Azumaya algebra

Azumaya algebra
Name	Azumaya algebra
Type	Algebraic structure
Introduced	1951
Introduced by	Goro Azumaya

Azumaya algebra.

An Azumaya algebra is a class of algebras that generalizes central simple algebras from field theory to schemes and rings, connecting ideas of Goro Azumaya, Richard Brauer, Emil Artin, John Tate, and Alexander Grothendieck. It appears naturally in the study of tensor categories related to Jean-Pierre Serre, Pierre Deligne, Jean-Louis Verdier, Alexander Grothendieck's cohomology theories and in formulations influenced by Oscar Zariski, André Weil, Claude Chevalley, and Henri Cartan. The notion plays a role in contexts involving Alexander Merkurjev, Dmitry Kaledin, Michel Demazure, David Mumford, and Michel Artin-style algebraic geometry.

Definition and basic properties

An Azumaya algebra over a commutative ring R is a sheaflike generalization characterized by local matrix algebra structure, formulated in the spirit of Goro Azumaya and later recast by Alexander Grothendieck and Jean-Pierre Serre. The definition uses local triviality: étale- or flat-locally on Spec R the algebra is isomorphic to a full matrix algebra M_n(R), a viewpoint advanced by Jean-Louis Verdier and influenced by descent techniques of Alexander Grothendieck and Jean Giraud. Key properties include centrality over R, separability conditions reflecting work by Emil Artin and John Tate, and finite presentation features related to David Mumford and Alexander Grothendieck's finiteness theorems. Structural invariants connect to the Brauer group studied by Richard Brauer, Claude Chevalley, Emil Artin, and Jean-Pierre Serre.

Examples and constructions

Basic examples arise from matrix algebras M_n(R) studied by Issai Schur, Richard Brauer, and Emil Artin. Crossed product constructions tie to Emil Artin and Richard Brauer's group cohomology approaches, using group actions as in work by James G. McKay and Jean-Pierre Serre. Azumaya algebras constructed from central simple algebras over fields relate to classical examples by Richard Brauer and Emil Artin. Étale forms constructed via torsors under projective linear groups involve techniques from Alexander Grothendieck, Jean Giraud, Pierre Deligne, and Jean-Pierre Serre. Twisted sheaves and gerbes inspired by Gérard Laumon, Laurent Lafforgue, Max Lieblich, and Kai Behrend produce examples used in moduli problems studied by David Mumford and Michael Artin.

Relationship to central simple algebras and Brauer group

Over a field, Azumaya algebras coincide with central simple algebras studied by Richard Brauer, Emil Artin, and Issai Schur. The passage from Azumaya classes over a scheme to Brauer classes involves cohomology groups investigated by Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, and Jean-Louis Verdier. The Brauer group itself was developed through contributions of Richard Brauer, Emil Artin, Claude Chevalley, and later reformulations by Alexander Grothendieck and Jean-Pierre Serre. Invariants like index and exponent reflect classical insights of Richard Brauer and arithmetic connections pursued by John Tate, Serge Lang, Kurt Hensel, and Alexander Merkurjev.

Étale and flat descent; sheaf-theoretic viewpoint

Descent theory for Azumaya algebras leverages étale topology and flat topology techniques advanced by Alexander Grothendieck, Jean Giraud, Jean-Pierre Serre, and Pierre Deligne. The sheaf-theoretic approach uses stacks and gerbes developed by Giraud, Grothendieck, Pierre Deligne, Jean-Louis Verdier, and later refined in geometric contexts by Max Lieblich and Kai Behrend. Étale local triviality connects to work on torsors under algebraic groups like Claude Chevalley's linear groups and Armand Borel's contributions to algebraic groups. Flat descent considerations relate to foundational ideas by Alexander Grothendieck, Jean-Pierre Serre, David Mumford, and Michael Artin.

Cohomological classification and obstruction theory

Cohomological classification of Azumaya algebras is expressed via second cohomology groups H^2 with coefficients in G_m, a viewpoint originated by Alexander Grothendieck and elaborated by Jean-Pierre Serre, Pierre Deligne, and Jean Giraud. Obstructions to lifting and splitting connect to spectral sequence techniques used by Jean-Louis Verdier and Pierre Deligne, and obstruction classes interact with theories by Michael Artin and David Mumford in deformation contexts. Relations with group cohomology derive from seminal work by Emil Artin, Richard Brauer, Serge Lang, and Alexander Merkurjev.

Module theory and Morita equivalence

Module categories over Azumaya algebras are Morita equivalent to module categories over the base ring, following Morita theory developed by Kiiti Morita and applied by Emil Artin and David Mumford. Equivalences of categories involve Tannakian-style ideas connected to Pierre Deligne and Jean-Pierre Serre. Projective module classifications use results by Jean-Pierre Serre, Hyman Bass, Daniel Quillen, and Bertrand Toen in higher categorical settings. Connections to derived categories and tilting theory draw on work by Amnon Neeman, Alastair King, Maxim Kontsevich, and Raphaël Rouquier.

Applications in algebraic geometry and deformation theory

Azumaya algebras appear in twisted sheaf theories by David Mumford, Pierre Deligne, Gérard Laumon, and Max Lieblich for moduli of vector bundles; they inform gerbe constructions used by Kai Behrend and Behrend-Noohi-style stacks. Deformation-theoretic roles reflect approaches by Michael Artin, Alexander Grothendieck, Pierre Deligne, Maxim Kontsevich, and Dmitry Kaledin in noncommutative deformations and derived algebraic geometry pursued by Jacob Lurie, Bertrand Toen, and Gabriele Vezzosi. Arithmetic applications tie to local and global class field techniques of John Tate, Serge Lang, Alexander Merkurjev, and Vladimir Voevodsky's motivic ideas. In geometric representation theory and mirror symmetry, Azumaya-related twists enter work by Edward Witten, Maxim Kontsevich, Nigel Hitchin, and Simon Donaldson.

Category:Algebraic structures