Exercise 17, § 5

Let A be a ring and P a faithful finitely generated projective A-module.

(a) Show that the A-algebra $B = \mathrm{End}_A(P)$ is an Azumaya algebra. (Observe first that the A-module B is isomorphic to $P \otimes_A P^*$ and hence finitely presented, then use , no. 7, Proposition 19.)

(b) Show that P is a left projective B-module (use Theorem 1 (d) and § 3, Exercise 8 (c)).

(c) Let $P'$ be another faithful finitely generated projective A-module. Show that $P \otimes_A P'$ is a faithful finitely generated projective A-module and that $\mathrm{End}_A(P \otimes_A P')$ is canonically identified with $\mathrm{End}_A(P) \otimes_A \mathrm{Eng}_A(P')$ (use Theorem 1 (e), § 2, no. 7, Propositions 18 and 19 and § 3, no. 3, Theorem 1). Consider the case where $P'$ is of rank 1.