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Concept
Coherent sheaf
Summary
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.
Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an , and so they are closed under operations such as taking , , and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank.
Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf.
A quasi-coherent sheaf on a ringed space is a sheaf of -modules which has a local presentation, that is, every point in has an open neighborhood in which there is an exact sequence
for some (possibly infinite) sets and .
A coherent sheaf on a ringed space is a sheaf satisfying the following two properties:
is of finite type over , that is, every point in has an open neighborhood in such that there is a surjective morphism for some natural number ;
for any open set , any natural number , and any morphism of -modules, the kernel of is of finite type.
Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of -modules.
When is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf of -modules is quasi-coherent if and only if over each open affine subscheme the restriction is isomorphic to the sheaf associated to the module over . When is a locally Noetherian scheme, is coherent if and only if it is quasi-coherent and the modules above can be taken to be finitely generated.
On an affine scheme , there is an equivalence of categories from -modules to quasi-coherent sheaves, taking a module to the associated sheaf . The inverse equivalence takes a quasi-coherent sheaf on to the -module of global sections of .
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