Invertible sheaf

 ( Geometry Of Divisors ) 

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 , they play a central role in the study of algebraic varieties.

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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:EGA 0_I, 5.4.Stacks Project, tag 01CR, [1].

• There exists a sheaf M such that $L\; \backslash otimes\_\{\backslash mathcal\{O\}\_X\}\; M\; \backslash cong\; \backslash mathcal\{O\}\_X$.
• The natural homomorphism $L\; \backslash otimes\_\{\backslash mathcal\{O\}\_X\}\; L^\backslash vee\; \backslash to\; \backslash mathcal\{O\}\_X$ is an isomorphism, where $L^\backslash vee$ denotes the dual sheaf $\backslash underline\{\backslash operatorname\{Hom\}\}(L,\; \backslash mathcal\{O\}\_X)$.
• The functor from O _X-modules to O _X-modules defined by $F\; \backslash mapsto\; F\; \backslash otimes\_\{\backslash mathcal\{O\}\_X\}\; L$ 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 , to the point where the two are sometimes conflated.

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

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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

$\backslash mathrm\{Pic\}(X)\backslash$

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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See also

• First Chern class
• Birkhoff-Grothendieck theorem

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