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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 Zig-zag lemma ： ウィキペディア英語版 Zig-zag lemma In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category. == Statement == In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), let $(\backslash mathcal,\backslash partial\_),\; (\backslash mathcal,\backslash partial\_\text{'})$ and $(\backslash mathcal,\backslash partial\_\text{'}\text{'})$ be chain complexes that fit into the following short exact sequence: : $0\; \backslash longrightarrow\; \backslash mathcal\; \backslash stackrel\; \backslash mathcal\; \backslash stackrel\; \backslash mathcal\backslash longrightarrow\; 0$ Such a sequence is shorthand for the following commutative diagram: commutative diagram representation of a short exact sequence of chain complexes where the rows are exact sequences and each column is a complex. The zig-zag lemma asserts that there is a collection of boundary maps : $\backslash delta\_n\; :\; H\_n(\backslash mathcal)\; \backslash longrightarrow\; H\_(\backslash mathcal),$ that makes the following sequence exact: long exact sequence in homology, given by the Zig-Zag Lemma The maps $\backslash alpha\_$ *^ and $\backslash beta\_$ *^ are the usual maps induced by homology. The boundary maps $\backslash delta\_n^$ are explained below. The name of the lemma arises from the "zig-zag" behavior of the maps in the sequence. In an unfortunate overlap in terminology, this theorem is also commonly known as the "snake lemma," although there is another result in homological algebra with that name. Interestingly, the "other" snake lemma can be used to prove the zig-zag lemma, in a manner different from what is described below. 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「Zig-zag lemma」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.