Sheaf of modules

Examples

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 Ω X and the canonical sheaf ω X is the n-th exterior power (determinant) of Ω X .

Operations

Let (X, O) be a ringed space. If F and G are O-modules, then their tensor product, denoted by

F ⊗ O G or F ⊗ G ,

is the O-module that is the sheaf associated to the presheaf U ↦ F ( U ) ⊗ O ( U ) G ( U ) . (To see that sheafification cannot be avoided, compute the global sections of O ( 1 ) ⊗ O ( − 1 ) = O where O(1) is Serre's twisting sheaf on a projective space.)

Similarly, if F and G are O-modules, then

H o m O ( F , G )

denotes the O-module that is the sheaf U ↦ Hom O | U ⁡ ( F | U , G | U ) . In particular, the O-module

H o m O ( F , O )

is called the dual module of F and is denoted by F ˇ . Note: for any O-modules E, F, there is a canonical homomorphism

E ˇ ⊗ F → H o m O ( E , F ) ,

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:

L ˇ ⊗ L ≃ O ,

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 H 1 ⁡ ( X , O ∗ ) (by the standard argument with Čech cohomology).

If E is a locally free sheaf of finite rank, then there is an O-linear map E ˇ ⊗ E ≃ End O ⁡ ( E ) → O 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

∧ k F

is the sheaf associated to the presheaf U ↦ ∧ O ( U ) k F ( U ) . If F is locally free of rank n, then ∧ n F is called the determinant line bundle (though technically invertible sheaf) of F, denoted by det(F). There is a natural perfect paring:

∧ r F ⊗ ∧ n − r F → det ⁡ ( F ) .

Let f: (X, O) →(X', O') be a morphism of ringed spaces. If F is an O-module, then the direct image sheaf f ∗ F 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 f ∗ G of G is the O-module given as the tensor product of modules:

f − 1 G ⊗ f − 1 O ′ O

where f − 1 G is the inverse image sheaf of G and f − 1 O ′ → O is obtained from O ′ → f ∗ O by adjuction.

There is an adjoint relation between f ∗ and f ∗ : for any O-module F and O'-module G,

Hom O ⁡ ( f ∗ G , F ) ≃ Hom O ′ ⁡ ( G , f ∗ F )

as abelian group. There is also the projection formula: for an O-module F and a locally free O'-module E of finite rank,

f ∗ ( F ⊗ f ∗ E ) ≃ f ∗ F ⊗ E .

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:

⊕ i ∈ I O → F → 0 .

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 Γ ( X , − ) 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. For each pair D ( f ) ⊂ D ( g ) , by the universal property of localization, there is a natural map

ρ g , f : M [ g − 1 ] → M [ f − 1 ]

which has the property that ρ g , f = ρ g , h ∘ ρ h , f . Then

D ( f ) ↦ M [ f − 1 ]

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 M ~ on X called the sheaf associated to M.

The most basic example is the structure sheaf on X; i.e., O X = A ~ . Moreover, M ~ has the structure of O X = A ~ and thus one gets the exact functor M ↦ M ~ from Mod_A, the category of modules over A to the category of modules over O X . It defines an equivalence from Mod_A to the category of quasi-coherent sheaves on X, with the inverse Γ ( X , − ) , 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,

M [ f − 1 ] ∼ = M ~ | D ( f ) .

For any prime ideal p of A, M ~ p ≃ M p as O_p = A_p-module.

( M ⊗ A N ) ∼ ≃ M ~ ⊗ A ~ N ~ .

If M is finitely presented, Hom A ⁡ ( M , N ) ∼ ≃ H o m A ~ ( M ~ , N ~ ) .

Hom A ⁡ ( M , N ) ≃ Γ ( X , H o m A ~ ( M ~ , N ~ ) ) , since the equivalence between Mod_A and the category of quasi-coherent sheaves on X.

( lim → ⁡ M i ) ∼ ≃ lim → ⁡ M i ~ ; 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). Then there is an O-module M ~ such that for any homogeneous element f of positive degree of R, there is a natural isomorphism

M ~ | { f ≠ 0 } ≃ ( M [ f − 1 ] 0 ) ∼

as sheaves of modules on the affine scheme { f ≠ 0 } = Spec ⁡ ( R [ f − 1 ] 0 ) ; in fact, this defines M ~ by gluing.

Example: Let R(1) be the graded R-module given by R(1)_n = R_n+1. Then O ( 1 ) = R ( 1 ) ~ is called Serre's twisting sheaf (the dual of the tautological line bundle.)

If F is an O-module on X, then, writing F ( n ) = F ⊗ O ( n ) , there is a canonical homomorphism:

( ⊕ n ≥ 0 Γ ( X , F ( n ) ) ) ∼ → F ,

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 H i ⁡ ( X , F ( n ) ) = 0 , i ≥ 1 , n ≥ n 0 .

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

0 → F → G → H → 0.

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 E x t O 1 ( H , F ) , where the identity element in E x t O 1 ( H , F ) corresponds to the trivial extension.

In the case where H is O, we have: for any i ≥ 0,

H i ⁡ ( X , F ) = E x t O i ( O , F ) ,

since both the sides are the right derived functors of the same functor Γ ( X , − ) = Hom O ⁡ ( O , − ) .

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

Ext O i ⁡ ( F , G ( n ) ) = Γ ( X , E x t O i ( F , G ( n ) ) ) , n ≥ n 0 .