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 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 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

$F\otimes _{O}G$ 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 $U\mapsto \wedge _{O(U)}^{k}F(U)$. 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 pairing:

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

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 and write $D(f)=\{f\neq 0\}=\operatorname {Spec} (A[f^{-1}])$. 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 ${\mathcal {O}}_{X}={\widetilde {A}}$-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,

• $M[f^{-1}]^{\sim }={\widetilde {M}}|_{D(f)}$.
• 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 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:

$(\oplus _{n\geq 0}\Gamma (X,F(n)))^{\sim }\to F$,

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

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

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 $\mathrm {Ext} _{O}^{1}(H,F)$, 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

$\operatorname {Ext} _{O}^{i}(F,G(n))=\Gamma (X,{\mathcal {E}}xt_{O}^{i}(F,G(n))),\,n\geq n_{0}$.

See also: local-to-global Ext spectral sequence

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

$X={\text{Proj}}\left({\frac {\mathbb {C} [x_{0},\ldots ,x_{n}]}{(f)(g_{1},g_{2},g_{3})}}\right)\subset \mathbb {P} ^{n}$

where $(f,g_{1},g_{2},g_{3})$ is a smooth complete intersection and ${\text{deg}}(f)=d$, ${\text{deg}}(g_{i})=e_{i}$. 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 this to compute ${\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})$.

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

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