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.

Definitions

A quasi-coherent sheaf on a ringed space

(

X

,

O

X

)

{\displaystyle (X,{\mathcal {O}}_{X})}

is a sheaf

F

{\displaystyle {\mathcal {F}}}

of

O

X

{\displaystyle {\mathcal {O}}_{X}}

-modules that has a local presentation, that is, every point in

X

{\displaystyle X}

has an open neighborhood

U

{\displaystyle U}

in which there is an exact sequence

O

X

⊕

I

|

U

→

O

X

⊕

J

|

U

→

F

|

U

→

0

{\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

I

{\displaystyle I}

and

J

{\displaystyle J}

.

A coherent sheaf on a ringed space

(

X

,

O

X

)

{\displaystyle (X,{\mathcal {O}}_{X})}

is a sheaf

F

{\displaystyle {\mathcal {F}}}

of

O

X

{\displaystyle {\mathcal {O}}_{X}}

-modules satisfying the following two properties:

F

{\displaystyle {\mathcal {F}}}

is of finite type over

O

X

{\displaystyle {\mathcal {O}}_{X}}

, that is, every point in

X

{\displaystyle X}

has an open neighborhood

U

{\displaystyle U}

in

X

{\displaystyle X}

such that there is a surjective morphism

O

X

n

|

U

→

F

|

U

{\displaystyle {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {F}}|_{U}}

for some natural number

n

{\displaystyle n}

;

for any open set

U

⊆

X

{\displaystyle U\subseteq X}

, any natural number

n

{\displaystyle n}

, and any morphism

φ

:

O

X

n

|

U

→

F

|

U

{\displaystyle \varphi :{\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {F}}|_{U}}

of

O

X

{\displaystyle {\mathcal {O}}_{X}}

-modules, the kernel of

φ

{\displaystyle \varphi }

is of finite type.

Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of

O

X

{\displaystyle {\mathcal {O}}_{X}}

-modules.

The case of schemes

When

X

{\displaystyle X}

is a scheme, the general definitions above are equivalent to more explicit ones.