Azumaya algebra

Summary

An Azumaya algebra over a commutative ring $R$ is an associative algebra that behaves like a central simple algebra over every point of $R$. It is a faithful, finitely generated projective $R$-module $A$ for which $A \otimes_R A^{\mathrm{op}} \to \End_R(A)$ is an isomorphism. Over a field the Azumaya algebras are exactly the central simple algebras. Azumaya algebras define the Brauer group of a commutative ring and, in geometry, of a scheme.