# Coherent sheaf

Generalization of vector bundles From Wikipedia, the free encyclopedia

Generalization of vector bundles From Wikipedia, the free encyclopedia

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 **quasi-coherent sheaf** on a ringed space is a sheaf of -modules that 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 of -modules 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 .

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

**Theorem** — Let be a scheme and an -module on it. Then the following are equivalent.

- is quasi-coherent.
- For each open affine subscheme of , is isomorphic as an -module to the sheaf associated to some -module .
- There is an open affine cover of such that for each of the cover, is isomorphic to the sheaf associated to some -module.
- For each pair of open affine subschemes of , the natural homomorphism

- is an isomorphism.

- For each open affine subscheme of and each , writing for the open subscheme of where is not zero, the natural homomorphism

- is an isomorphism. The homomorphism comes from the universal property of localization.

On an arbitrary ringed space, quasi-coherent sheaves do not necessarily form an abelian category. On the other hand, the quasi-coherent sheaves on any scheme form an abelian category, and they are extremely useful in that context.^{[2]}

On any ringed space , the coherent sheaves form an abelian category, a full subcategory of the category of -modules.^{[3]} (Analogously, the category of coherent modules over any ring is a full abelian subcategory of the category of all -modules.) So the kernel, image, and cokernel of any map of coherent sheaves are coherent. The direct sum of two coherent sheaves is coherent; more generally, an -module that is an extension of two coherent sheaves is coherent.^{[4]}

A submodule of a coherent sheaf is coherent if it is of finite type. A coherent sheaf is always an -module of *finite presentation*, meaning that each point in has an open neighborhood such that the restriction of to is isomorphic to the cokernel of a morphism for some natural numbers and . If is coherent, then, conversely, every sheaf of finite presentation over is coherent.

The sheaf of rings is called coherent if it is coherent considered as a sheaf of modules over itself. In particular, the **Oka coherence theorem** states that the sheaf of holomorphic functions on a complex analytic space is a coherent sheaf of rings. The main part of the proof is the case . Likewise, on a locally Noetherian scheme , the structure sheaf is a coherent sheaf of rings.^{[5]}

- An -module on a ringed space is called
**locally free of finite rank**, or a**vector bundle**, if every point in has an open neighborhood such that the restriction is isomorphic to a finite direct sum of copies of . If is free of the same rank near every point of , then the vector bundle is said to be of rank .

- Vector bundles in this sheaf-theoretic sense over a scheme are equivalent to vector bundles defined in a more geometric way, as a scheme with a morphism and with a covering of by open sets with given isomorphisms over such that the two isomorphisms over an intersection differ by a linear automorphism.
^{[6]}(The analogous equivalence also holds for complex analytic spaces.) For example, given a vector bundle in this geometric sense, the corresponding sheaf is defined by: over an open set of , the -module is the set of sections of the morphism . The sheaf-theoretic interpretation of vector bundles has the advantage that vector bundles (on a locally Noetherian scheme) are included in the abelian category of coherent sheaves.

- Locally free sheaves come equipped with the standard -module operations, but these give back locally free sheaves.
^{[vague]}

- Let , a Noetherian ring. Then vector bundles on are exactly the sheaves associated to finitely generated projective modules over , or (equivalently) to finitely generated flat modules over .
^{[7]} - Let , a Noetherian -graded ring, be a projective scheme over a Noetherian ring . Then each -graded -module determines a quasi-coherent sheaf on such that is the sheaf associated to the -module , where is a homogeneous element of of positive degree and is the locus where does not vanish.
- For example, for each integer , let denote the graded -module given by . Then each determines the quasi-coherent sheaf on . If is generated as -algebra by , then is a line bundle (invertible sheaf) on and is the -th tensor power of . In particular, is called the tautological line bundle on the projective -space.
- A simple example of a coherent sheaf on that is not a vector bundle is given by the cokernel in the following sequence

- this is because restricted to the vanishing locus of the two polynomials has two-dimensional fibers, and has one-dimensional fibers elsewhere.

- Ideal sheaves: If is a closed subscheme of a locally Noetherian scheme , the sheaf of all regular functions vanishing on is coherent. Likewise, if is a closed analytic subspace of a complex analytic space , the ideal sheaf is coherent.
- The structure sheaf of a closed subscheme of a locally Noetherian scheme can be viewed as a coherent sheaf on . To be precise, this is the direct image sheaf , where is the inclusion. Likewise for a closed analytic subspace of a complex analytic space. The sheaf has fiber (defined below) of dimension zero at points in the open set , and fiber of dimension 1 at points in . There is a short exact sequence of coherent sheaves on :

- Most operations of linear algebra preserve coherent sheaves. In particular, for coherent sheaves and on a ringed space , the tensor product sheaf and the sheaf of homomorphisms are coherent.
^{[8]} - A simple
**non-example of a quasi-coherent sheaf**is given by the extension by zero functor. For example, consider for

^{[9]}

- Since this sheaf has non-trivial stalks, but zero global sections, this cannot be a quasi-coherent sheaf. This is because quasi-coherent sheaves on an affine scheme are equivalent to the category of modules over the underlying ring, and the adjunction comes from taking global sections.

Let be a morphism of ringed spaces (for example, a morphism of schemes). If is a quasi-coherent sheaf on , then the inverse image -module (or **pullback**) is quasi-coherent on .^{[10]} For a morphism of schemes and a coherent sheaf on , the pullback is not coherent in full generality (for example, , which might not be coherent), but pullbacks of coherent sheaves are coherent if is locally Noetherian. An important special case is the pullback of a vector bundle, which is a vector bundle.

If is a quasi-compact quasi-separated morphism of schemes and is a quasi-coherent sheaf on , then the direct image sheaf (or **pushforward**) is quasi-coherent on .^{[2]}

The direct image of a coherent sheaf is often not coherent. For example, for a field , let be the affine line over , and consider the morphism ; then the direct image