Invertible module over a commutative ring

Let $R$ be a commutative unital ring. An $R$ -module $L$ is invertible if there is an $R$ -module $M$ and an isomorphism

L\otimes_R M\cong R,

where $\otimes_R$ is the tensor product. Thus $L$ has an inverse for tensor multiplication of modules up to isomorphism.

Local description

Equivalently, $L$ is a finitely generated projective module with $L_{\mathfrak p}\cong R_{\mathfrak p}$ for every prime ideal $\mathfrak p$ . The inverse is the dual $L^\vee=\operatorname{Hom}_R(L,R)$ , and evaluation gives $L\otimes_RL^\vee\cong R$ .

For a domain, an invertible fractional ideal is an example. It need not be generated by one element globally, even though it has one generator after localization at each prime.

Picard group

The isomorphism classes of invertible modules form the Picard group of $R$ . The identity class is represented by $R$ itself.

References

The Stacks Project, Picard groups of rings, Definition 15.119.1 and Lemma 15.119.2.