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An extended form of the Grothendieck–Serre conjecture I. A. Panin St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
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     § 1. Main resultsLet $R$ be a commutative unital ring. We recall that an $R$-group scheme $\mathbf{G}$ is said to be reductive (resp. semisimple or simple) if it is affine and smooth as an $R$-scheme and if, moreover, for every algebraically closed field $\Omega$ and every ring homomorphism $R\to\Omega$, the $\Omega$-group scheme $\mathbf{G}_\Omega$ is a connected reductive (resp. semisimple or simple) algebraic group over $\Omega$. The class of reductive group schemes contains the class of semisimple group schemes, which in turn contains the class of simple group schemes. This definition of a reductive $R$-group scheme coincides with Definition 2.7 in [1], Lecture XIX. The definition of a simple $R$-group scheme coincides with that of a simple semisimple $R$-group scheme of Grothendieck and Demazure ([1], Lecture XIX, Definition 2.7, and Lecture XXIV, subsection 5.3). We now state our first main result. It is based on [2], [3] and considerably extends the corresponding results of [2] and [3]. Theorem 1.1. Let $R$ be a regular semi-local integral domain containing a field, $K$ the fraction field of $R$, and $\mu\colon \mathbf{G} \to\mathbf{T}$ an $R$-group scheme morphism between reductive $R$-group schemes which is smooth as a scheme morphism. Suppose that $\mathbf{T}$ 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 is exact, where $\mathfrak p$ runs over the set of all prime ideals of height one in $R$, and each $r_{\mathfrak p}$ is a natural map (the projection to the quotient group). Remark 1.2. In [2] and [3], there was an additional assumption that the kernel of $\mu$ is a reductive $R$-group scheme. In particular, we required in [2] and [3] that the kernel of $\mu$ is geometrically connected. Theorem 1.1 imposes no restrictions of the kernel of $\mu$. We 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 1.4). 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 $\operatorname{Spec} R$. Hence Theorem 1.5 yields that the map $\mathbf{T}(R)/\mu(\mathbf{G}(R)) \to\mathbf{T}(K)/\mu(\mathbf{G}(K))$ is injective. Theorem 1.3. Let $R$ be a regular semi-local integral domain containing a field, $K$ the fraction field of $R$, and $\mathbf{G}_1$, $\mathbf{G}_2$ semi-local $R$-group schemes whose general fibres $\mathbf{G}_{1,K}$, $\mathbf{G}_{2,K}$ are isomorphic as algebraic $K$-groups. Then the $R$-group schemes $\mathbf{G}_1$ and $\mathbf{G}_2$ are isomorphic. To prove Theorem 1.3, we need to work with the automorphism group scheme of a semisimple $R$-group scheme. The latter group scheme is not geometrically connected in general. Hence Theorem 1.3 cannot be derived from [4] and [3]. In what follows we state Theorem 1.5 saying that an extended version of the Grothendieck–Serre conjecture holds for such rings $R$. We shall prove Theorem 1.5 and derive from it Theorem 1.3 and the first assertion of Theorem 1.1. The following definition is convenient in order to state Theorem 1.5. Definition 1.4 (quasi-reductiveness). Assume that $S$ is a Noetherian commutative ring. An $S$-group scheme $\mathbf{H}$ is said to be quasi-reductive if one can find a finite étale $S$-group scheme $\mathbf{C}$ and a smooth morphism of $S$-group schemes $\lambda\colon \mathbf{H} \to \mathbf{C}$ such that the kernel of $\lambda$ is a reductive $S$-group scheme and $\lambda$ is surjective locally in the étale topology on $S$. Reductive $S$-group schemes are clearly quasi-reductive, and all quasi-reductive $S$-group schemes are affine and smooth as $S$-schemes. There are two types of quasi-reductive $S$-group schemes which are interesting for us in this paper. The first type consists of the automorphism group schemes of semisimple reductive $R$-group schemes. The second type is obtained as follows. Take a reductive $S$-group scheme $\mathbf{G}$, an $S$-torus $\mathbf{T}$ and a smooth $S$-group morphism $\mu\colon \mathbf{G} \to \mathbf{T}$. One can verify 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 by means of a reductive $S$-group scheme $\mathbf{G}_0$. Assume that $U$ is an irreducible regular scheme and $\mathbf{H}$ is a quasi-reductive $U$-group scheme. Recall that a $U$-scheme $\mathcal{H}$ with a left 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 [5], § 6. Since $\mathbf{H}$ is $S$-smooth, such an $\mathbf{H}$-bundle is trivial locally in the étale topology on $U$, but not in the Zariski topology on $U$. Grothendieck [6] and Serre [7] conjectured that, for a reductive $U$-group scheme $\mathbf{H}$, a principal $\mathbf{H}$-bundle over $U$ is trivial locally in the Zariski topology on $U$ if it is trivial generically on $U$. Important results concerning this conjecture were obtained in [8]–[10]. A survey paper on this topic is [11]. The conjecture holds if $\Gamma(U,\mathcal{O}_U)$ contains a field (see [4] and [3]). It was proved in [12] that the conjecture holds in general for discrete valuation rings. This result was extended in [13] to the case of semi-local Dedekind integral domains assuming that $\mathbf{G}$ is simple simply connected and isotropic in a certain precise sense. The results in [12] and [13] were further extended in [14]. Namely, it was proved that the conjecture holds in general for semi-local Dedekind integral domains. The following result strengthens the main result of [3]. To state it, we use Definition 1.4. Theorem 1.5. Let $R$ be a regular semi-local integral domain containing a field, $K$ the fraction field of $R$, and $\mathbf{H}$ a quasi-reductive group scheme over $R$. Then the map induced by the inclusion of $R$ in $K$ has a trivial kernel. In other words, every principal $\mathbf{H}$-bundle over $R$ having a $K$-rational point is trivial under these assumptions on $R$ and $\mathbf{H}$. Corollary 1.6. Under the hypotheses of Theorem 1.5, the map induced by the inclusion of $R$ in $K$ is injective. Equivalently, if $\mathcal{H}_1$ and $\mathcal{H}_2$ are principal $\mathbf{H}$-bundles isomorphic over $\operatorname{Spec} K$, then they are isomorphic over $R$. Proof. Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be principal $\mathbf{H}$-bundles which are isomorphic over $\operatorname{Spec} K$. Let $\underline{\operatorname{Iso}}(\mathcal{H}_1,\mathcal{H}_2)$ be the scheme of isomorphisms of principal $\mathbf{H}$-bundles. This scheme is a principal $\underline{\operatorname{Aut}}\, \mathcal{H}_1$-bundle. It is trivial by Theorem 1.5 and we see that $\mathcal{H}_1\cong\mathcal{H}_2$ as principal $\mathbf{H}$-bundles. $\Box$ Theorems 1.5 and 1.3 are proved in § 2. Theorem 1.1 is proved in § 8. § 2. Proof of Theorems 1.5 and 1.3We begin with the following general lemma. Lemma 2.1. Let $X$ be a regular semi-local irreducible scheme, $\pi\colon X'\to X$ a finite morphism, and $\eta\in X$ the generic point of $X$. Then there is a bijection between the sections of $\pi$ over $X$ and the sections of $\pi$ over $\eta$. Proof. It suffices to check that each section $s\colon \eta\to X'$ of $\pi$ can be extended to a section of $\pi$ over $X$.
We write $\pi$ as a composite $X' \xrightarrow{i} \mathbf{A}^n_X \xrightarrow{p} X$, where $p$ is the projection and $i$ is a closed embedding. Let $s\colon \eta\to X'$ be a section of $\pi$. Since $X$ is regular and semi-local and $\pi$ is projective, one can find a closed subset $Z$ of codimension two in $X$ and a section $\varphi\colon X- Z \to X'$ of $\pi$ that extends $s$. Since $\Gamma(X, \mathcal O_X)=\Gamma(X-Z, \mathcal O_X)$, the section $\varphi$ extends to a section $\widetilde \varphi\colon X\to X'$ of $\pi$. $\Box$ Corollary 2.2. Let $X$ and $\eta\in X$ be as in the previous lemma and let $\mathbf{E}$ be a finite étale $X$-group scheme. Then the $\eta$-points of $\mathbf{E}$ coincide with the $X$-points of $\mathbf{E}$. Corollary 2.3. Under the hypotheses of Corollary 2.2, the kernel of the pointed set map $\mathrm H^1_{\unicode{x00E9}\textrm{t}}(X,\mathbf{E})\to \mathrm H^1_{\unicode{x00E9}\textrm{t}}(\eta,\mathbf{E})$ is trivial. Proof. Let $\mathcal{E}$ be a principal $\mathbf{E}$-bundle over $X$. Standard decent arguments show that the $X$-scheme $\mathcal{E}$ is finite and étale over $X$. Hence $\mathcal{E}(X)=\mathcal{E}(\eta)$. $\Box$ Proof of Theorem 1.5. Since $\mathbf{H}$ is a quasi-reductive $R$-group scheme, one can find a finite étale $R$-group scheme $\mathbf{C}$ and a smooth $R$-group scheme morphism $\lambda\colon \mathbf{H} \to \mathbf{C}$ whose kernel $\mathbf{G}$ is a reductive $R$-group scheme such that the morphism $\lambda$ is surjective locally in the étale topology on $\operatorname{Spec} R$. The sequence $1\to \mathbf{G}\to \mathbf{H}\to \mathbf{C}\to 1$ of étale sheaves on $\operatorname{Spec} R$ is exact. Thus, it induces a commutative diagram of pointed set maps with exact rows Note that $\alpha$ is bijective by Corollary 2.2, the kernel of $\delta$ is trivial by Corollary 2.3 and $\beta$ is injective by Corollary 1.2 in [3]. Now a simple diagram chase shows that $\operatorname{ker}(\gamma)=\ast$ . $\Box$ Remark 2.4. The statement and proof of Lemma 3.7 in [15] are inaccurate since the authors do not assume that the map $\mathrm H_{\unicode{x00E9}\textrm{t}}^1(R,\mathbf{G}^0)\to \mathrm H_{\unicode{x00E9}\textrm{t}}^1(K,\mathbf{G}^0_K)$ is injective. Proof of Theorem 1.3. The $R$-group scheme $\underline{\operatorname{Aut}}:= \underline{\operatorname{Aut}}_{\,R\text{-gr-sch}}(\mathbf{G}_1)$ is quasi-reductive by [1]. The $R$-scheme $\underline{\operatorname{Iso}}:= \underline{\operatorname{Iso}}_{\,R\text{-gr-sch}}(\mathbf{G}_1,\mathbf{G}_2)$ is a principal $\underline{\operatorname{Aut}}$-bundle. Any isomorphism $\varphi\colon \mathbf{G}_{1,K} \to \mathbf{G}_{2,K}$ of algebraic $K$-groups gives a section of $\underline{\operatorname{Iso}}$ over $K$. Hence $\underline{\operatorname{Iso}}_{\,K}$ is a trivial principal $\underline{\operatorname{Aut}}_{\,K}$-bundle. Therefore $\underline{\operatorname{Iso}}$ is a trivial principal $\underline{\operatorname{Aut}}$-bundle by Theorem 1.5. Thus, it has a section over $R$. This section determines an $R$-group scheme isomorphism $\mathbf{G}_1 \cong \mathbf{G}_2$. $\Box$
§ 3. Proof of the first assertion of Theorem 1.1Lemma 3.1. Let $X$ be a regular irreducible affine scheme, $\mathbf{G}$ a reductive $X$-group scheme, $\mathbf{T}$ an $X$-torus, and $\mu\colon\mathbf{G} \to \mathbf{T}$ an $X$-group scheme morphism which is smooth as a scheme morphism. Then the kernel of $\mu$ is a quasi-reductive $X$-group scheme. Proof. Consider the coradical $\operatorname{Corad}(\mathbf{G})$ of $\mathbf{G}$ together with the canonical $X$-group morphism $\alpha\colon \mathbf{G}\to \operatorname{Corad}(\mathbf{G})$. By the universal property of $\alpha$, there is a unique $X$-group morphism $\overline \mu\colon \operatorname{Corad}(\mathbf{G})\to T$ such that $\mu=\overline \mu\circ \alpha$. Since $\mu$ is surjective locally for the étale topology on $X$, so is $\overline \mu$. Let $\operatorname{ker}(\overline \mu)$ be the kernel of $\overline\mu$ and let $\mathbf{H}\colon =\alpha^{-1}(\operatorname{ker}(\overline\mu))$ be the scheme-theoretic preimage of $\operatorname{ker}(\overline \mu)$. It is clear that $\mathbf{H}$ is a closed $X$-subgroup scheme of $\mathbf{G}$ and $\mathbf{H}$ is the kernel of $\mu$. We need to verify that $\mathbf{H}$ is quasi-reductive.
The $X$-group scheme $\operatorname{ker}(\overline \mu)$ is of multiplicative type. Hence one can find a finite $X$-group scheme $\mathbf{M}$ of multiplicative type and a faithfully flat $X$-group morphism $\operatorname{can}\colon \operatorname{ker}(\overline \mu)\to \mathbf{M}$ with the following property. For every finite $X$-group scheme $\mathbf{M}'$ of multiplicative type and every $X$-group morphism $\varphi\colon \operatorname{ker}(\overline \mu)\to\mathbf{M}'$ there is a unique $X$-group morphism $\psi\colon\mathbf{M}\to \mathbf{M}'$ such that $\psi \circ \operatorname{can}=\varphi$. It is known that the kernel of $\operatorname{can}$ is an $X$-torus. Call it $\mathbf{T}^0$. Since $\mu$ is smooth, so is $\overline \mu$. Thus the $X$-group scheme $\operatorname{ker}(\overline \mu)$ is $X$-smooth. This yields that $M$ is étale over $X$. Put $\beta=\alpha|_{\mathbf{H}}\colon \mathbf{H}\to \operatorname{ker}(\overline \mu)$ and let $\mathbf{G}^0:=\beta^{-1}(\mathbf{T}^0)$ be the scheme-theoretic preimage of the $X$-torus $\mathbf{T}^0$. Clearly, $\mathbf{G}^0$ is a closed $X$-subgroup scheme of $\mathbf{H}$ and $\mathbf{G}^0$ is the kernel of the morphism $\operatorname{can}{\circ}\, \beta\colon \mathbf{H}\to \mathbf{M}$. Put $\gamma=\beta|_{\mathbf{G}^0}\colon \mathbf{G}^0\to \mathbf{T}^0$. The $X$-group scheme $\mathbf{M}$ is finite and étale over $X$. The morphism $\operatorname{can}$ is smooth. Moreover, $\beta$ is smooth as a base change of the smooth morphism $\alpha$. Thus the morphism $\lambda:=\operatorname{can}\,{\circ}\,\beta$ is smooth. It is also surjective locally in the étale topology on $X$ because $\operatorname{can}$ and $\beta$ have this property. By construction, $\mathbf{G}^0\,{=}\operatorname{ker}(\lambda)$. Thus, to prove that $\mathbf{H}$ is quasi-reductive, it suffices to verify that $\mathbf{G}^0$ is reductive. The $X$-group scheme $\mathbf{G}^0$ is affine over $X$ as a closed $X$-subgroup scheme of the reductive $X$-group scheme $\mathbf{G}$. We claim that $\mathbf{G}^0$ is smooth over $X$. Indeed, the morphism $\gamma$ is smooth as a base change of the smooth morphism $\alpha$. The $X$-group scheme $\mathbf{T}^0$ is $X$-smooth since it is an $X$-torus. Thus the $X$-scheme $\mathbf{G}^0$ is $X$-smooth. Write $X$ as $\operatorname{Spec} S$, where $S$ is a regular integral domain. We need to verify that, for every algebraically closed field $\Omega$ and every ring homomorphism $S\to\Omega$, the scalar extension $\mathbf{G}^0_\Omega$ of $\mathbf{G}^0$ is a connected reductive algebraic group over $\Omega$. Firstly, recall that $\operatorname{ker}(\alpha)$ is a semisimple $S$-group scheme. It is the $S$-group scheme $\mathbf{G}^{\mathrm{ss}}$ in the notation of [1]. Clearly, $\operatorname{ker}(\gamma)=\operatorname{ker}(\alpha)$. Thus, $\operatorname{ker}(\gamma)=\mathbf{G}^{\mathrm{ss}}$ is a semisimple $S$-group scheme. Since $\gamma$ is smooth, we have the following short exact sequence of $\Omega$-smooth algebraic groups for every algebraically closed field $\Omega$ and every ring homomorphism $S\to\Omega$
$$
\begin{equation*}
1\to \mathbf{G}^{\mathrm{ss}}_\Omega \to \mathbf{G}^0_\Omega \to \mathbf{T}^0_\Omega \to 1.
\end{equation*}
\notag
$$
The groups $\mathbf{T}^0_\Omega$ and $\mathbf{G}^{\mathrm{ss}}_\Omega$ are connected. Hence so is $\mathbf{G}^0_\Omega$. We already know that $\mathbf{G}^0_\Omega$ is affine.
We claim that the unipotent radical $\mathbf U$ of the group $\mathbf{G}^0_\Omega$ is trivial. Indeed, since there are no non-trivial $\Omega$-group morphisms $\mathbf U\to \mathbf{T}^0_\Omega$, we conclude that $\mathbf U\subset \mathbf{G}^{\mathrm{ss}}_\Omega$. Since $\mathbf{G}^{\mathrm{ss}}_\Omega$ is semisimple, $\mathbf U=\{1\}$. This completes the proof of reductivity of the $S$-group scheme $\mathbf{G}^0$. Hence the $S$-group scheme $\mathbf{H}$ is quasi-reductive. $\Box$ Proof of the first assertion of Theorem 1.1. Let $\mathbf{H}$ be the kernel of $\mu$. Since $\mu$ is smooth, the sequence of $R$-group schemes gives rise to a short exact sequence of sheaves in the étale topology on $\operatorname{Spec} R$. In its turn, this sequence of sheaves induces a long exact sequence of pointed sets. Hence the boundary map $\partial\colon \mathbf{T}(R) \to \mathrm{H}^1_{\unicode{x00E9}\textrm{t}}(R,\mathbf{H})$ fits in a commutative diagram The horizontal arrows clearly have trivial kernels. The kernel of the right vertical arrow is trivial by Lemma 3.1 and Theorem 1.5. Thus the kernel of the left vertical arrow is also trivial. Since this arrow is a group homomorphism, it is injective. $\Box$
§ 4. NormsWe recall a construction in [16]. Let $k\subset K\subset L$ be field extensions such that $L$ is finite separable over $K$. Let $K^{\mathrm{sep}}$ be a separable closure of $K$ and let $\sigma_i\colon L\to K^{\mathrm{sep}}$, $1\leqslant i\leqslant n$, be the distinct embeddings of $L$ in $K^{\mathrm{sep}}$ over $K$. Suppose that $C$ is a $k$-smooth commutative algebraic group scheme defined over $k$. One can define a norm map by putting ${\mathcal N}_{L/K}(\alpha)=\prod_i C(\sigma_i)(\alpha) \in C(K^{\mathrm{sep}})^{{\mathcal G}(K)} =C(K)$. Let $p\colon X\to Y$ be a finite flat morphism of affine schemes. Suppose that its rank is constant and equal to $d$. We write $S^d(X/Y)$ for the $d$th symmetric power of $X$ over $Y$. Suslin and Voevodsky ([17], § 6) constructed a canonical section for the projection $S^d(X/Y)\to Y$. We denote this section by $N_{X/Y}\colon Y\to S^d(X/Y)$.Let $k$ be a field, $\mathcal O$ the semi-local ring of finitely many closed points on a smooth irreducible affine $k$-variety, $C$ an affine smooth commutative $\mathcal O$-group scheme, $p\colon X\to Y$ a finite flat $\mathcal O$-morphism of constant degree $d$ between affine $\mathcal O$-schemes, and $f\colon X\to C$ an arbitrary $\mathcal O$-morphism. The norm $N_{X/Y}(f)$ of $f$ was defined in [16] as the composite map
$$
\begin{equation}
Y \xrightarrow{N_{X/Y}} S^d(X/Y) \to S^d_{\mathcal O}(X) \xrightarrow{S^d_{\mathcal o}(f)} S^d_{\mathcal O}(C)\xrightarrow{\times} C.
\end{equation}
\tag{2}
$$
Here we write “$\times$” for the group law on $C$. The norm maps $N_{X/Y}\colon C(X)\to C(Y)$ possess the following properties. $\mathrm{(i')}$ Base change. For every $\mathcal O$-morphism $f\colon Y'\to Y$ of affine $\mathcal O$-schemes, putting $X'=X\times_Y Y'$, we have a commutative diagram $\mathrm{(ii')}$ Multiplicativity. If $X=X_1 \amalg X_2$ and $X_i/Y$ is of constant degree $d_i$ for $i=1,2$, then the following diagram commutes: $\mathrm{(iii')}$ Normalization. If $X=Y$ and the morphism $X \to Y$ is the identity map, then $N_{X/Y}=\operatorname{id}_{C(X)}$. § 5. Unramified elementsLet $k$ be a field, $\mathcal O$ the semi-local ring of finitely many closed points on a smooth irreducible affine $k$-variety $X$, and $K$ the fraction field of $\mathcal O$, that is, $K=k(X)$. Let be a smooth $\mathcal O$-morphism of reductive $\mathcal O$-group schemes, where $\mathbf{T}$ is an $\mathcal O$-torus. In this section we work with the category of commutative Noetherian $\mathcal O$-algebras. Given a commutative Noetherian $\mathcal O$-algebra $S$, we put Given any element $\alpha \in \mathbf{T}(S)$, we write $\overline \alpha$ for its image in $\mathcal{F}(S)$. In this section we will write $\mathcal{F}$ for the functor (3). The following result is a particular case of the first part of Theorem 1.1 (this part was proved in § 3).Theorem 5.1. Let $S$ be an $\mathcal O$-algebra which is a discrete valuation ring with fraction field $L$. Then the map $\mathcal{F}(S) \to \mathcal{F}(L)$ is injective. Lemma 5.2. Let $\mu\colon \mathbf{G} \to \mathbf{T}$ be the above morphism of reductive $\mathcal O$-group schemes and let $\mathbf{H}=\operatorname{Ker}(\mu)$. Then for any $\mathcal O$-algebr a $L$, where $L$ is a field, the boundary map $\partial\colon \mathbf{T}(L)/{\mu (\mathbf{G}(L))} \to\mathrm{H}^1_{\unicode{x00E9}\textrm{t}}(L,\mathbf{H})$ is injective. Let $k$, $\mathcal O$ and $K$ be as above in this section. Given a field $\mathcal K$ containing $K$ and a discrete valuation $x\colon \mathcal K^* \to \mathbb Z$ vanishing on $K^{\times}$, we write $A_x\subset \mathcal K$ for the discrete valuation ring of $x$. It is clear that $\mathcal O \subset A_x$. Let $\widehat A_x$ and $\widehat {\mathcal K}_x$ be the completions of $A_x$ and $\mathcal K$ with respect to $x$, and let $i\colon \mathcal K \hookrightarrow \widehat {\mathcal K}_x$ be the inclusion. The map $\mathcal{F}(\widehat A_x)\to \mathcal{F}(\widehat{\mathcal K}_x)$ is injective by Theorem 5.1. We will identify $\mathcal{F}(\widehat A_x)$ with its image under this map. Put
$$
\begin{equation*}
\mathcal{F}_x(\mathcal K)=i_*^{-1}\bigl(\mathcal{F}(\widehat A_x)\bigr).
\end{equation*}
\notag
$$
The inclusion $A_x\hookrightarrow \mathcal K$ induces a map $\mathcal{F}(A_x) \to \mathcal{F}(\mathcal K)$, which is injective by Theorem 5.1. The image of $\mathcal{F}(A_x)$ in $\mathcal{F}(\mathcal K)$ is called the subgroup of all elements unramified at $x$. The groups $\mathcal{F}(A_x)$ and $\mathcal{F}_x(\mathcal K)$ are subgroups of $\mathcal{F}(\mathcal K)$. The following lemma shows that $\mathcal{F}_x(\mathcal K)$ coincides with the subgroup $\mathcal{F}(A_x)$ in $\mathcal{F}(\mathcal K)$ consisting of all elements unramified at $x$. Let $S$ be an $\mathcal O$-algebra which is an integral domain and suppose that $S$ is a regular ring. Let $L$ be the fraction field of $S$. For every prime ideal $\mathfrak p$ of height $1$ in $S$, the group homomorphism $\mathcal{F}(S_{\mathfrak p})\to \mathcal{F}(L)$ is injective by the first part of Theorem 1.1. We define the subgroup $\mathcal F_{\mathrm{nr},S}(L)$ of $S$-unramified elements of the group $\mathcal F (L)$ by putting
$$
\begin{equation}
\mathcal F_{\mathrm{nr},S}(L)= \bigcap_{\mathfrak p \in \operatorname{Spec}(S)^{(1)}} \mathcal F(S_{\mathfrak p}) \subseteq \mathcal{F}(L),
\end{equation}
\tag{4}
$$
where $\operatorname{Spec}(S)^{(1)}$ is the set of prime ideals of height $1$ in $S$. The image of $\mathcal F(S)$ in $\mathcal F(L)$ is clearly contained in $\mathcal F_{\mathrm{nr},S}(L)$. For every prime ideal $\mathfrak p$ of height $1$ in $S$ we shall construct a specialization map $s_{\mathfrak p}\colon \mathcal{F}_{\mathrm{nr}, S}(L) \to \mathcal{F} (l(\mathfrak p))$, where $L$ is the fraction field of $S$ and $l(\mathfrak p)$ is the residue field of $S$ at the prime ideal $\mathfrak p$. Definition 5.4. Let $Ev_{\mathfrak p}\colon \mathbf{T}(S_{\mathfrak p}) \to \mathbf{T}(l(\mathfrak p))$ and $ev_{\mathfrak p}\colon \mathcal{F}(S_{\mathfrak p}) \to \mathcal{F}(l(\mathfrak p))$ be the maps induced by the canonical $S$-algebra homomorphism $S_{\mathfrak p} \to l(\mathfrak p)$. We define a homomorphism $s_{\mathfrak p}\colon \mathcal{F}_{\mathrm{nr}, S}(L) \to \mathcal{F} (l(\mathfrak p))$ by putting $s_{\mathfrak p}(\alpha)= ev_{\mathfrak p}(\widetilde \alpha)$, where $\widetilde \alpha$ is a lift of $\alpha$ to $\mathcal{F}(S_{\mathfrak p})$. Theorem 5.1 yields 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 $\mathfrak p$. For every $\alpha \in \mathbf{T}(S_\mathfrak p)$ we clearly have $s_{\mathfrak p}(\overline \alpha)=\overline {Ev_{\mathfrak p}(\alpha)} \in \mathcal{F}(l(\mathfrak p))$. Let $k$, $\mathcal O$ and $K$ be as above in this section. The next two results can be proved by repeating verbatim the proofs of Theorem 6.5 in [2] and Corollary 6.6 in [2] respectively. Theorem 5.5 (homotopy invariance). Let $K(t)$ be the rational function field in one variable over $K$. Define the group $\mathcal{F}_{\mathrm{nr},K[t]}(K(t))$ by (4). Then we have an equality Corollary 5.6. Let $s_0, s_1\colon \mathcal{F}_{\mathrm{nr}, K[t]}(K(t)) \rightrightarrows \mathcal{F}(K)$ be the specialization maps at zero and at one (or at the prime ideals $(t)$ and $(t-1)$ respectively). 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 is of dimension one. Suppose that $0\neq f \in A$ and let $h \in B\cap fA$ be such that the induced map $B/hB\to A/fA$ is an isomorphism of rings. Suppose that $hA=fA\cap J''$ for some ideal $J'' \subseteq A$ coprime to the principal ideal $fA$. Let $E$ and $F$ be the fraction fields of $B$ and $A$ respectively. Let $\alpha \in \mathbf{T}(A_f)$ be such that the element $\overline \alpha \in \mathcal{F}(F)$ is $A$-unramified. Put $\beta= N_{F/E}(\alpha)$. Then the class $\overline \beta \in \mathcal{F}(E)$ of $\beta$ is $B$-unramified. § 6. A few recallsLet $X$ be an affine irreducible $k$-smooth $k$-variety, let $x_1,x_2,\dots,x_n$ be closed points in $X$ and let $\mathcal O$ the semi-local ring $\mathcal O_{X,\{x_1,x_2,\dots,x_n\}}$. We put $U=\operatorname{Spec}(\mathcal O)$ and write $\operatorname{can}\colon U\hookrightarrow X$ for the canonical embedding. Let $\mathbf{G}$ be a reductive $X$-group scheme, $\mathbf{G}_U=\operatorname{can}^*(\mathbf{G})$ the pull-back of $\mathbf{G}$ to $U$, $\mathbf{T}$ an $X$-torus, and $\mathbf{T}_U=\operatorname{can}^*(\mathbf{T})$ the pull-back of $\mathbf{T}$ to $U$. Let $\mu\colon \mathbf{G} \to \mathbf{T}$ be an $X$-group scheme morphism which is smooth as an $X$-scheme morphism. We put $\mu_U=\operatorname{can}^*(\mu)$. The following result is Theorem 4.1 in [2]. Theorem 6.1. Let $\mathrm{f}\in k[X]$ be a non-zero function vanishing at each point $x_i$. Then there is a diagram of the form with an irreducible affine scheme $\mathcal X'$, a smooth morphism $q_U$, a finite surjective $U$-morphism $\sigma$, an essentially smooth morphism $q_X$, and a function $f' \in q^*_X(\mathrm{f})k[\mathcal X']$ such that the following conditions hold. (a) If $\mathcal Z'$ is the closed subscheme of $\mathcal X'$ defined by the ideal $(f')$, then the morphism $\sigma|_{\mathcal Z'}\colon \mathcal Z' \to \mathbf{A}^1\times U$ is a closed embedding and the morphism $q_U|_{\mathcal Z'}\colon \mathcal Z' \to U$ is finite. $\mathrm{(a')}$ $q_U\circ \Delta'=\operatorname{id}_U$ and $q_X\circ \Delta'=\operatorname{can}$, and $\sigma\circ \Delta'=i_0$, where $i_0$ is the zero section of the projection $\operatorname{pr}_U$. (b) $\sigma$ is étale in a neighbourhood of $\mathcal Z'\cup \Delta'(U)$. (c) $\sigma^{-1}(\sigma(\mathcal Z'))=\mathcal Z'\coprod \mathcal Z''$ scheme-theoretically for some closed subscheme $\mathcal Z''$, and $\mathcal Z'' \cap \Delta'(U)=\varnothing$. (d) $\mathcal D_0:=\sigma^{-1}(\{0\} \times U)=\Delta'(U)\coprod \mathcal D'_0$ scheme-theoretically for some closed subscheme $\mathcal D'_0$, and $\mathcal D'_0 \cap \mathcal Z'=\varnothing$. (e) If $\mathcal D_1:=\sigma^{-1}(\{1\} \times U)$, then $\mathcal D_1 \cap \mathcal Z'=\varnothing$. (f) There is a monic polynomial $h \in \mathcal O[t]$ such that
$$
\begin{equation*}
(h)=\operatorname{Ker}\bigl[\mathcal O[t] \xrightarrow{\sigma^*} k[\mathcal X'] \xrightarrow{-} k[\mathcal X']/(f')\bigr],
\end{equation*}
\notag
$$
where the bar stands for the map sending every $g\in k[\mathcal X']$ to ${\overline g}\in k[\mathcal X']/(f')$. $\mathrm{(g)}$ There are $\mathcal X'$-group schemes isomorphisms $\Phi\colon q^*_U(\mathbf{G}_U)\to q^*_X(\mathbf{G})$ and $\Psi\colon q^*_U(\mathbf{T}_U)\to q^*_X(\mathbf{T})$ such that $(\Delta')^*(\Phi)= \operatorname{id}_{\mathbf{G}_U}$, $(\Delta')^*(\Psi)= \operatorname{id}_{\mathbf{T}_U}$ and $q^*_X(\mu) \circ \Phi=\Psi \circ q^*_U(\mu_U)$. Remark 6.2. The triple $(q_U\colon \mathcal X' \to U, f', \Delta')$ is a nice triple over $U$ since $\sigma$ is a finite surjective $U$-morphism. See Definition 3.1 in [18] for the definition of a nice triple. The morphism $q_X$ is not equal to $\operatorname{can}{\circ}\, q_U$ since $f' \in q^*_X(\mathrm{f})k[\mathcal X']$ and the morphism $q_U|_{\mathcal Z'}\colon \mathcal Z'=\{f'=0\} \to U$ is finite. To state a corollary of Theorem 6.1 (see Corollary 6.3), we note that parts (b) and (c) of Theorem 6.1 enable us to find an element $g \in I(\mathcal Z'')$ such that (1) $(f')+(g)=\Gamma(\mathcal X', \mathcal O_{\mathcal X'})$; (2) $\operatorname{Ker}((\Delta')^*)+(g)=\Gamma(\mathcal X', \mathcal O_{\mathcal X'})$; (3) $\sigma_g=\sigma|_{\mathcal X'_g}\colon \mathcal X'_g \to \mathbf{A}^1_U$ is étale. Here is the corollary. It is proved in [19], Corollary 7.2. Corollary 6.3. The function $f'$ in Theorem 6.1, the polynomial $h$ in part (f) of that theorem, the morphism $\sigma\colon \mathcal X' \to \mathbf{A}^1_U$ 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'_g}\colon \mathcal X'_g \to \mathbf{A}^1\times U $ is étale. (ii) The data $(\mathcal O[t],\sigma^*_g\colon \mathcal O[t] \to \Gamma(\mathcal X',\mathcal O_{\mathcal X'})_g, h)$ satisfy the hypotheses of Proposition 2.6 in [8], that is, $\Gamma(\mathcal X',\mathcal O_{\mathcal X'})_g$ is a finitely generated $\mathcal O[t]$-algebra, the element $(\sigma_g)^*(h)$ is not a zero divisor in $\Gamma(\mathcal X',\mathcal O_{\mathcal X'})_g$, and
$$
\begin{equation*}
\mathcal O[t]/(h)=\Gamma(\mathcal X',\mathcal O_{\mathcal X'})_g/h\Gamma(\mathcal X',\mathcal O_{\mathcal X'})_g.
\end{equation*}
\notag
$$
(iii) $(\Delta(U) \cup \mathcal Z') \subset \mathcal X'_g$ and $\sigma_g \circ \Delta=i_0\colon U\to \mathbf{A}^1\times U$. (iv) $\mathcal X'_{gh} \subseteq \mathcal X'_{gf'}\subseteq \mathcal X'_{f'}\subseteq \mathcal X'_{q^*_X(\mathrm{f})}$. (v) $\mathcal O[t]/(h)=\Gamma(\mathcal X',\mathcal O_{\mathcal X'})/(f')$, $h\Gamma(\mathcal X',\mathcal O_{\mathcal X'})=(f')\cap I(\mathcal Z'')$ and $(f') +I(\mathcal Z'')=\Gamma(\mathcal X',\mathcal O_{\mathcal X'})$. § 7. PurityLet $S$ be a regular ring, $\mathbf{G}$ a reductive $S$-group scheme, $\mathbf{T}$ an $S$-torus, $\mu\colon \mathbf{G}\to \mathbf{T}$ an $S$-group scheme morphism which is smooth as a scheme morphism. Suppose that $S$ is an integral domain containing a field. Let $L$ be its field of fractions. Given any $S$-algebra $S'$, we will write
$$
\begin{equation*}
\mathcal{F}(S') \quad \text{for} \quad \mathbf{T}(S')/\mu(\mathbf{G}(S'))
\end{equation*}
\notag
$$
in this section. For every $a\in \mathbf{T}(S')$, we denote the class of $a$ in $\mathcal{F}(S')$ by $\overline a$. Let $\mathfrak p$ be a prime ideal of height one in $S$. Then, by Theorem 1.1, the group $\mathcal{F}(S_{\mathfrak p})$ is a subgroup of $\mathcal{F}(L)$. We recall some notions. Given an element $a\in \mathbf{T}(L)$ and a prime ideal $\mathfrak p\subset S$ of height one, we say that $\overline a\in \mathcal{F}(L)$ is unramified at $\mathfrak p$ if $\overline a$ is in $\mathcal{F}(S_{\mathfrak p})$. We say that the element $\overline a\in \mathcal{F}(L)$ is $S$-unramified if, for every prime ideal $\mathfrak p$ of height one in $S$, the element $\overline a$ is in $\mathcal{F}(S_{\mathfrak p})$. The image of $\mathcal{F}(S)$ in $\mathcal{F}(L)$ is clearly contained in $\bigcap \mathcal{F}(S_{\mathfrak p})$, where the intersection is taken over all prime ideals of height one in $S$. We say that purity holds for the ring $S$ if
$$
\begin{equation*}
\operatorname{Im}[\mathcal{F}(S)\to \mathcal{F}(L)]=\bigcap \mathcal{F}(S_{\mathfrak p}).
\end{equation*}
\notag
$$
Equivalently, purity holds for $S$ if each $S$-unramified element of $\mathcal{F}(L)$ comes from $\mathcal{F}(S)$. It is clear that the sequence
$$
\begin{equation*}
\{1\} \to \mathcal{F}(S_{\mathfrak p}) \to \mathcal{F}(L) \xrightarrow{r_{\mathfrak p}} \mathbf{T}(L)/[\mathbf{T}(S_{\mathfrak p})\cdot \mu(\mathbf{G}(L))] \to \{1\}
\end{equation*}
\notag
$$
is exact, where $r_{\mathfrak p}$ is the factorization map. Thus, for an element $a\in \mathbf{T}(L)$, its class $\overline a$ in $\mathcal{F}(L)$ is unramified at $\mathfrak p$ if and only if $r_{\mathfrak p}(\overline a)= 0$. Hence purity holds for $S$ if and only if the sequence $\mathcal{F}(S)\to \mathcal{F}(L)\xrightarrow{\sum r_{\mathfrak p}} \bigoplus_{\mathfrak p} \mathbf{T}(L)/[\mathbf{T}(S_{\mathfrak p})\cdot \mu(\mathbf{G}(L))]$ is exact. Our aim in this section is to prove the following result. $(\ast)$ Purity holds for the ring $R$, the $R$-group schemes $\mathbf{G}$, $\mathbf{T}$ and the morphism $\mu$ occurred in Theorem 1.1. The proof of $(\ast)$ is divided into several steps. Claim 7.1. Let $X$ be a $k$-smooth irreducible affine $k$-variety, $\mathbf{G}$ a reductive $X$-group scheme, $\mathbf{T}$ an $X$-torus, and $\mu\colon \mathbf{G} \to \mathbf{T}$ an $X$-group scheme morphism which is smooth as an $X$-scheme morphism. Suppose that the $k$-algebra $R$ is the semi-local ring of finitely many closed points on $X$. Then purity holds for $R$. Proof. We repeat the proof of Theorem 1.1 in [2] verbatim with the following changes. Replace references to Corollary 4.3, (ii), (v) in [2] by references to parts (ii) and (v) of Corollary 6.3. Replace the reference to Lemma 6.7 in [2] by reference to Lemma 5.7. Replace the reference to Theorem 6.5 in [2] by reference to Theorem 5.5. Replace the reference to Corollary 6.6 in [2] by reference to Corollary 5.6. Replace the reference to Theorem 4.1 in [2] by reference to Theorem 6.1. Replace the reference to Definition 6.4 in [2] by reference to the remark at the end of Definition 5.4. $\Box$ Claim 7.2. Let $X$ be a $k$-smooth irreducible affine $k$-variety and let $\xi_1,\dots,\xi_n$ be points of $X=\operatorname{Spec}(k[X])$ such that, for every pair $r,s$, the point $\xi_r$ is not in the closure $\overline {\{\xi_s\}}$ of the point $\xi_s$. We write $R$ for the semi-local ring $\mathcal{O}_{X,\xi_1,\dots,\xi_n}$ of scheme points $\xi_1,\dots,\xi_n$ of $\operatorname{Spec}(k[X])$. Let $\mathbf{G}$ be a reductive $X$-group scheme, $\mathbf{T}$ an $X$-torus, and $\mu\colon\mathbf{G} \to \mathbf{T}$ an $X$-group scheme morphism which is smooth as an $X$-scheme morphism. Then purity holds for $R$. Proof. Take an element $a \in \mathbf{T}(k(X))$ whose class $\overline a$ is unramified at every irreducible divisor $D\,{\subset}\, X$ containing at least one of the points $\xi_r$. We have to prove that the element $\overline a\in \mathcal{F}(K)$ is in the image of $\mathcal{F}(R)$. Clearly, there is a non-zero element $f\in k[X]$ such that $a \in \mathbf{T}(k[X_f])$. We represent the divisor $\operatorname{div}(f)\in \operatorname{Div}(X)$ in the form $\operatorname{div}(f)=\sum m_iD_i + \sum n_jD'_j$ such that for every index $i$ there is an index $r$ with $\xi_r\in D_i$ and, for every index $j$ and every index $r$, the point $\xi_r$ does not belong to $D'_j$. There is an element $g\in k[X]$ such that, for every index $j$, the divisor $D'_j$ is contained in the closed set $\{g=0\}$ and $g$ does not belong to any prime ideal $\xi_r$ (we recall that the points of $X=\operatorname{Spec}(k[X])$ are the prime ideals in $k[X]$). Replacing $X$ by $X_g$, we see that $a \in \mathbf{T}(k[X_f])$, $\operatorname{div}(f)=\sum m_iD_i$ and $\overline a$ is unramified at every irreducible divisor $D_i$. Hence $\overline a$ is unramified at every prime ideal of height $1$ in $k[X]$. Our assumptions on the points $\xi_r$ guarantee that there are closed points $x_r\in \overline {\{\xi_s\}}$ such that, for every $r\neq s$, the point $x_r$ does not lie in the closure $\overline {\{\xi_s\}}$ of the point $\xi_s$. In particular, $x_r\neq x_s$ whenever $r\neq s$. The element $\overline a\in \mathcal{F}(k(X))$ is unramified at every prime ideal of height $1$ in $k[X]$. Thus, by Claim 7.1, the element $\overline a\in \mathcal{F}(k(X))$ is in the image of $\mathcal{F}(\mathcal{O}_{X,x_1,\dots,x_n})$. Therefore $\overline a$ is in the image of $\mathcal{F}(\mathcal{O}_{X,\xi_1,\dots,\xi_n})=\mathcal{F}(R)$. $\Box$ Proof. Clearly, we can assume that $k$ is a prime field and, therefore, $k$ is perfect. It follows from Popescu’s theorem [20], [21] that the $k$-algebra $R$ is a filtered inductive limit of $k$-smooth $k$-algebras $R_{\alpha}$. Modifying the inductive system of $k$-algebras $R_{\alpha}$ if necessary, we can assume that each $R_{\alpha}$ is an integral domain. For every maximal ideal $\mathfrak m_i$ in $R$ ($i = 1,\dots,n$) we put $\mathfrak p_i =\varphi^{-1}_{\alpha}(\mathfrak m_i)$. The homomorphism $\varphi_{\alpha}\colon R_{\alpha}\to R$ induces a homomorphism $\varphi'_{\alpha}\colon (R_{\alpha})_{\mathfrak p_1,\dots,\mathfrak p_n} \to R$ of semi-local rings. Till the end of the proof, we will write $A_{\alpha}$ for $(R_{\alpha})_{\mathfrak p_1,\dots,\mathfrak p_n}$ and $A$ for $R$ (to preserve the consistency of notation). Thus $A$ is a filtered inductive limit of regular semi-local $k$-algebras $A_{\alpha}$.
One can choose an index $\alpha$, a reductive $A_{\alpha}$-group scheme $\mathbf{G}_{\alpha}$, a torus $\mathbf{T}_{\alpha}$ over $A_{\alpha}$, and an $A_{\alpha}$-group scheme morphism $\mu_{\alpha}\colon \mathbf{G}_{\alpha} \to \mathbf{T}_{\alpha}$ which is smooth as an $A_{\alpha}$-scheme morphism in such a way that $\mathbf{G}=\mathbf{G}_{\alpha}\times_{\operatorname{Spec}(A_{\alpha})} \operatorname{Spec}(A)$, $\mathbf{T}=\mathbf{T}_{\alpha}\times_{\operatorname{Spec}(A_{\alpha})} \operatorname{Spec}(A)$, $\mu=\mu_{\alpha}\times_{\operatorname{Spec}(A_{\alpha})} \operatorname{Spec}(A)$. Replacing the index system by a co-final subsystem consisting of indices $\beta\geqslant \alpha$, we may and will suppose that the reductive group scheme $\mathbf{G}$, the torus $\mathbf{T}$ and the group scheme morphism $\mu\colon \mathbf{G}\to \mathbf{T}$ come from $A_{\alpha}$ and, moreover, $\mu$ is smooth as an $A_{\alpha}$-scheme morphism. These observations and Claim 7.2 yield the following intermediate result. $(\ast\ast)$ For these $\mathbf{G}$, $\mathbf{T}$ and $\mu\colon \mathbf{G}\to \mathbf{T}$ over $A_{\alpha}$, purity holds for every ring $A_{\beta}$ with $\beta\geqslant \alpha$. Let $K$ be the field of fractions of $A$. For every $\beta\geqslant \alpha$ let $K_\beta$ be the field of fractions of $A_\beta$. Given any $\beta\geqslant \alpha$, we write $\mathfrak a_{\beta}$ for the kernel of the map $\varphi'_{\beta}\colon A_{\beta} \to A$ and put $B_{\beta}=(A_{\beta})_{\mathfrak a_{\beta}}$. It is clear that $K_{\beta}$ is the ring of fractions of $B_{\beta}$ for every $\beta\geqslant \alpha$. The composite map $A_{\beta} \to A \to K$ factors through $B_{\beta}$. Since $A$ is a filtering direct limit of the rings $A_{\beta}$, we see that $K$ is a filtering direct limit of the rings $B_{\beta}$. We write $\psi_{\beta}$ for the canonical morphism $B_{\beta} \to K$. Lemma 7.4. For every $\beta\geqslant \alpha$, the map $\mathcal{F}(B_{\beta})\to \mathcal{F}(K_{\beta})$ is injective. Lemma 7.5. Let $a\in \mathcal{F}(K)$ be an $A$-unramified element. Then one can find an index $\beta\geqslant \alpha$ and an element $b_{\beta} \in \mathcal{F}(B_{\beta})$ such that $\psi_{\beta}(b_{\beta})=a$ and the class $b_{\beta} \in \mathcal{F}(K_{\beta})$ is $A_{\beta}$-unramified. Proof. Repeat verbatim the proof of Lemma 9.0.9 in [16]. It also holds in the semi-local case. $\Box$ We complete the proof of Claim 7.3 as follows. Let $a\in \mathcal{F}(K)$ be an $A$-unramified element. We need to check that it comes from $\mathcal{F}(A)$. By Lemma 7.5 one can find an index $\beta\geqslant \alpha$ and an element $b_{\beta} \in \mathcal{F}(B_{\beta})$ such that $\psi_{\beta}(b_{\beta})=a$ and the class $b_{\beta} \in \mathcal{F}(K_{\beta})$ is $A_{\beta}$-unramified. Fix $\beta$ and consider the following commutative diagram of $k$-algebras: The class $b_{\beta} \in \mathcal{F}(K_{\beta})$ is $A_{\beta}$-unramified. Hence, by $(\ast\ast)$, there is an element $a_{\beta} \in \mathbf{T}(A_{\beta})$ such that $b_{\beta}=\overline a_{\beta}$ in $\mathcal{F}(K_{\beta})$. Lemma 7.4 yields that $b_{\beta}=\overline a_{\beta}$ in $\mathcal{F}(B_{\beta})$. Hence the element $a\,{\in}\,\mathcal{F}(K)$ coincides with the image of the element $\varphi_{\beta} (\overline a_{\beta})$ in $\mathcal{F}(A)$. This proves Claim 7.3. Thus the sequence (1) is exact at its middle term. $\Box$§ 8. Proof of Theorem 1.1We begin this section with the following preliminary comment. Let $S$ be a regular semi-local domain and let $L$ be its field of fractions. Then, for every prime ideal $\mathfrak q\subset S$ of height $1$, the ring $S_{\mathfrak q}$ is a discrete valuation ring and $L^{\times}/S^{\times}_{\mathfrak q}=\mathbb Z$. Since $S$ is factorial, the sequence $0\to S^{\times}\to L^{\times}\to \bigoplus_{\mathfrak q} L^{\times}/S^{\times}_{\mathfrak q}\to 0$ is short exact, where $\mathfrak q$ runs over all prime ideals of height $1$ in $S$ and, for every such ideal $\mathfrak q$, the map $L^{\times}\to L^{\times}/S^{\times}_{\mathfrak q}$ is the factorization map. If $\mathcal T$ is a split $S$-torus, then it is clear that the sequence $0\to \mathcal T(S)\to \mathcal T(L)\to \bigoplus_{\mathfrak q} \mathcal T(L)/\mathcal T(S_{\mathfrak q})\to 0$ is also short exact. Proof of Theorem 1.1. The first assertion of Theorem 1.1 was proved in § 3. The exactness of (1) at its middle term was proved in § purity.
We now prove that the map $\sum r_{\mathfrak p}$ is surjective. It is clearly sufficient to prove the surjectiivity of the map $\mathbf T(K) \xrightarrow{\sum r'_{\mathfrak p}} \bigoplus_{\mathfrak p} \mathbf T(K)/\mathbf T(R_{\mathfrak p})$, where $\mathfrak p$ runs over all prime ideals of height $1$ in $R$ and $r'_{\mathfrak p}$ is the factorization map. We follow the arguments in [3], § 9. We prefer to switch to the scheme terminology. Put $X:=\operatorname{Spec}(R)$. Consider a finite étale Galois morphism $\pi\colon \widetilde X\to X$ such that the torus $\mathbf{T}$ splits over $\widetilde X$. Let $\operatorname{Gal}:=\operatorname{Aut}(\widetilde X/X)$ be its Galois group. The torus $\mathbf{T}$ splits over $\widetilde X$. Therefore, by the comment above, we have a short exact sequence of $\operatorname{Gal}$-modules
$$
\begin{equation*}
0\to \mathbf{T}(\widetilde{\mathcal{O}})\to \mathbf{T}(\widetilde K)\to \bigoplus_{y} \mathbf{T}(\widetilde K)/\mathbf{T}(\widetilde{\mathcal{O}}_{X,y})\to 0,
\end{equation*}
\notag
$$
where $\widetilde{\mathcal{O}}=\Gamma(\widetilde X, \mathcal O_{\widetilde X})$, $\widetilde K$ is the fraction field of $\widetilde{\mathcal{O}}$, $y$ runs over the set $X^{(1)}$ of points of codimension $1$ in $X$ and, for every $y\in X^{(1)}$, the ring $\widetilde{\mathcal{O}}_{X,y}$ is the semi-local ring $\mathcal{O}_{\widetilde X, \widetilde y}$ of the finite set $\widetilde y=\pi^{-1}(y)$ on the scheme $\widetilde X$. We write $\mathcal{O}$ for $R$ in order to be consistent with the notation above.
This short exact sequence of $\operatorname{Gal}$-modules yields the following long exact sequence of $\operatorname{Gal}$-cohomology groups:
$$
\begin{equation*}
\begin{aligned} \, 0 &\to \mathbf{T}(\mathcal{O})\xrightarrow{\mathrm{in}} \mathbf{T}(K)\to \bigoplus_{y} [\mathbf{T}(\widetilde K)/\mathbf{T}(\widetilde{\mathcal{O}}_{X,y})]^{\mathrm{Gal}} \\ &\to \mathrm{H}^1(\operatorname{Gal},\mathbf{T}(\widetilde{\mathcal{O}})) \xrightarrow{\mathrm{H}^1(\mathrm{in})} \mathrm{H}^1(\operatorname{Gal},\mathbf{T}(\widetilde K)). \end{aligned}
\end{equation*}
\notag
$$
We claim that the map $\mathrm{H}^1(\mathrm{in})$ is a monomorphism. Indeed, the group $\mathrm{H}^1(\operatorname{Gal}, \mathbf{T}(\widetilde{\mathcal{O}}))$ is a subgroup of $\mathrm H^1_{\unicode{x00E8}\textrm{t}}(X, \mathbf{T})$ while $\mathrm{H}^1(\operatorname{Gal},\mathbf{T}(\widetilde K))$ is a subgroup of $\mathrm H^1_{\unicode{x00E8}\textrm{t}}(\operatorname{Spec}K, \mathbf{T}_K)$. By Theorem 1.5, the map $\mathrm H^1_{\unicode{x00E8}\textrm{t}}(X, \mathbf{T})\to \mathrm H^1_{\unicode{x00E8}\textrm{t}}(\operatorname{Spec}K, \mathbf{T}_K)$ is a monomoprhism. Hence so is $\mathrm{H}^1(\mathrm{in})$. Thus we have a short exact sequence of the form $0\to \mathbf{T}(\mathcal{O})\xrightarrow{\mathrm{in}} \mathbf{T}(K)\to \bigoplus_{y} [\mathbf{T}(\widetilde K)/\mathbf{T}(\widetilde{\mathcal{O}}_{X,y})]^{\mathrm{Gal}}\to 0$.
There is a complex $0\to \mathbf{T}(\mathcal{O})\xrightarrow{\mathrm{in}} \mathbf{T}(K)\to \bigoplus_{y} \mathbf{T}(K)/T(\mathcal{O}_{X,y})$. Put $\alpha=\operatorname{id}_{\mathbf{T}(\mathcal{O})}$, $\beta=\operatorname{id}_{\mathbf{T}(K)}$ and $\gamma=\bigoplus_{y} \gamma_y $, where $\gamma_y\colon \mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})\to [\mathbf{T}(\widetilde K)/\mathbf{T}(\widetilde{\mathcal{O}}_{X,y})]^{\mathrm{Gal}}$ is induced by the inclusion $K\subset \widetilde K$. The maps $\alpha$, $\beta$ and $\gamma$ form a morphism of this complex to the short exact sequence above. We claim that this morphism is an isomorphism. This claim completes the proof of the theorem. To prove the claim, it suffices to establish that $\gamma$ is an isomorphism. Since the map $\mathbf{T}(K)\to \bigoplus_{y} [\mathbf{T}(\widetilde K)/\mathbf{T}(\widetilde{\mathcal{O}}_{X,y})]^{\mathrm{Gal}}$ is an epimorphism, so is $\gamma$. It remains to prove that $\gamma$ is an isomorphism. To do this, it is sufficient to check that the map $\mathbf{T}(K)/\mathbf{T}(\mathcal{O}_{X,y})\to \mathbf{T}(\widetilde K)/\mathbf{T}(\widetilde{\mathcal{O}}_{X,y})$ is a monomorphism for every point $y\in X^{(1)}$. Denoting the latter map by $\varepsilon_y$, we have to check the equality
$$
\begin{equation*}
\mathbf{T}(\mathcal{O}_{X,y})=\mathbf{T}(K)\cap \mathbf{T}(\widetilde{\mathcal{O}}_{X,y}).
\end{equation*}
\notag
$$
Since $\mathbf{T}$ is an affine $X$-scheme, we can embed it in an affine space and it suffices to check the equality $\mathcal{O}_{X,y}=K\cap\widetilde{\mathcal{O}}_{X,y}$. This equality holds because $\mathcal{O}_{X,y}$ is a discrete valuation ring: there is no ring sitting strictly between it and its field of fractions. The injectivity of $\varepsilon_y$ is proved. The surjectivity of the map $\sum r_{\mathfrak p}$ is proved. Theorem 1.1 is proved.
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\Bibitem{Pan22} |
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