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In mathematics, the inflation-restriction exact sequence is an exact sequence occurring in group cohomology and is a special case of the five-term exact sequence arising from the study of spectral sequences. Specifically, let ''G'' be a group, ''N'' a normal subgroup, and ''A'' an abelian group which is equipped with an action of ''G'', i.e., a homomorphism from ''G'' to the automorphism group of ''A''. The quotient group ''G/N'' acts on ''AN = ''. Then the inflation-restriction exact sequence is: ::0 → ''H'' 1(''G''/''N'', ''A''''N'') → ''H'' 1(''G'', ''A'') → ''H'' 1(''N'', ''A'')''G''/''N'' → ''H'' 2(''G''/''N'', ''A''''N'') →''H'' 2(''G'', ''A'') : In this sequence, there are maps * ''inflation'' ''H'' 1(''G''/''N'', ''A''''N'') → ''H'' 1(''G'', ''A'') * ''restriction'' ''H'' 1(''G'', ''A'') → ''H'' 1(''N'', ''A'')''G''/''N'' * ''transgression'' ''H'' 1(''N'', ''A'')''G''/''N'' → ''H'' 2(''G''/''N'', ''A''''N'') * ''inflation'' ''H'' 2(''G''/''N'', ''A''''N'') →''H'' 2(''G'', ''A'') The inflation and restriction are defined for general ''n'': * ''inflation'' ''H''''n''(''G''/''N'', ''A''''N'') → ''H''''n''(''G'', ''A'') * ''restriction'' ''H''''n''(''G'', ''A'') → ''H''''n''(''N'', ''A'')''G''/''N'' The transgression is defined for general ''n'' * ''transgression'' ''H''''n''(''N'', ''A'')''G''/''N'' → ''H''''n''+1(''G''/''N'', ''A''''N'') only if ''H''''i''(''N'', ''A'')''G''/''N'' = 0 for ''i'' ≤ ''n''-1.〔Gille & Szamuely (2006) p.67〕 The sequence for general ''n'' may be deduced from the case ''n''=1 by dimension-shifting or from the Lyndon–Hochschild–Serre spectral sequence.〔Gille & Szamuely (2006) p.68〕 ==References== * * * * * * 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「Inflation-restriction exact sequence」の詳細全文を読む スポンサード リンク
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