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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.
A quasi-coherent sheaf on a ringed space is a sheaf of -modules which 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 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.
Theorem — Let be a scheme and an -module on it. Then the following are equivalent.
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.
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.[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 .
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 which 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 is the zero object.
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 :
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 . 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-compactquasi-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 :
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 .
Serre gave an algebraic description of all coherent sheaves on projective space, more subtle than what happens for affine space. Namely, let be a Noetherian ring (for example, a field), and consider the polynomial ring as a graded ring with each having degree 1. Then every finitely generated graded -module has an associated coherent sheaf on over . Every coherent sheaf on arises in this way from a finitely generated graded -module . (For example, the line bundle is the sheaf associated to the -module with its grading lowered by .) But the -module that yields a given coherent sheaf on is not unique; it is only unique up to changing by graded modules that are nonzero in only finitely many degrees. More precisely, the abelian category of coherent sheaves on is the quotient of the category of finitely generated graded -modules by the Serre subcategory of modules that are nonzero in only finitely many degrees.
The tangent bundle of projective space over a field can be described in terms of the line bundle . Namely, there is a short exact sequence, the Euler sequence:
It follows that the canonical bundle (the dual of the determinant line bundle of the tangent bundle) is isomorphic to . This is a fundamental calculation for algebraic geometry. For example, the fact that the canonical bundle is a negative multiple of the ample line bundle means that projective space is a Fano variety. Over the complex numbers, this means that projective space has a Kähler metric with positive Ricci curvature.
Vector bundles on a hypersurface
Consider a smooth degree- hypersurface defined by the homogeneous polynomial of degree . Then, there is an exact sequence
where the second map is the pullback of differential forms, and the first map sends
Note that this sequence tells us that is the conormal sheaf of in . Dualizing this yields the exact sequence
hence is the normal bundle of in . If we use the fact that given an exact sequence
of vector bundles with ranks ,,, there is an isomorphism
of line bundles, then we see that there is the isomorphism
Chern classes and algebraic K-theory
A vector bundle on a smooth variety over a field has Chern classes in the Chow ring of , in for . These satisfy the same formal properties as Chern classes in topology. For example, for any short exact sequence
of vector bundles on , the Chern classes of are given by
It follows that the Chern classes of a vector bundle depend only on the class of in the Grothendieck group. By definition, for a scheme , is the quotient of the free abelian group on the set of isomorphism classes of vector bundles on by the relation that for any short exact sequence as above. Although is hard to compute in general, algebraic K-theory provides many tools for studying it, including a sequence of related groups for integers .
A variant is the group (or ), the Grothendieck group of coherent sheaves on . (In topological terms, G-theory has the formal properties of a Borel-Moore homology theory for schemes, while K-theory is the corresponding cohomology theory.) The natural homomorphism is an isomorphism if is a regular separated Noetherian scheme, using that every coherent sheaf has a finite resolution by vector bundles in that case. For example, that gives a definition of the Chern classes of a coherent sheaf on a smooth variety over a field.
More generally, a Noetherian scheme is said to have the resolution property if every coherent sheaf on has a surjection from some vector bundle on . For example, every quasi-projective scheme over a Noetherian ring has the resolution property.
Applications of resolution property
Since the resolution property states that a coherent sheaf on a Noetherian scheme is quasi-isomorphic in the derived category to the complex of vector bundles :
we can compute the total Chern class of with
For example, this formula is useful for finding the Chern classes of the sheaf representing a subscheme of . If we take the projective scheme associated to the ideal , then
since there is the resolution
Bundle homomorphism vs. sheaf homomorphism
When vector bundles and locally free sheaves of finite constant rank are used interchangeably,
care must be given to distinguish between bundle homomorphisms and sheaf homomorphisms. Specifically, given vector bundles , by definition, a bundle homomorphism is a scheme morphism over (i.e., ) such that, for each geometric point in , is a linear map of rank independent of . Thus, it induces the sheaf homomorphism of constant rank between the corresponding locally free -modules (sheaves of dual sections). But there may be an -module homomorphism that does not arise this way; namely, those not having constant rank.
In particular, a subbundle is a subsheaf (i.e., is a subsheaf of ). But the converse can fail; for example, for an effective Cartier divisor on , is a subsheaf but typically not a subbundle (since any line bundle has only two subbundles).
The category of quasi-coherent sheaves
Quasi-coherent sheaves on any scheme form an abelian category. Gabber showed that, in fact, the quasi-coherent sheaves on any scheme form a particularly well-behaved abelian category, a Grothendieck category. A quasi-compact quasi-separated scheme (such as an algebraic variety over a field) is determined up to isomorphism by the abelian category of quasi-coherent sheaves on , by Rosenberg, generalizing a result of Gabriel.
The fundamental technical tool in algebraic geometry is the cohomology theory of coherent sheaves. Although it was introduced only in the 1950s, many earlier techniques of algebraic geometry are clarified by the language of sheaf cohomology applied to coherent sheaves. Broadly speaking, coherent sheaf cohomology can be viewed as a tool for producing functions with specified properties; sections of line bundles or of more general sheaves can be viewed as generalized functions. In complex analytic geometry, coherent sheaf cohomology also plays a foundational role.
Among the core results of coherent sheaf cohomology are results on finite-dimensionality of cohomology, results on the vanishing of cohomology in various cases, duality theorems such as Serre duality, relations between topology and algebraic geometry such as Hodge theory, and formulas for Euler characteristics of coherent sheaves such as the Riemann-Roch theorem.