Sheaf of modules

{{Short description|Sheaf consisting of modules on a ringed space; generalizing vector bundles}}{{Technical|date=November 2023}}

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.Vakil, [http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf Math 216: Foundations of algebraic geometry], 2.5. Moreover, this category has enough injectives,{{harvnb|Hartshorne|loc=Ch. III, Proposition 2.2.}} 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 $\Gamma(X, -)$ .This cohomology functor coincides with the right derived functor of the global section functor in the category of abelian sheaves; cf. {{harvnb|Hartshorne|loc=Ch. III, Proposition 2.6.}}

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 $U \mapsto \operatorname{Hom}_{O|_U}(F|_U, G|_U)$ .There is a canonical homomorphism:

: $\mathcal{H}om_O(F, O)_x \to \operatorname{Hom}_{O_x} (F_x, O_x),$

which is an isomorphism if F is of finite presentation (EGA, Ch. 0, 5.2.6.) 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),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 $F \otimes G \simeq O$ and if F is coherent, then F, G are locally free of rank one. (cf. EGA, Ch 0, 5.4.3.) 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 $U \mapsto \bigwedge^k_{O(U)} F(U)$ . If F is locally free of rank n, then $\bigwedge^n F$ 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).{{harvnb|Hartshorne|loc=Ch III, Lemma 2.4.}} 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.see also: https://math.stackexchange.com/q/447234

Sheaf associated to a module

Let $M$ be a module over a ring $A$ . Put $X=\operatorname{Spec}(A)$ and write $D(f) = \{ f \ne 0 \} = \operatorname{Spec}(A[f^{-1}])$ . 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{{harvnb|Hartshorne|loc=Ch. II, Proposition 5.1.}} 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 $\varphi:M\to N$ ,

$M[f^{-1}]^{\sim} = \widetilde{M}|_{D(f)}$ .{{harvnb|EGA I|loc=Ch. I, Proposition 1.3.6.}}
For any prime ideal p of A, $\widetilde{M}_p \simeq M_p$ as O_p = A_p-module.
$(M \otimes_A N)^{\sim} \simeq \widetilde{M} \otimes_{\widetilde{A}} \widetilde{N}$ .{{harvnb|EGA I|loc=Ch. I, Corollaire 1.3.12.}}
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.
$(\varinjlim M_i)^{\sim} \simeq \varinjlim \widetilde{M_i}$ ;{{harvnb|EGA I|loc=Ch. I, Corollaire 1.3.9.}} 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 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

: $\widetilde{M}|_{\{f \ne 0\}} \simeq (M[f^{-1}]_0)^{\sim}$

as sheaves of modules on the affine scheme $\{f \ne 0\} = \operatorname{Spec}(R[f^{-1}]_0)$ ;{{harvnb|Hartshorne|loc=Ch. II, Proposition 5.11.}} 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:

: $\left(\bigoplus_{n \ge 0} \Gamma(X, F(n))\right)^{\sim} \to F,$

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:

{{math_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{{Cite web |title=Section 30.2 (01X8): Čech cohomology of quasi-coherent sheaves—The Stacks project |url=https://stacks.math.columbia.edu/tag/01X8 |access-date=2023-12-07 |website=stacks.math.columbia.edu}} 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, Hⁱ(X, F) is finitely generated over R₀, and

There is an integer n₀, depending on F, such that

$\operatorname{H}^i(X, F(n)) = 0, \, i \ge 1, n \ge n_0.$

{{harvnb|Costa|Miró-Roig|Pons-Llopis|2021|loc=Theorem 1.3.1}}{{cite book |doi=10.1017/CBO9781139044059.023 |chapter=Links with sheaf cohomology |title=Local Cohomology |series=Cambridge Studies in Advanced Mathematics |year=2012 |pages=438–479 |publisher=Cambridge University Press |isbn=9780521513630 }}{{harvnb|Serre|1955|loc=§.66 Faisceaux algébriques cohérents sur les variétés projectives.}}

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 n₀ such that

: $\operatorname{Ext}_O^i(F, G(n)) = \Gamma(X, \mathcal{Ext}_O^i(F, G(n))), \, n \ge n_0$ ,

where $\mathcal{Ext}_O$ denotes the derived functors of $\mathcal{Hom}_O$ .{{harvnb|Hartshorne|loc=Ch. III, Proposition 6.9.}}

= Locally free resolutions =

$\mathcal{Ext}(\mathcal{F},\mathcal{G})$ can be readily computed for any coherent sheaf $\mathcal{F}$ using a locally free resolution:{{cite book|last1=Hartshorne|first1=Robin|title=Algebraic Geometry|pages=233–235}} 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 $\mathcal{Ext}^i(\mathcal{O}_X,\mathcal{F})$ .

Notes

References

{{EGA|book=I}}
{{Hartshorne AG}}
{{cite book |doi=10.1515/9783110647686|url={{Google books|v9IuEAAAQBAJ|pg=PT22|plainurl=yes}}|title=Ulrich Bundles |year=2021 |last1=Costa |first1=Laura |last2=Miró-Roig |first2=Rosa María |last3=Pons-Llopis |first3=Joan |isbn=9783110647686 }}
{{cite book |doi=10.1017/CBO9781139044059.023 |chapter=Links with sheaf cohomology |title=Local Cohomology |series=Cambridge Studies in Advanced Mathematics |year=2012 |pages=438–479 |publisher=Cambridge University Press |isbn=9780521513630 }}
{{Citation|author1-first=Jean-Pierre|author1-last=Serre|author1-link=Jean-Pierre Serre|title=Faisceaux algébriques cohérents (§.66 Faisceaux algébriques cohérents sur les variétés projectives.)|journal=Annals of Mathematics|volume=61|pages=197–278|year=1955|issue=2|doi=10.2307/1969915|jstor=1969915|mr=0068874|url=https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf}}

Category:Sheaf theory