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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Factor system ： ウィキペディア英語版 Factor system In mathematics, a factor system is a function on a group giving the data required to construct an algebra. A factor system constitutes a realisation of the cocycles in the second cohomology group in group cohomology. Let ''G'' be a group and ''L'' a field on which ''G'' acts as automorphisms. A ''cocycle'' or ''factor system'' is a map ''c'':''G'' × ''G'' → ''L'' * satisfying :$c(h,k)^g\; c(hk,g)\; =\; c(h,kg)\; c(k,g)\; .$ Cocycles are ''equivalent'' if there exists some system of elements ''a'' : ''G'' → ''L'' * with :$c\text{'}(g,h)\; =\; c(g,h)\; (a\_g^h\; a\_h\; a\_^)\; .$ Cocycles of the form :$c(g,h)\; =\; a\_g^h\; a\_h\; a\_^$ are called ''split''. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group H2(''G'',''L'' *). ==Crossed product algebras== Let us take the case that ''G'' is the Galois group of a field extension ''L''/''K''. A factor system ''c'' in H2(''G'',''L'' *) gives rise to a ''crossed product algebra'' ''A'', which is a ''K''-algebra containing ''L'' as a subfield, generated by the elements λ in ''L'' and ''u''''g'' with multiplication :$\backslash lambda\; u\_g\; =\; u\_g\; \backslash lambda^g\; ,$ :$u\_g\; u\_h\; =\; u\_\; c(g,h)\; .$ Equivalent factor systems correspond to a change of basis in ''A'' over ''K''. We may write :$A\; =\; (L,G,c)\; .$ Every central simple algebra over ''K'' that splits over ''L'' arises in this way.〔Jacobson (1996) p.57〕 The tensor product of algebras corresponds to multiplication of the corresponding elements in H2. We thus obtain an identification of the Brauer group, where the elements are classes of CSAs over ''K'', with H2.〔Saltman (1999) p.44〕〔Jacobson (1996) p.59〕 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「Factor system」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.