Exercise 12, § 1

Let A be a Noetherian ring and P a projective A-module. Show that, if, for all $p \in \operatorname{Ass}(A)$, $P_p$ is a finitely generated A,-module, then P is a finitely generated A-module (embed A in the product $\prod_{p \in \operatorname{Ass}(A)} A$, (Exercise 11) and use Algebra, Chapter II, § 5, no. 5, Proposition 9).