## 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.

## Ag. Algebraic Geometry – Are Azumaya Algebras of the Trivial Brewer Class Isomorphic to \$ mathcal {E} nd ( mathcal {F}) \$?

To let $$X$$ be a scheme, let it be $$mathcal {A}$$ be a bunch of locally free algebras $$X$$, We say $$mathcal {A}$$ is an Azumaya algebra when the natural map $$mathcal {A} otimes_ { mathcal {O} _X} mathcal {A} ^ {opp} to mathcal {E} nd _ { mathcal {O} _X} ( mathcal {A} ),$$ $$a otimes b mapsto (x mapsto axb)$$ is an isomorphism.

Two Azumaya algebras $$mathcal {A}, mathcal {B}$$ are called Morita equivalent if locally free sheaves are present $$mathcal {F}, mathcal {G}$$, so that $$mathcal {A} otimes mathcal {E} and _ { mathcal {O} _X} ( mathcal {F}) cong mathcal {B} otimes mathcal {E} and _ { mathcal {O} _X} ( mathcal {G}).$$

To let $$mathcal {A}$$ be an Azumaya algebra corresponding to Morita $$mathcal {B}: = mathcal {O} _X$$does $$mathcal {A}$$ necessarily have the form $$mathcal {E} nd _ { mathcal {O} _X} ( mathcal {H})$$ for some locally free sheaf $$mathcal {H}$$ on $$X$$?