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.
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.
(To see that sheafification cannot be avoided, compute the global sections of
[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.
If E is a locally free sheaf of finite rank, then there is an O-linear map
For example, the k-th exterior power is the sheaf associated to the presheaf
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.
There is also the projection formula: for an O-module F and a locally free O'-module E of finite rank,
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 si of F such that the images of si in each stalk Fx generates Fx as Ox-module.
Another example: according to Cartan's theorem A, any coherent sheaf on a Stein manifold is spanned by global sections.
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.
)[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
One can show[8] it is in fact a B-sheaf (i.e., it satisfies the gluing axiom) and thus defines the sheaf
It defines an equivalence from ModA to the category of quasi-coherent sheaves on X, with the inverse
, There is a graded analog of the construction and equivalence in the preceding section.
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
is called Serre's twisting sheaf, which is the dual of the tautological line bundle if R is finitely generated in degree-one.
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
Then for any i, where the right-hand side is the i-th Čech cohomology.
Serre's vanishing theorem[13] 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.
Baer sum), which is isomorphic to the Ext group
Note: Some authors, notably Hartshorne, drop the subscript O.
Assume X is a projective scheme over a Noetherian ring.
using a locally free resolution:[18] given a complex then hence Consider a smooth hypersurface