Proposition 12

Let B be a ring and A a commutative subring of the centre of B such that B is a faithfully flat A-module. For an A-module F to be projective and finitely generated, it is necessary and sufficient that $B \otimes_A F$ be a finitely generated projective left B-module.

The condition is obviously necessary without any hypothesis on A or B (Algebra, Chapter II, § 5, no. 1, Corollary to Proposition 4); we prove that it is sufficient. If a finitely generated projective module admits a finite presentation (§ 2, no. 8, Lemma 8), the hypothesis implies that F admits a finite presentation by virtue of Proposition 11, hence, for every A-module M, there is a canonical isomorphism

(§ 2, no. 10, Proposition 11). Then let $v : M \to M''$ be a surjective A-module homomorphism and consider the commutative diagram

As $1_B \otimes v$ is surjective and $B \otimes_A F$ is assumed projective, $\operatorname{Hom}(1_{B \otimes F}, 1_B \otimes v)$ is surjective (Algebra, Chapter II, § 2, no. 2, Proposition 4) and so then is $1_B \otimes \operatorname{Hom}(1_F, v)$. But as B is a faithfully flat A-module, $\operatorname{Hom}(1_F, v)$ is itself surjective (no. 1, Proposition 2), hence F is a projective A-module (Algebra, Chapter II, § 2, no. 2, Proposition 4).