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 1964–65. There are now several points of access to the basic definitions.
and from Local class field theory, there is the following commutative diagram:
where the vertical maps are isomorphisms and the horizontal maps are injections.
It comes from the composition of the cup product in etale cohomology with the Hilbert's Theorem 90 isomorphism
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.