User:LkNsngth/Invertible module

In mathematics, particularly commutative algebra, an invertible module is intuitivley a module that has an inverse with respect to the tensor product. They form the foundation for definitions of "invertible sheaves" in algebraic geometry. Formally, if R is a ring and M is a finitely generatedR-module, then M is said to be invertible if I is locally a free module of rank 1. In other words $$ M_P\cong R_P $$ for all primes P of R. Now, if M is an invertible R-module, then its dual M* = Hom(M,R) is its inverse with respect to the tensor product, i.e. $$I\otimes _R I^*\cong R$$. The theory of invertible modules is closely related to the theory of codimension one varieties including the theory of divisors.