Invertible module

In mathematics, particularly commutative algebra, an invertible module is intuitively a module that has an inverse with respect to the tensor product. Invertible modules form the foundation for the definition of invertible sheaves in algebraic geometry.
Formally, a finitely generated module "M" over a ring "R" is said to be invertible if it is locally a free module of rank 1. In other words, formula_1 for all primes "P" of "R". Now, if "M" is an invertible "R"-module, then its dual is its inverse with respect to the tensor product, i.e. formula_2.
The theory of invertible modules is closely related to the theory of codimension one varieties including the theory of divisors.