# A short exact sequence

Ivan Panin111The author acknowledges support of the RFBR grant No.
19-01-00513.

###### Abstract

Let $R$ be a regular semi-local integral domain containing a field and $K$ be
its fraction field. Let $\mu:\mathbf{G}\to\mathbf{T}$ be an $R$-group schemes
morphism between reductive $R$-group schemes, which is smooth as a scheme
morphism. Suppose that $T$ is an $R$-torus. Then the map
$\mathbf{T}(R)/\mu(\mathbf{G}(R))\to\mathbf{T}(K)/\mu(\mathbf{G}(K))$ is
injective and certain purity theorem is true. These and other results are
derived from an extended form of Grothendieck–Serre conjecture proven in the
present paper for rings $R$ as above.

##  1 Main results

Let $R$ be a commutative unital ring. Recall that an $R$-group scheme
$\mathbf{G}$ is called reductive, (respectively, semi-simple or simple), if it
is affine and smooth as an $R$-scheme and if, moreover, for each algebraically
closed field $\Omega$ and for each ring homomorphism $R\to\Omega$ the scalar
extension $\mathbf{G}_{\Omega}$ is a connected reductive (respectively, semi-
simple or simple) algebraic group over $\Omega$. 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 reductive $R$-group scheme
coincides with [SGA3, Exp. XIX, Definition 2.7]. This notion of a simple
$R$-group scheme coincides with the notion of a simple semi-simple $R$-group
scheme from Demazure and Grothendieck [SGA3, Exp. XIX, Definition 2.7 and Exp.
XXIV, 5.3]. Here is our first main result based on results of [Pan2] and
[Pan3] and significantly extending the corresponding results of [Pan2] and
[Pan3].

######  Theorem 1.1.

Let $R$ be a regular semi-local integral domain containing a field. Let $K$ be
the fraction field of $R$. Let $\mu:\mathbf{G}\to\mathbf{T}$ be an $R$-group
scheme morphism between reductive $R$-group schemes, which is smooth as a
scheme morphism. Suppose $T$ is an $R$-torus. Then the map
$\mathbf{T}(R)/\mu(\mathbf{G}(R))\to\mathbf{T}(K)/\mu(\mathbf{G}(K))$ is
injective and the sequence

$\\{1\\}\to\mathbf{T}(R)/\mu(\mathbf{G}(R))\to\mathbf{T}(K)/\mu(\mathbf{G}(K))\xrightarrow{\sum
r_{\mathfrak{p}}}\bigoplus_{\mathfrak{p}}\mathbf{T}(K)/[\mathbf{T}(R_{\mathfrak{p}})\cdot\mu(\mathbf{G}(K))]\to\\{1\\}$
(1)

is exact, where $\mathfrak{p}$ runs over all height one prime ideals of $R$
and each $r_{\mathfrak{p}}$ is the natural map (the projection to the factor
group).

Let us comment on the first assertion of the theorem. Let $\mathbf{H}$ be the
kernel of $\mu$. It turns out that $\mathbf{H}$ is a quasi-reductive $R$-group
scheme (see Definition 1.3). There is a sequence of group sheaves
$1\to\mathbf{H}\to\mathbf{G}\to\mathbf{T}\to 1$, which is exact in the étale
topology on $SpecR$. Theorem 1.4 yields now the injectivity of the map
$\mathbf{T}(R)/\mu(\mathbf{G}(R))\to\mathbf{T}(K)/\mu(\mathbf{G}(K))$.

######  Theorem 1.2.

Let $R$ be a regular semi-local integral domain containing a field. Let $K$ be
the fraction field of $R$. Let $\mathbf{G}_{1}$ and $\mathbf{G}_{2}$ be two
semi-simple $R$-group schemes. Suppose the generic fibres $\mathbf{G}_{1,K}$
and $\mathbf{G}_{2,K}$ are isomorphic as algebraic $K$-groups. Then the
$R$-group schemes $\mathbf{G}_{1}$ and $\mathbf{G}_{2}$ are isomorphic.

To prove Theorem 1.2 we need to work with automorphism group scheme of a semi-
simple $R$-group scheme. The latter group scheme is not geometrically
connected in general. So, Theorem 1.2 can not be derived from [FP] and [Pan3].

We state right below a theorem, which asserts that an extended version of
Grothendieck–Serre conjecture holds for rings $R$ as above. This latter
theorem is proved in this paper. Theorem 1.2 and the first assertion of
Theorem 1 are derived from it. To state the mentioned theorem it is convenient
to give the following.

######  Definition 1.3 (quasi-reductive).

Assume that $S$ is a Noetherian commutative ring. An $S$-group scheme
$\mathbf{H}$ is called quasi-reductive if there is a finite étale $S$-group
scheme $\mathbf{C}$ and a smooth $S$-group scheme morphism
$\lambda:\mathbf{H}\to\mathbf{C}$ such that its kernel is a reductive
$S$-group scheme and $\lambda$ is surjecive locally in the étale topology on
$S$.

Clearly, reductive $S$-group schemes are quasi-reductive. Quasi-reductive
$S$-group schemes are affine and smooth as $S$-schemes. There are two types of
quasi-reductive $S$-group schemes, which we are focusing on in the present
paper. The first one is the automorphism group scheme of a semi-simple
$S$-group scheme. The second one is obtained as follows: take a reductive
$S$-group scheme $\mathbf{G}$, an $S$-torus $\mathbf{T}$ and a smooth
$S$-group morphism $\mu:\mathbf{G}\to\mathbf{T}$. Then one can check that the
kernel $\mathbf{H}$ of $\mu$ is quasi-reductive. It is an extension of a
finite étale $S$-group scheme $\mathbf{C}$ of multiplicative type via a
reductive $S$-group scheme $\mathbf{G}_{0}$.

Assume that $U$ is a regular scheme, $\mathbf{H}$ is a quasi-reductive
$U$-group scheme. Recall that a $U$-scheme $\mathcal{H}$ with an action of
$\mathbf{H}$ is called _a principal $\mathbf{H}$-bundle over $U$_, if
$\mathcal{H}$ is faithfully flat and quasi-compact over $U$ and the action is
simple transitive, that is, the natural morphism
$\mathbf{H}\times_{U}\mathcal{H}\to\mathcal{H}\times_{U}\mathcal{H}$ is an
isomorphism, see [Gr4, Section 6]. Since $\mathbf{H}$ is $S$-smooth, such a
bundle is trivial locally in étale topology but in general not in Zariski
topology. Grothendieck and Serre conjectured that for a reductive $U$-group
scheme $\mathbf{H}$ a principal $\mathbf{H}$-bundle $\mathcal{H}$ over $U$ is
trivial locally in Zariski topology, if it is trivial generically. A survey
paper on the topic is [P2].

The conjecture is true, if $\Gamma(U,\mathcal{O}_{U})$ contains a field (see
[FP] and [Pan3]). It is proved in [Ni2] that the conjecture is true in general
for discrete valuation rings. This result is extended in [PSt] to the case of
semi-local Dedekind integral domains assuming that $\mathbf{G}$ is simple
simply connected and isotropic in a certain precise sense. In [NG] results of
[Ni2] and [PSt] are extended further. It is proved there that the conjecture
is true in general for the case of semi-local Dedekind integral domains. The
following result is a further extension of the main theorem of [Pan3].

######  Theorem 1.4.

Let $R$ be a regular semi-local integral domain containing a field. Let $K$ be
the fraction field of $R$. Let $\mathbf{H}$ be a quasi-reductive group scheme
over $R$. Then the map

$\textrm{H}^{1}_{\text{\'{e}t}}(R,\mathbf{H})\to\textrm{H}^{1}_{\text{\'{e}t}}(K,\mathbf{H}),$

induced by the inclusion of $R$ into $K$ has a trivial kernel. In other words,
under the above assumptions on $R$ and $\mathbf{H}$, each principal
$\mathbf{H}$-bundle over $R$ having a $K$-rational point is trivial.

######  Corollary 1.5.

Under the hypothesis of Theorem 1.4, the map

$\textrm{H}^{1}_{\text{\'{e}t}}(R,\mathbf{H})\to\textrm{H}^{1}_{\text{\'{e}t}}(K,\mathbf{H}),$

induced by the inclusion of $R$ into $K$, is injective. Equivalently, if
$\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are two principal $\mathbf{H}$-bundles
isomorphic over $\text{\rm Spec}K$, then they are isomorphic.

###### Proof.

Let $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ be two principal
$\mathbf{H}$-bundles isomorphic over $\operatorname{Spec}K$. Let
$\operatorname{Iso}(\mathcal{H}_{1},\mathcal{H}_{2})$ be the scheme of
isomorphisms of principal $\mathbf{H}$-bundles. This scheme is a principal
$\text{\rm Aut}\mathcal{H}_{1}$-bundle. By Theorem 1.4 it is trivial, and we
see that $\mathcal{H}_{1}\cong\mathcal{H}_{2}$. ∎

Theorems 1.4 and 1.2 are proved in Section 2. Theorem 1.1 is proved in Section
8.

##  2 Proof of Theorems 1.4 and 1.2

We begin with the following general

######  Lemma 2.1.

Let $X$ be a regular semi-local integral domain. Let $\pi:X^{\prime}\to X$ be
a finite morphism. Let $\eta\in X$ be the generic point of $X$. Then sections
of $\pi$ over $X$ are in the bijection with sections of $\pi$ over $\eta$.

###### Proof.

It suffices to check that each section $s:\eta\to X^{\prime}$ of $\pi$ can be
extended to a section of $\pi$ over $X$.

Decompose $\pi$ as a composition
$X^{\prime}\xrightarrow{i}\mathbf{A}^{n}_{X}\xrightarrow{p}X$, where $p$ is
the projection and $i$ is a closed embedding. Let $s:\eta\to X^{\prime}$ be a
section of $\pi$. Since $X$ is a regular semi-local and $\pi$ is projective
there is a closed codimension two subset $Z$ in $X$ and a section
$\varphi:X-Z\to X^{\prime}$ of $\pi$ that extends $s$. Since
$\Gamma(X,\mathcal{O}_{X})=\Gamma(X-Z,\mathcal{O}_{X})$ the section $\varphi$
is extended to a section $\tilde{\varphi}$ of $\pi$. The lemma is proved. ∎

######  Corollary 2.2.

Let $X$, $\eta\in X$ be as in the previous lemma and $\mathbf{E}$ be a finite
étale group $X$-scheme. Then the $\eta$-points of $\mathbf{E}$ coincides with
the $X$-points of $\mathbf{E}$.

######  Corollary 2.3.

Under the hypothesis of Corollary 2.2 the kernel of the pointed set map
$H^{1}_{\text{\'{e}t}}(X,\mathbf{E})\to
H^{1}_{\text{\'{e}t}}(\eta,\mathbf{E})$ is trivial.

###### Proof.

Let $\cal E$ be a principal $\mathbf{E}$-bundle over $X$. The standard descent
arguments shows that the $X$-scheme $\cal E$ is finite and étale. Thus, ${\cal
E}(X)=\cal E(\eta)$. This proves the corollary. ∎

###### Proof of Theorem 1.4.

Since $\mathbf{H}$ is quasi-reductive $R$-group scheme, there is a finite
étale $R$-group scheme $\mathbf{C}$ and a smooth $R$-group scheme morphism
$\lambda:\mathbf{H}\to\mathbf{C}$ such that its kernel $\mathbf{G}$ is a
reductive $R$-group scheme and $\lambda$ is surjecive locally in the étale
topology on $S$. The sequence of the étale sheaves
$1\to\mathbf{G}\to\mathbf{H}\to\mathbf{C}\to 1$ is exact. Thus, it induces a
commutative diagram of pointed set maps with exact rows

---
$\textstyle{\mathbf{C}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\alpha}$$\textstyle{{H_{\text{{\'{e}t}}}^{1}(R,\mathbf{G})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{{H_{\text{{\'{e}t}}}^{1}(R,\mathbf{H})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{H_{\text{{\'{e}t}}}^{1}(R,\mathbf{C})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\mathbf{C}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{{H_{\text{{\'{e}t}}}^{1}(K,\mathbf{G})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H_{\text{{\'{e}t}}}^{1}(K,\mathbf{H})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H_{\text{{\'{e}t}}}^{1}(K,\mathbf{C})}}$

The map $\alpha$ is bijective by Corollary 2.2, the map $\delta$ has the
trivial kernel by Corollary 2.3, the map $\beta$ is injective by [Pan3,
Corollary 1.2] Now a simple diagram chase shows that $ker(\gamma)=\ast$. This
proves the theorem. ∎

######  Remark 2.4.

The statement of [Ch-G-P/O, Lemma] and its proof are non-acurate both, since
the authors do not assume the injectivity of the map
$H_{\text{{\'{e}t}}}^{1}(R,\mathbf{G}^{0})\to
H_{\text{{\'{e}t}}}^{1}(K,\mathbf{G}^{0}_{R})$.

###### Proof of Theorem 1.2.

The $R$-group scheme $\underline{\text{Aut}}:=\underline{\text{Aut}}_{R-gr-
sch}(\mathbf{G}_{1})$ is quasi-reductive by [D-G]. The $R$-scheme
$\underline{\text{Iso}}:=\underline{\text{Iso}}_{R-gr-sch}({\bf G_{1}},{\bf
G_{2}})$ is a principal $\underline{\text{Aut}}$-bundle. An isomorphism
$\varphi:\mathbf{G}_{1,K}\to\mathbf{G}_{2,K}$ of algebraic $K$-groups gives a
section of $\underline{\text{Iso}}$ over $K$. So, $\underline{\text{Iso}}_{K}$
is a trivial principal $\underline{\text{Aut}}_{K}$-bundle. Hence
$\underline{\text{Iso}}$ is a trivial principal
$\underline{\text{Aut}}$-bundle by Theorem 1.4. Thus, it has a section over
$R$. So, there is an $R$-group scheme isomorphism
$\mathbf{G}_{1}\cong\mathbf{G}_{2}$. ∎

##  3 Proof of the first assertion of Theorem 1.1

######  Lemma 3.1.

Let $X$ be a regular irreducible affine scheme. Let $\mathbf{G}$ be a
reductive $X$-group scheme and $\mathbf{T}$ be an $X$-torus. Let
$\mu:\mathbf{G}\to\mathbf{T}$ be an $X$-group schemes morphism, which is
smooth as a scheme morphism. Then the kernel of $\mu$ is a quasi-reductive
$X$-group scheme.

###### Proof.

Consider the coradical $Corad(\mathbf{G})$ of $\mathbf{G}$ together with the
canonical $X$-group morphism $\alpha:\mathbf{G}\to Corad(\mathbf{G})$. By the
universal property of the $X$-group morphism $can$ there is a unique $X$-group
morphism $\bar{\mu}:Corad(\mathbf{G})\to T$ such that
$\mu=\bar{\mu}\circ\alpha$. Since $\mu$ is surjective locally for the étale
topology, hence so is $\bar{\mu}$. Let $\text{ker}(\bar{\mu})$ be the kernel
of $\bar{\mu}$ and let $\mathbf{H}:=\alpha^{-1}(\text{ker}(\bar{\mu}))$ be the
scheme theoretic pre-image of $\text{ker}(\bar{\mu})$. Clearly, $\mathbf{H}$
is a closed $X$-subgroup scheme of $\mathbf{G}$, which is the kernel of $\mu$.
We must check that $\mathbf{H}$ is a quasi-reductive.

The $X$-group scheme $\text{ker}(\bar{\mu})$ is of multiplicative type. Hence
there is a finite $X$-group scheme $\mathbf{M}$ of multiplicative type and a
faithfully flat $X$-group scheme morphism
$can:\text{ker}(\bar{\mu})\to\mathbf{M}$, which has the following property:
for any finite $X$-group scheme $\mathbf{M}^{\prime}$ of multiplicative type
and an $X$-group morphism
$\varphi:\text{ker}(\bar{\mu})\to\mathbf{M}^{\prime}$ there is a unique
$X$-group morphism $\psi:\mathbf{M}\to\mathbf{M}^{\prime}$ with $\psi\circ
can=\varphi$. It is known that the kernel of $can$ is an $X$-torus. Call it
$\mathbf{T}^{0}$. Since $\mu$ is smooth, hence so is $\bar{\mu}$. Thus, the
$X$-group scheme $\text{ker}(\bar{\mu})$ is an $X$-smooth scheme. This yields
that $M$ is étale over $X$.

Let $\beta=\alpha|_{\mathbf{H}}:\mathbf{H}\to\text{ker}(\bar{\mu})$ and let
$\mathbf{G}^{0}:=\beta^{-1}(\mathbf{T}^{0})$ be the scheme theoretic pre-image
of $\mathbf{T}^{0}$. Clearly, $\mathbf{G}^{0}$ is a closed $X$-subgroup scheme
of $\mathbf{H}$, which is the kernel of the morphism
$can\circ\beta:\mathbf{H}\to\mathbf{M}$. Let
$\gamma=\beta|_{\mathbf{G}^{0}}:\mathbf{G}^{0}\to\mathbf{T}^{0}$.

The $X$-group scheme $\mathbf{M}$ is finite and étale. The morphism $can$ is
smooth. The morphism $\beta$ is smooth as a base change of the smooth morphism
$\alpha$. Thus, $\lambda:=can\circ\beta$ is smooth. It is also surjective
locally in the étale topology on $X$, because $can$ and $\beta$ have this
property. By the construction $\mathbf{G}^{0}=\text{ker}(\lambda)$. So, to
prove that $\mathbf{H}$ is quasi-reductive it remains to check the reductivity
of $\mathbf{G}^{0}$.

The $X$-group scheme $\mathbf{G}^{0}$ is affine as a closed $X$-subgroup
scheme of the reductive $X$-group scheme $\mathbf{G}$. Prove now that
$\mathbf{G}^{0}$ is smooth over $X$. Indeed, the morphism $\gamma$ is smooth
as a base change of the smooth morphism $\alpha$. The $X$-scheme
$\mathbf{T}^{0}$ is smooth, since it is an $X$-torus. Thus, the $X$-scheme
$\mathbf{G}^{0}$ is smooth.

Write $X$ as $SpecS$ for a regular integral domain $S$. It remains to verify
that for each algebraically closed field $\Omega$ and for each ring
homomorphism $S\to\Omega$ the scalar extension $\mathbf{G}^{0}_{\Omega}$ is a
connected reductive algebraic group over $\Omega$. Firstly, recall that
$\text{ker}(\alpha)$ is a semi-simple $S$-group scheme. It is the $S$-group
scheme $\mathbf{G}^{ss}$ under the notation of [D-G]. Clearly,
$\text{ker}(\gamma)=\text{ker}(\alpha)$. Thus,
$\text{ker}(\gamma)=\mathbf{G}^{ss}$ is a semi-simple $S$-group scheme. Since
the morphism $\gamma$ is smooth for each algebraically closed field $\Omega$
and for each ring homomorphism $S\to\Omega$ we have an exact sequence of
smooth algebraic groups over $\Omega$

$1\to\mathbf{G}^{ss}_{\Omega}\to\mathbf{G}^{0}_{\Omega}\to\mathbf{T}^{0}_{\Omega}\to
1.$

The groups $\mathbf{T}^{0}_{\Omega}$, $\mathbf{G}^{ss}_{\Omega}$ are
connected. Hence the group $\mathbf{G}^{0}_{\Omega}$ is connected too. We know
already that it is affine.

Finally, check that its unipotent radical $\mathbf{U}$ of
$\mathbf{G}^{0}_{\Omega}$ is trivial. Since there is no non-trivial
$\Omega$-group morphisms $\mathbf{U}\to\mathbf{T}^{0}_{\Omega}$, we conclude
that $\mathbf{U}\subset\mathbf{G}^{ss}_{\Omega}$. Since
$\mathbf{G}^{ss}_{\Omega}$ is semi-simple one has $\mathbf{U}=\\{1\\}$. This
completes the proof of the reductivity of the $R$-group scheme
$\mathbf{G}^{0}$. Thus, the $R$-group scheme $\mathbf{H}$ is quasi-reductive.
This proves the lemma. ∎

###### Proof of the first assertion of Theorem 1.1.

Let $\mathbf{H}$ be the kernel of $\mu$. Since $\mu$ is smooth, the group
scheme sequence

$1\to\mathbf{H}\to\mathbf{G}\to\mathbf{T}\to 1$

gives rise to an short exact sequence of group sheaves in the étale topology.
In turn that sequence of sheaves induces a long exact sequence of pointed
sets. So, the boundary map
$\partial:\mathbf{T}(R)\to\textrm{H}^{1}_{\text{\'{e}t}}(R,\mathbf{H})$ fits
in a commutative diagram

$\begin{CD}\mathbf{T}(R)/\mu(\mathbf{G}(R))@>{}>{}>\textrm{H}^{1}_{\text{\'{e}t}}(R,\mathbf{H})\\\
@V{}V{}V@V{}V{}V\\\
\mathbf{T}(K)/\mu(\mathbf{G}(K))@>{}>{}>\textrm{H}^{1}_{\text{\'{e}t}}(K,\mathbf{H}).\\\
\end{CD}$

Clearly, the horizontal arrows have trivial kernels. The right vertical arrow
has trivial kernel by Lemma 3.1 and Theorem 1.4. Thus the left vertical arrow
has trivial kernel too. Since it is a group homomorphism, it is injective. ∎

##  4 Norms

We recall here a construction from [P1]. 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 $L$. Let $C$ be a $k$-smooth
commutative algebraic group scheme defined over $k$. One can define a norm map

${\mathcal{N}}_{L/K}:C(L)\to C(K)$

by ${\mathcal{N}}_{L/K}(\alpha)=\prod_{i}C(\sigma_{i})(\alpha)\in
C(K^{sep})^{{\mathcal{G}}(K)}=C(K)\ .$ In [P1] following Suslin and Voevodsky
[SV, Sect.6] we generalize this construction to finite flat ring extensions.
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 $Y$.

Let $k$ be a field. Let $\mathcal{O}$ be the semi-local ring of finitely many
closed points on a smooth affine irreducible $k$-variety $X$. Let $C$ be an
affine smooth commutative $\mathcal{O}$-group scheme, Let $p:X\to Y$ be a
finite flat morphism of affine $\mathcal{O}$-schemes and $f:X\to C$ any
$\mathcal{O}$-morphism. In [P1] the norm $N_{X/Y}(f)$ of $f$ is defined as the
composite map

$Y\xrightarrow{N_{X/Y}}S^{d}(X/Y)\to
S^{d}_{\mathcal{O}}(X)\xrightarrow{S^{d}_{\mathcal{O}}(f)}S^{d}_{\mathcal{O}}(C)\xrightarrow{\times}C$
(2)

Here we write $"\times"$ for the group law on $C$. The norm maps $N_{X/Y}$
satisfy the following conditions

  * (i’)

Base change: for any map $f:Y^{\prime}\to Y$ of affine schemes, putting
$X^{\prime}=X\times_{Y}Y^{\prime}$ we have a commutative diagram

$\begin{CD}C(X)@>{(id\times f)^{*}}>{}>C(X^{\prime})\\\
@V{N_{X/Y}}V{}V@V{}V{N_{X^{\prime}/Y^{\prime}}}V\\\
C(Y)@>{f^{*}}>{}>C(Y^{\prime})\end{CD}$

  * (ii’)

multiplicativity: if $X=X_{1}\amalg X_{2}$ then the diagram commutes

$\begin{CD}C(X)@>{(id\times f)^{*}}>{}>C(X_{1})\times C(X_{2})\\\
@V{N_{X/Y}}V{}V@V{}V{N_{X_{1}/Y}N_{X_{2}/Y}}V\\\ C(Y)@>{id}>{}>C(Y)\end{CD}$

  * (iii’)

normalization: if $X=Y$ and the map $X\to Y$ is the identity then
$N_{X/Y}=id_{C(X)}$.

##  5 Unramified elements

Let $k$ be a field, $\mathcal{O}$ be the semi-local ring of finitely many
closed points on a $k$-smooth irreducible affine $k$-variety $X$. Let $K$ be
the fraction field of $\mathcal{O}$, that is $K=k(X)$. Let

$\mu:\mathbf{G}\to\mathbf{T}$

be a smooth $\mathcal{O}$-morphism of reductive $\mathcal{O}$-group schemes,
with a torus $\mathbf{T}$. We work in this section with the category of
commutative Noetherian $\mathcal{O}$-algebras. For a commutative
$\mathcal{O}$-algebra $S$ set

${\cal F}(S)=\mathbf{T}(S)/\mu(\mathbf{G}(S)).$ (3)

For an element $\alpha\in\mathbf{T}(S)$ we will write $\bar{\alpha}$ for its
image in ${\cal F}(S)$. In this section we will write ${\cal F}$ for the
functor  (3). The following result is a particular case of the first part of
Theorem 1.1 (those part is proved in Section 3).

######  Theorem 5.1.

Let $S$ be an $\mathcal{O}$-algebra which is discrete valuation ring with
fraction field $L$. Then the map ${\cal F}(S)\to{\cal F}(L)$ is injective.

######  Lemma 5.2.

Let $\mu:\mathbf{G}\to\mathbf{T}$ be the above morphism of our reductive group
schemes. Then for an $\mathcal{O}$-algebra $L$, where $L$ is a field, the
boundary map
$\partial:\mathbf{T}(L)/{\mu(\mathbf{G}(L))}\to\textrm{H}^{1}_{\text{\'{e}t}}(L,\mathbf{H})$
is injective.

###### Proof.

Repeat literally the proof of [Pan2, Lemma 6.2]. ∎

Let $k$, $\mathcal{O}$ and $K$ be as above in this Section. Let $\mathcal{K}$
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 $x$. Clearly,
$\mathcal{O}\subset A_{x}$. Let $\hat{A}_{x}$ and $\hat{\mathcal{K}}_{x}$ be
the completions of $A_{x}$ and $\mathcal{K}$ with respect to $x$. Let
$i:\mathcal{K}\hookrightarrow\hat{\mathcal{K}}_{x}$ be the inclusion. By
Theorem 5.1 the map ${\cal F}(\hat{A}_{x})\to{\cal F}(\hat{\mathcal{K}}_{x})$
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 Theorem 5.1. So both
groups ${\cal F}(A_{x})$ 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 unramified at $x$.

######  Lemma 5.3.

${\cal F}(A_{x})={\cal F}_{x}(\mathcal{K})$.

###### Proof.

Repeat literally the proof of [Pan2, Lemma 6.3]. ∎

Let $S$ be an $\mathcal{O}$-algebra which is an integral domain and suppose
$S$ is a regular ring. Let $L$ be its fraction field. For each height $1$
prime ideals $\mathfrak{p}$ in $S$ the group map ${\cal
F}(S_{\mathfrak{p}})\to{\cal F}(L)$ is injective by the first part of Theorem
1.1. Define the subgroup of $S$-unramified elements of $\mathcal{F}(L)$ as

$\mathcal{F}_{nr,S}(L)=\bigcap_{\mathfrak{p}\in
Spec(S)^{(1)}}\mathcal{F}(S_{\mathfrak{p}})\subseteq{\cal F}(L),$ (4)

where $Spec(S)^{(1)}$ is the set of height $1$ prime ideals in $S$. Obviously
the image of $\mathcal{F}(S)$ in $\mathcal{F}(L)$ is contained in
$\mathcal{F}_{nr,S}(L)$. For each height one prime $\mathfrak{p}$ in $S$ we
construct specialization maps $s_{\mathfrak{p}}:{\cal F}_{nr,S}(L)\to{\cal
F}(l(\mathfrak{p}))$, where $L$ is the field of fractions of $S$ and
$l(\mathfrak{p})$ is the residue field of $S$ at the prime $\mathfrak{p}$.

######  Definition 5.4.

Let
$Ev_{\mathfrak{p}}:\mathbf{T}(S_{\mathfrak{p}})\to\mathbf{T}(l(\mathfrak{p}))$
and $ev_{\mathfrak{p}}:{\cal F}(S_{\mathfrak{p}})\to{\cal F}(K(\mathfrak{p}))$
be the maps induced by the canonical $S$-algebra homomorphism
$S_{\mathfrak{p}}\to l(\mathfrak{p})$. Define a homomorphism
$s_{\mathfrak{p}}:{\cal F}_{nr,S}(L)\to{\cal F}(l(\mathfrak{p}))$ by
$s_{\mathfrak{p}}(\alpha)=ev_{\mathfrak{p}}(\tilde{\alpha})$, where
$\tilde{\alpha}$ is a lift of $\alpha$ to ${\cal F}(S_{\mathfrak{p}})$.
Theorem 5.1 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 $\alpha\in\mathbf{T}(S_{\mathfrak{p}})$ one has
$s_{\mathfrak{p}}(\bar{\alpha})=\overline{Ev_{\mathfrak{p}}(\alpha)}\in{\cal
F}(l(\mathfrak{p}))$.

Let $k$, $\mathcal{O}$ and $K$ be as above in this Section. The following two
results are proved using literally the same arguments as in the proof of
[Pan2, Thm. 6.5] and [Pan2, Cor. 6.6] respectively.

######  Theorem 5.5 (Homotopy invariance).

Let $K(t)$ be the rational function field in one variable over the field $K$.
Define ${\cal F}_{nr,K[t]}(K(t))$ by the formulae (4). Then one has an
equality

${\cal F}(K)={\cal F}_{nr,K[t]}(K(t)).$

######  Corollary 5.6.

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 $s_{0}=s_{1}$.

######  Lemma 5.7.

Let $B\subset A$ be a finite extension of $K$-smooth algebras, which are
integral domains and each has dimension one. Let $0\neq f\in A$ and let $h\in
B\cap fA$ be such that the induced map $B/hB\to A/fA$ is an isomorphism.
Suppose $hA=fA\cap J^{\prime\prime}$ for an ideal $J^{\prime\prime}\subseteq
A$ co-prime to the principal ideal $fA$.

Let $E$ and $F$ be the field of fractions of $B$ and $A$ respectively. Let
$\alpha\in\mathbf{T}(A_{f})$ be such that $\bar{\alpha}\in{\cal F}(F)$ is
$A$-unramified. Then, for $\beta=N_{F/E}(\alpha)$, the class
$\bar{\beta}\in{\cal F}(E)$ is $B$-unramified.

###### Proof.

Repeat literally the proof of [Pan2, Lemma 6.7]. ∎

##  6 Few recollections

Let $X$ be an affine $k$-smooth irreducible $k$-variety, and let
$x_{1},x_{2},\dots,x_{n}$ be closed points in $X$. Let $\mathcal{O}$ be the
semi-local ring $\mathcal{O}_{X,\\{x_{1},x_{2},\dots,x_{n}\\}}$. Let
$U=Spec(\mathcal{O})$ and $can:U\hookrightarrow X$ be the canonical embedding.
Let $\mathbf{G}$ be a reductive $X$-group scheme and let
$\mathbf{G}_{U}=can^{*}(\mathbf{G})$ be the pull-back of $\mathbf{G}$ to $U$.
Let $\mathbf{T}$ be an $X$-torus and let $\mathbf{T}_{U}=can^{*}(\mathbf{T})$
be the pull-back of $\mathbf{G}$ to $U$. Let $\mu:\mathbf{G}\to\mathbf{T}$ be
an $X$-group scheme morphism which is smooth as an $X$-scheme morphism. Let
$\mu_{U}=can^{*}(\mu)$. The following result is [Pan2, Theorem 4.1].

######  Theorem 6.1.

Given a non-zero function $\emph{f}\in k[X]$ vanishing at each point $x_{i}$,
there is a diagram of the form

|
---|---
$\textstyle{\mathbf{A}^{1}\times
U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{\rm
pr}_{U}}$$\textstyle{\mathcal{X}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{q_{U}}$$\scriptstyle{q_{X}}$$\textstyle{X}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$can$\scriptstyle{\Delta^{\prime}}$
(5)

with an irreducible affine scheme $\mathcal{X}^{\prime}$, a smooth morphism
$q_{U}$, a finite surjective $U$-morphism $\sigma$ and an essentially smooth
morphism $q_{X}$, and a function $f^{\prime}\in q^{*}_{X}(\emph{f}\
)k[\mathcal{X}^{\prime}]$, which enjoys the following properties:

  * (a)

if $\mathcal{Z}^{\prime}$ is the closed subscheme of $\mathcal{X}^{\prime}$
defined by the ideal $(f^{\prime})$, then the morphism
$\sigma|_{\mathcal{Z}^{\prime}}:\mathcal{Z}^{\prime}\to\mathbf{A}^{1}\times U$
is a closed embedding and the morphism
$q_{U}|_{\mathcal{Z}^{\prime}}:\mathcal{Z}^{\prime}\to U$ is finite;

  * (a’)

$q_{U}\circ\Delta^{\prime}=id_{U}$ and $q_{X}\circ\Delta^{\prime}=can$ and
$\sigma\circ\Delta^{\prime}=i_{0}$,
where $i_{0}$ is the zero section of the projection $\text{\rm pr}_{U}$;

  * (b)

$\sigma$ is étale in a neighborhood of
$\mathcal{Z}^{\prime}\cup\Delta^{\prime}(U)$;

  * (c)

$\sigma^{-1}(\sigma(\mathcal{Z}^{\prime}))=\mathcal{Z}^{\prime}\coprod\mathcal{Z}^{\prime\prime}$
scheme theoretically for some closed subscheme $\mathcal{Z}^{\prime\prime}$
and $\mathcal{Z}^{\prime\prime}\cap\Delta^{\prime}(U)=\emptyset$;

  * (d)

$\mathcal{D}_{0}:=\sigma^{-1}(\\{0\\}\times
U)=\Delta^{\prime}(U)\coprod\mathcal{D}^{\prime}_{0}$ scheme theoretically for
some closed subscheme $\mathcal{D}^{\prime}_{0}$ and
$\mathcal{D}^{\prime}_{0}\cap\mathcal{Z}^{\prime}=\emptyset$;

  * (e)

for $\mathcal{D}_{1}:=\sigma^{-1}(\\{1\\}\times U)$ one has
$\mathcal{D}_{1}\cap\mathcal{Z}^{\prime}=\emptyset$.

  * (f)

there is a monic polynomial $h\in\mathcal{O}[t]$ with
$(h)=Ker[\mathcal{O}[t]\xrightarrow{\sigma^{*}}k[\mathcal{X}^{\prime}]\xrightarrow{-}k[\mathcal{X}^{\prime}]/(f^{\prime})]$,
where the map bar takes any $g\in k[\mathcal{X}^{\prime}]$ to ${\bar{g}}\in
k[\mathcal{X}^{\prime}]/(f^{\prime})$;

  * (g)

there are $\mathcal{X}^{\prime}$-group scheme isomorphisms
$\Phi:q^{*}_{U}(\mathbf{G}_{U})\to q^{*}_{X}(\mathbf{G})$,
$\Psi:q^{*}_{U}(\mathbf{T}_{U})\to q^{*}_{X}(\mathbf{T})$ with
$(\Delta^{\prime})^{*}(\Phi)=id_{\mathbf{G}_{U}}$,
$(\Delta^{\prime})^{*}(\Psi)=id_{\mathbf{T}_{U}}$ and
$q^{*}_{X}(\mu)\circ\Phi=\Psi\circ q^{*}_{U}(\mu_{U})$.

######  Remark 6.2.

The triple $(q_{U}:\mathcal{X}^{\prime}\to U,f^{\prime},\Delta^{\prime})$ is a
nice triple over $U$, since $\sigma$ is a finite surjective $U$-morphism. See
[PSV, Defn.3.1] for the definition of a nice triple.

The morphism $q_{X}$ is not equal to $can\circ q_{U}$, since $f^{\prime}\in
q^{*}_{X}(\textrm{f})k[\mathcal{X}^{\prime}]$ and the morphism
$q_{U}|_{\mathcal{Z}^{\prime}}:\mathcal{Z}^{\prime}=\\{f^{\prime}=0\\}\to U$
is finite.

To formulate a consequence of the theorem 6.1 (see Corollary 6.3), note that
using the items (b) and (c) of Theorem 6.1 one can find an element $g\in
I(\mathcal{Z}^{\prime\prime})$ such that
(1)
$(f^{\prime})+(g)=\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})$,
(2)
$Ker((\Delta^{\prime})^{*})+(g)=\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})$,
(3)
$\sigma_{g}=\sigma|_{\mathcal{X}^{\prime}_{g}}:\mathcal{X}^{\prime}_{g}\to\mathbf{A}^{1}_{U}$
is étale.

Here is the corollary. It is proved in [Pan0, Cor. 7.2].

######  Corollary 6.3.

The function $f^{\prime}$ from Theorem 6.1, the polynomial $h$ from the item
$(\textrm{f}\ )$ of that Theorem, the morphism
$\sigma:\mathcal{X}^{\prime}\to\mathbf{A}^{1}_{U}$ and the function
$g\in\Gamma(\mathcal{X},\mathcal{O}_{\mathcal{X}})$ defined just above enjoy
the following properties:

  * (i)

the morphism
$\sigma_{g}=\sigma|_{\mathcal{X}^{\prime}_{g}}:\mathcal{X}^{\prime}_{g}\to\mathbf{A}^{1}\times
U$ is étale,

  * (ii)

data
$(\mathcal{O}[t],\sigma^{*}_{g}:\mathcal{O}[t]\to\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})_{g},h)$
satisfies the hypotheses of [C-T/O, Prop.2.6], i.e.
$\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})_{g}$ is a
finitely generated $\mathcal{O}[t]$-algebra, the element $(\sigma_{g})^{*}(h)$
is not a zero-divisor in
$\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})_{g}$ and
$\mathcal{O}[t]/(h)=\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})_{g}/h\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})_{g}$
,

  * (iii)

$(\Delta(U)\cup\mathcal{Z}^{\prime})\subset\mathcal{X}^{\prime}_{g}$ and
$\sigma_{g}\circ\Delta=i_{0}:U\to\mathbf{A}^{1}\times U$,

  * (iv)

$\mathcal{X}^{\prime}_{gh}\subseteq\mathcal{X}^{\prime}_{gf^{\prime}}\subseteq\mathcal{X}^{\prime}_{f^{\prime}}\subseteq\mathcal{X}^{\prime}_{q^{*}_{X}(\emph{f})}$
,

  * (v)

$\mathcal{O}[t]/(h)=\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})/(f^{\prime})$,
$h\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})=(f^{\prime})\cap
I(\mathcal{Z}^{\prime\prime})$ and
$(f^{\prime})+I(\mathcal{Z}^{\prime\prime})=\Gamma(\mathcal{X}^{\prime},\mathcal{O}_{\mathcal{X}^{\prime}})$.

##  7 Purity

Let $S$ be a regular ring, $\mathbf{G}$ a reductive $S$-group scheme,
$\mathbf{T}$ an $S$-torus, $\mu:\mathbf{G}\to\mathbf{T}$ an $S$-group scheme
morphisms which is smooth as a scheme morphism. Suppose $S$ is an integral
domain. Let $L$ be its field of fractions. For each $S$-algebra $S^{\prime}$
write ${\cal F}(S^{\prime})$ for the group
$\mathbf{T}(S^{\prime})/\mu(\mathbf{G}(S^{\prime}))$. For any
$a\in\mathbf{T}(S^{\prime})$ write $\bar{a}$ for the class of $a$ in ${\cal
F}(S^{\prime})$. Let $\mathfrak{p}$ be a height one prime ideal in $S$, then
by Theorem 1.1 the group ${\cal F}(S_{\mathfrak{p}})$ is a subgroup in ${\cal
F}(L)$.

Recall some notion. For an element $a\in\mathbf{T}(L)$ and a height one prime
$\mathfrak{p}\subset S$ we say that $\bar{a}\in{\cal F}(L)$ is unramified at
$\mathfrak{p}$, if $\bar{a}$ is in ${\cal F}(S_{\mathfrak{p}})$. We say that
the element $\bar{a}\in{\cal F}(L)$ is $S$-unramified if for any height one
prime ideal $\mathfrak{p}$ in $S$ the element $\bar{a}$ is in ${\cal
F}(S_{\mathfrak{p}})$. Clearly, the image of ${\cal F}(S)$ in ${\cal F}(L)$ is
in $\cap\ {\cal F}(S_{\mathfrak{p}})$, where the intersection is taken over
all height one primes of $S$. We say that purity holds for the ring $S$ if

$Im[{\cal F}(S)\to{\cal F}(L)]=\cap\ {\cal F}(S_{\mathfrak{p}}).$

Equivalently, purity holds for $S$ if each $S$-unramified element of ${\cal
F}(L)$ comes from ${\cal F}(S)$. Clearly, the sequence $\\{1\\}\to{\cal
F}(S_{\mathfrak{p}})\to{\cal
F}(L)\xrightarrow{r_{\mathfrak{p}}}\mathbf{T}(L)/[\mathbf{T}(R_{\mathfrak{p}})\cdot\mu(\mathbf{G}(L))]\to\\{1\\}$
is exact, where $r_{\mathfrak{p}}$ is the factorization map. Thus, an element
$a\in\mathbf{T}(L)$ its class $\bar{a}$ in ${\cal F}(L)$ is unramified at
$\mathfrak{p}$, if and only if $r_{\mathfrak{p}}(\bar{a})=0$. Hence purity
holds for $S$ if and only if the sequence ${\cal F}(S)\to{\cal
F}(L)\xrightarrow{\sum
r_{\mathfrak{p}}}\oplus_{\mathfrak{p}}\mathbf{T}(L)/[\mathbf{T}(R_{\mathfrak{p}})\cdot\mu(\mathbf{G}(L))]$
is exact. Our aim is to prove the following assertion:
$(\ast)$ Purity holds for the ring $R$, group schemes
$\mathbf{G}$,$\mathbf{T}$ and the morphism $\mu$ as in Theorem 1.1.

The proof is subdivided in few steps.
Claim 1. Let $X$ be a $k$-smooth irreducible affine $k$-variety. Let
$\mathbf{G}$ be a reductive $X$-group scheme, $\mathbf{T}$ be an $X$-torus and
$\mu:\mathbf{G}\to\mathbf{T}$ be an $X$-group scheme morphism which is smooth
as an $X$-scheme morphism. Suppose the $k$-algebra $R$ is the semi-local ring
of finitely many closed points on $X$. Then purity holds for $R$.

To prove this Claim we just repeat literally the proof of [Pan2, Theorem 1.1]
replacing references to [Pan2, Corollary 4.3, (ii),(v)] with the one to the
items (ii) and (v) of Corollary 6.3. Replacing also references to [Pan2, Lemma
6.7] with the one to Lemma 5.7. Replacing also references to [Pan2, Theorem
6.5] with the one to Theorem 5.5. Replacing also references to [Pan2,
Corollary 6.6] with the one to Corollary 5.6. Replacing also references to
[Pan2, Theorem 4.1] with the one to Theorem 6.1. Replacing also references to
[Pan2, Definition 6.4] with the one to the remark at the end of Definition
5.4. The Claim 1 is proved.

Claim 2. Let $X$ be a $k$-smooth irreducible affine $k$-variety and
$\xi_{1},...,\xi_{n}$ be points of the scheme $Spec(k[X])$ such that for each
pair $r,s$ the point $\xi_{r}$ is not in the closure
$\overline{\\{\xi_{s}\\}}$ of $\xi_{s}$. Let $R$ be the semi-local ring
$\mathcal{O}_{X,\xi_{1},...,\xi_{n}}$ of scheme points $\xi_{1},...,\xi_{n}$
of $Spec(k[X])$. Let $\mathbf{G}$ be a reductive $X$-group scheme,
$\mathbf{T}$ be an $X$-torus and $\mu:\mathbf{G}\to\mathbf{T}$ be an $X$-group
scheme morphism which is smooth as an $X$-scheme morphism. Then purity holds
for $R$.

To prove this Claim take an element $a\in\mathbf{T}(k(X))$ such that $\bar{a}$
is unramified at each irreducible divisor $D$ containing at least one of the
points $\xi_{r}$. We have to prove that the element $\bar{a}\in{\cal F}(K)$ is
in the image of ${\cal F}(R)$. Clearly, there is a non-zero $f\in k[X]$ such
that $a\in\mathbf{T}(k[X_{f}])$. Write down the divisor $div(f)\in Div(X)$ in
the form $div(f)=\Sigma m_{i}D_{i}+\Sigma n_{j}D^{\prime}_{j}$ such that for
each index $i$ there is an index $r$ with $\xi_{r}\in D_{i}$ and for any index
$j$ and any index $r$ the point $\xi_{r}$ does not belong to $D^{\prime}_{j}$.
There is an element $g\in k[X]$ such that for any index $j$ one has
$D^{\prime}_{j}$ is contained in the closed subset $\\{g=0\\}$ and $g$ does
not belong to any of $\xi_{r}$’r. Replacing $X$ with $X_{g}$ we see that
$a\in\mathbf{T}(k[X_{f}])$, $div(f)=\Sigma m_{i}D_{i}$ and $\bar{a}$ is
unramified at each irreducible divisor $D_{i}$. Hence $\bar{a}$ is unramified
at each height one prime ideal of $k[X]$. Our assumption on points $\xi_{r}$’s
yield the following: one can choose closed points
$x_{r}\in\overline{\\{\xi_{s}\\}}$ such that for each $r\neq s$ the point
$x_{r}$ is not in $\overline{\\{\xi_{s}\\}}$. Particularly, for each $r\neq s$
one has $x_{r}\neq x_{s}$. The element $\bar{a}$ is unramified at each height
one prime ideal of $k[X]$. Thus, by Claim 1 the element $\bar{a}$ is in the
image of ${\cal F}(\mathcal{O}_{X,x_{1},...,x_{n}})$. So, the element
$\bar{a}$ is in the image of ${\cal
F}(\mathcal{O}_{X,\xi_{1},...,\xi_{n}})={\cal F}(R)$. The Claim 2 is proved.

Claim 3. The assertion $(\ast)$ is true.
In the rest of the section we prove Claim 3. Clearly, we may assume that $k$
is a prime field and hence $k$ is perfect. It follows from Popescu’s theorem
[Pop, Swa, Spi] that $R$ is a filtered inductive limit of smooth $k$-algebras
$R_{\alpha}$. Modifying the inductive system $R_{\alpha}$ if necessary, we can
assume that each $R_{\alpha}$ is integral. For each maximal ideal
$\mathfrak{m}_{i}$ in $R$ ($i=1,...,n$) set
$\mathfrak{p}_{i}=\varphi^{-1}_{\alpha}(\mathfrak{m}_{i})$. The homomorphism
$\varphi_{\alpha}:R_{\alpha}\to R$ induces a homomorphism of semi-local rings
$\varphi^{\prime}_{\alpha}:(R_{\alpha})_{\mathfrak{p}_{1},...,\mathfrak{p}_{n}}\to
R$. Since this moment we will write $A_{\alpha}$ for
$(R_{\alpha})_{\mathfrak{p}_{1},...,\mathfrak{p}_{n}}$ and $A$ for $R$ (to
keep consistency of notation). Thus, $A$ is a filtered inductive limit of
regular semi-local $k$-algebras $A_{\alpha}$.

There exist an index $\alpha$, a reductive group scheme $\mathbf{G}_{\alpha}$,
a torus $\mathbf{T}_{\alpha}$ over $A_{\alpha}$ and an $A_{\alpha}$-group
scheme morphism $\mu_{\alpha}:\mathbf{G}_{\alpha}\to\mathbf{T}_{\alpha}$ which
is smooth as an $A_{\alpha}$-scheme morphism such that
$\mathbf{G}=\mathbf{G}_{\alpha}\times_{Spec(A_{\alpha})}Spec(A)$,
$\mathbf{T}=\mathbf{T}_{\alpha}\times_{Spec(A_{\alpha})}Spec(A)$,
$\mu=\mu_{\alpha}\times_{Spec(A_{\alpha})}Spec(A)$. Replacing the index system
with a co-final one consisting of indexes $\beta\geq\alpha$, we may and will
suppose that the reductive group scheme $\mathbf{G}$, the torus $\mathbf{T}$
and the group scheme morphism $\mu:\mathbf{G}\to\mathbf{T}$ comes from
$A_{\alpha}$, and $\mu$ is is smooth as an $A_{\alpha}$-scheme morphism. These
observations and Claim 2 yield the following intermediate result
$(\ast\ast)$ for these $\mathbf{G}$, $\mathbf{T}$ and
$\mu:\mathbf{G}\to\mathbf{T}$ over $A_{\alpha}$ purity holds for each ring
$A_{\beta}$ with $\beta\geq\alpha$.

Let now $K$ be the field of fractions of $A$ and, for each $\beta\geq\alpha$,
let $K_{\beta}$ be the field of fractions of $A_{\beta}$. For each index
$\beta\geq\alpha$ let $\mathfrak{a}_{\beta}$ be the kernel of the map
$\varphi^{\prime}_{\beta}:A_{\beta}\to A$ and
$B_{\beta}=(A_{\beta})_{\mathfrak{a}_{\beta}}$. Clearly, for each
$\beta\geq\alpha$, $K_{\beta}$ is the field of fractions of $B_{\beta}$. The
composition map $A_{\beta}\to A\to K$ factors through $B_{\beta}$. Since $A$
is a filtering direct limit of the $A_{\beta}$’s we see that $K$ is a
filtering direct limit of the $B_{\beta}$’s. We will write $\psi_{\beta}$ for
the canonical morphism $B_{\beta}\to K$.

######  Lemma 7.1.

For each index $\alpha$ the group map $W(B_{\alpha})\to W(K_{\alpha})$ is
injective.

###### Proof.

Just apply the first part of Theorem 1.1 to the $k$-algebra $B_{\alpha}$. ∎

######  Lemma 7.2.

Let $a\in W(K)$ be an $A$-unramified element. Then there exists an index
$\alpha$ and an element $b_{\alpha}\in W(B_{\beta})$ such that
$\psi_{\alpha}(b_{\alpha})=b$ and the class $b_{\alpha}\in W(K_{\beta})$ is
$A_{\alpha}$-unramified.

###### Proof.

Repeat literally the proof of [P1, Lemma 9.0.9]. It works for the semi-local
case as well. ∎

We complete the proof of Claim 3 as follows. Let $a\in W(K)$ be an
$A$-unramified element. We have to check that it comes from $W(A)$. By Lemma
7.2 there exists an index $\alpha$ and an element $b_{\alpha}\in
W(B_{\alpha})$ such that $\psi_{\alpha}(b_{\alpha})=b$ and the class
$b_{\alpha}\in W(K_{\beta})$ is $A_{\alpha}$-unramified. For this index
$\alpha$ consider a commutative diagram of $k$-algebras

---
$\textstyle{A_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\alpha}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{\alpha}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{\alpha}}$$\textstyle{K}$$\textstyle{K_{\alpha}.}$

The class $\bar{b}_{\beta}\in{\cal F}(K_{\beta})$ is $A_{\beta}$-unramified.
Hence by the statement $(\ast\ast)$ there exists an element
$a_{\beta}\in\mathbf{T}(A_{\beta})$ such that
$\bar{b}_{\beta}=\bar{a}_{\beta}$ in ${\cal F}(K_{\beta})$. By Lemma 7.1 one
has an equality $\bar{b}_{\beta}=\bar{a}_{\beta}$ in ${\cal F}(B_{\beta})$.
Hence $\bar{b}\in{\cal F}(K)$ coincides with the image of the element
$\varphi_{\beta}(\bar{a}_{\beta})$ in ${\cal F}(K)$. The Claim 3 is proved.
Thus, the sequence (1) is exact at its middle term.

##  8 Proof of Theorem 1.1

###### Proof of Theorem 1.1.

The proof of the first assertion of Theorem 1.1 is given in Section 3. The
exactness of the sequence (1) at its middle term is proved in Section 7.

Prove now the surjectivity of the map $\sum r_{\mathfrak{p}}$. Clearly, it is
sufficient to prove the surjectivity of the map
$\mathbf{T}(K)\xrightarrow{\sum
r^{\prime}_{\mathfrak{p}}}\bigoplus_{\mathfrak{p}}\mathbf{T}(K)/\mathbf{T}(R_{\mathfrak{p}})$,
where $\mathfrak{p}$ runs over the height $1$ primes of $R$ and
$r^{\prime}_{\mathfrak{p}}$ is the factorisation map. We follow arguments from
[Pan3, Section 9].

We prefer to switch to the scheme terminology. Set $X:=Spec(R)$. Consider a
finite étale Galois morphism $\pi:\tilde{X}\to X$ with irreducible $\tilde{X}$
such that the torus $\mathbf{T}$ splits over $\tilde{X}$. Let
$Gal:=Aut(\tilde{X}/X)$ be its Galois group. Since the torus $\mathbf{T}$
splits over $\tilde{X}$ we have a short exact sequence of $Gal$-modules

$0\to\mathbf{T}(\tilde{\mathcal{O}})\to\mathbf{T}(\tilde{K})\to\oplus_{y}\mathbf{T}(\tilde{K})/\mathbf{T}(\tilde{\mathcal{O}}_{X,y})\to
0,$

where $\tilde{\mathcal{O}}=\Gamma(\tilde{X},\mathcal{O}_{\tilde{X}})$,
$\tilde{K}$ is the fraction field of $\tilde{\mathcal{O}}$, $y$ runs over the
set $X^{(1)}$ of all codimension $1$ points of $X$ and for any $y\in X^{(1)}$
the ring $\tilde{\mathcal{O}}_{X,y}$ is the semi-local ring
$\mathcal{O}_{\tilde{X},\tilde{y}}$ of the finite set $\tilde{y}=\pi^{-1}(y)$
on the scheme $\tilde{X}$. Write $\mathcal{O}$ for $R$ to be consistent with
the above notation.

The above short exact sequence of $Gal$-modules gives rise to a long exact
sequence of $Gal$-cohomology groups of the form

$0\to\mathbf{T}(\mathcal{O})\xrightarrow{in}\mathbf{T}(K)\to\oplus_{y}[\mathbf{T}(\tilde{K})/\mathbf{T}(\tilde{\mathcal{O}}_{X,y})]^{Gal}\to
H^{1}(Gal,\mathbf{T}(\tilde{\mathcal{O}}))\xrightarrow{H^{1}(in)}H^{1}(Gal,\mathbf{T}(\tilde{K})).$

We claim that the map $H^{1}(in)$ is a monomorphism. Indeed, the group
$H^{1}(Gal,\mathbf{T}(\tilde{\mathcal{O}}))$ is a subgroup of the group
$H^{1}_{et}(X,\mathbf{T})$ and the group $H^{1}(Gal,\mathbf{T}(\tilde{K}))$ is
a subgroup of the group $H^{1}_{et}(Spec~{}K,\mathbf{T}_{K})$. By Theorem 1.4
the group map $H^{1}_{et}(X,\mathbf{T})\to
H^{1}_{et}(Spec~{}K,\mathbf{T}_{K})$ is a monomorphism. Thus, $H^{1}(in)$ is a
monomorphism also. So, we have a short exact sequence of the form
$0\to\mathbf{T}(\mathcal{O})\xrightarrow{in}\mathbf{T}(K)\to\oplus_{y}[\mathbf{T}(\tilde{K})/\mathbf{T}(\tilde{\mathcal{O}}_{X,y})]^{Gal}\to
0$.

There is also the complex
$0\to\mathbf{T}(\mathcal{O})\xrightarrow{in}\mathbf{T}(K)\to\oplus_{y}\mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})$.
Set $\alpha=id_{\mathbf{T}(\mathcal{O})}$, $\beta=id_{\mathbf{T}(K)}$ and let
$\gamma=\oplus_{y}\gamma_{y}$, where
$\gamma_{y}:\mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})\to[\mathbf{T}(\tilde{K})/\mathbf{T}(\tilde{\mathcal{O}}_{X,y})]^{Gal}$
is induced by the inclusion $K\subset\tilde{K}$. The maps $\alpha$, $\beta$
and $\gamma$ form a morphism between this complex and the above short exact
sequence. We claim that this morphism is an isomorphism. This claim completes
the proof of the theorem.

To prove this claim it is sufficient to prove that $\gamma$ is an isomophism.
Since the map
$\mathbf{T}(K)\to\oplus_{y}[\mathbf{T}(\tilde{K})/\mathbf{T}(\tilde{\mathcal{O}}_{X,y})]^{Gal}$
is an epimorphism, hence so is the map $\gamma$. It remains to prove that
$\gamma$ is a monomorphism. To do this it is sufficient to check that for any
point $y\in X^{(1)}$ the map
$\mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})\to\mathbf{T}(\tilde{K})/\mathbf{T}(\tilde{\mathcal{O}}_{X,y})$
is a monomorphism. We will write $\epsilon_{y}$ for the latter map. We prove
below that $ker(\epsilon_{y})$ is a torsion group and the group
$\mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})$ has no torsion. These two claims
show that the map $\epsilon_{y}$ is injective indeed.

To prove that $ker(\epsilon_{y})$ is a torsion group recall that there are
norm maps
$N_{\tilde{\mathcal{O}}_{X,y}/\mathcal{O}_{X,y}}:\mathbf{T}(\tilde{\mathcal{O}}_{X,y})\to\mathbf{T}(\mathcal{O}_{X,y})$
and $N_{\tilde{K}/K}:\mathbf{T}(\tilde{K})\to\mathbf{T}(K)$ (see Section 4).
These maps induce a homomorphism

$N_{y}:\mathbf{T}(\tilde{K})/\mathbf{T}(\tilde{\mathcal{O}}_{X,y})\to\mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})$

such that $N_{y}\circ\epsilon_{y}$ is the multiplication by the degree $d$ of
$\tilde{K}$ over $K$. Thus, $ker(\epsilon_{y})$ is killed by the
multiplication by $d$.

Show now that the group $\mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})$ has no
torsion. Take an element $a_{K}\in\mathbf{T}(K)$ and suppose that its class in
$\mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})$ is a torsion element. Let
$\tilde{a}_{K}$ be the image of $a_{K}$ in $\mathbf{T}(\tilde{K})$. Since
$\mathbf{T}$ splits over $\tilde{K}$ we see that
$\mathbf{T}(\tilde{K})/\mathbf{T}(\tilde{\mathcal{O}}_{X,y})$ is torsion free.
Thus, the class of $\tilde{a}_{K}$ in
$\mathbf{T}(\tilde{K})/\mathbf{T}(\tilde{\mathcal{O}}_{X,y})$ vanishes. So,
there is a unique element $\tilde{a}$ in
$\mathbf{T}(\tilde{\mathcal{O}}_{X,y})$ whose image in $\mathbf{T}(\tilde{K})$
is $\tilde{a}_{K}$. Moreover, $\tilde{a}$ is a $Gal$-invariant element in
$\mathbf{T}(\tilde{\mathcal{O}}_{X,y})$, because $\tilde{a}_{K}$ comes from
$\mathbf{T}(K)$. Since
$\mathbf{T}(\tilde{\mathcal{O}}_{X,y})^{Gal}=\mathbf{T}(\mathcal{O}_{X,y})$,
there is a unique element $a\in\mathbf{T}(\mathcal{O}_{X,y})$ whose image in
$\mathbf{T}(\tilde{\mathcal{O}}_{X,y})$ is $\tilde{a}$. Clearly, the image of
$a$ into $\mathbf{T}(K)$ is the element $a_{K}$. Thus, the class of $a_{K}$ in
$\mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})$ vanishes. So, the group
$\mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})$ is torsion free. The injectivity
of $\epsilon_{y}$ is proved. The surjectivity of the map $\sum
r_{\mathfrak{p}}$ is proved. Theorem 1.1 is proved. ∎

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