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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 Mapping cone (homological algebra) ： ウィキペディア英語版 Mapping cone (homological algebra) In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map ''f'' being acyclic means that the map is a quasi-isomorphism; if we pass to the derived category of complexes, this means that ''f'' is an isomorphism there, which recalls the familiar property of maps of groups, modules over a ring, or elements of an arbitrary abelian category that if the kernel and cokernel both vanish, then the map is an isomorphism. If we are working in a t-category, then in fact the cone furnishes both the kernel and cokernel of maps between objects of its core. ==Definition== The cone may be defined in the category of cochain complexes over any additive category (i.e., a category whose morphisms form abelian groups and in which we may construct a direct sum of any two objects). Let $A,\; B$ be two complexes, with differentials $d\_A,\; d\_B;$ i.e., :$A\; =\; \backslash dots\; \backslash to\; A^\; \backslash xrightarrow\; A^\; \backslash to\; \backslash cdots$ and likewise for $B.$ For a map of complexes $f\; :\; A\; \backslash to\; B,$ we define the cone, often denoted by $\backslash operatorname(f)$ or $C(f),$ to be the following complex: :$C(f)\; =\; A()\; \backslash oplus\; B\; =\; \backslash dots\; \backslash to\; A^n\; \backslash oplus\; B^\; \backslash to\; A^\; \backslash oplus\; B^n\; \backslash to\; A^\; \backslash oplus\; B^\; \backslash to\; \backslash cdots$ on terms, with differential : (acting as though on column vectors). Here $A()$ is the complex with $A()^n=A^$ and $d^n\_=-d^\_$. Note that the differential on $C(f)$ is different from the natural differential on $A()\; \backslash oplus\; B$, and that some authors use a different sign convention. Thus, if for example our complexes are of abelian groups, the differential would act as :$\backslash begin$ d^n_(a^, b^n) &=& \begin d^n_ & 0 \\ f()^n & d^n_B \end \begin a^ \\ b^n \end \\ &=& \begin - d^_A & 0 \\ f^ & d^n_B \end \begin a^ \\ b^n \end \\ &=& \begin - d^_A (a^) \\ f^(a^) + d^n_B(b^n) \end\\ &=& \left(- d^_A (a^), f^(a^) + d^n_B(b^n)\right). \end 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「Mapping cone (homological algebra)」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.