Editing Invertible sheaf

{{Short description|Type of sheaf of modules}}
In [[mathematics]], an '''invertible sheaf''' is a [[sheaf (mathematics)|sheaf]] on a [[ringed space]] that has an inverse with respect to [[tensor product]] of [[sheaf of modules|sheaves of modules]]. It is the equivalent in [[algebraic geometry]] of the topological notion of a [[line bundle]]. Due to their interactions with [[Cartier divisor]]s, they play a central role in the study of [[algebraic varieties]]. 

==Definition==
Let (''X'', ''O''<sub>''X''</sub>) be a ringed space.  Isomorphism classes of sheaves of ''O''<sub>''X''</sub>-modules form a [[monoid]] under the operation of tensor product of ''O''<sub>''X''</sub>-modules.  The [[identity element]] for this operation is ''O''<sub>''X''</sub> itself.  Invertible sheaves are the invertible elements of this monoid.  Specifically, if ''L'' is a sheaf of ''O''<sub>''X''</sub>-modules, then ''L'' is called '''invertible''' if it satisfies any of the following equivalent conditions:<ref>EGA 0<sub>I</sub>, 5.4.</ref><ref>[[Stacks Project]], tag 01CR, [https://stacks.math.columbia.edu/tag/01CR].</ref>
* There exists a sheaf ''M'' such that <math>L \otimes_{\mathcal{O}_X} M \cong \mathcal{O}_X</math>.
* The natural homomorphism <math>L \otimes_{\mathcal{O}_X} L^\vee \to \mathcal{O}_X</math> is an isomorphism, where <math>L^\vee</math> denotes the dual sheaf <math>\underline{\operatorname{Hom}}(L, \mathcal{O}_X)</math>.
* The functor from ''O''<sub>''X''</sub>-modules to ''O''<sub>''X''</sub>-modules defined by <math>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 bundle]]s, to the point where the two are sometimes conflated.

==Examples==
Let ''X'' be an affine scheme {{math|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 ideal]]s of [[algebraic number fields]], since these are rank one projective modules over the rings of integers of the number field.

==The Picard group==
{{main|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>\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==
* [[First Chern class]]
* [[Birkhoff-Grothendieck theorem]]

==References==
{{Reflist}}
*{{EGA|book=I}}

[[Category:Geometry of divisors]]
[[Category:Sheaf theory]]