Brauer group

From Academic Kids

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

For example when K is the real number field R, the Brauer group Br(R) is a cyclic group of order two: there are just two types of division algebra, R and the quaternion algebra H. The product in the Brauer group is based on the tensor product: the statement that H has order two in the group is equivalent to the existence of an isomorphism of R-algebras

<math> \mathbb{H} \otimes \mathbb{H} \cong M(4, \mathbb{R})<math>

of 16-dimensional algebras, where the RHS is the ring of 4×4 real matrices.

A central simple algebra (CSA) over K is a finite-dimensional (associative) algebra K which is a simple ring for which the center is exactly K. For example, the complex numbers C form a CSA over themselves, but not over R (the center is too large).

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

<math> A\otimes B <math>

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

<math>(a \otimes b) \cdot (c\otimes d) = ac\otimes bd.<math>.

This makes the tensor product into a K-algebra (see also tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterisation: a central simple algebra over K is a K-algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of K.

Given this closure property for CSAs, they form a monoid under tensor product. To get a group, apply the Artin-Wedderburn theorem (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. A generalisation via the theory of Azumaya algebras was introduced in algebraic geometry by Grothendieck.

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