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\begin{document}

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

\author{Ivan Panin \footnote{The author acknowledges support of the joint DFG - RFBR-project 09-01-91333-NNIO-a} }

\date{08.05.2009}

\maketitle

\begin{abstract}
Let $R$ be a regular semi-local ring containing an {\bf infinite perfect subfield} and
let $K$ be its field of fractions. Let $G$ be
a reductive $R$-group scheme satifying a mild "isotropy condition". Then each principal
$G$-bundle $P$ which becomes trivial over $K$ is trivial itself. If $R$ is of geometric type,
then it suffices to assume that $R$ is of geometric type over an infinite field.
Two main Theorems of Panin's, Stavrova's and Vavilov's
\cite{PSV}
state the same results for semi-simple simply connected $R$-group schemes.
Our proof is heavily based on those two Theorems of
\cite[Thm.1.1]{PSV}, on the main result of
\cite{C-T-S}
and {\bf on two purity Theorems} proven in the present preprint.

%Given a regular semi-local ring $R$ containing an {\bf infinite perfect subfield}, given a reductive $R$-group scheme $G$,
Those purity result look as follows.
Given an $R$-torus $C$ and {\bf a smooth} $R$-group scheme morphism
$\mu: G \to C$ one can
form a functor from $R$-algebras to abelian groups
$S \mapsto {\cal F}(S):=C(S)/\mu(G(S))$.
%Assuming  that $\mu$ is strictly flat (epimorphism locally for the \'{e}tale topology on $\text{Spec}(R)$)
We prove that this functor satisfies
a purity theorem for $R$.
If $R$ is of geometric type,
then it suffices to assume that $R$ is of geometric type over an infinite field.

Examples to mentioned purity results are considered in the very end of the preprint.
\end{abstract}

\section{Introduction}
\label{SectIntroduction}
Recall that an $R$-group scheme $G$ is called reductive
(respectively, semi-simple; respectively, 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; respectively, 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.

Let $R$ be a semi-local ring and $G$ be a reductive group scheme over $R$.
Recall an "isotropy condition " for $G$.

\begin{itemize}
\label{CondIsotropy}
\item[({\bf I})]
{\it Each $R$-simple component of the derived group $G_{der}$ of the group $G$ is isotropic over} $R$.
\end{itemize}

%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}
%$\Bbb G_{m,R}$).

%Let $R$ be a semi-local ring and $G$ be a reductive group scheme over $R$.
%Recall a "isotropy condition " for $G$.

%\begin{itemize}
%\label{CondIsotropy}
%\item[({\bf I})]
%{\it Each $R$-simple component of the derived group $G_{der}$ of the group $G$ is isotropic over} $R$.
%\end{itemize}
The hypotheses {\bf(I)} means more precicely the following : consider the derived group $G_{der}$
of the group $G$ and the simply-connected cover $G^{sc}_{der}$ of $G_{red}$; that group $G^{sc}_{der}$
is a product over $R$ (in a unique way) of $R$-indecomposable groups, which are required to be isotropic.

\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
a reductive $R$-group scheme satisfying the condition ({\bf I}).
% 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.

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.
\end{thm}
\par\smallskip
$\bullet$
{\it One can ask} whether two elements
$\xi, \zeta \in H^1_{\text{\rm\'et}}(R,G)$
which are
equal over $K$ are equal over $R$. Theorem
\ref{MainThm} {\it does not imply that in general}. However it {\it does imply that
provided that} at least one of the two group schemes $G(\xi)$ or $G(\zeta)$ is isotropic.
\par\smallskip
$\bullet$
Clearly, this Theorem extends as the geometric case of the main result of
J.-L. Colliot-Th\'{e}l\`{e}ne and J.-J. Sansuc
\cite{C-T-S},
so main results of
I.Panin, A.Stavrova and N.Vavilov
\cite[Thm.1.1, Thm. 1.2]{PSV}. However our proof of Theorem
\ref{MainThm}
{\bf is based heavily on those results
and on two purity results} (Theorems
\ref{PurityGeometric}
and
\ref{PurityGeneral})
proven in the
present preprint.
%???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
$\bullet$
The case of arbitrary reductive group scheme over a discrete valuation ring
is completely solved by Y.Nisnevich in
\cite{Ni}.
\par\smallskip
$\bullet$
The case when $G$ is an arbitrary tori over a regular local ring is done by
J.-L. Colliot-Th\'{e}l\`{e}ne and J.-J. Sansuc in
\cite{C-T-S}.
\par\smallskip
$\bullet$ For simple group schemes of {\bf
classical} series this result follows from more general results
established by the first author, A.~Suslin, M.~Ojanguren and
K.~Zainoulline \cite{PS}, \cite{OP}, \cite{Z},
\cite{OPZ},
\cite{Pa}.
In fact, unlike our Theorem \ref{MainThm}, {\bf no isotropy
hypotheses} was imposed there.
However our result {\bf is new, say for simple adjoint groups of exceptional type}
(and for many others).
\par\smallskip
$\bullet$
The case of {\bf 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{PaPS}.
\par\smallskip
$\bullet$ There exists a folklore result, concerning type
$G_2$. It gives affirmative answer in this case, also independent
of isotropy hypotheses, see %% the very end of
the paper by V.~Chernousov and the first author
\cite{ChP}.
\par\smallskip
\par\smallskip
$\bullet$ The case when 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
perfect, in \cite{R1};
O.~Gabber announced a proof for a general ground field $k$.
%when $k$ is infinite, and in \cite{Ga} in
%general.


%\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 of $\mathcal O$.
Let $G$ be a reductive $\mathcal O$-group scheme satisfying the condition ({\bf I}) above.
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.

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.
\end{thm}

To state our purity Theorems recall certain notions.
Let  ${\cal F}$ be  a covariant functor from the category of
commutative $R$-algebras to the category
of abelian groups. For any domain $R$ consider  the sequence
$${\cal F}(R)\to {\cal F}(K)\to \bigoplus_{\mathfrak p}\mathcal F(K)/{\cal F}(R_{\mathfrak p})$$
where $p$ runs over
the height $1$ primes of $R$ and
$K$ is the field of fractions of $R$.
We say that
{\it $\mathcal F$ satisfies purity for $R$} if this sequence --- which is
clearly a complex --- is exact. The purity for $R$ is equivalent to the following
property
$$
\bigcap_{\textrm{ht} \mathfrak p=1} Im[{\cal F}(R_{\mathfrak p}) \to {\cal F}(K)]=
Im[{\cal F}(R) \to {\cal F}(K)].
$$


%Let $k$ be a characteristic zero field and $\bar k$ be its algebraic closure.
%Let $A$ be a commutative ring.
%Recall that a reductive $A$-group scheme $G$
%is an $A$-smooth affine $A$-group scheme
%such that for each ring homomorphism
%$A \to \Omega$ with an algebraically closed field $\Omega$ the
%$\Omega$-group scheme $G_{\Omega}$ is a reductive linear algebraic $\Omega$-group.
%In particular, a a reductive $A$-group scheme $G$ is always $A$-smooth.
%The main purity result of this paper is the following purity theorem




\begin{thm}[A] %%%%% why Theorem (A) instead %%%%%%%%%%%%%of Theorem A as Cartan did?

\label{TheoremA}
\label{PurityGeometric}
Let $\mathcal O$ be a semi-local ring of finitely many points
on a $k$-smooth scheme $X$ with an infinite field $k$.
Let
$$\mu: G\to C$$
be a smooth $\mathcal O$-morphism of reductive algebraic
$\mathcal O$-group schemes, with a torus $C$.
The functor
$$\mathcal F: S\mapsto C(S)/\mu(G(S))$$
defined on the category of $\mathcal O$-algebras
satisfies purity for $\mathcal O$.
\end{thm}

\begin{thm}[B]
\label{PurityGeneral}
Let $R$ be a regular local
ring containing an infinite perfect field $k$.
Let
$$\mu: G\to C$$
be a smooth $R$-group scheme morphism to a tori $C$.
The functor
$$\mathcal F: S\mapsto C(S)/\mu(G(S))$$
defined on the category of $R$-algebras
satisfies purity for $R$.

If $K$ is the field of fractions of $R$, then
this statement can be restated in an explicit way
as follows: given an element $c \in C(K)$
suppose that for each height $1$ prime ideal
$\mathfrak p $ in $ A $ there exist
$a_{\mathfrak p } \in C(R_{\mathfrak p })$,
$g_{\mathfrak p } \in G(K)$
with
$a=a_{\mathfrak p } \cdot \mu(g_{\mathfrak p }) \in C(K)$.
Then there exist
$g_{\mathfrak m } \in G(K)$, $a_{\mathfrak m } \in C(R)$
such that
$$
a=a_{\mathfrak m } \cdot \mu(g_{\mathfrak m }) \in C(K).
$$
\end{thm}
After the pioneering articles
\cite{C-T/P/S}
and
\cite{R}
on purity theorems for algebraic groups,
various versions of purity theorems were proved in
\cite{C-TO},
\cite{PS},
\cite{Z},
\cite{Pa}.
The most general result in the so called constant case was given in
\cite[Exm.3.3]{Z}. This result follows now from our Theorem (A) by
 taking $G$ to be a $k$-rational reductive group, $C=\mathbb G_{m,k}$
and
$\mu: G \to \mathbb G_{m,k}$ a dominant $k$-group morphism. The papers
\cite{PS},
\cite{Z},
\cite{Pa}
contain results for the nonconstant case. However they only consider specific examples
of algebraic scheme morphisms $\mu: G \to C$.

It seems plausible to expect purity theorem in the following context.
Let $R$ be a regular local ring. Let
$\mu: G \to T$ be a smooth of reductive $R$-group schemes
with an $R$-torus $T$.
Let  ${\cal F}$ be the covariant functor from the category of
commutative rings to the category of abelian groups
given by
$S \mapsto T(S)/\mu(G(S))$. Then ${\cal F}$ should satisfies purity for $R$.


Another functor for which one should expect purity is the following one.
Let $R$ be a regular local ring, $G$ and $G'$ reductive $R$-group schemes,
$\pi: G \to G'$  a smooth central $R$-group scheme morphism.
Assume $Z=ker(\pi)$ is  finite
\'etale of multiplicative type over $R$.
The boundary operator
$\delta_{\pi,R}: G'(R) \to H^1_{\text{\'et}}(R,Z)$
makes sense and
 is a group homomorphism
\cite[Ch.II, \S 5.6, Cor.2]{Se}.
For an $R$-algebra $S$
set
\begin{equation}
\label{Functor2}
{\cal F}(S)= \textrm{H}^1_{\text{\'et}}(S,Z)/ \textrm{Im}(\delta_{\pi,S}).
\end{equation}
It seems plausible that the functor ${\cal F}$
satisfies purity for $R$.
{\bf Theorem}
\ref{PurityForSubgroup}
{{\bf states that this is the case if $R$ contains an infinite perfect field} $k$.
%of characteristic zero and $G$, $G'$ and $\pi$
%are  already defined  over $k$.




Note that
{\bf we use transfers} for the functor
$R \mapsto C(R)$, but {\bf we do not use at all the norm principle}
for the homomorphism $\mu: G \to C$.



The preprint is organized as follows. In Section
\ref{SectNorms}
we construct
norm maps following a method from
\cite[Sect.6]{SV}.
In Section
\ref{SectElemNisnevichSquare}
we prove the main geometric Lemma
\ref{Lemma2}.
%and
%\ref{Lemma1}.
In Section
\ref{SectUnramifiedElements}
we discuss
unramified elements.
A key point here is Lemma
\ref{KeyUnramifiedness}.
In Section
\ref{SectSpecializationMaps}
we discuss specialization maps.
A key point here is Corollary
\ref{TwoSpecializations}.
In Section
\ref{ElementaryFibrations}
certain Artin's results are generalized.
In Section
\ref{NiceTriples}
a convenient technical tool is introduced.
In Section
\ref{SectProofOfTheoremA}
Theorem (A)
is proved.
In Section
\ref{SectProofofTheoremB}
Theorem (B) is proved.
In Section
\ref{SectionOneMorePurityTheorem}
we consider the functor
(\ref{Functor2})
and prove a purity Theorem
\ref{PurityForSubgroup}
for that functor.
In Section
\ref{SectionGr_SerreConj}
Theorems
\ref{MainThm}
and
\ref{MainThmGeometric}
are proved.
Finally in Section
\ref{ExamplesOfPurityResults}
we collect several examples {\bf  illustrating our Purity Theorems}.

\subparagraph{Acknowledgments}
The author thanks very much M.Ojanguren for useful discussions concerning purity results
obtained in the present article. The author thanks very much N.Vavilov and A.Stavrova.
Long discussions with them over our joint work
\cite{PSV}
resulted unexpectedly to me with the present preprint.
%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.





\section{Norms}
\label{SectNorms}
Let $k\subset K\subset L$ be field extensions and assume that $L$
is finite separable over $K$. Let $K^{sep}$ be a separable closure
of $K$ and
$$\sigma_i:K\to K^{sep},\enspace 1\leq i\leq n$$
the different embeddings of $K$ into $L$. As in \S1, let $C$ be a
commutative algebraic group scheme defined over $k$. We can define
a norm map
$${\mathcal N}_{L/K}:C(L)\to C(K)$$
by
$${\mathcal N}_{L/K}(\alpha)=\prod_i C(\sigma_i)(\alpha) \in C(K^{sep})^{{\mathcal
G}(K)} =C(K)\;.
$$
Following Suslin and Voevodsky
\cite[Sect.6]{SV} we
generalize this construction to finite flat ring extensions.
\smallskip
Let $p:X\to Y$ be a finite flat morphism of affine schemes.
Suppose that its rank is constant, equal to $d$. Denote by
$S^d(X/Y)$ the $d$-th symmetric power of $X$ over $Y$.



\begin{lem} There is a canonical section
$$\mathcal N_{X/Y}: Y\to S^d(X/Y)$$
which satisfies the following three properties:
\begin{itemize}
\item
[(1)] Base change: for any map $f:Y'\to Y$ of affine schemes, putting
$X'=X\times_Y Y'$ we have a commutative diagram
$$\xymatrix{Y'\ar[rr]^{\mathcal N_{X'/Y'}}\ar[d]_{f}&&S^d(X'/Y')\ar[d]^{S^d(Id_X\times
f)}\\
             Y\ar[rr]^{\mathcal N_{X/Y}}&&S^d(X/Y)}$$
\item[(2)]
Additivity: If $f_1: X_1\to Y$ and $f_2:X_2\to Y$ are finite
flat morphisms of degree $d_1$ and $d_2$ respectively, then,
putting $X=X_1\coprod X_2$, $f=f_1\coprod f_2$ and $d=d_1+d_2$, we
have a commutative diagram
$$\xymatrix{S^{d_1}(X_1/Y)\times S^{d_2}(X_2/Y)\ar[rr]^(0.63)\sigma&&S^d(X/Y)\\
&&\\
&Y\ar[uul]^{\mathcal N_{X_1/Y}\times \mathcal N_{X_2/Y}}\ar[uur]_{\mathcal N_{X/Y}}& }$$
where $\sigma$ is the canonical imbedding.\\
\item[(3)]
Normalization: If $X=Y$ and $p$ is the identity, then
$\mathcal N_{X/Y}$ is the identity.
\end{itemize}
\end{lem}



\begin{proof} We construct a map $\mathcal N_{X/Y}$ and check that it has the
desired properties. Let $B=k[X]$ and $A=k[Y]$, so that $B$ is a
locally free $A$-module of finite rank $d$. Let $B^{\otimes
d}=B\otimes_A B\otimes_A\cdots \otimes_A B$ be the $d$-fold tensor product
of $B$ over $A$. The permutation group ${\frak S}_d$ acts on
$B^{\otimes d}$ by permuting the factors.  Let $S^d(B)\subseteq
B^{\otimes d}$ be the $A$-algebra of all the ${\frak
S}_d$-invariant elements of $B^{\otimes d}$. We consider
$B^{\otimes d}$ as an $S^d(B)$-module through the inclusion
$S^d(B)\subseteq B^{\otimes d}$ of $A$-algebras. Let $I$ be the
kernel of the canonical  homomorphism $B^{\otimes d}\to
\bigwedge^d(B  )$ mapping $b_1\otimes\cdots\otimes b_d$ to
$b_1\wedge\cdots\wedge b_d$. It is well-known (and easily checked
locally on $A$) that $I$ is generated by all  the elements $x\in
B^{\otimes d}$ such that $\tau(x)=x$ for some transposition
$\tau$. If $s$ is in $S^d(B)$, then $\tau(sx)=\tau(s)\tau(x)=sx$,
hence $sx$ is in $S^d(B)$ too. In other words, $I$ is an
$S^d(B)$-submodule of $B^{\otimes d}$. The induced $S^d(B)$-module
structure on $\bigwedge^d(B)$ defines an $A$-algebra homomorphism
$$\varphi:S^d(B)\to End_A(\bigwedge^d(B))\;.$$
Since $B$ is locally free of rank $d$ over $A$, $\bigwedge^d(B)$
is an invertible $A$-module and we can canonically identify
$End_A(\bigwedge^d(B)$ with $A$. Thus we have a map

$$\varphi:S^d(B)\to A$$
and we define
$$Tr_{X/Y}: Y\to S^d(X)$$
as the morphism of $Y$-schemes induced by $\varphi$. The
verification of properties (1), (2) and (3) is straightforward.
\end{proof}

\smallskip
Let now $C$ be a commutative $k$-group scheme  and $f:X\to C$ any
morphism. We define the norm $N_{X/Y}(f)$ of $f$ as the composite map
\begin{equation}
\label{NormMap}
Y \xra{Tr_{X/Y}}
S^d(X)\xra{S^d(f)} S^d(C)\xra{\times} C
\end{equation}
Here we write $"\times"$ for the group law on $C$.
The norm maps $N_{X/Y}$ satisfy the following conditions
\begin{itemize}
\item[(1)]
Base change: for any map $f:Y'\to Y$ of affine schemes, putting
$X'=X\times_Y Y'$ we have a commutative diagram
$$
\begin{CD}
C(X)                  @>{(id \times f)^{*}}>>            C(X^{\prime})      \\
@V{N_{X/Y}}VV @VV{N_{X^{\prime}/Y^{\prime}}}V    \\
C(Y)                  @>{f^{*}}>>            C(Y^{\prime})
\end{CD}
$$
\item[(2)]
multiplicativity: if $X=X_1 \amalg X_2$ then the diagram commutes
$$
\begin{CD}
C(X)                  @>{(id \times f)^{*}}>>            C(X_1) \times C(X_2)      \\
@V{N_{X/Y}}VV                              @VV{N_{X_1/Y}N_{X_2/Y}}V    \\
C(Y)                  @>{id}>>            C(Y)
\end{CD}
$$
\item[(3)]
normalization: if $X=Y$ and the map $X \to Y$ is the identity then $N_{X/Y}=id_{C(X)}$.
\end{itemize}

%\subsection{Geometric lemmas}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Geometric lemmas}%An elementary Nisnevich square}
\label{SectElemNisnevichSquare}
The aim of this Section is to prove an analog of three Lemmas from
\cite{Pa2}
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 equi-dimensional $k$-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 $S$
(it might be that $\mathfrak m_0=\mathfrak m_i$ for an index $i$).
Then there exists a non-zero divisor
$\bar s \in S$
such that $S$ is finite over $k[\bar s]$ and
\begin{itemize}
\item[(1)]
the ideals
$\mathfrak n_i:= \mathfrak m_i \cap k[\bar s]$ ($i=1,2,\dots, n$)
are all different from each other and different  from
$\mathfrak n_0:= \mathfrak m_0 \cap k[\bar s]$
provided that
$\mathfrak m_0$
is different  from
$\mathfrak m_i$'s;
\item[(2)]
the extension $S/k[\bar s]$ is \'etale at each
$\mathfrak m_i$'s and at $\mathfrak m_0$;
\item[(3)]
$k[\bar s]/\mathfrak n_i = S/\mathfrak m_i$ for each
$i=1,2,\dots, n$;
\item[(4)]
$\mathfrak n_0= \bar s k[\bar s]$.
\end{itemize}
\end{lem}

\begin{proof}
For each index $i$ let $x_i$ be the point on $\Spec(S)$ corresponding to the ideal
$\mathfrak m_i$.
Consider a closed imbedding
$\Spec(S) \hra \Aff^n_k$
and find a generic $k$-defined linear projection
$p: \Aff^n_k \to \Aff^1_k$ to
such that
for each index $i$ one has
\begin{itemize}
\item[(1)]
for each $i\geq 0, j \geq 0$ one has $p(x_i) \neq p(x_j)$ provided that $x_i \neq x_j$;
\item[(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[(3)]
the separable degree extension $k(x_i) / k(p(x_i))$ is one.
\end{itemize}
With this in hand one gets for each the equality $k(p(x_i)) = k(x_i)$.
In fact the extension $k(x_i) / k(p(x_i))$ is separable by the item (2).
By the item (3) we conclude that $k(p(x_i)) = k(x_i)$.
The 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 different of
$\mathfrak m_0, \mathfrak m_1, \dots, \mathfrak m_n$.
Let $N(f)=N_{S/k[s]}(f)$ be the norm of $f$. Then for an element
$\bar s \in S$ satisfying $(1)$ to $(4)$ of Lemma
\ref{Lemma3}
one has
\begin{itemize}
\item[(a)]
$N(f)=fg$ for an element $g \in S$;
\item[(b)]
$fS + gS= S$;
\item[(c)]
the map $k[\bar s]/(N(f)) \to S/(f)$ is an isomorphism.
\end{itemize}
\end{lem}

\begin{proof}
It is straightforward.

\end{proof}


\begin{lem}
\label{Lemma2}
Let $R$ be a semi-local essentially smooth $k$-algebra with maximal ideals
$\mathfrak p_i$'s, where $i$ runs from $1$ to $r$.
Let $R[t] \subset A$
be an $R[t]$-algebra which is smooth
as an $R$-algebra and is finite over the
polynomial algebra $R[t]$. Assume that for each index $i$ the
$R/\mathfrak p_i$-algebra
$A_i= A/\mathfrak p_i A$
is equi-dimensional 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 $u \in A$ satisfying the following conditions
\begin{itemize}
\item[(1)]
$A$ is finite projective as an $R[u]$-module;
\item[(2)]
$A/uA=A/I \times A/J$ for certain ideal $J$;
\item[(3)]
$J + fA = A$;
\item[(4)]
$(u-1)A + fA = A$;
\item[(5)]
if $N(f)=N_{A/R[u]}(f)$, then $N(f)=fg \in A$ for certain $g \in A$ \\
%(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[(6)]
$fA + gA = A$;
\item[(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 $R[t]$ we conclude that
it is a finite projective $R[t]$-module by a theorem of Grothendieck
\cite[Cor.17.18]{E}.
Since $A$ is finite over $R[t]$ and $A/fA$ is finite over $R$
we conclude that $R[t]/(N(f))$ is finite over $R$, hence
$R/(tN(f))$ is finite over $R$ too. So setting $v=tN(f)$
we get an integral extension $A$ over $R[v]$ and one has equalities
$$v=tN_{A/R[t]}(f)=(ft)g=tfg.$$

%Now replacing
%$t$ by $u$ we may assume
We claim that $A/R[v]$ is integral,
$\epsilon (v)=0$ and $u \in fA$.
In fact,
$u=tN_{A/R[t]}(f)=t(fg)$
and whence
$\epsilon (v)=\epsilon (t) \epsilon (fg) =0$. Finally,
and
$v \in fA$.

We use below ``bar" and lower script $i$
to denote the reduction modulo the ideal
$\mathfrak p_iA$. Let
$l_i=\bar R_i=R/\mathfrak p_i$.
By the  assumption of the lemma the $l_i$-algebra $\bar A_i$
is an $l_i$-smooth
equi-dimensional of dimension $1$.
Let
${\mathfrak m}^{(i)}_1, {\mathfrak m}^{(i)}_2, \dots, {\mathfrak m}^{(i)}_n$
be different 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 the conditions $(1)$ to $(4)$ of Lemma
\ref{Lemma3}.

Let $s \in A$ be a common lift of $\bar s_i$'s , that is for each index $i$ one has
$\overline s_i =\bar s_i$ in $\bar A_i$. Replacing $s$ by
$s - \epsilon (s)$
we may assume that
$\epsilon (s) = 0$
and still for each index $i$ one has
$\overline s_i = \bar s_i$.

Let
$s^n + p_1(v)s^{n-1} + \dots + p_n(v)=0$
be an integral dependence equation for $s$.
Let $N$ be an integer large than the
$max\{2, deg (p_i(v))\}$, where
$i=1, 2, \dots, n$. Then for any
$r \in k^{\times}$
the element
$u= s - rv^N$
is such that $v$ is integral over $R[u]$.
Thus for any
$r \in k^{\times}$
the ring $R$ is integral over $A[u]$.


On the other hand for each index $i$ the element
$\bar u_i = \bar s_i -r\bar {v_i}^N$
still satisfies the conditions
$(1)$ to $(4)$ of Lemma
\ref{Lemma3}
because for each index
$j=0, 1, 2, \dots, n$
one has
$\bar v_i \in \mathfrak m^{(i)}_j$.

{\bf We claim that the element $u \in R$
is the required element
(for almost all} $r \in k^{\times}$).

In fact, for almost all
$r \in k^{\times}$
the element $u$ satisfies the conditions
$(1)$ to $(4)$ of Lemma
\cite[Lemma 5.2]{OP}.
It remains to show that the conditions
$(5)$ to $(7)$ are satisfied for all
$r \in k^{\times}$.

Since $A$ is finite over $R[u]$ we conclude that
it is a finite projective $R[u]$-module by a theorem of Grothendieck
\cite[Cor.18.17]{E}
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 coefficient is
$\pm N(f)$ (the norm of $f$). Thus
$f^n - a_1f^{n-1} + \dots \pm N(f)=0$
and $N(f)=fg$ for an element $g \in R$.

To prove $(6)$ one has to check that this $g$ 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 $\bar f_i \bar A_i$. 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 $\bar g_i$ is a unit modulo the ideal $\bar f_i \bar R_j$
by the item $(b)$ of Lemma
\ref{Lemma4}.

To prove $(7)$ note that $R[u]/(N_{A/k[X]}(f))$ are $A/fA$
finite $A$-modules. So 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
$\mathfrak m^{(j)}_i$. For that 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 $\bar \varphi_i$ is an isomorphism by the item $(c)$ of Lemma
\ref{Lemma4}.
The lemma follows.

\end{proof}

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

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

\end{proof}

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



\section{Unramified elements}
\label{SectUnramifiedElements}
%Let $k$ be the characteristic zero field.
We work in this section with {\it the category of commutative $R$-algebras}.
Let  $\mathcal F$ be a covariant functor from the category of commutative
$R$-algebras to the category of abelian groups. Let $K$ be a field containing $k$ and
$S\subset K$ be a $k$-subalgebra whose field of fractions is $K$. We
define the {\it subgroup of $S$-unramified elements of $K$} as
\begin{equation}
\label{DefnUnramified}
\mathcal F_{nr,S}(K)=
\bigcap_{\mathfrak p \in Spec(S)^{(1)}} Im [ \mathcal F(S_{\mathfrak p})\to\mathcal F(K) ],
\end{equation}
where $Spec(S)^{(1)}$ is the set of hight $1$ prime ideals in $S$.
Obviously the image of $\mathcal F(R)$ in $\mathcal F(K)$ is contained in
$\mathcal F_{nr,S}(K)$. In most cases
$\mathcal F(S_{\mathfrak p})$
injects into
$\mathcal F(K)$
and
$\mathcal F_{nr,S}(K)$
is simply the intersection of all
$\mathcal F(S_p)$.

A basic functor we are interested in is the following one.
Let
$\mu: G\to C$
be the morphism of reductive $R$-group schemes from Theorem
\ref{TheoremA}. For a commutative $R$-algebra $S$ set
\begin{equation}
\label{MainFunctor}
{\cal F}(S)=C(S)/\mu(G(S)).
\end{equation}
For an element $\alpha \in C(S)$ we will write $\bar \alpha$ for its
image in ${\cal F}(S)$. In this section we will write  ${\cal F}$
for the functor
(\ref{MainFunctor}),
the only exception being Lemma
\ref{UnramifiednessLemma}.
We will repeatedly use  the following result
%\begin{thm}[\cite{C-TO}]
%\label{NisnevichThm}
%Let $H$ be a linear algebraic group over the field $k$. Let
%$R$ be a discrete valuation ring with a fraction field $K$.
%Then the map of pointed sets
%$$
%H^1_{\text{\'et}}(R,H) \to H^1_{\text{\'et}}(K,H)
%$$
%induced by the inclusion $R \subset K$ has the trivial kernel.
%\end{thm}

\begin{thm}[\cite{Ni}] %%%%%% This is due to Nisnevich, the label should be
%%%%%%%%%%%% put as a reference, not as an attribution
\label{NisnevichCor}
Let $R$ be a discrete valuation ring with  fraction field $K$.
The map
${\cal F}(R) \to {\cal F}(K)$
is injective.
\end{thm}

\begin{proof}
Let $H$ be the kernel of $\mu$. Then the boundary map
$\partial: C(R) \to \textrm{H}^1_{\text{\'et}}(R,H)$
fits in a commutative diagram
$$
\CD
C(R)/\mu (G(R)) @>>> C(K)/\mu (G(K)) \\
@VVV @VVV \\
\textrm{H}^1_{\text{\'et}}(R,H) @>>> \textrm{H}^1_{\text{\'et}}(K,H). \\
\endCD
$$
in which the vertical arrows have  trivial kernels.
The bottom arrow has  trivial kernel by a Theorem from
\cite{Ni}.
Thus the top arrow has  trivial kernel too.

\end{proof}

\begin{lem}
\label{BoundaryInj}
Let $\mu: G \to C$ be the morphism of our reductive group schemes.
Let $H= \ker (\mu)$.
Then for an $R$-algebra $K$, where $K$ is a field, the boundary map
$\partial: C(K)/{\mu (G(K))} \to \textrm{H}^1_{\text{\'et}}(K,H)$
is injective.
\end{lem}

\begin{proof}
For a $K$-rational point $t \in C$ set
$H_t = \mu^{-1}(t)$. The action by  left multiplication of $H$ on $H_t$
makes $H_t$ into a left principal homogeneous
$H$-space and moreover
$\partial (t) \in \textrm{H}^1_{\text{\'et}}(K,H)$
coincides with the isomorphism class of
$H_t$. Now suppose that $s,t \in C(K)$ are such that
$\partial (s)=\partial (t)$. This means that
$H_t$ and $H_s$ are
isomorphic as  principal homogeneous $H$-spaces.
We must check that for certain $g \in G(K)$ one has
$t=sg$.

Let $K^{sep}$ be a separable closure of $K$. Let
$\psi: H_s \to H_t$
be an isomorphism of  left $H$-spaces. For any
$r \in H_s(K^{sep})$
and
$h \in H_s(K^{sep})$
one has
$$
(hr)^{-1}\psi (hr)= r^{-1}h^{-1}h \psi (r)= r^{-1} \psi (r).
$$
Thus for any
$\sigma \in Gal(K^{sep}/K)$
and any
$r \in H_s(K^{sep})$
one has
$$
r^{-1}\psi(r)=(r^{\sigma})^{-1}\psi (r^{\sigma})=(r^{-1}\psi(r))^{\sigma}
$$
which means that the point
$u=r^{-1}\psi(r)$ is a $Gal(K^{sep}/K)$-invariant point of
$G(K^{sep})$. So $u \in G(K)$.
The following relation shows that the $\psi$ coincides with the right multiplication
by $u$. In fact, for any $r \in H_s(K^{sep})$ one has
$\psi (r)= rr^{-1}\psi (r)= ru$. Since $\psi$ is the right multiplication
by $u$ one has $t=s\mu(u)$, which proves the lemma.

\end{proof}


Let $K$ be the field of fractions of $R$ and $\mathcal K$ be a field containing $K$ and
$x: \mathcal K^* \to \mathbb Z$
be a discrete valuation vanishing on $K$.
Let $A_x$ be the valuation ring of $x$.
Let
$\hat A_x$ and $\hat {\mathcal K}_x$
be the completions of $A_x$ and $\mathcal K$ with respect to $x$.
Let
$i:K \hookrightarrow \hat K_x$
be the inclusion. By Lemma
\ref{NisnevichCor}
the map
${\cal F}(\hat A_x)\to {\cal F}(\hat{K}_x)$
is injective. We will identify
${\cal F}(\hat A_x)$
with its image under this map. Set
$$
{\cal F}_x(\mathcal K)=i_*^{-1}({\cal F}(\hat A_x)).
$$


The inclusion
$A_x\hookrightarrow \mathcal K$
induces a map
$
{\cal F}(A_x) \to {\cal F}(\mathcal K)
$
which is injective by Lemma
\ref{NisnevichCor}.
So both groups
${\cal F}(A)$ and ${\cal F}_x(\mathcal K)$
are subgroups of
${\cal F}(\mathcal K)$.
The following lemma shows that
${\cal F}_x(\mathcal K)$
coincides with the subgroup of $
{\cal F}(\mathcal K)$
consisting of all elements {\it unramified} at $x$.

\begin{lem}
\label{TwoUnramified}
${\cal F}(A_x)={\cal F}_x(\mathcal K)$.
\end{lem}

\begin{proof}
We only have to check the inclusion
$
{\cal F}_x(\mathcal K) \subseteq {\cal F}(A_x)
$.
Let
$
a_x \in {\cal F}_x(\mathcal K)
$
be an element. It determines the elements
$
a \in {\cal F}(\mathcal K)
$
and
$
\hat a \in {\cal F}(\hat A_x)
$
which coincide when regarded as elements of
$
{\cal F}(\hat{\mathcal K}_x)
$.
We denote this common element in
$
{\cal F}(\hat{\mathcal K}_x)
$
 by
$
\hat a_x
$.
Let $H=\ker (\mu)$ and let
$\partial: C(-) \to \textrm{H}^1_{\text{\'et}}(-,H)$
be the boundary map.

Let
$
\xi=\partial(a)\in \textrm{H}^1_{\text{\'et}}(\mathcal K,H)
$,
$
\hat \xi=\partial(\hat a)\in \textrm{H}^1_{\text{\'et}}(\hat A_x,H)
$
and
$\hat \xi_x=\partial(\hat a_x)\in \textrm{H}^1_{\text{\'et}}(\hat {\mathcal K_x},H)$
Clearly,
$
\hat \xi
$
and
$
\xi
$
both coincide with
$
\hat \xi_x
$
when regarded as elements of
$
\textrm{H}^1_{\text{\'et}}(\hat {\mathcal K_x},H)
$.
Thus one can glue
$
\xi
$
and
$
\hat \xi
$
to get a
$
\xi_x \in \textrm{H}^1_{\text{\'et}}(A_x,H)
$
which maps to
$
\xi
$
under the map induced by the inclusion
$
A_x \hookrightarrow \mathcal K
$
and maps to
$
\hat \xi
$
under the map induced by the inclusion
$
A_x \hookrightarrow \hat A_x
$.

We now show that
$
\xi _x
$
has the form
$
\partial (a_x^\prime)
$
for a certain
$
a_x^\prime \in {\cal F}(A_x)
$.
In fact, observe that the image
$
\zeta
$
of
$
\xi
$
in
$
\textrm{H}^1_{\text{\'et}}(\mathcal K, G)
$
is  trivial.
By Theorem
\cite{Ni}
the map
$$
\textrm{H}^1_{\text{\'et}}(A_x, G) \to \textrm{H}^1_{\text{\'et}}(\mathcal K,G)
$$
has  trivial kernel. Therefore the image
$
\zeta_x
$
of
$
\xi_x
$
in
$
\textrm{H}^1_{\text{\'et}}(A_x, G)
$
is trivial. Thus there exists an element
$
a_x^\prime \in {\cal F}(A_x)
$
with
$
\partial(a_x^\prime)=\xi_x \in \textrm{H}^1_{\text{\'et}}(A_x, H).
$

We now prove that
$
a_x^\prime
$
coincides with
$
a_x
$
in
$
{\cal F}_x(\mathcal K)
$.
Since
$
{\cal F}(A_x)
$
and
$
{\cal F}_x(\mathcal K)
$
are both subgroups of
$
{\cal F}(\mathcal K)
$,
it suffices to show that
$
a_x^\prime
$
coincides with the element
$
a
$
in
$
{\cal F}( \mathcal K)
$.
By Lemma
\ref{BoundaryInj}
the map
\begin{equation}
\label{BoundaryInjFormulae}
{\cal F}(\mathcal K)\xrightarrow{\partial} \textrm{H}^1_{\text{\'et}}(\mathcal K, H)
\end{equation}
is \underbar {injective}. Thus it suffices to check that
$
\partial(a_x^\prime)=\partial(a)
$
in
$
\textrm{H}^1_{\text{\'et}}(\mathcal K, H)
$.
This is indeed the case because
$
\partial(a_x^\prime)=\xi_x
$
and
$
\partial(a)=\xi
$,
and
$
\xi_x
$
coincides with
$
\xi
$
when regarded over
$
\mathcal K
$.
We have proved that
$
a_x^\prime \in {\cal F}(A_x)
$
coincides with
$
a_x
$
in
$
{\cal F}_x(\mathcal K)
$.
Thus the inclusion
$
{\cal F}_x(\mathcal K) \subseteq {\cal F}(A_x)
$
is proved, whence the lemma.

\end{proof}



\begin{lem}
\label{KeyUnramifiedness}
Let $A \subset B$ be a finite extension of Dedekind $K$-algebras.
Let $0 \neq f \in B$ be such that $B/fB$ is finite over $K$ and reduced.

Suppose $N_{B/A}(f)=fg \in B$ for a certain $g \in B$ coprime
with $f$. Suppose the composite map
$A/N(f)A \to B/N(f)B \to B/fB$
is an isomorphism.
Let $F$ and $E$ be the field of fractions of $A$ and $B$ respectively.
Let $\beta \in C(B_f)$ be such that
$\bar \beta \in {\cal F}(E)$
is $B$-unramified. Then, for
$\alpha= N_{E/F}(\beta)$,
the class
$\bar \alpha \in {\cal F}(F)$
is $A$-unramified.
\end{lem}

\begin{proof}
The only primes at which $\bar \alpha$ could be ramified are those which divide
$N(f)$. Let $\mathfrak p$ be one of them. Check that $\bar \alpha$ is
unramified at $\mathfrak p$.

To do this we consider all primes
$\mathfrak q_1, \mathfrak q_2, \dots, \mathfrak q_n$
in $B$ lying over
$\mathfrak p$.
Let
$\mathfrak q_1$
be the unique prime dividing $f$ and lying over
$\mathfrak p$.
Then
$$
\hat B_{\mathfrak p} = \hat B_{\mathfrak q_1} \times \prod_{i \neq 1} \hat B_{\mathfrak q_i}
$$
with
$\hat B_{\mathfrak q_1}=\hat A_{\mathfrak p}$.
If $F$, $E$ are the fields of fractions of $A$ and $B$
then
$$
E \otimes \hat F_{\mathfrak p}=\hat E_{\mathfrak q_1} \times \dots \times \hat E_{\mathfrak q_n}
$$
and
$\hat E_{\mathfrak q_1}=\hat F_{\mathfrak p}$.
We will write
$\hat E_i$ for $\hat E_{\mathfrak q_i}$
and
$\hat B_i$ for $\hat B_{\mathfrak q_i}$.
Let
$\beta \otimes 1= (\beta_1,\dots, \beta_n)
\in C(\hat E_1) \times \dots \times  C(\hat E_n)$.
Clearly for $i \geq 2$
$\beta_i \in C(\hat B_i)$
and
$\beta_1=\mu(\gamma_1)\beta^{\prime}_1$
with
$\beta^{\prime}_1 \in C(\hat B_1)=C(\hat A_{\mathfrak p})$
and
$\gamma_1 \in G(\hat E_1)=G(\hat F_{\mathfrak p})$.
Now
$\alpha \otimes 1 \in C(\hat F_{\mathfrak p})$
coincides with the product
$$
\beta_1N_{\hat L_2/\hat K_{\mathfrak p}}(\beta_2)\cdots N_{\hat L_n/\hat K_{\mathfrak p}}(\beta_n)=
\mu(\gamma_1)
[\beta^{\prime}_1N_{\hat L_2/\hat K_{\mathfrak p}}(\beta_2)\cdots N_{\hat L_n/\hat K_{\mathfrak p}}(\beta_n)].
$$
Thus
$\overline {\alpha \otimes 1}=\bar \beta^{\prime}_1\overline {N_{\hat L_2/\hat K_{\mathfrak p}}(\beta_2)}
\cdots \overline {N_{\hat L_n/\hat K_{\mathfrak p}}(\beta_n)} \in {\cal F}(\hat A_{\mathfrak p})$.
Let
$i: F \hra \hat F_{\mathfrak p}$
be the inclusion and
$i_*:{\cal F}(F) \to {\cal F}(\hat F_{\mathfrak p})$
be the induced map.
Clearly
$i_*(\bar \alpha)=\overline {\alpha \otimes 1}$
in
${\cal F}(\hat F_{\mathfrak p})$.
Now
Lemma
\ref{TwoUnramified}
shows that
the element
$\bar \alpha \in {\cal F}(F)$
belongs to
${\cal F}(A_{\mathfrak p})$.
Hence $\bar \alpha$ is $A$-unramified.

\end{proof}

\begin{lem}[Unramifiedness Lemma]
\label{UnramifiednessLemma}
Let  $\mathcal F$ be a covariant functor from the category of commutative
$R$-algebras to the category of abelian groups.
Let $S^{\prime}$ and $R^{\prime}$ be two $R$-algebras which are noetherian domains
with  fields of fractions
$K^{\prime}$ and $L^{\prime}$
respectively.
Let
$
S^{\prime} \xrightarrow{i} R^{\prime}
$
be an injective flat $R$-algebra
homo\-morph\-ism of finite type and
let
$
j :K^{\prime} \to L^{\prime}
$
be the induced inclusion of the field of fractions.
Then for each localization
$R^{\prime\prime} \supset R^{\prime}$ of $R^{\prime}$
the map
$$
j _*:{\cal F}(K^{\prime}) \to {\cal F}(L^{\prime})
$$
takes
$S^{\prime}$-unramified elements to
$R^{\prime\prime}$-unramified elements.
\end{lem}

\begin{proof}
Let
$v \in {\cal F}(K^{\prime})$
and let
$\mathfrak r$
be  height $1$ primes of $R^{\prime\prime}$. Then
$\mathfrak q = R^{\prime} \cap \mathfrak r$
is a height $1$ prime of $R^{\prime}$. Let
$\mathfrak p = S^{\prime} \cap \mathfrak q$.
Since the $S^{\prime}$-algebra $R^{\prime}$ is flat of finite type one has
$\text{ht}(\mathfrak q) \geq \text{ht}(\mathfrak p)$. Thus
$\text{ht}(\mathfrak p)$
is $1$ or $0$. The commutative diagram
$$
\CD
{\cal F}(K^{\prime}) @>>> {\cal F}(L^{\prime}) \\
@AAA @AAA \\
{\cal F}(S^{\prime}_{\mathfrak p}) @>>> {\cal F}(R^{\prime\prime}_{\mathfrak r}) \\
\endCD
$$
shows that the class
$j_* (v)$
is in the image of
${\cal F}(R^{\prime}_{\mathfrak r})$ and
hence the class
$j_* (v) \in {\cal F}(L)$
is $R^{\prime\prime}$-unramified.

\end{proof}

\section{Specialization maps}
\label{SectSpecializationMaps}
%Let $A$ be a local ring of a
%smooth algebraic variety $X$ over an infinite field $k$.
%Let
%$K$ be the field of fractions of $A$.
Let
$K$
be an infinite field.
Let
\begin{equation}
\label{MainRedGroupMorphism}
\mu: G\to C
\end{equation}
be a smooth $K$-morphism of reductive algebraic $K$-group schemes, with a torus $C$.
We work in this section with {\it the category of commutative $K$-algebras} and with
the functor
\begin{equation}
\label{MainFunctor}
\mathcal F: S\mapsto C(S)/\mu(G(S))
\end{equation}
defined on the category of $A$-algebras. So, we assume in this Section that
each ring from this Section is equipped with a distinguished $K$-algebra structure and
each ring homomorphism from this Section respects that structures.
%We work in this section with {\it the category of commutative $A$-algebras}.
%In this Section we consider the functor $R \mapsto {\cal F}(R)$ defined by
%(\ref{MainFunctor}).

For a regular ring $R$ with the fraction field $\cal K$
and each height one prime $\mathfrak p$ in $R$
we construct {\bf specialization maps}
$s_{\mathfrak p}: {\cal F}_{nr, R}(\cal K) \to \mathcal F (k({\mathfrak p}))$,
where $\cal K$ is the field of fractions of $R$ and
$k(\mathfrak p)$
is the residue field of
$R$ at the prime $\mathfrak p$.


\begin{defn}
\label{SpecializationDef}
Let
$Ev_{\mathfrak p}: C(R_{\mathfrak p}) \to C(k(\mathfrak p))$
and
$ev_{\mathfrak p}: {\cal F}(R_{\mathfrak p}) \to {\cal F}(k(\mathfrak p))$
be the maps induced by the canonical ring homomorphism
$R_{\mathfrak p} \to k(\mathfrak p)$.
Define a homomorphism
$s_{\mathfrak p}: {\cal F}_{nr, R}(\cal K) \to {\cal F}(k(\mathfrak p))$
by
$s_{\mathfrak p}(\alpha)= ev_{\mathfrak p}(\tilde \alpha)$,
where
$\tilde \alpha$
is a lift of $\alpha$ in
${\cal F}(R_{\mathfrak p})$.
Theorem
\ref{NisnevichCor}
shows that the map $s_{\mathfrak p}$ is well-defined.
It is called the specialization map. The map $ev_{\mathfrak p}$ is called the evaluation
map at the prime $\mathfrak p$.

Obviously for
$\alpha \in C(R_\mathfrak p)$
one has
$s_{\mathfrak p}(\bar \alpha)=\overline {Ev_{\mathfrak p}(\alpha)}
\in {\cal F}(k(\mathfrak p))$.
\end{defn}

\begin{lem}[\cite{H}]
\label{Harder}
Let $H$ be a smooth linear algebraic group over the field $K$.
Let $R$ be a $K$-algebra which is a Dedekind domain with field of fractions $\cal K$.
If $\xi \in \textrm{H}^1_{\text{\'et}}({\cal K},H)$ is an $R$-unramified element for the functor
$\textrm{H}^1_{\text{\'et}}(-,H)$
(see (\ref{DefnUnramified}) for the Definition), then $\xi$ can be lifted  to an element of
$\textrm{H}^1_{\text{\'et}}(R,H)$.
\end{lem}

\begin{proof}
Patching.

\end{proof}

\begin{thm}
\label{C-TOthm}
%Let $K$ be a field of characteristic zero and let $H$ be a linear algebraic group
%over $K$. Then
The canonical map
$\textrm{H}^1_{\text{\'et}}({K},H) \to \textrm{H}^1_{\text{\'et}}({K}[t],H)$
is bijective.
\end{thm}
\begin{proof}
Let $i: \{0\} \hra \Aff^1_K$ be the origin of the affine line.
Let
$\textrm{H}^1_{\text{\'et}}({K}[t],H) \xra{i^*} \textrm{H}^1_{\text{\'et}}({K},H)$
be the pull-back map. To prove the theorem we need the following

\begin{clm}
\label{InjectivityAndKernel}
The following two statements are equivalent.
\begin{itemize}
\item[(1)]
For any linear algebraic group $H$ over $K$ the map $i^*$ is injective.
\item[(2)]
For any linear algebraic group $H$ over $K$ the map $i^*$ has the trivial kernel.
\end{itemize}
\end{clm}
Clearly the first statement implies the second one. Now suppose the second statement
holds and prove the first one. For that consider
$\xi, \xi' \in \textrm{H}^1_{\text{\'et}}(K[t],H)$
and set
$\xi_0 = i^*(\xi), \xi'_0= i^*(\xi')$.
Let $H_0$ be the inner form of the group $H$ with respect to the $H$-torsor ${\cal H}_0$.
For any $K$-scheme $S$ there is a well-known bijection
$\phi_S: \textrm{H}^1_{\text{\'et}}(S,H) \to \textrm{H}^1_{\text{\'et}}(S,H_0)$
between non-pointed sets. It takes the $H$-torsor
${\cal H}_0 \times_K S$ to the trivial $H_0$-torsor $H_0 \times_K S$,
where ${\cal H}_0$ is an $H$-torsor over $K$ representing the class $\xi_0$.
These bijections
respect $K$-scheme morphisms.
Suppose
$\xi_0=\xi'_0$, then
$$
i^*(\phi_{\Aff1}(\xi))= \phi_K (\xi_0)= * =\phi_K(\xi'_0)=i^*(\phi_{\Aff1}(\xi'))
\in \textrm{H}^1_{\text{\'et}}(K,H_0).
$$
So
$\phi_{\Aff1}(\xi)= * = \phi_{\Aff1}(\xi') \in \textrm{H}^1_{\text{\'et}}(K[t],H_0)$.
Whence
$\xi = \xi'$ and the injectivity follows.
With this Claim in hand prove the theorem as follows.

The surjectivity of the map $i^*$ is obvious.
To prove its injectivity just use the Claim and
mimic the proof of
\cite[Thm.2.1, Property P2]{C-TO}. Let
$p: \Aff1_K \to \textrm{Spec}(K)$
be the structural morphism.
Since $p \circ i =id$ and $i^*$ is bijective
$p^*$ is  bijective.

\end{proof}


We need the following theorem.
\begin{thm}[Homotopy invariance]
\label{HomInvNonram}
Let $K$ be the field and
$\mu: G \to C$ be the algebraic group morphism
(\ref{MainRedGroupMorphism}).
%from Theorem
%\ref{TheoremA}.
Let
$K(t)$ be the rational function field in one variable.
Let $R \mapsto {\cal F}(R)$ be the functor defined by the formulae
(\ref{MainFunctor}). Then one has
$$
F(K)=F_{nr,K[t]}(K(t)).
$$
\end{thm}

\begin{proof}
The injectivity is clear, since the composition
$$
{\cal F}(K) \to {\cal F}_{nr,K[t]}(K(t)) \xrightarrow{s_0} {\cal F}(K)
$$
coincides with the identity (here
$
s_0
$
is the specialization map at the point zero defined in 4.6).

It remains to check the surjectivity. Let
$a\in {\cal F}_{nr}(K(t))$
and let
$H=ker(\mu)$.
Then by Lemma
\ref{NisnevichCor}
the element
$
\partial (a)\in H_{et}^1(K(t),H)
$
is a class which for every
$
x\in \Aff^1_K
$
belongs to the image of
$
H_{et}^1({\cal O}_x,H)
$.
Thus by Lemma
\ref{Harder},
$
\partial (a)
$
can be represented by an element
$
\xi \in H_{et}^1(K[t],H)
$,
where
$
K[t]
$
is the polynomial ring.
By Theorem
\ref{C-TOthm},
the map
$$
H_{et}^1(K,H) \to H_{et}^1(K[t],H)
$$
is an isomorphism. Then
$
\xi = \rho (\xi _0)
$
for an element
$
\xi _0\in H_{et}^1(K,H)
$.
Consider the diagram
$$
\SelectTips{cm}{}
\xymatrix @C=2pc @R=10pt {
{} & a \ar@{|->}[r] & {\xi} \ar@{|->}[r] & {\ast} & {} \\
1 \ar[r] & {{\cal F}(K(t))} \ar[r]^{\partial\qquad} & {H_{et}^1(K(t),H)} \ar[r] & {H_{et}^1(K(t),G)} \\
{} & {} & {} & {} & {} \\
1 \ar[r] & {{\cal F}(K)} \ar[r]^{\partial\qquad} \ar[uu]_{\epsilon} & {H_{et}^1(K,H)} \ar[r] \ar[uu]_{\rho} & {H_{et}^1(K,G)} \ar[uu]_{\eta}\\
{} & {a_0} \ar@{|->}[r] & {\xi_0 \ ,} & {} & {} }
$$
in which all the maps are canonical and all the vertical arrows
have trivial kernels. Since
$
\xi
$
goes to the trivial element in
$
H_{et}^1(K(t),G)
$,
one concludes that
$
\xi_0
$
goes to the trivial element in
$
H_{et}^1(K,G)
$.
Thus there exists an element
$
a_0 \in {\cal F}(K)
$
such that
$
\partial (a_0)=\xi _0.
$
The map
$
{\cal F}(K(t)) \to H_{et}^1(K(t),H)
$
is injective by Lemma
\ref{BoundaryInj}.
Thus
$
\epsilon(a_0)=a.
$

\end{proof}

\begin{cor}
\label{TwoSpecializations}
Let $S \mapsto {\cal F}(S)$ be the functor defined in
(\ref{MainFunctor}).
Let
$$
s_0, s_1: {\cal F}_{nr, K[t]}(K(t)) \to {\cal F}(K)
$$
be the specialization maps at zero and at one
(at the primes (t) and (t-1)). Then $s_0=s_1$.
\end{cor}

\begin{proof}
It is an obvious consequence of Theorem
\ref{HomInvNonram}.
\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 analog of \cite[Prop.7.1]{OP}

\begin{prop}
\label{PropEquatingGroups}
Let $S$ be a regular semi-local
irreducible scheme.
Let
$\mu_1: G_1\to C_1$ and
$\mu_2: G_2\to C_2$
be two smooth $S$-group scheme morphisms with tori $C_1$ and $C_2$.
Let $T \subset S$ be a non-empty closed
sub-scheme of $S$, and
$\varphi: G_1|_T \to G_2|_T$,
$\psi: C_1|_T \to C_2|_T$
be $S$-group scheme isomorphisms such that
$(\mu_2|_{T}) \circ \varphi = \psi \circ (\mu_1|_{T})$.
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 $\tilde S$-group scheme isomorphisms
$\Phi: \pi^*{G_1} \to \pi^*{G_2}$
and
$\Psi: \pi^*{C_1} \to \pi^*{C_2}$
such that
\begin{itemize}
\item[(i)]
$\delta^*(\Phi)=\varphi$,
\item[(ii)]
$\delta^*(\Psi)=\psi$,
\item[(iii)]
$\pi^*(\mu_2) \circ \Phi = \Psi \circ \pi^*(\mu_1): \pi^*(G_1) \to \pi^*(C_2)$.
\end{itemize}
\end{prop}

\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,Defn.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}
satisfying the following conditions
\begin{itemize}
\item[(i)]
$j$ is an open immersion dense at each fibre of $\overline p$, and $X=\overline X - Y$;
\item[(ii)]
$\overline p$ is a smooth projective such that all fibres are geometrically irreducible of dimension one;
\item[(iii)]
$q$ is a finite \'{e}tale morphism such that each fibre is non-empty.
\end{itemize}
\end{defn}

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

\begin{thm}
\label{GeneralSection}
Let $k$ be an infinite field, $V \subset \Pro^n_k$ a locally closed subscheme of pure dimension $r$.
Let $V^{\prime} \subset V$ be an open subscheme consisting of all points $x \in V$ such that
$V$ is $k$-smooth at $x$. Let
$p_1, p_2, \dots, p_m \in \Pro^n_k$
be a family of closed points with
$p_i \neq p_j$ for $i \neq j$. For a family $H_1(d), H_2(d), \dots, H_s(d)$ ($s \leq r$)
of hyperplanes of degree $d$ containing all points $p_i$ ($i=1,2, \dots, m$) set
$Y= H_1(d) \cap H_2(d) \cap \dots \cap H_s(d)$.

There exists an integer $d$ depending of 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 general enough, then
$Y$ crosses $V$
transversally at each point of
$Y \cap V^{\prime}$.

If $V$ is irreducible (geometrically irreducible)
and $s < r$ then for the same integer $d$ and for a general enough family
$H_1(d), H_2(d), \dots, H_s(d)$
the intersection $Y \cap V$ is additionally
irreducible (geometrically irreducible).

\end{thm}

Using this Bertini type 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 Zarisky 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 subscheme of the projective space $\Pro^{dim X -1}$. Additionally, if
$Z$ is a closed codimension one subvariety in $X$, then one can choose $X^0$ and $p$ such that
$p|_{Z \bigcap X^0} : Z \bigcap X^0 \to S$ is finite surjective.
\end{prop}

\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 such that the left hand side square is Cartesian.
\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]^{in} && \Pro^1 \times S &&  \ar[ll]_{i} \{\infty\} \times S &\\
    }
\end{equation}
where $j$ and $i$ are from Definition
\ref{DefnElemFib}
and
$pr_S \circ \pi = p$,
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
$X$ (resp. $\Aff^1 \times S$ ) is regarded as an $S$-scheme via the morphism
$p$ (resp. via the projection $pr_S$).
\end{prop}
We omit for now the proof of this proposition.



\section{Nice triples}
\label{NiceTriples}
We study in the present Section certain packages of geometric data and morphisms of that
packages. The concept of "nice triples" is very closed to the one of "standard triples "
\cite[Defn.4.1]{Vo} and is inspired by the latter one. 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.
Let
$\text{Spec}(\mathcal O_{X, \{x_1,x_2,\dots,x_n\}})$
be the respecting 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 nice triple over $U$
consists of the following family of data:
\begin{itemize}
\item[(i)]
a smooth morphism
$q_U: \mathcal X \to U$,
\item[(iii)]
%a closed codimension one subscheme
an element
$f \in \Gamma(\mathcal X, \mathcal O_{\mathcal X})$
%$\mathcal Z$,
\item[(ii)]
a section
$\Delta$
of the morphism
$q_U$.
\end{itemize}
These data must satisfy the following conditions:
\begin{itemize}
\item[(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[(b)]
the
$\Gamma(\mathcal X, \mathcal O_{\mathcal X})/f\cdot\Gamma(\mathcal X, \mathcal O_{\mathcal X})$
is a finite
$\Gamma(U, \mathcal O_{U})=O_{X, \{x_1,x_2,\dots,x_n\}}$-module,
%$\mathcal Z$ is finite and surjective over $U$;
\item[(c)]
there exists a finite surjective $U$-morphism
$\Pi: \mathcal X \to \Aff^1 \times U$,
\item[(d)]
$\Delta^*(f) \neq 0 \in \Gamma(U, \mathcal O_{U})$.
\end{itemize}

A 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[(1)]
$q^{\prime}_U = q_U \circ \theta$,
\item[(2)]
$f^{\prime} = \theta^{*}(f)\cdot g^{\prime}$
for an element
$g^{\prime} \in \Gamma(\mathcal X^{\prime}, \mathcal O_{\mathcal X^{\prime}})$ \\
%$\theta^{-1} (\mathcal Z) \subset \mathcal Z^{\prime}$
(in particular,
$\Gamma(\mathcal X^{\prime} , \mathcal O_{\mathcal X^{\prime} })/\theta^*(f)\cdot\Gamma(\mathcal X^{\prime} , \mathcal O_{\mathcal X^{\prime} })$
is a finite
$O_{X, \{x_1,x_2,\dots,x_n\}}$-module ),
%$\theta^{-1} (\mathcal Z)$ is finite surjective over $U$);
\item[(3)]
$\Delta = \theta \circ \Delta^{\prime}$.
\end{itemize}
(Stress that there are no conditions concerning an interaction of $\Pi^{\prime}$ and $\Pi$ ).
\end{defn}

\begin{thm}
\label{ThmEquatingGroups}
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 reductive
$\mathcal X$-group scheme and
$G_U:= \Delta^*(G_{\mathcal X})$
and
$G_{\text{const}}$
be the pull-back of $G_U$ to
$\mathcal X$.
Let $C_{\mathcal X}$ be an
$\mathcal X$-tori and
$C_U:= \Delta^*(C_{\mathcal X})$
and
$C_{\text{const}}$
be the pull-back of $C_U$ to
$\mathcal X$.
Then
there exist a morphism
$\theta: (\mathcal X^{\prime}, f^{\prime}, \Delta^{\prime}) \to
(\mathcal X, f, \Delta)$
of nice triples and isomorphisms
$$\Phi: \theta^*(G_{\text{const}}) \to \theta^*(G_{\mathcal X}), \ \Psi: \theta^*(C_{\text{const}}) \to \theta^*(C_{\mathcal X})$$
of
$\mathcal X^{\prime}$-group schemes such that
$(\Delta^{\prime})^*(\Phi)= id_{G_U}$,
$(\Delta^{\prime})^*(\Phi)= id_{G_U}$
and
\begin{equation}
\label{PhiAndPsiRelation}
\theta^*(\mu_2) \circ \Phi = \Psi \circ \theta^*(\mu_1)
\end{equation}
\end{thm}

This Theorem is a consequence of Proposition
\ref{PropEquatingGroups}.
The proof of this theorem is omitted for now.
%Section
%\ref{ProofOfThmEquatingGroups}.

\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 $\text{f} \in k[X]$.
% 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 $\text{f}$.

Replacing
$k$ with 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 $\Pro^{dim X-1}$ such that
$$ p|_{\{\text{f}=0\}\cap X^0}:\{\text{f}=0\}\cap X^0\to S $$
\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 $\{x_1,x_2,\dots,x_n\}$.
Now we replace $X$ by $p^{-1}(S)$ and $\text{f}$ 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
$$ \{x_1,\dots,x_n\}\subset\{\text{f}=0\}\subset X, $$
\noindent
$S$ is an open affine sub-scheme in the projective space
$\Pro^{dim X-1}$, and the restriction of
$p|_{\{\text{f}=0\}}:\{\text{f}=0\}\to S$ to the vanishing locus of $\text{f}$ is a
finite surjective morphism. In other words, $k[X]/(\text{f})$ is finite
as a $k[S]$-module.
\par
As an open affine sub-scheme of the projective space
$\Pro^{dim X-1}$ the scheme $S$ is regular. By Proposition
\ref{CartesianDiagram}
one can shrink $S$ such 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 $\{x_1,x_2,\dots,x_n\}$.
Now we replace $X$ by $p^{-1}(S)$ and $\text{f}$ by its
restriction to this new $X$.
\par
In this way we get an elementary fibration $p:X\to S$ such that
$$ \{x_1,\dots,x_n\}\subset\{\text{f}=0\}\subset X, $$
\noindent
$S$ is an open affine sub-scheme in the projective space
$\Pro^{dim X-1}$, and the restriction of
$p|_{\{\text{f}=0\}}:\{\text{f}=0\}\to S$ to the vanishing locus of $\text{f}$ is a
finite surjective morphism. And eventually there exists a finite surjective morphism
$$ \pi:X\to\Aff^1\times S $$
\noindent
such that $p=pr_S\circ\pi$.
\par
Now set $U:=\text{Spec}(\mathcal O_{X,\{x_1,x_2,\dots,x_n\}})$,
and denote by \ $can:U\hra X$ the canonical inclusion of schemes,
and $p_U= p\circ can:U\to S$. 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
$f:= q_X^*(\text{f})$.
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 $k[X]/\text{f}\cdot k[X]$ 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} Note that the schemes $\Delta(U)$ and
$\{f=0\}$ are both semi-local and that 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 $\text{f}$.


\section{Proof of Theorem (A)}
\label{SectProofOfTheoremA}
\begin{proof}


We begin with the following data.
Fix a smooth irreducible affine
$k$-scheme $X$, a finite family of points $x_1,x_2,\dots,x_r$ on
$X$, and set
$\mathcal O:=\mathcal O_{X,\{x_1,x_2,\dots,x_r\}}$
and
$U:= \text{Spec}(\mathcal O)$. Replacing $k$ with its algebraic closure
in $\Gamma(X, \mathcal O_X)$ we may and will assume that
$X$ is $k$-smooth and geometrically irreducible.
Further,
consider the reductive $U$-group scheme $G$, the $U$-tori $C$
and the smooth $U$-group scheme morphism
$$\mu : G \to C. $$
Let $K$ be the fraction field of $\mathcal O$.
Let
$\xi \in C(K)$
be such that
the element
$\bar \xi \in \mathcal F(K)$
is $\mathcal O$-unramified (see (\ref{DefnUnramified})).

%a principal $G$-bundle $P$ over $\mathcal O$ which is trivial over
%the field of fractions $K$ of $\mathcal O$.
%{\bf 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 $x_1,x_2,\dots,x_r$ are still in $X$;
\par\smallskip
(ii) The group schemes $G$, $C$ and the group morphism $\mu$ are defined over $X$,
$G$ is reductive over $X$, $C$ is a tori over $X$ and $\mu$ is a smooth $X$-group morphism.
We will often denote this $X$-group scheme $G$ by $G_X$
and write $G_U$ for the original $U$-group scheme $G$;
We will often denote this $X$-group scheme $C$ by $C_X$
and write $C_U$ for the original $U$-group scheme $C$;
\par\smallskip
(iii) The element $\xi$ is defined over $X_{\text{f}}$, that is
$\xi \in C(k[X]_{\text{f}})$
for a non-zero function
$\text{f} \in k[X]$
such that $\text{f}$ vanishes at each $x_i$'s and
the $k$-algebra $k[X]/(\textrm{f})$ is reduced;
\par\smallskip
(iv) we may assume further that
$\bar \xi \in {\cal F}(k[X]_\textrm{f})$
is $k[X]$-unramified.
\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}).
%
%Let $\xi \in C(K)$ be such that the
%$\bar \xi \in {\cal F}(K)$
%is $A$-unramified. We may assume that
%$\xi \in C(k[X]_\textrm{f})$
%%for an appropriative non-zero $\textrm{f} \in k[X]$
%with a reduced $k$-algebra $k[X]/(\textrm{f})$.
%{\bf
%Shrinking $X$ once more we may assume that}
%$\textrm{f}(x)=0$.

So, we are given now with a geometically irreducible $k$-smooth scheme $X$,
a finite family points
$x_1, x_2, \dots , x_r$ on $X$
and a non-zero function
$\text{f} \in k[X]$
vanishing at each $x_i$'s.
Beginning with these data the nice triple
(\ref{FirstNiceTriple})
$$(q_U:\mathcal X\to U,f,\Delta)$$
was constructed in Section
\ref{BasicTriple}.

Set $G_{\mathcal X}= q^*_X(G_X)$ and
$G_U= \Delta^*(G_{\mathcal X})=can^*(G_X)$.
Let
$G_{\text{const}}$
be the pull-back of $G_U$ to
$\mathcal X$.

Set $C_{\mathcal X}=q^*_X(C_X)$
and
$C_U:= \Delta^*(C_{\mathcal X})=can^*(C_X)$.
Let
$C_{\text{const}}$
be the pull-back of $C_U$ to
$\mathcal X$.



%\begin{rem}
%\label{DeltaIsWellDefined}
%{\bf Note as well that 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}
%$\{f=0\}$.
%\end{rem}

By Theorem
\ref{ThmEquatingGroups}
there exist a morphism
$\theta: (\mathcal X^{\prime}, f^{\prime}, \Delta^{\prime}) \to
(\mathcal X, f, \Delta)$
of nice triples and isomorphisms
$$\Phi: \theta^*(G_{\text{const}}) \to \theta^*(G_{\mathcal X}), \ \Psi: \theta^*(C_{\text{const}}) \to \theta^*(C_{\mathcal X})$$
of
$\mathcal X^{\prime}$-group schemes such that
$(\Delta^{\prime})^*(\Phi)= id_{G_U}$,
$(\Delta^{\prime})^*(\Phi)= id_{G_U}$
and
\begin{equation}
\label{PhiAndPsiRelation}
\theta^*(\mu_2) \circ \Phi = \Psi \circ \theta^*(\mu_1): \theta^*(G_{const}) \to \theta^*(C_{\mathcal X})
\end{equation}
The equality
(\ref{PhiAndPsiRelation})
implies that for each $U$-scheme
$\mathcal Y^{\prime}$
and each $U$-scheme morphism
$t: \mathcal Y^{\prime} \to \mathcal X^{\prime}$
{\bf the group isomorphism}
$\Psi(\mathcal Y^{\prime}): \theta^*(C_{const})(\mathcal Y^{\prime}) \to \theta^*(C_{\mathcal X})(\mathcal Y^{\prime})$
induces a group isomorphism
\begin{equation}
\label{EquatingFunctors}
C_U(\mathcal Y^{\prime})/\mu(G_U(\mathcal Y^{\prime})) =\mathcal F_{const}(\mathcal Y^{\prime}) \cong
\mathcal F_{\mathcal X}(\mathcal Y^{\prime})=
C(\mathcal Y^{\prime})/(\mu(G(\mathcal Y^{\prime})))
\end{equation}
Group isomorphisms
$\Psi(\mathcal Y^{\prime})$
form a natural functor transformation of functors defined on the category of $U$-schemes and $U$-scheme morphisms.
The same holds concerning isomorphisms
(\ref{EquatingFunctors}).

Taking the open inclusion
$\mathcal X^{\prime}_{f^{\prime}} \hra \mathcal X^{\prime}$
and the element
$\zeta:= (q^{\prime}_X)^*(\xi) \in \theta^*(C_{\mathcal X})(\mathcal X^{\prime}_{f^{\prime}})$
and applying the equality
(\ref{PhiAndPsiRelation})
we get an element
$\zeta_{U} \in \theta^*(C_{const})(\mathcal X^{\prime}_{f^{\prime}})$.

Let
$U= \text{Spec}(\mathcal O_{X, \{x_1,x_2,\dots,x_r\}})$
be as in Definition
\ref{DefnNiceTriple}.
Write
$R$ for $\mathcal O_{X, \{x_1,x_2,\dots,x_n\}}$.
It is a local essentially smooth $k$-algebra with maximal ideals
$\mathfrak m_i$'s where $i$ runs from $1$ to $r$.
Let
$(\mathcal X^{\prime}, f^{\prime}, \Delta^{\prime})$
be the above mentioned nice triple over $U$.
Show that it gives rise to certain data
subjecting the hypotheses of Lemma
\ref{Lemma2}.

Let
$A=\Gamma(\mathcal X^{\prime}, \mathcal O_{\mathcal X^{\prime}})$.
It is an $R$-algebra via the ring homomorphism
$(q^{\prime}_U)^*: R \to \Gamma(\mathcal X, \mathcal O_{\mathcal X})$.
Moreover this $R$-algebra is smooth.
The triple
$(\mathcal X^{\prime}, f^{\prime}, \Delta^{\prime})$
is a nice triple. Thus there exists a finite surjective $U$-morphism
$\Pi: \mathcal X^{\prime} \to \Aff^1_U$. It induces
the respecting
$R$-algebras inclusion
$R[t] \hra \Gamma(\mathcal X^{\prime}, \mathcal O_{\mathcal X^{\prime}})=A$
such that the
$A$
is finitely generated
as an
$R[t]$-module. For each index $i$ the
$R/\mathfrak m_i$-algebra
$A/\mathfrak m_i A$
is equi-dimensional of dimension one since
$(\mathcal X^{\prime}, f^{\prime}, \Delta^{\prime})$
is a nice triple.
Let $\epsilon= (\Delta^{\prime})^*: A \to R$ be an
$R$-algebra homomorphism induced by the section
$\Delta^{\prime}$ of the morphism $q^{\prime}_U$.
Clearly, that $\epsilon$ is an augmentation. Further,
$\epsilon(f^{\prime}) \neq 0 \in R$
since
$(\mathcal X^{\prime}, f^{\prime}, \Delta^{\prime})$
is a nice triple.
Set
$I=ker(\epsilon)$.
The $R$-module
$A/f^{\prime}A$
is finite since
$(\mathcal X^{\prime}, f^{\prime}, \Delta^{\prime})$
is a nice triple.
So, we are under the hypotheses of Lemma
\ref{Lemma2}. Thus we may use the conclusion of that Lemma.

So, there exists an element $u \in A$ subjecting (1) to (7)
of Lemma
\ref{Lemma2}.
The function $u$ defines a finite morphism
$\pi: {\cal X^{\prime}} \to U \times \Aff^1$.
Since ${\cal X^{\prime}}$ and
${\cal Y^{\prime}}:=U \times \Aff^1$
are regular schemes and $\pi$ is finite surjective
it is finite and flat by a theorem of Grothendieck
\cite[Cor.18.17]{E}.
So, for any map
$t:{\cal Y^{\prime\prime}}\to {\cal Y^{\prime}}$
of affine schemes, putting
${\cal X^{\prime\prime}}={\cal X^{\prime}}\times_{\cal Y^{\prime}} {\cal Y^{\prime\prime}}$
we have the norm map
given by
(\ref{NormMap})
$$
C({\cal X^{\prime\prime}}) \to C(\cal Y^{\prime\prime}).
$$
Set
${\cal D} = \textrm{Spec}(R/J)$
and
${\cal D}_1 = \textrm{Spec}(R/(s-1)R)$.
Clearly
${\cal D}_1$ is the scheme theoretic pre-image of
$U \times \{1\}$
and the disjoint union
${\cal D}_0:= {\cal D} \coprod \delta (U)$
is the scheme theoretic pre-image of
$U \times \{0\}$
under the morphism $\pi$.
In particular ${\cal D}$ is finite flat over $U$.
We will write $\Delta^{\prime}$ for $\Delta^{\prime} (U)$.
Recall that
$\xi \in C(k[X]_\textrm{f})$
and set
$\zeta= (q_X^{\prime})^*(\xi) \in C(A_{f^{\prime}})$.
Since the function
$f^{\prime} \in A$
is co-prime to the ideals
$J$ and $(s-1)R$ one can form the following element
\begin{equation}
\label{RightElement}
\xi^{\prime} = N_{{\cal D}_1/U}(\zeta|_{{\cal D}_1})N_{{\cal D}/U}(\zeta|_{{\cal D}})^{-1}
\in C(U):=C(R).
\end{equation}

\begin{clm}[Main]
\label{MainClaim}
One has $\bar \xi^{\prime}_K = \bar \xi_K$ in ${\cal F}(K)$,
where $K$ is the fraction field  of both $k[X]$ and $k[U]=A$.
\end{clm}
To complete the proof, it remains to prove the claim.
To do this it is convenient to fix some notations.
For an $A$-module $M$ set
$M_K= K \otimes_R M$,
for
an $U$-scheme ${\cal Z}$
set
${\cal Z}_K= \textrm{Spec}(K) \times_U {\cal Z}$,
for a
$U$-morphism
$\varphi: {\cal Z} \to {\cal W}$
set
$\varphi_K= \textrm{Spec}(K) \times_U \varphi$.
Clearly one has
$R_K=K$,
$U_K= \textrm{Spec}(K)$,
${\cal X^{\prime}}_K = \textrm{Spec}(A_K)$ and
$(U \times \Aff^1)_K$
is just the affine line
$\Aff^1_K$.
Its closed subschemes
$U_K \times \{1\}$
and
$U_K \times \{0\}$
coincide with the points
$1$ and $0$ of $\Aff^1_K$.
The morphism
$(q^{\prime}_U)_K: {\cal X^{\prime}}_K \to U_K=\textrm{Spec}(K)$
is smooth. In fact, the morphism $\theta$ is \'{e}tale and
$q_U: \mathcal X \to U$ is smooth.
The morphism $\pi_K$ is finite flat and fits in the commutative triangle
\begin{equation}
    \xymatrix{
    {\cal X^{\prime}}_K \ar[rr]^-{\pi_K} \ar[rd]_-{(q^{\prime}_U)_K} && \Aff^1_K \ar[ld]^-{pr_K} &   \\
    & \textrm{Spec}(K)  \\
    }
\end{equation}
One has
${\cal D}_K = \textrm{Spec}(R_K/J_K)$,
${\cal D}_{1,K} = \textrm{Spec}(R_K/(s-1)R_K)$.
Clearly
${\cal D}_{1,K}$ is the scheme theoretic pre-image of
the point $\{1\} \in \Aff^1_K$
and the disjoint union
${\cal D}_{0, K}:= {\cal D}_K \coprod \Delta_K$
is the scheme theoretic pre-image of the point
$\{0\} \in \Aff^1_K$
under the morphism
$\pi_K$.
Let
\begin{equation}
\label{ZetaU}
\zeta_u:= N_{K({\cal X^{\prime}}_K)/K(\Aff^1)}(\zeta)  \in C(K(\Aff^1))=C(K(u)).
\end{equation}
To prove Claim
\ref{MainClaim}
we need the following one:
\begin{clm}
\label{Claim}
The class
$\bar \zeta_u \in {\cal F}(K(s))$
is $K[s]$-unramified.
\end{clm}
Assuming this claim we complete the proof of Claim
\ref{MainClaim}.
By Claim
\ref{Claim}
we can apply the specialization maps to $\bar \zeta_u$.
By Corollary
\ref{TwoSpecializations}
the specializations at $0$ and $1$ of the element
$\bar \zeta_u$ coincide, that is
\begin{equation}
\label{Rel2}
s_1(\bar \zeta_u)=s_0(\bar \zeta_u) \in {\cal F}(K).
\end{equation}
By Corollary
\ref{NormAtZeroAndOne}
the function
$N_{A_K/K[u]}(f) \in K[s]$
does not vanishes as at $1$, so at $0$ and
by the condition $(5)$ of Lemma
\ref{Lemma2}
one has
$\zeta_u \in C(K[u]_{N(f)})$.
Using the relation between specialization and evaluation maps described in
Definition
\ref{SpecializationDef}
one has a chain of relations
\begin{equation}
\label{Rel3}
\overline {Ev_1(\zeta_u)} = s_1(\bar \zeta_u)= s_0(\bar \zeta_u)=
\overline {Ev_0(\zeta_u)} \in {\cal F}(K).
\end{equation}
Base change and the multiplicativity properties of the norm map
(\ref{NormMap})
imply relations
\begin{equation}
\label{Rel4}
Ev_1(\zeta_u)= N_{{\cal D}_{1,K}/U_K}(\zeta|_{{\cal D}_{1,K}})
\end{equation}
and
\begin{equation}
\label{Rel4}
Ev_0(\zeta_u)=
N_{{\cal D}_K \coprod \Delta_K/U_K}(\zeta|_{{\cal D}_K \coprod \Delta_K})=
N_{{\cal D}_K/U_K}(\zeta|_{{\cal D}_K})N_{\Delta_K/U_K}(\zeta|_{\Delta_K})
\end{equation}
So, we have a chain of relations in ${\cal F}(K)$
$$
\overline {N_{{\cal D}_{1,K}/U_K}(\zeta|_{{\cal D}_{1,K}})}=
\overline {Ev_1(\zeta_u)}=\overline {Ev_0(\zeta_u)}=
\overline {N_{{\cal D}_K/U_K}(\zeta|_{{\cal D}_K})}\cdot \overline {N_{\Delta_K/U_K}(\zeta|_{\Delta_K})}
$$
By the normalization property of the norm map
(\ref{NormMap})
one has
$N_{\Delta_K/U_K}(\zeta|_{\Delta_K})=\delta_K^*(\zeta)$.
Since
$\delta_K^*(\zeta)= \xi_K \in C(K)$
we have a chain of relations in ${\cal F}(K)$
$$
\bar \xi_K=
\overline {\delta_K^*(\zeta)}=
\overline {N_{\Delta_K/U_K}(\zeta|_{\Delta_K})}=
\overline {N_{{\cal D}_{1,K}/U_K}(\zeta|_{{\cal D}_{1,K}})} \cdot
(\overline {N_{{\cal D}_K/U_K}(\zeta|_{{\cal D}_K})})^{-1}
$$
By the base change property of the norm map
(\ref{NormMap}) one has
\begin{equation}
\label{Rel1}
\xi'_K=N_{{\cal D}_{1,K}/U_K}(\zeta|_{{\cal D}_{1,K}})N_{{\cal D}_K/U_K}(\zeta|_{{\cal D}_K})^{-1}
\in C(U_K):=C(K).
\end{equation}
So $\bar \xi'_K=\bar \xi_K$.
 Claim
\ref{MainClaim}
follows.

It remains to prove Claim
\ref{Claim}.
Recall that
the class
$\bar \xi \in {\cal F}(k[X]_\textrm{f})$
is $k[X]$-unramified.
Since the morphism
$p:X \to S$
is smooth
so is the morphism
$r: U \hra X \to S$.
Since the
$p_X: {\cal X} \to X$
is the base change of $r$ it coincides
with a composition map
${\cal X} \hra X \times_S X \xra{id \times pr_X} X$
where $id \times pr_X$ is a smooth morphism. Whence
$id \times pr_X$ is flat and of finite type.
By Lemma
\ref{UnramifiednessLemma}
the class
$\bar \zeta \in {\cal F}(A_f)$
is $R$-unramified.
Thus the same class
$\bar \zeta$ when regarded in
${\cal F}(K[{\cal X^{\prime}}_K]_f)$
is
$K[{\cal X^{\prime}}_K]$-unramified.

Now check that the inclusion of $K$-algebras
$K[u] \subset A_K=K[{\cal X^{\prime}}_K]$ and
the function $f^{\prime} \in A$ satisfy the hypotheses of Lemma
\ref{KeyUnramifiedness}.
We first check that the $K$-algebra $A_K/f^{\prime}A_K$ is reduced.
Recall that the ring
$k[X]/(\textrm{f})$ is reduced. As we mentioned
above, the morphism
$q^{\prime}_{\mathcal X^{\prime}}: {\cal X^{\prime}} \to X$
is essentially smooth,  hence the ring
$A/(f^{\prime})$
is reduced too and thus its localization
$A_K/f^{\prime}A_K$ is reduced.
Since the extension
$K[u] \subset A_K=K[{\cal X^{\prime}}_K]$
and the functions $f^{\prime} \in A$
satisfy
the conditions $(5)$ to $(7)$ of Lemma
\ref{Lemma2}
they also satisfy the hypotheses of Lemma
\ref{KeyUnramifiedness}
for the functor
$\mathcal F_{const}$ from equality
(\ref{EquatingFunctors}).
Thus by Lemma
\ref{KeyUnramifiedness}
the class $\bar \zeta_u$
is $K[u]$-unramified.
This implies Claim
\ref{Claim}. The proof of {\bf the Theorem A} is completed.


\end{proof}

\section{Proof of Theorem (B)}
\label{SectProofofTheoremB}
\begin{proof}
To prove Theorem (B)
we now recall a celebrated result of Dorin
Popescu (see \cite{P}
or, for a self-contained proof, \cite{Sw}).

Let $k$ be a field and $R$ a local $k$-algebra. We say that $R$ is
{\it geometrically regular} if $k'\otimes_kR$ is regular for any
finite extension $k'$ of $k$. A ring homomorphism $A\to R$ is
called {\it geometrically regular} if it is flat and for each
prime ideal $\mathfrak q$ of $R$ lying over $\mathfrak p$, $R_\mathfrak q/\mathfrak p R_\mathfrak q =
k(\mathfrak p)\otimes_AR_\mathfrak q$ is geometrically regular over
$k(\mathfrak p)=A_\mathfrak p/\mathfrak p_\mathfrak p$.

Observe that any regular semi- local ring containing a {\bf perfect} field $k$ is
geometrically regular over $k$.

\begin{thm}[Popescu's theorem] A  homomorphism $A\to R$  of
noetherian rings is geometrically regular if and only if $R$ is a
filtered direct limit of smooth $A$-algebras.
\end{thm}

{\it Proof of Theorem B.}
%{\bf We need in one more reduction: to the case of a finite prime subfield.
%A problem is that we CAN NOT use a norm principle}.

Let $R$ be a regular semi-local ring
containing an infinite {\bf perfect} field $k$. Since $k$ is perfect
one can apply Popescu's theorem. So, $R$ can be presented as a filtered
direct limit of smooth $k$-algebras $A_{\alpha}$ over the infinite field $k$.
We
first observe that we may replace the direct system of the
$A_\alpha$'s by a system of  essentially smooth semi-local
$k$-algebras. In fact, if
${\mathfrak  m_1}, {\mathfrak  m_2}, \dots , {\mathfrak  m_r}$
are all maximal ideal of $R$ and $S=R- (\cup \ {\mathfrak  m_i})$
we can replace each
$A_\alpha$
by
$\left(A_\alpha\right)_{S_{\alpha}}$,
where
$S_\alpha=S \cap A_\alpha$. Note
that in this case the canonical morphisms
$\varphi_\alpha:A_\alpha\to R$
are local (sends a maximal ideal to a maximal one) and every
$A_\alpha$
is a regular semi-local ring, in particular a factorial ring.

Let now $L$ be the field of fractions of $R$ and, for each
$\alpha$,  let $K_\alpha$ be the field of fractions of $A_\alpha$.
For each index $\alpha$ let
$\mathfrak a_{\alpha}$
be the kernel of the map
$\varphi_{\alpha}: A_{\alpha} \to R$
and
$B_{\alpha}=(A_{\alpha})_{\mathfrak a_{\alpha}}$.
Clearly, for each
$\alpha$,  $K_{\alpha}$
is the field of fractions of
$B_{\alpha}$.
The composition map
$A_{\alpha} \to R \to L$
factors through $B_{\alpha}$
and hence it also factors through the residue field $k_{\alpha}$
of $B_{\alpha}$.
Since $R$ is a filtering direct limit of the $A_{\alpha}$'s
we see that $L$ is a filtering direct limit of the $B_{\alpha}$'s.

Let $\xi \in C(L)$ be such that the class
$\bar \xi \in {\cal F}(L)$
is $R$-unramified. We need the following two lemmas.
\begin{lem}
\label{WeakGrothendieck}
%Let $B={\cal O}_{X,x}$ be the local ring of a $k$-smooth
%variety at a point $x$. Let $k(x)$ be its residue field
Let $B$ be a local ring which is a domain and let
$K$ be its field of fractions.
Let ${\mathfrak m}$ be a maximal ideal of $B$
and
$\bar B= B/{\mathfrak m}$.
For an element
$\theta \in {\cal F}(B)$
write
$\bar \theta$ for its image in
${\cal F}(\bar B)$
and
$\theta_K$
for its image in
${\cal F}(K)$.
Let
$\eta, \rho \in {\cal F}(B)$
be such that
$\eta_K = \rho_K \in {\cal F}(K)$.
Then
$\bar \eta = \bar \rho \in {\cal F}(\bar B)$.
\end{lem} %%%%%%% or \eta(x)=\rho(x)?

\begin{lem}
\label{LiftToFiniteLevel}
There exists an index $\alpha$ and an element
$\xi_{\alpha} \in C(B_{\alpha})$
such that
$\varphi(\xi_{\alpha})=\xi$
and the class
$\bar \xi_{\alpha} \in {\cal F}(K_{\alpha})$
is $A_{\alpha}$-unramified.
\end{lem}
Assuming these two claims we complete the proof as follows.
Consider a commutative diagram
$$
\xymatrix{
    A_{\alpha}  \ar[d]\ar[rr]^-{\varphi_{\alpha}}\ar[d]_-{} && R \ar[d]^-{} &   \\
     B_{\alpha} \ar[d]\ar[r]^{} & k_{\alpha} \ar[r]^{} & L \\
     K_{\alpha}.
    }
$$
By Lemma
\ref{LiftToFiniteLevel}
the class
$\bar \xi_{\alpha} \in {\cal F}(K_{\alpha})$
is $A_{\alpha}$-unramified.
Hence by Theorem $A$ there exists an element
$\eta \in C(A_{\alpha})$
such that
$\bar \xi_{\alpha}=\bar \eta \in {\cal F}(K_{\alpha})$.
By Lemma
\ref{WeakGrothendieck}
the elements
$\bar \xi_{\alpha}$
and
$\bar \eta$
have the same image in ${\cal F}(k_{\alpha})$.
Hence $\bar \xi \in {\cal F}(L)$ coincides with
the image of the element $\varphi_{\alpha} (\bar \eta)$
in ${\cal F}(L)$.
It remains to prove the two Lemmas.

{\it Proof of Lemma \ref{WeakGrothendieck}.}
Induction on $dim (B)$. The case of dimension $1$ follows from Theorem
\ref{NisnevichCor} applied to the local ring
$B$. To prove the general case choose an $f \in B$
such that $\eta = \rho \in {\cal F}(B_f)$. Let $\pi \in B$
be such that
$\pi$ is ô regular parameter in
$B$, having no common factors with $f$. Let
$\overline B= B/\pi B$. Then for the image
$\overline {\eta - \rho}$ of $\eta - \rho$
in
${\cal F}(\overline B)$
we have
$(\overline {\eta - \rho})_f = \overline {(\eta - \rho)_f} =0
\in {\cal F}(\overline B_{\overline f})$.
By the inductive hypotheses one has
$\overline {\eta - \rho}=0 \in {\cal F}(\bar B)$.
Thus $\bar \eta = \bar \rho \in {\cal F}(\bar B)$.

{\it Proof of Lemma \ref{LiftToFiniteLevel}.}
%We give a proof of Lemma
%$\ref{LiftToFiniteLevel}$
%only for the case when $R$ is local
%and left the case of semi-local $R$ to the reader.
%{\bf Dumayu privedennoe dok-vo bez izmenenii rabotaet v obschem sluchae}.

Choose an $f \in R$ such that $\xi$ is defined over $R_f$.
Then $\xi$ is ramified at most at those hight one primes
$\mathfrak p_1, \dots, \mathfrak p_r$
which contains $f$. Since the class
$\bar \xi \in {\cal F}(L)$
is $R$-unramified there exists, for any
$\mathfrak p_i$, an element
$\sigma_i \in G(L)$
and an element
$\xi_i \in C(R_{\mathfrak p_i})$
such that
$\xi=\mu(\sigma_i) \xi_i \in C(L)$.
We may assume that $\xi_i$ is defined over $R_{h_i}$
for some
$h_i \in R - \mathfrak p_i$
and that $\sigma_i$ is defined over $R_{g_i}$
for some $g_i \in R$.

We can find an index $\alpha$ such that $A_{\alpha}$
contains lifts
$f_{\alpha}, h_{1,\alpha}, \dots, h_{r,\alpha}, g_{1\alpha}, \dots, g_{r,\alpha}$
and moreover
\begin{itemize}
\item[(1)]
$C(A_{\alpha, f_{\alpha}})$ contains a lift $\xi_{\alpha}$ of $\xi$,
\item[(2)]
$C(A_{\alpha, h_{i,\alpha}})$ contains a lift of $\xi_{i,\alpha}$ of $\xi_i$,
\item[(3)]
$G(A_{\alpha, g_{i,\alpha}})$ contains a lift of $\sigma_{i,\alpha}$ of $\sigma_i$.
\end{itemize}
Since none of the
$f_{\alpha}, h_{1,\alpha}, \dots, h_{r,\alpha}, g_{1\alpha}, \dots, g_{r,\alpha}$
 vanishes in $R$, the elements
$\xi_{\alpha}, \xi_{1,\alpha}, \dots, \xi_{ir,\alpha}$
and
$\sigma_{1,\alpha}, \dots, \sigma_{r,\alpha}$
may be regarded as elements of
$C(B_{\alpha})$
and
$G(B_{\alpha})$
respectively.

We know that
$\xi_{i,\alpha} \mu (\sigma_{i,\alpha})$
and
$\xi_{\alpha}$
map to the same element in $C(L)$.
Hence replacing $\alpha$ by a larger index, we may assume that
$\xi_{\alpha}= \xi_{i,\alpha} \mu (\sigma_{i,\alpha}) \in C(B_{\alpha})$.
We claim that the class
$\bar \xi_{\alpha} \in {\cal F}(K_{\alpha})$
is $A_{\alpha}$-unramified.
To prove this note that
the only primes at which $\bar \xi_{\alpha}$ could be ramified are those which divide
$f_{\alpha}$. Let
$\mathfrak q_{\alpha}$
be one of them. Check that $\bar \xi_{\alpha}$ is
unramified at $\mathfrak q_{\alpha}$. Let $q_{\alpha} \in A_{\alpha}$ be
a prime element such that
$q_{\alpha}A_{\alpha}=\mathfrak q_{\alpha}$. Then
$q_{\alpha}r_{\alpha} = f_{\alpha}$
for an element
$r_{\alpha}$.
Thus $qr=f \in R$ for the images of $q_{\alpha}$ and $r_{\alpha}$ in $R$.
Since the homomorphism $\varphi_{\alpha}: A_{\alpha} \to R$ is local,
$q \in \mathfrak m_R$. The relation $qr=f$ shows that
$q \in \mathfrak p_i$ for some index $i$. Thus
$q_{\alpha} \in \varphi_{\alpha}^{-1}(\mathfrak p_i)$
and
$\mathfrak q_{\alpha} \subset \varphi_{\alpha}^{-1}(\mathfrak p_i)$.
On the other hand
$h_{i, \alpha} \in A_{\alpha} - \varphi_{\alpha}^{-1}(\mathfrak p_i)$,
because
$h_i \in R - \mathfrak p_i$. Thus
$h_{i, \alpha} \in A_{\alpha} - \mathfrak q_{\alpha}$.
Now the relation
$\xi_{\alpha}= \xi_{i,\alpha} \mu (\sigma_{i,\alpha}) \in C(B_{\alpha})$
with
$\xi_{i,\alpha} \in C(A_{\alpha, h_{i,\alpha}})$
shows that
$\bar \xi_{\alpha}$ is unramified at
$\mathfrak q_{\alpha}$.
Thus
$\bar \xi_{\alpha}$
is unramified at each hight one prime in $A_{\alpha}$ containing $f_{\alpha}$.
Since
$\xi_{\alpha} \in C(A_{\alpha, f_{\alpha}})$
we conclude that
$\bar \xi_{\alpha}$
is $A_{\alpha}$-unramified.
The lemma follows. The theorem is proved.

\end{proof}

\section{Applications}
\label{SectionOneMorePurityTheorem}
In this Section we prove a purity theorem for reductive groups.
Let $G$ be a reductive group over the characteristic zero field $k$ and
$Z \xra{i} G$ a closed central subgroup of $G$.
Let $G'=G/Z$ be the factor group,
$\pi: G \to G'$ be the projection. For any $k$-algebra $R$
consider the boundary operator
$\delta_{\pi,R}: G'(R) \to \textrm{H}^1_{\text{\'et}}(R,Z)$.
It is a group homomorphism
\cite[Ch.II, \S 5.6, Cor.2]{Se}.
Set
$$
{\cal F}(R)= \textrm{H}^1_{\text{\'et}}(R,Z)/ Im(\delta_{\pi,R}).
$$
\begin{thm}
\label{PurityForSubgroup}
Let $R$ be a regular local
ring containing the field $k$.
The functor
${\cal F}$
satisfies purity for $R$.
If $K$ is the fraction field of $R$ this statement can be restated in an explicit way
as follows:

given an element $\xi \in \textrm{H}^1_{\text{\'et}}(K,Z)$
suppose that for each height $1$ prime ideal
$\mathfrak p $ in $R$ there exist
$\xi_{\mathfrak p } \in \textrm{H}^1_{\text{\'et}}(R_{\mathfrak p }, Z)$,
$g_{\mathfrak p } \in G'(K)$
with
$\xi=\xi_{\mathfrak p } + \delta_{\pi}(g_{\mathfrak p }) \in \textrm{H}^1_{\text{\'et}}(K,Z)$.
Then there exist
$\xi_{\mathfrak m } \in \textrm{H}^1_{\text{\'et}}(R, Z)$, $g_{\mathfrak m } \in G'(K)$,
such that
$$
\xi=\xi_{\mathfrak m } + \delta_{\pi}(g_{\mathfrak m }) \in \textrm{H}^1_{\text{\'et}}(K,Z).
$$

\end{thm}

\begin{proof}
Since $G$ is reductive the group $Z$ is of  multiplicative type.
So we can find a commutative separable $k$ algebra $l$ and
a closed embedding
$Z \hra R_{l/k}(\mathbb G_{m,\ l})$
into the permutation torus
$T^{+}=R_{l/k}(\mathbb G_{m,\ l})$.
Let
$G^{+}=(G \times T^{+})/Z$
and
$T=T^{+}/Z$,
where
$Z$ is embedded in
$G \times T^{+}$
diagonally.
Clearly
$G^{+}/G=T$.
Consider a commutative diagram
$$
\xymatrix {
{}           & \{1\}                       & \{1\} \\
{}           & {G'} \ar[r]^{id}  \ar[u]          & {G'}   \ar[u]          \\
\{1\} \ar[r] & {G} \ar[r]^{j^+} \ar[u]^{\pi} & {G^+} \ar[r]^{\mu^+} \ar[u]^{\pi^+} & {T} \ar[r] & \{1\} \\
\{1\} \ar[r] & {Z} \ar[r]^{j} \ar[u]^{i} & {T^+} \ar[r]^{\mu} \ar[u]^{i^+} & {T} \ar[u]_{id} \ar[r] &  \{1\} \\
{}           & \{1\} \ar[u]                      & \{1\} \ar[u]\\
}
$$
with exact rows and columns. For a local $k$-algebra $A$ one has
$\textrm{H}^1_{\text{\'et}}(A,T^+)=\{*\}$
by Hilbert $90$
and this diagram gives rise to
a commutative diagram of pointed sets
$$
\xymatrix {
{}           & {}                            & {\textrm{H}^1_{\text{\'et}}(A,G')} \ar[r]^{id}            & {\textrm{H}^1_{\text{\'et}}(A,G')}            \\
{G^+(A)} \ar[r]^{\mu^+_A} & {T(A)} \ar[r]^{\delta^+_A}  & {\textrm{H}^1_{\text{\'et}}(A,G)} \ar[r]^{j^+_*} \ar[u]^{\pi_*} & {\textrm{H}^1_{\text{\'et}}(A,G^+)} \ar[u]^{\pi^+_*}  \\
{T^+(A)} \ar[r]^{\mu_A} \ar[u]^{i^+_*} & {T(A)} \ar[r]^{\delta_A} \ar[u]^{id} & {\textrm{H}^1_{\text{\'et}}(A,Z)} \ar[r]^{\mu} \ar[u]^{i_*} & {\{*\}} \ar[u]^{i^+_*}  \\
{}           & {} & {G'(A)} \ar[u]^{\delta_{\pi}}                      & {} \\
}
$$
with exact rows and columns. It follows that
$\pi^+_*$ has  trivial kernel and one has a chain of group isomorphisms
$$
\textrm{H}^1_{\text{\'et}}(A,Z) / Im (\delta_{\pi,A})= ker (\pi_*)= ker (j^+_*) = T(A)/\mu^+ (G^+(A)).
$$
Clearly these isomorphisms respect $k$-homomorphisms of local $k$-algebras.
The functor
$A \mapsto T(A)/\mu^+ (G^+(A))$
satisfies purity for the regular local $k$-algebra $R$ by Theorem (B).
Hence the functor
$A \mapsto \textrm{H}^1_{\text{\'et}}(A,Z) / Im (\delta_{\pi,A})$
satisfies purity for $R$.
\end{proof}

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

\begin{proof}[Proof of Theorem
\ref{MainThm}]
Consider a short exact sequence of reductive $R$-group schemes
\begin{equation}
\label{DerRedTori}
\{1\} \to G_{der} \xra{i} G \xra{\mu} C \to \{1\},
\end{equation}
where
$G_{der}$ is the derived $R$-group scheme of $G$ and $C$ be a tori over $R$.
Prove that
\begin{equation}
\label{Gr_SerreForG}
ker[H^1_{\text{\'et}}(R,G) \to H^1_{\text{\'et}}(K,G)]=*
\end{equation}
{\bf provided that} one has
\begin{equation}
\label{Gr_SerreForG_red}
ker[H^1_{\text{\'et}}(R,G_{der}) \to H^1_{\text{\'et}}(K,G_{der})]=*
\end{equation}

This sequence of $R$-group schemes gives a short exact sequence of the corresponding
sheaves in the \'{e}tale topology on the big \'{e}tale site. That sequence of sheaves
gives rise to a commutative diagram with exact arrows of pointed sets
%\begin{equation}
%\label{DerRedTori}
%\{1\} \to C(R)/\mu(G(R)) \to H^1_{\text{\'et}}(R,G_{der}) \to  H^1_{\text{\'et}}(R,G) \to H^1_{\text{\'et}}(R,C)
%\end{equation}

\begin{equation}
\label{RactangelDiagram}
    \xymatrix{
\{1\} \ar[r]  & C(R)/\mu(G(R)) \ar[r]^{\partial}\ar[d]_{\alpha} & H^1_{\text{\'et}}(R,G_{der}) \ar[r]^{i_*} \ar[d]^{\beta} &
H^1_{\text{\'et}}(R,G) \ar[r]^{\mu} \ar[d]_{\gamma} & H^1_{\text{\'et}}(R,C) \ar[d]^{\delta} &\\
\{1\}  \ar[r]  & C(R_{\mathfrak p})/\mu(G(R_{\mathfrak p})) \ar[r]^{\partial}\ar[d]_{\alpha_{\mathfrak p}} & H^1_{\text{\'et}}(R_{\mathfrak p},G_{der}) \ar[r]^{i_*} \ar[d]^{\beta_{\mathfrak p}} &
H^1_{\text{\'et}}(R_{\mathfrak p},G) \ar[r]^{\mu} \ar[d]_{\gamma_{\mathfrak p}} & H^1_{\text{\'et}}(R_{\mathfrak p},C) \ar[d]^{\delta_{\mathfrak p}} &\\
\{1\}  \ar[r] & C(K)/\mu(G(K)) \ar[r]^{\partial} & H^1_{\text{\'et}}(K,G_{der}) \ar[r]^{i_*} &
H^1_{\text{\'et}}(K,G) \ar[r]^{\mu}  & H^1_{\text{\'et}}(K,C) &\\
}
\end{equation}
Set
$\alpha_K=\alpha_{\mathfrak p} \circ \alpha$,
$\beta_K= \beta_{\mathfrak p} \circ \beta$,
$\gamma_K = \gamma_{\mathfrak p} \circ \gamma$,
$\delta_K = \delta_{\mathfrak p} \circ \delta$.
By a theorem of Nisnevich
\cite{Ni}
one has
\begin{equation}
\label{NisnevichForPurity}
ker(\alpha_{\mathfrak p})=ker(\beta_{\mathfrak p})=ker(\gamma_{\mathfrak p})=ker(\delta_{\mathfrak p})=*
\end{equation}
Let
$\xi \in ker(\gamma_K)$,
then
$\mu(\xi) \in ker(\delta_K)$.
By
\cite{C-T-S}
$ker(\delta_K)=*$,
whence
$\mu(\xi)=*$
and
$\xi=i_*(\zeta)$
for an
$\zeta \in H^1_{\text{\'et}}(R,G_{der})$.
By a Theorem of Nisnevich
\cite[]{Ni}
$ker(\gamma_{\mathfrak p})=*$.
Since
$\gamma_K(\xi)=*$
we see that
$\gamma(\xi)=*$.
Whence
$i_*(\beta(\zeta))=*$
and
$\beta(\zeta)= \partial (\epsilon_{\mathfrak p})$
for an
$\epsilon_{\mathfrak p} \in C(R_{\mathfrak p})/\mu(G(R_{\mathfrak p}))$.
A diagram chaise shows that there exists a unique element
$\epsilon_K \in C(K)/\mu(G(K))$
for each hight one prime ideal
$\mathfrak p$ of $R$
one has
$\alpha_{\mathfrak p}(\epsilon_{\mathfrak p})= \epsilon_K \in C(K)/\mu(G(K))$.
By Purity Theorem
\ref{PurityGeneral}
there exists an element
$\epsilon \in C(R)/\mu(G(R))$
such that
$\alpha_K(\epsilon)=\epsilon_K$.
The element
$\epsilon_K$
has the property that
$\partial (\epsilon_K)= \beta_K(\zeta)$.
Whence
$\partial (\epsilon) = \zeta$
and
$\xi=i(\partial(\epsilon))=*$.
The proof equality
(\ref{Gr_SerreForG})
is reduced to the proof of
the equality
(\ref{Gr_SerreForG_red}).

To prove
(\ref{Gr_SerreForG_red})
recall that
$G_{der}$
is a semi-simple $R$-group scheme.
Let
$G^{sc}_{red}$
be the corresponding simply-connected semi-simple $R$-group scheme
and let
$\pi: G^{sc}_{red} \to G$
be the corresponding $R$-group scheme morphism.
Let $Z^{\prime}=ker(\pi)$.
It is known that
$Z^{\prime}$
is contained in the center of
$G^{sc}_{red}$
and
$Z^{\prime}$
is a finite group scheme of multiplicative type. Consider
a short exact sequence of $R$-group schemes
\begin{equation}
\label{ZenterCsDer}
\{1\} \to Z^{\prime} \to G^{cs}_{der} \xra{\pi} G_{der} \to \{1\},
\end{equation}
By the main result of
\cite[Thm.1.1]{PSV}
one has
\begin{equation}
\label{Gr_SerreForG_red}
ker[H^1_{\text{\'et}}(R,G^{sc}_{der}) \to H^1_{\text{\'et}}(K,G^{sc}_{der})]=*
\end{equation}
The short exact sequence
(\ref{ZenterCsDer})
give rises to an exact sequence of pointed sets
\begin{equation}
\label{H1OfZenterCsDer}
H^1_{\text{\'et}}(R, Z^{\prime})/\partial (G_{red}) \to H^1_{\text{\'et}}(R,G^{sc}_{der}) \to H^1_{\text{\'et}}(R,G_{der})
\to H^2_{\text{\'et}}(R,Z^{\prime})
\end{equation}
and furthermore to a commutative diagram like
(\ref{RactangelDiagram}) with exact arrows.
Now a diagram chaise similar to the one above in this proof
completes the proof of Theorem
\ref{PurityGeneral}.

\end{proof}

\begin{proof}[Proof of Theorem
\ref{MainThmGeometric}]
Repeat literally the proof of Theorem
\ref{MainThm}.


\end{proof}



\section{Examples}
\label{ExamplesOfPurityResults}
We follow here the notation of The Book of Involutions
\cite{KMRT}.
The field $k$ is a characteristic zero field.
The functors
(\ref{Example1})
to
(\ref{Example12})
satisfy purity for regular local rings containing $k$
as follows either from Theorem
\ref{PurityForSubgroup}
or from
Theorem (B).

\begin{itemize}
\item[(1)]
Let $G$ be a simple algebraic group over the field $k$,
$Z$ a central subgroup, $G'=G/Z$, $\pi: G \to G'$
 the canonical morphism.
For any $k$-algebra $A$ let
$\delta_{\pi,R}: G'(R) \to \textrm{H}^1_{\text{\'et}}(R,Z)$
be the boundary operator. One has a functor
\begin{equation}
\label{Example1}
R \mapsto \textrm{H}^1_{\text{\'et}}(R,Z)/ \textrm{Im}(\delta_{\pi,R}).
\end{equation}

\item[(2)]
Let $(A,\sigma)$ be a finite separable $k$-algebra with an orthogonal involution.
Let
$\pi: \textrm{Spin}(A, \sigma) \to \textrm{PGO}^+(A, \sigma)$
be the canonical morphism of the spinor $k$-group scheme to
the projective orthogonal $k$-group scheme. Let
$Z=ker (\pi)$.
%THE DEFINITION IS GIVEN in
%\cite[just above (30.13)]{KMRT}.\\
For a $k$-algebra $R$ let
$\delta_R: \textrm{PGO}^+(A, \sigma)(R) \to \textrm{H}^1_{\text{\'et}}(R,Z)$
be the boundary operator.
%It coincides with
%the spinor norm transform described in
%\cite[Prop. (13.36)]{KMRT}.
One has a functor
\begin{equation}
\label{Example2}
R \mapsto \textrm{H}^1_{\text{\'et}}(R,Z)/Im(\delta_R).
\end{equation}
In $(2a)$ and $(2b)$ below we describe this functor somewhat more explicitly
following
\cite{KMRT}.

\item[(2a)]
Let $C(A, \sigma)$ be the Clifford algebra. Its center $l$ is an \'etale
quadratic $k$-algebra.
Assume that $deg(A)$ is divisible by $4$.
Let
$\Omega(A,\sigma)$ be the extended Clifford group
\cite[Definition given just below (13.19)]{KMRT}.
Let
$\underline \sigma$
be the canonical involution of
$C(A,\sigma)$
as it is described in
\cite[just above (8.11)]{KMRT}.
Then
$\underline \sigma$
is either orthogonal or symplectic by
\cite[Prop.8.12]{KMRT}.
Let
$
\underline \mu: \Omega(A,\sigma) \to R_{l/k}(\mathbb G_{m,l})
$
be the multiplier map defined in
\cite[just above (13.25)]{KMRT}
by
$\underline \mu (\omega)= \underline \sigma(\omega)\cdot \omega$.
Set
$R_l=R \otimes_k l$.
For a field or a local ring $R$ one has
$
\textrm{H}^1_{\text{\'et}}(R,Z)/Im(\delta_R)=R_l^{\times}/\underline \mu (\Omega(A,\sigma)(R))
$
by
\cite[the diagram in (13.32)]{KMRT}.
Consider the functor
\begin{equation}
\label{Example2a}
R \mapsto R_l^{\times}/\underline \mu (\Omega(A,\sigma)(R)).
\end{equation}
It coincides with the functor
$R \mapsto \textrm{H}^1_{\text{\'et}}(R,Z)/Im(\delta_R)$
on local rings containing $k$.


\item[(2b)]
Now let $deg(A)=2m$ with odd $m$. Let $\tau: l \to l$ be the involution of $l/k$.
The kernel of the morphism
$R_{l/k}(\mathbb G_{m,l}) \xra{id - \tau} R_{l/k}(\mathbb G_{m,l})$
coincides with $\mathbb G_{m,k}$. Thus $id-\tau$ %%%%% this is ugly; one is tempted to write i(d - 1)
induces a $k$-group scheme morphism
which we denote
$\overline {id-\tau}: R_{l/k}(\mathbb G_{m,l})/\mathbb G_{m,k} \hra R_{l/k}(\mathbb G_{m,l})$.
Let
$
\underline \mu: \Omega(A,\sigma) \to R_{l/k}(\mathbb G_{m,l})
$
be the multiplier map defined in
\cite[just above (13.25)]{KMRT}
by
$\underline \mu (\omega)= \underline \sigma(\omega)\cdot \omega$.
Let
$\kappa: \Omega(A,\sigma) \to R_{l/k}(\mathbb G_{m,l})/\mathbb G_{m,k}$
be the $k$-group scheme morphism described in
\cite[Prop.13.21]{KMRT}.
The composition
$\overline {(id-\tau)} \circ \kappa$
lands in
$R_{l/k}(\mathbb G_{m,l})$.
Let
$U \subset \mathbb G_{m,k} \times R_{l/k}(\mathbb G_{m,l})$
be a closed $k$-subgroup consisting of all $(\alpha, z)$ such that
$\alpha^4 = N_{l/k}(z)$.

Set
$\mu_*=(\underline \mu, [\overline {(id-\tau)} \circ \kappa]\cdot\underline \mu2): \Omega(A,\sigma)
\to \mathbb G_{m,k} \times R_{l/k}(\mathbb G_{m,l})$.
This $k$-group scheme morphism lands in $\textrm{U}$.
So we get a $k$-group scheme morphism
$\mu_*: \Omega(A,\sigma) \to \textrm{U}$.
On the level of $k$-rational points it coincides with the one described in
\cite[just above (13.35)]{KMRT}. For a field or a local ring one has
$$
\textrm{H}^1_{\text{\'et}}(R,Z)/Im(\delta_R)=
\textrm{U}(R)/[\{(N_{l/k}(\alpha), \alpha^4) | \alpha \in R_l^{\times}\}\cdot \mu_*(\Omega(A,\sigma)(R))].
$$
Consider the the functor
\begin{equation}
\label{Example2b}
R \mapsto \textrm{U}(R)/[\{(N_{l/k}(\alpha), \alpha^4) | \alpha \in R_l^{\times}\}\cdot \mu_*(\Omega(A,\sigma)(R))].
\end{equation}
It coincides with the functor
$R \mapsto \textrm{H}^1_{\text{\'et}}(R,Z)/Im(\delta_R)$
on local rings containing $k$.

\item[(3)]
Let
$\Gamma(A,\sigma)$
be the Clifford group $k$-scheme of $(A,\sigma)$.
Let
$\textrm{Sn}: \Gamma(A,\sigma) \to \mathbb G_{m,k}$
be the spinor norm map. It is dominant.
Consider the functor
\begin{equation}
\label{Example3}
R \mapsto R^{\times}/\textrm{Sn}(\Gamma(A,\sigma)(R)).
\end{equation}
Purity for this functor was originally  proved in
\cite[Thm.3.1]{Z}.
In fact,
$\Gamma(A,\sigma)$
is $k$-rational.

\item[(4)]
We follow here the Book of Involutions
\cite[\S 23]{KMRT}.
Let $A$ be a separable finite dimensional $k$-algebra with center $l$
and $k$-involution $\sigma$ such that $k$ coincides with all $\sigma$-invariant
elements of $l$, that is $k=l^{\sigma}$.
Consider the $k$-group schemes of
similitudes
of $(A,\sigma)$:
$$
\textrm{Sim}(A,\sigma)(R)=\{a \in A_R^{\times}\ | a\cdot \sigma_R (a) \in l_K^{\times} \}.
$$
We have a $k$-group scheme morphism
$
\mu: \textrm{Sim}(A,\sigma) \to \mathbb G_{m,k}, \ a \mapsto a \cdot \sigma (a).
$
It gives an exact sequence of algebraic $k$-groups
$$
\{1\} \to \textrm{Iso}(A,\sigma) \to \textrm{Sim}(A,\sigma) \to \mathbb G_{m,k} \to \{1\}.
$$
One has a the functor
\begin{equation}
\label{Example4}
R \mapsto R^{\times}/\mu (\textrm{Sim}(A,\sigma)(R)).
\end{equation}
Purity for this functor was originally  proved in
\cite[Thm.1.2]{Pa}.
Various particular cases are obtained
considering unitary, symplectic and orthogonal involutions.
\item[(4a)]
In the case of an orthogonal involution $\sigma$ the connected component
$\textrm{GO}^+(A, \sigma)$
\cite[(12.24)]{KMRT}
of the similitude $k$-group scheme
$\textrm{GO}(A, \sigma):=\textrm{Sim}(A,\sigma)$
has the index two in
$\textrm{GO}(A, \sigma)$.
The restriction of $\mu$ to
$\textrm{GO}^+(A, \sigma)$
is still a dominant morphism to
$\mathbb G_{m,k}$. One has a functor
\begin{equation}
\label{Example4a}
R \mapsto R^{\times}/\mu (\textrm{GO}^+(A,\sigma)(R))
\end{equation}
It seems that its purity
does not follow from
\cite[Thm.1.2]{Pa}. In fact we do not know
whether the norm principle holds for
$\mu: \textrm{GO}^+(A, \sigma) \to \mathbb G_{m,k}$
or not.

\item[(5)]
Let $A$ be a central simple algebra (csa) over $k$ and
$\textrm{Nrd}:GL_{1,A} \to \mathbb G_{m,k}$
 the reduced norm morphism. One has a functor
\begin{equation}
\label{Example5}
R \mapsto R^{\times}/\textrm{Nrd}(\textrm{GL}_{1,A}(R)).
\end{equation}
Purity for this functor was originally  proved in
\cite[Thm.5.2]{C-TO}.
\item[(6)]
Let $(A,\sigma)$ be a finite separable $k$-algebra with
a unitary involution such that its center $l$ is a quadratic extension of $k$.
Let $U(A,\sigma)$ be the unitary $k$-group scheme. Let
$U_l(1)$ be an algebraic tori given by $N_{l/k}=1$.
One has a functor
\begin{equation}
\label{Example6}
R \mapsto \textrm{U}_l(1)(R)/\textrm{Nrd}(\textrm{U}_{A,\sigma}(R)) =
\{\alpha \in R^{\times}_l | N_{l/k}(\alpha)=1 \} /\textrm{Nrd}(\textrm{U}_{A,\sigma}(R))
\end{equation}
where
$\textrm{Nrd}$
is the reduced norm map.
Purity for this functor was originally  proved in
\cite[Thm.3.3]{Z}.

\item[(7)]
With the notation of example $(5)$ choose an integer $d$ and consider
the $k$-group scheme morphism
$\mu: \textrm{GL}_{1,A} \times \mathbb G_{m,k} \to \mathbb G_{m,k}$
given by
$(\alpha,a) \mapsto \textrm{Nrd}(\alpha)\cdot a^d$.
One has a functor
\begin{equation}
\label{Example7}
R \mapsto R^{\times}/\mu [\textrm{GL}_{1,A} \times \mathbb G_{m,k})(R)]=
R^{\times}/\textrm{Nrd}(A^{\times}_R)\cdot {R^{\times}}^d
\end{equation}
Purity for this functor was originally  proved in
\cite[Thm.3.2]{Z}.

\item[(8)]
With the notation of example $(5)$ choose
 an integer $d$ and consider the functor
\begin{equation}
\label{Example8}
R \mapsto
U_l(1)(R)/
[\textrm{Nrd}(U_{A,\sigma}(R))\cdot \{\alpha \in R^{\times}_l | N_{l/k}(\alpha)=1 \}^d]
\end{equation}
Purity for this functor was originally  proved in
\cite[Thm.3.2]{Z}.

\item[(9)]
Let $G_1$,$G_2$,$C$ be affine $k$-group schemes.
  Assume that $C$ is commutative
and let
$\mu_1: G_1 \to C$,
$\mu_2: G_2 \to C$
be $k$-group scheme morphisms and
$\mu: G_1 \times G_2 \to C$ be given by
$\mu(g_1,g_2)= \mu_1(g_1)\cdot \mu_2(g_2)$.
One has a functor
\begin{equation}
\label{Example9}
R \mapsto C(R)/\mu [(G_1 \times G_2)(R)]= C(R)/[\mu_1(G_1(R))\cdot \mu_2(G_2(R))]
\end{equation}
In this style one could get a lot of curious examples of functors, one of which is given here.
\item[(10)]
Let $(A_1,\sigma_1)$ be a finite separable $k$-algebra with an orthogonal involution.
Let $A_2$ be a csa over $k$. Let $\mu_1$ be the multiplier map for
$(A_1,\sigma_1)$
and
$\textrm{Nrd}_2$
be the reduced norm for $A_2$.
One has a functor
\begin{equation}
\label{Example10}
R \mapsto
R^{\times}/[\mu_1(\textrm{GO}^+(A_1,\sigma_1)(R))\cdot {\textrm{Nrd}_2({A^{\times}_{2,R}})} \cdot {R^{\times}}^d].
\end{equation}

\item[(10)]
Let $A$ be a csa of degree $3$ over $k$, $\textrm{Nrd}$  the reduced norm
and $\textrm{Trd}$ be the reduced trace. Consider the cubic form
on the $27$-dimensional $k$-vector space
$A \times A \times A$
given by
$N:= \textrm{Nrd}(x)+\textrm{Nrd}(y)+\textrm{Nrd}(z)-\textrm{Trd}(xyz)$.
Let
$\textrm{Iso}(A,N)$
be the $k$-group scheme of isometries of $N$ and
$\textrm{Sim}(N)$
be the $k$-group scheme of similitudes of $N$.
It is known that
$\textrm{Iso}(N)$ is a normal algebraic subgroup in $\textrm{Sim}(A,N)$
and the factor group coincides with
$\mathbb G_{m,k}$. So we have a canonical $k$-group morphism
(the multiplier)
$\mu: \textrm{Sim}(N) \to \mathbb G_{m,k}$.
Now one has a functor
\begin{equation}
\label{Example11}
R \mapsto R^{\times}/\mu(\textrm{Sim}(N)(R)).
\end{equation}
Note that the connected component of $\textrm{Iso}(N)$ is
a simply connected algebraic $k$-group of the type $E_6$.

\item[(11)]
Let $(A,\sigma)$ be a csa of degree $8$ over $k$
with a symplectic involution. Let $V \subset A$ be the subspace
of all skew-symmetric elements. It is of dimension $28$.
Let
$\textrm{Pfd}$
be the reduced Pfaffian on $V$ and
$\textrm{Trd}$
be the reduced trace on $A$. Consider the degree $4$ form on
the space
$V \times V$
given by
$F:=\textrm{Pfr}(x)+\textrm{Pfr}(y)-1/4\textrm{Trd}((xy)2)-1/16{\textrm{Trd}(xy)}^2$.
Consider the symplectic form on
$V \times V$
given by
$\phi((x_1,y_1),(x_2,y_2))= \textrm{Trd}(x_1y_2-x_2y_1)$.
Let
$\textrm{Iso}(F)$
(resp. $\textrm{Iso}(\phi)$)
be the $k$-group scheme of isometries of the pair $F$
(resp. of $\phi)$).
Let
$\textrm{Sim}(F)$
(resp. $\textrm{Sim}(\phi)$)
be the $k$-group scheme of similitudes of $F$
(resp. of $\phi$).
Set
$G=\textrm{Iso}(F) \cap \textrm{Iso}(\phi)$
and
$G^+=\textrm{Sim}(F) \cap \textrm{Sim}(\phi)$.
It is known that
$G$ is a normal algebraic subgroup in $G^+$
and the factor group coincides with
$\mathbb G_{m,k}$. So we have a canonical $k$-group morphism %%%%%(the multiplier  ???) \\
$\mu: G^+ \to \mathbb G_{m,k}$.
Now one has a functor
\begin{equation}
\label{Example12}
R \mapsto R^{\times}/\mu(G^+(R)).
\end{equation}
Note that $G$ is a simply-connected group of the type $E_7$.

\end{itemize}


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\end{thebibliography}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this Section few geometric lemmas will be proved.
Let $k$ be a field of characteristic zero.
The main result of this Section are the lemmas
\ref{Lemma2}
and
\ref{Lemma1}.
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 $F$ be a field of characteristic zero and let $S$ be an $F$-smooth algebra
which is a domain of dimension $1$. Let $\mathfrak m_0$ be a maximal ideal with
$S/\mathfrak m_0=F$. Let
$\mathfrak m_1, \mathfrak m_2, \dots, \mathfrak m_n$
be different maximal ideals of $S$
(it might be that $\mathfrak m_0=\mathfrak m_i$ for an index $i$).
Then there exists an element
$\beta \in S$
such that $S$ is finite over $F[\beta]$ and
\begin{itemize}
\item[(1)]
the ideals
$\mathfrak n_i:= \mathfrak m_i \cap F[t]$ ($i=1,2,\dots, n$)
are all different from each other and different  from
$\mathfrak n_0:= \mathfrak m_0 \cap F[t]$
provided that
$\mathfrak m_0$
is different  from
$\mathfrak m_i$'s;
\item[(2)]
the extension $S/F[t]$ is \'etale at each
$\mathfrak m_i$'s and at $\mathfrak m_0$;
\item[(3)]
$F[\beta]/\mathfrak n_i = S/\mathfrak m_i$ for each
$i=1,2,\dots, n$;
\item[(4)]
$\mathfrak n_0= \beta F[\beta]$.
\end{itemize}
\end{lem}

\begin{proof}
Consider  a closed imbedding
$\Spec(S) \hra \Aff^n_F$
and use a generic linear projection to
$\Aff1_F$.

\end{proof}

\begin{lem}
\label{Lemma4}
Under the hypotheses of Lemma
\ref{Lemma3}
let $0 \neq f \in S$
be an element that does not belong to a maximal ideal different of
$\mathfrak m_0, \mathfrak m_1, \dots, \mathfrak m_n$.
Let $N(f)=N_{S/F[\beta]}$ be the norm of $f$. Then for an element
$\beta \in S$ satisfying $(1)$ to $(4)$ of Lemma
\ref{Lemma3}
one has
\begin{itemize}
\item[(a)]
$N(f)=fg$ for an element $g \in S$;
\item[(b)]
$fS + gS= S$;
\item[(c)]
the map $F[\beta]/(N(f)) \to S/(f)$ is an isomorphism.
\end{itemize}
\end{lem}

\begin{proof}
It is straightforward.

\end{proof}


\begin{lem}
\label{Lemma2}
Let $A$ be a local essentially smooth $k$-algebra with a maximal ideal
$\mathfrak m$.
Let $A[t] \subset R$
be an $A[t]$-algebra which is essentially smooth
as a $k$-algebra and is finite over the
polynomial algebra $A[t]$. Assume
$R/\mathfrak m R$
is a domain and assume there exists an element
$h \in R \setminus \mathfrak m R$
such that $R_h$ is $A$-smooth.
Let $\epsilon: R \to A$ be an $A$-augmentation and
$I=ker(\epsilon)$.
Given an $f \in R$ with
$0 \neq \epsilon (f) \in A$
one can find an $s \in R$ satisfying the following conditions
\begin{itemize}
\item[(1)]
$R$ is finite over $A[s]$;
\item[(2)]
$R/sR=R/I \times R/J$ for certain ideal $J$;
\item[(3)]
$J + fR = R$;
\item[(4)]
$(s-1)R + fR = R$;
\item[(5)]
if $N(f)=N_{R/A[s]}(f)$ then $N(f)=fg \in R$ for certain $g \in R$ \\
($R/A[t]$ is a projective module since $R$ and $A[t]$ are regular and $R$ is finite over $A[t]$);
\item[(6)]
$fR + gR = R$;
\item[(7)]
the composition map $A[s]/(N(f)) \to R/(N(f)) \to R/(f)$ is an isomorphism.
\end{itemize}

\end{lem}

\begin{proof}
Replacing $t$ by $t- \epsilon (t)$ we may assume that
$\epsilon (t)=0$.
Since $R$ is finite over $A[t]$ and $R/fR$ is finite over $A$
we conclude that $A[t]/(N(f))$ is finite over $A$, hence
$A/(tN(f))$ is finite over $R$ too. So setting $u=tN(f)$
we get an integral extension $A[t]/A[u]$.
Consider the characteristic polynomial of
the operator
$R \xra{tf} R$ as an $A[u]$-module operator. This polynomial vanishes on
$tf$ and its free coefficient is $N(tf)$ up to a sing. Thus
$N(tf)=tfg$ for an element $g \in R$.
Now replacing
$t$ by $u$ we may assume that $R/A[t]$ is still integral,
$\epsilon (t)=0$ and $t \in fR$.
In fact, $u=N(ft)=ftg$ and whence
$\epsilon (u)=\epsilon (t) \epsilon (fg) =0$
and
$u \in fR$.

We use below ``bar" to denote the reduction modulo the ideal
$\mathfrak mR$. Let
$F=\bar A=A/\mathfrak m$.
By the  assumption of the lemma the $F$-algebra $\bar R$
is an $F$-smooth domain of dimension $1$.
Let
$\mathfrak m_1, \mathfrak m_2, \dots, \mathfrak m_n$
be different maximal ideals of $\bar R$ dividing $\bar f$
and let
$\mathfrak m_0 = ker (\bar {\epsilon})$.
Let $\beta \in \bar R$ be such that the extension
$\bar {R}/F[\beta]$
satisfies the conditions $(1)$ to $(4)$ of Lemma
\ref{Lemma3}.

Let $\alpha \in R$ be a lift of $\beta$, that is
$\bar \alpha=\beta$ in $\bar R$. Replacing $\alpha$ by
$\alpha - \epsilon (\alpha)$
we may assume that
$\epsilon (\alpha) = 0$
and still
$\bar \alpha=\beta$.

Let
$\alpha^n + p_1(t)\alpha^{n-1} + \dots + p_n(t)=0$
be an integral dependence equation for $\alpha$.
Let $N$ be an integer large than the
$max\{2, deg (p_i(t))\}$, where
$i=1, 2, \dots, n$. Then for any
$r \in k^{\times}$
the element
$s= \alpha -rt^N$
is such that $t$ is integral over $A[s]$.
Thus for any
$r \in k^{\times}$
the ring $R$ is integral over $A[s]$.


On the other hand the element
$\bar s = \bar \alpha -r\bar {t}^N$
still satisfies the conditions
$(1)$ to $(4)$ of Lemma
\ref{Lemma3}
because for any
$i=1, 2, \dots, n$ one has
$\bar t \in \mathfrak m_i$.

We claim that the element $s \in R$ is the required element
(for almost all $r \in k^{\times}$).

In fact, for almost all
$r \in k^{\times}$
the element $s$ satisfies the conditions
$(1)$ to $(4)$ of Lemma
\cite[Lemma 5.2]{OP}.
It remains to show that the conditions
$(5)$ to $(7)$ are satisfied for all
$r \in k^{\times}$.

To prove $(5)$ consider the characteristic polynomial of the
operator $R \xra{f} R$ as an $A[s]$-module operator.
This polynomial vanishes on $f$ and its free coefficient is
$\pm N(f)$ (the norm of $f$). Thus
$f^n - a_1f^{n-1} + \dots \pm N(f)=0$
and $N(f)=fg$ for an element $g \in R$.

To prove $(6)$ one has to check that this $g$ is a unit
modulo the ideal $fR$. It suffices to check that
$\bar g \in \bar R$
is a unit modulo the ideal $\bar f \bar R$. The field
$F=A/\mathfrak m$, the $F$-algebra $S=\bar R$,
its maximal ideals
$\mathfrak m_0, \mathfrak m_1, \dots, \mathfrak m_n$
and the element satisfy the hypotheses of Lemma
\ref{Lemma4}
with $\beta$ replaced by $\bar s$.
Now $\bar g$ is a unit modulo the ideal $\bar f \bar R$
by the item $(b)$ of Lemma
\ref{Lemma4}.

To prove $(7)$ note that $R/fR$ and $A[s]/(N(f))$ are
finite $A$-modules. So it remains to check that the map
$\varphi: A[s]/(N(f)) \to R/fR$
is an isomorphism modulo the maximal ideal
$\mathfrak m$. For that it suffices to verify that the map
$\bar \varphi: F[\bar s]/(N(\bar f)) \to \bar R/\bar f \bar R$
is an isomorphism where
$N(\bar f):=N_{\bar R/F[\bar s]}$.
Now $\bar \varphi$ is an isomorphism by the item $(c)$ of Lemma
\ref{Lemma4}.
The lemma follows.

\end{proof}

\begin{cor}
\label{NormAtZeroAndOne}
Under the hypotheses of Lemma
\ref{Lemma2} let $K$ be the field of fractions of $A$,
$R_K=R \otimes_A K$ and $\epsilon_K=\epsilon \otimes_A id: R_K \to K$.
Consider the inclusion
$K[s] \subset R_K$. Then the norm
$N(f) \in K[s]$ does not vanish  at the points $1$ and $0$ of
the affine line $\Aff1_K$.
\end{cor}

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

\end{proof}

\begin{lem}
\label{Lemma1}
Let $X$ be a geometrically irreducible $k$-smooth affine variety of
dimension $d$. Let $x \in X$ be a closed point,
$0\neq f \in k[X]$
and $Z$ be the vanishing locus of $f$.
Then there exists a commutative diagram
\begin{equation}
\label{Triangle}
    \xymatrix{
    X \ar[rr]^-{\varrho}\ar[rd]_-{q} && \Aff^d \ar[ld]^-{pr} &   \\
    & \Aff^{d-1}  \\
    }
\end{equation}
with a linear projection $pr$ such that
\begin{itemize}
\item[(1)]
there exists a principle open set $X_{\textrm{h}} \subset X$ containing
the fibre $q^{-1}q(x)$ such that the map
$q: X_{\textrm{h}} \to \Aff^{d-1}$
is smooth;
\item[(2)]
the fibre $q^{-1}q(x)$ is geometrically irreducible;
\item[(3)]
the map $\varrho$ is finite;
\item[(4)]
the map $q|_Z: Z \to \Aff^{d-1}$ is finite;
\item[(5)]
k(x)=k(q(x)).
\end{itemize}
\end{lem}

\begin{proof}
Bertini plus generic projection. Consider a closed imbedding
$X \hra \Aff^n_k$, the closure $\bar X$ of $X$ in the projective
space $\Pro^n_k$ and the complement $X_{\infty}$ of $X$ in $\bar X$.
Let $\bar Z$ be the closure of $Z$ in $\Pro^n_k$ and
$Z_{\infty}$ be the complement of $Z$ in $\bar Z$.
Since $dim X=d$ one has $dim X_{\infty}=d-1$. Set
$\Pro^{n-1}_k=\Pro^n_k - \Aff^n_k$. Now extend the scalars  to an algebraic closure
$\bar k$ of $k$. Let
$x_1, x_2, \dots, x_n \in X(\bar k)$
be the points of
$x \otimes_k \bar k$.
Let
$\Pi \subset \Pro^{n-1}_{\bar k}$
be a  linear subspace of dimension $n-d$ which is general in the sense that
\begin{itemize}
\item[(i)]
$\Pi \cap X_{\infty,\ \bar k}$ consists of finitely many points;
\item[(ii)]
$\Pi \cap Z_{\infty,\ \bar k} = \emptyset$;
\item[(iii)]
if $<\Pi, x_i>$ is the space generated by $\Pi$ and by an $x_i$ then
the $<\Pi, x_i> \cap X_{\bar k}$ is $\bar k$-smooth and irreducible
for each $i=1, 2, \dots, n$;
\item[(iv)]
$<\Pi, x_i> \neq <\Pi, x_j>$ for $i \neq j$.
The requirement $(iii)$ can be satisfied by a Bertini type theorem
for a hyperplane section crossing a given rational point because we are
in characteristic zero.
\end{itemize}
Since the set of linear spaces $\Pi$ satisfying conditions $(i)$ to $(iii)$
is an open set of a Grassmannian we may choose a $\Pi$ which is already defined
over the ground field $k$. We may then choose
a hyperplaneplane $\pi \subset  \Pi$ of dimension $n-d-1$,  defined over $k$
and such that
\begin{equation}
\label{Condition5}
\pi \cap X_{\infty,\ \bar k} \neq \emptyset .
\end{equation}
\end{proof}
Now consider the linear projection
$\Pro^n_k \setminus \pi \to \Pro^n_d$.
Its restriction to $\bar X$ defines a finite morphism
$\bar \rho: \bar X \to \Pro^d_k$.
And its restriction to $X$ defines a finite morphism
$\rho: X \to \Aff^d_k$,
because it is a base change of $\bar \rho$ by means of
the open imbedding
$\Aff^d_k \hra \Pro^d_k$.

The linear projection with the center $\Pi$
defines a morphism
$q: X \to \Aff^{n-1}_k$
which clearly fits in the commutative diagram
(\ref{Triangle})
with a linear projection
$pr: \Aff^d_k \to \Aff^{d-1}_k$.
We claim that these
$\rho$, $q$ and $pr$
are the required morphisms.

To prove this recall that
$\varrho$ is finite. The morphism
$q|_Z: Z \to \Aff^{d-1}_k$
is finite too, because
$\Pi \cap Z_{\infty} = \emptyset$.
Condition $(5)$ is satisfied because
$p \otimes_k \bar k$
separates the closed points $x_i$'s of the scheme
$x \otimes_k \bar k$.
The fibre of the morphism
$p \otimes_k \bar k$
over the point $x_i$ coincides with the
intersection
$<\Pi, x_i> \cap X_{\bar k}$
for each $i=1, 2, \dots, n$.
After the scalar extension $k$ to $\bar k$ the scheme theoretic fibre
of $q$ over $q(x)$ becomes a disjoint union of the fibres of
$q \otimes_k \bar k$
over the points
$(q \otimes_k \bar k)(x_i)$.
Since $k(q(x))=k(x)$ and for each $i$ the intersection
$<\Pi, x_i> \cap X_{\bar k}$
is irreducible we conclude that the fibre $q^{-1}q(x)$
is irreducible too, and thus condition $(2)$ is verified.
Since for each $i$
the intersection
$<\Pi, x_i> \cap X_{\bar k}$
is smooth we conclude that the scheme theoretic fibre
of $q$ over $q(x)$
is smooth too.
Thus condition $(1)$ is satisfied.
The lemma is proved.


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

%\bye

%Here is a scketch of our method for the geometric case.
%Given a $k$-smooth variety $X$, a closed point $x \in X$ and an
%${\cal O}_{X,x}$-unramified class
%$\bar \xi \in {\cal F}(k(X))$
%one can find a representative $\xi$
% of $\bar \xi$ such that $\xi \in C(X_\textrm{f})$
%for an appropriate $\textrm{f} \in k[X]$ with $\textrm{f}(x) \neq 0$. Letting
%$U= \textrm{\Spec}({\cal O}_{X,x})$ and $i: U \hra X$
%one can find
% multi-valued morphisms %%%%%%%% what's that? %%%%%%%
%$h_0, h_1: U \to X_{\textrm{f}}$ and
%a multi-valued homotopy
%$H: \Aff^1_U \to X$
%between
%$i + h_0$ and $h_1$. %%%%%%%% what is $i$? %%%%%%%
%Now as a lift in $C(U)$ of  $\bar \xi$ we can choose %%% correct?%%%%%
%$$
%\xi' := h_1^*(\xi)\cdot h_0^*(\xi^{-1}) \in C(U).
%$$
%To check that
%$\overline {\xi'_{k(X)}}=\overline {\xi_{k(X)}} \in {\cal F}(k(X))={\cal F}(k(U))$
%we do the following.
%The restriction of $H$ to the generic point of $\Aff^1_U$
%defines a multi-valued map
%$H_{\eta}: \textrm{Spec}(k(\Aff^1_U)) \to X_{\textrm{f}}$
%and an element
%$\xi_t:= H^*_{\eta}(\xi) \in C(k(\Aff^1_U))$.
%Choosing $H$ carefully we may assume that the class
%$\overline {\xi_t} \in {\cal F}(k(\Aff^1_U))$
%is $\Aff^1_{k(U)}$-unramified and thus it is constant.
%Specializing
%$\overline {\xi_t}$
%at the points $1$ and $0$ we get
%a relation
%$\overline {\xi_{k(U)}} + \overline {h^*_0(\xi)}= \overline {h^*_1(\xi^{-1})}$
%in
%${\cal F}(k(U))$.
%Whence
%$\overline {\xi'_{k(U)}}=\overline {\xi_{k(U)}} \in {\cal F}(k(U))={\cal F}(k(X))$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

By Lemma
\ref{Lemma1}
we can find a commutative diagram
\begin{equation}
\label{Diagram1}
    \xymatrix{
    X \ar[rr]^-{\varrho}\ar[rd]_-{q} && \Aff^d \ar[ld]^-{pr} &   \\
    & \Aff^{d-1}  \\
    }
\end{equation}
with a linear projection $pr$ and a function
$\textrm{h} \in k[X]$
such that the conditions
$(1)$ to $(5)$ of are satisfied. Let
$U=\textrm{Spec}(A)$ the point $x \in X$ is the closed point of $U$.
Consider the Cartesian
square of schemes
$$
\xymatrix{
    {} & \mathcal X \ar[rr]^-{p_X}\ar[d]_-{p} && X \ar[d]^-{q} &   \\
    U \ar[ru]^{\delta}\ar[r]^{id} & U \ar[rr]^{r} && \Aff^{d-1}  \\
    }
$$
where
$r=q|_{U}$, ${\cal X}=U \times_{\Aff^{d-1}} X$, $p$
is the first projection and
$\delta$
is the diagonal map.

Applying base change by means of $r$ to
the diagram
(\ref{Diagram1})
we get a commutative triangle
\begin{equation}
    \xymatrix{
    {\cal X} \ar[rr]^-{\lambda}\ar[rd]_-{p} && U \times \Aff1 \ar[ld]^-{pr} &   \\
    & U  \\
    }
\end{equation}
with a finite surjective $\lambda$.
Set
$f=p_X^*(\textrm{f})$
and
$h=p_X^*(\textrm{h})$,
where
$\textrm{h}$ is from the condition $(1)$ of Lemma
\ref{Lemma1}.

Now check that the field $k$, the $k$-algebra $A$ with the maximal ideal
$\mathfrak m:= \mathfrak m_x$,
the ring $R$ of regular functions on the affine scheme
$\cal X$, the inclusion
$\lambda^*: A[t] \hra R$
induced by $\lambda$,
the function $h$, the $A$-augmentation
$\epsilon:= \Delta^* : R \to A$
and the function $f$
satisfy the assumption of Lemma
\ref{Lemma2}.

As a local ring of a $k$-smooth variety $A$ is essentially
$k$-smooth.
Since $r$ is essentially smooth and $X$ is $k$-smooth,
${\cal X}$ is essentially $k$-smooth and so is $R$.
Since $\varrho$ is finite $R$ is finite over $A[t]$ too.
Since $k(q(x))=k(x)$ and the fibre of $q$ over $q(x)$
is geometrically irreducible so is the fibre of $p$ over
$x \in U$. This implies that $R/\mathfrak m$ is a domain.
By base change, the open subscheme ${\cal X}_h$ contains the closed fibre of $p$ and
the restriction of $p$ to ${\cal X}_h$ is smooth.
The assumptions are satisfied.

Now we can find an
$s \in R$ satisfying the conditions $(1)$ to $(7)$ of Lemma
\ref{Lemma2}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
By Proposition
\ref{ArtinsNeighbor}
there exist a Zarisky 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 subscheme of the projective space
$\Pro^{dim X -1}$
such that
$p|_{\{f=0\} \bigcap X^0} : \{f=0\} \bigcap X^0 \to S$
is finite surjective. For each $i=1,2, \dots , n$ let $s_i=p(x_i) \in S$.
Shrinking $S$ we may assume that $S$ {\bf is affine} and still contains the family
$\{ s_1,s_2, \dots ,s_n \}$. Clearly in that case
$p^{-1}(S)$
contains the family
$\{ x_1,x_2, \dots ,x_n \}$.
Replace $X^0$ with
$p^{-1}(S)$
and write $X$ for $p^{-1}(S)$.
Replace $f$ with its restriction to that new $X$ and still write $f$ for that restriction.
%Respectively still write $Z$ for the vanish locus of $f$ on that new $X$.

So, we get an elementary fibration
$p: X \to S$
such that
$\{ x_1, \dots ,x_n \} \subset \{f=0\} \subset X$
and
$S$
is an open affine in the projective space
$\Pro^{dim X - 1}$
and the restriction of
$p|_{\{f=0\}}: \{f=0\} \to S$
to the vanishing of $f$ is a finite surjective morphism.
The latter means that the
$k[S]$-module $k[X]/(f)$ is finite.

The scheme $S$ is affine and regular since it is an open affine subscheme of the projective space
$\Pro^{dim X -1}$. By Proposition
\ref{CartesianDiagram}
there exists a finite surjective morphism
$$\pi: X \to \Aff^1 \times S$$
such that
$p = pr_S \circ \pi$.

Now let
$U:= \text{Spec}(\mathcal O_{X, \{x_1,x_2,\dots,x_n\}})$,
let
$can: U \hra X$
be the canonical inclusion of schemes,
$p_U= p \circ can: U \to S$,
$\mathcal X: = U \times_{S} X$.
Let
$q_U: \mathcal X \to U$
be the projection on $U$,
$q_X: \mathcal X \to X$
be the projection on $X$,
$\Delta: U \to \mathcal X$
be the diagonal morphism.
Summarizing we get a diagram of the form
\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}
such that
\begin{equation}
\label{DeltaQx}
q_X \circ \Delta = \text{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}.
In fact, $q_U$ is a base change via $p_U$ of the morphism $p$ and $p$ has the mentioned 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
$$\Pi: \mathcal X \to \Aff^1 \times U $$
such that}
$q_U = pr_U \circ \Pi$. Set
$\textit{f}:= q_X^{-1}(\text{f})$.
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
$k[X]/f \cdot k[X]$
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 $S$.

{\bf Clearly, the data
\begin{equation}
\label{FirstNiceTriple}
(q_U: \mathcal X \to U, f, \Delta )
\end{equation}
form a nice triple}.
Set $G_{\mathcal X}= q^*_X(G_X)$ and
$G_U= \Delta^*(G_{\mathcal X})=can^*(G_X)$.
Let
$G_{\text{const}}$
be the pull-back of $G_U$ to
$\mathcal X$.
Set $C_{\mathcal X}=q^*_X(C_X)$
and
$C_U:= \Delta^*(C_{\mathcal X})=can^*(C_X)$.
Let
$C_{\text{const}}$
be the pull-back of $C_U$ to
$\mathcal X$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
With these two Propositions in hand we may form a very nice diagram
(similar to a famous Quillen's and Voevodsy's ones).
Namely,
fix a smooth irreducible affine
$k$-scheme $X$, a finite family of points
$x_1,x_2, \dots , x_n$
on $X$,
a non-zero function
$f \in k[X]$.
% and a closed subset $Z$ of $X$ given by the equation $f=0$.
{\bf We may and will assume that the set
$\{x_1,x_2, \dots , x_n \}$} is contained in the vanishing locus of the function $f$.
Replacing $k$ with its algebraic closure in $k[X]$ we may assume
that $X$ is geometrically irreducible $k$-variety.
By Proposition
\ref{ArtinsNeighbor}
there exist a Zarisky 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 subscheme of the projective space
$\Pro^{dim X -1}$
such that
$p|_{\{f=0\} \bigcap X^0} : \{f=0\} \bigcap X^0 \to S$
is finite surjective. For each $i=1,2, \dots , n$ let $s_i=p(x_i) \in S$.
Shrinking $S$ we may assume that $S$ {\bf is affine} and still contains the family
$\{ s_1,s_2, \dots ,s_n \}$. Clearly in that case
$p^{-1}(S)$
contains the family
$\{ x_1,x_2, \dots ,x_n \}$.
Replace $X^0$ with
$p^{-1}(S)$
and write $X$ for $p^{-1}(S)$.
Replace $f$ with its restriction to that new $X$ and still write $f$ for that restriction.
%Respectively still write $Z$ for the vanish locus of $f$ on that new $X$.

So, we get an elementary fibration
$p: X \to S$
such that
$\{ x_1, \dots ,x_n \} \subset \{f=0\} \subset X$
and
$S$
is an open affine in the projective space
$\Pro^{dim X - 1}$
and the restriction of
$p|_{\{f=0\}}: \{f=0\} \to S$
to the vanishing of $f$ is a finite surjective morphism.
The latter means that the
$k[S]$-module $k[X]/(f)$ is finite.

The scheme $S$ is affine and regular since it is an open affine subscheme of the projective space
$\Pro^{dim X -1}$. By Proposition
\ref{CartesianDiagram}
there exists a finite surjective morphism
$$\pi: X \to \Aff^1 \times S$$
such that
$p = pr_S \circ \pi$.

Now let
$U:= \text{Spec}(\mathcal O_{X, \{x_1,x_2,\dots,x_n\}})$,
let
$can: U \hra X$
be the canonical inclusion of schemes,
$p_U= p \circ can: U \to S$,
$\mathcal X: = U \times_{S} X$.
Let
$q_U: \mathcal X \to U$
be the projection on $U$,
$q_X: \mathcal X \to X$
be the projection on $X$,
$\Delta: U \to \mathcal X$
be the diagonal morphism.
Summarizing we get a diagram of the form
\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}
such that
\begin{equation}
\label{DeltaQx}
q_X \circ \Delta = \text{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}.
In fact, $q_U$ is a base change via $p_U$ of the morphism $p$ and $p$ has the mentioned 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
$$\Pi: \mathcal X \to \Aff^1 \times U $$
such that}
$q_U = pr_U \circ \Pi$. Set
$\textit{f}:= q_X^{-1}(\text{f})$.
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
$k[X]/f \cdot k[X]$
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 $S$.

The data
\begin{equation}
\label{FirstNiceTriple}
(q_U: \mathcal X \to U, f, \Delta )
\end{equation}
{\bf is an example of a nice triple} which are studied in the following Section.

\begin{rem}
\label{DeltaIsWellDefined}
Note that 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
$\{f=0\}$.
\end{rem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $R$ be a semi-local ring and $G$ be a reductive group scheme over $R$.
Recall a "isotropy condition " for $G$.

\begin{itemize}
\label{CondIsotropy}
\item[({\bf I})]
{\it Each $R$-simple component of the derived group $G_{der}$ of the group $G$ is isotropic over} $R$.
\end{itemize}
The hypotheses {\bf(I)} means more precicely the following : consider the derived group $G_{der}$
of the group $G$ and the simply-connected cover $G^{sc}_{der}$ of $G_{red}$; that group $G^{sc}_{der}$
is a product over $R$ (in a unique way) of $R$-simple groups, which are required to be isotropic.

The following results are consequences of Purity Theorems mentioned above,
the main results of
\cite[Thm. 1.1, Thm. 1.2]{PSV}
ad the main result of
\cite{C-T-S}
\begin{thm}
\label{Gr_SerreReductiveGeometric}
Let $X$ be a smooth irreducible variety over an infinite field $k$.
Regard $X$ as a scheme and take a family $x_1, x_2, \cdots, x_n$ be of points of $X$.
Let
$\mathcal O_{X;x_1, \dots, x_n}$
be the semi-local ring of that family of pints.
Let $K$ be the field of fractions of $R$.
Let $G$ be a reductive group scheme over $R$ satisfying the hypotheses {\bf(I)}. 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}

\begin{thm}
\label{Gr_SerreReductiveGeneral}
Let $R$ be a semi-local regular domain containing an {\bf infinite perfect} field
and let $K$ be the field of fractions of $R$.
Let $G$ be a reductive group scheme over $R$ satisfying the hypotheses {\bf(I)}.
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}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
By  assumption there exists a smooth $d$-dimensional
$k$-algebra $S=k[t_1,t_2, \dots, t_n]$ and a prime ideal
$\mathfrak p$ in $R$ such that $A=S_{\mathfrak p}$.
We first reduce the proof to the case in which $\mathfrak p$
is maximal.
To do this,  choose a maximal ideal ${m}$
containing ${p}$. Since $k$ is of characteristic zero it is infinite.
By a standard
general position argument we can find $d$ algebraically
independent elements $X_1,\dots,X_d$ such that $S$ is finite over
$k[X_1,\dots,X_d]$ and \'etale at ${m}$. After a linear
change of coordinates we may assume that $S/{p}$ is finite
over $B=k[X_1,\dots,X_m]$, where $m$ is the dimension of
$S/{p}$. Clearly $S$ is smooth over $B$ at ${m}$ and thus, for
some $h\in S\setminus {m}$, the localization $S_h$ is smooth
over $B$. Let $\cal S$ be the set of nonzero elements of $B$,
$k'={\cal S}^{-1}B$ the field of fractions of $B$ and
$S'={\cal S}^{-1}S_h$. The
prime ideal ${p}'={\cal S}^{-1}{p}_h$ is maximal in $S'$, the
$k'$-algebra $S'$ is smooth and $S=S'_{{p}'}$.

From now on and till the end of this proof
we assume
that $A={\mathcal O}_{X,x}$ is the local ring of a closed point $x$ of
a smooth $d$-dimensional irreducible affine variety $X$ over $k$. Replacing
$k$ by its algebraic closure in $k[X]$ we may assume furthermore that
this $X$ is a $k$-smooth {\bf geometrically irreducible affine} variety over $k$.
