Sheaf of modules

• 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 ${\displaystyle \Omega _{X}}$ and the canonical sheaf ${\displaystyle \omega _{X}}$ is the n-th exterior power (determinant) of ${\displaystyle \Omega _{X}}$.
• A sheaf of algebras is a sheaf of module that is also a sheaf of rings.

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

${\displaystyle F\otimes _{O}G}$ or ${\displaystyle F\otimes G}$,

is the O-module that is the sheaf associated to the presheaf ${\displaystyle U\mapsto F(U)\otimes _{O(U)}G(U).}$ (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

${\displaystyle {\mathcal {H}}om_{O}(F,G)}$

denotes the O-module that is the sheaf ${\displaystyle U\mapsto \operatorname {Hom} _{O|_{U}}(F|_{U},G|_{U})}$.[4] In particular, the O-module

${\displaystyle {\mathcal {H}}om_{O}(F,O)}$

is called the dual module of F and is denoted by ${\displaystyle {\check {F}}}$. Note: for any O-modules E, F, there is a canonical homomorphism

${\displaystyle {\check {E}}\otimes F\to {\mathcal {H}}om_{O}(E,F)}$,

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),[5] then this reads:

${\displaystyle {\check {L}}\otimes L\simeq O,}$

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 ${\displaystyle \operatorname {H} ^{1}(X,{\mathcal {O}}^{*})}$ (by the standard argument with Čech cohomology).

If E is a locally free sheaf of finite rank, then there is an O-linear map ${\displaystyle {\check {E}}\otimes E\simeq \operatorname {End} _{O}(E)\to O}$ 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

${\displaystyle \wedge ^{k}F}$

is the sheaf associated to the presheaf ${\displaystyle U\mapsto \wedge _{O(U)}^{k}F(U)}$. If F is locally free of rank n, then ${\displaystyle \wedge ^{n}F}$ is called the determinant line bundle (though technically invertible sheaf) of F, denoted by det(F). There is a natural perfect pairing:

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

Let f: (X, O) →(X', O') be a morphism of ringed spaces. If F is an O-module, then the direct image sheaf ${\displaystyle f_{*}F}$ 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 ${\displaystyle f^{*}G}$ of G is the O-module given as the tensor product of modules:

${\displaystyle f^{-1}G\otimes _{f^{-1}O'}O}$

where ${\displaystyle f^{-1}G}$ is the inverse image sheaf of G and ${\displaystyle f^{-1}O'\to O}$ 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,

${\displaystyle \operatorname {Hom} _{O}(f^{*}G,F)\simeq \operatorname {Hom} _{O'}(G,f_{*}F)}$

as abelian group. There is also the projection formula: for an O-module F and a locally free O'-module E of finite rank,

${\displaystyle f_{*}(F\otimes f^{*}E)\simeq f_{*}F\otimes E.}$

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:

${\displaystyle \oplus _{i\in I}O\to F\to 0}$.

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.)[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 ${\displaystyle \Gamma (X,-)}$ in the category of O-modules coincides with the usual i-th sheaf cohomology in the category of abelian sheaves.[7]

Let M be a module over a ring A. Put X = Spec A and write ${\displaystyle D(f)=\{f\neq 0\}=\operatorname {Spec} (A[f^{-1}])}$. For each pair ${\displaystyle D(f)\subseteq D(g)}$, by the universal property of localization, there is a natural map

${\displaystyle \rho _{g,f}:M[g^{-1}]\to M[f^{-1}]}$

having the property that ${\displaystyle \rho _{g,f}=\rho _{g,h}\circ \rho _{h,f}}$. Then

${\displaystyle D(f)\mapsto M[f^{-1}]}$

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[8] it is in fact a B-sheaf (i.e., it satisfies the gluing axiom) and thus defines the sheaf ${\displaystyle {\widetilde {M}}}$ on X called the sheaf associated to M.

The most basic example is the structure sheaf on X; i.e., ${\displaystyle {\mathcal {O}}_{X}={\widetilde {A}}}$. Moreover, ${\displaystyle {\widetilde {M}}}$ has the structure of ${\displaystyle {\mathcal {O}}_{X}={\widetilde {A}}}$-module and thus one gets the exact functor ${\displaystyle M\mapsto {\widetilde {M}}}$ from Mod_A, the category of modules over A to the category of modules over ${\displaystyle {\mathcal {O}}_{X}}$. It defines an equivalence from Mod_A to the category of quasi-coherent sheaves on X, with the inverse ${\displaystyle \Gamma (X,-)}$, 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,

• ${\displaystyle M[f^{-1}]^{\sim }={\widetilde {M}}|_{D(f)}}$.[9]
• For any prime ideal p of A, ${\displaystyle {\widetilde {M}}_{p}\simeq M_{p}}$ as O_p = A_p-module.
• ${\displaystyle (M\otimes _{A}N)^{\sim }\simeq {\widetilde {M}}\otimes _{\widetilde {A}}{\widetilde {N}}}$.[10]
• If M is finitely presented, ${\displaystyle \operatorname {Hom} _{A}(M,N)^{\sim }\simeq {\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}})}$.[10]
• ${\displaystyle \operatorname {Hom} _{A}(M,N)\simeq \Gamma (X,{\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}}))}$, since the equivalence between Mod_A and the category of quasi-coherent sheaves on X.
• ${\displaystyle (\varinjlim M_{i})^{\sim }\simeq \varinjlim {\widetilde {M_{i}}}}$;[11] in particular, taking a direct sum and ~ commute.

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 ${\displaystyle {\widetilde {M}}}$ such that for any homogeneous element f of positive degree of R, there is a natural isomorphism

${\displaystyle {\widetilde {M}}|_{\{f\neq 0\}}\simeq (M[f^{-1}]_{0})^{\sim }}$

as sheaves of modules on the affine scheme ${\displaystyle \{f\neq 0\}=\operatorname {Spec} (R[f^{-1}]_{0})}$;[12] in fact, this defines ${\displaystyle {\widetilde {M}}}$ by gluing.

Example: Let R(1) be the graded R-module given by R(1)_n = R_n+1. Then ${\displaystyle O(1)={\widetilde {R(1)}}}$ 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 ${\displaystyle F(n)=F\otimes O(n)}$, there is a canonical homomorphism:

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

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

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,

(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

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

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

${\displaystyle 0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0.}$

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 ${\displaystyle \mathrm {Ext} _{O}^{1}(H,F)}$, where the identity element in ${\displaystyle \mathrm {Ext} _{O}^{1}(H,F)}$ corresponds to the trivial extension.

In the case where H is O, we have: for any i ≥ 0,

${\displaystyle \operatorname {H} ^{i}(X,F)=\mathrm {Ext} _{O}^{i}(O,F),}$

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 n₀ such that

${\displaystyle \operatorname {Ext} _{O}^{i}(F,G(n))=\Gamma (X,{\mathcal {E}}xt_{O}^{i}(F,G(n))),\,n\geq n_{0}}$.[13]

• D-module (in place of O, one can also consider D, the sheaf of differential operators.)
• fractional ideal
• holomorphic vector bundle
• generic freeness

• 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