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 A^o is the opposite algebra) is isomorphic to the matrix algebra End_R(A) ≈ M_r(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 O_X, according to the original Grothendieck seminar, is a sheaf A of O_X-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 O_X-algebras whose stalk A_x at each point x is an Azumaya algebra over the local ring O_X,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(E_i) is the endomorphism sheaf of E_i. 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.

References

• Knus, Max-Albert (1991), Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften 294, Berlin etc.: Springer-Verlag, ISBN 3-540-52117-8, Zbl 0756.11008
• Knus, Max-Albert; Ojanguren, Manuel (1974), Théorie de la descente et algèbres d'Azumaya, Lecture Notes in Mathematics 389, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0057799, MR 0417149, Zbl 0284.13002
• Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.