Sheaf of modules

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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 ${\displaystyle {\underline {\mathbf {Z} }}}$, 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.^[1] Moreover, this category has enough injectives,^[2] and consequently one can and does define the sheaf cohomology ${\displaystyle \operatorname {H} ^{i}(X,-)}$ as the i-th right derived functor of the global section functor ${\displaystyle \Gamma (X,-)}$.^[3]

• 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

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

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

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

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

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

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.)^[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 ${\displaystyle M}$ be a module over a ring ${\displaystyle A}$. Put ${\displaystyle X=\operatorname {Spec} (A)}$ 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, and any morphism ${\displaystyle \varphi :M\to N}$,

• ${\displaystyle M[f^{-1}]^{\sim }={\widetilde {M}}|_{D(f)}}$.^[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.
• A sequence of A-modules is exact if and only if the induced sequence by ${\displaystyle \sim }$ is exact. In particular, ${\displaystyle (\ker(\varphi ))^{\sim }=\ker({\widetilde {\varphi }}),(\operatorname {coker} (\varphi ))^{\sim }=\operatorname {coker} ({\widetilde {\varphi }}),(\operatorname {im} (\varphi ))^{\sim }=\operatorname {im} ({\widetilde {\varphi }})}$.

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:

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

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

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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 ${\displaystyle {\mathfrak {U}}}$ an open cover of X such that ${\displaystyle \operatorname {H} ^{i}(U_{i_{0}}\cap \cdots \cap U_{i_{p}},F)=0}$ for any i, p and ${\displaystyle U_{i_{j}}}$'s in ${\displaystyle {\mathfrak {U}}}$. Then for any i,

${\displaystyle \operatorname {H} ^{i}(X,F)=\operatorname {H} ^{i}(C^{\bullet }({\mathfrak {U}},F))}$

where the right-hand side is the i-th Čech cohomology.

Serre's vanishing theorem^[13] 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,

1. For each i, Hⁱ(X, F) is finitely generated over R₀, and
2. There is an integer n₀, depending on F, such that $${\displaystyle \operatorname {H} ^{i}(X,F(n))=0,\,i\geq 1,n\geq n_{0}.}$$

^[14][15][16]

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

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

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

${\displaystyle \operatorname {Ext} _{O}^{i}(F,G(n))=\Gamma (X,{\mathcal {E}}xt_{O}^{i}(F,G(n))),\,n\geq n_{0}}$.^[17]

${\displaystyle {\mathcal {Ext}}({\mathcal {F}},{\mathcal {G}})}$ can be readily computed for any coherent sheaf ${\displaystyle {\mathcal {F}}}$ using a locally free resolution:^[18] given a complex

${\displaystyle \cdots \to {\mathcal {L}}_{2}\to {\mathcal {L}}_{1}\to {\mathcal {L}}_{0}\to {\mathcal {F}}\to 0}$

then

${\displaystyle {\mathcal {RHom}}({\mathcal {F}},{\mathcal {G}})={\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}})}$

hence

${\displaystyle {\mathcal {Ext}}^{k}({\mathcal {F}},{\mathcal {G}})=h^{k}({\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}}))}$

Consider a smooth hypersurface ${\displaystyle X}$ of degree ${\displaystyle d}$. Then, we can compute a resolution

${\displaystyle {\mathcal {O}}(-d)\to {\mathcal {O}}}$

and find that

${\displaystyle {\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})=h^{i}({\mathcal {Hom}}({\mathcal {O}}(-d)\to {\mathcal {O}},{\mathcal {F}}))}$

Consider the scheme

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

where ${\displaystyle (f,g_{1},g_{2},g_{3})}$ is a smooth complete intersection and ${\displaystyle \deg(f)=d}$, ${\displaystyle \deg(g_{i})=e_{i}}$. We have a complex

${\displaystyle {\mathcal {O}}(-d-e_{1}-e_{2}-e_{3}){\xrightarrow {\begin{bmatrix}g_{3}\\-g_{2}\\-g_{1}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1}-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{1}-e_{3})\\\oplus \\{\mathcal {O}}(-d-e_{2}-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}g_{2}&g_{3}&0\\-g_{1}&0&-g_{3}\\0&-g_{1}&g_{2}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1})\\\oplus \\{\mathcal {O}}(-d-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}fg_{1}&fg_{2}&fg_{3}\end{bmatrix}}}{\mathcal {O}}}$

resolving ${\displaystyle {\mathcal {O}}_{X},}$ which we can use to compute ${\displaystyle {\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})}$.

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

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• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Costa, Laura; Miró-Roig, Rosa María; Pons-Llopis, Joan (2021). Ulrich Bundles. doi:10.1515/9783110647686. ISBN 9783110647686.
• "Links with sheaf cohomology". Local Cohomology. Cambridge Studies in Advanced Mathematics. Cambridge University Press. 2012. pp. 438–479. doi:10.1017/CBO9781139044059.023. ISBN 9780521513630.
• Serre, Jean-Pierre (1955), "Faisceaux algébriques cohérents (§.66 Faisceaux algébriques cohérents sur les variétés projectives.)" (PDF), Annals of Mathematics, 61 (2): 197–278, doi:10.2307/1969915, JSTOR 1969915, MR 0068874