Coherent sheaf
Encyclopedia
In mathematics
, especially in algebraic geometry
and the theory of complex manifold
s, coherent sheaves are a specific class of sheaves
having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings
that codifies this geometrical information. In addition, there is a related concept of quasi-coherent sheaves. Many results and properties in algebraic geometry and complex analytic geometry are formulated in terms of coherent sheaves and their cohomology
.
Coherent sheaves can be seen as a generalization of (sheaves of sections of) vector bundles. They form a category closed under usual operations such as taking kernels
, cokernel
s and finite direct sum
s. In addition, under suitable compactness
conditions they are preserved under maps of the underlying spaces and have finite dimensional cohomology spaces
.
is a sheaf
of 
-modules
with the following two properties:
The sheaf of rings
is coherent if it is coherent considered as a sheaf of modules over itself. Important examples of coherent sheaves of rings include the sheaf of germs of holomorphic functions on a complex manifold
and the structure sheaf of a Noetherian scheme from algebraic geometry.
A coherent sheaf is always a sheaf of finite presentation, or in other words each point
has an open neighbourhood 
such that the restriction 
of 
to 
is isomorphic to the cokernel of a morphism 
for some integers 
and 
. If 
is coherent, then the converse is true and each sheaf of finite presentation over 
is coherent.
For a sheaf of rings
, a sheaf 
of 
-modules is said to be quasi-coherent if it has a local presentation, i.e. if there exist an open cover by 
of the topological space and an exact sequence
where the first two terms of the sequence are direct sums (possibly infinite) of copies of the structure sheaf.
For an affine variety X with (affine) coordinate ring R, there exists a covariant equivalence of categories
between that of quasi-coherent sheaves and sheaf morphisms on the one hand, and R-modules and module homomorphisms on the other hand. In case the ring R is Noetherian
, coherent sheaves correspond exactly to finitely generated modules.
Coherence of sheaves is working in the background of some results in commutative algebra
, e.g. Nakayama's lemma, which in terms of sheaves says that if
is a coherent sheaf, then the fiber 
if and only if there is a neighborhood 
of 
so that 
.
The role played by coherent sheaves is as a class of sheaves, say on an algebraic variety
or complex manifold, that is more general than the locally free sheaf — such as invertible sheaf
, or sheaf of sections of a (holomorphic) vector bundle
— but still with manageable properties. The generality is desirable, to be able to take kernel
s and cokernels of morphisms, for example, without moving outside the given class of sheaves.
theory of coherent sheaves is called coherent cohomology. It is one of the major and most fruitful applications of sheaves, and its results connect quickly with classical theories.
Using a theorem of Schwartz on compact operators in Frechet spaces, Cartan and Serre proved that compact complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension.
This result had been proved previously by Kodaira for the particular case of locally free sheaves on Kähler manifolds. It plays a major role in the proof of the "GAGA" equivalence analytic <-> algebraic. An algebraic (and much easier) version of this theorem was proved by Serre
. Relative versions of this result for a proper morphism were proved by Grothendieck in the algebraic case and by Grauert
and Remmert
in the analytic case. For example Grothendieck's result concerns the functor
Rf* or push-forward, in sheaf cohomology. (It is the right derived functor of the direct image of a sheaf.) For a proper morphism in the sense of scheme theory, it was shown that this functor sends coherent sheaves to coherent sheaves. The result of Serre
is the case of a morphism to a point.
The duality theory in scheme theory that extends Serre duality
is called coherent duality
(or Grothendieck duality). Under some mild conditions of finiteness, the sheaf of Kähler differential
s on an algebraic variety is a coherent sheaf Ω1. When the variety is non-singular its 'top' exterior power acts as the dualising object; and it is locally free (effectively it is the sheaf of sections of the cotangent bundle
, when working over the complex numbers, but that is a statement that requires more precision since only holomorphic 1-forms count as sections). The successful extension of the theory beyond this case was a major step.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and the theory of complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s, coherent sheaves are a specific class of sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
that codifies this geometrical information. In addition, there is a related concept of quasi-coherent sheaves. Many results and properties in algebraic geometry and complex analytic geometry are formulated in terms of coherent sheaves and their cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
.
Coherent sheaves can be seen as a generalization of (sheaves of sections of) vector bundles. They form a category closed under usual operations such as taking kernels
Kernel (mathematics)
In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...
, cokernel
Cokernel
In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
s and finite direct sum
Direct sum
In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...
s. In addition, under suitable compactness
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
conditions they are preserved under maps of the underlying spaces and have finite dimensional cohomology spaces
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
.
Definition
A coherent sheaf on a ringed spaceRinged space
In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space...
is a sheafSheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of -modulesModule (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
with the following two properties:
-
is of finite type over 
, i.e., for any point 
there is an open neighbourhood 
such that the restriction 
of 
to 
is generated by a finite number of sections (in other words, there is a surjective morphism 
for some 
); and - for any open set
, any 
and any morphism 
of 
-modules, the kernel of 
is of finite type.
The sheaf of rings
is coherent if it is coherent considered as a sheaf of modules over itself. Important examples of coherent sheaves of rings include the sheaf of germs of holomorphic functions on a complex manifoldComplex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
and the structure sheaf of a Noetherian scheme from algebraic geometry.
A coherent sheaf is always a sheaf of finite presentation, or in other words each point
has an open neighbourhood such that the restriction of to is isomorphic to the cokernel of a morphism for some integers and . If is coherent, then the converse is true and each sheaf of finite presentation over is coherent.For a sheaf of rings
, a sheaf of -modules is said to be quasi-coherent if it has a local presentation, i.e. if there exist an open cover by of the topological space and an exact sequencewhere the first two terms of the sequence are direct sums (possibly infinite) of copies of the structure sheaf.
For an affine variety X with (affine) coordinate ring R, there exists a covariant equivalence of categories
Equivalence of categories
In category theory, an abstract branch of 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 from many areas of mathematics...
between that of quasi-coherent sheaves and sheaf morphisms on the one hand, and R-modules and module homomorphisms on the other hand. In case the ring R is Noetherian
Noetherian
In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...
, coherent sheaves correspond exactly to finitely generated modules.
Coherence of sheaves is working in the background of some results in commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, e.g. Nakayama's lemma, which in terms of sheaves says that if
is a coherent sheaf, then the fiber if and only if there is a neighborhood of so that .The role played by coherent sheaves is as a class of sheaves, say on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
or complex manifold, that is more general than the locally free sheaf — such as invertible sheaf
Invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
, or sheaf of sections of a (holomorphic) vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
— but still with manageable properties. The generality is desirable, to be able to take kernel
Kernel (mathematics)
In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...
s and cokernels of morphisms, for example, without moving outside the given class of sheaves.
Examples of coherent sheaves
- On noetherian schemes, the structure sheaf
itself. - Sheaves of sections in vector bundles.
- The Oka coherence theorem shows that the sheaf of holomorphic functions on a complex manifold is coherent.
- Ideal sheaves: If Z is a closed complex subspace of a complex analytic spaceAnalytic spaceAn analytic space in the theory of functions of several complex variables is a generalization of the concept of an analytic manifold: heuristically, an analytic space is a set which is locally isomorphic to an analytic variety, while an analytic manifold is locally isomorphic to a complex euclidean...
X, the sheaf IZ of all holomorphic functions vanishing on Z is coherent. - Structure sheaves of subspaces.
Coherent cohomology
The sheaf cohomologySheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
theory of coherent sheaves is called coherent cohomology. It is one of the major and most fruitful applications of sheaves, and its results connect quickly with classical theories.
Using a theorem of Schwartz on compact operators in Frechet spaces, Cartan and Serre proved that compact complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension.
This result had been proved previously by Kodaira for the particular case of locally free sheaves on Kähler manifolds. It plays a major role in the proof of the "GAGA" equivalence analytic <-> algebraic. An algebraic (and much easier) version of this theorem was proved by Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
. Relative versions of this result for a proper morphism were proved by Grothendieck in the algebraic case and by Grauert
Hans Grauert
Hans Grauert was a German mathematician. He is known for major works on several complex variables, complex manifolds and the application of sheaf theory in this area, which influenced later work in algebraic geometry. Together with Reinhold Remmert he established and developed the theory of...
and Remmert
Reinhold Remmert
Reinhold Remmert is a German mathematician born in Osnabrück, Germany. He studied mathematics, mathematical logic and physics in Münster. He established and developed the theory of complex spaces in joint work with Hans Grauert...
in the analytic case. For example Grothendieck's result concerns the functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
Rf* or push-forward, in sheaf cohomology. (It is the right derived functor of the direct image of a sheaf.) For a proper morphism in the sense of scheme theory, it was shown that this functor sends coherent sheaves to coherent sheaves. The result of Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
is the case of a morphism to a point.
The duality theory in scheme theory that extends Serre duality
Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...
is called coherent duality
Coherent duality
In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory....
(or Grothendieck duality). Under some mild conditions of finiteness, the sheaf of Kähler differential
Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.-Presentation:The idea was introduced by Erich Kähler in the 1930s...
s on an algebraic variety is a coherent sheaf Ω1. When the variety is non-singular its 'top' exterior power acts as the dualising object; and it is locally free (effectively it is the sheaf of sections of the cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
, when working over the complex numbers, but that is a statement that requires more precision since only holomorphic 1-forms count as sections). The successful extension of the theory beyond this case was a major step.