Exercise 7, § 5

Let $A$ be an absolutely flat (commutative) ring (cf. § 4, Exercise 16) and let $X$ be its spectrum. (a) Let $F$ be a closed subset of $X$ and $a$ the corresponding ideal of $A$ (loc. cit.). Show that $M = A/a$ is a monogenous flat $A$-module such that $M$ is a free $A_m$-module for all $m \in X$, but that $M$ is projective only if $F$ is open. (b) Let $E$ be the module the direct sum of the $A/m$, for $m \in X$. Show that is a flat $A$-module such that $E_m$ is a free $A_m$-module of rank 1 for all $m \in X$; for $E$ to be a projective $A$-module, it is necessary and sufficient that $X$ be finite (note that, if $A/m$ is a projective $A$-module, $m$ is finitely generated).