Kummer theory: Difference between revisions

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In [[mathematics]], an '''invertible sheaf''' is a [[coherent sheaf]] ''S'' on a [[ringed space]] ''X'', for which there is an inverse ''T'' with respect to [[tensor product]] of ''O''<sub>''X''</sub>-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==


An '''invertible sheaf''' is a [[coherent sheaf]] ''S'' on a [[ringed space]] ''X'', for which there is an inverse ''T'' with respect to [[tensor product]] of ''O''<sub>''X''</sub>-modules, that is, we have
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:<math>S \otimes T\ </math>
 
isomorphic to ''O''<sub>''X''</sub>, which acts as [[identity element]] for the tensor product. The most significant cases are those coming from [[algebraic geometry]] and [[complex manifold]] theory. The invertible sheaves in those theories are in effect the [[line bundle]]s appropriately formulated.
 
In fact, the abstract definition in [[scheme theory]] of invertible sheaf can be replaced by the condition of being ''locally free, of rank 1''. That is, the condition of a tensor inverse then implies, locally on ''X'', that ''S'' is the sheaf form of a free rank 1 module over a [[commutative ring]]. Examples come from [[fractional ideal]]s in [[algebraic number theory]], so that the definition captures that theory. More generally, when ''X'' is an [[affine scheme]] ''Spec(R)'', the invertible sheaves come from [[projective module]]s over ''R'', of rank 1.
 
==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>\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==
*Section 0.5.4 of {{EGA|book=I}}
 
[[Category:Geometry of divisors]]
[[Category:Sheaf theory]]

Latest revision as of 14:11, 10 August 2014


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