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In mathematics, especially in the areas of abstract algebra dealing with group cohomology or relative homological algebra, Shapiro's lemma, also known as the Eckmann–Shapiro lemma, relates extensions of modules over one ring to extensions over another, especially the group ring of a group and of a subgroup. It thus relates the group cohomology with respect to a group to the cohomology with respect to a subgroup. Shapiro's lemma is named after Arnold Shapiro, who proved it in 1961;〔.〕 however, Beno Eckmann had discovered it earlier, in 1953.〔.〕 ==Statement for rings== Let ''R'' → ''S'' be a ring homomorphism, so that ''S'' becomes a left and right ''R''-module. Let ''M'' be a left ''S''-module and ''N'' a left ''R''-module. By restriction of scalars, ''M'' is also a left ''R''-module. * If ''S'' is projective as a right ''R''-module, then: : See . The projectivity conditions can be weakened into conditions on the vanishing of certain Tor- or Ext-groups: see . 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「Shapiro's lemma」の詳細全文を読む スポンサード リンク
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