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
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
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 In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the exis ...
is a powerful technique, in particular for studying the sections of a given coherent sheaf.
Definitions
A quasi-coherent sheaf on a
ringed space
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
is a sheaf
of
-
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
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
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
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
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fr ...
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 A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
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
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
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
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ext ...
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
In mathematics, the Oka coherence theorem, proved by , states that the sheaf \mathcal := \mathcal_ of germs of holomorphic functions on \mathbb^n over a complex manifold is coherent.In paper it was called the idéal de domaines indéterminé ...
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
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 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 ]