# 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 the restriction 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 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. 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 $\Omega _{X}$.
* A sheaf of algebras is a sheaf of modules 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
    $F\otimes _{O}G$ 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
    $\bigwedge ^{k}F$
is the sheaf associated to the presheaf U↦⋀$_{O(U)}$$^k$F(U){\textstyle U\mapsto \bigwedge _{O(U)}^{k}F(U)}. If F is locally free of rank n, then ⋀$^n$F{\textstyle \bigwedge ^{n}F} is called the determinant line bundle (though technically invertible sheaf) of F, denoted by det(F). There is a natural perfect pairing:
    $\bigwedge ^{r}F\otimes \bigwedge ^{n-r}F\to \det(F).$
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:
    $\bigoplus _{i\in I}O\to F\to 0.$
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 $X=\operatorname {Spec} (A)$ and write $D(f)=\{f\neq 0\}=\operatorname {Spec} (A[f^{-1}])$. For each pair $D(f)\subseteq D(g)$, by the universal property of localization, there is a natural map
    $\rho _{g,f}:M[g^{-1}]\to M[f^{-1}]$
having 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 ${\mathcal {O}}_{X}={\widetilde {A}}$-module 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, and any morphism $\varphi :M\to N$,
* $M[f^{-1}]^{\sim }={\widetilde {M}}|_{D(f)}$.
* 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.
* A sequence of A-modules is exact if and only if the induced sequence by $\sim$ is exact. In particular, $(\ker(\varphi ))^{\sim }=\ker({\widetilde {\varphi }}),(\operatorname {coker} (\varphi ))^{\sim }=\operatorname {coker} ({\widetilde {\varphi }}),(\operatorname {im} (\varphi ))^{\sim }=\operatorname {im} ({\widetilde {\varphi }})$.

## 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 ${\widetilde {M}}$ such that for any homogeneous element f of positive degree of R, there is a natural isomorphism
    ${\widetilde {M}}|_{\{f\neq 0\}}\simeq (M[f^{-1}]_{0})^{\sim }$
as sheaves of modules on the affine scheme $\{f\neq 0\}=\operatorname {Spec} (R[f^{-1}]_{0})$; 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, 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 $F(n)=F\otimes O(n)$, there is a canonical homomorphism:
    $\left(\bigoplus _{n\geq 0}\Gamma (X,F(n))\right)^{\sim }\to F,$
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:
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 vanishing theorem states that if X is a projective variety and F a coherent sheaf on it, then, for sufficiently large n, the Serre twist F(n) is generated by finitely many global sections. Moreover,
* For each i, H$^i$(X, F) is finitely generated over R$_0$, and
* There is an integer n$_0$, depending on F, such that $\operatorname {H} ^{i}(X,F(n))=0,\,i\geq 1,n\geq n_{0}.$

## 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 $\operatorname {Ext} _{O}^{1}(H,F)$, where the identity element in $\operatorname {Ext} _{O}^{1}(H,F)$ corresponds to the trivial extension.
In the case where H is O, we have: for any i ≥ 0,
    $\operatorname {H} ^{i}(X,F)=\operatorname {Ext} _{O}^{i}(O,F),$
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 n$_0$ such that
    $\operatorname {Ext} _{O}^{i}(F,G(n))=\Gamma (X,{\mathcal {E}}xt_{O}^{i}(F,G(n))),\,n\geq n_{0}$.

### Locally free resolutions

${\mathcal {Ext}}({\mathcal {F}},{\mathcal {G}})$ can be readily computed for any coherent sheaf ${\mathcal {F}}$ using a locally free resolution: given a complex
    $\cdots \to {\mathcal {L}}_{2}\to {\mathcal {L}}_{1}\to {\mathcal {L}}_{0}\to {\mathcal {F}}\to 0$
then
    ${\mathcal {RHom}}({\mathcal {F}},{\mathcal {G}})={\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}})$
hence
    ${\mathcal {Ext}}^{k}({\mathcal {F}},{\mathcal {G}})=h^{k}({\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}}))$

### Examples
#### Hypersurface

Consider a smooth hypersurface $X$ of degree $d$. Then, we can compute a resolution
    ${\mathcal {O}}(-d)\to {\mathcal {O}}$
and find that
    ${\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})=h^{i}({\mathcal {Hom}}({\mathcal {O}}(-d)\to {\mathcal {O}},{\mathcal {F}}))$

#### Union of smooth complete intersections

Consider the scheme
    $X={\text{Proj}}\left({\frac {\mathbb {C} [x_{0},\ldots ,x_{n}]}{(f)(g_{1},g_{2},g_{3})}}\right)\subseteq \mathbb {P} ^{n}$
where $(f,g_{1},g_{2},g_{3})$ is a smooth complete intersection and $\deg(f)=d$, $\deg(g_{i})=e_{i}$. We have a complex
    ${\mathcal {O}}(-d-e_{1}-e_{2}-e_{3}){\xrightarrow {\begin{bmatrix}g_{3}\\-g_{2}\\-g_{1}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1}-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{1}-e_{3})\\\oplus \\{\mathcal {O}}(-d-e_{2}-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}g_{2}&g_{3}&0\\-g_{1}&0&-g_{3}\\0&-g_{1}&g_{2}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1})\\\oplus \\{\mathcal {O}}(-d-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}fg_{1}&fg_{2}&fg_{3}\end{bmatrix}}}{\mathcal {O}}$
resolving ${\mathcal {O}}_{X},$ which we can use to compute ${\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})$.