text:
\begin{document}

\title[Proof of Grothendieck--Serre conjecture]{Proof of Grothendieck-Serre conjecture
on principal bundles over regular local rings containing a finite field}

\keywords{Reductive group schemes; Principal bundles\newline
The author acknowledges support of the
RNF-grant 14-11-00456.}

\begin{abstract}
Let $R$ be a regular local ring, containing {\bf a finite field}. Let $\bG$ be a reductive group scheme over~$R$. We prove that a principal $\bG$-bundle over~$R$ is trivial, if it is trivial over the fraction field of $R$. In other words, if $K$ is the fraction field of $R$, then the map of non-abelian cohomology pointed sets
\[
  H^1_{\text{\'et}}(R,\bG)\to H^1_{\text{\'et}}(K,\bG),
\]
induced by the inclusion of $R$ into $K$, has a trivial kernel.

Certain arguments used in the present preprint do not work if the ring $R$ contains
a characteristic zero field.
In that case and, more generally, in the case when the regular local ring $R$ contains {\bf an infinite field} this result is proved in \cite{FP}.

\end{abstract}

\author{Ivan Panin}
\email{paniniv@gmail.com}
\address{Steklov Institute of Mathematics at St.-Petersburg, Fontanka 27, St.-Petersburg 191023, Russia}

\maketitle

\section{Introduction}

Assume that $U$ is a regular scheme, $\bG$ is a reductive $U$-group scheme. Recall that a $U$-scheme $\cG$ with an action of $\bG$ is called \emph{a principal $\bG$-bundle over $U$}, if $\cG$ is faithfully flat and quasi-compact over $U$ and the action is simple transitive, that is, the natural morphism $\bG\times_U\cG\to\cG\times_U\cG$ is an isomorphism, see~\cite[Section~6]{FGA}. It is well known that such a bundle is trivial locally in \'etale topology but in general not in Zariski topology. Grothendieck and Serre conjectured that $\cG$ is trivial locally in Zariski topology, if it is trivial generically. More precisely
\begin{conjecture}
Let $R$ be a regular local ring, let $K$ be its field of fractions. Let $\bG$ be a reductive group scheme over $U:=\spec R$, let $\cG$ be a principal $\bG$-bundle. If $\cG$ is trivial over $\spec K$, then it is trivial. Equivalently, the map of non-abelian cohomology pointed sets
\[
  H^1_{\text{\'et}}(R,\bG)\to H^1_{\text{\'et}}(K,\bG),
\]
induced by the inclusion of $R$ into $K$, has a trivial kernel.
\end{conjecture}
The main result of this paper is a proof of this conjecture for regular semi-local domains $R$, containing {\bf a finite field}.
Our proof was inspired by the preprint
[FP], where the conjecture is proven for semi-local regular domains containing an {\bf infinite} field.
{\it Thus,
the conjecture holds for
semi-local regular domains containing a field}.

The proof in the present preprint uses

\cite[Thm.1.1]{Pan1},
\cite[Thm.1.0.1]{Pan2},
the key ideas of the paper
\cite{FP}
and a Bertini type theorem from
\cite{Poo}.

Our result implies that two principal $\bG$-bundles over $U$ are isomorphic, if they are isomorphic over $\spec K$ as proved in the next section. This result is new even for constant group schemes (that is, for group schemes coming from the ground field).

Recall that a part of the Gersten conjecture asserts that the natural homomorphism of $\mathrm K$-groups $\mathrm K_q(R)\to\mathrm K_q(K)$ is injective. Very roughly speaking, the Grothendieck--Serre conjecture is a non-abelian version of this part of the Gersten conjecture.

\subsection{History of the topic}
Here is a list of known results in the same vein, corroborating the Grothendieck--Serre conjecture.

\smallskip $\bullet$ The case, where the group scheme $\bG$ comes from an infinite ground field, is completely solved by J.-L.~Colliot-Th\'el\`ene, M.~Ojanguren, and M.~S.~Raghunatan in~\cite{C-TO} and \cite{R1,R2}; O.~Gabber announced a proof for group schemes coming from arbitrary ground fields.

\smallskip $\bullet$ The case of an arbitrary reductive group scheme over a discrete valuation ring or over a henselian ring is completely solved by Y.~Nisnevich in~\cite{Ni1}. He also proved the conjecture for two-dimensional local rings in the case, when $\bG$ is quasi-split in~\cite{Ni2}.

\smallskip $\bullet$ The case, where $\bG$ is an arbitrary reductive group scheme over a regular semi-local domain containing an infinite field,
was settled by R.~Fedorov and I.~Panin in~\cite{FP}.

\smallskip $\bullet$ The case, where $\bG$ is an arbitrary torus over a regular local ring, was settled by J.-L.~Colliot-Th\'{e}l\`{e}ne and J.-J.~Sansuc in~\cite{C-T-S}.

\smallskip $\bullet$ For some simple group schemes of classical series the conjecture is solved in works of the author, A.~Suslin, M.~Ojanguren, and K.~Zainoulline; see~\cite{Oj1}, \cite{Oj2}, \cite{PS}, \cite{OP2}, \cite{Z}, \cite{OPZ}.

\smallskip $\bullet$ Under an isotropy condition on $\bG$ and assuming that the ring contains an infinite field the conjecture is proved in a series of preprints~\cite{PSV} and~\cite{Pa2}.

\smallskip $\bullet$ The case of strongly inner simple adjoint group schemes of the types $E_6$ and $E_7$ is done by the second author,
V.~Petrov, and A.~Stavrova in~\cite{PPS}. No isotropy condition is imposed there, however it is supposed that the ring contains an infinite field.

\smallskip $\bullet$ The case, when $\bG$ is of the type $F_4$ with trivial $g_3$-invariant and the field is of characteristic zero, is settled by V.~Chernousov in~\cite{Chernous}; the case, when $\bG$ is of the type $F_4$ with trivial $f_3$-invariant and the field is infinite and perfect, is settled by V.~Petrov and A.~Stavrova in~\cite{PetrovStavrova}.

\subsection{Acknowledgments}

The author thanks A.Suslin for his interest to the topic of the present preprint.

\section{Main results}\label{Introduction}
Let $R$ be a commutative unital ring. Recall that an $R$-group scheme $\bG$ is called \emph{reductive},
if it is affine and smooth as an $R$-scheme and if, moreover,
for each algebraically closed field $\Omega$ and for each ring homomorphism $R\to\Omega$ the scalar extension $\bG_\Omega$ is
a connected reductive algebraic group over $\Omega$. This definition of a reductive $R$-group scheme
coincides with~\cite[Exp.~XIX, Definition~2.7]{SGA3-3}.
A~well-known conjecture due to J.-P.~Serre and A.~Grothendieck
(see~\cite[Remarque, p.31]{Se}, \cite[Remarque 3, p.26-27]{Gr1}, and~\cite[Remarque~1.11.a]{Gr2})
asserts that given a regular local ring $R$ and its field of fractions~$K$ and given a reductive group scheme $\bG$ over $R$, the map
\[
  H^1_{\text{\'et}}(R,\bG)\to H^1_{\text{\'et}}(K,\bG),
\]
induced by the inclusion of $R$ into $K$, has a trivial kernel. The following theorem, which is the main result of the present paper,
asserts that this conjecture holds,
provided that $R$ contains {\bf a finite field}.
If $R$ contains an infinite field, then the conjecture is proved in [FP].

\begin{theorem}\label{MainThm1}
Let $R$ be a regular semi-local domain containing {\bf a finite field}, and let $K$ be its field of fractions. Let $\bG$ be
a reductive group scheme over $R$. Then the map
\[
  H^1_{\text{\'et}}(R,\bG)\to H^1_{\text{\'et}}(K,\bG),
\]
\noindent
induced by the inclusion of $R$ into $K$, has a trivial kernel. In other words, under the above assumptions on $R$ and $\bG$, each principal $\bG$-bundle over $R$ having a $K$-rational point is trivial.
\end{theorem}

Theorem~\ref{MainThm1} has the following
\begin{corollary*}
Under the hypothesis of Theorem~\ref{MainThm1}, the map
\[
  H^1_{\text{\'et}}(R,\bG)\to H^1_{\text{\'et}}(K,\bG),
\]
\noindent
induced by the inclusion of $R$ into $K$, is injective. Equivalently, if $\cG_1$ and $\cG_2$ are two principal bundles isomorphic over $\spec K$, then they are isomorphic.
\end{corollary*}
\begin{proof}
Let $\cG_1$ and $\cG_2$ be two principal $\bG$-bundles isomorphic over $\spec K$. Let $\Iso(\cG_1,\cG_2)$ be the scheme of isomorphisms. This scheme is a principal $\Aut\cG_2$-bundle. By Theorem~\ref{MainThm1} it is trivial, and we see that $\cG_1\cong\cG_2$.
\end{proof}

Note that, while Theorem~\ref{MainThm1} was previously known for reductive group schemes $\bG$ coming from the ground field
(an unpublished result due to O.Gabber), in many cases the corollary is a new result even for such group schemes.

For a scheme $U$ we denote by $\A^1_U$ the affine line over $U$ and by $\P^1_U$ the projective line over $U$. Let $T$ be a $U$-scheme. By a principal $\bG$-bundle over $T$ we understand a principal $\bG\times_UT$-bundle.

In Section~\ref{sect:redtopsv} we deduce Theorem~\ref{MainThm1} from the following result of independent interest (cf.~\cite[Thm.1.3]{PSV}).

\begin{theorem}\label{th:psv}
Let $R$ be the semi-local ring of finitely many closed points on an irreducible smooth affine variety over {\bf a finite field} $k$,
set $U=\spec R$. Let $\bG$ be a simple simply-connected group scheme over $U$ (see~\cite[Exp.~XXIV, Sect.~5.3]{SGA3-3} for the definition).
Let $\cE_t$ be a principal $\bG$-bundle over the affine line $\A^1_U=\spec R[t]$, and let $h(t)\in R[t]$ be a monic polynomial.
Denote by $(\A^1_U)_h$ the open subscheme in $\A^1_U$ given by $h(t)\ne0$ and assume that
the restriction of $\cE_t$ to $(\A^1_U)_h$ is a trivial principal $\bG$-bundle.
Then for each section $s:U\to\A^1_U$ of the projection $\A^1_U\to U$ the $\bG$-bundle $s^*\cE_t$ over $U$ is trivial.
\end{theorem}

The derivation of Theorem~\ref{MainThm1} from Theorem~\ref{th:psv} is based on
\cite[Thm.1.0.1]{Pan2} and~\cite[Thm.1.1]{Pan1}.

Let $Y$ be a semi-local scheme. We will call a simple $Y$-group scheme
quasi-split if its restriction to each connected component of $Y$ contains {\bf a Borel subgroup scheme}.

\begin{theorem}\label{MainThm2}
Let $R$, $U$, and $\bG$ be as in Theorem~\ref{th:psv}.
Let $Z\subset\P^1_U$ be a closed subscheme finite over $U$.
Let $Y\subset\P^1_U$ be a closed subscheme finite and

\'etale over $U$ and such that\\

(i) $\bG_Y:=\bG\times_UY$ is quasi-split, \\
(ii) $Y\cap Z=\emptyset$ and $Y \cap \{\infty\}\times U=\emptyset= Z \cap \{\infty\}\times U$, \\
(iii) for any closed point $u \in U$ one has $Pic(\P^1_u - Y_u)=0$, where $Y_u:=\P^1_u\cap Y$.\\

Let $\cG$ be a~principal $\bG$-bundle over

$\P^1_U$ such that its restriction to

$\P^1_U- Z$ is trivial.
Then the restriction of $\cG$ to

$\P^1_U-Y$ is also trivial.\\
In particular, the principal $\bG$-bundle $\cG$ is trivial locally for the Zarisky topology.
\end{theorem}

The proof of this result is inspired by \cite[Thm.3]{FP}.

\subsection{Organization of the paper}
In Section~\ref{sect:redtopsv}, we reduce Theorem~\ref{MainThm1} to Theorem~\ref{th:psv}. In Section~\ref{sect:reducing},
we reduce Theorem~\ref{th:psv} to Theorem~\ref{MainThm2}. This reduction is based
on~\cite[Thm.1.0.1]{Pan2}, \cite[Thm.1.1]{Pan1},
on a theorem of D.~Popescu~\cite{P}
and on
Proposition \ref{SchemeY}.
The latter proposition is a new ingredient
comparing with respecting arguments from
\cite[Section 4]{FP}.

In Section~\ref{sect:proof2} we prove Theorem~\ref{MainThm2}.

We give an outline of the proof in Section~\ref{sect:outline}.
We use the technique of henselization.

In Section~\ref{sect:application} we give an application of Theorem~\ref{MainThm1}.

In the Appendix we recall the definition of henselization from~\cite[Section~0]{Gabber}.

\section{Reducing Theorem~\ref{MainThm1} to Theorem~\ref{th:psv}}\label{sect:redtopsv}
In what follows ``$\bG$-bundle'' always means ``principal $\bG$-bundle''.  Now we assume that Theorem~\ref{th:psv} holds. We start with the following particular case of Theorem~\ref{MainThm1}.

\begin{proposition}\label{pr:geometric}
Let $R$, $U=\spec R$, and $\bG$ be as in Theorem~\ref{th:psv}. Let $\cE$ be a principal $\bG$-bundle over $U$, trivial at the generic point of $U$. Then $\cE$ is trivial.
\end{proposition}

\begin{proof}
Under the hypothesis of the proposition, the following data are constructed in
\cite[Thm.1.1]{Pan1}:\\
\stepzero\noindstep a principal $\bG$-bundle $\cE_t$ over $\A^1_U$;\\
\noindstep a monic polynomial $h(t)\in R[t]$.\\
Moreover these data satisfies the following conditions:\\
(1) the restriction of $\cE_t$ to $(\A^1_U)_h$ is a trivial principal $\bG$-bundle;\\
(2) there is a section $s:U\to\A^1_U$ such that $s^*\cE_t=\cE$.

Now it follows from Theorem~\ref{th:psv} that $\cE$ is trivial.
\end{proof}

\begin{proposition}\label{pr:reductivegeometric}
Let $U$ be as in Theorem~\ref{th:psv}. Let $\bG$ be a reductive group scheme over $U$.
Let $\cE$ be a principal $\bG$-bundle over $U$ trivial at the generic point of $U$. Then $\cE$ is trivial.
\end{proposition}

\begin{proof}
Firstly, using~\cite[Thm.1.0.1]{Pan2}, we can assume that~$\bG$ is semi-simple and simply-connected. Secondly, standard arguments (see for instance~\cite[Section~9]{PSV}) show that we can assume that $\bG$ is simple and simply-connected. (Note that for this reduction it is necessary to work with semi-local rings.) Now the proposition is reduced to Proposition~\ref{pr:geometric}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{MainThm1}]
Let us prove a general statement first. Let $k'$ be {\bf a finite field}, $X$ be a $k'$-smooth irreducible affine variety, $\bH$ be a reductive group scheme over $X$. Denote by $k'[X]$ the ring of regular functions on $X$ and by $k'(X)$ the field of rational functions on $X$. Let $\cH$ be a principal $\bH$-bundle over $X$ trivial over $k'(X)$. Let $\mathfrak p_1,\dots,\mathfrak p_n$ be prime ideals in $k'[X]$, and let $\cO_{\mathfrak p_1,\dots,\mathfrak p_n}$ be the corresponding semi-local ring.
\begin{lemma}\label{lm:primemax}
The principal $\bH$-bundle $\cH$ is trivial over $\cO_{\mathfrak p_1,\dots,\mathfrak p_n}$.
\end{lemma}
\begin{proof}
For each $i=1,2,\ldots,n$ choose a maximal ideal $\mathfrak m_i\subset k'[X]$ containing $\mathfrak p_i$. One has inclusions of $k'$-algebras
\[
\cO_{\mathfrak m_1,\dots,\mathfrak m_n}\subset\cO_{\mathfrak p_1,\dots,\mathfrak p_n}\subset k'(X).
\]
By Proposition~\ref{pr:reductivegeometric} the principal $\bH$-bundle $\cH$ is trivial over $\cO_{\mathfrak m_1,\dots,\mathfrak m_n}$. Thus it is trivial over $\cO_{\mathfrak p_1,\dots,\mathfrak p_n}$.
\end{proof}

Let us return to our situation. Let $\fm_1,\ldots,\fm_n$ be all the maximal ideals of $R$.
Let $\cE$ be a $\bG$-bundle over $R$ trivial over the fraction field of $R$.
Clearly, there is a non-zero $f\in R$ such that $\cE$ is trivial over $R_f$.

Let $k$ be the prime field of $R$.

Note that $k$ is perfect. It follows from Popescu's theorem (\cite{P,Sw}) that~$R$ is a filtered inductive limit of smooth $k$-algebras $R_\alpha$.
Modifying the inductive system $R_\alpha$ if necessary, we can assume that each $R_\alpha$ is integral.
There are an index $\alpha$, a reductive group scheme $\bG_{\alpha}$ over $R_{\alpha}$,
a principal $\bG_{\alpha}$-bundle $\cE_{\alpha}$ over $R_{\alpha}$,
and an element $f_{\alpha }\in R_{\alpha}$ such that
$\bG=\bG_{\alpha}\times_{\spec R_{\alpha}}\spec R$, $\cE$ is isomorphic to
$\cE_{\alpha}\times_{\spec R_{\alpha}}\spec R$ as principal $\bG$-bundle,
$f$ is the image of $f_{\alpha}$ under the homomorphism
$\phi_{\alpha}: R_{\alpha}\to R$, $\cE_{\alpha}$ is trivial over $(R_{\alpha})_{f_{\alpha}}$.

For each maximal ideal $\mathfrak m_i$ in~$R$ ($i=1,\dots, n$) set $\mathfrak p_i=\phi_{\alpha}^{-1}(\mathfrak m_i)$.
The homomorphism $\phi_\alpha$ induces a homomorphism of semi-local rings $(R_{\alpha})_{\mathfrak p_1,\dots,\mathfrak p_n}\to R$.
By Lemma~\ref{lm:primemax} the principal $\bG_{\alpha}$-bundle $\cE_{\alpha}$ is trivial over
$(R_{\alpha})_{\mathfrak p_1,\dots,\mathfrak p_n}$. Whence the $\bG$-bundle $\cE$ is trivial over $R$.
\end{proof}

\section{Reducing Theorem~\ref{th:psv} to Theorem~\ref{MainThm2}}\label{sect:reducing}
Now we assume that Theorem~\ref{MainThm2} is true. Let $k$, $U$ and $\bG$ be as in Theorem~\ref{th:psv}.
Let $u_1,\ldots,u_n$ be all the closed points of $U$. Let $k(u_i)$ be the residue field of $u_i$. Consider the reduced closed subscheme $\bu$ of $U$,
whose points are $u_1$, \ldots, $u_n$. Thus
\[
 \bu\cong\coprod_i\spec k(u_i).
\]
Set $\bG_\bu=\bG\times_U\bu$. By $\bG_{u_i}$ we denote the fiber of $\bG$ over $u_i$;
it is a simple simply-connected algebraic group over $k(u_i)$.

\begin{proposition}
\label{SchemeY}
Let $Z\subset\A^1_U$ be a closed subscheme finite over $U$.
There is a closed subscheme $Y\subset\A^1_U$ which is \'etale and finite over $U$ and such that \\

(i) $\bG_Y:=\bG\times_UY$ is quasi-split, \\
(ii) $Y\cap Z=\emptyset$,\\
(iii) for any closed point $u \in U$ one has $Pic(\P^1_u - Y_u)=0$, where $Y_u:=\P^1_u\cap Y$.\\
(Note that $Y$ and $Z$ are closed in $\P^1_U$ since they are finite over $U$).

\end{proposition}

\begin{proof}

For every $u_i$ in $\bu$ choose a Borel subgroup $\bB_{u_i}$ in $\bG_{u_i}$.
{\it The laller is possible since the fields $k(u_i)$ are finite.}
Let $\cB$ be the $U$-scheme of Borel subgroup schemes of $\bG$.

It is a smooth projective $U$-scheme (see~\cite[Cor.~3.5, Exp.~XXVI]{SGA3-3}).
The subgroup $\bB_{u_i}$ in $\bG_{u_i}$ is a $k(u_i)$-rational point~$b_i$ in the fibre of $\cB$ over the point $u_i$.
Using a variant of Bertini theorem
({\bf see \cite[Thm.1.2]{Poo}}),
we can find a closed subscheme $Y^{\prime}$ of $\cB$ such that
$Y^{\prime}$ is \'etale over $U$ and all the $b_i$'s are in $Y$
(take an embedding of $\cB$ into a projective space $\P^N_U$ and
intersect $\cB$  with appropriately chosen family of hypersurfaces containing the points $b_i$.
Arguing as in the proof of \cite[Lemma~7.2]{OP2},
we get a scheme $Y^{\prime}$ finite and \'etale over $U$).

For any closed point $u_i$ in $U$ the fibre $Y^{\prime}_{u_i}$ of $Y^{\prime}$ over $u_i$
contains a $k(u_i)$-rational point (it is the point $b_i$).

To continue the proof of the Proposition we need the following
\begin{lemma}
\label{F1F2}
Let $U$ be as in the Proposition.
Let $Z\subset\A^1_U$ be a closed subscheme finite over $U$.
Let $Y^{\prime} \to U$ be a finite \'{e}tale morphism such that
for any closed point $u_i$ in $U$ the fibre $Y^{\prime}_{u_i}$ of $Y^{\prime}$ over $u_i$
contains a $k(u_i)$-rational point. Then there are finite field extensions
$k_1$ and $k_2$ of the finite field $k$ such that \\
(i) the degrees $[k_1: k]$ and $[k_2: k]$ are coprime,\\
(ii) $k(u_i) \otimes_k k_r$ is a field for $r=1$ and $r=2$,\\
(iii) the degrees $[k_1: k]$ and $[k_2: k]$ are strictly greater than any of the degrees
$[k(z): k]$ , where $z$ runs over all closed points of $Z$,\\
(iv) there is a closed embedding of $U$-schemes
$Y^{\prime\prime}=((Y^{\prime}\otimes_k k_1) \coprod (Y^{\prime}\otimes_k k_2)) \xrightarrow{i} \A^1_U$,\\
(v) for $Y=i(Y^{\prime\prime})$ one has $Y \cap Z = \emptyset$,\\
(vi) for any closed point $u_i$ in $U$ one has
$Pic(\P^1_{u_i}-Y_{u_i})=0$.
\end{lemma}
To prove this Lemma note that it's easy to find field extensions $k_1$ and $k_2$
subjecting (i) to (iii). To satisfy (iv) it suffices to require that
for any closed point $u_i$ in $U$ and for $r=1$ and $r=2$
the number of closed points in
$Y^{\prime}_{u_i}\otimes_k k_r$
is the same as the number of closed points in
$Y^{\prime}_{u_i}$,
and to require that for any integer $n>0$ and any closed point $u_i$ in $U$
the number of points
$y \in Y^{\prime\prime}_{u_i}$
with
$[k(y): k(u_i)]=n$
is not more than the number of points
$x \in \A^1_{u_i}$
with
$[k(x): k(u_i)]=n$.
Clearly, these requirements can be satisfied,
which proves the item (iv).

The condition (v) holds for any closed $U$-embedding
$i: Y^{\prime\prime} \hookrightarrow \A^1_U$ from item (iv),
since the property (iii).
The condition (vi) holds since the property (i).

Now complete the proof of Proposition \ref{SchemeY}. Take the $U$-scheme
$Y^{\prime} \subset \bB$ as in the beginning of the proof.
This $U$-scheme $Y^{\prime}$ satisfies the assumption of Lemma
\ref{F1F2}. Take the closed subscheme $Y$ of $\A^1_U$ as in the item (v)
of the Lemma. For this $Y$ the conditions (ii) and (iii) of the Proposition are
obviously satisfied. The condition (i) is satisfied too, since already
it is satisfied for the $U$-scheme $Y^{\prime}$. The Proposition follows.

\end{proof}

\begin{proof}[Proof of Theorem~\ref{th:psv}]
Set $Z:=\{h=0\}\cup s(U)\subset\A^1_U$.
Clearly, $Z$ is finite over $U$.
Since the principal $\bG$-bundle $\cE_t$ is trivial over $(\A^1_U)_h$
it is trivial over $\A^1_U-Z$.
Note that $\{h=0\}$ is closed in $\P^1_U$ and finite over $U$ because $h$ is monic.
Further, $s(U)$ is also closed in $\P^1_U$ and finite over $U$ because it is a zero set of a degree one monic polynomial.
Thus $Z\subset\P^1_U$ is closed and finite over $U$.

Since the principal $\bG$-bundle $\cE_t$ is trivial over $(\A^1_U)_h$, and
$\bG$-bundles can be glued in Zariski topology,
there exists a principal $\bG$-bundle $\cG$ over $\P^1_U$ such that\\
\indent (i) its restriction to $\A^1_U$ coincides with $\cE_t$;\\
\indent (ii) its restriction to $\P^1_U-Z$ is trivial.

Now choose $Y$ in $\A^1_U$ as in Proposition
~\ref{SchemeY}.
Clearly, $Y$ is finite \'{e}tale over $U$ and closed in $\P^1_U$.
Moreover,
$Y \cap \{\infty\}\times U=\emptyset= Z \cap \{\infty\}\times U$
and
$Y\cap Z= \emptyset$.
Applying Theorem~\ref{MainThm2} with this choice of $Y$ and $Z$,
we see that the restriction of $\cG$ to $\P^1_U-Y$ is a trivial $\bG$-bundle.
Since $s(U)$ is in $\A^1_U-Y$
and $\cG|_{\A^1_U}$ coincides with $\cE_t$,
we conclude that $s^*\cE_t$ is a trivial principal $\bG$-bundle over $U$.
\end{proof}

\section{Proof of Theorem~\ref{MainThm2}}\label{sect:proof2}
We will be using notation from Theorem~\ref{MainThm2}.
Let $\bu$

be as in Section~\ref{sect:reducing}. For $u\in\bu$ set $\bG_u=\bG|_u$.

\begin{proposition}\label{pr:trivclsdfbr}
Let $\cE$ be a $\bG$-bundle over $\P^1_U$ such that $\cE|_{\P^1_u}$ is a trivial $\bG_u$-bundle for all $u\in\bu$.
Assume that there exists a closed subscheme $T$ of $\P^1_U$ finite over $U$ such that the restriction of $\cE$ to $\P^1_U-T$ is trivial
and $(\infty \times U) \cap T = \emptyset$. Then $\cE$ is trivial.
\end{proposition}
\begin{proof}
This follows from Theorem~9.6 of~\cite{PSV}, since $\cE|_{(\infty \times U)}$ is a trivial $\bG$-bundle.
\end{proof}

\subsection{An outline of the
proof of Theorem~\ref{MainThm2}}\label{sect:outline}

Our proof of this Theorem almost literally coincides with the proof of
\cite[Thm.3]{FP}. Our arguments are simpler at certain points.

An outline of the proof.

Denote by $Y^h$ the henselization of the pair $(\A_U^1,Y)$, it is a scheme over $\A_U^1$. Let $s:Y\to Y^h$ be the canonical closed embedding, see Section~\ref{sect:distinguishedLimit} for more details. Set $\dot Y^h:=Y^h-s(Y)$. Let $\cG'$ be a $\bG$-bundle over $\P^1_U-Y$. Denote by $\Gl(\cG',\phi)$ the $\bG$-bundle over $\P_U^1$ obtained by gluing $\cG'$ with the trivial $\bG$-bundle $\bG\times_U Y^h$ via a $\bG$-bundle isomorphism $\phi:\bG\times_U\dot Y^h\to\cG'|_{\dot Y^h}$.

Note that the $\bG$-bundle $\cG$ can be presented in the form $\Gl(\cG',\phi)$, where $\cG'= \cG|_{\P^1_U-Y}$. The idea is to show that

\vskip4pt
\newlength{\shortwidth}
\setlength{\shortwidth}{\textwidth}
\addtolength{\shortwidth}{-.5cm}

\noindent ($*$) \emph{\hfill \parbox{\shortwidth}{There is $\alpha\in\bG(\dot Y^h)$ such that
the $\bG_\bu$-bundle $\Gl(\cG',\phi\circ\alpha)|_{\P^1_\bu}$ is trivial (here $\alpha$ is regarded as
an automorphism of the $\bG$-bundle $\bG\times_U\dot Y^h$ given by the right translation by the element $\alpha$).}}

\vskip4pt

If we find $\alpha$ satisfying condition ($*$), then Proposition~\ref{pr:trivclsdfbr}, applied to $T=Y\cup Z$,
shows that the $\bG$-bundle $\Gl(\cG',\phi\circ\alpha)$ is trivial over $\P^1_U$.
On the other hand, its restriction to $\P^1_U-Y$ coincides with the $\bG$-bundle $\cG'=\cG|_{\P^1_U-Y}$.
\emph{Thus $\cG|_{\P^1_U-Y}$ is a trivial $\bG$-bundle}.

To prove ($*$) it suffices to show that\\
\indent (i) the bundle $\cG|_{\P^1_\bu-Y_\bu}$ is trivial;\\
\indent (ii) each element $\gamma_\bu\in\bG_\bu(\dot Y_\bu^h)$ can be written in the form
\[
 \alpha|_{\dot Y_\bu^h}\cdot\beta_\bu|_{\dot Y_\bu^h}
\]
for certain elements $\alpha\in\bG(\dot Y^h)$ and $\beta_\bu\in\bG_\bu(Y_\bu^h)$.

A realization of this plan in details is given below in the paper.

\subsection{Henselization of affine pairs}\label{sect:distinguishedLimit}
We will use the theory of henselian pairs and, in particular,
a notion of a henselization $A^h_I$ of a commutative ring $A$ at an ideal $I$ (see~Appendix and~\cite[Section~0]{Gabber}).
We refer to
\cite[subsection 5.2]{FP}
for the geometric counterpart.
Let $S=\spec A$ be a scheme and $T=\spec(A/I)$ be a closed subscheme.
Let $(T^h,\pi: T^h \to S,s: T \to T^h)$ be
\emph{the henselization of the pair $(S,T)$} (cf. Definition~\ref{def:henzelisation}).
By definition the scheme $T^h$ is {\it affine} and the composite morphism
$\pi \circ s: T \to S$ is the closed embedding $T \hookrightarrow S$.
{\it Recall} that the pair $(T^h,s(T))$ is henselian, which means that for any affine \'etale morphism $\pi:Z\to T^h$,
any section $\sigma$ of $\pi$ over $s(T)$ uniquely extends to a section of $\pi$ over $T^h$.
It is known that $\pi^{-1}(T)=s(T)$.

In the notation of~\cite[Section~0]{Gabber} we have $T^h=\spec A^h_I$, $\pi:T^h\to S$ is induced by the structure of $A$-algebra on $A^h_I$.

{\it Recall three properties of henselization of affine pairs}\\
\indent (i) Let $T$ be a semi-local scheme. Then the henselization commutes with restriction to closed subschemes.
In more details, if $S'\subset S$ is a closed subscheme,
then

there is a natural morphism
$(T\times_SS')^h\to T^h\times_SS'$.
This morphism is an isomorphism and the canonical section $s':T\times_SS'\to(T\times_SS')^h$ coincides under this identification with
\[
    s\times_S\Id_{S'}:T\times_SS'\to T^h\times_SS'.
\]

\indent (ii) If $T=\coprod_i T_i$ is a disjoint union, then $T^h=\coprod_i T_i^h$.\\
\indent (iii) If we replace in a pair $(S,T)$ the scheme $S$ by an \'{e}tale affine neighborhood of $T$, then the
$(T^h,\pi,s)$ {\it remains the same}. In more details,
given a pair $(S,T)$ as above we write temporarily
$(S^{hen}_T, \pi_{S,T}, s_{S,T})$ for $(T^h,\pi,s)$.
If $p: W \to S$ is an \'{e}tale morphism and $t: T \hookrightarrow W$ is such that $p \circ t: T \hookrightarrow S$ coincides
with the closed embedding $T$ into $S$,
then there is a canonical {\it isomorphism}
$\rho: W^{hen}_T \to S^{hen}_T$
of the $S$-schemes
$(W^{hen}_T, \pi_{W,T})$ and $(S^{hen}_T, \pi_{S,T})$
such that
$\rho \circ s_{W,T}=s_{S,T}$.

\subsection{Gluing principal $\bG$-bundles}
Recall that $U=\spec R$, where $R$ is the semi-local ring of finitely many closed points on an irreducible $k$-smooth affine variety over a finite field $k$.
Also, $\bG$ is a simple simply-connected group scheme over $U$, and
$Y$ is a closed subscheme of $\P_U^1$ finite and \'etale over $U$.

{\it We

will assume below in the preprint that} $Y\subset\A_U^1$
(as in the hypotheses of Theorem \ref{MainThm2}).

Let $(Y^h,\pi,s)$ be the henselization of the pair $(\A_U^1,Y)$
and let
$\dot Y^h=Y^h-s(Y)$ and let $in: \A_U^1 \hookrightarrow \P_U^1$ be the open inclusion.

\begin{proposition}\cite{FP}
\label{pr:affine}\stepzero

The schemes $Y^h$ and $\dot Y^h$ are affine.
\end{proposition}

Let us make a general remark. Let $\cF$ be a $\bG$-bundle over a $U$-scheme $T$. By definition, a trivialization of $\cF$ is a $\bG$-equivariant isomorphism $\bG\times_UT\to\cF$. Equivalently, it is a section of the projection $\cF\to T$. If $\phi$ is such a trivialization and $f:T'\to T$ is a $U$-morphism, we get a trivialization $f^*\phi$ of $f^*\cF$. Sometimes we denote this trivialization by $\phi|_{T'}$. We also sometimes call a trivialization of $f^*\cF$ \emph{a trivialization of $\cF$ on $T'$}.

The main cartesian square we will work with is
\begin{equation}
\begin{CD}\label{eq:distinguishedLimit}
\dot Y^h @>>> Y^h\\
@VVV @VV{in\circ\pi}V\\
\P^1_U - Y @>>>\P^1_U.
\end{CD}
\end{equation}

Let $\cA$ be  the category  of pairs $(\cE,\psi)$, where $\cE$ is a $\bG$-bundle on $\P_U^1$, $\psi$ is a trivialization of $\cE|_{Y^h}:=(in\circ\pi)^*\cE$. A morphism between $(\cE,\psi)$ and $(\cE',\psi')$ is an isomorphism $\cE\to\cE'$ compatible with trivializations.

Similarly, let $\cB$ be the category of pairs $(\cE,\psi)$, where $\cE$ is a $\bG$-bundle on $\P_U^1-Y$, $\psi$ is a trivialization of $\cE|_{\dot Y^h}$.

Consider the restriction functor $\Psi:\cA\to\cB$.
\begin{proposition}\cite{FP}
\label{pr:gluing2}
    The functor $\Psi$ is an equivalence of categories.
\end{proposition}

\begin{construction}\cite{FP}
\label{equally_well}
By Proposition~\ref{pr:gluing2} we can choose a functor quasi-inverse to $\Psi$. Fix such a functor $\Theta$. Let $\Lambda$ be the forgetful functor from $\cA$ to the category of $\bG$-bundles over $\P_U^1$. For $(\cE,\psi)\in\cB$ set
\[
    \Gl(\cE,\psi)=\Lambda(\Theta(\cE,\psi)).
\]
Note that $\Gl(\cE,\psi)$ comes with a canonical trivialization over $Y^h$.

Conversely, if $\cE$ is a principal $\bG$-bundle over $\P_U^1$ such that its restriction to $Y^h$ is trivial, then $\cE$ can be represented as $\Gl(\cE',\psi)$, where $\cE'=\cE|_{\P_U^1-Y}$, $\psi$ is a trivialization of $\cE'$ on $\dot Y^h$.
\end{construction}

Let $\bu$ be as in Section~\ref{sect:reducing}, $Y_\bu:=Y\times_U\bu$.
Let $(Y_\bu^h,\pi_\bu,s_\bu)$ be the henselization of $(\A^1_\bu,Y_\bu)$.
Using property~(i) of henselization, we get $Y_\bu^h=Y^h\times_U\bu$.
Thus we have a natural closed embedding $Y_\bu^h\to Y^h$.
Set $\dot Y_\bu^h=Y_\bu^h-s_\bu(Y_\bu)$.
We get a closed embedding
\begin{equation}
\label{dotYdotYh}
\dot Y_\bu^h\hookrightarrow \dot Y^h.
\end{equation}
Thus the pull-back of the cartesian square
(\ref{eq:distinguishedLimit})
by means of the closed embedding $\bu\hookrightarrow U$ has the form
\[
\begin{CD}
\dot Y_\bu^h @>>>Y_\bu^h\\
@VVV @VV in_\bu\circ\pi_\bu V\\
\P^1_\bu - Y_\bu @>>>\P^1_\bu,
\end{CD}
\]
where $in_\bu:\A_\bu^1\to\P_\bu^1$.

Similarly to the above, we can define categories $\cA_\bu$ and $\cB_\bu$ and an equivalence of categories
$\Psi_\bu:\cA_\bu\to\cB_\bu$.

Let $\Theta_\bu$ be a functor quasi-inverse to $\Psi_\bu$ and $\Lambda_\bu$ be the forgetful functor from $\cA_\bu$
to the category of $\bG_\bu$-bundles over $\P_\bu^1$.
Let $\cE_\bu$ be a principal $\bG_\bu$-bundle over $\P^1_\bu-Y_\bu$ and $\psi_\bu$ be a trivialization of $\bG_\bu$ on $\dot Y_\bu^h$.
Set
\[
\Gl_\bu(\cE_\bu,\psi_\bu)=\Lambda_\bu(\Theta_\bu(\cE_\bu,\psi_\bu)).
\]
\begin{lemma}\cite{FP}
\label{basechange_limit}
Let $(\cE,\psi)\in\cB$, and let $\Gl(\cE,\psi)$ be the $\bG$-bundle obtained by Construction~\ref{equally_well}. Then
\[
 \Gl_\bu(\cE|_{\P^1_\bu-Y_\bu},\psi|_{\dot Y_\bu^h}) \ \text{and} \ \Gl(\cE,\psi)|_{\P^1_\bu}
\]
are isomorphic as $\bG_\bu$-bundles over $\P^1_\bu$.
\end{lemma}

\begin{lemma}\cite{FP}
\label{coboundary_limit}
For any $(\cE_\bu,\psi_\bu)\in\cB_\bu$ and any $\beta_\bu\in\bG_\bu(Y_\bu^h)$ the $\bG_\bu$-bundles
\[
 \Gl_\bu(\cE_\bu,\psi_\bu)\ \text{and} \ \Gl_\bu(\cE_\bu,\psi_\bu\circ\beta_\bu|_{\dot Y_\bu^h})
\]
are isomorphic (here $\beta_\bu|_{\dot Y_\bu^h}$ is regarded as an automorphism of the $\bG_\bu$-bundle $\bG_\bu\times_\bu\dot Y_\bu^h$ given by the right translation by $\beta_\bu|_{\dot Y_\bu^h}$).
\end{lemma}

\subsection{Proof of Theorem~\ref{MainThm2}: presentation of $\cG$ in the form $\Gl(\cG',\phi)$}\label{sect:presentation}
Let $U$, $\bG$, $Z$, $Y$ and $\cG$ be as in Theorem~\ref{MainThm2}.

\begin{proposition}\cite{FP}
\label{presentation}
The $\bG$-bundle $\cG$ over $\P^1_U$ is of the form $\Gl(\cG',\phi)$ for the $\bG$-bundle $\cG':=\cG|_{\P^1_U-Y}$ and a
trivialization $\phi$ of $\cG'$ over $\dot Y^h$.
\end{proposition}
\begin{proof}
In view of Construction~\ref{equally_well},
it is enough to prove that the restriction of the principal $\bG$-bundle $\cG$ to $Y^h$ is trivial.
Let us choose a closed subscheme $Z'\subset\A^1_U$ such that $Z'$ contains $Z$, $Z'\cap Y=\emptyset$, and $\A^1_U-Z'$ is affine.
Then $\A^1_U-Z'$ is an affine neighborhood of $Y$. By the property (iii) from subsection
\ref{sect:distinguishedLimit}
the henselization of the pair $(\A^1_U-Z',Y)$ coincides with
the henselization of the pair $(\A^1_U,Y)$.
Since $\cG$ is trivial over $\A^1_U-Z'$, its pull-back to $Y^h$ is trivial too.
The proposition is proved.
\end{proof}

\emph{Our aim is to modify the trivialization $\phi$ via an element
\[
   \alpha\in\bG(\dot Y^h)
\]
so that the $\bG$-bundle $\Gl(\cG',\phi\circ\alpha)$ becomes trivial over} $\P^1_U$.

\subsection{Proof of Theorem~\ref{MainThm2}: proof of property~(i) from the outline}\label{sect:properties_i}
Now we are able to prove property (i) from the outline of the proof. In fact, we will prove the following
modification of
\cite[Lemma 5.11]{FP}.
\begin{lemma}\label{podpravka}
Let $\Gl(\cG',\phi)$ be the presentation of the $\bG$-bundle $\cG$ over~$\P^1_U$ given in
Proposition~\ref{presentation}.
Set $\phi_\bu:=\phi|_{\dot Y_\bu^h}$.
Then there is $\gamma_\bu\in\bG_\bu(\dot Y_\bu^h)$
such that the $\bG_\bu$-bundle
$\Gl_\bu(\cG'|_{\P^1_\bu-Y_\bu},\phi_\bu\circ\gamma_\bu)$
is trivial.
\end{lemma}

\begin{proof}
We show first that $\cG|_{\P^1_\bu-Y_\bu}$ is trivial. One has
\[
    \P^1_\bu= \coprod_{u\in\bu}\P^1_u

\]
For $u\in\bu$ set $Y_u:=Y\times_Uu$, $\bG_u:=\bG\times_Uu$, and $\cG_u:=\cG\times_Uu$.

Take $u\in\bu$. By our assumption on $Y$,
$Pic(\P^1_u-Y_u)=0$.

The $\bG_u$-bundle $\cG_u$ is trivial over $\A^1_u-Z_u$.
Thus, by~\cite[Corollary~3.10(a)]{GilleTorseurs},

it is trivial over $\P^1_u-Y_u$.

We see that
$\cG'|_{\P^1_\bu-Y_\bu}=\cG|_{\P^1_\bu-Y_\bu}$
is trivial.
Choosing a trivialization, we may identify $\phi_\bu$ with an element of
$\bG_\bu(\dot Y_\bu^h)$. Set $\gamma_\bu=\phi_\bu^{-1}$.
By the very choice of~$\gamma_\bu$ the $\bG_\bu$-bundle
$\Gl_\bu(\cG'|_{\P^1_\bu-Y_\bu},\phi_\bu\circ\gamma_\bu)$
is trivial.
\end{proof}

\subsection{Proof of Theorem~\ref{MainThm2}: reduction to property (ii) from the outline}\label{sect:properties_ii}
The aim of this section is to deduce Theorem~\ref{MainThm2} from the following
\begin{proposition}\cite{FP}
\label{alpha}
Each element $\gamma_\bu\in\bG_\bu(\dot Y_\bu^h)$ can be written in the form
\[
 \alpha|_{\dot Y_\bu^h}\cdot\beta_\bu|_{\dot Y_\bu^h}
\]
for certain elements $\alpha\in\bG(\dot Y^h)$ and $\beta_\bu\in\bG_\bu(Y_\bu^h)$.
\end{proposition}
\begin{proof}[Deduction of Theorem~\ref{MainThm2} from Proposition~\ref{alpha}]\cite{FP}
Let $\Gl(\cG',\phi)$ be the presentation of the $\bG$-bundle $\cG$ from Proposition~\ref{presentation}.
Let $\gamma_\bu\in\bG_\bu(\dot Y_\bu^h)$ be the element from Lemma~\ref{podpravka}.
Let $\alpha\in\bG(\dot Y^h)$ and $\beta_\bu\in\bG_\bu(Y_\bu^h)$ be the elements from Proposition~\ref{alpha}. Set
\[
 \cG^{new}=\Gl(\cG',\phi\circ\alpha).
\]
\emph{Claim.} The $\bG$-bundle $\cG^{new}$ is trivial over $\P^1_U$.\\ Indeed, by Lemmas~\ref{basechange_limit} and~\ref{coboundary_limit} one has a chain of isomorphisms of $\bG_\bu$-bundles
\begin{multline*}
  \cG^{new}|_{\P^1_\bu}\cong
  \Gl_\bu(\cG'|_{\P^1_\bu-Y_\bu},\phi_\bu \circ \alpha|_{\dot Y_\bu^h})
\cong\\
\Gl_\bu(\cG'|_{\P^1_\bu-Y_\bu},\phi_\bu \circ
\alpha|_{\dot Y_\bu^h}\circ\beta_\bu|_{\dot Y_\bu^h})=
\Gl_\bu(\cG'|_{\P^1_\bu-Y_\bu}, \phi_\bu \circ \gamma_\bu),
\end{multline*}
which is trivial by the choice of $\gamma_\bu$.
The $\bG$-bundles $\cG|_{\P^1_U-Y}$ and $\cG^{new}|_{\P^1_U-Y}$ coincide
by the very construction of~$\cG^{new}$.
By Proposition~\ref{pr:trivclsdfbr},
applied to $T=Z\cup Y$, the $\bG$-bundle $\cG^{new}$ is trivial.
Whence the claim.

The claim above implies that the $\bG$-bundle $\cG|_{\P^1_U-Y}= \cG^{new}|_{\P^1_U-Y}$ is trivial.
Theorem~\ref{MainThm2} is proved.
\end{proof}

\subsection{End of proof of Theorem~\ref{MainThm2}: proof of property (ii) from the outline}
\emph{In the remaining part of Section~\ref{sect:proof2} we will prove Proposition~\ref{alpha}. This will complete the proof of Theorem~\ref{MainThm2}}.

By our assumption on $Y$, the group scheme $\bG_Y=\bG\times_UY$ is {\bf quasi-split}.
Thus we can and will choose a Borel subgroup scheme $\bB^+$ in $\bG_Y$.

Since $Y$ is an affine scheme,
by~\cite[Exp.~XXVI, Cor.~2.3, Th~4.3.2(a)]{SGA3-3}
there is an opposite to $\bB^+$ Borel subgroup scheme $\bB^-$ in $\bG_Y$. Let $\bU^+$ be the unipotent radical of
$\bB^+$, and let $\bU^-$ be the unipotent radical of $\bB^-$.

\begin{definition}\label{EYi}
We will write $\bE$ for the functor, sending a $Y$-scheme $T$ to the subgroup $\bE(T)$ of the group $\bG_Y(T)=\bG(T)$ generated by the subgroups $\bU^+(T)$ and $\bU^-(T)$ of the group $\bG_Y(T)=\bG(T)$.
\end{definition}

\begin{lemma}
\label{lm:surjectivity}
The functor $\bE$ has the property that for every closed subscheme $S$ in an affine $Y$-scheme $T$ the induced map $\bE(T)\to\bE(S)$ is surjective.
\end{lemma}
\begin{proof}
The restriction maps $\bU^\pm(T)\to\bU^\pm(S)$ are surjective, since $\bU^\pm$ are isomorphic to vector bundles as $Y$-schemes
(see~\cite[Exp.~XXVI, Cor.~2.5]{SGA3-3}).
\end{proof}

Recall that $(Y^h,\pi,s)$ is the henselization of the pair $(\A_U^1,Y)$. Also, $in:\A_U^1\to\P_U^1$ is the embedding. Denote the projection $\A_U^1\to U$ by $pr$ and the projection $\A_Y^1\to Y$ by $pr_Y$.
\begin{lemma}\cite{FP}
\label{lm:retraction}
There is a morphism $r:Y^h\to Y$ making the following diagram commutative
\begin{equation}
\label{eq:retraction}
\begin{CD}
Y^h @>r>> Y\\
@V{in\circ\pi}VV @VV{pr|_{Y}}V\\
\P^1_U @>pr>> U
\end{CD}
\end{equation}
and such that $r\circ s=\Id_Y$.
\end{lemma}

We view $Y^h$ as a $Y$-scheme via $r$. Thus various subschemes of $Y^h$ also become $Y$-schemes. In particular, $\dot Y^h$ and $\dot Y_\bu^h$ are $Y$-schemes, and we can consider
\[
\bE(\dot Y^h)\subset\bG(\dot Y^h) \ \text{ and } \ \bE(\dot Y_\bu^h)\subset\bG(\dot Y_\bu^h)=\bG_\bu(\dot Y_\bu^h).
\]

\begin{lemma}\label{nastia}
\[
    \bG_\bu(\dot Y_\bu^h)=\bE(\dot Y_\bu^h)\bG_\bu(Y_\bu^h).
\]
\end{lemma}
\begin{proof}
Firstly, one has $Y_\bu=\coprod_{u\in\bu}\coprod_{y\in Y_u}y$. (Note that $Y_u$ is a finite scheme.) Thus by property (ii) of henselization, we have
\[
    Y_\bu^h=\coprod_{u\in\bu}\coprod_{y\in Y_u}y^h,\qquad
    \dot Y_\bu^h=\coprod_{u\in\bu}\coprod_{y\in Y_u}\dot y^h,
\]
where $(y^h,\pi_y,s_y)$ is the henselization of the pair

$(\A_u^1,y)$,
$\dot y^h:=y^h-s_y(y)$.
We see that $y^h$ and $\dot y^h$ are subschemes of $Y^h$, so we can view them as $Y$-schemes, and
$\bG_{y^h}:=\bG_Y\times_Y{y^h}$ is {\bf quasi-split}.
Also, $\bE(\dot y^h)$ makes sense as a subgroup of
$\bG(\dot y^h)=\bG_u(\dot y^h)=\bG_{y^h}(\dot y^h)$.

One has
\[
\begin{split}
   \bG_\bu(\dot Y_\bu^h)&=\prod_{u\in\bu}\prod_{y\in Y_u}\bG_u(\dot y^h)=\prod_{u\in\bu}\prod_{y\in Y_u}\bG_{y^h}(\dot y^h),\\
   \bE(\dot Y_\bu^h)&=\prod_{u\in\bu}\prod_{y\in Y_u}\bE(\dot y^h),\\
   \bG_\bu(Y_\bu^h)&=\prod_{u\in\bu}\prod_{y\in Y_u}\bG_u(y^h)=\prod_{u\in\bu}\prod_{y\in Y_u}\bG_{y^h}(y^h).
\end{split}
\]
Thus it suffices for each $u\in\bu$ and each $y\in Y_u$ to check the equality
\[
   \bG_{y^h}(\dot y^h)=\bE(\dot y^h)\bG_{y^h}(y^h).
\]
This equality holds by Fait 4.3 and Lemma 4.5 of~\cite{Gille:BourbakiTalk}.
In fact, $y^h=\spec\mathcal O$,
where $\mathcal O=k(u)[t]_{\mathfrak m_y}^h$ is a henselian discrete valuation ring,
and $\mathfrak m_y\subset k(u)[t]$ is the maximal ideal defining the point $y\in\A^1_u$.

Further, $\dot y^h=\spec L$, where $L$ is the fraction field of $\mathcal O$.
The lemma is proved.
\end{proof}

We have the closed embedding
(\ref{dotYdotYh})
and the scheme
$\dot Y^h$
is affine by Proposition~\ref{pr:affine}.
Recall that we regard
$\dot Y^h$ as a $Y$-scheme via the morphism
$r|_{\dot Y^h}$.
Thus
by Lemma~\ref{lm:surjectivity}

the restriction map $\bE(\dot Y^h)\to\bE(\dot Y_\bu^h)$ is surjective.
Since $\bE(\dot Y^h)\subset\bG(\dot Y^h)$, the proposition
\ref{alpha}
follows. \emph{This completes the proof of Theorem~\ref{MainThm2}}.

\section{An application}\label{sect:application}
The following result is a straightforward consequence of Theorem~\ref{MainThm1} and an exact sequence for \'etale cohomology.
Recall that by our definition a reductive group scheme has geometrically connected fibres.
\begin{theorem}\label{Norms}
Let $R$ be as in Theorem~\ref{MainThm1} and $\bG$ be a reductive $R$-group scheme. Let $\mu:\bG\to\bT$ be a group scheme morphism to an $R$-torus $\bT$ such that $\mu$ is locally in the \'{e}tale topology on $\spec R$ surjective. Assume further that the $R$-group scheme $\bH:=\Ker(\mu)$ is reductive. Let $K$ be the fraction field of $R$. Then the group homomorphism
\[
\bT(R)/\mu(\bG(R))\to\bT(K)/\mu(\bG(K))
\]
is injective.
\end{theorem}
This theorem extends all the known results of this form proven in~\cite{C-TO}, \cite{PS}, \cite{Z}, \cite{OPZ}.

\appendix
\section{\cite{FP}}
For a commutative ring $A$ we denote by $\Rad(A)$ its Jacobson ideal. The following definition one can find in~\cite[Section 0]{Gabber}.
\begin{definition}
If $I$ is an ideal in a commutative ring $A$, then the pair $(A,I)$ is called \emph{henselian\/}, if $I\subset\Rad(A)$ and for every two relatively prime monic polynomials $\bar g, \bar h\in\bar A[t]$, where $\bar A=A/I$, and monic lifting $f\in A[t]$ of $\bar g\bar h$, there exist monic liftings $g,h\in A[t]$ such that $f=gh$. (Two polynomials are called relatively prime, if they generate the unit ideal.)
\end{definition}

\begin{lemma}\cite{FP}
\label{basechhens}
Let $(A,I)$ be a henselian pair with a semi-local ring $A$ and $J\subset A$ be an ideal. Then the pair $(A/J,(I+J)/J)$ is henselian.
\end{lemma}

The following definition one can find in~\cite[Section 0]{Gabber}.
\begin{definition}\label{def:henzelisation}
The henselization of any pair $(A,I)$ is the pair $(A_I^h,I^h)$ (over $(A,I)$) defined as follows
\[
(A_I^h,I^h):=\text{the filtered inductive limit over the category $\mathcal N$ of }(A',\Ker(\sigma)),
\]
where $\mathcal N$ is the filtered category of pairs $(A',\sigma)$ such that $A'$ is an \'{e}tale $A$-algebra and $\sigma\in\text{Hom}_{A-alg}(A',A/I)$.
\end{definition}


TITLE: What is the first modern maths text?
QUESTION [3 upvotes]: I am aware that this question might appear subjective to some, but inasmuch as there is a definite difference between the algorithmic/algebraic type of mathematics practised by Euler (just like there is a definite difference between that mathematics and the Euclidean geometrical tradition), it appears reasonable to suppose that this shift was caused by some major work(s) which spread through to the rest of mathematics.
My questions are:

What was this particular work (or works) which started the modern abstract style of mathematics?
Is the difference in style to be attributed to the new branches of mathematics themselves, or is it more a difference of method (modus operandi)? In other words, is it possible to treat say Euclidean geometry qua Euclidean geometry but using the modern style, or is the ancient geometrical method somehow embedded/inseparable from Euclidean geometry?

REPLY [1 votes]: I don't know about first, but I'd venture to say Moderne Algebra by van der Waerden, because of its impact.
Wikipedia says this about it:

The book was one of the first textbooks to use an abstract axiomatic approach to groups, rings, and fields, and was by far the most successful, becoming the standard reference for graduate algebra for several decades. It "had a tremendous impact, and is widely considered to be the major text on algebra in the twentieth century"

Also, A Course of Pure Mathematics by Hardy.