Invertible Sheaf

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_X-modules, that is, we have

isomorphic to O_X, 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 bundles 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 ideals 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 modules over R, of rank 1.