In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf of abelian groups F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) â F(V) are compatible with the restriction maps O(U) â 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 prime spectrum of a ring R, then any R-module defines an O<sub>X</sub>-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<sub>X</sub>-module in a natural way. O-modules arising in such a fashion are examples of 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 abelian category. Moreover, this category has enough injectives, and consequently one can and does define the sheaf cohomology as the i-th right derived functor of the global section functor .
Let (X, O) be a ringed space. If F and G are O-modules, then their tensor product, denoted by
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 of finite rank. In particular, if L is locally free of rank one (such L is called an invertible sheaf or a line bundle), 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 ÃÂech 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 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 (though technically invertible sheaf) of F, denoted by det(F). There is a natural perfect pairing:
Let f: (X, O) âÂÂ(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 âÂÂf<sub>*</sub>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<nowiki>'</nowiki>-module G,
as abelian group. There is also the projection formula: for an O-module F and a locally free O<nowiki>'</nowiki>-module E of finite rank,
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<sub>i</sub> of F such that the images of s<sub>i</sub> in each stalk F<sub>x</sub> generates F<sub>x</sub> as O<sub>x</sub>-module.
An example of such a sheaf is that associated in algebraic geometry to an R-module M, R being any commutative ring, on the spectrum of a ring Spec(R). Another example: according to Cartan's theorem A, any coherent sheaf 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) â 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.
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 category of abelian groups. One can show it is in fact a 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 Mod<sub>A</sub>, the category of modules over A to the category of modules over . It defines an equivalence from Mod<sub>A</sub> to the category of quasi-coherent sheaves on X, with the inverse , the global section functor. When X is Noetherian, the functor is an equivalence from the category of finitely generated A-modules to the category of coherent sheaves on X.
The construction has the following properties: for any A-modules M, N, and any morphism ,
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 R<sub>0</sub>-algebra (R<sub>0</sub> means the degree-zero piece) and M a graded R-module. Let X be the Proj of R (so X is a 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 ; in fact, this defines by gluing.
Example: Let R(1) be the graded R-module given by R(1)<sub>n</sub> = R<sub>n+1</sub>. Then is called Serre's twisting sheaf, which is the dual of the 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.
Sheaf cohomology has a reputation for being difficult to calculate. Because of this, the next general fact is fundamental for any practical computation:
Serre's vanishing theorem states that if X is a projective variety and F a coherent sheaf on it, then, for sufficiently large n, the Serre twist F(n) is generated by finitely many global sections. Moreover, <ol type="a"> <li> For each i, H<sup>i</sup>(X, F) is finitely generated over R<sub>0</sub>, and</li> <li> There is an integer n<sub>0</sub>, depending on F, such that </li> </ol>
Let (X, O) be a ringed space, and let F, H be sheaves of O-modules on X. An extension of H by F is a short exact sequence of O-modules
As with group extensions, if we fix F and H, then all equivalence classes of extensions of H by F form an abelian group (cf. Baer sum), which is isomorphic to the Ext group , where the identity element in corresponds to the trivial extension.
In the case where H is O, we have: for any i âÂÂ¥ 0,
since both the sides are the right derived functors of the same functor
Note: Some authors, notably Hartshorne, drop the subscript O.
Assume X is a projective scheme over a Noetherian ring. Let F, G be coherent sheaves on X and i an integer. Then there exists n<sub>0</sub> such that
where denotes the derived functors of .
can be readily computed for any coherent sheaf using a locally free resolution: given a complex
then
hence
Consider a smooth hypersurface of degree . Then, we can compute a resolution
and find that
Consider the scheme
where is a smooth complete intersection and , . We have a complex
resolving which we can use to compute .