3.1 Two facts about the Witt ring

Let $F$ be a field of characteristic $\neq 2$ . We let $W(F)$ and $I(F)$ denote the Witt ring of $F$ and its fundamental ideal, respectively. For a nondegenerate quadratic form $q$ over $F$ , $[q]$ denotes the corresponding class in $W(F)$ . As usual, the quadratic form $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}$ with $a_{i}\in F^{\times }$ will be denoted by $\langle a_{1},\ldots ,a_{n}\rangle$ , whereas $\langle \!\langle a_{1},\ldots ,a_{d}\rangle \!\rangle$ will be used to denote the $d$ -fold Pfister form $\langle 1,-a_{1}\rangle \otimes \cdots \otimes \langle 1,-a_{d}\rangle$ . Clearly, for any $d\geqslant 1$ , the $d$ th power $I(F)^{d}$ is additively generated by the classes of $d$ -fold Pfister forms.

Proof. The fact that $[\unicode[STIX]{x1D706}q]\in I(F)^{d}$ is obvious. Furthermore, we have

$$\begin{eqnarray}[q]-[\unicode[STIX]{x1D706}q]=[q\bot (-\unicode[STIX]{x1D706}q)]=[\langle 1,-\unicode[STIX]{x1D706}\rangle \otimes q]\in I(F)^{d+1},\end{eqnarray}$$

as required. ◻

Now, let $F$ be a field complete with respect to a discrete valuation $v$ such that the characteristic of the residue field $F^{(v)}$ is $\neq 2$ . We let $U(F)$ denote the group of units in $F$ , and fix a uniformizer $\unicode[STIX]{x1D70B}$ . Then one can define the first and second residue maps

$$\begin{eqnarray}\unicode[STIX]{x2202}_{i}:W(F)\rightarrow W(F^{(v)}),\quad i=1,2,\end{eqnarray}$$

which are homomorphisms of additive groups uniquely characterized by the conditions

$$\begin{eqnarray}\unicode[STIX]{x2202}_{1}\langle u\rangle =\langle \bar{u}\rangle ,\quad \unicode[STIX]{x2202}_{1}\langle \unicode[STIX]{x1D70B}u\rangle =0\quad \text{and}\quad \unicode[STIX]{x2202}_{2}\langle u\rangle =0,\quad \unicode[STIX]{x2202}_{2}\langle \unicode[STIX]{x1D70B}u\rangle =\langle \bar{u}\rangle ,\end{eqnarray}$$

for any $u\in U(F)$ (where $\bar{u}$ denote the image of $u$ in $F^{(v)}$ ) (see [Reference LamLam05, ch. VI, § 1] or [Reference MilnorMil70, § 5] for the details). Let $W_{0}(F)$ be the subring of $W(F)$ generated by the classes of $\langle u\rangle$ for $u\in U(F)$ . Then $W_{0}(F)=\operatorname{Ker}\unicode[STIX]{x2202}_{2}$ , whereas $\unicode[STIX]{x2202}_{1}$ yields an isomorphism between $W_{0}(F)$ and $W(F^{(v)})$ . Thus, we have the following split exact sequence

$$\begin{eqnarray}0\rightarrow W_{0}(F)\longrightarrow W(F)\stackrel{\unicode[STIX]{x2202}_{2}}{\longrightarrow }W(F^{(v)})\rightarrow 0.\end{eqnarray}$$

Let $I_{0}(F)=W_{0}(F)\,\cap \,I(F)$ . Milnor [Reference MilnorMil70, § 5] shows that for any $d\geqslant 1$ , the restriction of $\unicode[STIX]{x2202}_{2}$ to $I^{d}(F)$ (the $d$ th power of $I(F)$ ) yields the exact sequence

$$\begin{eqnarray}0\rightarrow I_{0}^{d}(F)\longrightarrow I^{d}(F)\stackrel{\unicode[STIX]{x2202}_{2}}{\longrightarrow }I^{d-1}(F^{(v)})\rightarrow 0.\end{eqnarray}$$

This, in particular, gives the following result.

3.2 The Milnor isomorphism and unramified classes

Again let $F$ be a field of characteristic $\neq 2$ . It is a consequence of Voevodsky’s proof of the Milnor conjecture (see [Reference Orlov, Vishik and VoevodskyOVV07, Reference VoevodskyVoe03, Reference VoevodskyVoe11]) that for $d\geqslant 1$ , there are natural isomorphisms of Abelian groups

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{F,d}:I(F)^{d}/I(F)^{d+1}\longrightarrow H^{d}(F,\unicode[STIX]{x1D707}_{2}).\end{eqnarray}$$

Explicitly, $\unicode[STIX]{x1D6FE}_{F,d}$ is defined by sending the class of the Pfister form $\langle \!\langle a_{1},\ldots ,a_{d}\rangle \!\rangle$ to the cup-product $\unicode[STIX]{x1D712}_{a_{1}}\,\cup \,\cdots \,\cup \,\unicode[STIX]{x1D712}_{a_{d}}$ , where for $a\in F^{\times }$ , we let $\unicode[STIX]{x1D712}_{a}$ be the corresponding 1-cocycle given by Kummer theory.

Proof. It immediately follows from Lemma 3.1 that $[q]\in I(F)^{d}$ and $\unicode[STIX]{x1D6FE}_{F,d}([q])=\unicode[STIX]{x1D6FE}_{F,d}([\unicode[STIX]{x1D706}q])$ . Thus, it is enough to show that if $[q]\in I(F)^{d}\,\cap \,W_{0}(F)$ , then $\unicode[STIX]{x1D6FE}_{F,d}([q])$ is unramified at $v$ . According to Lemma 3.2, we have $I(F)^{d}\,\cap \,W_{0}(F)=I_{0}(F)^{d}$ , which is additively generated by the classes of Pfister forms $\langle \!\langle a_{1},\ldots ,a_{d}\rangle \!\rangle$ with $a_{i}\in U(F)$ . However, for any such form, the corresponding class

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{F,d}([\langle \!\langle a_{1},\ldots ,a_{d}\rangle \!\rangle ])=\unicode[STIX]{x1D712}_{a_{1}}\cup \cdots \cup \unicode[STIX]{x1D712}_{a_{d}}\end{eqnarray}$$

is clearly unramified, and our claim follows. ◻

3.3 Proof of Theorem 2.1

Let $\{q_{i}\}\text{}_{i\in I}$ be a family of $n$ -dimensional nondegenerate quadratic forms over $K$ such that:

• ∙ for each $i\in I$ , the spinor group $G_{i}=\operatorname{Spin}_{n}(q_{i})$ has good reduction at all $v\in V$ ; and

• ∙ for $i,j\in I$ , $i\neq j$ , the forms $q_{i}$ and $q_{j}$ are not similar (i.e.  $q_{i}$ is not equivalent to any nonzero scalar multiple of $q_{j}$ ).

We wish to show that $I$ is finite and

(1) $$\begin{eqnarray}|I|\leqslant d_{0}d_{1}\cdots d_{\ell },\end{eqnarray}$$

where

$$\begin{eqnarray}d_{0}=|\text{Pic}(V)/2\cdot \operatorname{Pic}(V)|\quad \text{and}\quad d_{i}=|H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}|\quad \text{for }i=1,\ldots ,\ell =[\log _{2}n]+1.\end{eqnarray}$$

We begin by observing that the first of the above conditions implies that for each $i\in I$ and any $v\in V$ , there exists $\unicode[STIX]{x1D706}_{v}^{(i)}\in K_{v}^{\times }$ such that the form $\unicode[STIX]{x1D706}_{v}^{(i)}q_{i}\in W_{0}(K_{v})$ in the above notation. In addition, we may assume without loss of generality that for each $i\in I$ , we have $\unicode[STIX]{x1D706}_{v}^{(i)}=1$ for almost all $v$ ; then $\unicode[STIX]{x1D706}^{(i)}:=(\unicode[STIX]{x1D706}_{v}^{(i)})_{v\in V}\in \mathbb{I}(K,V)$ . Using Lemma 2.2, we see that

$$\begin{eqnarray}\mathbb{I}(K,V)/\mathbb{I}(K,V)^{2}\mathbb{I}_{0}(K,V)K^{\times }\simeq \operatorname{Pic}(V)/2\cdot \operatorname{Pic}(V).\end{eqnarray}$$

Thus, there exists a subset $J_{0}\subset I$ of size ${\geqslant}|I|/d_{0}$ (if $I$ is infinite then this simply means that $J_{0}$ is also infinite) such that all $\unicode[STIX]{x1D706}^{(i)}$ for $i\in J_{0}$ have the same image in $\mathbb{I}(K,V)/\mathbb{I}(K,V)^{2}\mathbb{I}_{0}(K,V)K^{\times }$ . Now, fix $j_{0}\in J_{0}$ . For any $j\in J_{0}$ , we can write

$$\begin{eqnarray}\unicode[STIX]{x1D706}^{(j)}=\unicode[STIX]{x1D706}^{(j_{0})}(\unicode[STIX]{x1D6FC}^{(j)})^{2}\unicode[STIX]{x1D6FD}^{(j)}\unicode[STIX]{x1D6FF}^{(j)}\quad \text{with }\unicode[STIX]{x1D6FC}^{(j)}\in \mathbb{I}(K,V),~\unicode[STIX]{x1D6FD}^{(j)}\in \mathbb{I}_{0}(K,V),~\unicode[STIX]{x1D6FF}^{(j)}\in K^{\times },\end{eqnarray}$$

(assuming that the elements $\unicode[STIX]{x1D6FC}^{(j_{0})}$ , $\unicode[STIX]{x1D6FD}^{(j_{0})}$ and $\unicode[STIX]{x1D6FF}^{(j_{0})}$ are all trivial), and then set

$$\begin{eqnarray}\tilde{q}_{j}=\unicode[STIX]{x1D6FF}^{(j)}q_{j}\quad \text{and}\quad \tilde{\unicode[STIX]{x1D706}}^{(j)}=\unicode[STIX]{x1D706}^{(j)}(\unicode[STIX]{x1D6FF}^{(j)})^{-1}.\end{eqnarray}$$

We note that the forms $\tilde{q}_{j}$ for $j\in J_{0}$ remain pairwise nonsimilar, in particular, inequivalent. Furthermore, for any $v\in V$ we have $\tilde{\unicode[STIX]{x1D706}}_{v}^{(j)}\tilde{q}_{j}=\unicode[STIX]{x1D706}_{v}^{(j)}q_{j}$ , hence $[\tilde{\unicode[STIX]{x1D706}}_{v}^{(j)}\tilde{q}_{j}]\in W_{0}(K_{v})$ , and

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D706}}^{(j)}=\tilde{\unicode[STIX]{x1D706}}^{(j_{0})}(\unicode[STIX]{x1D6FC}^{(j)})^{2}\unicode[STIX]{x1D6FD}^{(j)}.\end{eqnarray}$$

Then for any $v\in V$ , the form

$$\begin{eqnarray}q(j,v):=\tilde{\unicode[STIX]{x1D706}}_{v}^{(j_{0})}(\tilde{q}_{j}\bot (-\tilde{q}_{j_{0}}))=(\unicode[STIX]{x1D6FC}_{v}^{(j)})^{-2}(\unicode[STIX]{x1D6FD}^{(j)})^{-1}(\tilde{\unicode[STIX]{x1D706}}_{v}^{(j)}\tilde{q}_{j})\bot \tilde{\unicode[STIX]{x1D706}}_{v}^{(j_{0})}(-\tilde{q}_{j_{0}})\end{eqnarray}$$

is equivalent to

$$\begin{eqnarray}(\unicode[STIX]{x1D6FD}_{v}^{(j)})^{-1}(\tilde{\unicode[STIX]{x1D706}}_{v}^{(j)}\tilde{q}_{j})\bot \tilde{\unicode[STIX]{x1D706}}_{v}^{(j_{0})}(-\tilde{q}_{j_{0}}).\end{eqnarray}$$

As $\unicode[STIX]{x1D6FD}_{v}^{(j)}\in U(K_{v})$ , we see that $[q(j,v)]\in W_{0}(K_{v})\,\cap \,I(K_{v})$ . Now, invoking Lemma 3.3 with $d=1$ , we obtain that $\unicode[STIX]{x1D6FE}_{K_{v},1}([\tilde{q}_{j}]-[\tilde{q}_{j_{0}}])\in H^{1}(K_{v},\unicode[STIX]{x1D707}_{2})$ is unramified at $v$ . This being true for all $v\in V$ , we conclude that

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{K,1}([\tilde{q}_{j}]-[\tilde{q}_{j_{0}}])\in H^{1}(K,\unicode[STIX]{x1D707}_{2})_{V}.\end{eqnarray}$$

Then one can find a subset $J_{1}\subset J_{0}$ of size

$$\begin{eqnarray}{\geqslant}|J_{0}|/d_{1}\geqslant |I|/(d_{0}d_{1})\end{eqnarray}$$

(again, if $I$ is infinite this simply means that $J_{1}$ is also infinite) such that the elements $[\tilde{q}_{j}]-[\tilde{q}_{j_{0}}]\in I(K)$ for $j\in J_{1}$ have the same image under $\unicode[STIX]{x1D6FE}_{K,1}$ . Fix $j_{1}\in J_{1}$ . Then for any $j\in J_{1}$ we have

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{K,1}([\tilde{q}_{j}]-[\tilde{q}_{j_{1}}])=\unicode[STIX]{x1D6FE}_{K,1}(([\tilde{q}_{j}]-[\tilde{q}_{j_{0}}])-([\tilde{q}_{j_{1}}]-[\tilde{q}_{j_{0}}]))=0,\end{eqnarray}$$

implying that $[\tilde{q}_{j}]-[\tilde{q}_{j_{1}}]\in I(K)^{2}$ . Furthermore, we observe that

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D706}}^{(j)}=\tilde{\unicode[STIX]{x1D706}}^{(j_{1})}(\bar{\unicode[STIX]{x1D6FC}}^{(j)})^{2}\bar{\unicode[STIX]{x1D6FD}}^{(j)}\quad \text{with }\bar{\unicode[STIX]{x1D6FC}}^{(j)}\in \mathbb{I}(K,V),~\bar{\unicode[STIX]{x1D6FD}}^{(j)}\in \mathbb{I}_{0}(K,V),\end{eqnarray}$$

using which one shows that for each $v\in V$ the class of the form $\tilde{\unicode[STIX]{x1D706}}_{v}^{(j_{1})}(\tilde{q}_{j}\bot (-\tilde{q}_{j_{1}}))$ lies in $W_{0}(K_{v})\,\cap \,I(K_{v})^{2}$ . Now, Lemma 3.3 with $d=2$ yields that

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{K,2}([\tilde{q}_{j}]-[\tilde{q}_{j_{1}}])\in H^{2}(K,\unicode[STIX]{x1D707}_{2})_{V}.\end{eqnarray}$$

Then there exists a subset $J_{2}\subset J_{1}$ of size ${\geqslant}|I|/(d_{0}d_{1}d_{2})$ such that the elements $[\tilde{q}_{j}]-[\tilde{q}_{j_{1}}]$ for $j\in J_{2}$ have the same image under $\unicode[STIX]{x1D6FE}_{K,2}$ . Consequently, fixing $j_{2}\in J_{2}$ , we will have $[\tilde{q}_{j}]-[\tilde{q}_{j_{2}}]\in I(K)^{3}$ for all $j\in J_{2}$ . Proceeding inductively, we construct a nested sequence of subsets

$$\begin{eqnarray}I\supset J_{0}\supset J_{1}\supset J_{2}\supset \cdots \supset J_{\ell }\end{eqnarray}$$

such that:

1. (a) $|J_{m}|\geqslant |I|/(d_{0}d_{1}\cdots d_{m})$ ; and

2. (b) fixing $j_{m}\in J_{m}$ , we will have $[\tilde{q}_{j}]-[\tilde{q}_{j_{m}}]\in I(K)^{m+1}$ for all $j\in J_{m}$ ,

for any $m=1,\ldots ,\ell$ . However, according to [Reference LamLam05, ch. X, Hauptsatz 5.1], the dimension of any positive-dimensional anisotropic form in $I(K)^{\ell +1}$ is ${\geqslant}2^{\ell +1}>2^{\log _{2}n+1}=2n$ . Thus, the fact that

$$\begin{eqnarray}[\tilde{q}_{j}]-[\tilde{q}_{j_{\ell }}]=[\tilde{q}_{j}\bot (-\tilde{q}_{j_{\ell }})]\in I(K)^{\ell +1}\end{eqnarray}$$

implies that the form $\tilde{q}_{j}\bot (-\tilde{q}_{j_{\ell }})$ is hyperbolic, and consequently $\tilde{q}_{j}$ and $\tilde{q}_{j_{\ell }}$ are equivalent. As $j\in J_{\ell }$ was arbitrary and the forms $\tilde{q}_{j}$ for $j\in J_{\ell }$ are pairwise inequivalent, we see that $J_{\ell }$ actually reduces to a single element. Then the inequality in part (a) yields the required estimation (1).◻

3.4 Another application of the method

The method developed to prove Theorem 2.1 can be used in various situations. Here we would like to indicate one application to the analysis of the global-to-local map in Galois cohomology.

Theorem 3.4. Let $K$ be a field equipped with a set $V$ of discrete valuations satisfying conditions $(\text{A})$ and $(\text{B})$ and let $n\geqslant 5$ be an integer. Assume that:

$(\bullet )$

for each $i=1,\ldots ,[\log _{2}n]+1=\ell$ , the kernel $\unicode[STIX]{x1D6FA}_{i}$ of the diagonal map

$$\begin{eqnarray}H^{i}(K,\unicode[STIX]{x1D707}_{2})\longrightarrow \mathop{\prod }_{v\in V}H^{i}(K_{v},\unicode[STIX]{x1D707}_{2})\end{eqnarray}$$

is finite of order $\unicode[STIX]{x1D714}_{i}$ .

Then for a nondegenerate $n$ -dimensional quadratic form $q$ over $K$ and the diagonal map

$$\begin{eqnarray}\unicode[STIX]{x1D70B}:H^{1}(K,\operatorname{SO}_{n}(q))\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},\operatorname{SO}_{n}(q)),\end{eqnarray}$$

the set $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D70B}(h))$ is finite of size ${\leqslant}\unicode[STIX]{x1D714}_{1}\cdots \unicode[STIX]{x1D714}_{\ell }$ for any $h\in H^{1}(K,\operatorname{SO}_{n}(q))$ . In particular, $\unicode[STIX]{x1D70B}$ is a proper map.

Proof. Let $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D70B}(h))=\{h_{i}\}\text{}_{i\in I}$ and let $q_{i}$ be the quadratic form obtained from $q$ by twisting using $h_{i}$ . It is well-known that the $K$ -equivalence classes of nondegenerate $n$ -dimensional quadratic forms over $K$ are in a natural bijective correspondence with the elements of $H^{1}(K,\text{O}_{n}(q))$ . As the map $H^{1}(K,\operatorname{SO}_{n}(q))\rightarrow H^{1}(K,\text{O}_{n}(q))$ is injective (cf. [Reference Platonov and RapinchukPR94, § 6.6]), the forms $q_{i}$ for $i\in I$ are pairwise inequivalent. On the other hand, by our construction, the form $q_{i}$ for any $i\in I$ is equivalent over $K_{v}$ to the form $q_{0}$ obtained from $q$ by twisting using $h$ , for any $v\in V$ . Thus, for any $i,j\in I$ , the class $[q_{i}]-[q_{j}]$ is trivial in $W(K_{v})$ for any $v\in V$ . Thus, using the compatibility of the map $\unicode[STIX]{x1D6FE}_{F,d}$ with base change, we conclude that if $[q_{i}]-[q_{j}]\in I(K)^{d}$ , then $\unicode[STIX]{x1D6FE}_{K,d}([q_{i}]-[q_{j}])\in \unicode[STIX]{x1D6FA}_{d}$ .

We now proceed as in the proof of Theorem 2.1. Fix $j_{0}\in I$ . Then there exists a subset $J_{1}\subset I$ of size ${\geqslant}|I|/\unicode[STIX]{x1D714}_{1}$ such that all elements $[q_{j}]-[q_{j_{0}}]\in I(K)$ for $j\in J_{1}$ have the same image under $\unicode[STIX]{x1D6FE}_{K,1}$ . Fix $j_{1}\in J_{1}$ . Then for any $j\in J_{1}$ we have

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{K,1}([q_{j}]-[q_{j_{1}}])=\unicode[STIX]{x1D6FE}_{K,1}(([q_{j}]-[q_{j_{0}}])-([q_{j_{1}}]-[q_{j_{0}}]))=0,\end{eqnarray}$$

which means that $[q_{j}]-[q_{j_{1}}]\in I(K)^{2}$ . Inductively, we construct a sequence of subsets $I\supset J_{1}\supset J_{2}\supset \cdots \supset J_{\ell }$ such that:

$(\text{a}^{\prime })$

$|J_{m}|\geqslant |I|/\unicode[STIX]{x1D714}_{1}\cdots \unicode[STIX]{x1D714}_{m}$ ;

$(\text{b}^{\prime })$

fixing $j_{m}\in J_{m}$ , we will have $[q_{j}]-[q_{j_{m}}]\in I(K)^{m+1}$ for all $j\in J_{m}$ ,

for any $m=1,\ldots ,\ell$ . As in the proof of Theorem 2.1, we see that the fact that $[q_{j}]\,-\,[q_{j_{\ell }}]\in I(K)^{\ell +1}$ implies that $q_{j}$ and $q_{j_{\ell }}$ are actually equivalent. This means that $J_{\ell }$ reduces to a single element, and then the inequality in $(\text{a}^{\prime })$ yields the required estimation.◻

We observe that for any $i$ , we have the inclusion $\unicode[STIX]{x1D6FA}_{i}\subset H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ in the above notation. In particular, the finiteness of $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ implies that of $\unicode[STIX]{x1D6FA}_{i}$ , which enables us to apply Theorem 3.4.

4.1 The proof

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be any divisorial set of places of $K$ associated with a (normal) model $\mathfrak{X}$ of finite type on which $2$ is invertible (thus, $\operatorname{char}K^{(v)}\neq 2$ for all $v\in V$ ). According to Theorem 2.1, it is enough to establish the finiteness of the quotient $\operatorname{Pic}(V)/2\cdot \operatorname{Pic}(V)$ and of the unramified cohomology groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for all $i\geqslant 1$ . However, it easily follows from the definitions that the group $\operatorname{Pic}(V)$ coincides with the usual Picard group $\operatorname{Pic}(\mathfrak{X})$ of the scheme $\mathfrak{X}$ . Thus, the finiteness of $\operatorname{Pic}(V)/2\cdot \operatorname{Pic}(V)$ is a consequence of the following well-known statement.

Sketch of proof.

As $n$ is invertible on $\mathfrak{X}$ , we have the following Kummer sequence of étale sheaves on $\mathfrak{X}$

$$\begin{eqnarray}1\rightarrow \unicode[STIX]{x1D707}_{n,\mathfrak{X}}\longrightarrow \mathbb{G}_{m,\mathfrak{X}}\stackrel{[n]}{\longrightarrow }\mathbb{G}_{m,\mathfrak{X}}\rightarrow 1,\end{eqnarray}$$

which gives rise to the long exact sequence

$$\begin{eqnarray}H^{1}(\mathfrak{X},\unicode[STIX]{x1D707}_{n,\mathfrak{X}})\longrightarrow H^{1}(\mathfrak{X},\mathbb{G}_{m,\mathfrak{X}})\stackrel{[n]}{\longrightarrow }H^{1}(\mathfrak{X},\mathbb{G}_{m,\mathfrak{X}})\longrightarrow H^{2}(\mathfrak{X},\unicode[STIX]{x1D707}_{n,\mathfrak{X}})\end{eqnarray}$$

in étale cohomology. As $\operatorname{Pic}(\mathfrak{X})=H^{1}(\mathfrak{X},\mathbb{G}_{m,\mathfrak{X}})$ (cf. [Reference MilneMil80, ch. III, Proposition 4.9]), we have a surjection $H^{1}(\mathfrak{X},\unicode[STIX]{x1D707}_{n,\mathfrak{X}}){\twoheadrightarrow}_{n\!}\operatorname{Pic}(\mathfrak{X})$ and an injection $\operatorname{Pic}(\mathfrak{X})/n\cdot \operatorname{Pic}(\mathfrak{X}){\hookrightarrow}H^{2}(\mathfrak{X},\unicode[STIX]{x1D707}_{n,\mathfrak{X}})$ . Thus, it is enough to prove the finiteness of $H^{i}(\mathfrak{X},\unicode[STIX]{x1D707}_{n,\mathfrak{X}})$ for $i=1,2$ . In fact, these groups are finite for all $i$ . For $\mathfrak{X}$ of finite type over a finite field, this is a consequence of [Reference DeligneSGA4½, Théorème 1.1 in ‘Théorèmes de finitude’] and the Leray spectral sequence [Reference MilneMil80, ch. III, Theorem 1.18] in general and is [Reference MilneMil80, ch. VI, Corollary 5.5] for $\mathfrak{X}$ smooth. For $\mathfrak{X}$ of finite type over $\mathbb{Z}$ , this is [Reference Chernousov, Rapinchuk and RapinchukCRR16a, Theorem 10.2].◻

It should be noted that Kahn [Reference KahnKah06] in fact showed that the group $\operatorname{Pic}(\mathfrak{X})$ is finitely generated.

Combining the finiteness of $_{2\!}\operatorname{Pic}(V)$ with the fact that the group of $V$ -units

$$\begin{eqnarray}U(V)=\{a\in K^{\times }\mid v(a)=0\text{ for all }v\in V\}\end{eqnarray}$$

is finitely generated (cf. [Reference SamuelSam66]) and applying [Reference Chernousov, Rapinchuk and RapinchukCRR16a, Proposition 5.1(a)], we obtain that $H^{1}(K,\unicode[STIX]{x1D707}_{2})_{V}$ is finite. The finiteness of $H^{2}(K,\unicode[STIX]{x1D707}_{2})_{V}=_{2\!}\operatorname{Br}(K)_{V}$ was established in [Reference Chernousov, Rapinchuk and RapinchukCRR16a, § 10]. The finiteness of $H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V}$ is a new result (see Corollary 6.2 of the more general Theorem 6.1) whose proof requires separate arguments in the characteristic zero and positive characteristic cases. The proof for the characteristic zero case that is given in § 6 was suggested by one of the referees; it uses several powerful results, first and foremost, those of Kato [Reference KatoKat86] on cohomological Hasse principles. We have included our original proof of Theorem 6.1 in Appendix A. It uses considerably less input; in particular, it does not rely on Kato’s local–global principle but is based instead on a modification of Jannsen’s [Reference JannsenJan89–90] proof of the latter. The reason we decided to keep this argument in the paper is that it appears to be more amenable to generalizations in the spirit of Jannsen’s proof [Reference JannsenJan16] of Kato’s local–global principle for higher-dimensional varieties that generalized his original argument [Reference JannsenJan89–90]. The finiteness of $H^{3}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for two-dimensional global fields of positive characteristic will be established in § 7 (we note that this result is implicitly contained in [Reference RapinchukRap18, Proposition 4.4 and Remark 4.5]).

As the cohomological dimension of a two-dimensional global field of positive characteristic is ${\leqslant}3$ (cf. [Reference SerreSer97, ch. II, 4.2]), the groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})$ vanish for all $i\geqslant 4$ . We will now show that in characteristic zero, the finiteness of $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for $i\geqslant 4$ easily follows from the results of Poitou–Tate (cf. [Reference SerreSer97, ch. II, 6.3]), which will complete the proof of Theorem 1.1.

To apply the proposition in our situation, we observe that given a divisorial set of places $V$ of a two-dimensional global field $K$ of characteristic zero, there exists a smooth geometrically connected curve $C$ defined over a number field $k$ such that $K=k(C)$ and for the set of geometric places $V_{0}$ of $K$ associated with the closed points of $C$ , we have the inclusion

$$\begin{eqnarray}H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}\subset H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V_{0}}.\end{eqnarray}$$

For the proof of the proposition, we first need to review the exact sequence for the étale cohomology of a curve, which will also be used in Appendix A.

4.2 The fundamental exact sequence

Let $C$ be a geometrically integral smooth affine curve over an arbitrary field $k$ and let $n\geqslant 2$ be an integer prime to $\operatorname{char}k$ . Then for any integer $d$ , we consider the Hochschild–Serre spectral sequence in étale cohomology

$$\begin{eqnarray}E_{2}^{p,q}=H^{p}(k,H^{q}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{n}^{\otimes d}))\Rightarrow H^{p+q}(C,\unicode[STIX]{x1D707}_{n}^{\otimes d}).\end{eqnarray}$$

As $C$ is affine, we have $H^{p}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{n}^{\otimes d})=0$ for $p\geqslant 2$ (cf. [Stacks19, Lemma 65.3], [Reference MilneMil80, ch. V, § 2]). Thus, for each $\ell \geqslant 1$ , the spectral sequence yields the following short exact sequence:

(2) $$\begin{eqnarray}H^{\ell }(k,\unicode[STIX]{x1D707}_{n}^{\otimes d})\stackrel{\unicode[STIX]{x1D704}_{k,n}^{d,\ell }}{\longrightarrow }H^{\ell }(C,\unicode[STIX]{x1D707}_{n}^{\otimes d})\stackrel{\unicode[STIX]{x1D714}_{k,n}^{d,\ell }}{\longrightarrow }H^{\ell -1}(k,H^{1}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{n}^{\otimes d})).\end{eqnarray}$$

Furthermore, let $V_{0}$ denote the set of places of $K=k(C)$ associated with the closed points of $C$ . Then, because $n$ is prime to $\operatorname{char}k$ , it follows from the Bloch–Ogus spectral sequence (see [Reference Bloch and OgusBO74, Reference Colliot-Thélène, Hoobler and KahnCHK97, Reference Jannsen, Sato and SaitoJSS14]) that for each $\ell \geqslant 1$ , the natural map $H^{\ell }(C,\unicode[STIX]{x1D707}_{n}^{\otimes d})\rightarrow H^{\ell }(K,\unicode[STIX]{x1D707}_{n}^{\otimes d})_{V_{0}}$ is surjective. (This is a consequence of the fact that the $E_{2}$ -terms in the Bloch–Ogus spectral sequence satisfy $E_{2}^{p,q}=0$ for all $p>\dim C=1$ and all $q$ , whereas $E_{2}^{0,q}=H^{q}(K,\unicode[STIX]{x1D707}_{n}^{\otimes d})_{V_{0}}$ .)

Let

$$\begin{eqnarray}M(n,d)=H^{1}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{n}^{\otimes d}).\end{eqnarray}$$

Equivalently,

(3) $$\begin{eqnarray}M(n,d)=\operatorname{Hom}(\unicode[STIX]{x1D70B}_{1}(C\otimes _{k}\bar{k},\bar{\unicode[STIX]{x1D702}}),\unicode[STIX]{x1D707}_{n}^{\otimes d})\quad \text{where }\bar{\unicode[STIX]{x1D702}}=\operatorname{Spec}\bar{K}.\end{eqnarray}$$

It follows that we have the following equality for the Tate twist

$$\begin{eqnarray}M(n,d)(\ell )=M(n,d+\ell ).\end{eqnarray}$$

4.3 Proof of Proposition 4.2

We may assume that $C$ is affine, and it is enough to show that $H^{i}(C,\unicode[STIX]{x1D707}_{n}^{\otimes d})$ is finite for all $i\geqslant 4$ . According to Poitou–Tate, for a number field $k$ and any finite Galois module $M$ , the natural homomorphism

$$\begin{eqnarray}H^{i}(k,M)\longrightarrow \mathop{\prod }_{v\in V_{\infty }^{k}}H^{i}(k_{v},M),\end{eqnarray}$$

where $V_{\infty }^{k}$ is the set of all real places of $k$ , is an isomorphism for any $i\geqslant 3$ (see [Reference SerreSer97, ch. II, 6.3, Theorem B]); in particular, the group $H^{i}(k,M)$ is finite. As $H^{1}(C\otimes _{k}\bar{k},\unicode[STIX]{x1D707}_{n}^{\otimes d})$ is finite by [Reference Artin, Deligne, Grothendieck and VerdierSGA4, Exposé XVI, Théorème 5.2], it follows that the left-most and the right-most terms in the exact sequence (2) are finite. Thus, the middle term is also finite for all $i\geqslant 4$ , as required.◻

5.1 The finiteness of genus

Theorem 1.2 will be derived from the following result.

To prove Theorem 5.1, we need the following general statement.

Let $V$ be a divisorial set of places of our two-dimensional global field $K$ of characteristic $\neq \,2$ . Using the fact that $V$ satisfies condition (A), it is not difficult to see that the set $v\in V$ where $G=\operatorname{Spin}_{n}(q)$ does not have good reduction, is finite. Picking a divisorial set of places in the complement of this finite set, we may assume without loss of generality that $G$ has good reduction at all $v\in V$ . Then according to Theorem 5.2, every $G^{\prime }\in \mathbf{gen}_{K}(G)$ has good reduction at all $v\in V$ . Thus, Theorem 1.1 yields the finiteness of the number of $K$ -isomorphism classes of the spinor groups $G^{\prime }=\operatorname{Spin}_{n}(q^{\prime })$ lying in $\mathbf{gen}_{K}(G)$ , which is precisely the assertion of Theorem 5.1.

Now, Theorem 1.2 for quadratic forms of odd dimension immediately follows from Theorem 5.1. To consider the case of isotropic quadratic forms $q$ of even dimension, we need to combine the latter with part (a) of the following proposition, which implies that every group in $\mathbf{gen}_{K}(G)$ is of the form $\operatorname{Spin}_{n}(q^{\prime })$ .

(Note that $d$ is a power of $2$ .)

We will first establish one general fact that involves the notion of generic tori, which we now recall for the reader’s convenience. Let $\mathscr{G}$ be a semi-simple algebraic group over a field $\mathscr{K}$ . Fix a maximal $\mathscr{K}$ -torus $\mathscr{T}$ of $\mathscr{G}$ , and let $\unicode[STIX]{x1D6F7}(\mathscr{G},\mathscr{T})$ and $W(\mathscr{G},\mathscr{T})$ denote the corresponding root system and the Weyl group. The natural action of the absolute Galois group $\operatorname{Gal}(\mathscr{K}^{\text{sep}}/\mathscr{K})$ , where $\mathscr{K}^{\text{sep}}$ is a fixed separable closure of $\mathscr{K}$ , on the character group $X(\mathscr{T})$ of $\mathscr{T}$ gives rise to a group homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{\mathscr{T}}:\operatorname{Gal}(\mathscr{K}^{\text{sep}}/\mathscr{K})\rightarrow \operatorname{Aut}(\unicode[STIX]{x1D6F7}(\mathscr{G},\mathscr{T}))\end{eqnarray}$$

that factors through the Galois group $\operatorname{Gal}(\mathscr{K}_{\mathscr{T}}/\mathscr{K})$ of the minimal splitting field of $\mathscr{T}$ in $\mathscr{K}^{\text{sep}}$ inducing an injective homomorphism $\bar{\unicode[STIX]{x1D703}}_{\mathscr{T}}:\operatorname{Gal}(\mathscr{K}_{\mathscr{T}}/\mathscr{K})\rightarrow \operatorname{Aut}(\unicode[STIX]{x1D6F7}(\mathscr{G},\mathscr{T}))$ . We say that $\mathscr{T}$ is generic over $\mathscr{K}$ if $\unicode[STIX]{x1D703}_{\mathscr{T}}(\operatorname{Gal}(\mathscr{K}^{\text{sep}}/\mathscr{K}))\supset W(\mathscr{T},\mathscr{G})$ .

Proof. Let $\unicode[STIX]{x1D712}_{0}=1,\unicode[STIX]{x1D712}_{1},-\unicode[STIX]{x1D712}_{1},\ldots ,\unicode[STIX]{x1D712}_{t},-\unicode[STIX]{x1D712}_{t}$ be all the weights for the action of $\mathscr{S}(\mathscr{K})$ on $W=\mathscr{D}^{m}$ , and denote by $W_{\unicode[STIX]{x1D712}}$ the weight subspace corresponding to the character $\unicode[STIX]{x1D712}$ of $\mathscr{S}$ . Set

$$\begin{eqnarray}W_{0}=W_{\unicode[STIX]{x1D712}_{0}}\quad \text{and}\quad W_{i}=W_{\unicode[STIX]{x1D712}_{i}}\,\oplus \,W_{-\unicode[STIX]{x1D712}_{i}}\quad \text{for }i=1,\ldots ,t.\end{eqnarray}$$

Then $W$ has the following orthogonal decomposition

$$\begin{eqnarray}W=W_{0}\bot W_{1}\bot \cdots \bot W_{t}.\end{eqnarray}$$

Let

$$\begin{eqnarray}m_{0}=\dim _{\mathscr{D}}W_{0}\quad \text{and}\quad m_{i}=\dim _{\mathscr{D}}W_{\unicode[STIX]{x1D712}_{i}}=\dim _{\mathscr{D}}W_{-\unicode[STIX]{x1D712}_{i}}\quad \text{for }i=1,\ldots ,t,\end{eqnarray}$$

so that $m_{i}\leqslant r\leqslant m/2$ and $\dim _{\mathscr{D}}W_{i}=2m_{i}$ . Then

$$\begin{eqnarray}C_{\mathscr{H}}(\mathscr{S})=\mathscr{F}_{0}\times \mathscr{F}_{1}\times \cdots \times \mathscr{F}_{t},\end{eqnarray}$$

where $\mathscr{F}_{0}=\operatorname{SU}_{m_{0}}(\mathscr{D},h|W_{0})$ and $\mathscr{F}_{i}$ for $i>0$ is the stabilizer of the subspaces $W_{\unicode[STIX]{x1D712}_{i}}$ and $W_{-\unicode[STIX]{x1D712}_{i}}$ in $\operatorname{SU}_{2m_{i}}(\mathscr{D},h|W_{i})$ . As $m_{i}\geqslant 1$ and $t>0$ as $\mathscr{S}$ is nontrivial, we see that $m_{0}\leqslant m-2$ , and therefore $\mathscr{F}_{0}$ is of type $\mathsf{D}_{\ell }$ with

$$\begin{eqnarray}\ell =dm_{0}/2\leqslant d(m-2)/2\end{eqnarray}$$

(obviously, not a triality form). Furthermore, for $i>0$ , the group $\mathscr{F}_{i}$ is isomorphic to $\operatorname{GL}(W_{\unicode[STIX]{x1D712}_{i}})$ (or $\operatorname{GL}(W_{-\unicode[STIX]{x1D712}_{i}})$ ). Thus, its semi-simple part is $\operatorname{SL}(W_{\unicode[STIX]{x1D712}_{i}})$ , which is an inner form of type $\mathsf{A}_{\ell -1}$ with

$$\begin{eqnarray}\ell =dm_{i}\leqslant dr.\Box\end{eqnarray}$$

We will now complete the proof of Proposition 5.4. Assume the contrary, i.e.  $\mathscr{H}$ contains a maximal $\mathscr{K}$ -torus $\mathscr{T}=\mathscr{T}_{1}\mathscr{T}_{2}$ as in the proposition. Applying the lemma to $\mathscr{S}=\mathscr{T}_{1}$ , we obtain that

$$\begin{eqnarray}\mathscr{T}=\mathscr{T}_{1}\mathscr{T}_{2}\subset \mathscr{S}_{0}\mathscr{H}_{1}\cdots \mathscr{H}_{s},\end{eqnarray}$$

where $\mathscr{S}_{0}$ is the central torus of $C_{\mathscr{H}}(\mathscr{T}_{1})$ and the $\mathscr{H}_{i}$ are semisimple $\mathscr{K}$ -groups of the types specified in the lemma. Clearly, $\mathscr{T}\not \subset \mathscr{S}_{0}$ as otherwise $\mathscr{H}$ would be quasi-split over $\mathscr{K}$ , which it is not because $d>1$ . As $\mathscr{T}_{1}\subset \mathscr{S}_{0}$ , we conclude that $\mathscr{T}_{2}\not \subset \mathscr{S}_{0}$ . It follows that there exists $i\in \{1,\ldots ,s\}$ such that

$$\begin{eqnarray}\mathscr{T}_{2}\not \subset \hat{\mathscr{H}}_{i}:=\mathscr{S}_{0}\mathscr{H}_{1}\cdots \mathscr{H}_{i-1}\mathscr{H}_{i+1}\cdots \mathscr{H}_{s}.\end{eqnarray}$$

This means that for the quotient map $\unicode[STIX]{x1D70B}_{i}:C_{\mathscr{H}}(\mathscr{T}_{1})\rightarrow C_{\mathscr{H}}(\mathscr{T}_{1})/\hat{\mathscr{H}}_{i}=:\bar{\mathscr{H}}_{i}$ , we have $\unicode[STIX]{x1D70B}_{i}(\mathscr{T}_{2})\neq \{e\}$ . Being generic over $\mathscr{K}$ , the torus $\mathscr{T}_{2}$ is $\mathscr{K}$ -irreducible, i.e. contains no proper $\mathscr{K}$ -defined subtori. Thus, the restriction of $\unicode[STIX]{x1D70B}_{i}$ gives an isogeny of $\mathscr{T}_{2}$ onto a $\mathscr{K}$ -subtorus $\bar{\mathscr{T}}_{i}$ of $\bar{\mathscr{H}}_{i}$ . Moreover, because $\dim \mathscr{T}_{2}=\operatorname{rk}\mathscr{H}-1$ and $\operatorname{rk}\bar{\mathscr{H}}_{i}\leqslant \operatorname{rk}\mathscr{H}-1$ , we see that $\bar{\mathscr{T}}_{i}$ is actually a maximal $\mathscr{K}$ -torus of $\bar{\mathscr{H}}_{i}$ . Let $\mathscr{L}$ be the common minimal splitting field of $\mathscr{T}_{2}$ and $\bar{\mathscr{T}}_{i}$ . As $\mathscr{T}_{2}$ is $\mathscr{K}$ -generic in a group of type $\mathsf{D}_{n/2-1}$ where $n=dm$ , we have

(4) $$\begin{eqnarray}|\text{Gal}(\mathscr{L}/\mathscr{K})|\geqslant 2^{n/2-2}\cdot (n/2-1)!.\end{eqnarray}$$

Now, assume that $\mathscr{H}_{i}$ is an inner form of type $\mathsf{A}_{\ell -1}$ with $\ell \leqslant dr$ . As $r\leqslant m/2$ , we obtain

$$\begin{eqnarray}|\text{Gal}(\mathscr{L}/\mathscr{K})|\leqslant |W(\bar{\mathscr{H}}_{i},\bar{\mathscr{T}}_{i})|=\ell !\leqslant (n/2)!.\end{eqnarray}$$

Comparing with (4), we obtain the inequality

$$\begin{eqnarray}2^{n/2-2}\leqslant n/2,\end{eqnarray}$$

which fails for all $n\geqslant 10$ .

Next, let $\mathscr{H}_{i}$ be of type $\mathsf{D}_{\ell }$ with $\ell \leqslant d(m-2)/2\leqslant n/2-2$ as $d\geqslant 2$ . As it is not a triality form when $\ell =4$ , we have

$$\begin{eqnarray}|\text{Gal}(\mathscr{L}/\mathscr{K})|\leqslant 2\cdot |W(\bar{\mathscr{H}}_{i},\bar{\mathscr{T}}_{i})|=2^{\ell }\cdot \ell !\leqslant 2^{n/2-2}\cdot (n/2-2)!.\end{eqnarray}$$

Thus, comparing with (4), we obtain the inequality

$$\begin{eqnarray}2^{n/2-2}\cdot (n/2-1)!\leqslant 2^{n/2-2}\cdot (n/2-2)!\end{eqnarray}$$

which never holds for $n\geqslant 6$ .

Thus, the assumption that $\mathscr{H}$ contains a maximal $\mathscr{K}$ -torus $\mathscr{T}=\mathscr{T}_{1}\mathscr{T}_{2}$ leads to a contradiction in both cases, proving our claim.◻

5.2 Properness of the global-to-local map

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places. As we already mentioned, for each $i\geqslant 1$ , the kernel

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{i}=\ker \biggl(H^{i}(K,\unicode[STIX]{x1D707}_{2})\longrightarrow \mathop{\prod }_{v\in V}H^{i}(K_{v},\unicode[STIX]{x1D707}_{2})\biggr)\end{eqnarray}$$

is contained in $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ , hence is finiteFootnote ² (cf. § 4.1). Applying now Theorem 3.4, we conclude that for $q$ a nondegenerate quadratic form in $n\geqslant 5$ variables and $G=\operatorname{SO}_{n}(q)$ , the map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{G,V}:H^{1}(K,G)\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},G)\end{eqnarray}$$

is proper, which is precisely the assertion of Theorem 1.3.

To put this result in a more general context, we recall the following general observation [Reference Chernousov, Rapinchuk and RapinchukCRR15, Reference RapinchukRap14]: Let $G$ be an absolutely almost simple simply connected algebraic group over a field $K$ , and let $V$ be a set of discrete valuations of $K$ that satisfies condition (A). Assume that for any finite subset $S\subset V$ , the set of $K$ -isomorphism classes of inner $K$ -forms of $G$ having good reduction at all $v\in V\setminus S$ is finite. Then for the corresponding adjoint group $\overline{G}$ , the map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{\overline{G},V}:H^{1}(K,\overline{G})\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},\overline{G})\end{eqnarray}$$

is proper. This means, in particular, that for $n$ odd, Theorem 1.3 directly follows from Theorem 1.1. On the other hand, for $n$ even, the latter implies neither Theorem 1.3 nor the corresponding fact for the simply connected and adjoint groups $\widetilde{G}=\operatorname{Spin}_{n}(q)$ and $\overline{G}=\operatorname{PSO}_{n}(q)$ (thus, in this case Theorem 1.3 is an independent result). Later, we will see that the analogues of Theorem 1.3 are valid for special unitary groups of Hermitian forms over a quadratic extension of the base (two-dimensional global) field and for the group of type $\mathsf{G}_{2}$ over such a field (note that all these groups are simply connected); see Theorems 8.4 and 9.1(iii). We close this subsection with some additional results on the properness of the global-to-local map for simply connected groups.

Theorem 5.7. Let $K$ be a two-dimensional global field, let $V$ be its divisorial set of places and let $m$ be a square-free integer prime to $\operatorname{char}K$ . Then for a central simple $K$ -algebra $A$ of degree $m$ and $G=\operatorname{SL}_{1,A}$ , the map

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{G,V}:H^{1}(K,G)\longrightarrow \mathop{\prod }_{v\in V}H^{1}(K_{v},G)\end{eqnarray}$$

is proper.

Proof. We may assume without loss of generality that $\operatorname{char}K^{(v)}$ is prime to $m$ for all $v\in V$ . It is well-known (cf. [Reference Knus, Merkurjev, Rost and TignolKMRT98, 29.4]) that for any extension $F/K$ there is a bijection

$$\begin{eqnarray}\unicode[STIX]{x1D708}_{F}:H^{1}(F,G)\rightarrow F^{\times }/\operatorname{Nrd}_{A_{F}/F}(A_{F}^{\times })\quad \text{where }A_{F}=A\,\otimes _{K}\,F,\end{eqnarray}$$

and this bijection is functorial in $F$ . Thus, letting $A_{v}=A_{K_{v}}$ for $v\in V$ , we need to show that the map

$$\begin{eqnarray}\unicode[STIX]{x1D704}:K^{\times }/\operatorname{Nrd}_{A/K}(A^{\times })\rightarrow \mathop{\prod }_{v\in V}K_{v}^{\times }/\operatorname{Nrd}_{A_{v}/K_{v}}(A_{v}^{\times })\end{eqnarray}$$

has finite kernel. For any field extension $F/K$ , we have a homomorphism

$$\begin{eqnarray}F^{\times }\rightarrow H^{3}(F,\unicode[STIX]{x1D707}_{m}^{\otimes 2}),\quad a\mapsto [A_{F}]\cup \unicode[STIX]{x1D712}_{m,a},\end{eqnarray}$$

where $[A_{F}]$ is the class of $A_{F}$ in $_{m}\operatorname{Br}(F)=H^{2}(F,\unicode[STIX]{x1D707}_{m})$ . As $m$ is square-free, according to [Reference Merkurjev and SuslinMS82, Theorem 12.2], the kernel of this map coincides with $\operatorname{Nrd}_{A_{F}/F}(A_{F}^{\times })$ , so we have an injective homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{F}:F^{\times }/\operatorname{Nrd}_{A_{F}/F}(A_{F}^{\times })\rightarrow H^{3}(F,\unicode[STIX]{x1D707}_{m}^{\otimes 2}).\end{eqnarray}$$

Then we have the following commutative diagram

where $\unicode[STIX]{x1D6E5}_{V}=\prod _{v\in V}\unicode[STIX]{x1D6FF}_{K_{v}}$ . The fact that the unramified cohomology group $H^{3}(K,\unicode[STIX]{x1D707}_{m}^{\otimes 2})_{V}$ is finite (Theorem 6.1) implies that $\ker \unicode[STIX]{x1D702}$ is finite. Then using the injectivity of $\unicode[STIX]{x1D6FF}_{K}$ , we conclude that $\ker \unicode[STIX]{x1D704}$ is finite as well, as needed.◻

We now turn to spinor groups. Let $q$ be a nondegenerate quadratic form of dimension at least three over a field $K$ of characteristic $\neq 2$ and let $\unicode[STIX]{x1D70B}:\operatorname{Spin}_{n}(q)\rightarrow \operatorname{SO}_{n}(q)$ be the canonical $K$ -isogeny. For a field extension $F/K$ , we let

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{F}:H^{1}(F,\operatorname{Spin}_{n}(q))\rightarrow H^{1}(F,\operatorname{SO}_{n}(q))\end{eqnarray}$$

be the map induced by $\unicode[STIX]{x1D70B}$ . Assume now that $K$ is equipped with a set $V$ of discrete valuations such that the map $\unicode[STIX]{x1D703}:H^{1}(K,\operatorname{SO}_{n}(q))\rightarrow \prod _{v\in V}H^{1}(K_{v},\operatorname{SO}_{n}(q))$ is proper. Then, analyzing the commutative diagram

where $\unicode[STIX]{x1D6F1}=\prod _{v\in V}\unicode[STIX]{x1D70B}_{K_{v}}$ , and using twisting, we see that in order to prove that $\tilde{\unicode[STIX]{x1D703}}$ is proper, it is enough to prove that for any nondegenerate $n$ -dimensional quadratic form $q^{\prime }$ over $K$ , the set

(5) $$\begin{eqnarray}\unicode[STIX]{x0428}(\operatorname{Spin}_{n}(q^{\prime }),V)\,\cap \,\ker \unicode[STIX]{x1D70B}_{K}^{\prime }\end{eqnarray}$$

(where $\unicode[STIX]{x1D70B}_{K}^{\prime }$ is the map analogous to $\unicode[STIX]{x1D70B}_{K}$ for the quadratic form $q^{\prime }$ ) is finite. For a field extension $F/K$ , combining the standard exact sequence of cohomology corresponding to the exact sequence

$$\begin{eqnarray}1\rightarrow \unicode[STIX]{x1D707}_{2}\longrightarrow \operatorname{Spin}_{n}(q^{\prime })\stackrel{\unicode[STIX]{x1D70B}^{\prime }}{\longrightarrow }\operatorname{SO}_{n}(q^{\prime })\rightarrow 1\end{eqnarray}$$

with the description of the spinor norm (cf. [Reference Knus, Merkurjev, Rost and TignolKMRT98, 13.30, 13.31]), one easily obtains a (functorial) identification of $\ker \unicode[STIX]{x1D70B}_{F}^{\prime }$ with the quotient $F^{\times }/\operatorname{Sn}(q^{\prime },F)$ , where $\operatorname{Sn}(q^{\prime },F)$ denotes the image of the spinor norm $\operatorname{SO}_{n}(q^{\prime })_{F}$ . Furthermore, one observes that this identification induces an injection of the set (5) into the kernel of the map

$$\begin{eqnarray}\unicode[STIX]{x1D708}_{q^{\prime },V}:K^{\times }/\operatorname{Sn}(q^{\prime },K)\longrightarrow \mathop{\prod }_{v\in V}K_{v}^{\times }/\operatorname{Sn}(q^{\prime },K_{v}).\end{eqnarray}$$

Thus, if $\unicode[STIX]{x1D703}$ is known to be proper, then to establish the properness of $\tilde{\unicode[STIX]{x1D703}}$ , it is sufficient to prove that the kernel of $\unicode[STIX]{x1D708}_{q^{\prime },V}$ is finite for all nondegenerate $n$ -dimensional quadratic forms $q^{\prime }$ over $K$ . (Of course, $\ker \unicode[STIX]{x1D708}_{q^{\prime },V}$ is automatically trivial if $q^{\prime }$ is $K$ -isotropic, so the properness of $\unicode[STIX]{x1D703}$ immediately implies that of $\tilde{\unicode[STIX]{x1D703}}$ if every $n$ -dimensional quadratic form over $K$ is isotropic, i.e.  $n$ is greater than the $u$ -invariant of $K$ .)

Here we would like to point out the following partial result on the finiteness of the set (5).

Proposition 5.8. Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be its divisorial set of places. If $q$ is a quadratic Pfister form over $K$ of dimension $n=2^{d}$ , then the intersection

$$\begin{eqnarray}\unicode[STIX]{x0428}(\operatorname{Spin}_{n}(q),V)\cap \ker \unicode[STIX]{x1D70B}_{K}\end{eqnarray}$$

is finite.

Proof. According to the discussion prior to the statement of the proposition, it is enough to show that the map

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{q,V}:K^{\times }/\operatorname{Sn}(q,K)\rightarrow \mathop{\prod }_{v\in V}K_{v}^{\times }/\operatorname{Sn}(q,K_{v})\end{eqnarray}$$

has finite kernel. By our assumption, $q$ is a Pfister form $\langle \!\langle a_{1},\ldots ,a_{d}\rangle \!\rangle$ for some $a_{1},\ldots ,a_{d}\in K^{\times }$ . For any field extension $F/K$ , using the fact that the nonzero values of $q$ over $F$ form a subgroup of $F^{\times }$ (see [Reference LamLam05, ch. X, Theorem 1.8]), it is easy to see that $a\in F^{\times }$ belongs to $\operatorname{Sn}(q,F)$ if and only if the Pfister form $q\bot (-aq)=\langle \!\langle a_{1},\ldots ,a_{d},a\rangle \!\rangle$ is isotropic, hence hyperbolic (cf. [Reference LamLam05, ch. X, Theorem 1.7]). The latter is equivalent to the symbol

$$\begin{eqnarray}(a_{1},\ldots ,a_{d},a)\in k_{d+1}(F)=K_{d+1}(F)/2\cdot K_{d+1}(F),\end{eqnarray}$$

or the cup-product

$$\begin{eqnarray}\unicode[STIX]{x1D712}_{a_{1}}\cup \cdots \cup \unicode[STIX]{x1D712}_{a_{d}}\cup \unicode[STIX]{x1D712}_{a}\in H^{d+1}(F,\unicode[STIX]{x1D707}_{2}),\end{eqnarray}$$

being trivial. Thus, the map

$$\begin{eqnarray}F^{\times }\rightarrow H^{d+1}(F,\unicode[STIX]{x1D707}_{2}),\quad a\mapsto \unicode[STIX]{x1D712}_{a_{1}}\cup \cdots \cup \unicode[STIX]{x1D712}_{a_{d}}\cup \unicode[STIX]{x1D712}_{a},\end{eqnarray}$$

gives rise to an injective map

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{F}:F^{\times }/\operatorname{Sn}(q,F)\rightarrow H^{d+1}(F,\unicode[STIX]{x1D707}_{2}).\end{eqnarray}$$

We then have the following commutative diagram

where $\unicode[STIX]{x1D6E5}_{V}=\prod _{v\in V}\unicode[STIX]{x1D6FF}_{K_{v}}$ . As we already mentioned several times, the finiteness of the unramified cohomology group $H^{d+1}(K,\unicode[STIX]{x1D707}_{2})_{V}$ (see § 4.1), implies the finiteness of $\ker \unicode[STIX]{x1D704}_{d+1}$ . Thus, the injectivity of $\unicode[STIX]{x1D6FF}_{K}$ yields the finiteness of $\ker \unicode[STIX]{x1D702}_{q,V}$ , as required.◻