Invertible sheaf

Last updated December 09, 2024

In mathematics, an invertible sheaf is a sheaf on a ringed space that has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties.

Definition

Let (X, O_X) be a ringed space. Isomorphism classes of sheaves of O_X-modules form a monoid under the operation of tensor product of O_X-modules. The identity element for this operation is O_X itself. Invertible sheaves are the invertible elements of this monoid. Specifically, if L is a sheaf of O_X-modules, then L is called invertible if it satisfies any of the following equivalent conditions:^[1]^[2]

There exists a sheaf M such that $L\otimes _{{\mathcal {O}}_{X}}M\cong {\mathcal {O}}_{X}$ .
The natural homomorphism $L\otimes _{{\mathcal {O}}_{X}}L^{\vee }\to {\mathcal {O}}_{X}$ is an isomorphism, where $L^{\vee }$ denotes the dual sheaf ${\underline {\operatorname {Hom} }}(L,{\mathcal {O}}_{X})$ .
The functor from O_X-modules to O_X-modules defined by $F\mapsto F\otimes _{{\mathcal {O}}_{X}}L$ is an equivalence of categories.

Every locally free sheaf of rank one is invertible. If X is a locally ringed space, then L is invertible if and only if it is locally free of rank one. Because of this fact, invertible sheaves are closely related to line bundles, to the point where the two are sometimes conflated.

Examples

Let X be an affine scheme $Spec R$ . Then an invertible sheaf on X is the sheaf associated to a rank one projective module over R. For example, this includes fractional ideals of algebraic number fields, since these are rank one projective modules over the rings of integers of the number field.

The Picard group

Quite generally, the isomorphism classes of invertible sheaves on X themselves form an abelian group under tensor product. This group generalises the ideal class group. In general it is written

\mathrm {Pic} (X)\

with Pic the Picard functor. Since it also includes the theory of the Jacobian variety of an algebraic curve, the study of this functor is a major issue in algebraic geometry.

The direct construction of invertible sheaves by means of data on X leads to the concept of Cartier divisor.

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References

↑ EGA 0_I, 5.4.
↑ Stacks Project, tag 01CR, .

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.

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[1] EGA 0_I, 5.4.

[2] Stacks Project, tag 01CR, .

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