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.

# Tag: Azumaya

## 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 $?