Invertible sheaf
In mathematics, an invertible sheaf is a sheaf on a ringed space which 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, OX) be a ringed space. Isomorphism classes of sheaves of OX-modules form a monoid under the operation of tensor product of OX-modules. The identity element for this operation is OX itself. Invertible sheaves are the invertible elements of this monoid. Specifically, if L is a sheaf of OX-modules, then L is called invertible if it satisfies any of the following equivalent conditions:[1][2]
- There exists a sheaf M such that [math]\displaystyle{ L \otimes_{\mathcal{O}_X} M \cong \mathcal{O}_X }[/math].
- The natural homomorphism [math]\displaystyle{ L \otimes_{\mathcal{O}_X} L^\vee \to \mathcal{O}_X }[/math] is an isomorphism, where [math]\displaystyle{ L^\vee }[/math] denotes the dual sheaf [math]\displaystyle{ \underline{\operatorname{Hom}}(L, \mathcal{O}_X) }[/math].
- The functor from OX-modules to OX-modules defined by [math]\displaystyle{ F \mapsto F \otimes_{\mathcal{O}_X} L }[/math] 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
- [math]\displaystyle{ \mathrm{Pic}(X)\ }[/math]
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.
See also
- Vector bundles in algebraic geometry
- Line bundle
- First Chern class
- Picard group
- Birkhoff-Grothendieck theorem
References
- Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS 4. doi:10.1007/bf02684778. http://www.numdam.org/item/PMIHES_1960__4__5_0.
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