Exercise 8, § 5

Let $A$ be a ring and $M$ a projective $A$-module of rank $n$. Show that there exists a finitely generated (commutative) $A$-algebra $B$ such that the $A$-module

B is faithfully flat and $M_{(B)} = M \otimes_A B$ is a free B-module of rank $n$ (use Theorem 1 (d) and Proposition 3). Obtain the converse.