In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheafF 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 the restriction of s for any f in O(U) and s in F(U).
The standard case is when X is a scheme and O its structure sheaf. If O is the constant sheaf, then a sheaf of O-modules is the same as a
sheaf of abelian groups
(i.e., an abelian sheaf).
If X is the
quasi-coherent sheaves
, 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
enough injectives,[2] and consequently one can and does define the sheaf cohomology
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 of O, since for each open subset U of X, F(U) is an ideal of the ring O(U).
A sheaf of algebras 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 .[4] 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),[5]
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 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
. (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.)[6]
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.[7]
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
having the property that . Then
is a contravariant functor from the category whose objects are the sets D(f) and morphisms the inclusions of sets to the
B-sheaf
(i.e., it satisfies the gluing axiom) and thus defines the sheaf on X called the sheaf associated to M.
The most basic example is the structure sheaf on X; i.e., . Moreover, has the structure of -module and thus one gets the exact functor from ModA, the category of modules over A to the category of modules over . It defines an equivalence from ModA to the category of
, since the equivalence between ModA and the category of quasi-coherent sheaves on X.
;[11] in particular, taking a direct sum and ~ commute.
A sequence of A-modules is exact if and only if the induced sequence by is exact. In particular, .
Sheaf associated to a graded module
There is a graded analog of the construction and equivalence in the preceding section. Let R be a graded ring generated by degree-one elements as R0-algebra (R0 means the degree-zero piece) and M a graded R-module. Let X be the
projective scheme
if R is Noetherian). Then there is an O-module such that for any homogeneous element f of positive degree of R, there is a natural isomorphism
as sheaves of modules on the affine scheme ;[12] in fact, this defines by gluing.
Example: Let R(1) be the graded R-module given by R(1)n = Rn+1. Then is called
tautological line bundle
if R is finitely generated in degree-one.
If F is an O-module on X, then, writing , there is a canonical homomorphism:
which is an isomorphism if and only if F is quasi-coherent.
Computing sheaf cohomology
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Sheaf cohomology has a reputation for being difficult to calculate. Because of this, the next general fact is fundamental for any practical computation:
Theorem — Let X be a topological space, F an abelian sheaf on it and an open cover of X such that for any i, p and 's in . Then for any i,
^This cohomology functor coincides with the right derived functor of the global section functor in the category of abelian sheaves; cf. Hartshorne, Ch. III, Proposition 2.6.
which is an isomorphism if F is of finite presentation (EGA, Ch. 0, 5.2.6.)
^For coherent sheaves, having a tensor inverse is the same as being locally free of rank one; in fact, there is the following fact: if and if F is coherent, then F, G are locally free of rank one. (cf. EGA, Ch 0, 5.4.3.)