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In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over ''K'' and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is named after the algebraist Richard Brauer. The group may also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras. == Construction == A central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'', which is a simple ring, and for which the center is exactly ''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 ). 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 integers ''m'' and ''n'' at least 1, we get the Brauer equivalence and the Brauer classes. Given central simple algebras ''A'' and ''B'', one can look at the 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 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, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra ''A'' is the one containing the opposite algebra ''A''op (the opposite ring with the same action by ''K'' since the image of ''K'' → ''A'' is in the center of ''A''). In other words, for a CSA ''A'' we have ''A'' ⊗ ''A''op = M(''n''2,''K''), where ''n'' is the degree of ''A'' over ''K''. (This provides a substantial reason for caring about the notion of an opposite algebra: it provides the inverse in the Brauer group.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Brauer group」の詳細全文を読む スポンサード リンク
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