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 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.[1] Moreover, this category has enough injectives,[2] and consequently one can and does define the sheaf cohomology H^i(X, -) as the i-th right derived functor of the global section functor \Gamma(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).[3] 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),[4] 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 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.)[5] Since a flasque sheaf is acyclic, this implies that the i-th right derived functor of the global section functor \Gamma(X, -) coincides with the usual i-th cohomology.[6]

Sheaf associated to a module

See also: coherent sheaf

Let R be a ring and M an R-module. Let also X = \operatorname{Spec}(R). Then there is an \mathcal{O}_X-module \widetilde{M} whose stalk \widetilde{M}_p at a prime ideal p is isomorphic to the localization Mp of M at p as Op = Rp-module. Also, \Gamma(X, \widetilde{M}) = M.

The functor M \mapsto \widetilde{M} is an equivalence from the category of R-modules to the category of quasi-coherent \mathcal{O}_X-modules with X = \operatorname{Spec}R; the inverse is the global section functor F \mapsto \Gamma(X, F).[7] Suppose X is a Noetherian scheme or equivalently R is a Noetherian ring. Then the functor M \mapsto \widetilde{M} is an equivalence from the category of finitely generated R-modules to the category of coherent \mathcal{O}_X-modules.

There is a graded analog of the above construction and equivalence. 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);[8] in fact, this defines \widetilde{M} by gluing.

Example: Let R(1) be the graded R-module given by R(1)n = Rn+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), the direct sum

\Gamma_*(F) = \oplus_{n \ge 0} \Gamma(X, F(n))

is a graded R-module. Then there is a canonical homomorphism:

\widetilde{\Gamma_*(F)} \to F,

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

Serre's theorem A states that if R is a Noetherian ring and F is coherent, 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

H^1(X, F) = \mathrm{Ext}_O^1(O,F),[9]

so the group of extensions of O by F is also isomorphic to the first sheaf cohomology group with coefficients in F.

Note: Some authors, notably Hartshorne, drop the subscript O.

Examples

See also

Notes

  1. ↑ Vakil, Math 216: Foundations of algebraic geometry, 2.5.
  2. ↑ Hartshorne, Ch. III, Proposition 2.2.
  3. ↑ There is a canonical homomorphism:
    \mathcal{H}om_O(F, O)_x \to \operatorname{Hom}_{O_x} (F_x, O_x),
    which is an isomorphism if F is of finite presentation (EGA, Ch. 0, 5.2.6.)
  4. ↑ 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 F \otimes G \simeq O and if F is coherent, then F, G are locally free of rank one. (cf. EGA, Ch 0, 5.4.3.)
  5. ↑ Hartshorne, Ch III, Lemma 2.4.
  6. ↑ see also: http://math.stackexchange.com/questions/447220/hartshornes-weird-definition-of-right-derived-functors-and-prop-iii-2-6/447234#447234
  7. ↑ Hartshorne, Ch II, Corollary 5.5.
  8. ↑ Hartshorne, Ch. II, Proposition 5.11.
  9. ↑ since both the sides are the right derived functors of the same functor \Gamma(X, -) = \operatorname{Hom}_O(O, -).

References