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In mathematics, a cohomological invariant of an algebraic group ''G'' over a field is an invariant of forms of ''G'' taking values in a Galois cohomology group. ==Definition== Suppose that ''G'' is an algebraic group defined over a field ''K'', and choose a separably closed field containing ''K''. For a finite extension ''L'' of ''K'' in let Γ''L'' be the absolute Galois group of ''L''. The first cohomology H1(''L'', ''G'') = H1(Γ''L'', ''G'') is a set classifying the forms of ''G'' over ''L'', and is a functor of ''L''. A cohomological invariant of ''G'' of dimension ''d'' taking values in a Γ''K''-module ''M'' is a natural transformation of functors (of ''L'') from H1(L, ''G'') to H''d''(L, ''M''). In other words a cohomological invariant associates an element of an abelian cohomology group to elements of a non-abelian cohomology set. More generally, if ''A'' is any functor from finitely generated extensions of a field to sets, then a cohomological invariant of ''A'' of dimension ''d'' taking values in a Γ-module ''M'' is a natural transformation of functors (of ''L'') from ''A'' to H''d''(L, ''M''). The cohomological invariants of a fixed group ''G'' or functor ''A'', dimension ''d'' and Galois module ''M'' form an abelian group denoted by Inv''d''(''G'',''M'') or Inv''d''(''A'',''M''). 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「cohomological invariant」の詳細全文を読む スポンサード リンク
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