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 ${\displaystyle A\otimes _{R}A^{\circ }}$ (where A^o is the opposite algebra) is isomorphic to the matrix algebra End_R(A) ≈ M_r(R) via the map sending ${\displaystyle a\otimes b}$ to the endomorphism x → axb of A.

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