nLab
Azumaya algebra

Given a commutative unital ring $R$, an Azumaya $R$-algebra is a (noncommutative in general) $R$-algebra $A$ which is finitely generated faithful projective as an $R$-module and the canonical morphism $A{\otimes }_R{A}^{\mathrm{op}}\to {\mathrm{End}}_R(A)$ is an isomorphism. This definition extends the notion of a central simple algebra? over a field.

More generally, Grothendieck defines an Azumaya algebra over a scheme $X$ as a sheaf $𝒜$ of ${𝒪}_X$-algebras such that for each point $x\in X$, the corresponding stalk ${𝒜}_x$ is an Azumaya ${𝒪}_{X,x}$-algebra.

The Brauer group? $\mathrm{Br}(X)$ classifies Azumaya algebras over $X$ up to a suitably defined equivalence relation: $𝒜\sim ℬ$ if $𝒜{\otimes }_{{𝒪}_X}\mathrm{End}(ℰ)\cong 𝒜{\otimes }_{{𝒪}_X}\mathrm{End}(ℱ)$ for some locally free sheaves of ${𝒪}_X$-modules $ℰ$ and $ℱ$ of finite rank. The group operation of $\mathrm{Br}(X)$ is induced by the tensor product. The Brauer group can be reexpressed in terms of second nonabelian cohomology; indeed a sheaf of Azumaya algebras over $X$ determines an ${𝒪}_X^{*}$-gerbe (or $U(1)$-gerbe in the manifold context).

Brauer groups and Azumaya algebras are closely related to Morita theory? and they make sense in the context of algebras and bimodules in the context of braided monoidal categories. Karoubi K-theory involves an element in a Brauer group and in the original Karoubi–Donovan paper is related to a twisting with a “local system” which involves Azumaya algebras.

Literature (alphabetically)

• Category Cafe 2006: Picard and Brauer 2-groups
• G. Cortiñas, C. Weibel, Homology of Azumaya algebras, Proc. AMS 121, 1, pp. 1994 (jstor)
• J. W. Duskin, The Azumaya complex of a commutative ring, in Categorical Algebra and its Appl., Lec. Notes in Math. 1348 (1988) doi:10.1007/BFb0081352
• A. Grothendieck, Le groupe de Brauer I, II, III, in Dix exposes sur la cohomologie des schemas (I: Algèbres d’Azumaya et interprétations diverses) North-Holland Pub. Co., Amsterdam (1969)
• Karoubi, Donovan, Graded Brauer groups and $K$-theory with local coefficients (pdf)
• M-A. Knus, M. Ojanguren, Théorie de la descente et algèbres d’Azumaya, Lec. Notes in Math. 389, Springer 1974, doi:10.1007/BFb0057799, MR0417149
• J. Milne, Étale cohomology, Princeton Univ. Press
• R. Street, Descent, Oberwolfach preprint
• Enrico M. Vitale, A Picard-Brauer exact sequence of categorical groups, pdf
• wikipedia page