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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,\mathcal_X) is a sheaf \mathcal of \mathcal_X-modules with the following two properties:
# \mathcal is of ''finite type'' over \mathcal_X, i.e., for any point x\in X there is an open neighbourhood U\subset X such that the restriction \mathcal|_U of \mathcal to U is generated by a finite number of sections (in other words, there is a surjective morphism \mathcal_X^n|_U \to \mathcal|_U for some n\in\mathbb); and
# for any open set U\subset X, any n\in\mathbb and any morphism \varphi\colon \mathcal_X^n|_U \to \mathcal|_U of \mathcal_X-modules, the kernel of \varphi is finitely generated.
The sheaf of rings \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\in X has an open neighbourhood U such that the restriction \mathcal|_U of \mathcal to U is isomorphic to the cokernel of a morphism \mathcal_X^n|_U \to \mathcal_X^m|_U for some integers n and m. If \mathcal_X is coherent, then the converse is true and each sheaf of finite presentation over \mathcal_X is coherent.
A sheaf \mathcal of \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
:\mathcal_X^|_ \to \mathcal_X^|_ \to \mathcal|_ \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 \mathcal_X-module. Then:
*''F'' is quasi-coherent if there are open affine cover U_i = \operatorname A_i of ''X'' and ''A''''i''-modules ''M''''i'' such that F|_ \simeq \widetilde_i as O_-modules, where \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)
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