Azumaya algebra

A central simple algebra lives over a field; it is locally, after extending to the algebraic closure, just a matrix algebra. Azumaya algebras are what you get when you let the base be a commutative ring rather than a field — a family of central simple algebras varying over a space, that becomes matrix-like everywhere locally but may twist nontrivially as you move around. They are the right setting for the Brauer group of a ring or a scheme.

Precisely, an Azumaya algebra over a commutative ring R is an R-algebra A that is a faithful, finitely generated projective R-module and whose natural map A tensor_R A-op -> End_R(A) is an isomorphism. Equivalently A is central separable: its center is exactly R and it is separable as an R-algebra. When R is a field this recovers exactly the central simple algebras.

The defining isomorphism A tensor A-op ≅ End_R(A) is the engine: it generalizes the field fact that A tensor A-op is a matrix algebra, and it makes the Brauer equivalence classes of Azumaya algebras into a group Br(R) under tensor product, with inverse given by the opposite algebra. Locally, after passing to each residue field or completing at each prime, an Azumaya algebra of constant rank n^2 becomes the matrix ring M_n.

Azumaya algebras are indispensable in algebraic geometry, where the Brauer group of a scheme — assembled from Azumaya algebras over the structure sheaf, or cohomologically from the étale cohomology group H^2 with values in the multiplicative group — obstructs the existence of fine moduli spaces and underlies the Brauer-Manin obstruction in arithmetic. They are the genuinely global, relative version of the classical central simple algebra story.