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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 Coherent sheaf ： ウィキペディア英語版 Coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometrical information. Coherent sheaves can be seen as a generalization of vector bundles, or of locally free sheaves of finite rank. Unlike vector bundles, they form a "nice" category closed under usual operations such as taking kernels, cokernels and finite direct sums. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Many results and properties in algebraic geometry and complex analytic geometry are formulated in terms of coherent or quasi-coherent sheaves and their cohomology. == Definitions == A ''coherent sheaf'' on a ringed space $(X,\backslash mathcal\_X)$ is a sheaf $\backslash mathcal$ of $\backslash mathcal\_X$-modules with the following two properties: # $\backslash mathcal$ is of ''finite type'' over $\backslash mathcal\_X$, i.e., for any point $x\backslash in\; X$ there is an open neighbourhood $U\backslash subset\; X$ such that the restriction $\backslash mathcal|\_U$ of $\backslash mathcal$ to $U$ is generated by a finite number of sections (in other words, there is a surjective morphism $\backslash mathcal\_X^n|\_U\; \backslash to\; \backslash mathcal|\_U$ for some $n\backslash in\backslash mathbb$); and # for any open set $U\backslash subset\; X$, any $n\backslash in\backslash mathbb$ and any morphism $\backslash varphi\backslash colon\; \backslash mathcal\_X^n|\_U\; \backslash to\; \backslash mathcal|\_U$ of $\backslash mathcal\_X$-modules, the kernel of $\backslash varphi$ is finitely generated. The sheaf of rings $\backslash mathcal\_X$ is coherent if it is coherent considered as a sheaf of modules over itself. Important examples of coherent sheaves of rings include the sheaf of germs of holomorphic functions on a complex manifold (Oka coherence theorem) and the structure sheaf of a Noetherian scheme〔 from algebraic geometry. A submodule of a coherent sheaf is coherent if it is of finite type. A coherent sheaf is always a sheaf of ''finite presentation'', or in other words each point $x\backslash in\; X$ has an open neighbourhood $U$ such that the restriction $\backslash mathcal|\_U$ of $\backslash mathcal$ to $U$ is isomorphic to the cokernel of a morphism $\backslash mathcal\_X^n|\_U\; \backslash to\; \backslash mathcal\_X^m|\_U$ for some integers $n$ and $m$. If $\backslash mathcal\_X$ is coherent, then the converse is true and each sheaf of finite presentation over $\backslash mathcal\_X$ is coherent. A sheaf $\backslash mathcal$ of $\backslash mathcal\_$-modules is said to be quasi-coherent if it has a local presentation, i.e. if there exist an open cover by $U\_i$ of the topological space $X$ and an exact sequence :$\backslash mathcal\_X^|\_\; \backslash to\; \backslash mathcal\_X^|\_\; \backslash to\; \backslash mathcal|\_\; \backslash to\; 0$ where the first two terms of the sequence are direct sums (possibly infinite) of copies of the structure sheaf. Note: Some authors, notably Hartshorne, use a different but essentially equivalent definition of coherent and quasi-coherent sheaves on a scheme (cf. #Properties). Let ''X'' be a scheme and ''F'' an $\backslash mathcal\_X$-module. Then: *''F'' is quasi-coherent if there are open affine cover $U\_i\; =\; \backslash operatorname\; A\_i$ of ''X'' and ''A''''i''-modules ''M''''i'' such that $F|\_\; \backslash simeq\; \backslash widetilde\_i$ as $O\_$-modules, where $\backslash widetilde\_i$ are sheaves associated to $M\_i$. *When ''X'' is a Noetherian scheme, ''F'' is coherent if it is quasi-coherent and the $M\_i$ above can be taken to be finitely generated. 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「Coherent sheaf」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.