In mathematics, a sheaf of ''O''-modules or simply an ''O''-module over 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 ...
(''X'', ''O'') is a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper se ...
''F'' such that, for any open subset ''U'' of ''X'', ''F''(''U'') is an ''O''(''U'')-module and the restriction maps ''F''(''U'') → ''F''(''V'') are compatible with the restriction maps ''O''(''U'') → ''O''(''V''): the restriction of ''fs'' is the restriction of ''f'' times that of ''s'' for any ''f'' in ''O''(''U'') and ''s'' in ''F''(''U'').
The standard case is when ''X'' is a
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 ...
and ''O'' its structure sheaf. If ''O'' is the
constant sheaf
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific const ...
, then a sheaf of ''O''-modules is the same as a sheaf of abelian groups (i.e., an abelian sheaf).
If ''X'' is the
prime spectrum
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
of a ring ''R'', then any ''R''-module defines an ''O''
''X''-module (called an associated sheaf) in a natural way. Similarly, if ''R'' is a
graded ring and ''X'' is the
Proj
PROJ (formerly PROJ.4) is a library for performing conversions between cartographic projections. The library is based on the work of Gerald Evenden at the United States Geological Survey (USGS), but since 2019-11-26 is an Open Source Geospatial Fo ...
of ''R'', then any graded module defines an ''O''
''X''-module in a natural way. ''O''-modules arising in such a fashion are examples of
quasi-coherent 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 refe ...
, and in fact, on affine or projective schemes, all quasi-coherent sheaves are obtained this way.
Sheaves of modules over a ringed space form an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
. Moreover, this category has
enough injectives
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories ...
, and consequently one can and does define the
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally whe ...
as the ''i''-th
right derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in va ...
of the
global section functor .
Definition
Examples
*Given a ringed space (''X'', ''O''), if ''F'' is an ''O''-submodule of ''O'', then it is called the sheaf of ideals or
ideal sheaf In algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal in a ring. The ideal sheaves on a geometric object are closely connected to its subspaces.
Definition
Let ''X'' be a to ...
of ''O'', since for each open subset ''U'' of ''X'', ''F''(''U'') is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
of the ring ''O''(''U'').
*Let ''X'' be a
smooth variety of dimension ''n''. Then the
tangent sheaf of ''X'' is the dual of the
cotangent sheaf In algebraic geometry, given a morphism ''f'': ''X'' → ''S'' of schemes, the cotangent sheaf on ''X'' is the sheaf of \mathcal_X-modules \Omega_ that represents (or classifies) ''S''- derivations in the sense: for any \mathcal_X-modules ''F'', t ...
and the
canonical sheaf In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, i ...
is the ''n''-th exterior power (
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
) of
.
*A
sheaf of algebras In algebraic geometry, a sheaf of algebras on a ringed space ''X'' is a sheaf of commutative rings on ''X'' that is also a sheaf of \mathcal_X-modules. It is quasi-coherent if it is so as a module.
When ''X'' is a scheme, just like a ring, one ...
is a sheaf of module that is also a sheaf of rings.
Operations
Let (''X'', ''O'') be a ringed space. If ''F'' and ''G'' are ''O''-modules, then their tensor product, denoted by
:
or
,
is the ''O''-module that is the sheaf associated to the presheaf
(To see that sheafification cannot be avoided, compute the global sections of
where ''O''(1) is
Serre's twisting sheaf on a projective space.)
Similarly, if ''F'' and ''G'' are ''O''-modules, then
:
denotes the ''O''-module that is the sheaf
. In particular, the ''O''-module
:
is called the dual module of ''F'' and is denoted by
. Note: for any ''O''-modules ''E'', ''F'', there is a canonical homomorphism
:
,
which is an isomorphism if ''E'' is a
locally free sheaf
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 refer ...
of finite rank. In particular, if ''L'' is locally free of rank one (such ''L'' is called an
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 ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
or a
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
), then this reads:
:
implying the isomorphism classes of invertible sheaves form a group. This group is called the
Picard group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
of ''X'' and is canonically identified with the first cohomology group
(by the standard argument with
Čech cohomology).
If ''E'' is a locally free sheaf of finite rank, then there is an ''O''-linear map
given by the pairing; it is called the
trace map In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor pro ...
of ''E''.
For any ''O''-module ''F'', the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
,
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
and
symmetric algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
of ''F'' are defined in the same way. For example, the ''k''-th exterior power
:
is the sheaf associated to the presheaf
. If ''F'' is locally free of rank ''n'', then
is called the
determinant line bundle (though technically
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 ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
) of ''F'', denoted by det(''F''). There is a natural perfect pairing:
:
Let ''f'': (''X'', ''O'') →(''X'', ''O'') be a morphism of ringed spaces. If ''F'' is an ''O''-module, then the
direct image sheaf is an ''O''-module through the natural map ''O'' →''f''
*''O'' (such a natural map is part of the data of a morphism of ringed spaces.)
If ''G'' is an ''O''-module, then the module inverse image
of ''G'' is the ''O''-module given as the tensor product of modules:
:
where
is the
inverse image sheaf of ''G'' and
is obtained from
by
adjuction.
There is an adjoint relation between
and
: for any ''O''-module ''F'' and ''O
'''-module ''G'',
:
as abelian group. There is also the
projection formula: for an ''O''-module ''F'' and a locally free ''O
'''-module ''E'' of finite rank,
:
Properties
Let (''X'', ''O'') be a ringed space. An ''O''-module ''F'' is said to be generated by global sections if there is a surjection of ''O''-modules:
:
Explicitly, this means that there are global sections ''s''
''i'' of ''F'' such that the images of ''s''
''i'' in each stalk ''F''
''x'' generates ''F''
''x'' as ''O''
''x''-module.
An example of such a sheaf is that associated in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
to an ''R''-module ''M'', ''R'' being any
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, on the
spectrum of a ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the ...
''Spec''(''R'').
Another example: according to
Cartan's theorem A
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf on a Stein manifold . They are significant both as applied to several complex variables, and in the general development of s ...
, any
coherent sheaf
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 ref ...
on a
Stein manifold In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a Ste ...
is spanned by global sections. (cf. Serre's theorem A below.) In the theory of
schemes, a related notion is
ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...
. (For example, if ''L'' is an ample line bundle, some power of it is generated by global sections.)
An injective ''O''-module is
flasque (i.e., all restrictions maps ''F''(''U'') → ''F''(''V'') are surjective.) Since a flasque sheaf is acyclic in the category of abelian sheaves, this implies that the ''i''-th right derived functor of the global section functor
in the category of ''O''-modules coincides with the usual ''i''-th sheaf cohomology in the category of abelian sheaves.
Sheaf associated to a module
Let
be a module over a ring
. Put
and write
. For each pair
, by the universal property of localization, there is a natural map
: