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\def\PDGauthors{Mikhail Borovoi, Appendix by Zev Rosengarten}

\title[Galois cohomology] {Criterion for surjectivity of localization in Galois cohomology of a reductive group over a number field}

\author{\firstname{Mikhail} \lastname{Borovoi}\IsCorresp}
\address{Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 6997801 Tel Aviv, Israel}
\email{borovoi@tauex.tau.ac.il}
\thanks{The author was partially supported by the Israel Science Foundation (grant 1030/22).}

\contrib[appendixwriter]{\firstname{Zev} \lastname{Rosengarten}}
\address{Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, 91904 Jerusalem, Israel}
\email{zev.rosengarten@mail.huji.ac.il}

%\contrib[appendixwriter]{\firstname{Aev} \lastname{Sosengarten}}
%\address[2]{Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, 91904 Jerusalem, Israel}

\subjclass{11E72, 20G10, 20G20, 20G25, 20G30}
%\dedicatory{With an appendix by Zev Rosengarten}

%~\keywords{Linear algebraic group, number field, Galois cohomology, localization map, abelianization}

\begin{abstract}
Let $G$ be a connected reductive group over a number field $F$, and let $S$ be a set (finite or infinite) of places of $F$. We give a necessary and sufficient condition for the surjectivity of the localization map from $H^1(F,G)$ to the ``direct sum'' of the sets $H^1(F_v,G)$ where $v$ runs over $S$. In the appendices, we give a new construction of the abelian Galois cohomology of a reductive group over a field of arbitrary characteristic.
\end{abstract}

\begin{altabstract}
Soit $G$ un groupe réductif connexe sur un corps de nombres $F$, et soit $S$ un ensemble (fini ou infini) de places de $F$. On donne une condition nécessaire et suffisante pour la surjectivité de l'application de localisation de $H^1(F,G)$ vers la \og somme directe\fg{} des ensembles $H^1(F_v,G)$, où $v$ parcourt $S$. Dans les appendices on donne une nouvelle construction de la cohomologie galoisienne ab\'elienne d'un groupe r\'eductif sur un corps de caract\'eristique quelconque.
\end{altabstract}


\begin{document}
\maketitle

\section{Introduction}\label{s:intro}

\subsec{}
Let $G$ be a (connected) reductive group over a number field $F$ (we follow the convention of SGA3, where reductive groups are assumed to be connected). Let $\Fbar$ be a fixed algebraic closure of $F$. We denote by $\Vm(F)$ the set of places of $F$. For $v\in\Vm(F)$, we denote by $F_v$ the completion of $F$ at $v$. We refer to Serre's book~\cite{Serre} for the definition of the first Galois cohomology set $H^1(F,G)$.

In general, $H^1(F,G)$ is just a pointed set and has no natural groups structure. Let $H^1_\ab(F,G)$ denote the \emph{abelian Galois cohomology group} of $G$ introduced in~\cite[Section~2]{Borovoi-Memoir}; see also Labesse~\cite[Section~1.3]{Labesse}. This is an abelian group depending functorially on $G$ and $F$. There is a canonical \emph{abelianization map}
\[
\ab\colon H^1(F,G)\to H^1_\ab(F,G).
\]
We give a new, better construction of $H^1_\ab(F,G)$ in Appendix~\ref{s:ab}.

Let $S\subseteq \Vm(F)$ be a subset (finite or infinite). We consider the localization map
\begin{align}\label{e:loc-S-ab-prod}
H^1_\ab(F,G)\to\prod_{v\in S} H^1_\ab(F_v,G).
\end{align}
In fact this map takes values in the subgroup $\bigoplus_{v\in S} H^1_\ab(F_v,G)\subseteq \prod_{v\in S} H^1_\ab(F_v,G)$; see~\cite[Corollary~4.6]{Borovoi-Memoir}. Thus we obtain a localization map
\begin{equation}\label{e:loc-S-ab}
\loc_S^\ab\colon H^1_\ab(F,G)\to\bigoplus_{v\in S} H^1_\ab(F_v,G).
\end{equation}
Similarly, consider the localization map
\begin{equation*}
H^1(F,G)\to\prod_{v\in S} H^1(F_v,G).
\end{equation*}
In fact it takes values in the subset $\bigoplus_{v\in S} H^1(F_v,G)$ consisting of the families $(\xi_v)_{v\in S}$ with $\xi_v\in H^1(F_v,G)$ and such that $\xi_v=1$ for all $v$ except maybe finitely many of them. This well-known fact follows, for instance, from the corresponding assertion for~\eqref{e:loc-S-ab-prod} together with~\cite[Theorem~5.11 and Corollary~5.4.1]{Borovoi-Memoir}. Thus we obtain a localization map
\begin{equation}\label{e:loc-S}
\loc_S\colon H^1(F,G)\to\bigoplus_{v\in S} H^1(F_v,G).
\end{equation}
We wish to find conditions under which the localization maps~\eqref{e:loc-S-ab} and~\eqref{e:loc-S} are surjective.
\esubsec

\subsec{} We denote by $M=\pi_1(G)$ the \emph{algebraic fundamental group of $G$} (also known as the Borovoi fundamental group of $G$) introduced in~\cite[Section~1]{Borovoi-Memoir}, and also introduced by Merkurjev~\cite[Section~10.1]{Merkurjev} and Colliot-Th\'el\`ene~\cite[Proposition-Definition~6.1]{CT-flasque}. See Subsection~\ref{ss:pi1} for our definition of $\pi_1 (G)$. This is a finitely generated abelian group, on which the absolute Galois group $\Gal(\Fbar/F)$ naturally acts. Let $E/F$ be a finite Galois extension in $\Fbar$ such that $\Gal(\Fbar/E)$ acts on $M$ trivially and that $E$ has no real places. Then the Galois group $\G\coloneqq\Gal(E/F)$ naturally acts on $M$ and on the set of places $\Vm(E)$ of the field $E$.
\esubsec

\subsec{} We denote by $\Ch^1_S(F,G)$ the cokernel of the homomorphism~\eqref{e:loc-S-ab}, that is,
\[
\Ch^1_S(F,G)=\coker\bigg[\loc_S^\ab \colon H^1_\ab(F,G)\to\bigoplus_{v\in S} H^1_\ab(F_v,G)\bigg].
\]
After explaining our notation in Section~\ref{s:notation}, we compute in Section~\ref{s:main} the finite abelian group $\Ch^1_S(F,G)$ in terms of the action of $\G$ on $M$ and on $\Vm(E)=\Vm_f(E)\cup\Vm_\C(E)$; see Corollary~\ref{c:main-ab}. See Subsection~\ref{ss:mt1} for the notations $\Vm_f$ and $\Vm_\C$.

Concerning the map $\loc_S$ of~\eqref{e:loc-S}, in Section~\ref{s:main} we compute the image of this map; see Main Theorem~\ref{t:main}. Using this result, we give a \emph{criterion} (necessary and sufficient condition) for the map $\loc_S$ to be surjective; see Corollary~\ref{c:main}. This is also a criterion for the vanishing of $\Ch^1_S(F,G)$. Again, our criterion is given in terms of the action of $\G$ on $M$ and on $\Vm(E)=\Vm_f(E)\cup\Vm_\C(E)$. Using this criterion, we give a simple proof of the result of Borel and Harder~\cite[Theorem~1.7]{Borel-Harder} (see also Prasad and Rapinchuk~\cite[Proposition~1]{Prasad-Rapinchuk}\hs) on the surjectivity of the map $\loc_S$ when $G$ is semisimple and there exists a finite place $v_0$ of $F$ outside $S$; see Proposition~\ref{p:semisimple} below.

Let $\Gamma$ be a finite group. In Section~\ref{s:exact}, we construct an exact sequence arising from a short exact sequence of $\Gamma$-modules. In Section~\ref{s:PR}, using this exact sequence and Main Theorem~\ref{t:main}, we generalize a result of Prasad and Rapinchuk giving a sufficient condition for the surjectivity of the localization map $\loc_S$ when $G$ is reductive, in terms of the radical (largest central torus) of $G$; see Theorem~\ref{t:PR-generalized}. As a particular case, we obtain the following corollary.

\begin{coro}[of Theorem~\ref{t:PR-generalized}]\label{c:Prasad-Rapinchuk}
Let $G$ be a reductive group over a number field $F$, and let $C$ denote the radical of $G$ (the identity component of the center of $G$). Let $S\subset \Vm(F)$ be a set of places of $F$. Assume that the $F$-torus $C$ splits over a finite Galois extension of $F$ of prime degree $p$ and that there exists a finite place $v_0$ in the complement $\SC\coloneqq \Vm(F)\smallsetminus S$ of $S$ such that $C$ does not split over $F_{v_0}$\hs. Then the localization map $\loc_S$ of~\eqref{e:loc-S} is surjective.
\end{coro}

For $p=2$ this assertion was earlier proved by Prasad and Rapinchuk~\cite[Proposition~2(b)]{Prasad-Rapinchuk}.
\esubsec

\subsec{} Let $G$ be a reductive group over a field $F$ of characteristic 0. In~\cite{Borovoi-Memoir}, the author defined the abelian group $H^1_\ab(F,G)$ \emph{as a set} in a canonical way as the Galois hypercohomology of a certain crossed module. However, the definition of the structure of abelian group on $H^1_\ab(F,G)$ in~\cite{Borovoi-Memoir} was complicated. In Appendix~\ref{s:ab}, we define $H^1_\ab(F,G)$ (in arbitrary characteristic) following the letter of Breen to the author~\cite{Breen} and the article by Noohi~\cite{Noohi} (written at the author's request), as the Galois hypercohomology $\H^1(F,G_\ab)$ of a certain \emph{stable} crossed module, that is, a crossed module endowed with a symmetric braiding. The structure of abelian group comes from the symmetric braiding. Note that our specific crossed module and specific symmetric braiding were constructed by Deligne~\cite{Deligne}.

In Appendix~\ref{s:Zev}, Zev Rosengarten shows that certain equivalences of crossed modules of algebraic groups over a field $F$ of arbitrary characteristic induce equivalences on $\Fs$-points where $\Fs$ is a separable closure of $F$. This permits us to use in Appendix~\ref{s:ab} the \emph{Galois} hypercohomology of these crossed modules rather than fppf hypercohomology.
\esubsec


\section{Notation}\label{s:notation}

\subsec{} Let $A$ be an abelian group. We denote by $A_\Tors$ the torsion subgroup of $A$. We set $A_\tf=A/A_\Tors$\hs, which is a torsion-free group.
\esubsec

\subsec{} Let $\G$ be a finite group, and let $B$ be a $\G$-module. We denote by $B_\G$ the group of coinvariants of $\G$ in $B$, that is,
\[
B_\G=B\hs \big/\Bigg\{\sum_{\gamma\in \G}\Big(\,{}^{\gamma^{-1}}\!b_\gamma-b_\gamma\,\Bigr)
\ \Bigg|\ b_\gamma\in B\Bigg\}.
\]
We write $B_\Gt\coloneqq (B_\G)_\Tors$ (which is the torsion subgroup of $B_\G$),
\ $B_\Gtf=B_\G/ B_\Gt$\ (which is a torsion-free group).
\esubsec

\subsec{}\label{ss:pi1}
Let $G$ be a reductive group over a field $F$. Let $[G,G]$ denote the commutator subgroup of $G$, which is semisimple. Let $G^\ssc$ denote the universal cover of $[G,G]$, which is simply connected; see~\cite[Proposition~(2.24)$\MK$(ii)]{Borel-Tits-C} or~\cite[Corollary~A.4.11]{CGP}. Following Deligne~\cite[Section~0.2]{Deligne}, we consider the composite homomorphism
\[
\rho\colon G^\ssc\onto [G,G]\into G,
\]
which in general is neither injective nor surjective.

For a maximal torus $T\subseteq G$, we write $T^\ssc=\rho^{-1}(T)\subseteq G^\ssc$ and consider the natural homomorphism
\[
\rho\colon T^\ssc\to T.
\]
We consider the algebraic fundamental group $M=\pi_1 (G)$ of $G$ defined by
\[
\pi_1 (G)=\X_*(T)/\rho_*\X_*(T^\ssc)
\]
where $\X_*$ denotes the cocharacter group. The Galois group $\Gal(\Fs/F)$ naturally acts on $M$, and the $\Gal(\Fs/F)$-module $M$ is well defined (does not depend on the choice of $T$ up to a transitive system of isomorphisms); see~\cite[Lemma~1.2]{Borovoi-Memoir}.
\esubsec

\subsec{}\label{ss:mt1}
From now on (except for the appendices), $F$ is a number field. We denote by $\Vm(F)$, $\Vm_f(F)$, $\Vm_\infty(F)$, $\Vm_\R(F)$, and $\Vm_\C(F)$ the sets of all places of $F$, of finite places, of infinite places, of real places, and of complex places, respectively.

Let $E/F$ be a finite Galois extension of number fields with Galois group $\G=\Gal(E/F)$; then $\G$ acts on $\Vm(E)$. If $w\in\Vm(E)$, we write $\Gw$ for the stabilizer of $w$ in $\G$; then $\Gw\cong \Gal(E_w/F_v)$ where $v\in\Vm(F)$ is the restriction of $w$ to $F$.
\esubsec


\section{Main theorem}\label{s:main}

In this section we state and prove Main Theorem~\ref{t:main} computing the images of the localization maps~\eqref{e:loc-S-ab} and~\eqref{e:loc-S}. We deduce Corollary~\ref{c:main-ab} computing the group $\Ch^1_S(F,G)$, and Corollary~\ref{c:main} giving a necessary and sufficient condition for the surjectivity of the localization map~\eqref{e:loc-S}.

\subsec{}\label{ss:local-results}
Let $G$ be a reductive group over a number field $F$, and let $v\in\Vm_f(F)$ be a finite place of $F$. In~\cite{Borovoi-Memoir} we computed $H^1_\ab(F_v,G)$. Write $M=\pi (G)$. Let $E/F$ be a finite Galois extension in $\Fbar$ such that $\Gal(\Fbar/E)$ acts on $M$ trivially and that $E$ has no real places. Write $\G=\Gal(E/F)$.
\esubsec

\begin{theo}[{\cite[Proposition~4.1(i) and Corollary~5.4.1]{Borovoi-Memoir}}]\label{t:4.1-Memoir}
With the notation and assumptions of Subsection~\ref{ss:local-results}, for any \emph{finite} place $v$ of $F$ there is a canonical isomorphism of abelian groups
\[
\alpha_v^\ab\colon H^1_\ab(F_v,G)\isoto M_\Gwt
\]
where $w$ is a place of $E$ over $v$, and a canonical bijection
\[
\ab_v\colon H^1(F_v,G)\to H^1_\ab(F_v,G).
\]
\end{theo}


\subsec{} Let $v$ be a finite place of $F$. We have a surjective (even bijective) map
\[
\alpha_v\colon H^1(F_v,G)\labelto{\ab_v} H^1_\ab(F_v,G)\labelto{\alpha_v^\ab} M_\Gwt\hs.
\]
We consider two composite maps with the same image
\begin{align*}
\lambda_v^\ab\colon\, &H^1_\ab(F_v,G)\labelto{\alpha_v^\ab} M_\Gwt\labelto{\omega_v} M_\Gt\hs,\\
\lambda_v\colon\, &H^1(F_v,G)\labelto{\alpha_v} M_\Gwt\labelto{\omega_v} M_\Gt\hs,
\end{align*}
where\, $\omega_v\colon M_\Gwt\to M_\Gt$\, is the homomorphism induced by the inclusion $\Gw\into\G$. Since the maps $\alpha_v^\ab$ and $\alpha_v$ are surjective (even bijective), and $\omega_v$ is a homomorphism, we see that the set\, $\im\lambda_v^\ab=\im\lambda_v$\, is a subgroup of $M_\Gt$\hs, namely,\, $\im\lambda_v^\ab=\im\lambda_v=\im\hs\omega_v$\hs.

Let $v\in\Vm_\C(F)$ be a complex place. We have zero maps
\[
\lambda_v^\ab\colon H^1_\ab(F_v,G)=\{1\}\to \{0\}\subseteq M_\Gt\hs,\qquad
\lambda_v\colon H^1(F_v,G)=\{1\}\to \{0\}\subseteq M_\Gt\hs.
\]
Clearly, the set\, $\im\lambda^\ab_v=\im\lambda_v$\, is a subgroup of $M_\Gt$\hs, namely, the subgroup $\{0\}$.
\esubsec


\subsec{}\label{ss:real}
Let $v\in\Vm_\R(F)$ be a real place; then $\Gw$ is a group of order 2, $\Gw=\{1,\gamma\}$ where $\gamma=\gamma_w$ induces the nontrivial automorphism of $E_w$ over $F_v$\hs. We consider the Tate cohomology group
\[
\wh H^{-1}(\Gw,M)=\{m\in M\mid \upgam m=-m\}\hs/\hss\{m'-\upgam m'\mid m'\in M\}.
\]
We see immediately that the abelian group $\wh H^{-1}(\Gw,M)$ naturally embeds into $M_{\Gw}$\hs. If $m\in M$ is a $(-1)$-cocycle, that is, $\upgam m=-m$, then $2m=m+m=m-\upgam m$, whence $2\cdot\wh H^{-1}(\Gw,M)=0$. We conclude that $\wh H^{-1}(\Gw,M)$ naturally embeds into $M_{\Gw,\Tors}$\hs.

There is a canonical surjective map of Kottwitz~\cite[Theorem~1.2]{Kottwitz-86} (see also~\cite[Theorem~5.4]{Borovoi-Memoir})
\[
\ab_v\colon H^1(F_v, G)\onto H^1_\ab(F_v,G),
\]
a canonical isomorphism of~\cite[Proposition~8.21]{BT-arXiv}
\[
H^1_\ab(F_v,G)\isoto \wh H^{-1}(\Gw, M),
\]
and a canonical embedding
\[
\wh H^{-1}(\Gw,M)\into M_\Gwt\hs.
\]
Thus we obtain composite maps
\begin{align*}
\alpha_v^\ab\colon &H^1_\ab(F_v,G)\isoto\wh H^{-1}(\Gw, M)\into M_\Gwt\hs,\\
\alpha_v\colon H^1(F_v, G)\onto &H^1_\ab(F_v,G)\isoto\wh H^{-1}(\Gw, M)\into M_\Gwt\hs,
\end{align*}
with the same image\, $\im \alpha_v^\ab=\im\alpha_v$\hs,\, which is a subgroup of $M_\Gwt$\hs. Consider the composite maps with the same image
\begin{align*}
\lambda_v^\ab\colon\, &H^1_\ab(F_v, G)\labelto{\alpha_v^\ab} M_\Gwt\labelto{\omega_v} M_\Gt\hs,\\
\lambda_v\colon\, &H^1(F_v, G)\labelto{\alpha_v} M_\Gwt\labelto{\omega_v} M_\Gt\hs.
\end{align*}
Since the set\, $\im\alpha_v^\ab=\im\alpha_v$\, is a subgroup of $M_\Gwt$\hs, and $\omega_v$ is a homomorphism, we conclude that the set\, $\im\lambda_v^\ab=\im\lambda_v$\, is a subgroup of $M_\Gt$\hs, namely,\, $\im\lambda_v^\ab=\im\lambda_v=\omega_v\big(\wh H^{-1}(\Gw,M)\hs\big)$.
\esubsec

\begin{lemm}\label{ss:loc-sum}
Let $S\subseteq \Vm(F)$ be any subset, finite or infinite. Consider the summation maps
\begin{align*}
\Sigma_S^\ab\colon &\bigoplus_{v\in S} H^1_\ab(F_v,G)\,\lra\, M_\Gt,\quad
\ \xi_S^\ab= \big(\xi_v^\ab\big)_{v\in S}\,\longmapsto\, \sum_{v\in S}\lambda_v^\ab(\xi_v^\ab),\\
\Sigma_S\colon &\bigoplus_{v\in S} H^1(F_v,G)\,\lra\, M_\Gt,\quad
\ \ \xi_S= \big(\xi_v\big)_{v\in S}\,\longmapsto\, \sum_{v\in S}\lambda_v(\xi_v).
\end{align*}
Then the sets \,$\im\Sigma_S^\ab$ and \,$\im \Sigma_S$ are subgroups of $M_\Gt$\hs, and they are equal.
\end{lemm}

\begin{proof}
Indeed, we have
\[
\im\Sigma^\ab_S=\im\Sigma_S=\bigl\langle \im\lambda_v\bigr\rangle_{v\in S}
\quad\ \text{where}\ \,\im\lambda_v=
\begin{cases}
\im\omega_v &\text{if $v\in\Vm_f(F)$,}\\
\omega_v\big(\wh H^{-1}(\Gw,M)\hs\big) &\text{if $v\in\Vm_\R(F)$,}\\
0 &\text{if $v\in\Vm_\C(F)$.}
\end{cases}
\]
Here we write\, $\bigl\langle \im\lambda_v\bigr\rangle_{v\in S}$\, for the subgroup of $M_\Gt$ generated by the subgroups\, $\im\lambda_v$\, for $v\in S$.
\end{proof}

\begin{theo}\label{t:Kottwitz}
The following sequences are exact:
\begin{align}
&H^1_\ab(F,G)\labeltooo{\loc_{\Vm}^\ab}\bigoplus_{v\in \Vm} H^1_\ab(F_v,G)
\labeltooo{\Sigma_{\Vm}^\ab} M_\Gt\hs,\label{a:1}
\\
&H^1(F,G)\labeltooo{\loc_{\Vm}}\bigoplus_{v\in \Vm} H^1(F_v,G)
\labeltooo{\Sigma_{\Vm}} M_\Gt\hs,\label{a:2}
\end{align}
where for brevity we write $\Vm$ for $\Vm(F)$.
\end{theo}

Here~\eqref{a:1} is an exact sequence of abelian groups, and~\eqref{a:2} is an exact sequence of pointed sets.

\begin{proof}
In view of~\cite[Proposition~4.8]{Borovoi-Memoir}, exact sequence~\eqref{a:1} is actually a part of the exact sequence~\cite[(4.3.1)]{Borovoi-Memoir}. For~\eqref{a:2}, see~\cite[Proposition~2.6]{Kottwitz-86} or~\cite[Theorem~5.15]{Borovoi-Memoir}.
\end{proof}

\begin{enonce}{Main Theorem}\label{t:main}
Let $G$ be a reductive group over a number field $F$. Let $S\subseteq \Vm\coloneqq \Vm(F)$ be a subset. Write $\SC=\Vm\smallsetminus S$, the complement of $S$ in $\Vm$. Then:
\begin{align}
\im\hs \loc_S^\ab &=\Biggl\{\xi_S^\ab\in \bigoplus_{v\in S} H^1_\ab(F_v,G)
\ \Bigg|\ \Sigma_S^\ab(\xi_S^\ab)\in
\im\Sigma_S^\ab\hs\cap\hs\im\Sigma_\SC^\ab\Biggr\},\label{a:main-1}
\\
\im\hs \loc_S &=\Biggl\{\xi_S\in \bigoplus_{v\in S} H^1(F_v,G)
\ \Bigg|\ \Sigma_S(\xi_S)\in\im\Sigma_S\hs\cap\hs\im\Sigma_\SC\Biggr\}\label{a:main-2}
.
\end{align}
\end{enonce}

\begin{proof}
By Lemma~\ref{ss:loc-sum}, the sets $\im\Sigma_\SC^\ab$ and $\im\Sigma_\SC$ are (equal) subgroups of $M_\Gt$\hs, and therefore it suffices to prove~\eqref{a:main-1} with $\bigl(-\im\Sigma_\SC^\ab\bigr)$ instead of \,$\im\Sigma_\SC^\ab$\,, and to prove~\eqref{a:main-2} with $\bigl(-\im\Sigma_\SC\bigr)$ instead of \,$\im\Sigma_\SC$\hs. Now the corresponding assertions follow easily from the exactness of~\eqref{a:1} and~\eqref{a:2}, respectively.

For the reader's convenience, we provide an easy proof of~\eqref{a:main-2} with $\bigl(-\im\Sigma_\SC\bigr)$ instead of \,$\im\Sigma_\SC$\hs. Let
\[
\xi_S=\big(\xi_v\big)_{v\in S}\,\in\,\im\loc_S\,\subseteq\, \bigoplus_{v\in S} H^1(F_v,G),
\]
that is, $\xi_S=\loc_S(\xi)$ for some $\xi\in H^1(F,G)$. Write $\eta_\SC=\bigl(\eta_v\bigr)_{v\in\SC}=\loc_\SC(\xi)$. Since the sequence~\eqref{a:2} is exact, we have $(\Sigma_\Vm\circ\loc_\Vm)(\xi)=0$, whence
\[
\Sigma_S(\xi_S)+\Sigma_\SC(\eta_\SC)=0\quad\ \text{and}\quad\ \Sigma_S(\xi_S)=-\Sigma_\SC(\eta_\SC).
\]
We conclude that $\Sigma_S(\xi_S)\in \im\Sigma_S\cap\bigl(-\im \Sigma_\SC\bigr)$, as required.

Conversely, let an element $\xi_S=\bigl(\xi_v\bigr)_{v\in S}\in \bigoplus_{v\in S} H^1(F_v,G)$ be such that
\[
\Sigma_S(\xi_S)\in \im\Sigma_S\hs\cap\bigl(-\im\Sigma_\SC\bigr).
\]
Write $a=\Sigma_S(\xi_S)$. Then $-a\in \im\Sigma_\SC$\hs, that is,
\[
-a=\Sigma_\SC(\eta_\SC)\quad\text{for some}\quad \eta_\SC=
\big(\eta_v\big)_{v\in\SC}\hs\in\hs \bigoplus_{v\in \SC} H^1(F_v,G).
\]
Define
\[
\zeta_{\Vm}=\big(\zeta_v\big)_{v\in \Vm}\,\in\, \bigoplus_{v\in \Vm} H^1(F_v,G), \quad\ \zeta_v=
\begin{cases}
\xi_v &\text{if $v\in S$,}\\
\eta_v &\text{if $v\in \SC$.}
\end{cases}
\]
Then
\[
\Sigma_{\Vm}(\zeta_{\Vm})=a+(-a)=0.
\]
Since the sequence~\eqref{a:2} is exact, we have $\zeta_\Vm=\loc_\Vm(\zeta)$ for some $\zeta\in H^1(F,G)$. Then $\loc_S(\zeta)=\xi_S$, whence $\xi_S\in\im\loc_S$, as required.
\end{proof}

\begin{coro}\label{c:main-ab}
The homomorphism
\[
\chi_S^\ab\colon \bigoplus_{v\in\Vm} H^1_\ab(F_v,G)\labelto{\Sigma_S^\ab} \im\Sigma_S^\ab
\lra \im \Sigma_S^\ab/\big(\im\Sigma_S^\ab\cap\im\Sigma_\SC^\ab\big)
\]
induces a canonical isomorphism
\[
\Ch^1_S(F,G)\isoto \im \Sigma_S^\ab/\big(\im\Sigma_S^\ab\cap\im\Sigma_\SC^\ab\big).
\]
\end{coro}

\begin{proof}
The homomorphism $\chi_S^\ab$ is clearly surjective, and by Theorem~\ref{t:main} its kernel is the image $\im\loc_S^\ab$ of the localization homomorphism $\loc_S^\ab$ of~\eqref{e:loc-S-ab}. The corollary follows.
\end{proof}

\begin{coro}\label{c:main}
The localization map $\loc_S$ of~\eqref{e:loc-S} is surjective if and only if
\begin{equation}\label{e:main-formula}
\im \Sigma_S\,\subseteq\, \im\Sigma_\SC\hs.
\end{equation}
\end{coro}

\begin{proof}
Consider the map
\[
\chi_S\colon \bigoplus_{v\in S} H^1(F_v,G)\labelto{\Sigma_S}
\im\Sigma_S \lra \im \Sigma_S/\big(\im\Sigma_S\cap\im\Sigma_\SC\big).
\]
By Lemma~\ref{ss:loc-sum} the sets \ $\im \Sigma_S= \im \Sigma_S^\ab$ \ and
\ $\im\Sigma_S\cap\im\Sigma_\SC=\im\Sigma_S^\ab\cap\im\Sigma_\SC^\ab$ \ are abelian groups. The morphism of pointed sets $\chi_S$ is clearly surjective, and by Theorem~\ref{t:main} its kernel is $\im\loc_S$. We see that the following assertions are equivalent:
\begin{enumerate}\alphenumi
\item \label{a} the map $\loc_S$ is surjective, that is, \,$\im\loc_S=\bigoplus_{v\in S} H^1(F_v,G)$;
\item \label{b} $\ker \chi_S=\bigoplus_{v\in S} H^1(F_v,G)$;
\item \label{c} $\#(\im\chi_S) = 1$;
\item \label{d} $\im\Sigma_S\cap\im\Sigma_\SC=\im\Sigma_S$\hs;
\item \label{e} $\im\Sigma_S\subseteq\im \Sigma_\SC$\hs.
\end{enumerate}
This completes the proof.
\end{proof}

\begin{rema}
Since by Lemma~\ref{ss:loc-sum} we have $\im\Sigma_S^\ab=\im\Sigma_S$\, and\, $\im\Sigma_\SC^\ab=\im\Sigma_\SC$,\, we see from \eqref{d} in the proof above and from Corollary~\ref{c:main-ab} that the localization map $\loc_S$ of~\eqref{e:loc-S} is surjective if and only if $\Ch^1_S(F,G)=\{1\}$.
\end{rema}

\begin{coro}
Let $v_0\in\Vm(F)$, $S=\Vm(F)\smallsetminus\{v_0\}$. Then the localization map $\loc_S$ of~\eqref{e:loc-S} is surjective if and only if
\begin{equation}\label{e:lambda-lambda0}
\im\lambda_v\subseteq\im\lambda_{v_0}\quad\ \text{for all}\ \, v\in\Vm(F).
\end{equation}
\end{coro}

\begin{proof}
Indeed, in our case condition~\eqref{e:lambda-lambda0} is equivalent to~\eqref{e:main-formula}, and we conclude by Corollary~\ref{c:main}.
\end{proof}

\begin{coro}
For a subset $S\subset \Vm(F)$, let $v_0\in\SC$, and assume that
\begin{equation}\label{e:lambda-lambda0-bis}
\im\lambda_v\subseteq \im\lambda_{v_0}\quad\ \text{for all}\ \, v\in S.
\end{equation}
Then the localization map $\loc_S$ of~\eqref{e:loc-S} is surjective.
\end{coro}

\begin{proof}
Indeed, \eqref{e:lambda-lambda0-bis} implies~\eqref{e:main-formula}, and we conclude by Corollary~\ref{c:main}.
\end{proof}

\begin{coro}\label{c:v0-sur}
Let $v_0\in \SC$, and assume that the map $\lambda_{v_0}\colon H^1(F_{v_0},G)\to M_\Gt$ is surjective. Then the localization map $\loc_S$ of~\eqref{e:loc-S} is surjective.
\end{coro}

\begin{proof}
Indeed, then
\[
\im\Sigma_S\subseteq M_\Gt=\im\lambda_{v_0}\subseteq \im\Sigma_\SC\hs,
\]
and we conclude by Corollary~\ref{c:main}.
\end{proof}

\begin{prop}[{Borel and Harder~\cite[Theorem~1.7]{Borel-Harder}}]\label{p:semisimple}
Let $G$ be a \emph{semisimple} group over a number field $F$, and let $S\subset \Vm(F)$ be a subset such that the complement $\SC$ of $S$ contains a \emph{finite} place $v_0\in\Vm_f(F)$. Then the localization map $\loc_S$ of~\eqref{e:loc-S} is surjective.
\end{prop}

\begin{proof}
Since $G$ is semisimple, the $\G$-module $M$ is finite, and so are the groups $M_\G$ and $M_\Gw$ where $w$ is a place of $E$ over $v_0$\hs. It follows that
\[
M_\Gwt=M_\Gw\quad\ \text{and}\quad\ M_\Gt=M_\G\hs.
\]
The natural homomorphism $M_\Gw\to M_\G$ is clearly surjective. Therefore, the homomorphism
\begin{equation*}
\omega_{v_0}\colon\, M_\Gwt=M_\Gw\,\lra\, M_\G= M_\Gt
\end{equation*}
is surjective. Since $v_0$ is finite, we have $\im\lambda_{v_0}=\im\omega_{v_0}$, whence the map $\lambda_{v_0}$ is surjective. We conclude by Corollary~\ref{c:v0-sur}.
\end{proof}



\section{Exact sequence}\label{s:exact}
In this section we construct an exact sequence that we shall use in Section~\ref{s:PR}.

\begin{theo}\label{t:exact}
A finite group $\G$ and a short exact sequence of $\G$-modules
\begin{equation}\label{e:1-2-3}
0\to B_1\labelto i B_2\labelto j B_3\to 0
\end{equation}
give rises to an exact sequence
\begin{multline}\label{e:1-2-3-long}
(B_1)_\Gt \labelto{i_*} (B_2)_\Gt \labelto{j_*} (B_3)_\Gt\\ \labelto\delta 
\Q/\Z\otimes_\Z (B_1)_\G \labelto{i_*} \Q/\Z\otimes_\Z(B_2)_\G
\labelto{j_*} \Q/\Z\otimes_\Z(B_3)_\G\to 0
\end{multline}
depending functorially on $\G$ and on the sequence~\eqref{e:1-2-3}.
\end{theo}

\subsec{}\label{ss-delta}
We specify the homomorphism $\delta$. Let $x_3\in B_3$ be such that the image $(x_3)_\G$ of $x_3$ in $(B_3)_\G$ is contained in $(B_3)_\Gt$\hs. This means that there exist $n\in\Z_{>0}$ and $y_{3,\g}\in B_3$ such that
\[
nx_3=\sum_{\g\in\G}\big(\upgam y_{3,\g}-y_{3,\g}\big).
\]
We lift $x_3$ to some $x_2\in B_2$\hs, we lift each $y_{3,\g}$ to some $y_{2,\g}\in B_2$\hs, and we consider the element
\begin{equation*}
z_2=nx_2-\sum_{\g\in\G}\big(\upgam y_{2,\g}-y_{2,\g}\big).
\end{equation*}
Then $j(z_2)=0\in B_3$\hs, whence $z_2=i(z_1)$ for some $z_1\in B_1$\hs. We consider the image $(z_1)_\Gtf$ of $z_1\in B_1$ in $(B_1)_\Gtf$\hs, and we put
\[
\delta\big(\hs(x_3)_\G\big)= \frac1n\otimes (z_1)_\Gtf \hs
\in \hs\Q/\Z\otimes_\Z (B_1)_\Gtf=\Q/\Z\otimes_\Z (B_1)_\G
\]
where we write $\frac1n$ for the image in $\Q/\Z$ of $\frac1n\in \Q$.


Below we give the proof of Theorem~\ref{t:exact} suggested by Vladimir Hinich (private communication). For another proof, due to Alexander Petrov, see~\cite{SashaP}.
\esubsec

%\setcounter{equation}{0}
%\subsec{Proof Theorem~\ref{t:exact} due to Vladimir Hinich}
\begin{proof}[Proof of Theorem~\ref{t:exact} due to Vladimir Hinich]
The functor from the category of $\G$-modules to the category of abelian groups
\[
B\rsa \Q/\Z\otimes_\Z B_\G
\]
is the same as
\[
B\rsa\Q/\Z\otimes_\Lam\! B
\]
where $\Lam=\Z[\G]$ is the group ring of $\G$. From the short exact sequence of $\G$-modules~\eqref{e:1-2-3}, we obtain a long exact sequence
\begin{multline*}
\dots\to \Tor_1^\Lam(\Q/\Z,B_1) \labelto{i_*} \Tor_1^\Lam(\Q/\Z,B_2) \labelto{j_*} \Tor_1^\Lam(\Q/\Z,B_3)\\ \labelto\delta 
\Q/\Z\otimes_\Lam\! B_1\labelto{i_*} \Q/\Z\otimes_\Lam\! B_2\labelto{j_*} \Q/\Z\otimes_\Lam\! B_3\to 0
\end{multline*}
depending functorially on $\G$ and on~\eqref{e:1-2-3}; see Weibel~\cite{Weibel}. Now Theorem~\ref{t:exact} follows from the next proposition.
%\qed
\end{proof}

\begin{prop}\label{p:Q/Z-B-BGt}
For a finite group $\G$ and a $\G$-module $B$, there is a canonical and functorial isomorphism
\[
\Tor_1^\Lam (\Q/\Z,B)\isoto B_\Gt\hs
\]
where $\Lam=\Z[\G]$.
\end{prop}

\begin{proof}
Consider the short exact sequence
\[
0\to \Z\to \Q\to\Q/\Z\to 0
\]
regarded as a short exact sequence of $\G$-modules with trivial action of $\G$. Tensoring with $B$, we obtain a long exact sequence
\begin{equation}\label{e:long-tensor}
\dots\to\Tor_1^\Lam(\Q,B)\to \Tor_1^\Lam(\Q/\Z,B)\to \Z\otimes_\Lam\! B
\to \Q\otimes_\Lam\! B \to \Q/\Z\otimes_\Lam\! B\to 0.
\end{equation}
We have canonical isomorphisms
\[
\Z\otimes_\Lam\! B=B_\G\quad\ \text{and}\quad\
\ker\big[\Z\otimes_\Lam\! B\to \Q\otimes_\Lam\! B\big]= B_\Gt\hs.
\]
By Lemma~\ref{l:Tor-Q} below, we have $\Tor_1^\Lam(\Q,B)=0$, and the proposition follows from~\eqref{e:long-tensor}.
\end{proof}

\begin{lemm}\label{l:Tor-Q}
For a finite group $\G$ and any $\G$-module $B$, we have
\[
\Tor_1^\Lam (\Q,B)=0
\]
where $\Lam=\Z[\G]$.
\end{lemm}

\begin{proof}
Let
\[
P_\bullet:\quad\dots\to P_2\to P_1\to P_0\to\Z\to 0
\]
be a $\Lam$-free resolution of the trivial $\G$-module $\Z$, for example, the standard complex; see Atiyah and Wall~\cite[Section~2]{AW}. Tensoring with $\Q$ over $\Z$, we obtain a flat resolution of $\Q$
\[
\dots\to \Q\otimes_\Z P_2\to \Q\otimes_\Z P_1\to \Q\otimes_\Z P_0\to\Q\to 0.
\]
Tensoring with $B$ over $\Lam=\Z[\G]$, we obtain the complex $(\Q\otimes_\Z P_\bullet)\otimes_\Lam B$\hs:
\begin{equation}\label{e:QPB}
\quad\dots\to (\Q\otimes_\Z P_2)\otimes_\Lam B\to (\Q\otimes_\Z P_1)\otimes_\Lam B
\to (\Q\otimes_\Z P_0)\otimes_\Lam B\to\Q\otimes_\Lam B\to 0.
\end{equation}
By definition, $\Tor_1^\Lam(\Q,B)$ is the first homology group of this complex.

However, we can obtain the complex~\eqref{e:QPB} from $P_\bullet$ by tensoring first with $B$ over $\Lam$, and after that with $\Q$ over $\Z$:
\[
\Q\otimes_\Z\big(P_\bullet\otimes _\Lam B\big)\hs\cong\hs (\Q\otimes_\Z P_\bullet)\otimes_\Lam B.
\]
Since $\Q$ is a flat $\Z$-module, we obtain canonical isomorphisms
\[
\Tor_1^\Lam(\Q,B)\cong \Q\otimes_\Z \Tor_1^\Lam(\Z,B)=\Q\otimes_\Z H_1(\G,B).
\]

Now, since the group $\G$ is finite, the abelian group $H_1(\G,B)$ is killed by multiplication by $\#\G$; see, for instance, Atiyah and Wall~\cite[Section~6, Corollary~1 of Proposition~8]{AW}. It follows that $\Q\otimes_\Z H_1(\G,B)=0$. Thus $\Tor_1^\Lam (\Q,B)=0$, which completes the proofs of Lemma~\ref{l:Tor-Q}, Proposition~\ref{p:Q/Z-B-BGt}, and Theorem~\ref{t:exact}.
\end{proof}

Alternatively, one can check directly that the map $\delta$ constructed in Subsection~\ref{ss-delta} is well-defined (does not depend on the choices made) and that the sequence~\eqref{e:1-2-3-long} is exact.


\section{Surjectivity for a reductive group with nice radical}\label{s:PR}

In this section we prove the following theorem that gives a sufficient condition for the surjectivity of the localization map~\eqref{e:loc-S} for a reductive $F$-group $G$ in terms of the radical (largest central torus) of $G$.

\begin{theo}\label{t:PR-generalized}
Let $G$ be a reductive group over a number field $F$, and let $C$ denote the radical of $G$. Write $\ov G=G/C$, which is a semisimple group, and consider the short exact sequence of fundamental groups~\cite[Lemma~1.5]{Borovoi-Memoir}
\[
0\to M_C\to M\to \ov M\to 0
\]
where
\[
M_C=\pi_1 (C)=\X_*(C),\quad M=\pi_1 (G),\quad \ov M=\pi_1 (\ov G).
\]
We define $\Gamma=\Gal(E/F)$ for $M$ as in Subsection~\ref{ss:local-results}. Let $S\subset \Vm(F)$ be a subset, and assume that $\SC$ contains a \emph{finite} place $v_0$ such that
\begin{equation}\label{e:Z-Gw-G}
\im\hs [\Gw\to \Aut M_C] = \im\hs [\G\to \Aut M_C]
\end{equation}
where $w$ is a place of $E$ over $v_0$\hs. Then the localization map $\loc_S$ of~\eqref{e:loc-S} is surjective.
\end{theo}

\begin{proof}
It follows from~\eqref{e:Z-Gw-G} that $(M_C)_\Gw=(M_C)_\G$\hs, whence
\[
(M_C)_\Gwt=(M_C)_\Gt\quad\ \text{and}\quad\ \Q/\Z\otimes_\Z (M_C)_\Gw=\Q/\Z\otimes_\Z(M_C)_\G\hs.
\]
Using Theorem~\ref{t:exact}, we construct an exact commutative diagram
\[
\xymatrix{ (M_C)_\Gwt \ar[r]\ar@{=}[d] &M_\Gwt\ar[r]\ar[d]^-\omega &\ov M_\Gwt\ar[r]\ar[d]^-{\ov\omega} &\Q/\Z\otimes_\Z\!(M_C)_\Gw \ar@{=}[d] \\
(M_C)_\Gt \ar[r] &M_\Gt\ar[r] &\ov M_\Gt\ar[r] &\Q/\Z\otimes_\Z\!(M_C)_\G }
\]
Since $\ov G$ is semisimple, its algebraic fundamental group $\ov M$ is finite, and therefore the homomorphism $\ov\omega$ in the diagram above is surjective; see the proof of Proposition~\ref{p:semisimple}. By a four lemma, the homomorphism
\[
\omega=\omega_{v_0}\colon\hs M_\Gwt\to M_\Gt
\]
is surjective as well. Since $v_0$ is finite, the map
\[
\alpha_{v_0}\colon\, H^1(F_{v_0},G)\to H^1_\ab(F_{v_0},G)\to M_\Gwt
\]
is bijective, and therefore the map
\[
\lambda_{v_0}\colon\, H^1(F_{v_0},G)\to H^1_\ab(F_{v_0},G)\to M_\Gwt \labelt\omega M_\Gt
\]
is surjective. We conclude by Corollary~\ref{c:v0-sur}.
\end{proof}

\begin{coro}[{Prasad and Rapinchuk~\cite[Proposition~2(a)]{Prasad-Rapinchuk}}\hs]
Let $G$ be a reductive group over a number field $F$, and let $C$ denote the radical of $G$. Assume that the $F$-torus $C$ is split and that $\SC$ contains a finite place $v_0$. Then the localization map $\loc_S$ of~\eqref{e:loc-S} is surjective.
\end{coro}

\begin{proof}
We define $E$, $\G$, and $\Gw$ for $M=\pi_1 (G)$ as in Subsection~\ref{ss:local-results}. Then $\im\hs [\G\to \Aut M_C]=\{1\}$, and hence~\eqref{e:Z-Gw-G} holds. We conclude by Theorem~\ref{t:PR-generalized}.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{c:Prasad-Rapinchuk}]
We define $E$, $\G$, and $\Gw$ for $M=\pi_1 (G)$ as in Subsection~\ref{ss:local-results}. We have
\[
\im \big[\Gw\to \Aut M_C\big]\hs\subseteq\hs\im\big[\G\to \Aut M_C\big],
\quad\ \#\im\big[\Gw\to \Aut M_C\big]\ \big |\ p,
\quad\ \im\big[\Gw\to \Aut M_C\big]\neq \{1\}.
\]
It follows that~\eqref{e:Z-Gw-G} holds. We conclude by Theorem~\ref{t:PR-generalized}.
\end{proof}



\appendix




\section{Abelianization}\label{s:ab}

\subsec{}\label{ss:G-Gss-Gsc}
Let $G$ be a reductive group over a field $F$ \emph{of arbitrary characteristic}. We consider the homomorphism $\rho\colon G^\ssc\to G$ of Subsection~\ref{ss:pi1}.

The group $G$ acts by conjugation on itself on the left, and by functoriality $G$ acts on $G^\ssc$. We obtain an action
\[
\theta\colon G\times G^\ssc\to G^\ssc,\quad\ (g,s)\mapsto {}^g\! s.
\]
On $\Fbar$-points, if $s\in G^\ssc(\Fbar)$, $g_1\in G(\Fbar)$, $g_1=\rho(s_1)\cdot z_1$ with $s_1\in G^\ssc(\Fbar)$, $z_1\in Z_G(\Fbar)$, then
\[
\theta(g_1,s)={}^{g_1}\hm s= s_1 s s_1^{-1}.
\]
Since the groups $G$ and $G^\ssc$ are smooth, this formula uniquely determines $\theta$. The action $\theta$ has the following properties:
\begin{align*}
^{\rho(s)}\hm s'&=s\hs s' s^{-1},\\
\rho({}^{g_1}\hm s')&=g_1\hs \rho(s') g_1^{-1}
\end{align*}
for $g_1\in G(\Fbar),\ s,s'\in G^\ssc(\Fbar)$. In other words, $(G^\ssc,G,\rho,\theta)$ is a (left) \emph{crossed module of algebraic groups}; see for instance~\cite[Definition~3.2.1]{Borovoi-Memoir}. We write it as $\big(\GscR \theta\big)$, and we regard it as a complex in degrees $-1,\, 0.$ On $\Fs$-points we obtain a $\Gal(\Fs/F)$-equivariant crossed module $\big(\GscF \theta\big)$ where $\Fs$ is the separable closure of $F$ in $\Fbar$.
\esubsec

\subsec{} Deligne~\cite[Section~2.0.2]{Deligne} noticed that the commutator map
\[
[{\,\cdot\,,\cdot\,}]\colon G\times G\to G,\quad\ g_1,g_2\mapsto [g_1,g_2]\coloneqq g_1 g_2 g_1^{-1} g_2^{-1}
\]
lifts to a certain map (morphism of $F$-varieties)
\[
\{{\,\cdot\,,\cdot\,}\}\colon G\times G\to G^\ssc,\quad\ g_1,g_2\mapsto \{g_1,g_2\}
\]
as follows. The commutator map
\[
G^\ssc\times G^\ssc\to G^\ssc,\quad\ s_1,s_2\mapsto [s_1,s_2]\coloneqq s_1 s_2 s_1^{-1} s_2^{-1}
\]
clearly factors via a morphism of $F$-varieties
\[
(G^\ssc)^\ad\times (G^\ssc)^\ad\to G^\ssc
\]
where $(G^\ssc)^\ad=G^\ssc/Z_{G^\ssc}$ and $Z_{G^\ssc}$ denotes the center of $G^\ssc$. Identifying $(G^\ssc)^\ad$ with $G^\ad\coloneqq G/Z_G$\hs, we obtain the desired morphism of $F$-varieties
\[
\{{\,\cdot\,,\cdot\,}\}\colon G\times G\to G^\ad\times G^\ad\to G^\ssc.
\]
On $\Fbar$-points, if $g_1,g_2\in G(\Fbar),\ g_1=\rho(s_1) z_1\hs,\ g_2=\rho(s_2) z_2$ where $s_1,s_2\in G^\ssc(\Fbar),\ z_1,z_2\in Z_G(\Fbar)$, then
\[
\{g_1,g_2\}=[s_1,s_2]=s_1s_2s_1^{-1}s_2^{-1}.
\]
Since $G$ and $G^\ssc$ are smooth, this formula uniquely determines $\{{\,\cdot\,,\cdot\,}\}$. The constructed map $\{{\,\cdot\,,\cdot\,}\}$ satisfies the following equalities of Conduch\'e~\cite[(3.11)]{Conduche}):
\begin{align*}
&\rho\big(\{g_1,g_2\}\big)=[g_1,g_2];\\
&\big\{\rho(s_1),\rho(s_2)\big\}=[s_1,s_2];\\
&\{g_1,g_2\}=\{g_2,g_1\}^{-1};\\
&\{g_1g_2,\hs g_3\}=\{g_1 g_2 g_1^{-1},\,g_1 g_3 g_1^{-1}\}\hs\{g_1,g_3\}.
\end{align*}
In other words, the map $\{\,,\}$ is a \emph{symmetric braiding} of the crossed module $(G^\ssc,G, \rho,\theta)$. We denote by $G_\ab$ the corresponding \emph{stable} (=symmetrically braided) crossed module:
\[
G_\ab=\big(\GscR \theta,\{{\,\cdot\,,\cdot\,}\}\hs\big).
\]

Let $\varphi\colon G\to H$ be a homomorphism of reductive $F$-groups. It induces a homomorphism $\varphi^\ssc\colon G^\ssc\to H^\ssc$. It is easy to see that
\[
{}^{\varphi(g)}\varphi^\ssc(s)=\varphi^\ssc(\hs^g\hm s)\quad\ \text{for all }\ \,g\in G(\Fbar),\, s\in G^\ssc(\Fbar).
\]
Thus we obtain a morphism of crossed modules
\[
(G^\ssc\to G,\hs \theta_G)\,\to\,(H^\ssc\to H,\hs \theta_H)
\]
with obvious notations. Moreover, we have
\[
\big\{\varphi(g_1),\varphi(g_2)\big\}_H= \varphi^\ssc\big(\{g_1,g_2\}_G\big) \quad\ \text{for all}\ \,g_1,g_2\in G(\Fbar)
\]
with obvious notations; see~\cite{Borovoi-MO} for a proof. Thus we obtain a morphism of stable crossed modules
\begin{equation}\label{e:varphi-stable}
\big(G^\ssc\!\to G,\,\theta_G,\{{\,\cdot\,,\cdot\,}\}_G\hs\big)\,\lra\, \big(H^\ssc\!\to H,\, \theta_H,\{{\,\cdot\,,\cdot\,}\}_H\hs\big).
\end{equation}
\esubsec

%\begin{enonce}[definition]{}
\subsec{}
\label{ss:ab}
In this appendix, we denote by $H^1$ and $\H^1$ the first \emph{Galois} cohomology and hypercohomology. One can define the first Galois (hyper)cohomology of the $\Gal(\Fs/F)$-equivariant crossed module
\begin{equation}\label{e:BN}
\H^1(F,\hs \GscR\theta)\coloneqq
\H^1(\Gal(\Fs/F),\GscF \theta\hs\big);
\end{equation}
see~\cite[Section~3]{Borovoi-Memoir} or Noohi~\cite[Section~4]{Noohi}. A priori it is just a pointed set. However, using the symmetric braiding $\{{\,\cdot\,,\cdot\,}\}$, one can define a structure of abelian group on the pointed set~\eqref{e:BN}; see Noohi~\cite[Corollaries~4.2 and 4.5]{Noohi}. We denote the obtained abelian group by
\[
H^1_\ab(F,G)=\H^1(F, G_\ab)\coloneqq \H^1\bigl(\Gal(\Fs/F),\hs \GscF \theta,\hs\{{\,\cdot\,,\cdot\,}\}\hs\big).
\]
A homomorphism of reductive $F$-groups $\varphi\colon G\to H$ induces a morphism of stable crossed modules~\eqref{e:varphi-stable}, which in turn induces a homomorphism of abelian groups
\[
\varphi_\ab\colon H^1_\ab(F,G)\to H^1_\ab(F, H).
\]
Thus $G\rightsquigarrow H^1_\ab(F,G)$ is a functor from the category of reductive $F$-group to the category of abelian groups.
%\end{enonce}
\esubsec

\subsec{} The morphism of crossed modules (but not of stable crossed modules)
\[
i_G\colon (1\to G)\,\into\, (G^\ssc\to G)
\]
induces a morphism of pointed sets
\[
(i_G)_*\colon \H^1(F, 1\to G)\to \H^1(F,\hs G^\ssc\!\labelt\rho G).
\]
The \emph{abelianization map} is the composite morphism of pointed sets
\[
\ab\colon\, H^1(F,G)=\H^1(F, 1\to G)\,\labeltoo{(i_G)_*}\,
\H^1\big(F, \GscR\theta\big)=\H^1(F,G_\ab)\eqqcolon H^1_\ab(F,G).
\]
Here $\H^1\big(F, \GscR\theta\big)$ and $\H^1(F,G_\ab)$ are the same sets, but $\H^1(F,G_\ab)$ is endowed with the structure of abelian group coming from the symmetric braiding $\{{\,\cdot\,,\cdot\,}\}$.
\esubsec

%\begin{enonce}[definition]{}
\subsec{}
For a maximal torus $T\subseteq G$, we consider the homomorphism
\[
\rho\colon T^\ssc\to T
\]
of Subsection~\ref{ss:pi1}, which we regard as a stable crossed module with the trivial action $\theta_T$ of $T$ on $T^\ssc$ and the trivial symmetric braiding \,$\{{\,\cdot\,,\cdot\,}\}_T\colon T\times T\to T^\ssc$. We may and shall identify the first Galois hypercohomology of this stable crossed module with the usual first Galois hypercohomology of the complex $T^\ssc\!\labelt\rho T$ in degrees $-1,\,0$:
\[
\H^1\bigl(F,\, T^\ssc\!\labelt\rho T,\,\theta_T, \{{\,\cdot\,,\cdot\,}\}_T\bigr)=\H^1(F,T^\ssc\!\labelt\rho T).
\]
The morphism of stable crossed modules
\begin{equation}\label{e:iota}
j_T\colon\, \bigl(T^\ssc\!\labelt\rho T,\,\theta_T, \{{\,\cdot\,,\cdot\,}\}_T\bigr)\,\into\, \bigl(\hs G^\ssc\!\labelt\rho G,\,\theta, \{{\,\cdot\,,\cdot\,}\}\bigr)
\end{equation}
is an \emph{equivalence} (quasi-isomorphism), that is, it induces isomorphisms of $F$-group schemes
\[
\ker[T^\ssc\!\to T]\isoto \ker[G^\ssc\!\to G]\quad\ \text{and}
\quad \coker[T^\ssc\!\to T]\isoto \coker[G^\ssc\!\to G].
\]
Following an idea sketched by Labesse and Lemaire~\cite{LL}, we observe that~\eqref{e:iota} induces iso\-morphisms on groups of $\Fs$-points
\begin{align*}
\ker\bigl[T^\ssc(\Fs)\to T(\Fs)\big]&\isoto \ker\bigl[G^\ssc(\Fs)\to G(\Fs)\big]\\
\coker\bigl[T^\ssc(\Fs)\to T(\Fs)\big]&\isoto \coker\bigl[G^\ssc(\Fs)\to G(\Fs)\big]
\end{align*}
(in arbitrary characteristic); see Theorem~\ref{t:Zev} in Appendix~\ref{s:Zev} below. It follows that the induced map on Galois gypercohomology
\begin{equation*}
(j_T)_*\colon\, \H^1(F,T^\ssc\!\to T)\,\lra\,
\H^1\bigl(F,\GscR \theta, \{{\,\cdot\,,\cdot\,}\}\bigr)\eqqcolon H^1_\ab(F,G)
\end{equation*}
is an isomorphism of abelian groups; see Noohi~\cite[Proposition~5.6]{Noohi}. This shows that the abelian group structure on the pointed set $\H^1\big(F, \GscR\theta\big)$ defined using the bijection $(j_T)_*$ (as in~\cite[Section~3.8]{Borovoi-Memoir}\hs) coincides with the abelian group structure defined by the symmetric braiding $\{{\,\cdot\,,\cdot\,}\}$.
%\end{enonce}
\esubsec

\begin{rema}
Gonz\'alez-Avil\'es~\cite{GA} defined the abelian fppf cohomology group $H^1_{\fppf,\hs\ab}(X,G)$ and the abelianization map
\[
\ab\colon H^1_\fppf(X,G)\to H^1_{\fppf,\hs\ab}(X,G)
\]
for a reductive group scheme $G$ over an arbitrary base scheme $X$, which includes the case of a reductive group over a field $F$ of arbitrary characteristic. However, his definition uses the center $Z_G$ of $G$, and hence it is functorial only with respect to the \emph{normal} homomorphisms $G_1\to G_2$ (homomorphisms with normal image, hence sending $Z_{G_1}$ to $Z_{G_2}$)\hs), whereas our definition above (over a field only) is functorial with respect to all homomorphisms.
\end{rema}


\section{Equivalence on \texorpdfstring{$\Fs$}{Fs}-points in arbitrary characteristic\\ \emph{written by Zev Rosengarten}}\label{s:Zev}

In this appendix we prove the following theorem:

\begin{theo}\label{t:Zev}
Let $F$ be a field of arbitrary characteristic and let $\Fs$ be a fixed separable closure of $F$. Let
\[
\rho\colon G^\ssc\onto [G,G]\into G
\]
be as in Subsection~\ref{ss:pi1}. Let $T\subseteq G$ be a maximal torus. We write $T^\ssc=\rho^{-1}(T)$. Then the morphism of crossed modules
\[
\bigl(T^\ssc(\Fs)\to T(\Fs)\bigr)\,\lra\, \bigl(G^\ssc(\Fs)\to G(\Fs)\bigr)
\]
is an equivalence (quasi-isomorphism).
\end{theo}

\begin{proof}
We must show that the maps
\begin{equation}\label{e:ker}
i_{\ker}\colon \ker\bigl[T^\ssc(\Fs) \to T(\Fs)\bigr] \,\lra\, \ker\bigl[G^\ssc(\Fs) \to G(\Fs)\bigr]
\end{equation}
and
\begin{equation}\label{e:coker}
i_{\cok}\colon \coker\bigl[T^\ssc(\Fs) \to T(\Fs)\bigr] \,\lra\, \coker\bigl[G^\ssc(\Fs) \to G(\Fs)\bigr]
\end{equation}
are isomorphisms.

For~\eqref{e:ker}, the injectivity is obvious. Moreover, any element of $\ker\big[G^\ssc(\Fs) \to G(\Fs)\big]$ lies in the preimage $T^\ssc$ of $T$, hence it is an element of $T^\ssc(\Fs)$ and of $\ker\big[T^\ssc(\Fs) \to T(\Fs)\big]$, which gives the surjectivity of $i_{\ker}$.

We prove the injectivity of~\eqref{e:coker}. Let $[t]\in \coker\big[T^\ssc(\Fs) \to T(\Fs)\big]$, $t\in T(\Fs)$, and $[t]\in\ker i_\cok$\hs; then $t=\rho(s)$ for some $s\in G^\ssc(\Fs)$. Since $T^\ssc=\rho^{-1}(T)$, we see that $s\in T^\ssc(\Fs)$, whence $[t]=1$, as required.

We prove the surjectivity of~\eqref{e:coker}. Let $C \subseteq G$ denote the radical (largest central torus) of $G$. Then the map
\[
\psi\colon C \times G^\ssc \to G,\qquad (c,s)\mapsto c\cdot \rho(s)\ \,\text{for}\ c\in C,\, s\in G^\ssc
\]
is surjective with central kernel $Z\cong \rho^{-1}(C\cap [G,G])$ (which might be non-smooth). We have an exact commutative diagram of $F$-group schemes
\[
\xymatrix@R=6mm{ 1 \ar[r] &Z\ar[r]\ar@{=}[d] &C \times T^\ssc\ar[r]^-{\psi_T}\ar[d] &T\ar[r]\ar[d] &1 \\
1 \ar[r] &Z\ar[r] &C \times G^\ssc\ar[r]^-\psi &G \ar[r] &1 }
\]
in which the maps on $\Fs$-points
\[
\psi_T\colon C(\Fs)\times T^\ssc(\Fs)\to T(\Fs)\quad\ \text{and}
\quad\ \psi\colon C(\Fs)\times G^\ssc(\Fs)\to G(\Fs)
\]
might not be surjective. This diagram gives rise to an exact commutative diagram of fppf cohomology groups
\[
\xymatrix@R=6mm{ C(\Fs) \times T^\ssc(\Fs)\ar[r]^-{\psi_T}\ar[d] &T(\Fs)\ar[r]\ar[d] &H_\fppf^1(\Fs, Z)\ar[r]\ar@{=}[d] &H_\fppf^1(\Fs,C \times T^\ssc)=1\ar[d]\\
C(\Fs) \times G^\ssc(\Fs)\ar[r]^-\psi &G(\Fs)\ar[r] &H_\fppf^1(\Fs, Z)\ar[r] &H_\fppf^1(\Fs,C \times G^\ssc)=1 }
\]
in which the rightmost term in both rows is trivial because $\Fs$ is separably closed and the $F$-groups $C \times T^\ssc$, \,$C \times G^\ssc$ are smooth. The latter diagram shows that
\[
G(\Fs) = T(\Fs)\cdot \psi\big(\hs C(\Fs) \times G^\ssc(\Fs)\hs\big)= T(\Fs)\cdot C(\Fs)\cdot\rho\big(G^\ssc(\Fs)\big) = T(\Fs)\cdot \rho\big(G^\ssc(\Fs)\big),
\]
whence the surjectivity of~\eqref{e:coker}.
\end{proof}

\subsection*{Acknowledgements}

The author is very grateful to Vladimir Hinich for his short proof of Theorem~\ref{t:exact} (together with Proposition~\ref{p:Q/Z-B-BGt} and Lemma~\ref{l:Tor-Q}) included in this paper instead of the original lengthy proof of the author, and to Zev Rosengarten for his Appendix~\ref{s:Zev}. Both the author of the article and the author of the appendix thank the anonymous referee for their insightful comments and suggestions.

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