# Sheaf of modules – Wikipedia

Sheaf consisting of modules on a ringed space; generalizing vector bundles

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[edit]

- 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 - A sheaf of algebras is a sheaf of module that is also a sheaf of rings.

## Operations[edit]

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

is the *O*-module that is the sheaf associated to the presheaf

(To see that sheafification cannot be avoided, compute the global sections of

${displaystyle O(1)otimes O(-1)=O}$ where *O*(1) is Serre’s twisting sheaf on a projective space.)

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

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

${textstyle Umapsto bigwedge _{O(U)}^{k}F(U)}$. 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

${displaystyle f^{-1}G}$ is the inverse image sheaf of *G* and

is obtained from

${displaystyle O’to f_{*}O}$by adjuction.

There is an adjoint relation between

${displaystyle f_{*}}$and

${displaystyle f^{*}}$: 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[edit]

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[edit]

Let

${displaystyle M}$be a module over a ring

${displaystyle A}$. Put

${displaystyle X=operatorname {Spec} (A)}$and write

${displaystyle D(f)={fneq 0}=operatorname {Spec} (A[f^{-1}])}$. For each pair

${displaystyle D(f)subseteq D(g)}$, by the universal property of localization, there is a natural map

having the property that

${displaystyle rho _{g,f}=rho _{g,h}circ rho _{h,f}}$. 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,

${displaystyle {widetilde {M}}}$has the structure of

${displaystyle {mathcal {O}}_{X}={widetilde {A}}}$-module and thus one gets the exact functor

${displaystyle Mmapsto {widetilde {M}}}$ 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*,

- For any prime ideal
*p*of*A*, - If
*M*is finitely presented,

## Sheaf associated to a graded module[edit]

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

${displaystyle {fneq 0}=operatorname {Spec} (R[f^{-1}]_{0})}$;^{[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[edit]

This section
needs expansion. You can help by adding to it. (January 2016) |

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,

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

## Sheaf extension[edit]

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

${displaystyle operatorname {Ext} _{O}^{1}(H,F)}$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

${displaystyle Gamma (X,-)=operatorname {Hom} _{O}(O,-).}$

**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 *n*_{0} such that

### Locally free resolutions[edit]

${displaystyle {mathcal {Ext}}({mathcal {F}},{mathcal {G}})}$

can be readily computed for any coherent sheaf

${displaystyle {mathcal {F}}}$ using a locally free resolution:^{[14]} given a complex

then

hence

### Examples[edit]

#### Hypersurface[edit]

Consider a smooth hypersurface

${displaystyle X}$of degree

${displaystyle d}$. Then, we can compute a resolution

and find that

#### Union of smooth complete intersections[edit]

Consider the scheme

where

${displaystyle (f,g_{1},g_{2},g_{3})}$is a smooth complete intersection and

${displaystyle deg(f)=d}$,

${displaystyle deg(g_{i})=e_{i}}$. We have a complex

resolving

${displaystyle {mathcal {O}}_{X},}$which we can use to compute

${displaystyle {mathcal {Ext}}^{i}({mathcal {O}}_{X},{mathcal {F}})}$.

## See also[edit]

**^**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:which is an isomorphism if

*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**^**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.- ^
^{a}^{b}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.

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