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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 Sheaf cohomology ： ウィキペディア英語版 Sheaf cohomology In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf ''F''. This is the main step, in numerous areas, from sheaf theory as a description of a geometric problem, to its use as a tool capable of calculating dimensions of important geometric invariants. Its development was rapid in the years after 1950, when it was realised that sheaf cohomology was connected with more classical methods applied to the Riemann-Roch theorem, the analysis of a linear system of divisors in algebraic geometry, several complex variables, and Hodge theory. The dimensions or ranks of sheaf cohomology groups became a fresh source of geometric data, or gave rise to new interpretations of older work. ==One of motivations== The short exact sequence of sheaves $\backslash mathcal,\backslash mathcal,\backslash mathcal$ on a topological space ''X'' is the exact sequence of form :$0\backslash \; \backslash rightarrow\backslash mathcal\backslash \; \backslash stackrel\backslash \; \backslash mathcal\backslash \; \backslash stackrel\backslash \; \backslash mathcal\backslash \; \backslash rightarrow\backslash \; 0$ . Namely, $\backslash phi$ is injective, $\backslash psi$ surjective and $\backslash text\backslash phi=\backslash text\backslash psi$ . This sequence is exact if and only if $\backslash phi$ is injective and　$\backslash mathcal\backslash cong\backslash mathcal/\backslash mathcal$ . From this short exact sequence we can obtain the sequence of the sections of sheaves: :$0\backslash \; \backslash rightarrow\backslash \; \backslash Gamma(\backslash mathcal,X)\backslash \; \backslash stackrel\backslash \; \backslash Gamma(\backslash mathcal,X)\backslash \; \backslash stackrel\backslash \; \backslash Gamma(\backslash mathcal,X)$ . However, in general, $\backslash psi\_$ * is not always surjective. One of motivations of sheaf cohomology is to extend this sequence with a long exact sequence of sheaves. For a typical example there are Cousin problems. 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「Sheaf cohomology」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.