A local-to global principle for splitting of Azumaya algebras

Question

Let $S$ be a finitely generated domain with the field of fractions $F.$ Let X be a smooth, geometrically connected affine variety over $S.$ Let $A$ be an Azumaya algebra over $X.$ Assume that for all large enough primes $p,$ $A_p$ splits over $X_p$-the reduction modulo $p$ of $X.$ Does this assumption imply that $A_{\overline{F}}$ splits over $X_{\overline{F}}?$ My naive guess is that the answer should be "yes". Any suggestions or references would be greatly appreciated.

Answer 1

Let $n$ be the order of $A$ in the Brauer group of $X$, then $A_{\overline F}$ splits if and only if the corresponding class $[A_{\overline F}]\in H^2_{et}(X_{\overline F}, \mu_n)$ is 0. If $X$ is smooth and proper and $n$ is invertible on $S$ then by Deligne's theorem in SGA $4\frac{1}{2}$ the pushforward $R^2p_*\mu_n$ is a local system on $S$ where $p: X\rightarrow S$ is the projection. In particular the specialization map $H^2_{et}(X_{\overline F}, \mu_n)\rightarrow H^2_{et}(X_{s}, \mu_n)$ is an isomorphism for any geometric point $s\rightarrow S$ of some characteristic $p$. The image of $[A_{\overline F}]$ under specialization is exactly $[A_s]$, thus in fact $[A_{\overline F}]=0$ if and only if $[A_s]=0$. This means that it is enough to check that $A$ splits on a single geometric fiber of characteristic $p>n$. Using excision this can be generalized to $X$ being a complement in a smooth proper scheme of a normal crossing divisor.

Now returning to your original question, by Nagata any $X$ can be compactified by a proper $S$-scheme $\overline X\supset X$, moreover by Hironaka the singularities of $\overline X_F$ can be resolved such that the complement $\overline X_F\setminus X_F$ is a normal crossing divisor. "Spreading out" one gets that for any $X\rightarrow S$, the base change of $X_{S'}$ to a Zariski open subset $S'\subset S$ is also a complement of a normal crossing divisor in a smooth proper $S'$-scheme, thus the argument above applies.