Quasi-coherent sheaf
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 abelian category, and so they are closed under operations such as taking kernels, images, 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 quasicoherent sheaf on a ringed space (X,OX) is a sheaf F of OX-modules which has a local presentation, that is, every point in X has an open neighborhood U in which there is an exact sequence
for some sets I and J (possibly infinite).
A coherent sheaf on a ringed space (X,OX) is a quasicoherent sheaf F satisfying the following two properties:
Morphisms between (quasi)coherent sheaves are the same as morphisms of sheaves of OX-modules.
When X is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf F of OX-modules is quasicoherent if and only if over each open affine subscheme U=Spec(R) the restriction F|U is isomorphic to a sheaf associated to the module M=Γ(U, F) over R. When X is a locally Noetherian scheme, F is coherent if and only if it is quasi-coherent and the modules M above can be taken to be finitely generated.
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