Proposition 7

Let $A$ be a commutative ring, $B$ a ring, $\rho : A \to B$ a central homomorphism and $E, F$ two $A$-modules.

(i) If B is a projective (resp. finitely generated projective) A-module, the homomorphism (17) is injective* (resp. bijective).

(ii) If E is a finitely generated projective A-module, the homomorphism* (17) is bijective.

As (16) is bijective, the proposition follows from § 4, no. 2. Proposition 2, applied to the canonical homomorphism (15).