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 $\underline{\mathbf{Z}}$, then a sheaf of O-modules are the same as a sheaf of abelian groups (i.e., 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 $\operatorname{H}^i(X, -)$ as the i-th right derived functor of the global section functor $\Gamma(X, -)$.

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 $\Omega_X$ and the canonical sheaf $\omega_X$ is the n-th exterior power (determinant) of $\Omega_X$.

Operations

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

$F \otimes_O G$ or $F \otimes G$,

is the O-module that is the sheaf associated to the presheaf $U \mapsto F(U) \otimes_{O(U)} G(U).$ (To see that sheafification cannot be avoided, compute the global sections of $O(1) \otimes O(-1) = O$ where O(1) is Serre's twisting sheaf on a projective space.)

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

$\mathcal{H}om_O(F, G)$

denotes the O-module that is the sheaf $U \mapsto \operatorname{Hom}_{O|_U}(F|_U, G|_U)$. In particular, the O-module

$\mathcal{H}om_O(F, O)$

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

$\check{E} \otimes F \to \mathcal{H}om_O(E, F)$,

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:

$\check{L} \otimes L \simeq O,$

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 $\operatorname{H}^1(X, \mathcal{O}^*)$ (by the standard argument with Čech cohomology).

If E is a locally free sheaf of finite rank, then there is an O-linear map $\check{E} \otimes E \simeq \operatorname{End}_O(E) \to O$ 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

$\wedge^k F$

is the sheaf associated to the presheaf $U \mapsto \wedge^k_{O(U)} F(U)$. If F is locally free of rank n, then $\wedge^n F$ is called the determinant line bundle (though technically invertible sheaf) of F, denoted by det(F). There is a natural perfect paring:

$\wedge^r F \otimes \wedge^{n-r} F \to \operatorname{det}(F).$

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:

$f^{-1} G \otimes_{f^{-1} O'} O$

where $f^{-1} G$ is the inverse image sheaf of G and $f^{-1} O' \to O$ is obtained from $O' \to f_* O$ by adjuction.

There is an adjoint relation between $f_*$ and $f^*$: for any O-module F and O'-module G,

$\operatorname{Hom}_{O}(f^* G, F) \simeq \operatorname{Hom}_{O'}(G, f_*F)$

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

$f_*(F \otimes f^*E) \simeq f_* F \otimes E.$

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:

$\oplus_{i \in I} O \to F \to 0$.

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 $\Gamma(X, -)$ 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 $D(f) \subset D(g)$, by the universal property of localization, there is a natural map

$\rho_{g, f}: M[g^{-1}] \to M[f^{-1}]$

which has the property that $\rho_{g, f} = \rho_{g, h} \circ \rho_{h, f}$. Then

$D(f) \mapsto M[f^{-1}]$

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 $\widetilde{M}$ on X called the sheaf associated to M.

The most basic example is the structure sheaf on X; i.e., $\mathcal{O}_X = \widetilde{A}$. Moreover, $\widetilde{M}$ has the structure of $\mathcal{O}_X = \widetilde{A}$ and thus one gets the exact functor $M \mapsto \widetilde{M}$ from Mod_A, the category of modules over A to the category of modules over $\mathcal{O}_X$. It defines an equivalence from Mod_A to the category of quasi-coherent sheaves on X, with the inverse $\Gamma(X, -)$, 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,

• $M[f^{-1}]^{\sim} = \widetilde{M}|_{D(f)}$.
• For any prime ideal p of A, $\widetilde{M}_p \simeq M_p$ as O_p = A_p-module.
• $(M \otimes_A N)^{\sim} \simeq \widetilde{M} \otimes_{\widetilde{A}} \widetilde{N}$.
• If M is finitely presented, $\operatorname{Hom}_A(M, N)^{\sim} \simeq \mathcal{H}om_{\widetilde{A}}(\widetilde{M}, \widetilde{N})$.
• $\operatorname{Hom}_A(M, N) \simeq \Gamma(X, \mathcal{H}om_{\widetilde{A}}(\widetilde{M}, \widetilde{N}))$, since the equivalence between Mod_A and the category of quasi-coherent sheaves on X.
• $(\varinjlim M_i)^{\sim} \simeq \varinjlim \widetilde{M_i}$; 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 $\widetilde{M}$ such that for any homogeneous element f of positive degree of R, there is a natural isomorphism

$\widetilde{M}|_{\{f \ne 0\}} \simeq (M[f^{-1}]_0)^{\sim}$

as sheaves of modules on the affine scheme $\{f \ne 0\} = \operatorname{Spec}(R[f^{-1}]_0)$; in fact, this defines $\widetilde{M}$ by gluing.

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

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

$(\oplus_{n \ge 0} \Gamma(X, F(n)))^{\sim} \to F$,

which is an isomorphism if and only if F is quasi-coherent.

Computing sheaf cohomology

Main article: 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:

Theorem — Let X be a topological space, F an abelian sheaf on it and $\mathfrak{U}$ an open cover of X such that $\operatorname{H}^i(U_{i_0} \cap \cdots \cap U_{i_p}, F) = 0$ for any i, p and $U_{i_j}$'s in $\mathfrak{U}$. Then for any i,

$\operatorname{H}^i(X, F) = \operatorname{H}^{i}(C^{\bullet}(\mathfrak{U}, F))$

where the right-hand side is the i-th Čech cohomology.

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

$\operatorname{H}^i(X, F(n)) = 0, \, i \ge 1, n \ge n_0$.

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

$0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0.$

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 $\mathrm{Ext}_O^1(H,F)$, where the identity element in $\mathrm{Ext}_O^1(H,F)$ corresponds to the trivial extension.

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

$\operatorname{H}^i(X, F) = \mathrm{Ext}_O^i(O,F),$

since both the sides are the right derived functors of the same functor $\Gamma(X, -) = \operatorname{Hom}_O(O, -).$

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

$\operatorname{Ext}_O^i(F, G(n)) = \Gamma(X, \mathcal{E}xt_O^i(F, G(n))), \, n \ge n_0$.

See also: local-to-global Ext spectral sequence

See also

• D-module (in place of O, one can also consider D, the sheaf of differential operators.)
• fractional ideal
• holomorphic vector bundle
• generic freeness