- Invertible sheaf
In
mathematics , an invertible sheaf is acoherent sheaf "S" on aringed space "X", for which there is an inverse "T" with respect totensor product of "O""X"-modules. That is, we have:"S" ⊗ "T"
isomorphic to "O""X", which acts as
identity element for the tensor product. The most significant cases are those coming fromalgebraic geometry andcomplex manifold theory. The invertible sheaves in those theories are in effect theline 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 acommutative ring . Examples come fromfractional ideal s inalgebraic number theory , so that the definition captures that theory. More generally, when "X" is anaffine scheme "Spec(R)", the invertible sheaves come fromprojective module s over "R", of rank 1.Quite generally, the isomorphism classes of invertible sheaves on "X" themselves form an
abelian group under tensor product. This group generalises theideal class group . In general it is written:"Pic"("X")
with "Pic" the
Picard functor . Since it also includes the theory of theJacobian variety of analgebraic 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 .
Wikimedia Foundation. 2010.
