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.

Definitions

A quasi-coherent sheaf on a ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$ is a sheaf ${\displaystyle {\mathcal {F}}}$ of ${\displaystyle {\mathcal {O}}_{X}}$-modules that has a local presentation, that is, every point in ${\displaystyle X}$ has an open neighborhood ${\displaystyle U}$ in which there is an exact sequence

${\displaystyle {\mathcal {O}}_{X}^{\oplus I}|_{U}\to {\mathcal {O}}_{X}^{\oplus J}|_{U}\to {\mathcal {F}}|_{U}\to 0}$

for some (possibly infinite) sets ${\displaystyle I}$ and ${\displaystyle J}$.

A coherent sheaf on a ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$ is a sheaf ${\displaystyle {\mathcal {F}}}$ of ${\displaystyle {\mathcal {O}}_{X}}$-modules satisfying the following two properties:

1. ${\displaystyle {\mathcal {F}}}$ is of finite type over ${\displaystyle {\mathcal {O}}_{X}}$, that is, every point in ${\displaystyle X}$ has an open neighborhood ${\displaystyle U}$ in ${\displaystyle X}$ such that there is a surjective morphism ${\displaystyle {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {F}}|_{U}}$ for some natural number ${\displaystyle n}$;
2. for any open set ${\displaystyle U\subseteq X}$, any natural number ${\displaystyle n}$, and any morphism ${\displaystyle \varphi :{\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {F}}|_{U}}$ of ${\displaystyle {\mathcal {O}}_{X}}$-modules, the kernel of ${\displaystyle \varphi }$ is of finite type.

Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of ${\displaystyle {\mathcal {O}}_{X}}$-modules.

The case of schemes

When ${\displaystyle X}$ is a scheme, the general definitions above are equivalent to more explicit ones. A sheaf ${\displaystyle {\mathcal {F}}}$ of ${\displaystyle {\mathcal {O}}_{X}}$-modules is quasi-coherent if and only if over each open affine subscheme ${\displaystyle U=\operatorname {Spec} A}$ the restriction ${\displaystyle {\mathcal {F}}|_{U}}$ is isomorphic to the sheaf ${\displaystyle {\tilde {M}}}$ associated to the module ${\displaystyle M=\Gamma (U,{\mathcal {F}})}$ over ${\displaystyle A}$. When ${\displaystyle X}$ is a locally Noetherian scheme, ${\displaystyle {\mathcal {F}}}$ is coherent if and only if it is quasi-coherent and the modules ${\displaystyle M}$ above can be taken to be finitely generated.

On an affine scheme ${\displaystyle U=\operatorname {Spec} A}$, there is an equivalence of categories from ${\displaystyle A}$-modules to quasi-coherent sheaves, taking a module ${\displaystyle M}$ to the associated sheaf ${\displaystyle {\tilde {M}}}$. The inverse equivalence takes a quasi-coherent sheaf ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle U}$ to the ${\displaystyle A}$-module ${\displaystyle {\mathcal {F}}(U)}$ of global sections of ${\displaystyle {\mathcal {F}}}$.

Here are several further characterizations of quasi-coherent sheaves on a scheme.^[1]

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