## brauer groups – 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.

## Brauer group of $mathbb{Z}_{(p)}$ – MathOverflow

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Let $$R$$ be a Noetherian local ring and let $$R^h$$ be its Henselization. What can we say about the kernel and range of the map
$$Br(R) rightarrow Br(R^h)$$
Are there interesting situations when the map is injective? Do we know more when $$R$$ is a discrete valuation ring?
Let $$A$$ be a discrete valuation ring with perfect residue field $$k$$ of characteristic $$p$$ and field of fractions $$K$$ of characteristic $$0$$. Let $$G$$ and $$H$$ be two finite groups and assume that $$K$$ is sufficiently large, so that the irreducible characters of both groups form a $$K$$-basis of the respective space of class functions with values in $$K$$.
If $$G$$ and $$H$$ have the same character table over $$K$$, i.e. if we have a bijection between the conjugacy classes of $$G$$ and $$H$$ and a bijection between the irreducible characters over $$K$$ under which the character table is preserved, do $$G$$ and $$H$$ then also have the same Brauer character table?