# Sheaf of modules

In mathematics, a **sheaf of O-modules** or simply an

**over a ringed space (**

*O*-module*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 , 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.[1] Moreover, this category has enough injectives,[2] and consequently one can and does define the sheaf cohomology as the *i*-th right derived functor of the global section functor .[3]

## 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 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.

## Operations

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

denotes the *O*-module that is the sheaf .[4] In particular, the *O*-module

is called the **dual module** of *F* and is denoted by . 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),[5] 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 (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*.

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 . 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:

where is the inverse image sheaf of *G* and is obtained from by adjuction.

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,

## 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.)[6] 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.[7]

## 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[8] 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 Mod_{A}, the category of modules over *A* to the category of modules over . It defines an equivalence from Mod_{A} 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*,

- .[9]
- For any prime ideal
*p*of*A*, as*O*_{p}=*A*_{p}-module. - .[10]
- If
*M*is finitely presented, .[10] - , since the equivalence between Mod
_{A}and the category of quasi-coherent sheaves on*X*. - ;[11] 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 *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 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 ;[12] in fact, this defines by gluing.

**Example**: Let *R*(1) be the graded *R*-module given by *R*(1)_{n} = *R*_{n+1}. Then 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 , 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:

**Theorem** — Let *X* be a topological space, *F* an abelian sheaf on it and an open cover of *X* such that for any *i*, *p* and 's in . Then for any *i*,

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*, H^{i}(*X*,*F*) is finitely generated over*R*_{0}, and - (b) (Serre's theorem B) There is an integer
*n*_{0}, depending on*F*, such that- .

## 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 , 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 *n*_{0} such that

- .[13]

### Locally Free Resolutions

can be readily computed for any coherent sheaf using a locally free resolution[14]: given a complex

then

hence

### Examples

#### Hypersurface

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 .

## 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

## Notes

- 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:
*F*is of finite presentation (EGA, Ch. 0, 5.2.6.) - 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.

## References

- 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