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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 Ext functor ： ウィキペディア英語版 In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics. The name "Ext" comes from group theory, as the Ext functor is used in group cohomology to classify abelian group extensions.== Definition and computation ==Let ''R'' be a ring and let Mod''R'' be the category of modules over ''R''. Let ''B'' be in Mod''R'' and set ''T''(''B'') = Hom''R''(''A,B''), for fixed ''A'' in Mod''R''. This is a left exact functor and thus has right derived functors ''RnT''. The Ext functor is defined by:\operatorname_R^n(A,B)=(R^nT)(B).This can be calculated by taking any injective resolution:0 \rightarrow B \rightarrow I^0 \rightarrow I^1 \rightarrow \dots, and computing:0 \rightarrow \operatorname_R(A,I^0) \rightarrow \operatorname_R(A,I^1) \rightarrow \dots.Then (''RnT'')(''B'') is the homology of this complex. Note that Hom''R''(''A,B'') is excluded from the complex.An alternative definition is given using the functor ''G''(''A'')=Hom''R''(''A,B''). For a fixed module ''B'', this is a contravariant left exact functor, and thus we also have right derived functors ''RnG'', and can define:\operatorname_R^n(A,B)=(R^nG)(A).This can be calculated by choosing any projective resolution:\dots \rightarrow P^1 \rightarrow P^0 \rightarrow A \rightarrow 0, and proceeding dually by computing:0\rightarrow\operatorname_R(P^0,B)\rightarrow \operatorname_R(P^1,B) \rightarrow \dots.Then (''RnG'')(''A'') is the homology of this complex. Again note that Hom''R''(''A,B'') is excluded.These two constructions turn out to yield isomorphic results, and so both may be used to calculate the Ext functor.== Ext and extensions == In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics. The name "Ext" comes from group theory, as the Ext functor is used in group cohomology to classify abelian group extensions. == Definition and computation == Let ''R'' be a ring and let Mod''R'' be the category of modules over ''R''. Let ''B'' be in Mod''R'' and set ''T''(''B'') = Hom''R''(''A,B''), for fixed ''A'' in Mod''R''. This is a left exact functor and thus has right derived functors ''RnT''. The Ext functor is defined by :$\backslash operatorname\_R^n(A,B)=(R^nT)(B).$ This can be calculated by taking any injective resolution :$0\; \backslash rightarrow\; B\; \backslash rightarrow\; I^0\; \backslash rightarrow\; I^1\; \backslash rightarrow\; \backslash dots,$ and computing :$0\; \backslash rightarrow\; \backslash operatorname\_R(A,I^0)\; \backslash rightarrow\; \backslash operatorname\_R(A,I^1)\; \backslash rightarrow\; \backslash dots.$ Then (''RnT'')(''B'') is the homology of this complex. Note that Hom''R''(''A,B'') is excluded from the complex. An alternative definition is given using the functor ''G''(''A'')=Hom''R''(''A,B''). For a fixed module ''B'', this is a contravariant left exact functor, and thus we also have right derived functors ''RnG'', and can define :$\backslash operatorname\_R^n(A,B)=(R^nG)(A).$ This can be calculated by choosing any projective resolution :$\backslash dots\; \backslash rightarrow\; P^1\; \backslash rightarrow\; P^0\; \backslash rightarrow\; A\; \backslash rightarrow\; 0,$ and proceeding dually by computing :$0\backslash rightarrow\backslash operatorname\_R(P^0,B)\backslash rightarrow\; \backslash operatorname\_R(P^1,B)\; \backslash rightarrow\; \backslash dots.$ Then (''RnG'')(''A'') is the homology of this complex. Again note that Hom''R''(''A,B'') is excluded. These two constructions turn out to yield isomorphic results, and so both may be used to calculate the Ext functor. == Ext and extensions == 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「In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics. The name "Ext" comes from group theory, as the Ext functor is used in group cohomology to classify abelian group extensions.== Definition and computation ==Let ''R'' be a ring and let Mod''R'' be the category of modules over ''R''. Let ''B'' be in Mod''R'' and set ''T''(''B'') = Hom''R''(''A,B''), for fixed ''A'' in Mod''R''. This is a left exact functor and thus has right derived functors ''RnT''. The Ext functor is defined by:\operatorname_R^n(A,B)=(R^nT)(B).This can be calculated by taking any injective resolution:0 \rightarrow B \rightarrow I^0 \rightarrow I^1 \rightarrow \dots, and computing:0 \rightarrow \operatorname_R(A,I^0) \rightarrow \operatorname_R(A,I^1) \rightarrow \dots.Then (''RnT'')(''B'') is the homology of this complex. Note that Hom''R''(''A,B'') is excluded from the complex.An alternative definition is given using the functor ''G''(''A'')=Hom''R''(''A,B''). For a fixed module ''B'', this is a contravariant left exact functor, and thus we also have right derived functors ''RnG'', and can define:\operatorname_R^n(A,B)=(R^nG)(A).This can be calculated by choosing any projective resolution:\dots \rightarrow P^1 \rightarrow P^0 \rightarrow A \rightarrow 0, and proceeding dually by computing:0\rightarrow\operatorname_R(P^0,B)\rightarrow \operatorname_R(P^1,B) \rightarrow \dots.Then (''RnG'')(''A'') is the homology of this complex. Again note that Hom''R''(''A,B'') is excluded.These two constructions turn out to yield isomorphic results, and so both may be used to calculate the Ext functor.== Ext and extensions == 」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.