Sheaf of modules

In mathematics, a sheaf of "O"-modules or simply an "O"-module over a ringed space ("X", "O") is a sheaf "F" 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 formula_1, 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""X"-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""X"-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 formula_2 as the "i"-th right derived functor of the global section functor formula_3.
Operations.
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 formula_9 (To see that sheafification cannot be avoided, compute the global sections of formula_10 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 formula_12. In particular, the "O"-module
is called the dual module of "F" and is denoted by formula_14. 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 formula_17 (by the standard argument with Čech cohomology).
If "E" is a locally free sheaf of finite rank, then there is an "O"-linear map formula_18 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 formula_20. If "F" is locally free of rank "n", then formula_21 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 formula_23 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 formula_24 of "G" is the "O"-module given as the tensor product of modules:
where formula_26 is the inverse image sheaf of "G" and formula_27 is obtained from formula_28 by adjuction.
There is an adjoint relation between formula_29 and formula_30: 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 ("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""i" of "F" such that the images of "s""i" in each stalk "F""x" generates "F""x" as "O""x"-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") → "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 formula_3 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 formula_35. For each pair formula_36, by the universal property of localization, there is a natural map
having the property that formula_38. 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 formula_40 on "X" called the sheaf associated to "M".
The most basic example is the structure sheaf on "X"; i.e., formula_41. Moreover, formula_40 has the structure of formula_41-module and thus one gets the exact functor formula_44 from Mod"A", the category of modules over "A" to the category of modules over formula_45. It defines an equivalence from Mod"A" to the category of quasi-coherent sheaves on "X", with the inverse formula_3, 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",
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 "R"0-algebra ("R"0 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 formula_40 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 formula_55; in fact, this defines formula_40 by gluing.
Example: Let "R"(1) be the graded "R"-module given by "R"(1)"n" = "R""n"+1. Then formula_57 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 formula_58, 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"("n") is generated by finitely many global sections. Moreover,
Sheaf extension.
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 formula_62, where the identity element in formula_62 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 formula_65
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"0 such that
Locally Free Resolutions.
formula_67 can be readily computed for any coherent sheaf formula_68 using a locally free resolution: given a complex
then
hence
Examples.
Hypersurface.
Consider a smooth hypersurface formula_72 of degree formula_73. Then, we can compute a resolution
and find that
Union of Smooth Complete Intersections.
Consider the scheme
where formula_77 is a smooth complete intersection and formula_78, formula_79. We have a complex
resolving formula_81 which we can use to compute formula_82.