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 that 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 OX-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 OX-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.
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 trace map of E.
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) →(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:
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 moduleEdit
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 ModA, the category of modules over A to the category of modules over . It defines an equivalence from ModA 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,
, since the equivalence between ModA and the category of quasi-coherent sheaves on X.
; in particular, taking a direct sum and ~ commute.
Sheaf associated to a graded moduleEdit
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 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.
Serre's theorem A states that if X is a projective variety and F a coherent sheaf on it, then, for sufficiently large n, F(n) is generated by finitely many global sections. Moreover,
For each i, Hi(X, F) is finitely generated over R0, and
(Serre's theorem B) There is an integer n0, depending on F, such that
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 ≥ 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 n0 such that
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.)