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.[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 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 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 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
- 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
.[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 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 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 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
.[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 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 moduli of stable vector bundles in many specific cases, such as on principally polarized abelian varieties[17] and K3 surfaces.[19]
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
.[20] 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.[21] 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
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 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
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
Notes
References
- Antieau, Benjamin (2016), "A reconstruction theorem for abelian categories of twisted sheaves", Journal für die reine und angewandte Mathematik, 712: 175–188, arXiv:1305.2541, doi:10.1515/crelle-2013-0119, MR 3466552
- Danilov, V. I. (2001) [1994], "Coherent algebraic sheaf", Encyclopedia of Mathematics, EMS Press
- Grauert, Hans; Remmert, Reinhold (1984), Coherent Analytic Sheaves, Springer-Verlag, doi:10.1007/978-3-642-69582-7, ISBN 3-540-13178-7, MR 0755331
- Eisenbud, David (1995), Commutative Algebra with a View toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 978-0-387-94268-1, MR 1322960
- Fulton, William (1998), Intersection Theory, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1700-8, ISBN 978-0-387-98549-7, MR 1644323
- Sections 0.5.3 and 0.5.4 of Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
- Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
- Mumford, David (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians (2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 354063293X. 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
- Serre, Jean-Pierre (1955), "Faisceaux algébriques cohérents", Annals of Mathematics, 61: 197–278, doi:10.2307/1969915, MR 0068874
External links
- The Stacks Project Authors, The Stacks Project
- Part V of Vakil, Ravi, The Rising Sea