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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 Cochain complex ： ウィキペディア英語版 Chain complex In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of a topological space. More generally, homological algebra includes the study of chain complexes in the abstract, without any reference to an underlying space. In this case, chain complexes are studied axiomatically as algebraic structures. Applications of chain complexes usually define and apply their homology groups (cohomology groups for cochain complexes); in more abstract settings various equivalence relations are applied to complexes (for example starting with the chain homotopy idea). Chain complexes are easily defined in abelian categories, also. ==Formal definition== A chain complex $(A\_\backslash bullet,\; d\_\backslash bullet)$ is a sequence of abelian groups or modules ... ''A''2, ''A''1, ''A''0, ''A''-1, ''A''-2, ... connected by homomorphisms (called boundary operators) ''d''''n'' : ''A''''n''→''A''''n''−1, such that the composition of any two consecutive maps is zero: ''d''''n'' ∘ ''d''''n''+1 = 0 for all ''n''. They are usually written out as: ::$\backslash cdots\; \backslash to$ A_ \xrightarrow A_ \xrightarrow \to \cdots \xrightarrow A_1 \xrightarrow A_0 \xrightarrow A_ \xrightarrow \xrightarrow, $A^$, $A^0$, $A^1$, $A^2$, ... connected by homomorphisms $d^n\backslash colon\; A^n\; \backslash to\; A^$ such that the composition of any two consecutive maps is zero: $d^\; d^n\; =\; 0$ for all ''n'': ::$$ \cdots \to A^ \xrightarrow \xrightarrow A^1 \xrightarrow A^2 \to \cdots \to A^ \xrightarrow A^ \to \cdots. The index $n$ in either $A\_n$ or $A^n$ is referred to as the degree (or dimension). The only difference in the definitions of chain and cochain complexes is that, in chain complexes, the boundary operators decrease dimension, whereas in cochain complexes they increase dimension. A bounded chain complex is one in which almost all the ''A''''i'' are 0; ''i.e.'', a finite complex extended to the left and right by 0's. An example is the complex defining the homology theory of a (finite) simplicial complex. A chain complex is bounded above if all degrees above some fixed degree ''N'' are 0, and is bounded below if all degrees below some fixed degree are 0. Clearly, a complex is bounded both above and below if and only if the complex is bounded. Leaving out the indices, the basic relation on ''d'' can be thought of as ::$d\; d\; =\; 0.$ The elements of the individual groups of a chain complex are called chains (or cochains in the case of a cochain complex.) The image of ''d'' is the group of boundaries, or in a cochain complex, coboundaries. The kernel of ''d'' (i.e., the subgroup sent to 0 by ''d'') is the group of cycles, or in the case of a cochain complex, cocycles. From the basic relation, the (co)boundaries lie inside the (co)cycles. This phenomenon is studied in a systematic way using (co)homology groups. 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「Chain complex」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.