Coherent sheaf
In mathematics, especially in
Coherent sheaves can be seen as a generalization of
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
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
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.
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.[2]
On any ringed space , the coherent sheaves form an abelian category, a
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]
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.[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 schemeover 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 bundleon 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 ashort exact sequenceof 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 homomorphismsare coherent.[8]
- 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
If is a
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.[11] 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
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
Examples of vector bundles
For a morphism of schemes , let be the diagonal morphism, which is a closed immersion if is
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
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 .[14] 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
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
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
showing that
Serre construction and vector bundles
One useful technique for constructing rank 2 vector bundles is the Serre construction[16][17]pg 3 which establishes a correspondence between rank 2 vector bundles on a smooth projective variety and codimension 2 subvarieties using a certain -group calculated on . This is given by a cohomological condition on the line bundle (see below).
The correspondence in one direction is given as follows: for a section we can associated the vanishing locus . If is a codimension 2 subvariety, then
- It is a local complete intersection, meaning if we take an affine chart then can be represented as a function , where and
- The line bundle is isomorphic to the canonical bundle on
In the other direction,[18] for a codimension 2 subvariety and a line bundle such that
there is a canonical isomorphism
,
which is functorial with respect to inclusion of codimension subvarieties. Moreover, any isomorphism given on the left corresponds to a locally free sheaf in the middle of the extension on the right. That is, for that is an isomorphism there is a corresponding locally free sheaf of rank 2 that fits into a short exact sequence
This vector bundle can then be further studied using cohomological invariants to determine if it is stable or not. This forms the basis for studying
Chern classes and algebraic K-theory
A vector bundle on a smooth variety over a field has
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
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
over .
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
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
The quasi-coherent sheaves on any fixed 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.[22] 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.[23]
Coherent cohomology
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.
See also
- Picard group
- Divisor (algebraic geometry)
- Reflexive sheaf
- Quot scheme
- Twisted sheaf
- Essentially finite vector bundle
- Bundle of principal parts
- Gabriel–Rosenberg reconstruction theorem
- Pseudo-coherent sheaf
- Quasi-coherent sheaf on an algebraic stack
Notes
- ^ Mumford 1999, Ch. III, § 1, Theorem-Definition 3.
- ^ a b Stacks Project, Tag 01LA.
- ^ Stacks Project, Tag 01BU.
- ^ Serre 1955, §13
- ^ Grothendieck & Dieudonné 1960, Corollaire 1.5.2
- ^ Hartshorne 1977, Exercise II.5.18
- ^ Stacks Project, Tag 00NV.
- ^ Serre 1955, §14
- ^ Hartshorne 1977
- ^ Stacks Project, Tag 01BG.
- ^ Hartshorne 1977, Example III.12.7.2
- ^ Grothendieck & Dieudonné 1960, Ch. 0, 5.2.7
- ^ Eisenbud 1995, Exercise 20.13
- ^ Hartshorne 1977, Corollary II.5.16
- ^ Stacks Project, Tag 01YR.
- ^ Serre, Jean-Pierre (1960–1961). "Sur les modules projectifs". Séminaire Dubreil. Algèbre et théorie des nombres (in French). 14 (1): 1–16.
- ^ ISSN 0092-7872.
- ^ Hartshorne, Robin (1978). "Stable Vector Bundles of Rank 2 on P3". Mathematische Annalen. 238: 229–280.
- ISBN 978-0-521-13420-0.
- ^ Fulton 1998, §3.2 and Example 8.3.3
- ^ Fulton 1998, B.8.3
- ^ Stacks Project, Tag 077K.
- ^ Antieau 2016, Corollary 4.2
References
- Antieau, Benjamin (2016), "A reconstruction theorem for abelian categories of twisted sheaves", MR 3466552
- Danilov, V. I. (2001) [1994], "Coherent algebraic sheaf", Encyclopedia of Mathematics, EMS Press
- MR 0755331
- MR 1322960
- MR 1644323
- Sections 0.5.3 and 0.5.4 of MR 0217083.
- MR 0463157
- MR 1748380.
- Onishchik, A.L. (2001) [1994], "Coherent analytic sheaf", Encyclopedia of Mathematics, EMS Press
- Onishchik, A.L. (2001) [1994], "Coherent sheaf", Encyclopedia of Mathematics, EMS Press
- MR 0068874
External links
- The Stacks Project Authors, The Stacks Project
- Part V of Vakil, Ravi, The Rising Sea