# A local-to global principle for splitting of Azumaya algebras

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.

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.
• Are you using $0 \to \text{Pic}(X_{\overline{F}}) \to H^2_{et}(X_{\overline{F}}, \mu_n) \to Br(X_{\overline{F}})[n] \to 0$ and if so why "if and only if" in the first sentence? Mar 31, 2021 at 1:27
• @Johan Ah, hm.. I was rather thinking about the boundary map $H^1_{et}(X_{\overline F},PGL_n) \rightarrow H^2_{et}(X,\mu_n)$ coming from the fiber sequence $BSL_n\rightarrow BPGL_n\rightarrow B^2\mu_n$. On the other hand I guess one could use the exact sequence in your comment equally well by lifting the class of $A$ in $Br(X_{\overline F})[n]$ to any class in $H^2_{et}(X_{\overline F},\mu_n)$ and playing the same game.. Mar 31, 2021 at 9:06
• @Johan I finally understood that you are right and there is something essential missing from the argument (I also deleted what I wrote before since it was wrong). My proof works well for classes in $H^2_{et}(X_{\overline F},\mu_n)$, but then returning to the Azumaya algebra question, a priori all what I get is that the class in the specialization is an image of a class in $\mathrm{Pic}(X_s)/n\mathrm{Pic}(X_s)$. Mar 31, 2021 at 22:26