Brauer groups

Brauer groups
Name	Brauer groups
Field	Algebra, Number theory
Introduced	1920s
Notable	Richard Brauer, Emil Artin, Helmut Hasse

Brauer groups are algebraic invariants classifying equivalence classes of central simple algebras over a base, connecting algebra, number theory, and algebraic geometry. They arose in the work of Richard Brauer and were developed through interactions with Emil Artin, Helmut Hasse, and later contributors such as Jean-Pierre Serre and Alexander Grothendieck. Brauer groups encode obstructions to splitting algebras, relate to class field theory, and admit both cohomological and geometric incarnations.

Definition and basic examples

A classical definition identifies equivalence classes of finite-dimensional central simple algebras over a field, with the group law given by tensor product and inverse given by opposite algebra; typical examples include matrix algebras M_n(F) and division algebras such as quaternion algebras like the Hamiltonians related to William Rowan Hamilton. For global fields like Q or number fields studied by Carl Friedrich Gauss and Leopold Kronecker, one encounters nontrivial classes arising from cyclic algebras connected to Ludwig Faddeev-style constructions and explicit norm residue symbols used by Helmut Hasse in local reciprocity. Over algebraically closed fields such as C the group is trivial, while over finite fields appearing in the work of Évariste Galois and Galois theory the group often vanishes, illustrating contrasts visible in examples like the real numbers R where the quaternion division algebra gives a nontrivial two-torsion element linked to results of Ferdinand Georg Frobenius.

Cohomological and algebraic formulations

Cohomologically, one identifies the group with the Galois cohomology group H^2(G_K, K^s×) for the absolute Galois group G_K of a field K, an approach popularized by Jean-Pierre Serre and used in the proofs by John Tate and Serre of duality theorems. The Merkurjev–Suslin theorem, proved by Alexander Merkurjev and Andrei Suslin, connects the n-torsion of the group to Milnor K-theory K^M_2(K) and symbols considered by André Weil and Emil Artin, while the norm residue isomorphism conjecture resolved by Vladimir Voevodsky and collaborators generalizes these links. Algebraic descriptions use equivalence relations on central simple algebras developed in Richard Brauer's original papers and elaborated in later texts by Ira G. (I.G.) Rosenberg? and David J. Saltman.

Central simple algebras and division algebras

Central simple algebras (CSAs) over a field K, classified up to Brauer equivalence, are matrix algebras over division algebras by the Wedderburn theorem proved in work extending ideas of Joseph Wedderburn and Emil Artin. Division algebras like cyclic algebras constructed via cyclic extensions associated to Artin–Schreier theory or cyclic Galois groups studied by Emil Artin provide explicit representatives. The index and exponent invariants of a CSA reflect arithmetic properties studied in the works of Helmut Hasse, John Tate, and Alexander Merkurjev, with deep results such as Albert's theorem and the work of A.A. Albert on equivalence classes of central division algebras.

Brauer group of a field and local/global theory

For local fields like Q_p and completions considered in the analysis of Kurt Hensel, local class field theory and Hasse invariants classify Brauer groups via invariants in Q/Z, while for global fields such as number fields treated by Ernst Eduard Kummer and Richard Dedekind the Albert–Brauer–Hasse–Noether theorem relates the global Brauer group to local invariants and reciprocity laws central to Helmut Hasse and Emil Artin's program. The behavior under extension and corestriction maps ties into reciprocity maps appearing in the work of John Tate and Emmy Noether; explicit computations for cyclotomic fields engage results of Kummer and Leopold Kronecker on ramification and norms. Examples include nontrivial elements detected by Hilbert symbols used by David Hilbert in local reciprocity.

Connections with Galois cohomology and class field theory

The identification with H^2 of the absolute Galois group places Brauer groups at the heart of Galois cohomology as developed by Jean-Pierre Serre and John Tate, linking them to Tate duality and Poitou–Tate exact sequences used in arithmetic geometry by Gerd Faltings and Alexander Grothendieck. Class field theory, cultivated by Emil Artin and Helmut Hasse, uses Brauer group pairings to express reciprocity laws and dualities; the Brauer–Manin obstruction, introduced in works building on Yuri Manin's ideas, employs Brauer group elements to obstruct rational points on varieties studied by Peter Swinnerton-Dyer and Birch and Swinnerton-Dyer conjecture-related investigations. Connections extend to motivic cohomology and the norm residue isomorphism proven by Vladimir Voevodsky and Markus Rost.

Generalizations: Brauer group of a scheme and Azumaya algebras

Grothendieck extended the notion to schemes via equivalence classes of Azumaya algebras, leading to the cohomological Brauer group H^2_et(X, G_m) in étale cohomology pioneered by Alexander Grothendieck and Jean-Louis Verdier. Azumaya algebras generalize central simple algebras on schemes and appear in moduli problems treated by researchers such as Maxim Kontsevich and Andrei Suslin; the theory interacts with sheaf theory and the étale site introduced by Grothendieck and Michael Artin. Geometric applications include obstructions to the existence of universal bundles on moduli spaces studied by David Mumford and the role of twisted sheaves in derived categories explored by Alexander Polishchuk and Dmitri Orlov.

Category:Algebraic number theory