| Izvestiya: Mathematics |
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On Grothendieck–Serre conjecture in mixed characteristic for I. A. Panin St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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    § 1. IntroductionA well-known conjecture due to J.-P. Serre and A. Grothendieck (see [27], Remarque, p. 3, [9], Remarque 3, pp. 26, 27, and [10], Remarque 1.11) asserts that, for any regular local ring $R$ and any reductive group scheme $G$ over $R$ rationally trivial $G$-homogeneous spaces are trivial. In the present paper, we are interested in the case $G=\operatorname{SL}_1(D)$ of norm one elements of an Azumaya $R$-algebra $D$. In this case, the conjecture can be restated. Namely, the short exact sequence $\{1\}\to \operatorname{SL}_1(D)\to \operatorname{GL}_1(D)\xrightarrow{\operatorname{Nrd}} \mathbb G_{m,R} \to \{1\}$ gives rise to the exact sequence of the étale cohomology. That sequence together with the equality $\operatorname{H}^1_{\mathrm{\unicode{x00E9}t}}(R,{\operatorname{GL}_1(D)})=\{*\}$ yields the equality $R^{\times}/\operatorname{Nrd}({D^{\times}})=\operatorname{H}^1_{\mathrm{\unicode{x00E9}t}}(R,{\operatorname{SL}_1(D)})$. Particularly, the $\operatorname{H}^1_{\mathrm{\unicode{x00E9}t}}(R,{\operatorname{SL}_1(D)})$ is a group rather than a pointed set. Thus, the conjecture asserts the injectivity of the group homomorphism
$$
\begin{equation*}
R^{\times}/\operatorname{Nrd}({D^{\times}})\to K^{\times}/\operatorname{Nrd}({D^{\times}_K})
\end{equation*}
\notag
$$
induced by inclusion of the ring $R$ into its fraction field $K$. So, our results on the conjecture correspond to the case when $R$ is an unramified regular local ring of mixed characteristic and $G$ is the group $\operatorname{SL}_1(D)$ of norm one elements of an Azumaya $R$-algebra $D$. The results extend to the mixed characteristic case the ones proved in [23] by A. Suslin and the author. Note that our Theorem 3.2 is much stronger than Theorem 8.1 in [21]. Namely, in Theorem 3.2 the ring $R$ is an arbitrary regular local of mixed characteristic $(0,p)$ such that the ring $R/pR$ is also regular. Opposed to this, in [21] only rings $R$ of “geometric” nature subjecting to a relative form of the Noether normalization lemma are regarded. The latter condition is rather restrictive even for rings of geometric “nature”. This is the basic reason, why the limit arguments using the D. Popesco theorem are not allowed to extend Theorem 8.1 in [21] to the case of arbitrary unramified regular local of mixed characteristic $(0,p)$. To avoid the Noether normalization lemma requirements we use in the present paper a geometric presentation lemma due to Česnavičius (see [1], Proposition 4.1). The latter is really the crucial point. Namely, this allows one to apply the limit arguments to get Theorem 3.2. Our approach is as follows. Using the D. Popescu theorem, we reduce the question to the case when $R$ is essensially smooth over $\mathbb Z_{(p)}$ and $D$ is defined over $R$. Next, using a geometric presentation lemma due to Česnavičius (see [1], Proposition 4.1) and the method as in [23], we prove a purity result for the functor $K_1(-,D)$. Finally, a diagram chasing shows the result mentioned above. A rather good survey of the topic is given in [18]. We point out that the conjecture is solved in the case, when $R$ contains a field. More precisely, it is solved when $R$ contains an infinite field, by R. Fedorov and the author in [5]. It is solved by the author in the case, when $R$ contains a finite field in [17] (see also [16]). When $R$ contains a field, then certain extended form of the Grothendieck–Serre conjecture is proved in Theorem 1.5 in [19]. It has two interesting consequences (see Theorems 1.1, 1.3 in [19]). Theorem 1.3 in [19] states that two semi-simple group $R$-schemes are isomorphic if and only if their fibres over the fraction field $K$ of $R$ are isomorphic as algebraic groups over $K$. Theorem 1.1 in [19] is an essential extension of Theorem 1.1 in [20]. The case of mixed characteristic is widely open. Here are several references. $\bullet$ The case when the group scheme is $\operatorname{PGL}_n$ and the ring $R$ is an arbitrary regular local ring is done by Grothendieck [10] in 1968. $\bullet$ The case of an arbitrary reductive group scheme over a discrete valuation ring or over a henselian ring is solved by Nisnevich [11] in 1984. $\bullet$ The case, where $\mathbf{G}$ is an arbitrary torus over a regular local ring, was settled by Colliot-Thélène and Sansuc [2] in 1987. $\bullet$ The case, when $\mathbf{G}$ is quasi-split reductive group scheme over arbitrary two-dimensional local rings, is solved by Nisnevich [12] in 1989. $\bullet$ The case when $\mathbf{G}$ the unitary group scheme $\operatorname{U}^{\epsilon}_{A,\sigma}$ is solved by S. Gille and the author recently in [8]; here, $R$ is an unramified regular local ring of characteristic $(0,p)$ with $p\neq 2$ and $(A,\sigma)$ is an Azumaya $R$-algebra with involution. $\bullet$ In [22], the conjecture is solved for any semi-local Dedekind domain providing that $\mathbf{G}$ is simple simply-connected and $\mathbf{G}$ contains a torus $\mathbb G_{m,R}$. $\bullet$ The latter result is extended in [13] to arbitrary reductive group schemes $\mathbf{G}$ over any semi-local Dedekind domain. $\bullet$ There are also two very interesting recent publications [6], [7] by Fedorov. $\bullet$ The case, when $\mathbf{G}$ is quasi-split reductive group scheme over an unramified regular local ring is solved recently by Česnavičius in [1]. Finally, we point out that [3], [14], [15] indicate significant progress in the topic. § 2. AgreementsThrough the paper, $A$ is a d.v.r., $m_A\subseteq A$ is its maximal ideal; $\pi \in m_A$ is a generator of the maximal ideal; $k(v)$ is the residue field $A/m_A$; $p>0$ is the characteristic of the field $k(v)$; it is supposed in this paper that the fraction field of $A$ has characteristic zero; $d\geqslant 1$ is an integer; $X$ is an irreducible $A$-smooth affine $A$-scheme of relative dimension $d$. So, all open subschemes of $X$ are regular and all its local rings are regular. If $x_1,\dots,x_n$ are closed points in the scheme $X$, then we write $\mathcal O$ for the semi-local ring $\mathcal O_{X,\{x_1,\dots,x_n\}}$, $\mathcal K$ for the fraction field of $\mathcal O$, $U$ for $\operatorname{Spec}(\mathcal O)$, $\eta$ for $\operatorname{Spec}(\mathcal K)$. Recall that a regular local ring $R$ of mixed characteristic $(0,p)$ is called unramified if the ring $R/pR$ is regular; recall from [29], 0382, that a Noetherian algebra over a field $k$ is geometrically regular if its base change to every finite purely inseparable (equivalently, to every finitely generated) field extension of $k$ is regular; one says that a regular local $A$-algebra $R$ is geometrically regular if the $k(v)$-algebra $R/mR$ is geometrically regular. § 3. Main resultsTheorem 3.1. Let $R$ be an unramified regular semi-local domain of mixed characteristic $(0,p)$, $D$ an Azumaya $R$-algebra, $K$ the fraction field of $R$, $\operatorname{Nrd}\colon D^{\times} \to R^{\times}$ the reduced norm homomorphism. Let $a \in R^{\times}$ be a unit. Suppose the equation $\operatorname{Nrd}=a$ has a solution over $K$, then it has a solution over $R$. The following result is an extension of Theorem 3.1. Theorem 3.2. Let $R$ be a geometrically regular semi-local integral $A$-algebra. Let $D$ be an Azumaya $R$-algebra, $K$ the fraction field of $R$, $\operatorname{Nrd}\colon D^{\times} \to R^{\times}$ the reduced norm homomorphism. Let $a \in R^{\times}$ be a unit. Suppose the equation $\operatorname{Nrd}=a$ has a solution over $K$, then it has a solution over $R$. Corollary 3.3. Let $R$ be a geometrically regular semi-local integral $A$-algebra, $K$ the fraction field of $R$. Let $a$, $b$, $c$ be units in $R$. Consider the equation If it has a solution over $K$, then it has a solution over $R$. Remark 3.4. Let $a$, $b$, $c$ be units in $R$ as in the corollary. Let $D$ be the generalised quaternion $R$-algebra given by generators $u$, $w$ and relations $u^2=a$, $w^2=b$, $uw=-wu$. Then the reduced norm $\operatorname{Nrd}\colon D\to R$ takes a quaternion $\alpha+\beta u+ \gamma w+ \delta uw$ to the element $\alpha^2 - a\beta^2 - b\gamma^2 + ab\delta^2$. This is why the corollary is a consequence of Theorem 3.2. Remark 3.5. The form $\langle 1,-a,-b,ab\rangle$ is a two-fold Pfister form. Corollary 3.3 was extended in [25] to arbitrary Pfister forms. § 4. Česnavičius geometric presentation propositionLet $A$ be a d.v.r. and $X$ be an $A$-scheme as in § 2. The following result is due to Česnavičius (see [1], Proposition 4.1). Theorem 4.1 (see [1], Proposition 4.1). Let $x_1,\dots,x_n$ be closed points in the scheme $X$. Let $Z$ be a closed subset in $X$ of codimension at least $2$ in $X$. Then there are an affine neighborhood $X^{\circ}$ of points $x_1,\dots,x_n$, an open affine subscheme $S\subseteq \mathbf{A}^{d-1}_A$ and a smooth $A$-morphism of pure relative dimension $1$ such that $Z^{\circ}$ is $S$-finite, where $Z^{\circ}=Z\cap X^{\circ}$. § 5. Proof of the geometric case of Theorem 3.2Let $A$, $p>0$, $d\geqslant 1$, $X$, $x_1,\dots,x_n\in X$, $\mathcal O$ and $U$ be as in § 2. Let $\mathcal K$ denote the fraction field of the ring $\mathcal O$. Theorem 5.1. Let $D$ be an Azumaya $\mathcal O$-algebra and $\operatorname{Nrd}_D\colon D^{\times}\to \mathcal O^{\times}$ be the reduced norm homomorphism. Let $a \in \mathcal O^{\times}$. If $a$ is a reduced norm for the central simple $\mathcal K$-algebra $D\otimes_{\mathcal O} \mathcal K$, then $a$ is a reduced norm for the algebra $D$. Proposition 5.2. Let $D$ be an Azumaya $\mathcal O$-algebra. Let $K_*$ be the Quillen $K$-functor. Then, for each integer $n\geqslant 0$, the sequence is exact Particularly, it is exact for $n=1$. Reducing Theorem 5.1 to Proposition 5.2. Consider the commutative diagram of groups By Proposition 5.2 the complex on the top is exact. The bottom map $\eta^*$ is injective. The right-hand side vertical map $\operatorname{Nrd}$ is injective. Thus, the map
$$
\begin{equation*}
\mathcal O^{\times}/\operatorname{Nrd}(K_1(D))\to \mathcal K^{\times}/\operatorname{Nrd}(K_1(D\otimes_{\mathcal O} \mathcal K))
\end{equation*}
\notag
$$
is injective. The image of the left vertical map coincides with $\operatorname{Nrd}(D^{\times})$ and the image of the middle vertical map coincides with $\operatorname{Nrd}((D\otimes_{\mathcal O} \mathcal K)^{\times})$. Thus, the map
$$
\begin{equation*}
\mathcal O^{\times}/\operatorname{Nrd}(D^{\times})\to \mathcal K^{\times}/\operatorname{Nrd}((D\otimes_{\mathcal O} \mathcal K)^{\times})
\end{equation*}
\notag
$$
is injective. The derivation of Theorem 5.1 from Proposition 5.2 is completed. To prove Proposition 5.2, we need some preparation. For each integer $c\geqslant 0$, we let $\mathcal{M}(D)_{\geqslant c}$ denote the abelian category of finitely generated left $D$-modules supported (as $\mathcal O$-modules) in codimension at least $c$. Following [26], we put $K'_n(D)_{\geqslant c}=K_n(\mathcal{M}(D)_{\geqslant c})$. Following again [26], the abelian category inclusion $\mathcal{M}(D)_{\geqslant (c+1)}\subseteq \mathcal{M}(D)_{\geqslant c}$ gives rise to the long exact localization sequence
$$
\begin{equation}
\begin{aligned} \, \cdots \to K'_n(D)_{\geqslant c+1} &\xrightarrow{\operatorname{ext}^n_{c+1}} K'_n(D)_{\geqslant c} \xrightarrow{\operatorname{res}_n} \bigoplus_{y\in U^{(c)}}K_n(D\otimes_{\mathcal O} k(y)) \nonumber \\ &\xrightarrow{\partial_n} K'_{n-1}(D)_{\geqslant c+1}\to \cdots. \end{aligned}
\end{equation}
\tag{4}
$$
Lemma 5.3. Vanishing of the map $\operatorname{ext}_2^n$ yields Proposition 5.2. Proof. To prove the lemma, consider sequences (4) with $c=1$. If the map $\operatorname{ext}_{2}^n$ vanishes, then the map $K'_n(D)_{\geqslant 1} \xrightarrow{\operatorname{res}_n} \bigoplus_{y\in U^{(c)}}K_n(D\otimes_{\mathcal O} k(y))$ is injective. By the definition one has $d_n=\operatorname{res}_n\circ \,\partial_n\colon K_n(D\otimes_{\mathcal O} \mathcal K)\to \bigoplus_{y\in U^{(1)}}K_{n-1}(D\otimes_{\mathcal O} k(y))$. Therefore, $\operatorname{Ker}(d_n)=\operatorname{Ker}(\partial_n)$. The exactness of sequence (4) for $c=0$ shows that $\operatorname{Ker}(\partial_n)=\operatorname{Im}[\operatorname{res}_n\colon K_n(D)\to K_n(D\otimes_{\mathcal O} \mathcal K)]$. The lemma is proved. To prove the proposition it is convenient to switch to geometric notation. For each closed subset $Z$ in $U$, we write $\mathcal{M}(Z;D)$ for the abelian category of finitely generated left $D$-modules whose support (as $\mathcal O$-modules) is in $Z$. Put $K'_n(Z;D)=K_n(\mathcal{M}(Z;D))$. To be consistent, we put $K'_{n}(U;D)_{\geqslant c}=K'_{n}(D)_{\geqslant c}$, and so on. Proof of Proposition 5.2. By Lemma 5.3, to prove this proposition it sufficient to prove vanishing of the support extension map $\operatorname{ext}_{2}^n\colon K'_{n}(U;D)_{\geqslant 2}\to K'_{n}(U;D)_{\geqslant 1}$. We claim that $\operatorname{ext}_{2}^n=0$. Consider $a\in K'_n(U;D)_{\geqslant 2}$. We may assume that $a\in K'_n(Z;D)$ for a closed $Z$ in $U$ with $\operatorname{codim}_U(Z)\geqslant 2$. Enlarging $Z$ we may assume that each its irreducible component contains at least one of the point $x_i$’s and still $\operatorname{codim}_U(Z)\geqslant 2$. Our aim is to find a closed subset $Z_{\mathrm{ext}}$ in $U$ containing $Z$ such that the element $a$ vanishes in $K'_n(Z_{\mathrm{ext}};D)$ and $\operatorname{codim}_U(Z_{\mathrm{ext}})\geqslant 1$. We will follow the method of [23].
Let $\overline Z$ be the closure of $Z$ in $X$. Shrinking $X$ and $\overline Z$ accordingly, we may and will suppose that $D$ is an Azumaya algebra over $X$ and there is an element $\widetilde a\in K'_n(\overline Z;D)$ such that $\widetilde a|_Z=a$. By Theorem 4.1, there are an affine neighborhood $X^{\circ}$ of points $x_1,\dots,x_n$, an open affine subscheme $S\subseteq \mathbf{A}^{d-1}_A$, a smooth $A$-morphism of pure relative dimension 1 such that $Z^{\circ}/S$ is finite, where $Z^{\circ}=\overline Z\cap X^{\circ}$. We put $a^{\circ}=\widetilde a|_{Z^{\circ}}$.We put $s_i=q(x_i)$. Consider the semi-local ring $\mathcal O_{S,s_1,\dots,s_n}$. We put $B=\operatorname{Spec} \mathcal O_{S,s_1,\dots,s_n}$ and $X_B=q^{-1}(B)\subset X^{\circ}$, and define $Z_B=Z^{\circ}\cap X_B$. We let $q_B$ denote $q|_{X_B}\colon X_B\to B$. Note that $Z_B$ is finite over $B$. Since $B$ is semi-local, hence so is $Z_B$. Let $W\subset X_B$ be an open containing $Z_B$. We let $\mathcal Z_W$ denote $Z_B\times_B W$. Let $\Pi\colon \mathcal Z_W\to W$ be the projection to $W$ and $q_Z\colon \mathcal Z_W\to Z_B$ be the projection to $Z_B$. Since $Z_B/B$ is finite, so is the morphism $\Pi$. Since $\Pi$ is finite, the subset $Z_{\mathrm{new}}:=\Pi(\mathcal Z_W)$ of $W$ is closed in $W$. Since $W$ contains $Z_B$, we have an inclusion $i\colon Z_B\hookrightarrow Z_{\mathrm{new}}$ of closed subsets in $W$. Since $q_Z\colon \mathcal Z_W\to Z_B$ is smooth of relative dimension 1, $Z_{\mathrm{new}}$ has codimension at least one in $W$. Clearly, $Z_B$ contains all the points $x_1,\dots,x_n$. This shows that $W$ contains $U$ and $Z_B$ contains $Z$. One can check that $Z=Z_B\cap U$. Let $j\colon Z\to Z_B$ be the inclusion. We put $Z_{\mathrm{ext}}=U\cap Z_{\mathrm{new}}$. Since $Z$ is in $U\cap Z_{\mathrm{new}}=Z_{\mathrm{ext}}$, hence we have an inclusion $\operatorname{in}\colon Z\hookrightarrow Z_{\mathrm{ext}}$ of closed subsets in $U$. The inclusion $U\xrightarrow{\mathrm{can}} W$ is a flat morphism. Hence the inclusions $\operatorname{inj}\colon Z_{\mathrm{ext}}\to Z_{\mathrm{new}}$ and $j\colon Z\to Z_B$ are also flat morphisms. Thus the homomorphisms $\operatorname{inj}^*\colon K'_n(Z_{\mathrm{new}};D)\to K'_n(Z_{\mathrm{ext}};D)$ and $j^*\colon K'_n(Z_B;D)\to K'_n(Z;D)$ are well-defined. Moreover, $\operatorname{inj}^*\circ\, i_*=\operatorname{in}_*\circ\, j^*$. As explained just above, the generic point of $W$ is not in $Z_{\mathrm{new}}$. Thus, the generic point of $U$ is not in $Z_{\mathrm{ext}}$ and $Z_{\mathrm{ext}}$ has codimension at least one in $W$. Recall the Azumaya algebra $D$ is an Azumaya algebra over $X$. We still will write $D$ for $D|_W$. We are also given an element $a_B:=a^{\circ}|_{Z_B}\in K'_n(Z_B;D)$ such that $j^*(a_B)=a$ in $K'_n(Z;D)$. Claim. There is an open $W$ in $X_B$ containing $Z_B$ such that, for the closed inclusion $i\colon Z_B\hookrightarrow Z_{\mathrm{new}}$, the map $i_*\colon K'_n(Z_B;D)\to K'_n(Z_{\mathrm{new}};D)$ vanishes. Given this claim,let us complete the proof of the proposition as follows: $\operatorname{in}_*(a)=\operatorname{in}_*(j^*(a_B))=\operatorname{inj}^*(i_*(a_B))=inj^*(0)=0$ in $K'_n(Z_{\mathrm{ext}};D)$ and $Z_{\mathrm{ext}}\subsetneqq U$. In the rest of the proof, we prove the claim. So, let $W\subset X_B$ be an open set containing $Z_B$. Let $\mathcal Z_W=Z_B\times_B W$ and the projections $\Pi\colon \mathcal Z_W\to W$, $q_Z\colon \mathcal Z_W\to Z_B$ be as above in this proof. The closed embedding $\operatorname{in}\colon Z_B\hookrightarrow W$ defines a section $\Delta=(\operatorname{id}\times \operatorname{in})\colon Z_B\to \mathcal Z_W$ of the projection $q_Z$. We also have the equality $\operatorname{in}=\Pi \circ \Delta$. We put $D_Z=D|_{Z_B}$ and write ${_Z}D_W$ for the Azumaya algebra $\Pi^*(D)\otimes q^*_Z(D^{\mathrm{op}}_Z)$ over $\mathcal Z_W$. The $\mathcal O_{\mathcal Z_W}$-module $\Delta_*(D_Z)$ has an obvious left ${_Z}D_W$-module structure. And it is equipped with an obvious epimorphism $\pi\colon {_Z}D_X\to \Delta_*(D_Z)$ of the left ${_Z}D_W$-modules. Following [23], one can see that $I:=\operatorname{Ker}(\pi)$ is a left projective ${_Z}D_W$-module. Hence the left ${_Z}D_W$-module $\Delta_*(D_Z)$ defines an element $[\Delta_*(D_Z)]=[{_Z}D_W]-[I]$ in $K_0(\mathcal Z_W;{_Z}D_W)$. This element has rank zero. Hence by [4] it vanishes semi-locally on $\mathcal Z_W$. Thus, there is a neighborhood $\mathcal W$ of $Z_B\times_B Z_B$ in $\mathcal Z_W$ such that $\Delta_*(D_Z)$ vanishes in $K_0(\mathcal W;{_Z}D_W)$. It is easy to see that $\mathcal W$ contains a neighborhood of $Z_B\times_B Z_B$ of the form $\mathcal Z_{W'}$, where $W'\subset X_B$ is an open containing $Z_B$. Thus, replacing notation we may suppose that $W\subset X_B$ an open containing $Z_B$ and the element $[\Delta_*(D_Z)]=[{_Z}D_W]-[I]$ vanishes in $K_0(\mathcal Z_W;{_Z}D_W)$. It remains to check that, for this specific $W$, the map $i_*\colon K'_n(Z_B;D)=K'_n(Z_B;D_Z)\to K'_n(Z_{\mathrm{new}};D)$ vanishes. To do this, we note that the functor $(P,M)\mapsto P\bigotimes_{q^*_Z(D_Z)} M$ induces a bilinear pairing
$$
\begin{equation*}
\bigcup_{q^*_Z(D_Z)}\colon K_0(\mathcal Z_W;{_Z}D_W)\times K'_n(\mathcal Z_W;q^*_Z(D_Z))\to K'_n(\mathcal Z_W;\Pi^*(D)).
\end{equation*}
\notag
$$
Thus, each element $\alpha\in K_0(\mathcal Z_W;{_Z}D_W)$ defines a group homomorphism
$$
\begin{equation*}
\alpha_*=\Pi_*\circ \biggl( \alpha \bigcup_{q^*_Z(D_Z)} - \biggr)\circ q^*_Z\colon K'_n(Z_B;D)=K'_n(Z_B;D_Z)\to K'_n(Z_{\mathrm{new}};D),
\end{equation*}
\notag
$$
which takes an element $b\in K'_n(Z_B;D_Z)$ to $\Pi_*\bigl(\alpha \bigcup_{q^*_Z(D_Z)} q^*_Z(b)\big)$ in $K'_n(Z_{\mathrm{new}};D)$. Similarly to [23], the map $[\Delta_*(D_Z)]_*$ coincides with the map $i_*\colon K'_n(Z_B;D)=K'_n(Z_B;D_Z)\to K'_n(Z_{\mathrm{new}};D)$. The equality $0=[\Delta_*(D_Z)]\in K_0(\mathcal Z_W;{_Z}D_W)$ proved just above shows that the map $i_*$ vanishes. This proves the claim, and, therefore, the proposition. Let $A$, $p>0$, $d\geqslant 1$, $X$ be as in § 2. Let $y_1,\dots,y_n\in X$ be points that are not necessarily closed. We let $\mathcal O_y$ denote the semi-local ring $\mathcal O_{X,\{y_1,\dots,y_n\}}$ and $U_y$ be $\operatorname{Spec} \mathcal O_y$. We also denote by $\mathcal K$ the fraction field of the ring $\mathcal O_y$. The following result can be derived from Theorem 5.1 in a standard way. Theorem 5.4. Let $D$ be an Azumaya $\mathcal O_y$-algebra and $\operatorname{Nrd}_D\colon D^{\times}\to \mathcal O^{\times}_y$ be the reduced norm homomorphism. Let $a \in \mathcal O^{\times}_y$. If $a$ is a reduced norm for the central simple $\mathcal K$-algebra $D\otimes_{\mathcal O_y} \mathcal K$, then $a$ is a reduced norm for the algebra $D$. Proof of Theorem 5.4. One can choose closed points $x_1,\dots,x_n$ in $X$ such that $x_i$ is in the closure of $y_i$, $D=\widetilde D\otimes_{\mathcal O} \mathcal O_y$ for an Azumaya $\mathcal O$-algebra $\widetilde D$, where $\mathcal O:=\mathcal O_{X,\{x_1,\dots,x_n\}}$, $a\in \mathcal O^{\times}$. In particular, $\mathcal O\subseteq \mathcal O_y$. Now by Theorem 5.1 the element $a$ is a reduced norm from $\widetilde D$. Thus, $a$ is a reduced norm from $D$. Proof of Theorem 3.2. By Popescu theorem [24], [28], the ring R is a filtered direct limit of smooth $A$-algebras. Thus, a limit argument allows us to assume that $R$ is the semilocalization of a smooth $A$-algebra at finitely many primes. Now Theorem 5.4 completes the proof.
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\Bibitem{Pan25} |
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