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, M P ≅ R P {\displaystyle 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. M ⊗ R M ∗ ≅ R {\displaystyle M\otimes _{R}M^{*}\cong R} . The theory of invertible modules is closely related to the theory of codimension one varieties including the theory of divisors.
See also Picard group
References Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Springer, ISBN 978-0-387-94269-8