Exercise 5, § 5

Let $A$ be a reduced ring and $M$ a finitely generated $A$-module. Suppose that there exists an integer $n > 0$ such that, for every homomorphism $\phi$ of $A$ to a field $K$, $[M \otimes_A K; K_\phi] = n$. Show that $M$ is a projective module of rank $n$. (Reduce it to the case where $A$ is a local ring and use § 2, no. 2, Proposition 7.)