Proposition 3

Let A be a commutative ring. Every finitely generated and faithful projective A-module is generating. More generally, a finitely generated projective A-module P is a generating $A_P$-module.

Let P be a finitely generated projective A-module. We have $\tau (P)P = P$ (VIII, p. 80).

If the A-module P is faithful, then the ideal $\tau (P)$ is equal to A (Lemma 1) and the A-module P is generating (Theorem 1); this leads to the first assertion. The second follows by Lemma 2 below.