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, ${\displaystyle M_{P}\cong R_{P}}$ 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. ${\displaystyle M\otimes _{R}M^{*}\cong R}$.

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