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In mathematics, the Brauer–Wall group or super Brauer group or graded Brauer group for a field ''F'' is a group BW(''F'') classifying finite-dimensional graded central division algebras over the field. It was first defined by as a generalization of the Brauer group. The Brauer group of a field ''F'' is the set of the similarity classes of finite dimensional central simple algebras over ''F'' under the operation of tensor product, where two algebras are called similar if the commutants of their simple modules are isomorphic. Every similarity class contains a unique division algebra, so the elements of the Brauer group can also be identified with isomorphism classes of finite dimensional central division algebras. The analogous construction for Z/2Z-graded algebras defines the Brauer–Wall group BW(''F'').〔Lam (2005) pp.98–99〕 ==Properties== * The Brauer group B(''F'') injects into BW(''F'') by mapping a CSA ''A'' to the graded algebra which is ''A'' in grade zero. * showed that there is an exact sequence :: 0 → B(''F'') → BW(''F'') → Q(''F'') → 0 :where Q(''F'') is the group of graded quadratic extensions of ''F'', defined as an extension of Z/2 by ''F'' */''F'' *2 with multiplication (''e'',''x'')(''f'',''y'') = (''e'' + ''f'', (−1)''ef''''xy''). The map from W to BW is the Clifford invariant defined by mapping an algebra to the pair consisting of its grade and determinant. * There is a map from the additive group of the Witt–Grothendieck ring to the Brauer–Wall group obtained by sending a quadratic space to its Clifford algebra. The map factors through the Witt group,〔Lam (2005) p.113〕 which has kernel I3, where I is the fundamental ideal of W(''F'').〔Lam (2005) p.115〕 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–Wall group」の詳細全文を読む スポンサード リンク
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