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.

An Azumaya algebra over a commutative local ring R is an R-algebra A that is free and of finite rank r≥1 as an R-module, such that the tensor product A \otimes_R A^\circ (where Ao is the opposite algebra) is isomorphic to the matrix algebra EndR(A) ≈ Mr(R) via the map sending a \otimes b to the endomorphism x → axb of A.

An Azumaya algebra on a scheme X with structure sheaf OX, according to the original Grothendieck seminar, is a sheaf A of OX-algebras that is étale locally isomorphic to a matrix algebra sheaf; one should, however, add the condition that each matrix algebra sheaf is of positive rank. Milne, Étale Cohomology, starts instead from the definition that it is a sheaf A of OX-algebras whose stalk Ax at each point x is an Azumaya algebra over the local ring OX,x in the sense given above. Two Azumaya algebras A1 and A2 are equivalent if there exist locally free sheaves E1 and E2 of finite positive rank at every point such that

A_1\otimes\mathrm{End}(E_1) \simeq A_2\otimes\mathrm{End}(E_2),

where End(Ei) is the endomorphism sheaf of Ei. The Brauer group of X (an analogue of the Brauer group of a field) is the set of equivalence classes of Azumaya algebras. The group operation is given by tensor product, and the inverse is given by the opposite algebra.

There have been significant applications of Azumaya algebras in diophantine geometry, following work of Yuri Manin. The Manin obstruction to the Hasse principle is defined using the Brauer group of schemes.

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