Definition 111.22.1. Let $A$ be a ring. Recall that a *finite locally free* $A$-module $M$ is a module such that for every ${\mathfrak p} \in \mathop{\mathrm{Spec}}(A)$ there exists an $f\in A$, $f \not\in {\mathfrak p}$ such that $M_ f$ is a finite free $A_ f$-module. We say $M$ is an *invertible module* if $M$ is finite locally free of rank $1$, i.e., for every ${\mathfrak p} \in \mathop{\mathrm{Spec}}(A)$ there exists an $f\in A$, $f \not\in \mathfrak p$ such that $M_ f \cong A_ f$ as an $A_ f$-module.

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