Sheaf of modules


In mathematics, a sheaf of O-modules or simply an O-module over a ringed space is a sheaf F such that, for any open subset U of X, F is an O-module and the restriction maps F → F are compatible with the restriction maps O → O: the restriction of fs is the restriction of f times that of s for any f in O and s in F.
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.
If X is the prime spectrum of a ring R, then any R-module defines an OX-module 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.
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.

Examples

Let 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
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, 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 .
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 of F, denoted by det. There is a natural perfect pairing:
Let f: → 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
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: for an O-module F and a locally free O'-module E of finite rank,

Properties

Let 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 si of F such that the images of si in each stalk Fx generates Fx as Ox-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.
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.
An injective O-module is flasque → F 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 M be a module over a ring A. Put X = Spec A 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 and morphisms the inclusions of sets to the category of abelian groups. One can show it is in fact a B-sheaf 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,
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 and M a graded R-module. Let X be the Proj of R. 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 be the graded R-module given by Rn = Rn+1. 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.

Computing sheaf cohomology

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 theorem A states that if X is a projective variety and F a coherent sheaf on it, then, for sufficiently large n, F is generated by finitely many global sections. Moreover,

Sheaf extension

Let 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, 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

Locally Free Resolutions

can be readily computed for any coherent sheaf using a locally free resolution: given a complex
then
hence

Examples

Hypersurface

Consider a smooth hypersurface of degree. Then, we can compute a resolution
and find that

Union of Smooth Complete Intersections

Consider the scheme
where is a smooth complete intersection and,. We have a complex
resolving which we can use to compute.