Brauer group

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer. The Brauer group arose out of attempts to classify division algebras over a field. It can also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles. A central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A such that A is a simple ring and the center of A is equal to K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras. For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The finite-dimensional division algebras with center R (that means the dimension over R is finite) are the real numbers and the quaternions by a theorem of Frobenius, while any matrix ring over the reals or quaternions – M(n, R) or M(n, H) – is a CSA over the reals, but not a division algebra (if n > 1). We obtain an equivalence relation on CSAs over K by the Artin–Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n, D) for some division algebra D. If we look just at D, that is, if we impose an equivalence relation identifying M(m, D) with M(n, D) for all positive integers m and n, we get the Brauer equivalence relation on CSAs over K. The elements of the Brauer group are the Brauer equivalence classes of CSAs over K. Given central simple algebras A and B, one can look at their tensor product A ⊗ B as a K-algebra (see tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterization: a central simple algebra A over K is a K-algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of K.

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