rm\acute{e}t}}$ and functoriality of the trace map.
%
%Of course, such commutativity has nothing to do with $\mathbf{G}_m$, and we therefore set ourselves in the generality of having abelian sheaves $A$ on $\overline{X}$, and $B, B'$ on $\overline{X'}$, claiming the following diagram commutes:
%
%\begin{displaymath}
%\xymatrix{\Hom_{\overline{X'}}(\overline{\nu}^*A, B) & \times & \Hom_{\overline{X'}}(B,B') \ar[d]_{\overline{\nu}_*} \ar[r] & \Hom_{\overline{X'}}(\overline{\nu}^*A,B')\ar[d]^{\simeq}\\
% \Hom_{\overline{X}}(A, \overline{\nu}_*B)\ar[u]_{\simeq} & \times & \Hom_{\overline{X}}(\overline{\nu}_*B,\overline{\nu}_*B') \ar[r] & \Hom_{\overline{X}}(A,\overline{\nu}_*B')}
%\end{displaymath}
%
%the pairings being given by composition.
%
%Pick $\varphi\in\Hom_{\overline{X}}(A,\overline{\nu}_*B)$ and $\psi\in\Hom_{\overline{X'}}(B,B')$.
%$\varphi$ corresponds to a unique $\varphi'\in\Hom_{\overline{X}}(\overline{\nu}^*A, B)$ by the $(\overline{\nu}^*,\overline{\nu}_*)$ adjunction. Let us call $\varepsilon$ the natural unit map $A\to \overline{\nu}_*\overline{\nu}^*A$. $\varphi$ and $\varphi'$ satisfy:
%$$\overline{\nu}_*(\varphi') \circ\varepsilon = \varphi$$
%by adjointness. It follows that the composition:
%$$A\xrightarrow{\varepsilon} \overline{\nu}_*\overline{\nu}^*A\xrightarrow{\overline{\nu}_*(\psi\circ\varphi')} \overline{\nu}_*B'$$
%equals $\overline{\nu}_*(\psi)\circ\varphi : A\to\overline{\nu}_*B'$. It follows that the above diagram commutes.
%
%Now we fix abelian sheaves $A$ on $\overline{X}$ and $B$ on $\overline{X'}$. Let $A\to I^{\bullet}$ and $B\to J^{\bullet}$ be injective resolutions. We set up the following diagram:
%
%\begin{displaymath}
%\xymatrix{\Hom_{\overline{X'}}(\underline{\bZ}, J^{\bullet}[m]) & \times & \Hom_{\overline{X'}}(J^{\bullet}[m], \overline{\nu}^*I^{\bullet}[m+n])\ar[d]^{\overline{\nu}_*}\ar[r] & \Hom_{\overline{X'}}(\underline{\bZ}, \overline{\nu}^*I^{\bullet}[m+n])\ar[d]^{\simeq}\\
%\Hom_{\overline{X}}(\underline{\bZ}, \overline{\nu}_*J^{\bullet}[m])\ar[u]^{\omega_{\nu, J^{\bullet}[m]}} & \times & \Hom_{\overline{X}}(\overline{\nu}_*J^{\bullet}[m], \overline{\nu}_*\overline{\nu}^*I^{\bullet}[m+n])\ar[r]\ar[d]^{\rm trace} & \Hom_{\overline{X}}(\underline{\bZ}, \overline{\nu}_*\overline{\nu}^*I^{\bullet}[m+n])\ar[d]^{\rm trace}\\
%\Hom_{\overline{X}}(\underline{\bZ}, \overline{\nu}_*J^{\bullet}[m])\ar@{=}[u] & \times & \Hom_{\overline{X}}(\overline{\nu}_*J^{\bullet}[m], I^{\bullet}[m+n])\ar[r] & \Hom_{\overline{X}}(\underline{\bZ}, I^{\bullet}[m+n])}
%\end{displaymath}
%By the above discussion, the upper part of the diagram commutes. Likewise, the lower part commutes by functoriality of the trace map. $\overline{\nu}_*$ preserves injectives by right adjointness to $\overline{\nu}^*$, exact. Therefore, $\overline{\nu}_*B\to\overline{\nu}_*J^{\bullet}$ is an injective resolution of $\overline{\nu}_*B$. On passing to homotopy classes of maps, we conclude the following diagram commutes:
%\begin{displaymath}
%\xymatrix{\Ext^m_{\overline{X'}}(\underline{\bZ}, B) & \times & \Ext^{n}_{\overline{X'}}(B,\overline{\nu}^*A) \ar[d]_{\textup{N}} \ar[r] & \Ext^{m+n}_{\overline{X'}}(\underline{\bZ},\overline{\nu}^*A)\ar[d]^{\textup{N}}\\
% \Ext^m_{\overline{X}}(\underline{\bZ}, \overline{\nu}_*B)\ar[u]_{\ \ \ \simeq}^{\omega_{\nu, B}} & \times & \Ext^{n}_{\overline{X}}(\overline{\nu}_*B,A)\ar[r] & \Ext^{m+n}_{\overline{X}}(\underline{\bZ},A)}
%\end{displaymath}
%We apply this to $B = \widehat{F'}$ and $A = \overline{j}_*\mathbf{G}_{m, X}$, given that $\overline{\nu}^*A = \overline{j'}_*\mathbf{G}_{m, X'}$, and with $m =p, n= 3-p$, and we conclude.
Finally, we address the case $p<0$, so assume $F'$ is constructible. Letting $\eta' \in U'$ be the generic point,
by the {\em definition} of the Yoneda pairing in negative degrees via cup products in Tate
cohomology (using Corollary \ref{euler char disguise}!) we are reduced to checking that
for $v \in X_{\infty}$ and $w \in X'_{\infty}$ over $v$, the diagram
\begin{displaymath}
\xymatrix{\rH^{p-1}_T(I_w, F_{\eta'}') & \times & \rH^{3-p}_T(I_w, (F_{\eta'}')^{\vee}) \ar[d]_{\textup{N}}\ar[r] &
\rH^2_T(I_w,\mathbf{G}_{m,\eta'}) \ar[d]^{\textup{N}}\\
\rH^{p-1}_T(I_v, (\nu_*F')_{\eta}) \ar[u] & \times & \rH_T^{3-p}(I_v, (\nu_*F')_{\eta}^{\vee}) \ar[r] & \rH^2_T(I_v,\mathbf{G}_{m,\eta})}
\end{displaymath}
commutes, where the left map is the projection onto the $w$-summand as in Shapiro's Lemma and
we have used Remark \ref{finiteeuler} to identify the map between $\rH^{3-p}_T$'s.
%
%We have
%$$(\nu_*F')_{\eta} = \textup{Ind}_{\Gal(\overline{\eta}/\eta)}^{\Gal(\overline{\eta'}/\eta')}(F'_{\eta'})$$
%and the map $\omega_{\nu, F'[p]}$:
%$$\rH^{p-1}_T(I_v, \textup{Ind}(F'_{\eta'}))\to\rH^{p-1}_T(I_w, F'_{\eta'})$$
%is exactly the projection onto the $w$-summand as in Shapiro's Lemma.
The $\rH^i_T$'s on the bottom are identified as a direct sum of analogous terms for $F'_{\eta'}$ relative to the real
places of $K$ over $v$, and the two norm maps are isomorphisms onto the $w$-factors of such direct sums via
the {\em isomorphism} $I_w \simeq I_v$. The pushforward of the duality between $F'_{\eta'}$ and $(F'_{\eta'})^{\vee}$
has $K_v$-pullback that is the direct sum of the duality pairing for $F'_{\eta'}$ over $K'_{v'}$ indexed by the real places
$v'$ over $v$, so the desired commutativity is now obvious.
\end{proof}
\subsection{A duality comparison}
The construction of Tate's 9-term exact sequence via local cohomology requires
identifying many of the abstract maps with something concrete. This is recorded in Lemma \ref{maps},
whose proof has a loose end that we address here:
\begin{prop}\label{tatedual1} In the setting of Lemma $\ref{maps}$, the map
${\rm{H}}^1(G_S, M^{\vee}) \rightarrow \bigoplus_{s \in S} \rH^1(K_s, M)^D$ is the one
defined by the local duality pairings $\rH^1(K_s, M) \times \rH^1(K_s, M^{\vee}) \rightarrow \bQ/\bZ$.
\end{prop}
\begin{proof}\footnote{To be filled in, treating real $s$ separately.}
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{10}
{\footnotesize
\bibitem[Art]{Art} Artin, M. {\em Grothendieck Topologies}. Mimeographed Notes, Harvard Univ. 1962.
\bibitem[SGA4]{SGA} Artin, M., Grothendieck, A. et Verdier, J. L. {\em Th\'eorie des topos et cohomologie \'etale des sch\'emas I - III}, Lecture Notes in Math. 269, 270, 305, Springer Verlag.
\bibitem[CF]{CF} Cassels, J. W. S. and Frohlich, A. {\em Algebraic Number Theory}. Academic Press, London - New York 1967.
\bibitem[Ma]{Mazur} Mazur, B. {\em Notes on \'etale cohomology of number fields}. Ann. Scient. Ecole Norm. Sup. 4e s\'er., 6 (1973).
%\bibitem[Mil1]{milneet} Milne, J. {\em \'Etale cohomology}, Princeton Univ. Press.
\bibitem[Mi]{Milne} Lectures in \'etale cohomology. Available online.
\bibitem[Tate]{Tate} Tate, J. {\em Duality Theorems in Galois cohomology over number fields}. Proc. Intern. Congress 1962, Stockholm 1963, pp. 288-295.
\bibitem[Hab]{Hab} Haberland, K. {\em Galois cohomology of algebraic number fields}. Appendix 2 by Thomas Zink. Available at \texttt{http://math.stanford.edu/$\sim$conrad/BSDseminar/refs/ZinkTate.pdf}.
}
\end{thebibliography}
\ \\ \ \\
{\small
\textsc{Department of Mathematics, Stanford University, Building 380, Sloan Hall, Stanford, CA 94305 USA}
{\em E-mail address:} \texttt{conrad@math.stanford.edu}\\
{\em E-mail address:} \texttt{amasullo@stanford.edu}
}
\end{document}