Sheaf of modules – Wikipedia

Sheaf consisting of modules on a ringed space; generalizing vector bundles

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

${displaystyle {underline {mathbf {Z} }}}$

, 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]

Examples[edit]

• 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.

Operations[edit]

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

${displaystyle Fotimes _{O}G}$

or

${displaystyle Fotimes G}$

,

is the O-module that is the sheaf associated to the presheaf

${displaystyle Umapsto F(U)otimes _{O(U)}G(U).}$

(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

${displaystyle Umapsto operatorname {Hom} _{O|_{U}}(F|_{U},G|_{U})}$

.^[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

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

,

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:

${displaystyle {check {L}}otimes Lsimeq O,}$

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

${displaystyle {check {E}}otimes Esimeq operatorname {End} _{O}(E)to O}$

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

${textstyle Umapsto bigwedge _{O(U)}^{k}F(U)}$

. 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:

${displaystyle bigwedge ^{r}Fotimes bigwedge ^{n-r}Fto det(F).}$

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:

${displaystyle f^{-1}Gotimes _{f^{-1}O’}O}$

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,

${displaystyle f_{*}(Fotimes f^{*}E)simeq f_{*}Fotimes E.}$

Properties[edit]

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:

${displaystyle bigoplus _{iin I}Oto Fto 0.}$

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]

Sheaf associated to a module[edit]

Let

${displaystyle M}$

be a module over a ring

${displaystyle A}$

. Put

${displaystyle X=operatorname {Spec} (A)}$

and write

${displaystyle D(f)={fneq 0}=operatorname {Spec} (A[f^{-1}])}$

. 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

${displaystyle Mmapsto {widetilde {M}}}$

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.

• 
  ${displaystyle (Motimes _{A}N)^{sim }simeq {widetilde {M}}otimes _{widetilde {A}}{widetilde {N}}}$

  .^[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.

Sheaf associated to a graded module[edit]

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

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

as sheaves of modules on the affine scheme

${displaystyle {fneq 0}=operatorname {Spec} (R[f^{-1}]_{0})}$

;^[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

${displaystyle F(n)=Fotimes O(n)}$

, there is a canonical homomorphism:

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

,

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

Computing sheaf cohomology[edit]

This section needs expansion. You can help by adding to it. (January 2016)

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,

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

   

Sheaf extension[edit]

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

${displaystyle 0rightarrow Frightarrow Grightarrow Hrightarrow 0.}$

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

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

.^[13]

Locally free resolutions[edit]

${displaystyle {mathcal {Ext}}({mathcal {F}},{mathcal {G}})}$

can be readily computed for any coherent sheaf

${displaystyle {mathcal {F}}}$

using a locally free resolution:^[14] 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

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

Examples[edit]

Hypersurface[edit]

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

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

Union of smooth complete intersections[edit]

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

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

resolving

${displaystyle {mathcal {O}}_{X},}$

which we can use to compute

${displaystyle {mathcal {Ext}}^{i}({mathcal {O}}_{X},{mathcal {F}})}$

.

See also[edit]

References[edit]