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).
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.
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 .
- Given a ringed space (X, O), if F is an O-submodule of O, then it is called the sheaf of ideals or ideal sheaf of O, since for each open subset U of X, F(U) is an ideal of the ring O(U).
- Let X be a smooth variety of dimension n. Then the tangent sheaf of X is the dual of the cotangent sheaf and the canonical sheaf is the n-th exterior power (determinant) of .
- A sheaf of algebras is a sheaf of module that is also a sheaf of rings.
Let (X, O) be a ringed space. If F and G are O-modules, then their tensor product, denoted by
- or ,
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
is called the dual module of F and is denoted by . Note: for any O-modules E, F, there is a canonical homomorphism
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:
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,
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 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(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 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(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,
- For any prime ideal p of A, as Op = Ap-module.
- If M is finitely presented, .
- , 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 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 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
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(n) is generated by finitely many global sections. Moreover,
- (a) For each i, Hi(X, F) is finitely generated over R0, and
- (b) (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
Locally Free Resolutions
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 .
- Vakil, Math 216: Foundations of algebraic geometry, 2.5.
- Hartshorne, Ch. III, Proposition 2.2.
- This cohomology functor coincides with the right derived functor of the global section functor in the category of abelian sheaves; cf. Hartshorne, Ch. III, Proposition 2.6.
- There is a canonical homomorphism:
- 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.)
- Hartshorne, Ch III, Lemma 2.4.
- see also: https://math.stackexchange.com/q/447234
- Hartshorne, Ch. II, Proposition 5.1.
- EGA I, Ch. I, Proposition 1.3.6.
- EGA I, Ch. I, Corollaire 1.3.12.
- EGA I, Ch. I, Corollaire 1.3.9.
- Hartshorne, Ch. II, Proposition 5.11.
- Hartshorne, Ch. III, Proposition 6.9.
- Hartshorne, Robin. Algebraic Geometry. pp. 233–235.
- Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
- Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157