Coherent sheaf

  (Redirected from Locally free sheaves)

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.

DefinitionsEdit

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 quasi-coherent sheaf   satisfying the following two properties:

  1.   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  ;
  2. 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 schemesEdit

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.

PropertiesEdit

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]

Basic constructions of coherent sheavesEdit

  • 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   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  :
 
  • 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.

FunctorialityEdit

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   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 sheavesEdit

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 stalk   is a free module over the local ring   for every point   in  .[12]

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.[13]

Examples of vector bundlesEdit

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  .[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 Serre subcategory of modules that are nonzero in only finitely many degrees.[15]

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 hypersurfaceEdit

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

 

Chern classes and algebraic K-theoryEdit

A vector bundle   on a smooth variety   over a field has Chern classes in the Chow ring of  ,   in   for  .[16] 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.[17] 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 propertyEdit

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 homomorphismEdit

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 sheavesEdit

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.[18] 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.[19]

Coherent cohomologyEdit

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 alsoEdit

NotesEdit

  1. ^ Mumford, Ch. III, § 1, Theorem-Definition 3 .
  2. ^ a b Stacks Project, Tag 01LA.
  3. ^ Stacks Project, Tag 01BU.
  4. ^ Serre (1955), section 13.
  5. ^ Grothendieck, EGA I, Corollaire 1.5.2.
  6. ^ Hartshorne (1977), Exercise II.5.18.
  7. ^ Stacks Project, Tag 00NV.
  8. ^ Serre (1955), section 14.
  9. ^ Hartshorne, Robin. Algebraic Geometry.
  10. ^ Stacks Project, Tag 01BG.
  11. ^ Hartshorne (1977), Example III.12.7.2.
  12. ^ Grothendieck, EGA I, Ch. 0, 5.2.7.
  13. ^ Eisenbud (1995), Exercise 20.13.
  14. ^ Hartshorne (1977), Corollary II.5.16.
  15. ^ Stacks Project, Tag 01YR.
  16. ^ Fulton (1998), section 3.2 and Example 8.3.3.
  17. ^ Fulton (1998), B.8.3.
  18. ^ Stacks Project, Tag 077K.
  19. ^ Antieau (2016), Corollary 4.2.

ReferencesEdit

External linksEdit