Azumaya algebra

In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring. The notion was developed further in ring theory, and in algebraic geometry, where Alexander Grothendieck made it the basis for his geometric theory of the Brauer group in Bourbaki seminars from 1964-5. There are now several points of access to the basic definitions.

For R a local ring, an Azumaya algebra is an R-algebra A which is free and of finite rank r as an R-module, and for which the natural action of A on itself by left-multiplication, and of A^o (the opposite ring) on A by right-multiplication, makes the tensor product isomorphic to the r×r matrix algebra over R.

For the scheme theory definition, on a scheme X with structure sheaf O_X the definition as in the original Grothendieck seminar is of a sheaf of O_X-algebras A that is locally isomorphic to a matrix algebra sheaf. Milne, Étale Cohomology, starts instead from the definition that the stalks A_x are Azumaya algebras over the local rings O_X,x at each point, in the sense given above. The Brauer group under this definition is defined as eqivalence classes of Azumaya algebras, where two algebras A₁ and A₂ are equivalent if there exist finite rank locally free sheaves E₁ and E₂ such that

$A_{1}\otimes \mathrm {End} (E_{1})\simeq A_{2}\otimes \mathrm {End} (E_{2}).$

Here End(E_i) denotes the endomorphism sheaf of E_i, which is a global matrix algebra. The group operation is given by tensor product, and the inverse by the opposite algebra.

There have been substantive applications of these global Azumaya algebras in diophantine geometry, following work of Yuri Manin. This has helped to clarify the area of obstructions to the Hasse principle.