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 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.
The case of schemes
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..
Properties 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..
On any ringed space , the coherent sheaves form an abelian category, a full subcategory of the category of -modules.. (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.
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.
Basic constructions of coherent sheaves 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. (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.
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 .. 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. A simple non-example of a quasi-coherent sheaf is given by the extension by zero functor. For example, consider for
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.
Functoriality 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 .. 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 .
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 is the sheaf on associated to the polynomial ring , which is not coherent because has infinite dimension as a -vector space. On the other hand, the direct image of a coherent sheaf under a proper morphism is coherent, by results of Grauert and Grothendieck.
Local behavior of coherent sheaves An important feature of coherent sheaves is that the properties of at a point control the behavior of in a neighborhood of , more than would be true for an arbitrary sheaf. For example, Nakayama's lemma says (in geometric language) that if is a coherent sheaf on a scheme , then the fiber of at a point (a vector space over the residue field ) is zero if and only if the sheaf is zero on some open neighborhood of . A related fact is that the dimension of the fibers of a coherent sheaf is upper-semicontinuous. Thus a coherent sheaf has constant rank on an open set, while the rank can jump up on a lower-dimensional closed subset.
In the same spirit: a coherent sheaf on a scheme is a vector bundle if and only if its stalk is a free module over the local ring for every point in .
On a general scheme, one cannot determine whether a coherent sheaf is a vector bundle just from its fibers (as opposed to its stalks). On a reduced locally Noetherian scheme, however, a coherent sheaf is a vector bundle if and only if its rank is locally constant.
Examples of vector bundles For a morphism of schemes , let be the diagonal morphism, which is a closed immersion if is separated over . Let be the ideal sheaf of in . Then the sheaf of differentials can be defined as the pullback of to . Sections of this sheaf are called 1-forms on over , and they can be written locally on as finite sums for regular functions and . If is locally of finite type over a field , then is a coherent sheaf on .
If is smooth over , then (meaning ) is a vector bundle over , called the cotangent bundle of . Then the tangent bundle is defined to be the dual bundle . For smooth over of dimension everywhere, the tangent bundle has rank .
If is a smooth closed subscheme of a smooth scheme over , then there is a short exact sequence of vector bundles on :
which can be used as a definition of the normal bundle to in .
For a smooth scheme over a field and a natural number , the vector bundle of i-forms on is defined as the -th exterior power of the cotangent bundle, . For a smooth variety of dimension over , the canonical bundle means the line bundle . Thus sections of the canonical bundle are algebro-geometric analogs of volume forms on . For example, a section of the canonical bundle of affine space over can be written as
where is a polynomial with coefficients in .
Let be a commutative ring and a natural number. For each integer , there is an important example of a line bundle on projective space over , called . To define this, consider the morphism of -schemes
given in coordinates by . (That is, thinking of projective space as the space of 1-dimensional linear subspaces of affine space, send a nonzero point in affine space to the line that it spans.) Then a section of over an open subset of is defined to be a regular function on that is homogeneous of degree , meaning that
as regular functions on (. For all integers and , there is an isomorphism of line bundles on .
In particular, every homogeneous polynomial in of degree over can be viewed as a global section of over . Note that every closed subscheme of projective space can be defined as the zero set of some collection of homogeneous polynomials, hence as the zero set of some sections of the line bundles . This contrasts with the simpler case of affine space, where a closed subscheme is simply the zero set of some collection of regular functions. The regular functions on projective space over are just the "constants" (the ring ), and so it is essential to work with the line bundles .