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\begin{document}

\title
{On Grothendieck---Serre's conjecture concerning
principal $G$-bundles over reductive group schemes:I}

\author{I.~Panin\footnote{The author acknowledges support of the
RFFI project 09-01-91333-NNIO-a}\ ,
A.~Stavrova\footnote{The author acknowledges support of the
RFFI project 09-01-00878}\ ,
N.~Vavilov\footnote{The author acknowledges support of the
RFFI projects 08-01-00756, 09-01-00762, 09-01-00784, 09-01-00878
and 09-01-91333} }

\date{07.05.2009}

\maketitle

\begin{abstract}
Let $R$ be a semi-local regular domain containing an infinite
perfect subfield and let $K$ be its field of fractions. Let $G$ be a
reductive semi-simple simply connected $R$-group scheme such that
each of its $R$-indecomposable factors is isotropic.
%%contains a split torus} $\Bbb G_{m,R}$.
We prove that in this case the kernel of the map
$$ H^1_{\text{\'et}}(R,G)\to H^1_{\text{\'et}}(K,G) $$
\noindent
induced by the inclusion of $R$ into $K$ is trivial. In other
words, under the above assumptions every principal $G$-bundle $P$
which has a $K$-rational point is itself trivial. This confirms a
conjecture posed by Serre and Grothendieck. Our proof is based on
a combination of methods of Raghunathan's paper \cite{R1},
Ojanguren---Panin's paper \cite{OP1} and Panin's preprint
\cite{Pa1}.
\par
If $R$ is the semi-local ring of several points on
a $k$-smooth scheme, then it suffices to require that $k$ is infinite
and keep the same assumption concerning $G$.

\end{abstract}


\section{Introduction}
\label{Introduction}

Recall that an $R$-group scheme $G$ is called reductive
(respectively, semi-simple or simple), if it is affine and smooth
as an $R$-scheme and if, moreover, for each ring homomorphism
$s:R\to\Omega(s)$ to an algebraically closed field $\Omega(s)$,
its scalar extension $G_{\Omega(s)}$
%% of $G$ via the homomorphism $s$
is a reductive (respectively, semi-simple or simple) algebraic
group over $\Omega(s)$. The class of reductive group schemes
contains the class of semi-simple group schemes which in turn
contains the class of simple group schemes. This notion of a
simple $R$-group scheme coincides with the notion of a {simple
semi-simple $R$-group scheme from Demazure---Grothendieck
\cite[Exp.~XIX, Defn.~2.7 and Exp.~XXIV, 5.3]{D-G}.} {\it
Throughout the paper $R$ denotes an integral domain and $G$
denotes a semi-simple $R$-group scheme, unless explicitly stated
otherwise}.
\par
Such an $R$-group scheme $G$ is called {\it simply-connected\/}
(respectively, {\it adjoint\/}), provided that for an inclusion
$s:R\hra\Omega(s)$ of $R$ into an algebraically closed field
$\Omega(s)$ the scalar extension $G_{\Omega(s)}$ is a
simply-connected (respectively, adjoint) $\Omega(s)$-group scheme.
This definition coincides with the one from \cite[Exp.~XXII.
Defn.4.3.3]{D-G}.
\par
%To each such $R$-group scheme $G$ and each ring homomorphism
%$s: R \to \Omega(s)$ to an algebraically closed field
%$\Omega(s)$ one can canonically assign the root system
%$\Phi(G_{\Omega(s)})$ of the group $G_{\Omega(s)}$.
%If $R$ is a domain, then the root system does not depend of
%a choice of that homomorphisms $s$. So, for a domain $R$ we call
\par
A well-known conjecture due to J.-P.~Serre and A.~Grothendieck
\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 $G$ over $R$ the
map
$$ H^1_{\text{\'et}}(R,G)\to H^1_{\text{\'et}}(K,G), $$
\noindent
induced by the inclusion of $R$ into $K$, has trivial kernel. The
following theorem, which is the main result of the present paper,
asserts that for isotropic groups this is indeed the case
({\it recall that a simple
$R$-group scheme is called isotropic if it contains a split tori}
$\Bbb G_{m,R}$).
\begin{thm}
\label{MainThm} Let $R$ be regular semi-local domain containing an
infinite perfect field and let $K$ be its field of fractions. Let $G$ be
an isotropic simple simply-connected group scheme over $R$.
% containing a split rank $1$ torus $\Bbb G_m$.
Then the map
$$ H^1_{\text{\rm\'et}}(R,G)\to H^1_{\text{\rm\'et}}(K,G), $$
\noindent
induced by the inclusion $R$ into $K$, has trivial kernel.
\end{thm}
In other words, under the above assumptions on $R$ and $G$ each
principal $G$-bundle $P$ over $R$ which has a $K$-rational point
is itself trivial. More generally, it is natural to ask, whether
two elements $\xi,\zeta\in H^1_{\text{\rm\'et}}(R,G)$ which are
equal over $K$ are already equal over $R$. In general, this does
not follow from Theorem \ref{MainThm}. However, this is indeed the
case provided at least one of the group schemes $G(\xi)$ or
$G(\zeta)$ is isotropic.
%???Query: Does it imply that two $G$-bundles
%isomorphic over $K$ are indeed isomorphic. N.V.???
\par
Theorem \ref{MainThm} can be extended to the case of semi-simple
group schemes as well. However, in this generality its statement
is a bit more technical and we postpone it till
Section~\ref{SectSemi-Simple} (see
Theorem~\ref{MainThmSemi-Simple}). Let us list other known results
in the same vein, corroborating Serre---Grothendieck's conjecture.
\par\smallskip
$\bullet$ For simple simply connected group schemes of {\it
classical\/} series this result follows from more general results
established by the first author, A.~Suslin, M.~Ojanguren and
K.~Zainoulline \cite{PS}, \cite{OP1}, \cite{Z}, \cite{OPZ},
%\cite{Pa1}.
In fact, unlike our Theorem \ref{MainThm}, {\it no isotropy
hypotheses} was imposed there. However, for the {\it exceptional\/}
group schemes {\it our theorem is new\/}. Our proof is based on different
ideas and treats classical and exceptional types in a unified way.
\par\smallskip
$\bullet$ The case of an arbitrary reductive group scheme over a
discrete valuation ring is completely solved by Y.~Nisnevich in
\cite{Ni}.
\par\smallskip
$\bullet$ The case where $G$ 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}.
\par\smallskip
$\bullet$
In \cite{Pa2} I.Panin extended Theorem
\ref{MainThm}
to the case of {\it an arbitrary reductive} group scheme satisfying a mild "isotropy"
condition; {\bf however his proof is heavily based on} Theorem
\ref{MainThm}
and on the main result of
\cite{C-T-S}
concerning the case of tori.
\par\smallskip
$\bullet$
The case of {\it arbitrary simple adjoint} group schemes of type $E_6$ and $E_7$
is done by the first author, V.Petrov and A.Stavrova in
\cite{PPS}. {\it No isotropy condition is imposed there}.
\par\smallskip
$\bullet$ There exists a folklore result, concerning a simple $R$-group scheme of type
$G_2$, where $R$ is a regular semi-local ring containing an infinite perfect field or
the semilocal ring of finitely many points on a $k$-smooth scheme with an infinite field $k$.
That result gives affirmative answer in this case and also independent
of isotropy hypotheses, see %% the very end of
the paper by V.~Chernousov and the first author \cite{ChP}.
\par\smallskip
$\bullet$ The case where the group scheme $G$ comes from the
ground field $k$ is completely solved by
J.-L.~Colliot-Th\'el\`ene, M.~Ojanguren, M.~S.~Raghunatan and
O.~Gabber: in \cite{C-TO} when $k$ is infinite; in \cite{R1} when
$k$ is perfect; O.~Gabber \cite{Ga} announced a proof for an
arbitrary ground field $k$.
\par\smallskip
$\bullet$
{\it The remaining unsolved open question} in case of simple simply-connected group scheme of {\it classical type} is this:
prove the conjecture for the spinor group of an algebra with an orthogonal involution.
The case of the spinor group of a quadratic space is done in
\cite{OPZ}.

\par\smallskip
%\begin{cor}
%\label{MainCor}
%Let $R$ be regular semi-local domain containing an infinite field and let $K$ be
%the quotient field $K$ of $R$. Let $G$ be a semi-simple simply-connected group
%scheme $G$. Suppose each presque-$R$-simple component of the $R$-group scheme $G$ contains a split rank $1$ torus
%$\Bbb G_m$. Then
%the map
%$$H^1_{\text{\'et}}(R, G) \to H^1_{\text{\'et}}(K, G)$$
%induced by the inclusion $R$ into $K$ has the trivial kernel.

%In the other words under the above mentioned assumptions on $R$ and $G$
%each principle $G$-bundle $P$ over $R$ which has a $K$-rational point
%is trivial.
%\end{cor}
A geometric counterpart of Theorem
\ref{MainThm}
is the following result
\begin{thm}
\label{MainThmGeometric} Let $k$ be an infinite field. Let
$\mathcal O$ be the semi-local ring of finitely many points on a
smooth irreducible $k$-variety $X$ and let $K$ be its field of
fractions. Let $G$ be an isotropic simple simply-connected group
scheme over $\mathcal O$.
% containing a split rank $1$ torus $\Bbb G_m$.
Then the map
$$ H^1_{\text{\rm\'et}}(\mathcal O,G)\to H^1_{\text{\rm\'et}}(K,G), $$
\noindent
induced by the inclusion $\mathcal O$ into $K$, has trivial kernel.
\end{thm}

\section{An outline of the proof}
\label{Outline}

Since the proof of Theorem \ref{MainThm} is long and technical, in
this section we briefly describe its general outline.  The actual
proofs are postponed till Section \ref{ProofOfMainThm}. Our proof
consists of three parts.
\par\smallskip
(1) Reduction of the general case to Theorem
\ref{MainThmGeometric}.
%the case, where the ring $R$
%is the semi-local ring $\mathcal O_{X, \{x_1,x_2,\dots,x_n\}}$ of
%a smooth $k$-variety $X$ over an infinite field $k$ at a finite
%family of points $x_1,x_2,\dots,x_n\in X$.
\par\smallskip
Theorem \ref{MainThmGeometric} is usually called the {\it
geometric case\/} of Theorem \ref{MainThm}. The reduction itself
is rather standard, the arguments are very similar to those in
\cite[Sect.7]{OP2}, and we mostly skip them.
\par\smallskip
In turn, analysis of the geometric case is subdivided into two
parts:
\par\smallskip
(2) a {\it geometric part\/}, and
\par\smallskip
(3) a {\it group part}.
\par\smallskip

%To prove this result we prove the following Lemma and its version for semi-local rings. The Lemma inspired by
%\cite[Cor.1.8]{R1}
%
%\begin{lem}
%Let $R$ and $G$ be as above. Let $\textit{m}$ be the maximal ideal of $R$ and
%$l=R/\textit{m}$
%be the residue field.
%Let $S \subseteq R[t]$ be the multiplicative system of all unitary
%polynomial in $R[t]$ and let
%$$\pi: A=R[t]_S \to l[t]_S=l(t)=L$$
%be the canonical projection, induced by the projection
%$R \to l$. Let $\mathcal O=l[t]_{(1/t)}$
%be the localization of the ring $l[1/t]$ at the maximal ideal $1/t \cdot l[1/t]$.
%Let $U^+$ and $U^-$ be $R$-group schemes, which are the unipotent radicals of two opposite
%parabolic subgroups of $G$ over $R$.
%
%Let $\pi: G(A) \to G(L)$ be a group homomorphism induced by the projection
%$\pi: A \to L$. Then $G(L)=\pi (<U^+(A), U^-(A)>) \cdot G(\mathcal O)$.
%\end{lem}

The geometric part of the proof starts with the following data.
%Sections \ref{ElementaryFibrations} and
%\ref{NiceTriples}, is organised as follows.
Fix a smooth affine $k$-scheme $X$, a finite family of points
$x_1,x_2,\dots,x_n$ on $X$, and set $\mathcal O:=\mathcal
O_{X,\{x_1,x_2,\dots,x_n\}}$ and $U:=\text{Spec}(\mathcal O)$.
Further, consider a simple simply-connected $U$-group scheme $G$
and a principal $G$-bundle $P$ over $\mathcal O$ which is trivial
over the field of fractions $K$ of $\mathcal O$. We may and will
assume that for certain $\ttf\in\mathcal O$ the principal
$G$-bundle $P$ is trivial over $\mathcal O_{\ttf}$. Shrinking $X$
if necessary, we may secure the following properties
%(see Section
%\ref{BasicTriple}).
\par\smallskip
(i) The points $x_1,x_2,\dots,x_n$ are still in $X$.
\par\smallskip
(ii) The group scheme $G$ is a simple group scheme defined over
$X$. We often denote this $X$-group scheme by $G_X$ and write
$G_U$ for the original $G$.
\par\smallskip
(iii) The principal $G$-bundle $P$ is defined over $X$ and
$\ttf\in k[X]$.
\par\smallskip
(iv) The restriction $P_{\ttf}$ of the bundle $P$ to a principal
open sub-scheme $X_{\ttf}$ is trivial for a non-zero $\ttf\in
k[X]$ and $\ttf$ vanishes at each $x_i$'s.
\par\smallskip\noindent
In the sequel we may shrink $X$ a little further, if necessary,
always looking after verification of (i) to (iv). This way we
construct in Section \ref{BasicTriple} a basic nice triple
(\ref{FirstNiceTriple}), see Definition \ref{DefnNiceTriple}.
%% and looking after $x_1,x_2,\dots,x_n$ still sitting in $X$,
Beginning with that basic nice triple we construct in Section
\ref{MainConstruction} the following data:
%% (which is still the geometric part)}
\par\smallskip
(a) a unitary polynomial $h(t)$ in $R[t]$,
\par\smallskip
(b) a principal $G_U$-bundle $P_t$ over $\Aff^1_U$,
\par\smallskip
(c) a diagram of the form
\begin{equation}
\label{SquareDiagram}
    \xymatrix{
     \Aff^1_U   \ar[drr]_{\pr}  &&  Y \ar[ll]_{\sigma} \ar[d]_{q_U} \ar[rr]^{q_X} && X  &\\
    && U \ar[urr]_{\can} \ar@/_0.8pc/[u]_{\delta} &\\
    }
\end{equation}
%\begin{equation}
%\label{SquareDiagram}
%    \xymatrix{
%     \mathcal X  \ar[d]_{q_U} \ar[rr]^{q_X} &&  X   & \\
%     U \ar[urr]_{can} \ar@/_0.8pc/[u]_{\Delta} &\\
%    }
%\end{equation}
\par\smallskip
(d) an isomorphism $\Phi:q^*_U(G_U)\to q^*_X(G)$ of $Y$-group
schemes.
\par\smallskip\noindent
By Theorem \ref{MainData} one may chose these objects to enjoy
the following properties:
\par\smallskip
(1*) $q_U=\pr\circ\sigma$,
\par\smallskip
(2*) $\sigma$ is \'etale,
\par\smallskip
(3*) $q_U\circ\delta=\id_U$,
\par\smallskip
(4*) $q_X\circ\delta=\can$,
\par\smallskip
(5*) the restriction of $P_t$ to $(\Aff^1_U)_h$ is a trivial
$G_U$-bundle,
%% where $G_U$ is from the item $(ii)$ above;
\par\smallskip
(6*) $(\sigma)^*(P_t)$ and $q_X^*(P)$ are isomorphic as principal
$G_U$-bundles. Here $q_X^*(P)$ is regarded as a principal
$G_U$-bundle via the group scheme isomorphism $\Phi$ from Item
(d).
\par\smallskip
{\it This completes the geometric part}. In the group part we
prove the following result, see the end of Section
\ref{SecProjectiveLine}.
\begin{thm}
\label{TrivialityOfP_t} Let $\mathcal O$ and $G$ be the same as in Theorem
\ref{MainThmGeometric}. Let $P$ be a principal $G$-bundle over the
polynomial ring $\mathcal O[t]$ in one variable $t$ such that for a monic
polynomial $h(t)\in \mathcal O[t]$ the bundle $P_{h(t)}$ over the ring
$\mathcal O[t]_{h(t)}$ is trivial. Then the $G$-bundle $P$ is trivial.
\end{thm}
%This result is based on the following equality
%\begin{equation}
%\label{*Key}
%G_l(l(t^{-1}))=G_l(l(t^{-1}))^{+} \cdot G_l(R)
%%\end{equation}
%which holds for {\bf simple simply-connected}
%$l$-group scheme $G_l$
%and which is proved in the proof Proposition
%\ref{Key}. The equality
%(\ref{*Key})
%replaces the arguments from
%\cite[Proof of Thm.3.3]{C-TO},
%where the surjectivity of certain group homomorphism comes as an immediate consequence
%of rather specific geometry. Namely, the surjectivity they were interested in is a direct consequence
%of the fact the composition of $k$-algebra homomorphisms
%$$k[t]_{\bar g} \to B[t]_{g\bar g} \to k[t]_{\bar g}$$
%is the identity.
%
%One way of thinking about the equality
%(\ref{*Key}) is this:
%one has a surjective stabilization for the unstable functor
%$SK_1(G_l(l(t^{-1}))):=G_l(l(t^{-1}))^{+}\backslash{G_l(l(t^{-1}))}$.
%Namely,
%the
%$SK_1(G^{an}_l(l(t^{-1}))$
%surjects onto
%$SK_1(G_l(l(t^{-1})))$,
%where
%%$G^{an}_l$
%is the anisotropic kernel of $G_l$.
%And additionally, for each anisotropic reductive $l$-group $H$ one has
%another equality (see
%\cite[Prop.1.2]{R1}). Namely,
%$H_l(l(t^{-1}))=H_l(A)$.
%In, particuar, the latter hold for
%$H_l=G^{an}_l$.

%More specifically, we prove Corollary
%\ref{TrivialOnFibraIsTrivial}, which deals only with the case of
%Theorem \ref{TrivialityOfP_t}, where the ring $R$ is of geometric
%type. The general case can be deduced from the geometric one by
%standard arguments using the remarkable theorem of D.~Popescu, see
%% celebrated
%\cite{P}). Again, details of such a reduction closely follow the
%work by the first author and M.~Ojanguren \cite{OP} and will be
%mostly omitted.

%With that end
%let us explain how {\it the geometric case}
It remains to explain how Theorem \ref{MainThmGeometric} follows
from Theorem \ref{TrivialityOfP_t}.
%and Theorem \ref{MainData}.
%% ������: Fix a $k$-smooth affine scheme $X$, points
% $x_1,x_2,\dots,x_n$ on $X$, and set $U:=\mathcal
% O_{X,\{x_1,x_2,\dots,x_n\}}$. Let $G$ be a simple simply-connected
% $U$-group scheme and let $P$ be a principal $G$ bundle over $U$.
% Shrinking $X$ we may ensure Conditions (i) to (iv). Assuming $(i)$
%to $(iv)$ and shrinking $X$ a little bit more and keeping
% $x_1,x_2,\dots,x_n$ in $X$ we construct in Remark
% \ref{MainConstruction} data (a) to (d) subject to Conditions (1^*)
% to (6^*).
We have to check that the $G$-bundle $P$ is trivial
over $U$.
\par
By assumption the group scheme $G_U$ is isotropic.
%% contains a split torus $\Bbb
%% G_m$ by the assumption of Theorem \ref{MainThm}.
The restriction of the principal $G_U$-bundle $P_t$ to $(\Aff^1_U)_h$
is trivial by Condition (5*). Then the principal $G_U$-bundle
$P_t$ is itself trivial by Theorem \ref{TrivialityOfP_t}.
\par
On the other hand, the $G_U$-bundle $(\sigma)^*(P_t)$ is isomorphic
to $q_X^*(P)$ as principal $G_U$-bundle by (6*).
%% , where $q_X^*(P)$ is regarded as a principal $G_U$-bundle
%% via the group scheme isomorphism $\Phi$ from the item $(d)$.
Therefore, $q^*_X(G)$ is trivial as a $q^*_X(G)$-bundle. It
follows that $\delta^*(q^*_X(P))=\can^*(P)$ is trivial as a
$\delta^*(q^*_X(G))=\can^*(G)$-bundle. Thus, $\can^*(P)=P_U$ is
trivial as a principal $\can^*(G)=G_U$-bundle.
%% {\bf The Theorem \ref{MainThm} is reduced to Theorems
%% \ref{TrivialityOfP_t} and \ref{MainData} and Remark
%% \ref{MainConstruction}}. \\

%{\bf Sections $2$ to $3$ contains the geometric part of our proof of Theorem \ref{MainThm},
%Section \ref{GroupPart} contains the Group part.}

\section{Elementary fibrations}
\label{ElementaryFibrations}

In this Section we extend a result of Artin from
\cite{LNM305} concerning existence of nice neighborhoods.
The following notion is due to Artin~\cite[Exp. XI, D\'ef. 3.1]{LNM305}.
\begin{defn}
\label{DefnElemFib} An elementary fibration is a morphism of
schemes $p:X\to S$ which can be included in a commutative diagram
\begin{equation}
\label{SquareDiagram}
    \xymatrix{
     X\ar[drr]_{p}\ar[rr]^{j}&&
\overline X\ar[d]_{\overline p}&&Y\ar[ll]_{i}\ar[lld]_{q} &\\
     && S  &\\    }
\end{equation}
of morphisms satisfying the following conditions:
\begin{itemize}
\item[{\rm(i)}] $j$ is an open immersion dense at each fibre of
$\overline p$, and $X=\overline X-Y$; \item[{\rm(ii)}] $\overline
p$ is smooth projective all of whose fibres are geometrically
irreducible of dimension one; \item[{\rm(iii)}] $q$ is finite
\'{e}tale all of whose fibres are non-empty.
\end{itemize}
\end{defn}

The following Bertini type theorem is an extension of Artin's
result \cite[Exp.XI,Thm.2.1]{LNM305}

\begin{thm}
\label{GeneralSection} Let $k$ be an infinite field, and let
$V\subset\Pro^n_k$ be a locally closed sub-scheme of pure
dimension $r$. Further, let $V^{\prime}\subset V$ be an open
sub-scheme consisting of all points $x\in V$ such that $V$ is
$k$-smooth at $x$. Finally, let $p_1,p_2,\dots,p_m\in\Pro^n_k$ be
a family of pair-wise distinct closed points.
%% with $p_i \neq p_j$ for $i \neq j$.
For a family $H_1(d),H_2(d),\dots,H_s(d)$, with $s\le r$,
%, where $s\leq r$, of
of hyper-planes of degree $d$ containing all points $p_i$, $1\le
i\le m$, set
$$ Y=H_1(d)\cap H_2(d)\cap\dots\cap H_s(d). $$

Then there exists an integer $d$ depending on the family
$p_1,p_2,\dots,p_m$ such that if the family
$H_1(d),H_2(d),\dots,H_s(d)$ with $s\leq r$ is sufficiently
general, then $Y$ crosses $V$ transversally at each point of $Y
\cap V^{\prime}$.
\par
If, moreover, $V$ is irreducible {\rm(}respectively, geometrically
irreducible{\rm)} and $s<r$ then for the same integer $d$ and for
a sufficiently general family $H_1(d),H_2(d),\dots,H_s(d)$ the
intersection $Y\cap V$ is irreducible {\rm(}respectively,
geometrically irreducible{\rm)}.

\end{thm}

Using this theorem, one can prove the following result, which is a
slight extension of Artin's result \cite[Exp.XI,Prop.3.3]{LNM305}

\begin{prop}
\label{ArtinsNeighbor} Let $k$ be an infinite field, $X/k$ be a
smooth geometrically irreducible variety, $x_1,x_2,\dots,x_n\in X$
be closed points. Then there exists a Zariski open neighborhood
$X^0$ of the family $\{x_1,x_2,\dots,x_n\}$ and an elementary
fibration $p:X^0\to S$, where $S$ is an open sub-scheme of the
projective space $\Pro^{\mydim X-1}$.
\par
If, moreover, $Z$ is a closed co-dimension one subvariety in $X$,
then one can choose $X^0$ and $p$ in such a way that $p|_{Z\bigcap
X^0}:Z\bigcap X^0\to S$ is finite surjective.
\end{prop}

We also omit for now the proof of the following proposition.

\begin{prop}
\label{CartesianDiagram} Let $p: X \to S$ be an elementary
fibration. If $S$ is a regular semi-local scheme, then there
exists a commutative diagram of $S$-schemes
\begin{equation}
\label{RactangelDiagram}
    \xymatrix{
X\ar[rr]^{j}\ar[d]_{\pi}&&\overline X\ar[d]^{\overline \pi}&&
Y\ar[ll]_{i}\ar[d]_{}&\\
\Aff^1\times S\ar[rr]^{\text{\rm in}}&&\Pro^1\times S&&
\ar[ll]_{i}\{\infty\}\times S &\\  }
\end{equation}
\noindent
such that the left hand side square is Cartesian. Here, $j$ and
$i$ are the same as in Definition \ref{DefnElemFib}, while $\pr_S
\circ\pi=p$, where $\pr_S$ is the projection $\Aff^1\times S\to
S$.

In particular, $\pi:X\to\Aff^1\times S$ is a finite surjective
morphism of $S$-schemes, where $X$ and $\Aff^1\times S$ are
regarded as $S$-schemes via the morphism $p$ and the projection
$\pr_S$, respectively.
\end{prop}



\section{Nice triples}
\label{NiceTriples}

In the present section we introduce and study certain collections
of geometric data and their morphisms. The concept of a {\it nice
triple\/} is very similar to that of a {\it standard triple\/}
introduced by Voevodsky \cite[Defn.4.1]{Vo}, and was in fact
inspired by that last notion. Let $k$ be an infinite field, $X/k$
be a smooth geometrically irreducible variety, and let
$x_1,x_2,\dots,x_n\in X$ be its closed points. Further, let
$\mathcal O=\mathcal O_{X,\{x_1,x_2,\dots,x_n\}}$ be the
corresponding geometric semi-local ring.
%$U:= \text{Spec}(\mathcal O_{X, \{x_1,x_2,\dots,x_n\}})$.
\begin{defn}
\label{DefnNiceTriple} Let $U:=\text{\Spec}(\mathcal
O_{X,\{x_1,x_2,\dots,x_n\}})$.
%$J= \cap^n_{i=1} \mathfrak m_i, l=\mathcal O_{X, \{x_1,x_2,\dots,x_n\}}/J$.
A {\bf nice triple} over $U$ consists of the following data:
\begin{itemize}
\item[\rm{(i)}] a smooth morphism $q_U:\mathcal X\to U$,
\item[\rm{(ii)}] an element $f\in\Gamma(\mathcal X,\mathcal
O_{\mathcal X})$,
%a closed codimension one sub-scheme
%$\mathcal Z$,
\item[\rm{(iii)}] a section $\Delta$ of the morphism $q_U$,
\end{itemize}
subject to the following conditions:
\begin{itemize}
\item[\rm{(a)}] each component of each fibre of the morphism $q_U$
has dimension one,
%and can be decomposed as
%$pr_U \circ \Pi$,
%where
%$\Pi: \mathcal X \to \Aff^1 \times U$
%is a finite surjective morphism;
\item[\rm{(b)}] the module $\Gamma(\mathcal X,\mathcal O_{\mathcal
X})/f\cdot\Gamma(\mathcal X,\mathcal O_{\mathcal X})$ is finite as
a $\Gamma(U,\mathcal O_{U})=\mathcal O$-module,
%$\mathcal Z$ is finite and surjective over $U$;
\item[\rm{(c)}] there exists a finite surjective $U$-morphism
$\Pi:\mathcal X\to\Aff^1\times U$, \item[\rm{(d)}]
$\Delta^*(f)\neq 0\in\Gamma(U,\mathcal O_{U})$.
\end{itemize}
\end{defn}
\begin{defn}
\label{DefnMorphismNiceTriple} A {\bf morphism} of two nice
triples $(\mathcal
X^{\prime},f^{\prime},\Delta^{\prime})\to(\mathcal X,f,\Delta)$ is
an \'{e}tale morphism of $U$-schemes $\theta:\mathcal
X^{\prime}\to\mathcal X$ such that
\begin{itemize}
\item[\rm{(1)}] $q^{\prime}_U=q_U\circ\theta$, \item[\rm{(2)}]
$f^{\prime}=\theta^{*}(f)\cdot g^{\prime}$ for an element
$g^{\prime}\in\Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal
X^{\prime}})$,
%$\theta^{-1} (\mathcal Z) \subset \mathcal Z^{\prime}$
%$\theta^{-1} (\mathcal Z)$ is finite surjective over $U$);
\item[\rm{(3)}] $\Delta=\theta\circ\Delta^{\prime}$.
\end{itemize}
\end{defn}
Two observations are in order here.
\par\smallskip
$\bullet$ Item (2) implies in particular that $\Gamma(\mathcal
X^{\prime},\mathcal O_{\mathcal
X^{\prime}})/\theta^*(f)\cdot\Gamma(\mathcal X^{\prime},\mathcal
O_{\mathcal X^{\prime}})$ is a finite
$\mathcal O$-module.
\par\smallskip
$\bullet$ It should be emphasized that no conditions are imposed
on the inter-relation of $\Pi^{\prime}$ and $\Pi$.
\par\smallskip
Let us state two crucial results which will be used in our main
construction. Their proofs are postponed till Sections
\ref{SecEquatingGroups} and \ref{SectElemNisnevichSquare}.
\begin{thm}
\label{EquatingGroups} Let $U$ be as in Definition
\ref{DefnNiceTriple}. Let $(\mathcal X,f,\Delta)$ be a nice triple
over $U$. Let $G_{\mathcal X}$ be a simple simply-connected
$\mathcal X$-group scheme, and let $G_U:=\Delta^*(G_{\mathcal
X})$. Finally, let $G_{\const}$ be the pull-back of $G_U$ to
$\mathcal X$. Then there exists a morphism $\theta:(\mathcal
X^{\prime},f^{\prime},\Delta^{\prime})\to(\mathcal X,f,\Delta)$ of
nice triples and an isomorphism
$$ \Phi: \theta^*(G_{\const}) \to \theta^*(G_{\mathcal X}) $$
\noindent
of $\mathcal X^{\prime}$-group schemes such that
$(\Delta^{\prime})^*(\Phi)=\id_{G_U}$.
\end{thm}

The proof of this theorem is sketched in Section
\ref{SecEquatingGroups}. Let $U$ be as in Definition
\ref{DefnNiceTriple}. Let $(\mathcal X,f,\Delta)$ be a nice triple
over $U$. Then for each finite surjective $U$-morphism
$\sigma:\mathcal X\to\Aff^1\times U$ and the corresponding
$\mathcal O$-algebra inclusion
$\mathcal O[t]\hra\Gamma(\mathcal X,\mathcal O_{\mathcal X})$
%the element $t$ is a non-zero divisor in
%$\Gamma(\mathcal X, \mathcal O_{\mathcal X})$,
the algebra $\Gamma(\mathcal X,\mathcal O_{\mathcal X})$ is
finitely generated as an $\mathcal O[t]$-module. Since both rings
$\mathcal O[t]$ and $\Gamma(\mathcal X,\mathcal O_{\mathcal X})$
are regular, the algebra $\Gamma(\mathcal X,\mathcal O_{\mathcal
X})$ is finitely generated and projective as an $\mathcal
O[t]$-module by theorem \cite[Cor.18.17]{E}. Take the
characteristic polynomial $t^r - a_{n-1}t^{r-1} + \dots \pm N(f)$
of the $\mathcal O[t]$-module endomorphism $\Gamma(\mathcal
X,\mathcal O_{\mathcal X}) \xra{f} \Gamma(\mathcal X,\mathcal
O_{\mathcal X})$ and set
\begin{equation}
g_{f,\sigma}:=f^{r-1}-a_{n-1}f^{r-2}+\dots\pm a_1\in
\Gamma(\mathcal X,\mathcal O_{\mathcal X}).
\end{equation}
\noindent
\begin{lem}
\label{FandG} $f\cdot g_{f,\sigma}=\pm N(f)\in\Gamma(\mathcal
X,\mathcal O_{\mathcal X})$.
\end{lem}
In fact, the characteristic polynomial of the operator
$\Gamma(\mathcal X,\mathcal O_{\mathcal X}) \xra{f}
\Gamma(\mathcal X,\mathcal O_{\mathcal X})$ vanishes on $f$.

\begin{thm}
\label{ElementaryNisSquare} Let $U$ be as in Definition
\ref{DefnNiceTriple}. Let $(\mathcal X,f,\Delta)$ be a nice triple
over $U$. There exists a distinguished finite surjective morphism
$$ \sigma:\mathcal X\to\Aff^1\times U $$
\noindent
of $U$-schemes which enjoys the following properties.
\begin{itemize}
\item[\rm{(1)}] $\sigma$ is \'{e}tale along the closed subset
$\{f=0\}\cup\Delta(U)$. %%
\item[\rm{(2)}] For $g_{f,\sigma}$ and $N(f)$ defined by the
distinguished $\sigma$, one has
$$ \sigma^{-1}\Big(\sigma\big(\{f=0\}\big)\Big)=
\{N(f)=0\}=\{f=0\}\sqcup\{g_{f,\sigma}=0\}. $$
%%{\rm(the disjoint union )
%follows:
%let $N(f)$ is the determinant of
%take the characteristic polynomial
%$t^r - a_{n-1}t^{r-1} + \dots \pm N(f)=0$
%of
%the $\mathcal O[t]$-module endomorphism
%$\Gamma(\mathcal X, \mathcal O_{\mathcal X}) \xra{f} \Gamma(\mathcal X, \mathcal O_{\mathcal X})$
%and set
%$g:=f^{r-1}-a_{n-1}f^{r-2} + \dots \pm a_1 \in \Gamma(\mathcal X, \mathcal O_{\mathcal X})$.
%take the characteristic polynomial of that endomorphism vanishes on $f$, which means that
%$f^r -a_{n-1}f^{r-1} + \dots \pm N(f)=0$
%and
%$N(f)= f \cdot g$
%for an element
%$g \in \mathcal O[t]$.
\item[\rm{(3)}] Denote by $\mathcal X^0\hra\mathcal X$ the largest
open sub-scheme, where the morphism $\sigma$ is \'{e}tale. Write
$g$ for $g_{f,\sigma}$ in this item. Then the square
\begin{equation}
\label{SquareDiagram2}
    \xymatrix{
     \mathcal X^{0}_{N(f)}=\mathcal X^{0}_{fg}  \ar[rr]^{\inc} \ar[d]_{\sigma^0_{fg}} &&
     \mathcal X^{0}_g \ar[d]^{\sigma^0_g}  &\\
     (\Aff^1 \times U)_{N(f)} \ar[rr]^{\text{\rm in}} && \Aff^1 \times U &\\
    }
\end{equation}
is an elementary Nisnevich square. More precisely, this square is
Cartesian and the morphism of the reduced closed sub-schemes
$$ \sigma^0_g|_{\{f=0\}_{\red}}:\{f=0\}_{\red}\to\{N(f)=0\}_{\red} $$
\noindent
of the schemes $\mathcal X^{0}_g$ and $\Aff^1\times U$ is an
isomorphism.
%%
\item[\rm{(4)}] One has $\Delta(U)\subset\mathcal X^{0}_g$.
\end{itemize}
\end{thm}
A sketch of the proof of this Theorem is given in Section
\ref{SectElemNisnevichSquare}.
\par
Using Theorems \ref{EquatingGroups} and \ref{ElementaryNisSquare}
in the next Section we construct
%% \ref{MainConstruction}
data (a) to (d) from the Introduction subject to Conditions (1*)
to (6*).

\section{A basic nice triple}
\label{BasicTriple}

With Propositions \ref{ArtinsNeighbor} and \ref{CartesianDiagram}
at our disposal we may form {\it a basic nice triple}, namely the
triple (\ref{FirstNiceTriple}). This is the main aim of the
present Section.
%a very
%nice diagram similar to the famous diagrams of Quillen and
%Voevodsky.
Namely, fix a smooth irreducible affine $k$-scheme $X$,
a finite family of points $x_1,x_2, \dots , x_n$ on $X$, and a
non-zero function $\ttf\in k[X]$.
% and a closed subset $Z$ of $X$ given by the equation $f=0$.
We {\it always assume} that the set $\{x_1,x_2,\dots,x_n\}$ is
contained in the vanishing locus of the function $\ttf$.
\par
Replacing $k$ by its algebraic closure in $k[X]$, we may assume
that $X$ is a geometrically irreducible $k$-variety.
%It is easy to see that $X$ remains smooth over $k$.
By Proposition \ref{ArtinsNeighbor} there exist a Zariski open
neighborhood $X^0$ of the family $\{x_1,x_2,\dots,x_n\}$ and an
elementary fibration $p:X^0\to S$, where $S$ is an open sub-scheme
of the projective space $\Pro^{\mydim X-1}$, such that
$$ p|_{\{\ttf=0\}\cap X^0}:\{\ttf=0\}\cap X^0\to S $$
\noindent
is finite surjective. Let $s_i=p(x_i)\in S$, for each $1\le i\le
n$. Shrinking $S$, we may assume that $S$ is {\it affine \/} and
still contains the family $\{s_1,s_2,\dots,s_n\}$. Clearly, in
this case $p^{-1}(S)\subseteq X^0$ contains the family
$\{x_1,x_2,\dots,x_n\}$. We replace $X$ by $p^{-1}(S)$ and $\ttf$
by its restriction to this new $X$.
%(we continue to denote this restriction by $f$).
%Respectively still write $Z$ for the vanish locus of $f$ on that new $X$.
\par
In this way we get an elementary fibration $p:X\to S$ such that
$$ \{x_1,\dots,x_n\}\subset\{\ttf=0\}\subset X, $$
\noindent
$S$ is an open affine sub-scheme in the projective space
$\Pro^{\mydim X-1}$, and the restriction of
$p|_{\{\ttf=0\}}:\{\ttf=0\}\to S$ to the vanishing locus of $\ttf$
is a finite surjective morphism. In other words, $k[X]/(\ttf)$ is
finite as a $k[S]$-module.
\par
As an open affine sub-scheme of the projective space $\Pro^{\mydim
X-1}$ the scheme $S$ is regular. By Proposition
\ref{CartesianDiagram} one can shrink $S$ in such a way that $S$
is still affine, contains the family $\{s_1,s_2,\dots,s_n\}$ and
there exists a finite surjective morphism
$$ \pi:X\to\Aff^1\times S $$
\noindent
such that $p=\pr_S\circ\pi$. Clearly, in this case
$p^{-1}(S)\subseteq X$ contains the family
$\{x_1,x_2,\dots,x_n\}$. We replace $X$ by $p^{-1}(S)$ and $\ttf$
by its restriction to this new $X$.
\par
In this way we get an elementary fibration $p:X\to S$ such that
$$ \{x_1,\dots,x_n\}\subset\{\ttf=0\}\subset X, $$
\noindent
$S$ is an open affine sub-scheme in the projective space
$\Pro^{\mydim X-1}$, and the restriction of
$p|_{\{\ttf=0\}}:\{\ttf=0\}\to S$ to the vanishing locus of $\ttf$
is a finite surjective morphism. Eventually we conclude that there
exists a finite surjective morphism
$$ \pi:X\to\Aff^1\times S $$
\noindent
such that $p=\pr_S\circ\pi$.
\par
Now, set $U:=\text{Spec}(\mathcal O_{X,\{x_1,x_2,\dots,x_n\}})$,
denote by $\can:U\hra X$ the canonical inclusion of schemes, and
let $p_U=p\circ\can:U\to S$. Further, we consider the fibre
product $\mathcal X:=U\times_{S}X$. Then the canonical projections
$q_U:\mathcal X\to U$ and $q_X:\mathcal X\to X$ and the diagonal
morphism $\Delta:U\to\mathcal X$ can be included in the following
diagram
\begin{equation}
\label{SquareDiagram}
    \xymatrix{
     \mathcal X\ar[d]_{q_U}\ar[rr]^{q_X} &&  X   & \\
     U \ar[urr]_{\can} \ar@/_0.8pc/[u]_{\Delta} &\\
    }
\end{equation}
where
\begin{equation}
\label{DeltaQx} q_X\circ\Delta=\can
\end{equation}
and
\begin{equation}
\label{DeltaQu} q_U\circ\Delta=\id_U.
\end{equation}
Note that $q_U$ is {\it a smooth morphism with geometrically
irreducible fibres of dimension one}. Indeed, observe that $q_U$
is a base change via $p_U$ of the morphism $p$ which has the
desired properties. Taking the base change via $p_U$ of the finite
surjective morphism $\pi:X\to\Aff^1\times S$ we get {\it a finite
surjective morphism
$$ \Pi:\mathcal X\to\Aff^1\times U $$
\noindent
such that\/} $q_U=\pr_U\circ\Pi$. Set $f:=
q_X^*(\ttf)$. The $\mathcal
O_{X,\{x_1,x_2,\dots,x_n\}}$-module $\Gamma(\mathcal X,\mathcal
O_{\mathcal X})/f\cdot\Gamma(\mathcal X,\mathcal O_{\mathcal X})$
%and
%$\mathcal Z:= q_X^{-1}(Z)$
is finite, since the $k[S]$-module $k[X]/\ttf\cdot k[X]$ is finite.
%That is
%$\mathcal Z$
%is the vanishing locus of
%$\textit{f}$ .
%The locus $\mathcal Z$ {\it is finite and surjective over} $U$,
%since $Z$ is finite surjective over $S$.

Note that the data
\begin{equation}
\label{FirstNiceTriple} (q_U:\mathcal X\to U,f,\Delta)
\end{equation}
form an example of a {\it nice triple\/}, as defined in
Definition
\ref{DefnNiceTriple}.

\begin{clm}
\label{DeltaIsWellDefined} The schemes $\Delta(U)$ and $\{f=0\}$
are both semi-local and the set of closed points of $\Delta(U)$ is
contained in the set of closed points of $\{f=0\}$.
\end{clm}
This holds since the set $\{x_1,x_2,\dots,x_n\}$ is contained in
the vanishing locus of the function $\ttf$.
%\begin{equation}
%\xymatrix{U_K\ar[r]^{\bar\pi}\ar[d]&U\ar[d]\\
%{\spe K}\ar[r]^\pi\ar[ur]^y\ar@/^0.8pc/[u]^{\tilde y}
%\ar@/_0.8pc/[u]_{\tilde x}&{\spe k,}\ar@/_0.8pc/[u]_x}
%\end{equation}


\section{Main Construction}
\label{MainConstruction}
The main result of this Section is Theorem
\ref{MainData}.

Fix a smooth affine $k$-scheme $X$, a finite family of points
$x_1,x_2,\dots,x_n$ on $X$, and set $\mathcal O:=\mathcal
O_{X,\{x_1,x_2,\dots,x_n\}}$ and $U:= \text{Spec}(\mathcal O)$.
Further, consider a simple simply-connected $U$-group scheme $G$
and a principal $G$-bundle $P$ over $\mathcal O$ which is trivial
over the field of fractions $K$ of $\mathcal O$. We may and will
assume that for certain $\text{f} \in \mathcal O$ the principal
$G$-bundle $P$ is trivial over $\mathcal O_{\text{f}}$. Shrinking
$X$ if necessary, we may secure the following properties
%(see Section
%\ref{BasicTriple}).
\par\smallskip
(i) The points $x_1,x_2,\dots,x_n$ are still in $X$.
\par\smallskip
(ii) The group scheme $G$ is defined over $X$ and it is a simple
group scheme. We will often denote this $X$-group scheme by $G_X$
and write $G_U$ for the original $G$.
\par\smallskip
(iii) The principal $G$-bundle $P$ is defined over $X$ and the function $\text{f}$ belongs to $k[X]$.
\par\smallskip
(iv) The restriction $P_{\text{f}}$ of the bundle $P$ to a
principal open sub-set $X_{\text{f}}$ is trivial and $\text{f}$
vanishes at each $x_i$'s.
\par\smallskip
In particular, now we are given the smooth irreducible affine
$k$-scheme $X$, the finite family of points $x_1,x_2,\dots,x_n$ on
$X$, and the non-zero function $\ttf\in k[X]$ vanishing at each
point $x_i$. Recall, that starting from these data we
constructed at the very end
of Section \ref{BasicTriple} the nice triple
(\ref{FirstNiceTriple}).
We did that shrinking $X$ and secure properties (1) to (4) at the same time.
\par
Let $G$ be the simple simply-connected $X$-group scheme, $P$ be
the principal $G$-bundle over $X$. The restriction $P_{\ttf}$ of
the bundle $P$ to the principal open sub-scheme $X_{\ttf}$ is
trivial by Item (iv) above.
%Recall, that starting
%from these data at the very end of Section
%\ref{BasicTriple} we constructed the nice triple
%(\ref{FirstNiceTriple}).
Set $G_{\mathcal X}:=(q_X)^*(G)$. By Theorem \ref{EquatingGroups}
there exists a morphism of nice triples
$$ \theta: (\mathcal X^{\prime},f^{\prime},\Delta^{\prime})\to (\mathcal X,f,\Delta) $$
\noindent
and an isomorphism
\begin{equation}
\label{KeyEquation} \Phi: \theta^*(G_{\const}) \to
\theta^*(G_{\mathcal X})=: G_{\mathcal X^{\prime}}
%=(q_X \circ \theta)^*(G)=(q^{\prime}_X)^*(G)=: G_{\mathcal X^{\prime}} \ (see (\ref{QprimeX}))
\end{equation}
of $\mathcal X^{\prime}$-group schemes such that
$(\Delta^{\prime})^*(\Phi)=\id_{G_U}$.
\par
Set
\begin{equation}
\label{QprimeX} q^{\prime}_X=q_X\circ\theta:\mathcal X^{\prime}\to X.
\end{equation}
%$f^{\prime}= (q^{\prime}_X)^*(\text{f})$
Recall that
\begin{equation}
\label{QprimeU} q^{\prime}_U=q_U\circ\theta:\mathcal X^{\prime}\to
U,
\end{equation}
since
$\theta$
is a morphism of nice triples.
Consider $(q^{\prime}_X)^*(P)$ as a principal
$(q^{\prime}_U)^*(G_U)=G_{\const}$-bundle via the isomorphism
$\Phi$. Recall that $P$ is trivial as a $G$-bundle over
$X_{\text{f}}$. Therefore, $(q^{\prime}_X)^*(P)$ is trivial as a
principal $G_{\mathcal X^{\prime}}$-bundle over $\mathcal
X^{\prime}_{\theta^*(f)}$.
%Clearly, the vanishing locus of the function
%$(q^{\prime}_X)^*(\text{f})$
%coincides with
%the closed sub-scheme
%$\theta^{-1}(\mathcal Z)$
%of the scheme
%$\mathcal X^{\prime}$.
Since $\theta$ is a nice triple morphism one has
%$\theta^{-1}(\mathcal Z) \subset \mathcal Z^{\prime}$,
$f^{\prime}=\theta^*(f)\cdot g^{\prime}$, and thus the principal
$G_{\mathcal X^{\prime}}$-bundle
$(q^{\prime}_X)^*(P)=(q_X\circ\theta)^*(G)$ is trivial over
$\mathcal X^{\prime}_{f^{\prime}}$.
\par
We can conclude that $(q^{\prime}_X)^*(P)$ is trivial over
$\mathcal X^{\prime}_{f^{\prime}}$, when regarded as a principal
$G_{{\const}}$-bundle via the isomorphism $\Phi$.
\par
By Theorem \ref{ElementaryNisSquare} there exists a finite
surjective morphism $\sigma:\mathcal X^{\prime}\to\Aff^1\times U$
of $U$-schemes satisfying (1) to (3) from that Theorem. In
particular, one has
$$ \sigma^{-1}\Big(\sigma\big(\{f^{\prime}=0\}\big)\Big)=
N(f^{\prime})=\{f^{\prime}=0\}\sqcup\{g^{\prime}_\sigma=0\} $$
\noindent
with $N(f^{\prime})$ and $g^{\prime}$ defined in the item (2) of
Theorem \ref{ElementaryNisSquare}. Thus, replacing for brevity
$g^{\prime}_\sigma$ by $g^{\prime}$, one gets the following
elementary Nisnevich square
\begin{equation}
\label{ElemNisSquareDiagram}
    \xymatrix{
(\mathcal X^{\prime})^{0}_{N(f^{\prime})}= (\mathcal
X^{\prime})^{0}_{f^{\prime}g^{\prime}}\ar[rr]^{\inc}
\ar[d]_{\sigma^0_{f^{\prime}g^{\prime}}} &&
(\mathcal X^{\prime})^0_{g^{\prime}}\ar[d]^{\sigma^0_{g^{\prime}}}  &\\
(\Aff^1\times U)_{N(f^{\prime})}\ar[rr]^{\text{\rm in}}&&\Aff^1\times U &\\
}
\end{equation}
Regarded as a principal $G_U$-bundle via the isomorphism $\Phi$,
the bundle $(q^{\prime}_X)^*(P)$ over $\mathcal X^{\prime}$
becomes trivial over $\mathcal X^{\prime}_{f^{\prime}}$, and a
fortiori over $(\mathcal X^{\prime})^{0}_{f^{\prime}g^{\prime}}$.
Now, taking the trivial $G_U$-bundle over $(\Aff^1\times
U)_{N(f^{\prime})}$ and the isomorphism
\begin{equation}
\label{Skleika}
\psi: (\mathcal X^{\prime})^{0}_{N(f^{\prime})} \times_U G_U \to
(q^{\prime}_X)^*(P)|_{(\mathcal X^{\prime})^{0}_{N(f^{\prime})}}
\end{equation}
of principal $G_U$-bundles, we get a principal $G_U$-bundle $P_t$
over $\Aff^1\times U$ such that
\begin{itemize}
\item[(1)] it is trivial over $(\Aff^1\times U)_{N(f^{\prime})}$,
%%
\item[(2)] $(\sigma)^*(P_t)$ and $(q^{\prime}_X)^*(P)$ are
isomorphic as principal $G_U$-bundles. Here $(q^{\prime}_X)^*(P)$
is regarded as a principal $G_U$-bundle via the $\mathcal
X^{\prime}$-group scheme isomorphism $\Phi$ from
(\ref{KeyEquation}); %%
\item[(3)] over $(\mathcal X^{\prime})^{0}_{N(f^{\prime})}$ the
two $G_U$-bundles are identified via the isomorphism $\psi$ from
(\ref{Skleika}).
\end{itemize}

%Note that
%$\sigma (\mathcal Z^{\prime})$
%is a divisor on
%$\Aff^1 \times U$. In fact,
%$\mathcal Z^{\prime}$
%is a divisor on
%%$\mathcal X^{\prime}$
%by the item $(b)$ of Definition
%\ref{DefnNiceTriple} and the morphism
%$\sigma$ is finite surjective.
%Clearly,
%$\sigma (\mathcal Z^{\prime})$
%is finite surjective over $U$.
%The scheme $U$ is semi-local regular. Thus the ring
%$\mathcal O_{X, \{x_1,x_2,\dots,x_n\}}[t]$
%is factorial.So, there exists an element
%$h(t) \in \mathcal O_{X, \{x_1,x_2,\dots,x_n\}}[t]$
%with the vanishing locus equal to
%$\sigma (\mathcal Z^{\prime})$.
%The polynomial $h(t)$ is unitary, since
%the divisor
%$\sigma (\mathcal Z^{\prime})$
%is finite surjective over $U$.
%{\bf Whence,
%$\sigma (\mathcal Z^{\prime})= \{h(t)=0\}$,
%where $h(t) \in \mathcal O_{X, \{x_1,x_2,\dots,x_n\}}[t]$
%is a unitary polynomial}.

Finally, form the following diagram
\begin{equation}
\label{DeformationDiagram2}
    \xymatrix{
\Aff^1_U\ar[drr]_{\pr}&&(\mathcal
X^{\prime})^0_{g^{\prime}}\ar[d]^{}
\ar[ll]_{\sigma^0_{g^{\prime}}}\ar[d]_{q^{\prime}_U}
\ar[rr]^{q^{\prime}_X}&&X  &\\
&&U\ar[urr]_{\can}\ar@/_0.8pc/[u]_{\Delta^{\prime}} &\\
    }
\end{equation}
This diagram is well-defined since by Item (4) of Theorem
\ref{ElementaryNisSquare} the image of the morphism
$\Delta^{\prime}$ lands in $(X^{\prime})^0_{g^{\prime}}$.
%It follows from Item (4) of Theorem \ref{ElementaryNisSquare},
%that this diagram is well-defined.

%We need to check that
%$\Delta^{\prime} : U \to \mathcal X^{\prime}_0$
%lands into
%$(\mathcal X^{\prime})^{0}_{g^{\prime}}$.
%This follows easy from three facts:
%\begin{itemize}
%\item
%schemes
%$\Delta^{\prime}(U)$,
%$\{f^{\prime}=0\}$
%%$\mathcal Z^{\prime}$
%and
%$\{g^{\prime}=0\}$
%$\mathcal Z^{\prime}_1$
%are semi-local; $\Delta^{\prime}(U) \cup \{f^{\prime}=0\} \subset (\mathcal X^{\prime})^0$,
%\item
%the set of closed points of
%$\Delta^{\prime}(U)$
%is contained in the set of closed points of
%$\{f^{\prime}=0\}$ (see Remark
%\ref{DeltaIsWellDefined}),
%%$\mathcal Z^{\prime}$,
%\item
%$\{f^{\prime}=0\} \cap \{g^{\prime}=0 \} = \emptyset$.
%\end{itemize}

\begin{thm}
\label{MainData} The unitary polynomial $N(f^{\prime})$, the
principal $G_{\const}$-bundle $P_t$ over $\Aff^1_U$, the diagram
{\rm(\ref{DeformationDiagram2})} and the isomorphism
{\rm(\ref{KeyEquation})} constructed above
%%{\rm\ref{MainConstruction}}
satisfy Conditions {\rm(1*)} to {\rm(6*)} from the Introduction.
\end{thm}

\begin{proof} By the very choice of $\sigma$ it is an $U$-scheme
morphism, which proves (1*). Since $(\mathcal
X^{\prime})^0\hra\mathcal X^{\prime}$ is the largest open
sub-scheme where the morphism $\sigma$ is \'etale, one gets (2*).
Property (3*) holds for $\Delta^{\prime}$ since $(\mathcal
X^{\prime},\mathcal Z^{\prime},\Delta^{\prime})$ is a nice triple
and, in particular, $\Delta^{\prime}$ is a section of
$q^{\prime}_U$. Property (4*) can be established as follows:
$$ q^{\prime}_X\circ\Delta^{\prime}=
(q_X\circ\theta)\circ\Delta^{\prime}=q_X\circ\Delta=\can. $$
\noindent
The first equality here holds by the definition of $q^{\prime}_X$,
see (\ref{QprimeX}); the second one holds, since $\theta$ is a
morphism of nice triples; the third one follows from
(\ref{DeltaQx}). Property (5*) is just Property (1) in the above
construction of $P_t$.
%recall that the principal
%$G_U$-bundle $P_t$ is trivial over
%$(\Aff^1 \times U)_{N(f^{\prime})}$
%by the very construction of $P_t$, which is done just above. Finally,
Property (6*) is precisely Property (2) in our construction of
$P_t$.
\end{proof}

%\begin{thm}
%\label{RelativeSpecialization}
%Let
%$(\mathcal X, \mathcal Z, \Delta)$
%be a nice triple over the semi-local scheme
%$U$ from Definition
%\ref{DefnNiceTriple}. Let
%$G$ be a simple adjoint
%$\mathcal X$-group scheme containing a split torus $\Bbb G_m$.
%Let $P$ be a principal $G$-bundle over
%$\mathcal X$
%such that
%$P|_{\mathcal X - \mathcal Z}$
%is a trivial principal
%$G$-bundle. Then there exists a morphism
%$\theta: (\mathcal X^{\prime}, \mathcal Z^{\prime}, \Delta^{\prime}) \to
%(\mathcal X, \mathcal Z, \Delta)$
%of triples such that the principal
%$(\theta^*)(G)$-bundle is trivial.
%
%In particular, the principal
%$\Delta^*(G)$-bundle
%$\Delta^*(P)$
%is trivial.
%\end{thm}

\section{Group of points of isotropic simple groups}

In this section we establish several results concerning groups of
points of simple groups, in particular, \label{GroupPart} Lemma
\ref{Surjectivity}, Proposition \ref{Key} and Lemma
\ref{AlphaUtimesbeta}, which play crucial role in the rest of the
paper.

\begin{defn}\label{E_P}
Let $A$ be an arbitrary commutative ring, and let $G$ be a
reductive group scheme over $A$. Assume that $G$ has a proper
parabolic subgroup $P=P^+$ over $A$, and denote by $U^+$ its
unipotent radical.
%By \cite[Exp.~XXVI Cor.~2.3]{SGA}, there exists a Levi subgroup $L$ of $P$, and by
%\cite[Exp. XXVI Th. 4.3.2]{SGA} there exists a unique parabolic subgroup $P^-$ opposite to $P^+$,
By \cite[Exp.~XXVI Cor.~2.3, Th. 4.3.2]{D-G} there exists a
parabolic subgroup $P^-$ of $G$ opposite to $P^+$, and
by~\cite[Exp.~XXVI Cor.~1.8]{D-G} any two such subgroups are
conjugate by an element of $U^+(A)$. Let $U^-$ be the unipotent
radical of $P^-$. For any $A$-algebra $B$ we define the
$P$-elementary subgroup $E_P(B)$ of the group $G(B)$ as follows:
$$ E_P(B)=\langle U^+(B),U^-(B)\rangle. $$
\end{defn}
\begin{lem}
\label{Surjectivity} Let $B\to\bar B$ be a surjective $A$-algebra
homomorphism. Then the induced homomorphism of elementary groups
$E_P(B)\to E_P(\bar B)$ is also surjective.
%% $$ \langle U^{+}_l(A), U^{-}_l(A)\rangle \ \to \
%% \langle U^{+}_l(\overline A), U^{-}_l(\overline A)\rangle $$
%% \noindent
\end{lem}
\begin{proof}
By~\cite[Exp.XXVI Cor. 2.5]{D-G} the $A$-schemes $U^+$ and $U^-$
are isomorphic to $A$-vector bundles of finite rank. Thus, the
maps $U^{\pm}(B)\to U^{\pm}(\bar B)$ are surjective.
\end{proof}

Let $l$ be an infinite field and $G_l$ be {\bf an isotropic simple
simply-connected} $l$-group scheme. Recall that an isotropic scheme
contains an $l$-split rank one torus $\Bbb G_{m,l}$. Choose and
fix two opposite parabolic subgroups $P_l=P^+_l$ and $P^-_l$ of
the $l$-group scheme $G_l$. Let $U^+_l$ and $U^-_l$ be their
unipotent radicals. We will be interested mostly in the group of
points $G_l(l(t))$. The following definition originates
from~\cite[MainTheorem]{T}.
\begin{defn}
\label{G+} Define $G_l(l(t))^+$ as a subgroup of the group
$G_l(l(t))$ generated by $l(t)$-points of unipotent radicals of
all parabolic subgroups of $G_l$ defined over the field $l$.
\end{defn}

\begin{rem}
\label{AppearanceT-} Clearly, $l(t)=l(t^{-1})$. Thus,
$$ G_l(l(t))=G_l(l(t^{-1}))\qquad \text{and}\qquad
G_l(l(t))^{+}=G_l(l(t^{-1}))^{+}. $$
\end{rem}

By definition the group $G_l(l(t))^+$ is generated by unipotent
radicals of {\it all\/} $l$-parabolic subgroups, and thus contains
the elementary group $E_{P_l}(l(t))$, introduced in
Definition~\ref{E_P}. In fact they coincide.
\begin{prop}
\label{G+AndU+U-} The group $G_l(l(t))^+$ is generated by
$l(t)$-points of unipotent radicals of any two opposite parabolic
subgroups of the $l$-group scheme $G_{l}$. In particular, one has
the equality
\begin{equation}
\label{U+U-} G_l(l(t^{-1}))^{+}=\Big\langle U^{+}_l(l(t^{-1})),
U^{-}_l(l(t^{-1}))\Big\rangle=E_{P_l}(l(t^{-1})).
\end{equation}
\end{prop}
\begin{proof}
Set $G_{l(t)}=G_l\times_{\Spec\,l}\Spec\,l(t)$. The group
$G_l(l(t))^{+}$ is contained in the subgroup of
$G_l(l(t))=G_{l(t)}(l(t))$ generated by $l(t)$-points of unipotent
radicals of all parabolic subgroups of the group scheme $G_{l(t)}$
defined over the field $l(t)$. By~\cite[Prop.6.2.(v)]{BT73} the
latter group is generated by $l(t)$-points of unipotent radicals
of any two opposite parabolic subgroups of $G_{l(t)}$, in
particular, by $l(t)$-points of $U_{l(t)}^+$ and $U^-_{l(t)}$.
Since $U^\pm_{l(t)}(l(t))=U^\pm_{l}(l(t))$, we have~\eqref{U+U-}.
\end{proof}

\begin{rem}
\label{ConvinienceOfU+U-} For any ring $A$ and an $A$-algebra $B$,
and any reductive $A$-group scheme $G$ we can define the group
$G_A(B)^+$ as in the Definition~\ref{G+}, that is, as the subgroup
generated by $B$-points of unipotent radicals of all $A$-parabolic
subgroups of $G$. The question, whether this subgroup coincides
with $E_P(B)$ for an $A$-parabolic subgroup $P$ of $G$, is in
general rather subtle. See the paper {\rm \cite{PSt}} by V.~Petrov
and the second author for details.
\end{rem}

%\begin{rem}
%\label{ConvinienceOfU+U-} For each field extension $L/l$ one can
%define the group $G_l(L)^{+}$ literally repeating the Definition
%\ref{G+}. However, the {\it $P_l$-elementary subgroup\/}
%$$ E_{P_l}(A)=\langle U^{+}_l(A),U^{-}_l(A)\rangle $$
%\noindent
%of the group $G_l(A)$ is well-defined for an arbitrary $l$-algebra
%$A$. Furthermore, it is functorial: an $l$-algebra homomorphism
%$A\to A^{\prime}$ induces a group homomorphism
%$$ E_{P_l}(A)\to E_{P_l}(A'). $$
%%% $$ \langle U^{+}_l(A), U^{-}_l(A)\rangle \ \to \ \langle
%%% U^{+}_l(A^{\prime}), U^{-}_l(A^{\prime})\rangle. $$
%\noindent
%Functoriality and other nice properties of the elementary group
%%% $\langle U^{+}_l(A), U^{-}_l(A)\rangle$
%make it more useful. This will be amply illustrated below.
%\par
%As another side remark let us observe that the main result of the
%paper {\rm \cite[]{PS}} by V.~Petrov and the third author asserts
%that for simple groups of relative rank at least $2$ {\rm(}and in
%fact under somewhat weaker assumptions{\rm)} the elementary group
%$E_{P_l}(A)$ does not depend on the choice of an $l$-parabolic.
%Recall, that the groups of relative rank at least $2$ countain an
%$l$-split rank two torus $\Bbb G^2_{m,l}$.
%\end{rem}


The following result is crucial for the sequel.
\begin{prop}
\label{Key}
One has equality
\begin{equation}
\label{RaghunathanBT} G_l(l(t^{-1}))=G_l(l(t^{-1}))^{+}\cdot
G_l(R )
\end{equation}
where $G_l(l(t^{-1}))^{+}$ is the group defined in Definition
\ref{G+} {\rm(}see also Remark \ref{AppearanceT-}{\rm)} and
$R=l[t^{-1}]_{(t^{-1})}$ is the localization of $l[t^{-1}]$ at the
prime ideal $(t^{-1})$.
\end{prop}

\begin{proof}
Let $S_l$ be a maximal $l$-split torus in $G_l$ and let
$\Cent_{G_l}(S_l)$ be its scheme theoretic centralizer in $G_l$.
Then by \cite[Cor.1.7]{R1} one has
\begin{equation}
G_l((l(t^{-1})))=G_l(l(t^{-1}))^{+}\cdot
S_l(l(t^{-1}))\cdot\Cent_{G_l}(S_l)(R ).
\end{equation}
We show that $S_l(l(t^{-1}))$ is contained in
$G_l(l(t^{-1}))$. Choose a maximal torus $T_l$ in $G_l$
containing $S_l$. Let
$\Psi:=\Phi(S,G_l)$ be the {\it relative\/} root system of $G_l$
with respect to $S_l$. Further, let $l^{\sep}$ be the separable
closure of $l$. Denote by $\Phi$ the root system of $G_{l^{\sep}}$
with respect to $T_{l^{\sep}}$ --- the {\it absolute\/} root
system of $G_l$. Let
$$\Psi^{\prime}=\{\alpha\in\Psi\mid 2\alpha\notin\Psi\}. $$
\par
A semi-simple $l$-split $l$-subgroup scheme $H_l$ of $G_l$ is
constructed in \cite[Thm.7.2]{BT65}. That semi-simple $l$-split
$l$-subgroup scheme $H_l$ contains the torus $S_l$ as its maximal
$l$-split torus, the root system of $H_l$ with respect to $S_l$
coincides with $\Psi^{\prime}$ and for each
$\alpha\in\Psi^{\prime}$ the root subgroup $U_{\alpha,l}$ of the
$l$-group scheme $H_l$ is contained in the unipotent radical of an
appropriate $l$-parabolic subgroup $P_l$ of the $l$-group scheme
$G_l$, and consequently, in $G_l(l(t^{-1}))^+$.
\par
Further, by \cite[(4.6) (Corollary J.Humphreys)]{BT72} the
semi-simple $l$-split $l$-group scheme $H_l$ is simply-connected
provided that $G_l$ is simply-connected.
%% Thus, $H_l$ is simply-connected, since $G_l$ is
%% semi-simple simply-connected.
Since $H_l$ is split, it follows that the group of points
$H_l(l(t^{-1}))$ is generated by the root subgroups
$U_{\alpha,l}(l(t^{-1}))$, $\alpha\in\Psi^\prime$ (e.g.~\cite[Exp.
XXII Cor. 5.7.6]{D-G}). Since $S_l$ is a maximal split torus of
$H_l$, using the above inclusions $U_{\alpha,l}(l(t^{-1}))\le
G_l(l(t^{-1}))^{+}$, $\alpha\in\Psi^\prime$, we conclude that
$S_l(l(t^{-1}))\le G_l(l(t^{-1}))^{+}$.
\par
This completes the proof of the proposition, since the centralizer
$\Cent_{G_l}(S_l)$ is an $l$-subgroup scheme of $G_l$.

\end{proof}

%% Now, let $l[t]^{\bullet}$ be the set of all non-zero polynomials
%% in one variable $t$.
Let $f(t)\in l[t]$ be a polynomial of degree $n=\deg(f)$ in $t$
such that $f(0)\neq 0$. We consider the reciprocal polynomial
$$ f^*(t^{-1})=f(t)/t^n\in l[t^{-1}], $$
\noindent
clearly, $f^*(0)\neq 0$. Conversely, if $g(t^{-1})\in l[t^{-1}]$
is a polynomial of degree $n=\deg(g)$ in $t^{-1}$ such that
$g(0)\neq 0$, the reciprocal polynomial
$$ g^*(t)=g(t^{-1})\cdot t^n\in l[t] $$
\noindent
is defined by a similar formula.
%% is such that $g^*(0)\neq 0$.
The above correspondences are mutually inverse. Further, when
$f(t)\in l[t]$ runs over all polynomials in $t$ with $f(0)\neq 0$,
then the reciprocal polynomial $f^*(t^{-1})$ runs over all
polynomials $g(t^{-1})\in l[t^{-1}]$ with $g(0)\neq 0$.
\par
Now let us return to the setting considered in (\ref{U+U-}) and
(\ref{RaghunathanBT}) and Remark \ref{ConvinienceOfU+U-}. Each
non-constant $f(t)\in l[t]$ admits a unique factorisation of the
form $f(t)=t^r\cdot g(t)$, where $g(t)\in l[t]$ and $g(0)\neq 0$.
Clearly, for each $h(t)\in l[t]$ with $h(0)\neq 0$ one gets the
following inclusions
$$ G_l\big(l[t]_{f(t)}\big)\le G_l\big(l[t^{-1},t]_{g^*}\big)\le
G_l\big(l[t^{-1},t]_{g^*h^*}\big). $$
\noindent
This leads us to the following Lemma.

\begin{lem}
\label{AlphaUtimesbeta} For each $\alpha\in G_l(l[t]_{f(t)})$ one
can find a polynomial $h(t)\in l[t]$, $h(0)\neq 0$, and elements
$$ u\in E_{P_l}\big(l[t^{-1},t]_{g^*h^*}\big),\qquad
%% \langle U^+_l(l[t^{-1},t]_{g^*h^*}), \
%% U^+_l(l[t^{-1},t]_{g^*h^*})\rangle
 \beta\in G_l\big(l[t^{-1}]_{g^*h^*}\big) $$
\noindent
such that
%$\alpha \in G(l[t^{-1},t]_{g^*h^*})$ and
\begin{equation}
\label{AlphaEqUtimesBeta} \alpha = u\beta\in
G_l\big(l[t^{-1},t]_{g^*h^*}\big).
\end{equation}
\noindent
The chain of $l$-algebra inclusions
$l[t]_{fh}\subseteq l[t]_{tfh}=l[t,t^{-1}]_{gh}=
l[t^{-1},t]_{g^*h^*}$
shows that
%$$u \in E_{P_l}(l[t]_{tfh})$
%and
$$ u\in E_{P_l}\big(l[t]_{tfh}\big),\quad\text{and}\quad
%% \langle U^+_l(l[t]_{tfh}),U^+_l(l[t]_{tfh})\rangle,
\alpha\in G_l\big(l[t]_{tfh}\big). $$
\end{lem}
\begin{proof}
As observed above, inclusions $\alpha\in
G_l\big(l[t^{-1},t]_{g^*}\big)\le
G_l\big(l[t^{-1},t]_{g^*h^*}\big)$ are obvious. The equalities
(\ref{RaghunathanBT}) and (\ref{U+U-}) imply that there exists a
polynomial $h(t)\in l[t]$, $h(0)\neq 0$, and elements
$$ u\in E_{P_l}\big([t^{-1},t]_{g^*h^*}\big),\qquad
%% \langle U^+_l(l[t^{-1},t]_{g^*h^*},
%% U^-_l(l[t^{-1},t]_{g^*h^*})\rangle ,\
\beta\in G_l\big(l[t^{-1}]_{g^*h^*}\big), $$
\noindent
such that $\alpha=u\beta$ in $G_l\big(l[t^{-1},t]_{g^*h^*}\big)$.
The last claim follows from the obvious $l$-algebra inclusions
$$
l[t]_{fh}\subseteq l[t]_{tfh}=l[t,t^{-1}]_{gh}=
l[t^{-1},t]_{g^*h^*}.
$$
\end{proof}


\section{Principal $G$-bundles on a projective line}
\label{SecProjectiveLine}

The main result of the present section is
Corollary
\ref{TrivialOnFibraIsTrivial}.
%Theorem \ref{NewBundle}.
Let $B$ be a commutative ring, and let $\Aff^1_B$ and $\Pro^1_B$
be the affine line and the projective line over $B$, respectively.
Usually we identify the affine line with a sub-scheme of the
projective line as follows
$\Aff^1_B=\Pro^1_B-(\{\infty\}\times\text{Spec}(B))$, where
$\infty=[0:1]\in\Pro^1$. Let $G$ be a semi-simple $B$-group
scheme, let $P$ a principal $G$-bundle over $\Aff^1_B$, and let
$p:P\to\Aff^1_B$ be the corresponding canonical projection.

For a monic polynomial
$$ f=f(t)=t^n+a_{n-1}t^{n-1}+\dots+a_0\in B[t] $$
\noindent
we set $P_f=p^{-1}((\Aff^1_B)_f)$. Clearly, it is a principal
$G$-bundle over $(\Aff^1_B)_f$. Further, we denote by
$$ F(t_0,t_1)=t_1^n+a_{n-1}t_1^{n-1}t_0+\dots+a_0t_0^n $$
\noindent
the corresponding homogeneous polynomial in two variables. Note
that the intersection of the principal open set in $\Pro^1_B$
defined by the inequality $F\neq 0$ with the affine line
$\Aff^1_B$ equals the principal open subset $(\Aff^1_B)_f$. As in
the previous section in the case where $a_0\neq 0$ we consider the
reciprocal polynomial
$f^{*}(t^{-1})\in B[t^{-1}]$ equal to $f(t)/t^n$. %% $f^*$
\begin{defn}
\label{PofPhiAndf} Let $\varphi: G_{(\Aff^1_B)_f} \to P_f$ be a
principal $G$-bundle isomorphism. We write $P(\varphi, f)$ for a
principal $G$-bundle over the projective line $\Pro^1_B$ obtained
by gluing $P$ and $G_{(\Pro^1_B)_F}$ over $(\Aff^1_B)_f$ via the
principal $G$-bundle isomorphism $\varphi$.
\end{defn}
\begin{rem}
\label{PropertiesPofPhiAndf} For any $\varphi$ and $f$ the
principal $G$-bundles $P(\varphi,f)$ and $P(\varphi,fg)$ coincide
for each monic polynomial $g\in B[t]$.
\par
For any $\varphi$ and $f$, any monic polynomial $h(t)\in B[t]$
such that $h(0)\in B^{*}$ is invertible, and any $\beta\in
G(B[t^{-1}]_{h^*})$ the principal $G$-bundles
$P(\varphi,f)$ and $P(\varphi\circ\beta,tfh)$ are isomorphic. In fact,
they differ by a co-boundary.
\end{rem}
Now, let $l$ be an infinite field and $G_l$ be a semi-simple
$l$-group scheme. Let $f\in l[t]$ be a polynomial. Let $P$ be a
principal $G_l$-bundle over $\Aff^1_l$ such that $P_f$ is trivial
over $\Aff^1_f$. Let $\varphi:G_{\Aff^1_{f}}\to P_f$ be a
principal $G_l$-bundle isomorphism. Let $P(\varphi,f)$ be the
corresponding principal $G$-bundle over $\Pro^1_l$.
\begin{lem}
\label{RR} In the above notation there exists an $\alpha\in
G(l[t]_f)$ such that the principal $G$-bundle
$P(\varphi\circ\alpha,f)$ is trivial over $\Pro^1_l$.
\end{lem}
\begin{proof}
By the main theorem of \cite{RR} we may assume that there is an
isomorphism $G_{\Aff^1_l}=P$ over $\Aff^1_l$. In this case the
above isomorphism $\varphi$ coincides with the right
multiplication by an element $\beta\in G_l(l[t]_f)$. Clearly,
$P(\beta\circ\beta^{-1},f)$ is trivial over $\Pro^1_l$. Thus,
$P(\beta\circ\alpha,f)$ is trivial for $\alpha=\beta^{-1}$.
\end{proof}

%Let $k$ be an infinite field and $X$ be a $k$-smooth irreducible affine variety.
%Replacing $k$ by its algebraic closure in $k[X]$ we may assume that $X$ is smooth affine and geometrically
%irreducible over $k$.
%Let
%$x_1,x_2, \dots , x_n$
%be a finite family of points
%on $X$. Let $G$ be a simple simply-connected group scheme over $G$
%which contains a split rank one torus $\Bbb G_{m,X}$.
%For each $i=1,2, \dots, n$ let $G(x_i)$ be the fibre of $G$ over the point $x_i$,
%that is
%$G(x_i)=G \times_X x_i$.
%Note that for each
%$i=1,2, \dots, n$
%the $k(x_i)$-group scheme $G(x_i)$ is a
%simple simply-connected $k(x_i)$-group scheme
%which contains a split rank one torus
%$\Bbb G_{m,x_i}$.

\begin{cor}
\label{BundlesOnP1}
Let
%% $l/k$ and
$G_l$ be an isotropic simply-connected semi-simple $l$-group
scheme and
%% In particular, $G_l$ contains $\Bbb G_{m,l}$.
let $P$ be a $G_l$-bundle over $\Aff^1_l$. Further, let $f(t)\in
l[t]$ be a non-constant polynomial, $\varphi: G_{\Aff^1_f}\to
P_{\Aff^1_f}$ be a principal $G_l$-bundle isomorphism and let
$P(\varphi,f)$ be the corresponding principal $G_l$-bundle on the
projective line $\Pro^1_l$. Then there exist $h(t)\in l[t]$ and
$u\in G(l[t]_{tfh})^+$ such that the principal $G_l$-bundle
$P(\varphi \circ u,tfh)$ is trivial over $\Pro^1_l$.
\end{cor}

\begin{proof}
By Lemma \ref{RR} there exists an $\alpha \in G_l(l[t]_f)$ such
that the principal $G_l$-bundle $P(\phi \circ \alpha , f)$ is
trivial.
\par
Let $f(t)=t^rg(t)$ be the unique factorisation such that $g(t)\in
l[t]$ and $g(0)\neq 0$. By Lemma \ref{AlphaUtimesbeta} for each
$h(t)\in S$, $h(0)\neq 0$, one has the inclusion $\alpha\in
G_l(l[t^{-1},t]_{g^*h^*})$. Moreover, there exist an element
$h(t)\in l[t]$ with $h(0)\neq 0$ and elements
$$ u \in G_l(l[t]_{tfh})^{+},\qquad
\beta\in G_l(l[t^{-1}]_{g^*h^*}) $$
\noindent
such that
\begin{equation}
\label{AlphaEqUtimesBeta} \alpha=u\beta\in
G_l(l[t^{-1},t]_{g^*h^*}).
\end{equation}
The following chain of principal $G_l$-bundle isomorphisms
completes the proof
$$ G_l \times_{\text{Spec}(l)} \Pro^1_l =P(\varphi \circ \alpha, f)=
P(\varphi \circ \alpha,\ tfg) = P(\varphi \circ u \circ \beta,
tfh) \cong P(\varphi \circ u, \ tfh). $$
\noindent
All the equalities are obvious. The last isomorphism holds since
$\beta\in G_l(l[t^{-1}]_{g^*h^*})$.
\end{proof}

Let $k$ be an infinite field and let $X$ be a $k$-smooth
irreducible affine variety. Replacing $k$ by its algebraic closure
in $k[X]$ we may assume that $X$ is smooth affine and
geometrically irreducible over $k$. Let $x_1,x_2,\dots,x_n$ be a
finite family of points on $X$. Let $\mathcal O=\mathcal
O_{X,\{x_1,x_2,\dots,x_n\}}$ be the semi-local ring of the family
$x_1,x_2,\dots,x_n\in X$ and let $U$ for $\text{Spec}(\mathcal
O)$. Let $G$ be a simple group scheme over $\mathcal O$. By
$G(x_i)$, $1\le i\le n$, we denote the fibre of $G$ over the point
$x_i$, in other words, $G(x_i)=G\times_X x_i$. Note that for each
$i=1,2,\dots,n$ the $k(x_i)$-group scheme $G(x_i)$ is an isotropic
simple simply-connected $k(x_i)$-group scheme.
%% which contains a split rank one torus $\Bbb G_{m,x_i}$.
\par
Let $\frak{m}_i\subseteq  k[X]$ be the maximal ideal corresponding
to the point $x_i$. Let $J$ be the intersection of all
$\frak{m}_i$, $1\le i\le n$. Then $l=k[X]/J=l_1\times
l_2\times\dots\times l_n$, where $l_i=k(x_i)$.
%% Set $l=l_1\timesl_2\times\dots\times l_n$, then $k[X]/J = l$.
Let $G_l=G\times_X\Spec(l)$ be the fibre of $G$ over $\Spec(l)$.
%Let
%$\mathcal O:= \mathcal O_{X, \{x_1,x_2,\dots,x_n\}}$
%be semi-local ring of the family
In the sequel we write $\Pro^1$ and $\Aff^1$ for $\Pro^1_{\mathcal
O}$ and $\Aff^1_{\mathcal O}$ respectively, whereas $\Pro^1_l$
and $\Aff^1_l$ denote the projective line and the affine line over
$l$.

Let $f\in \mathcal O[t]$ be a monic polynomial and let $P$ be a
principal $G_{\mathcal O}$-bundle over $\Aff^1$ such that
$P_{\Aff^1_f}$ is trivial. Let $\varphi: G_{\Aff^1_f} \to
P_{\Aff^1_f}$ be a principal $G$-bundle isomorphism and let
$P(\varphi, f)$ be the corresponding principal $G$-bundle on
$\Pro^1_{\mathcal O}$ (see Definition \ref{PofPhiAndf}).

\begin{thm}
\label{NewBundle} Assume that the $\mathcal O$-group scheme $G$ is
isotropic simple and simply-connected.
%% and $G$ contains an $\mathcal O$-split torus $\Bbb G_{m,\mathcal O}$.
Then there exists a monic polynomial $h(t)\in\mathcal O[t]$ and an
element $\alpha\in G(\mathcal O[t]_{tfh})$ such that the principal
$G$-bundle $P(\varphi\circ\alpha,tfh)$ satisfies the condition
\begin{itemize}
\item[{\rm(}i{\rm)}] $P(\varphi\circ\alpha,tfh)|_{\Pro^1_l}$ is a
trivial principal $G_l$-bundle over the projective line
$\Pro^1_l$.
\end{itemize}
\end{thm}

\begin{proof}
Denote by $\overline P(\overline\varphi,\overline f)$ the
restriction of $P(\varphi,f)$ to the projective line $\Pro^1_l$.
By
%Lemma
%\ref{AlphaUtimesbeta}
%and
Corollary
\ref{BundlesOnP1}
there exists a monic polynomial
$\overline h(t)\in l[t]$, such that $\overline h(0)\in l^{\times}$ and
an element
$$ u\in E_{P_l}\big(l[t]_{t\overline f\,\overline h}\big)\le
%% \ \langle U^+_l(l[t]_{t\overline f \overline h}),
%% U^-_l(l[t]_{t\overline f\,\overline h})\rangle \ \subset
G_l\big(l[t]_{t\overline f\,\overline h}\big) $$
\noindent
such that the principal $G_l$-bundle $\overline
P(\overline\varphi\circ u,t\overline f\,\overline h)$ is trivial
over $\Pro^1_l$.
\par
Choose a monic polynomial $h(t)\in\mathcal O[t]$ of degree equal
to the degree of $\overline h(t)$ and such that $h(t)$ modulo $J$
coincides with $\overline h(t)$. Clearly, the $\mathcal
O$-algebras homomorphism $\mathcal O[t]_{tfh}\to l[t]_{t\overline
f\,\overline h}$ is surjective. By Lemma \ref{Surjectivity} it
induces a surjective group homomorphisms
$$ E_{P_{\mathcal O}}\big(\mathcal O[t]_{tfh}\big)\to E_{P_{\mathcal O}}\big(l[t]_{t\overline f\,\overline h}\big)=
E_{P_l}\big(l[t]_{t\overline f\,\overline h}\big). $$
%% \langle U^+(\mathcal O[t]_{tfh}), U^+(\mathcal O[t]_{tfh})\rangle
%% \to \ \langle U^+_l(l[t]_{t\overline f \overline h}),\
%% U^-(l[t]_{tfh})\rangle.
\noindent
Thus, there exists an $\alpha\in G(\mathcal O[t]_{tfh})$ such that
$\overline\alpha=u$. In other words, $\alpha$ equals
$$ u\in E_{P_l}(l[t]_{t\overline f\,\overline h})\le
%%\langle U^+_l(l[t]_{t\overline f\overline h}),
%% U^-_l(l[t]_{t\overline f\overline h})\rangle \subset
G_l(l[t]_{t\overline f\,\overline h}) $$
\noindent
modulo $J$.
\par
Take the $G$-bundle $P(\varphi \circ \alpha, tfh)$. We claim that
its restriction to the projective line $\Pro^1_l$ is trivial.
Indeed, one has the following chain of equalities and isomorphisms
of principal $G_l$-bundles over $\Pro^1_l$:
$$ \overline P(\overline{\varphi\circ\alpha},\overline{tfh})=
\overline P(\overline\varphi\circ\overline\alpha,t\overline
f\,\overline h)=\overline P(\overline\varphi\circ u,t\overline
f\,\overline h), $$
\noindent
where the principal $G_l$-bundle $\overline
P(\overline\varphi\circ u,t\overline f\overline h)$ is trivial
over $\Pro^1_l$.
\end{proof}
We keep the same notation as in Theorem \ref{NewBundle}, see also
the text immediately preceding its statement.
\begin{thm}
\label{ConstantOnFibreIsConstant} Let $G$ be an simple
simply-connected $\mathcal O$-group scheme. Let $P$ be a principal
$G$-bundle over $\Pro^1$ whose restriction to the closed fibre
$P_{\Pro^1_l}$ is trivial. Then $P$ is of the form:
$P=\pr^*(P_0)$, where $P_0$ is a principal $G$-bundle over
$\Spec(\mathcal O)$ and $\pr:\Pro^1\to\Spec(\mathcal O)$ is the
canonical projection.
\end{thm}
The proof of this Theorem is rather standard,
and for the most part follows \cite{R2}. However, our group scheme
$G$ does not come from the ground field $k$. Therefore, we have to
somewhat modify Raghunathan's arguments. We skip details for now.
%% rather than literally repeat them .
Let us state an important corollary of the above theorems.
\begin{cor}[{\bf=Theorem \ref{TrivialityOfP_t}}]
\label{TrivialOnFibraIsTrivial} Let $G$ be an $\mathcal O$-group
scheme satisfying the same assumptions as in Theorem
\ref{NewBundle}. Further, let $P$ be a principal $G$-bundle over
$\Aff^1$. Assume, that there exists a monic polynomial
$f\in\mathcal O[t]$ such that the principal $G$-bundle
$P_{\Aff^1_f}$ is trivial.
%Let $\varphi: G_{\Aff^1_f} \to P_{\Aff^1_f}$
%be a principal $G$-bundle isomorphism and let
%$P(\varphi, f)$ be the corresponding principal $G$-bundle on
%$\Pro^1_{\mathcal O}$ (see Definition
%\ref{PofPhiAndf}).
Then the principal $G$-bundle $P$ is trivial. In other words,
there exists a $G$-bundle isomorphism
$$ G\times_U\Aff^1\cong P. $$
\end{cor}

\begin{proof}[Proof of Corollary
\ref{TrivialOnFibraIsTrivial}]
%%{ConstantOnFibreIsConstant}]
Let $f\in\mathcal O[t]$ be a monic polynomial such that the
principal $G$-bundle $P_{\Aff^1_f}$ is trivial. Choose a principal
$G$-bundle isomorphism $\varphi:G_{\Aff^1_f}\to P_{\Aff^1_f}$. By
Theorem \ref{NewBundle} there exists a monic polynomial
$h(t)\in\mathcal O[t]$ and an element $\alpha\in G(\mathcal
O[t]_{tfh})$ such that the restriction
$P(\varphi\circ\alpha,tfh)|_{\Pro^1_l}$ of the principal
$G$-bundle $P(\varphi\circ\alpha,tfh)$ to the projective line
$\Pro^1_l$ is a trivial principal $G_l$-bundle.
\par
By Theorem \ref{ConstantOnFibreIsConstant} the principal
$G$-bundle $P(\varphi\circ\alpha,tfh)$ is of the form:
$P(\varphi\circ\alpha,tfh)=\pr^*(P_0)$, where $P_0$ is a principal
$G$-bundle over $\text{Spec}(\mathcal O)$. Note that
$$ G|_{\{\infty\}\times U}\cong P(\varphi\circ\alpha,
tfh)|_{\{\infty\}\times U}, $$
\noindent
where $U=\text{Spec}(\mathcal O)$ (that is the restriction of
$P(\varphi\circ\alpha, tfh)$ to $\{\infty\}\times U$ is trivial).
Thus
$$ G_{\Pro^1}\cong P(\varphi\circ\alpha,tfh). $$
\noindent
Since the original principal $G$-bundle $P$ over $\Aff^1$ is
isomorphic to $P(\varphi\circ\alpha,tfh)|_{\Aff^1}$, it follows
that $P$ is trivial. This finishes the proof.

\end{proof}

%\begin{proof}[Proof of Theorem \ref{TrivialityOfP_t}]
%It follows immediately from Corollary
%\ref{TrivialOnFibraIsTrivial}
%% \ref{ConstantOnFibreIsConstant}
%with the use of the celebrated Popescu theorem \cite{P}, see also
%\cite[Section 7, Popescu's Theorem]{OP}.
%\end{proof}


\section{Equating Groups}
\label{SecEquatingGroups} The aim of this Section is to sketch a
proof of Theorem \ref{EquatingGroups}. The following Proposition
is a straightforward analogue of \cite[Prop.7.1]{OP}
\def\tilde{\widetilde}
\begin{prop}
\label{PropEquatingGroups} Let $S$ be a regular semi-local
irreducible scheme and let $G_1,G_2$ be two semi-simple
simply-connected $S$-group schemes. Further, let $T\subset S$ be a
closed sub-scheme of $S$ and $\varphi:G_1|_T\to G_2|_T$ be an
$S$-group scheme isomorphism. Then there exists a finite \'{e}tale
morphism $\tilde S\xra{\pi}S$ together with its section
$\delta:T\to\tilde S$ over $T$ and an $\tilde S$-group scheme
isomorphism $\Phi:\pi^*{G_1}\to\pi^*{G_2}$ such that
$\delta^*(\Phi)=\varphi$.
\end{prop}

%To prove this proposition we used a variant of Bertini's theorem proved in
%\cite[Lemma 7.2]{OP}(see
%also [8]).
%
%\begin{lem}
%\label{Lemma_7_2}
%Let $S=\text{Spec}(R)$ be a regular
%semi-local scheme and $T$ a closed
%sub-scheme of $S$. Let $\overline X$ be a closed sub-scheme of
%$\Bbb P^d_S=\text{Proj}(S[X_0,\dots,X_d])$ and
%$X=\overline X\cap\Bbb \Aff^d_S$, where $\Aff^d_S$ is the affine space defined by
%$X_0\neq0$. Let
%$X_{\infty}=\overline X\setminus X$ be the intersection of $\overline X$ with the
%hyperplane at infinity
%$X_0=0$. Assume that over $T$ there exists a section $\delta:T\to X$
%of the canonical projection $X \to S$. Assume further that
%%\smallskip
%\begin{itemize}
%\item[(1)]
%$X$ is smooth and equidimensional over $S$,
%of relative dimension $r$;
%\item[(2)]
%For every closed
%point $s\in S$ the closed fibres of $X_\infty$ and $X$
%satisfy
%$$\text{dim}(X_\infty(s))<\text{dim}(X(s))=r\;.$$
%%\end{itemize}
%\smallskip
%Then there exists a closed sub-scheme $\tilde S \subset X$ which is finite
%\'etale over
%$S$ and contains $\delta(T)$.
%\end{lem}
\par\smallskip\noindent
{\bf Proof of Theorem \ref{EquatingGroups}.} We can start by
almost literally repeating arguments from the proof of \cite[Lemma
8.1]{OP}, which involve the following purely geometric lemma
\cite[Lemma 8.2]{OP}.
\par
For reader's convenience below we state that Lemma adapting
notation to the ones of Section \ref{NiceTriples}.
%and
%\cite[Lemma 8.3]{OP}.
%One should replace notation as follows:
%the finite surjective map
%$q: \mathcal X \to \Aff^1 \times U$
%from
%\cite[Lemma 8.2]{OP}
%replace with our finite surjective map
%$\Pi: \mathcal X \to \Aff^1 \times U$
%from the condition of Theorem
%\ref{EquatingGroups}.
%The following lemma is proved in
%\cite[Lemma 8.2]{OP}
Namely, let $U$ be as in Definition \ref{DefnNiceTriple} and let
$(\mathcal X,f,\Delta)$ be a nice triple over $U$. Further, let
$G_{\mathcal X}$ be a simple simply-connected $\mathcal X$-group
scheme, $G_U:=\Delta^*(G_{\mathcal X})$, and let $G_{\const}$ be
the pull-back of $G_U$ to $\mathcal X$. Finally, by the definition
of a nice triple there exists a finite surjective morphism
$\Pi:\mathcal X\to\Aff^1\times U$ of $U$-schemes.
\begin{lem}
\label{Lemma_8_2} Let $\mathcal Y$ be a closed nonempty sub-scheme
of $\mathcal X$, finite over $U$. Let $\mathcal V$ be an open
subset of $\mathcal X$ containing $\Pi^{-1}(\Pi(\mathcal Y))$. There
exists an open set $\mathcal W \subseteq \mathcal V$ still
containing $q_U^{-1}(q_U(\mathcal Y))$ and endowed with a finite
surjective morphism $\mathcal W\to\Aff^1\times U$ {\rm(}in general
$\neq\Pi${\rm)}.
\end{lem}
Let $\Pi:\mathcal X\to\Aff^1\times U$ be the above finite
surjective $U$-morphism.
%% from the hypotheses of Lemma \ref{Lemma_8_2}.
The following diagram summarises the situation:
$$ \xymatrix{
{}&\mathcal Z \ar[d]&{}\\
{\mathcal X - \mathcal Z\ }\ar@{^{(}->}@<-2pt>[r]&\mathcal X\ar@<2pt>[d]^{q_U}\ar[r]^>>>>{\Pi}& \Aff^1 \times U\\
{}& U\ar@<2pt>[u]^{\Delta}&{} } $$
\noindent
Here $\mathcal Z$ is the closed sub-scheme defined by the equation
$f=0$. By assumption, $\mathcal Z$ is finite over $U$. Let
$\mathcal Y=\Pi^{-1}(\Pi(\mathcal Z \cup \Delta(U)))$. Since
$\mathcal Z$ and $\Delta(U)$ are both finite over $U$ and since
$\Pi$ is a finite morphism of $U$-schemes, $\mathcal Y$ is also
finite over $U$. Denote by $y_1,\dots,y_m$ its closed points and
let $S=\text{Spec}(\mathcal O_{\mathcal X,y_1,\dots,y_m})$. Set
$T=\Delta(U)\subseteq S$. Further, let $G_U=\Delta^*(G_{\mathcal
X})$ be as in the hypotheses of Theorem
\ref{EquatingGroups} and
let
$G_{\const}$
be the pull-back of
$G_U$ to $\mathcal X$. Finally,
let
$\varphi:G_{\const}|_T \to G_{\mathcal X}|_T$
be the canonical
isomorphism. Recall that by assumption $\mathcal X$ is $U$-smooth,
and thus $S$ is regular.
\par
By Proposition
\ref{PropEquatingGroups} there exists a finite
\'etale covering $\theta_0:\tilde S\to S$, a section
$\delta:T\to\tilde S$ of $\theta_0$ over $T$ and an isomorphism
$$ \Phi_0:\theta^*_0(G_{\const,S})\to\theta^*_0(G_{\mathcal X}|_S) $$
\noindent
such that $\delta^*\Phi_0=\varphi$. We can extend these data to a
neighborhood $\mathcal V$ of $\{y_1,\dots,y_n\}$ and get the
diagram
\begin{equation}
\xymatrix{
     {}  &  \tilde S \ar[d]^{\theta_0} \ar@{^{(}->}@<-2pt>[r]  & \tilde {\mathcal V}  \ar[d]_{\theta} &\\
     T \ar@{^{(}->}@<-2pt>[r] \ar[ur]^{
\delta} & S \ar@{^{(}->}@<-2pt>[r]  &   \mathcal V \ar@{^{(}->}@<-2pt>[r]  &  \mathcal X &\\
    }
\end{equation}
\noindent
where $\pi:\tilde{\mathcal V}\to\mathcal V$ finite \'etale, and an
isomorphism $\Phi:\theta^*(G_{\const})\to\theta^*(G_{\mathcal
X})$.
\par
Since $T$ isomorphically projects onto $U$, it is still closed
viewed as a sub-scheme of $\mathcal V$. Note that since $\mathcal
Y$ is semi-local and $\mathcal V$ contains all of its closed
points, $\mathcal V$ contains $\Pi^{-1}(\Pi(\mathcal Y))=\mathcal
Y$. By Lemma \ref{Lemma_8_2} there exists an open subset $\mathcal
W\subseteq\mathcal V$ containing $\mathcal Y$ and endowed with a
finite surjective $U$-morphism $\Pi^*:\mathcal W\to\Aff^1\times
U$.
\par
Let $\mathcal X^{\prime}=\theta^{-1}(\mathcal W)$,
$f^{\prime}=\theta^{*}(f)$, $q^{\prime}_U=q_U\circ\theta$, and let
$\Delta^{\prime}:U\to\mathcal X^{\prime}$ be the section of
$q^{\prime}_U$ obtained as the composition of $\delta$ with
$\Delta$. We claim that the triple $(\mathcal
X^{\prime},f^{\prime},\Delta^{\prime})$ is a nice triple. Let us
verify this. Firstly, the structure morphism
$q^{\prime}_U:\mathcal X^{\prime} \to U$ coincides with the
composition
$$
\mathcal X^{\prime}\xra{\theta}
\mathcal W\hra\mathcal X\xra{q_U} U.
$$
\noindent
Thus, it is smooth. The element $f^{\prime}$ belongs to the ring
$\Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal X^{\prime}})$,
the morphism $\Delta^{\prime}$ is a section of $q^{\prime}_U$.
Each component of each fibre of the morphism $q_U$ has dimension
one, the morphism $\mathcal X^{\prime}\xra{\theta}\mathcal
W\hra\mathcal X$ is \'{e}tale. Thus, each component of each fibre
of the morphism $q^{\prime}_U$ is also of dimension one. Since
$\{f=0\} \subset {\mathcal W}$ and $\theta: \mathcal X^{\prime}
\to {\mathcal W}$ is finite, $\{f^{\prime}=0\}$ is finite over
$\{f=0\}$ and hence  also over $U$. In other words, the $\mathcal
O$-module $\Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal
X^{\prime}})/f^{\prime}\cdot\Gamma(\mathcal X^{\prime},\mathcal
O_{\mathcal X^{\prime}})$ is finite. The morphism $\theta:
\mathcal X^{\prime}\to\mathcal W$ is finite and surjective.
We have constructed above the finite surjective morphism
$\Pi^*:\mathcal W\to\Aff^1\times U$. It follows that
$\Pi^*\circ\theta:\mathcal X^{\prime}\to\Aff^1\times U$ is finite
and surjective.
\par
Clearly, the \'{e}tale morphism $\theta:\mathcal X^{\prime}
\to\mathcal X$ is a morphism of nice triples, with $g=1$.
\par
Denote the restriction of $\Phi$ to $\mathcal X^{\prime}$ simply
by $\Phi$. The equality $(\Delta^{\prime})^*{\Phi}=\id_{G_U}$
holds by the very construction of the isomorphism $\Phi$. Theorem
follows.
%The scheme $\mathcal X^{\prime}$ is smooth over $k$ because
%$\mathcal X^{\prime} \to \mathcal X$
%is \'etale and $\mathcal X$ is smooth over $k$.
%Since $\wt\x\to \x$ is
%\'etale, it is flat and therefore, $f$ being regular in $k[\x]$, $\wt
%f$ is regular in $k[\wt{\x}]$.



%We already mentioned the
%existence of a finite surjective morphism of $U$-schemes
%$\Pi^* \circ \theta: \mathcal X^{\prime} \to \Aff^1 \times U$.
%It remains to check that $q^{\prime}_U$ is a smooth morphism. This is the case
%since so are $q_U$ and $\theta$.
%REMARK: THE FACT THAT $q^{\prime}_U$ has geometrically irreducible fibres IS NOT CLEAR.
%IN THE NEW REDUCTION OF THE TEXT WE NEED NOT the even irreducibility of closed fibres !!!


\section{An elementary Nisnevich square}
\label{SectElemNisnevichSquare} The aim of this Section is to
sketch a proof of Theorem \ref{ElementaryNisSquare}. We will use
analogues of three lemmas from \cite{Pa} making them
characteristic free. Lemma \ref{Lemma2} is a refinement of
\cite[Lemma 2]{OP}.
%and Lemma
%\ref{Lemma1}
%is a refinement of Quillen's trick.


\begin{lem}
\label{Lemma3} Let $k$ be an infinite field and let $S$ be an
$k$-smooth equidimensional $k$-algebra of dimension $1$. Let $f\in
S$ be a non-zero divisor
\par
Let $\mathfrak m_0$ be a maximal ideal with $S/\mathfrak m_0=k$.
Let $\mathfrak m_1,\mathfrak m_2,\dots,\mathfrak m_n$ be pair-wise
distinct maximal ideals of $S$, {\rm(}possibly $\mathfrak
m_0=\mathfrak m_i$ for some $i${\rm)}. Then there exists a
non-zero divisor $\bar s\in S$ such that $S$ is finite over
$k[\bar s]$ and
\begin{itemize}
\item[{\rm(1)}] the ideals $\mathfrak n_i:=\mathfrak m_i\cap
k[\bar s]$, $1\le i\le n$, are pair-wise distinct. If $\mathfrak
m_0$ is distinct from all $\mathfrak m_i$'s, then $\mathfrak n_i$
are all distinct from $\mathfrak n_0:=\mathfrak m_0\cap k[\bar
s]$; %%
\item[{\rm(2)}] the extension $S/k[\bar s]$ is \'etale at each
$\mathfrak m_i$'s and at $\mathfrak m_0$; %%
\item[{\rm(3)}] $k[\bar s]/\mathfrak n_i=S/\mathfrak m_i$ for each
$i=1,2,\dots,n$; %%
\item[{\rm(4)}] $\mathfrak n_0=\bar s k[\bar s]$.
\end{itemize}
\end{lem}
\begin{proof}
Let $x_i$, $1\le i\le n$, be the point on $\Spec(S)$ corresponding
to the ideal $\mathfrak m_i$. Consider a closed embedding
$\Spec(S)\hra\Aff^n_k$ and find a generic linear projection
$p:\Aff^n_k\to\Aff^1_k$ defined over $k$ and such that for each
$i$ the following holds:
\begin{itemize}
\item[{\rm(1)}] for all $i,j\geq 0$ one has $p(x_i)\neq p(x_j)$,
provided that $x_i\neq x_j$; %%
\item[{\rm(2)}] for each index $i\geq 0$ the map $p|_{\Spec(S)}:
\Spec(S)\to\Aff^1$ is \'{e}tale at the point $x_i$; %%
\item[{\rm(3)}] the separable degree of the extension
$k(x_i)/k(p(x_i))$ is one. %%
\end{itemize}
These items imply equalities $k(p(x_i))=k(x_i)$, for all $i$.
Indeed, the extension $k(x_i)/k(p(x_i))$ is separable by (2). By
(3) we conclude that $k(p(x_i))=k(x_i)$. Lemma follows.

\end{proof}

\begin{lem}
\label{Lemma4} Under the hypotheses of Lemma \ref{Lemma3} let
$f\in S$ be a non-zero divisor which does not belong to a maximal
ideal distinct from $\mathfrak m_0,\mathfrak m_1,\dots,\mathfrak
m_n$. Let $N(f)=N_{S/k[s]}(f)$ be the norm of $f$. Then for an
element $\bar s\in S$ satisfying {\rm(1)} to {\rm(4)} of Lemma
\ref{Lemma3} one has
\begin{itemize}
\item[{\rm(a)}] $N(f)=fg$ for an element $g\in S$; %%
\item[{\rm(b)}] $fS+gS=S$; %%
\item[{\rm(c)}] the map $k[\bar s]/(N(f))\to S/(f)$ is an
isomorphism.
\end{itemize}
\end{lem}
\begin{proof}
Straightforward.

\end{proof}

\begin{lem}
\label{Lemma2} Let $R$ be a semi-local essentially smooth
$k$-algebra with maximal ideals $\mathfrak p_i$, $1\le i\le r$.
Let $A \supseteq R[t]$ be an $R[t]$-algebra smooth as an $R$-algebra and
finite over $R[t]$. Assume that for each $i$ the $R/\mathfrak
p_i$-algebra $A_i=A/\mathfrak p_i A$ is equidimensional of
dimension one.
%$R/\mathfrak m_i$
%is a domain and assume there exists an element
%$h \in R \setminus \mathfrak m R$
%such that $R_h$ is $A$-smooth.
Let $\epsilon:A\to R$ be an $R$-augmentation and
$I=\Ker(\epsilon)$. Given an $f\in A$ with
$$ 0\neq\epsilon(f)\in\bigcap\limits^r_{i=1}\mathfrak p_i \subset R$$
\noindent
and such that the $R$-module $A/fA$ is finite, one can find an
element $u\in A$ satisfying the following conditions
\begin{itemize}
\item[{\rm(1)}] $A$ is a finite projective module over $R[u]$; %%
\item[{\rm(2)}] $A/uA=A/I\times A/J$ for some ideal $J$; %%
\item[{\rm(3)}] $J+fA=A$; %%
\item[{\rm(4)}] $(u-1)A+fA=A$; %%
\item[{\rm(5)}] if $N(f)=N_{A/R[u]}(f)$, then $N(f)=fg\in A$ for
some $g\in A$ %%
%(by \cite[Cor.18.17]{E} the $R[u]$-module $A$ is a projective module
%since $A$ and $R[u]$ are regular and $A$ is finite over $R[u]$);
\item[{\rm(6)}] $fA+gA=A$; %%
\item[{\rm(7)}] the composition map $\varphi:R[u]/(N(f))\to
A/(N(f))\to A/(f)$ is an isomorphism.
\end{itemize}
\end{lem}

\begin{proof}
Replacing $t$ by $t-\epsilon(t)$ we may assume that
$\epsilon(t)=0$. Since $A$ is finite over $R[t]$ from a theorem of
Grothendieck \cite[Cor.17.18]{E} it follows that it is a finite
projective $R[t]$-module.

Since $A$ is finite over $R[t]$ and $A/fA$ is finite over $R$ we
conclude that $R[t]/(N(f))$ is finite over $R$, and thus
$R/(tN(f))$ is finite over $R$. Setting $v=tN(f)$, we get an
integral extension $A$ over $R[v]$ and, moreover,
$$ v=tN_{A/R[t]}(f)=(ft)g=tfg. $$
%Now replacing
%$t$ by $u$ we may assume
We claim that $A/R[v]$ is integral, $\epsilon (v)=0$ and $v\in
fA$. Indeed, $v=tN_{A/R[t]}(f)=t(fg)$ and thus
$\epsilon(v)=\epsilon(t)\epsilon(fg)=0$. Finally, by the very
definition $v\in fA$.
\par
Below, we use bar to denote the reduction modulo an ideal, at that
subscript $i$ indicates reduction modulo $\mathfrak p_iA$. Let
$l_i=\bar R_i=R/\mathfrak p_i$. By the assumption of the lemma the
$l_i$-algebra $\bar A_i$ is $l_i$-smooth equidimensional of
dimension $1$. Let ${\mathfrak m}^{(i)}_1,{\mathfrak
m}^{(i)}_2,\dots,{\mathfrak m}^{(i)}_n$ be distinct maximal ideals
of $\bar A_i$ dividing $\bar f_i$ and let $\mathfrak
m^{(i)}_0=\Ker(\bar{\epsilon}_i)$. Let $\bar s_i\in\bar A_i$ be
such that the extension $\bar{A}_i/l_i[\bar s_i]$ satisfies
Conditions (1) to (4) of Lemma \ref{Lemma3}.
\par
Let $s\in A$ be a common raising of $\bar s_i$'s, in other words,
$\overline s=\bar s_i$ in $\bar A_i$, for all $i$. Replacing $s$
by $s-\epsilon(s)$ we may assume that $\epsilon(s)=0$ and, as
above, $\overline s=\bar s_i$, for all $i$.
\par
Let $s^n+p_1(v)s^{n-1}+\dots+p_n(v)=0$ be an integral dependence
for $s$. Let $N$ be an integer larger than
$\max\{2,\deg(p_i(t))\}$, where $i=1,2,\dots,n$. Then for any
$r\in k^{*}$ the element $u=s-rv^N$ has the following property:
$v$ is integral over $R[u]$. Thus, for any $r\in k^{*}$ the ring
$R$ is integral over $A[u]$.
\par
On the other hand, one has $\bar v_j\in\mathfrak m^{j}_i$ for all
$j$ and all $1\le i\le n$. It follows, that the element $\bar
u_j=\bar s_j-r\bar{v_j}^N$ still satisfies Conditions (1) to (4)
of Lemma \ref{Lemma3}.
\par
We claim that the element $u\in R$  has the required properties,
for almost all $r\in k^{*}$.
\par
In fact, for almost all $r\in k^{*}$ the element $u$ satisfies
Conditions (1) to (4) of Lemma \cite[Lemma 5.2]{OP2}. It remains to
show that Conditions (5) to (7) hold for all $r\in k^{*}$.
\par
Since $A$ is finite over $R[u]$ a theorem of Grothendieck
\cite[Cor.17.18]{E} implies that it is a finite projective
$R[u]$-module. To prove (5), consider the characteristic
polynomial of the operator $A\xra{f} A$ as an $R[u]$-module
operator. This polynomial vanishes on $f$ and its free term equals
$\pm N(f)$, the norm of $f$. Thus, $f^n-a_1f^{n-1}+\dots\pm
N(f)=0$ and $N(f)=fg$ for some $g\in R$.
\par
To prove (6), one has to verify that the above $g$ is a unit
modulo the ideal $fA$. It suffices to check that for each index
$i$ the element $\bar g_i\in\bar A_i$ is a unit modulo the ideal
$\bar f_i\bar A_i$. With that end observe that the field
$l_i=R/\mathfrak p_i$, the $l_i$-algebra $S_i=\bar A_i$, its
maximal ideals $\mathfrak m^{(i)}_0,\mathfrak
m^{(i)}_1,\dots,\mathfrak m^{(i)}_n$ and the element $\bar u_i$
satisfy the hypotheses of Lemma \ref{Lemma4}, with $u$ replaced by
$\bar u_i$. Now, by Item (b) of Lemma \ref{Lemma4} the reduction
$\bar g_i$ is a unit modulo the ideal $\bar f_i\bar R_i$.
\par
To prove (7), observe that $R[u]/(N_{A/k[X]}(f))$ and $A/fA$ are
finite $A$-modules. Thus, it remains to check that the map
$\varphi:R[u]/(N_{A/k[X]}(f))\to A/fA$ is an isomorphism modulo
each maximal ideal $\mathfrak m^{(j)}_i$. With that end it
suffices to verify that the map $\bar\varphi_i:l_i[\bar
u_i]/(N(\bar f_i))\to\bar A_i/\bar f_i\bar A_i$ is an isomorphism
for each index $i$, where $N(\bar f_i):=N_{\bar A_i/l_i[\bar
u]}(\bar f_i)$. Now, by Item (c) of Lemma \ref{Lemma4} the map
$\bar\varphi_j$ is an isomorphism. Lemma follows.

\end{proof}

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


%\begin{cor}
%\label{NormAtZeroAndOne} Under the hypotheses of Lemma
%$\ref{Lemma2}$ let $K$ be the quotient field of $R$,
%$A_K=A\otimes_R K$ and $\epsilon_K=\epsilon\otimes_R\id:A_K\to K$.
%Consider the inclusion $K[s]\subset A_K$. Then the norm $N(f)\in
%K[s]$ does not vanish at the points $1$ and $0$ of the affine line
%$\Aff^1_K$.
%\end{cor}

%\begin{proof}
%Condition (4) of \ref{Lemma2} implies that $N(f)$ does not vanish
%at the point $1$. Since $\epsilon_K (f)\neq 0\in K$, Conditions
%(2) and (3) imply that $N(f)$ does not vanish at $0$ either.
%\end{proof}

%Now we are all set to finish the proof of Theorem
%\ref{ElementaryNisSquare}.

\begin{proof}[Proof of Theorem \ref{ElementaryNisSquare}]
Let $U=\text{Spec}(\mathcal O_{X,\{x_1,x_2,\dots,x_r\}})$ be as in
Definition \ref{DefnNiceTriple}. Write $R$ for $\mathcal
O_{X,\{x_1,x_2,\dots,x_r\}}$. It is a semi-local
essentially smooth $k$-algebra with maximal ideals $\mathfrak
p_i$, $1\le i\le r$. Let $(\mathcal X,f,\Delta)$ be a nice triple
over $U$. We show that it gives rise to certain data subject to
the hypotheses of Lemma \ref{Lemma2}.
\par
Let $A=\Gamma(\mathcal X,\mathcal O_{\mathcal X})$. It is an
$R$-algebra via the ring homomorphism $q^*_U:R\to\Gamma(\mathcal
X,\mathcal O_{\mathcal X})$. Furthermore, it is smooth as an
$R$-algebra. The triple $(\mathcal X,f,\Delta)$ is a nice triple.
Thus, there exists a finite surjective $U$-morphism $\Pi:\mathcal
X\to\Aff^1_U$. It induces an $R$-algebra inclusion
$R[t]\hra\Gamma(\mathcal X,\mathcal O_{\mathcal X})=A$ such that
$A$ is finitely generated as an $R[t]$-module. Since $(\mathcal
X,f,\Delta)$ is a nice triple, $R/\mathfrak p_i$-algebra
$A/\mathfrak p_i A$ is equidimensional of dimension one, for all
$i$. Let $\epsilon=\Delta^*:A\to R$ be an $R$-algebra homomorphism
induced by the section $\Delta$ of the morphism $q_U$. Clearly,
this $\epsilon$ is an augmentation, let $I=\Ker(\epsilon)$.
Further, since $(\mathcal X,f,\Delta)$ is a nice triple,
$\epsilon(f)\neq 0\in R$ and $A/fA$ is finite as an $R$-module.
%% since $(\mathcal X,f,\Delta)$.
Summarising the above, we can conclude that we find ourselves in
the setting of Lemma \ref{Lemma2}, and may use the conclusion of
that Lemma.
\par
Thus, there exists an element $u\in A$ subject to Conditions (1)
through (7) of Lemma \ref{Lemma2}. This $u$ induces an $R$-algebra
inclusion $R[u]\hra A$ such that $A$ is finite as an
$R[u]$-module. Let
$$ \sigma:\mathcal X\to\Aff^1\times U $$
\noindent
be the $U$-scheme morphism induced by the above inclusion $R[u]\hra
A$. Clearly, $\sigma$ is finite and surjective. Let $N(f)\in
R[u]\subseteq A$ and $g_{f,\sigma}\in A$ be elements defined
just above Lemma
\ref{FandG}. Write $g$ for $g_{f,\sigma}$ in this proof.
%in Theorem
%\ref{ElementaryNisSquare}.
\par
We claim that this morphism $\sigma$ and the chosen
elements $N(f)$ and $g$ satisfy conclusions (1) to (4) of Theorem
\ref{ElementaryNisSquare}. Let us verify this claim. Since $A$ is
finite as an $R[t]$-module and both rings $R[t]$ and $A$ are
regular, the $R[t]$-module $A$ is finitely generated and
projective, see \cite[Corollary 18.17]{E}). Thus, $\sigma$ is
\'{e}tale at a point $x\in \mathcal X$ if and only if the
$k(\sigma(x))$-algebra $k(\sigma(x))\otimes_{R[t]} A$ is
\'{e}tale. If the point $x$ belongs to the closed sub-scheme
$\text{Spec}(A/\mathfrak p_i A)$ for some maximal ideal $\mathfrak p_i$ of $R$, then
$$ k(\sigma(x))\otimes_{R[t]}A=k(\sigma(x))\otimes_{(R/\mathfrak
p_i)[t]}A/\mathfrak p_iA. $$
\noindent
We can conclude that $\sigma$ is \'{e}tale at a specific point $x$
if and only if the $(R/\mathfrak p_i)[t]$-algebra $A/\mathfrak
p_iA$ is \'etale at the point $x$. It follows from the proof of
Lemma \ref{Lemma2} that the morphism $\sigma$ induces a morphism
$\text{Spec}(A/\mathfrak p_i A)\xra{\sigma_i}\Aff^1_{l_i}$ on the
closed fibre $\text{Spec}(A/\mathfrak p_i A)$ for each $i$. This
induced morphism is \'{e}tale along the vanishing locus of the
function $\bar f_i$
and along each point $\overline
\Delta_i(\Spec\,l_i)$.
In fact, for the vanishing locus of the function
$\bar f_i$
this follows from items (6) and (7) of Lemma
\ref{Lemma2}. It follows from the hypotheses of Lemma
\ref{Lemma2}
that the function $f$ vanishes at each maximal ideal containing $I$.
Thus $\sigma$ is \'{e}tale along the closed sub-scheme
$\mathcal X$ defined by the ideal $I$, that is along $\Delta$.

Item (1) of Theorem
\ref{ElementaryNisSquare} follows.
\par
The first of the following equalities
$$ \sigma^{-1}(\sigma(\{f=0\}))=\{N(f)=0\}=\{f=0\}\sqcup\{g_{f,\sigma}=0\} $$
\noindent
is a commonplace. %% general well-known fact.
The second one follows from the equality $N(f)=\pm f \cdot
g_{f,\sigma}$, proved in Lemma \ref{FandG}
%just above Theorem
%\ref{ElementaryNisSquare}
and Item (6)
of Lemma \ref{Lemma2}.
\par
Write $g$ for $g_{f,\sigma}$. Clearly, the square
(\ref{SquareDiagram2}) is Cartesian and the morphism $\sigma^0_g$
is \'{e}tale. The scheme $\mathcal X^{0}_g$ contains a closed
sub-scheme $\Delta(U)$, and hence is non-empty. Item (7) of Lemma
\ref{Lemma2} shows that the morphism of the reduced closed
sub-schemes
$$ \sigma^0_g|_{\{f=0\}_{\red}}: \{f=0\}_{\red} \to \{N(f)=0\}_{\red} $$
\noindent
is an isomorphism. Thus, we have checked Item (3) of Theorem
\ref{ElementaryNisSquare}.
\par
It remains only to check Item (4). We already know that
$\{f=0\}\subset\mathcal X^{0}_g$. By Claim
\ref{DeltaIsWellDefined} both schemes $\Delta(U)$ and $\{f=0\}$
are semi-local and the set of closed points of $\Delta(U)$ is
contained in the set of closed points of the closed set $\{f=0\}$.
Thus, $\Delta(U)\subset\mathcal X^{0}_g$. This concludes the proof
of Item (4) of Theorem \ref{ElementaryNisSquare} and thus of the
theorem itself.

\end{proof}


\section{Proof of Theorems \ref{MainThm} and \ref{MainThmGeometric}}
\label{ProofOfMainThm}

%the case, where the ring $R$
%is the semi-local ring $\mathcal O_{X, \{x_1,x_2,\dots,x_n\}}$ of
%a smooth $k$-variety $X$ over an infinite field $k$ at a finite
%family of points $x_1,x_2,\dots,x_n\in X$.
%\par\smallskip
%Theorem
%\ref{MainThmGeometric}.

\begin{proof}[Proof of Theorem \ref{MainThmGeometric}]
%is subdivided into two
%parts:
%\par\smallskip
%(2) a {\it geometric part\/}, and
%\par\smallskip
%(3) a {\it group part}.
%\par\smallskip


%To prove this result we prove the following Lemma and its version for semi-local rings. The Lemma inspired by
%\cite[Cor.1.8]{R1}
%
%\begin{lem}
%Let $R$ and $G$ be as above. Let $\textit{m}$ be the maximal ideal of $R$ and
%$l=R/\textit{m}$
%be the residue field.
%Let $S \subseteq R[t]$ be the multiplicative system of all unitary
%polynomial in $R[t]$ and let
%$$\pi: A=R[t]_S \to l[t]_S=l(t)=L$$
%be the canonical projection, induced by the projection
%$R \to l$. Let $\mathcal O=l[t]_{(1/t)}$
%be the localization of the ring $l[1/t]$ at the maximal ideal $1/t \cdot l[1/t]$.
%Let $U^+$ and $U^-$ be $R$-group schemes, which are the unipotent radicals of two opposite
%parabolic subgroups of $G$ over $R$.
%
%Let $\pi: G(A) \to G(L)$ be a group homomorphism induced by the projection
%$\pi: A \to L$. Then $G(L)=\pi (<U^+(A), U^-(A)>) \cdot G(\mathcal O)$.
%\end{lem}

%The geometric part of the proof
We start with the following data.
%Sections \ref{ElementaryFibrations} and
%\ref{NiceTriples}, is organised as follows.
Fix a smooth affine $k$-scheme $X$, a finite family of points
$x_1,x_2,\dots,x_n$ on $X$, and set $\mathcal O:=\mathcal
O_{X,\{x_1,x_2,\dots,x_n\}}$ and $U:=\text{Spec}(\mathcal O)$.
Further, consider a simple simply-connected $U$-group scheme $G$
and a principal $G$-bundle $P$ over $\mathcal O$ which is trivial
over the field of fractions $K$ of $\mathcal O$. {\it We have to
check that the principal $G$-bundle $P$ is trivial\/}. We may and
will assume that for certain $\text{f}\in\mathcal O$ the principal
$G$-bundle $P$ is trivial over $\mathcal O_{\text{f}}$.
\par
Shrinking $X$ if necessary, we may secure the following properties
%(see Section
%\ref{BasicTriple}).
\par\smallskip
(i) The points $x_1,x_2,\dots,x_n$ are still in $X$;
\par\smallskip
(ii) The group scheme $G$ is defined over $X$ and it is a simple
and simply-connected group scheme. We will often denote this
$X$-group scheme by $G_X$ and write $G_U$ for the original $G$;
\par\smallskip
(iii) The principal $G$-bundle $P$ is defined over $X$; $\text{f}
\in k[X]$ and $\text{f}$ vanishes at each $x_i$'s;
\par\smallskip
(iv) The restriction $P_{\text{f}}$ of the bundle $P$ to the
principal open sub-scheme $X_{\text{f}}$ is trivial.
\par\smallskip\noindent
We may shrink $X$ a little further, if necessary,
always looking after verification of (i) to (iv).
This way we constructed in Section
\ref{BasicTriple}
a basic nice triple
(\ref{FirstNiceTriple}).
%% and looking after $x_1,x_2,\dots,x_n$ still sitting in $X$,
Beginning with that basic nice triple we constructed in Section
\ref{MainConstruction}
the following data:
%% (which is still the geometric part)}
\par\smallskip
(a) a unitary polynomial $h(t)$ in $R[t]$,
\par\smallskip
(b) a principal $G_U$-bundle $P_t$ over $\Aff^1_U$,
\par\smallskip
(c) a diagram of the form
\begin{equation}
\label{SquareDiagram}
    \xymatrix{
     \Aff^1_U   \ar[drr]_{\pr}  &&  Y \ar[ll]_{\sigma} \ar[d]_{q_U} \ar[rr]^{q_X} && X  &\\
    && U \ar[urr]_{\can} \ar@/_0.8pc/[u]_{\delta} &\\
    }
\end{equation}
with $Y=\mathcal X^{\prime}$,
%\begin{equation}
%\label{SquareDiagram}
%    \xymatrix{
%     \mathcal X  \ar[d]_{q_U} \ar[rr]^{q_X} &&  X   & \\
%     U \ar[urr]_{can} \ar@/_0.8pc/[u]_{\Delta} &\\
%    }
%\end{equation}
\par\smallskip
(d) an isomorphism $\Phi:q^*_U(G_U)\to q^*_X(G)$ of $Y$-group
schemes.
\par\smallskip\noindent
By Theorem
\ref{MainData}
these data may be chosen to be subject to the following
properties:
\par\smallskip
(1*) $q_U =\pr\circ\sigma$,
\par\smallskip
(2*) $\sigma$ is \'etale,
\par\smallskip
(3*) $q_U \circ\delta =\id_U$,
\par\smallskip
(4*) $q_X \circ \delta=\can$,
\par\smallskip
(5*) the restriction of $P_t$ to $(\Aff^1_U)_h$ is a trivial
$G_U$-bundle,
%% where $G_U$ is from the item $(ii)$ above;
\par\smallskip
(6*) $(\sigma)^*(P_t)$ and $q_X^*(P)$ are isomorphic as principal
$G_U$-bundles. Here $q_X^*(P)$ is regarded as a principal
$G_U$-bundle via the group scheme isomorphism $\Phi$ from the item
(d).
\par\smallskip

%{\it This completes the geometric part}.
%In the group part we prove the following result, see the end of
%Section \ref{SecProjectiveLine}.
%\begin{thm}
%\label{TrivialityOfP_t} Let $R$ and $G$ be the same as in Theorem
%\ref{MainThmGeometric}. Let $P$ be a principal $G$-bundle over the
%polynomial ring $R[t]$ in one variable $t$ such that for a monic
%polynomial $f(t)\in R[t]$ the bundle $P_{f(t)}$ over the ring
%$R[t]_{f(t)}$ is trivial. Then the $G$-bundle $P$ is trivial.
%\end{thm}

%It remains to explain how Theorem
%\ref{MainThmGeometric}
%follows
%from Theorem
%\ref{TrivialityOfP_t}.
%
%This result is based on the following equality
%\begin{equation}
%\label{*Key}
%G_l(l(t^{-1}))=G_l(l(t^{-1}))^{+} \cdot G_l(R)
%%\end{equation}
%which holds for {\bf simple simply-connected}
%$l$-group scheme $G_l$
%and which is proved in the proof Proposition
%\ref{Key}. The equality
%(\ref{*Key})
%replaces the arguments from
%\cite[Proof of Thm.3.3]{C-TO},
%where the surjectivity of certain group homomorphism comes as an immediate consequence
%of rather specific geometry. Namely, the surjectivity they were interested in is a direct consequence
%of the fact the composition of $k$-algebra homomorphisms
%$$k[t]_{\bar g} \to B[t]_{g\bar g} \to k[t]_{\bar g}$$
%is the identity.
%
%One way of thinking about the equality
%(\ref{*Key}) is this:
%one has a surjective stabilization for the unstable functor
%$SK_1(G_l(l(t^{-1}))):=G_l(l(t^{-1}))^{+}\backslash{G_l(l(t^{-1}))}$.
%Namely,
%the
%$SK_1(G^{an}_l(l(t^{-1}))$
%surjects onto
%$SK_1(G_l(l(t^{-1})))$,
%where
%%$G^{an}_l$
%is the anisotropic kernel of $G_l$.
%And additionally, for each anisotropic reductive $l$-group $H$ one has
%another equality (see
%\cite[Prop.1.2]{R1}). Namely,
%$H_l(l(t^{-1}))=H_l(A)$.
%In, particuar, the latter hold for
%$H_l=G^{an}_l$.

%More specifically, we prove Corollary
%\ref{TrivialOnFibraIsTrivial}, which deals only with the case of
%Theorem \ref{TrivialityOfP_t}, where the ring $R$ is of geometric
%type. The general case can be deduced from the geometric one by
%standard arguments using the remarkable theorem of D.~Popescu, see
%% celebrated
%\cite{P}). Again, details of such a reduction closely follow the
%work by the first author and M.~Ojanguren \cite{OP} and will be
%mostly omitted.

%With that end
%let us explain how {\it the geometric case}

%and Theorem \ref{MainData}.
%% ������: Fix a $k$-smooth affine scheme $X$, points
% $x_1,x_2,\dots,x_n$ on $X$, and set $U:=\mathcal
% O_{X,\{x_1,x_2,\dots,x_n\}}$. Let $G$ be a simple simply-connected
% $U$-group scheme and let $P$ be a principal $G$ bundle over $U$.
% Shrinking $X$ we may ensure Conditions (i) to (iv). Assuming $(i)$
%to $(iv)$ and shrinking $X$ a little bit more and keeping
% $x_1,x_2,\dots,x_n$ in $X$ we construct in Remark
% \ref{MainConstruction} data (a) to (d) subject to Conditions (1^*)
% to (6^*).
{\it Recall, that we have to check that the $G$-bundle $P$ is
trivial over $U$\/}. By assumption the group scheme $G_U$ is
simple and simply-connected and isotropic.
%% contains a split torus $\Bbb
%% G_m$ by the assumption of Theorem \ref{MainThm}.
The restriction of the principal $G_U$-bundle $P_t$ to $(\Aff^1_U)_h$
is trivial by Condition (5*). Then the principal $G_U$-bundle
$P_t$ is itself trivial by Theorem \ref{TrivialityOfP_t}.
\par
On the other hand, the $G_U$-bundle $(\sigma)^*(P_t)$ is isomorphic
to $q_X^*(P)$ as principal $G_U$-bundle by (6*).
%% , where $q_X^*(P)$ is regarded as a principal $G_U$-bundle
%% via the group scheme isomorphism $\Phi$ from the item $(d)$.
Therefore, $q^*_X(G)$ is trivial as a $q^*_X(G)$-bundle. It
follows that $\delta^*(q^*_X(P))=\can^*(P)$ is trivial as a
$\delta^*(q^*_X(G))=\can^*(G)$-bundle. Thus, $\can^*(P)=P_U$ is
trivial as a principal $\can^*(G)=G_U$-bundle.

\end{proof}

\begin{proof}[Proof of Theorem \ref{MainThm}]
Theorem follows from Theorem \ref{MainThmGeometric} using
arguments similar to those in \cite[Sect.7]{OP2}, and we skip them.
Let us only indicate here that arguments in \cite[Sect.7]{OP2} are
based on a Theorem due to D.~Popescu \cite{P} (see also \cite{Sw}
for a self-contained exposition of that theorem).

\end{proof}
%% {\bf The Theorem \ref{MainThm} is reduced to Theorems
%% \ref{TrivialityOfP_t} and \ref{MainData} and Remark
%% \ref{MainConstruction}}. \\

%{\bf Sections $2$ to $3$ contains the geometric part of our proof of Theorem \ref{MainThm},
%Section \ref{GroupPart} contains the Group part.}


\section{Semi-simple case}
\label{SectSemi-Simple}


In the present Section we extend Theorem \ref{MainThm} to the case
of semi-simple simply-connected groups. By~\cite[Exp. XXIV 5.3,
Prop. 5.10]{D-G} the category of semi-simple simply-connected
group schemes over a Noetherian domain $R$ is semi-simple. In
other words, each object has a unique decomposition into a product
of indecomposable objects. Indecomposable objects can be described
as follows. Take a domain $R^{\prime}$ such that $R\subseteq
R^{\prime}$ is a finite \'{e}tale extension and a simple
simply-connected group scheme $G^{\prime}$ over $R^{\prime}$. Now,
applying the Weil restriction functor $R_{R^{\prime}/R}$ to the
$R$-group scheme $G^{\prime}$ we get a simply-connected $R$-group
scheme $R_{R^{\prime}/R}(G^{\prime})$, which is
%we get $R$-group schemes which are
%% The latter $R$-group scheme is
an indecomposable object in the above category. Conversely, each
indecomposable object can be constructed in this way.
\begin{thm}
\label{MainThmSemi-Simple} Let $R$ be a regular semi-local domain
containing an infinite perfect subfield and let $K$ be the
quotient field of $R$. Let $G$ be a semi-simple simply-connected
group scheme $G$ all of whose indecomposable factors are
isotropic.
%% contains a split rank $1$ torus $\Bbb G_m$.
Then the map
$$ H^1_{\text{\rm\'et}}(R,G)\to H^1_{\text{\rm\'et}}(K,G) $$
\noindent
induced by the inclusion of $R$ into $K$ has trivial kernel.
\end{thm}
In other words, under the above assumptions on $R$ and $G$ each
principal $G$-bundle $P$ over $R$ which has a $K$-rational point
is itself trivial.
\begin{proof}
Take a decomposition of $G$ into indecomposable factors
$G=G_1\times G_2\times\dots\times G_r$. Clearly, it suffices to
check that for each index $i$ the kernel
$$ H^1_{\text{\'et}}(R,G_i)\to H^1_{\text{\'et}}(K,G_i) $$
\noindent
is trivial. We know that there exists a finite \'{e}tale extension
$R^{\prime}_i/R$ such that $R^{\prime}_i$ is a domain and the Weil
restriction $R_{R^{\prime}_i/R}(G^{\prime}_i)$ coincides with
$G_i$.
\par
The Faddeev---Shapiro Lemma \cite[Exp. XXIV Prop. 8.4]{D-G} states
that there is a canonical isomorphism preserving the distinguished
point
$$ H^1_{\text{\'et}}\big(R, R_{R^{\prime}_i/R}(G^{\prime}_i)\big)
\cong H^1_{\text{\'et}}\big(R^{\prime},G_i\big). $$
\noindent
To complete the proof it only remains to apply Theorem
\ref{MainThm} to the semi-local regular ring $R^{\prime}_i$, its
quotient field $K_i$ and the simple $R^{\prime}_i$-group scheme
$G^{\prime}_i$.
\end{proof}

\begin{thm}
\label{MainThmSemi-SimpleGeometric} Let $k$ be an infinite field.
Let $\mathcal O$ be the semi-local ring of finitely many points on
a smooth irreducible $k$-variety $X$ and let $K$ be its field of
fractions. Let $G$ be a semi-simple simply-connected group scheme
$G$ all of whose indecomposable factors are isotropic.
%% contains a split rank $1$ torus $\Bbb G_m$.
Then the map
$$ H^1_{\text{\rm\'et}}(R,G)\to H^1_{\text{\rm\'et}}(K,G) $$
\noindent
induced by the inclusion of $R$ into $K$ has trivial kernel.
\end{thm}

\begin{proof}
Use Theorem
\ref{MainThmGeometric}
and argue literally as in the proof of Theorem
\ref{MainThmSemi-Simple}

\end{proof}

\subparagraph{Acknowledgments} The authors are very grateful to
Konstantin Pimenov and Victor Petrov for useful discussions on the subject of the
present article.
%The  author thanks very much for the support the
%RTN-Network HPRN-CT-2002-00287,
%the RFFI-grant 03-01-00633a,
%the Swiss National Science Foundation
%and INTAS-03-51-3251.

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

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\end{document}

\begin{lem}
\label{Lemma3} Let $k$ be an infinite field and let $S$ be an
$k$-smooth equidimensional $k$-algebra of dimension $1$. Let $f
\in S$ be a non-zero divisor

Let $\mathfrak m_0$ be a maximal ideal with $S/\mathfrak m_0=k$.
Let $\mathfrak m_1,\mathfrak m_2,\dots,\mathfrak m_n$ be different
maximal ideals of $S$, {\rm(}possibly $\mathfrak m_0=\mathfrak
m_i$ for some $i${\rm)}. Then there exists a non-zero divisor
$\bar s\in S$ such that $S$ is finite over $k[\bar s]$ and
\begin{itemize}
\item[{\rm(1)}] the ideals $\mathfrak n_i:=\mathfrak m_i\cap
k[\bar s]$, $1\le i\le n$, are pair-wise distinct. If $\mathfrak
m_0$ is distinct from all $\mathfrak m_i$'s, then $\mathfrak n_i$
are all distinct from $\mathfrak n_0:=\mathfrak m_0\cap k[\bar
s]$; %%
\item[{\rm(2)}] the extension $S/k[\bar s]$ is \'etale at
each $\mathfrak m_i$'s and at $\mathfrak m_0$; %%
\item[{\rm(3)}] $k[\bar s]/\mathfrak n_i=S/\mathfrak m_i$ for each
$i=1,2,\dots,n$; %%
\item[{\rm(4)}] $\mathfrak n_0=\bar sk[\bar s]$.
\end{itemize}
\end{lem}
\begin{proof}
Let $x_i$, $1\le i\le n$, be the point on $\Spec(S)$ corresponding
to the ideal $\mathfrak m_i$. Consider a closed embedding
$\Spec(S)\hra\Aff^n_k$ and find a generic linear projection
$p:\Aff^n_k\to\Aff^1_k$ defined over $k$ and such that for each
$i$ the following holds:
\begin{itemize}
\item[{\rm(1)}] for all $i,j\geq 0$ one has $p(x_i)\neq p(x_j)$,
provided that $x_i\neq x_j$; %%
\item[{\rm(2)}] for each index $i\geq 0$ the map $p|_{\Spec(S)}:
\Spec(S)\to\Aff^1$ is \'{e}tale at the point $x_i$; %%
\item[{\rm(3)}] the separable degree of the extension
$k(x_i)/k(p(x_i))$ is one.
\end{itemize}
These items imply equalities $k(p(x_i))=k(x_i)$, for all $i$.
Indeed, the extension $k(x_i)/k(p(x_i))$ is separable by (2). By
(3) we conclude that $k(p(x_i))=k(x_i)$. Lemma follows.
\end{proof}

\begin{lem}
\label{Lemma4} Under the hypotheses of Lemma \ref{Lemma3} let
$f\in S$ be a non-zero divisor which does not belong to a maximal
ideal distinct from $\mathfrak m_0,\mathfrak m_1,\dots,\mathfrak
m_n$. Let $N(f)=N_{S/k[s]}(f)$ be the norm of $f$. Then for an
element $\bar s\in S$ satisfying {\rm(1)} to {\rm(4)} of Lemma
\ref{Lemma3} one has
\begin{itemize}
\item[{\rm(a)}] $N(f)=fg$ for an element $g\in S$; %%
\item[{\rm(b)}] $fS+gS=S$; %%
\item[{\rm(c)}] the map $k[\bar s]/(N(f))\to S/(f)$ is an
isomorphism.
\end{itemize}
\end{lem}
\begin{proof}
Straightforward.
\end{proof}

\begin{lem}
\label{Lemma2} Let $R$ be a semi-local essentially smooth
$k$-algebra with maximal ideals $\mathfrak p_i$'s, $1\le i\le r$.
Let $A\supseteq R[t]$ be an $R[t]$-algebra smooth as an $R$-algebra and
finite over $R[t]$. Assume that for each $i$ the $R/\mathfrak
p_i$-algebra $A/\mathfrak p_i A$ is equidimensional of dimension
one.
%$R/\mathfrak m_i$
%is a domain and assume there exists an element
%$h \in R \setminus \mathfrak m R$
%such that $R_h$ is $A$-smooth.
Let $\epsilon:A\to R$ be an $R$-augmentation and
$I=\Ker(\epsilon)$. Given an $f\in A$ with $0\neq\epsilon(f)\in A$
and such that the $R$-module $A/fA$ is finite, one can find an
element $u\in A$ satisfying the following conditions
\begin{itemize}
\item[{\rm(1)}] $A$ is a finite projective module over $R[u]$; %%
\item[{\rm(2)}] $A/uA=A/I\times A/J$ for some ideal $J$; %%
\item[{\rm(3)}] $J+fA=A$; %%
\item[{\rm(4)}] $(u-1)A+fA=A$; %%
\item[{\rm(5)}] if $N(f)=N_{A/R[u]}(f)$ then $N(f)=fg\in A$ for
some $g\in A$;
%{\rm(}observe that $A/R[u]$ is a projective module
%since $A$ and $R[u]$ are regular and $A$ is finite over
%$R[u]${\rm)}; %%
\item[{\rm(6)}] $fA+gA=A$; %%
\item[{\rm(7)}] the composition map $\varphi:A[u]/(N(f))\to
A/(N(f))\to A/(f)$ is an isomorphism.
\end{itemize}
\end{lem}

\begin{proof}
Replacing $t$ by $t-\epsilon(t)$ we may assume that
$\epsilon(t)=0$. Since $A$ is finite over $R[t]$ and $A/fA$ is
finite over $R$ we conclude that $R[t]/(N(f))$ is finite over $R$,
and thus $R/(tN(f))$ is finite over $R$. Setting $u=tN(f)$, we get
an integral extension $A$ over $R[u]$ and, moreover,
$$ u=tN_{A/R[t]}(f)=(ft)g=tfg. $$
\par
Now, replacing $t$ by $u$ if necessary, we may assume that
$A/R[t]$ is integral, $\epsilon(t)=0$ and $t\in fR$. Indeed,
$u=tN_{A/R[t]}(f)=t(fg)$ and thus $u\in fR$. Applying $\epsilon$
to we see that $\epsilon(u)=\epsilon(t)\epsilon(fg)=0$.
\par
\def\bar{\overline}
Below, we use bar to denote the reduction modulo an ideal, at that
subscript $i$ indicates reduction modulo $\mathfrak p_iA$. Let
$l_i=\bar R_i=R/\mathfrak p_i$. By the assumption of the lemma the
$l_i$-algebra $\bar A_i$ is $k$-smooth equidimensional of
dimension $1$. Let ${\mathfrak m}^{(i)}_1,{\mathfrak
m}^{(i)}_2,\dots,{\mathfrak m}^{(i)}_n$ be ??all?? distinct
maximal ideals of $\bar A_i$ dividing $\bar f_i$ and let
$\mathfrak m^{(i)}_0=\Ker(\bar{\epsilon}_i)$. Let $\bar s_i\in\bar
A_i$ be such that the extension $\bar{A}_i/l_i[\bar s_i]$
satisfies Conditions (1) to (4) of Lemma \ref{Lemma3}.
\par
Let $s\in A$ be a common raising of $\bar s_i$'s, in other words,
$\overline s=\bar s_i$ in $\bar A_i$, for all $i$. Replacing $s$
by $s-\epsilon(s)$ we may assume that $\epsilon(s)=0$ and, as
above, $\overline s=\bar s_i$, for all $i$.
\par
Let $s^n+p_1(t)s^{n-1}+\dots+p_n(t)=0$ be an integral dependence
for $s$. Let $N$ be an integer larger than the
$\max\{2,\deg(p_i(t))\}$, where $i=1,2,\dots,n$. Then for any
$r\in k^{*}$ the element $u=s-rt^N$ has the following property:
$t$ is integral over $R[u]$. Thus, for any $r\in k^{*}$ the ring
$R$ is integral over $A[u]$.
\par
On the other hand, one has $\bar t_j\in\mathfrak m^{j}_i$ for all
$j$ and all $1\le i\le n$. It follows, that the element $\bar
u_j=\bar s_j-r\bar{t_j}^N$ still satisfies the conditions (1) to
(4) of Lemma \ref{Lemma3}.
\par
We claim that the element $u\in R$ has the required properties,
for almost all $r\in k^{\times}$.
\par
In fact, for almost all $r\in k^{*}$ the element $u$ satisfies
Conditions (1) to (4) of Lemma \cite[Lemma 2]{OP}. It remains to
show that Conditions (5) to (7) hold for all $r\in k^{*}$.
\par
To prove (5), consider the characteristic polynomial of the
operator $A\xra{f} A$ as an $R[s]$-module operator. This
polynomial vanishes on $f$ and its free term equals $\pm N(f)$,
the norm of $f$. Thus, $f^n-a_1f^{n-1}+\dots\pm N(f)=0$ and
$N(f)=fg$ for some $g\in R$.
\par
To prove (6), one has to verify that the above $g$ is a unit
modulo the ideal $fA$. It suffices to check that for each index
$j$ the element $\bar g_j\in\bar A_j$ is a unit modulo the ideal
$\bar f_j\bar A_j$. With that end observe that the field
$l_j=R/\mathfrak p_j$, the $l_j$-algebra $S_j=\bar A_j$, its
maximal ideals $\mathfrak m^{(j)}_0,\mathfrak
m^{(j)}_1,\dots,\mathfrak m^{(j)}_n$ and the element $\bar u_j$
satisfy the hypotheses of Lemma \ref{Lemma4}, with $u$ replaced by
$\bar u_j$. Now, by the item (b) of Lemma \ref{Lemma4} the
reduction $\bar g_j$ is a unit modulo the ideal $\bar f_j\bar
R_j$.
\par
To prove (7), observe that $R/fR$ and $A[u]/(N_{A/k[X]}(f))$ are
finite $A$-modules. Thus, it remains to check that the map
$\varphi:A[u]/(N_{A/k[X]}(f))\to R/fR$ is an isomorphism modulo
each maximal ideal $\mathfrak m^{(j)}_i$. With that end it
suffices to verify that the map $\bar\varphi_j:l_j[\bar
u_j]/(N(\bar f_j))\to\bar R_j/\bar f_j\bar R_j$ is an isomorphism
for each index $j$, where $N(\bar f):=N_{\bar R/k[\bar u]}(\bar
f)$. Now, by the item (c) of Lemma \ref{Lemma4} the map
$\bar\varphi_j$ is an isomorphism. Lemma follows.
\end{proof}

Then for each finite surjective $U$-morphism
$\sigma:\mathcal X\to\Aff^1\times U$ and the corresponding
$\mathcal O$-algebra inclusion $\mathcal O[t]\hra\Gamma(\mathcal
X,\mathcal O_{\mathcal X})$
%the element $t$ is a non-zero divisor in
%$\Gamma(\mathcal X, \mathcal O_{\mathcal X})$,
the algebra $\Gamma(\mathcal X,\mathcal O_{\mathcal X})$ is
finitely generated and projective as an $\mathcal O[t]$-module
since both rings are regular and the morphism $\sigma$ is finite.
Take the characteristic polynomial $t^r - a_{n-1}t^{r-1} + \dots
\pm N(f)=0$ of the $\mathcal O[t]$-module endomorphism
$\Gamma(\mathcal X,\mathcal O_{\mathcal X}) \xra{f\cdot }
\Gamma(\mathcal X,\mathcal O_{\mathcal X})$ and set
\begin{equation}
g_{f,\sigma}:=f^{r-1}-a_{n-1}f^{r-2}+\dots\pm a_1\in
\Gamma(\mathcal X,\mathcal O_{\mathcal X}).
\end{equation}
\noindent
Then $fg_{f,\sigma}=\pm N(f)$ since the characteristic polynomial vanishes on
$f$.
\par
