Brauer group

In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field (mathematics) K . It is an abelian group with elements Isomorphism classes of division algebras over K , such that the center of an algebra is exactly K . The group is named for the algebraist Richard Brauer.

For example when K is the : the statement that H has order two in the group is equivalent to the existence of an isomorphism of R-algebras

: mathbb{H} otimes mathbb{H} cong M(4, mathbb{R})

of 16-dimensional algebras, where the Left-hand side and right-hand side of an equation is the ring of 4×4 real matrices.

A central simple algebra (CSA) over K is a finite-dimensional (associative) algebra over a field A , which is a simple ring, and for which the center of an algebra is exactly K . For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large). That is why division algebras over R are not 1-1 with Br(R).

Given two such central simple algebras A and B , one defines a product on

: Aotimes B

(taken as vector spaces over K ) using the bilinearity of the definition

:(a otimes b) cdot (cotimes d) = acotimes bd.

This makes the tensor product into a K -algebra (see also of K .

Given this closure property for CSAs, they form a monoid under tensor product. To get a group, apply the Artin-Wedderburn theorem (Joseph Wedderburn s part, in fact), to express any CSA as M( n ,D), an n × n matrix ring over a division algebra D. If we look just at D, rather than the value of n , the monoid becomes a group. That is, if we impose an equivalence relation identifying M( m ,D) with M( n ,D) for all integers m and n at least 1, we get a congruence relation; and the congruence classes are all invertible.

In the further theory, the Brauer groups of local fields are computed (they all turn out to be subgroups of Q/Z, for p-adic fields); and the results applied to global fields. This gives one approach to class field theory. It also has been applied to Diophantine equations.

In the general theory the Brauer group is expressed by factor sets; and expressed in terms of Galois cohomology via

:Br(K) cong H^2(Gal(K^s/K), {K^s}^*).

Here, not assuming K to be a perfect field, K s is the separable closure. When K is perfect this is the same as an algebraic closure; otherwise the Galois group must be defined in terms of K s / K even to make sense.

A generalisation via the theory of Azumaya algebras was introduced in algebraic geometry by Alexander Grothendieck.