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 of ...
(''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 ser ...
''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 cons ...
, 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 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 refer ...
, 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. 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 categori ...
, 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 when i ...
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 var ...
of the
global section functor
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
.
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 t ...
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 considere ...
of the ring ''O''(''U'').
*Let ''X'' be a
smooth variety of dimension ''n''. Then the
tangent 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'', th ...
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'', th ...
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, ...
is the ''n''-th exterior power (
determinant) 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 ca ...
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
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not fun ...
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 of finite rank. In particular, if ''L'' is locally free of rank one (such ''L'' is called an
invertible sheaf 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 organisin ...
), then this reads:
:
implying the isomorphism classes of invertible sheaves form a group. This group is called the
Picard group of ''X'' and is canonically identified with the first cohomology group
(by the standard argument with
ÄŒech 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 produ ...
of ''E''.
For any ''O''-module ''F'', the
tensor algebra,
exterior algebra and
symmetric algebra 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
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 ...
(though technically
invertible sheaf) 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 In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topolo ...
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 In algebraic geometry, the projection formula states the following:http://math.stanford.edu/~vakil/0708-216/216class38.pdf
For a morphism f:X\to Y of ringed spaces, an \mathcal_X-module \mathcal and a locally free sheaf, locally free \mathcal_Y-mo ...
: 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 sp ...
, 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, 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 refe ...
on a
Stein manifold is spanned by global sections. (cf. Serre's theorem A below.) In the theory of
schemes, a related notion is
ample line bundle. (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
: