The Stacks project

Exercise 111.40.3. Algebra.

1. Show that an invertible ${\mathcal O}_ X$-module on an affine scheme $\mathop{\mathrm{Spec}}(A)$ corresponds to an $A$-module $M$ which is (i) finite, (ii) projective, (iii) locally free of rank 1, and hence (iv) flat, and (v) finitely presented. (Feel free to quote things from last semesters course; or from algebra books.)

2. Suppose that $A$ is a domain and that $M$ is a module as in (a). Show that $M$ is isomorphic as an $A$-module to an ideal $I \subset A$ such that $IA_{\mathfrak p}$ is principal for every prime ${\mathfrak p}$.