Exercise 8, § 14

Let A be a commutative ring and S an A-algebra; suppose that S is a projective, faithful, and finitely generated A-module. Denote the A-algebra $S\otimes_AS^o$ by E.

a) Prove that the following conditions are equivalent:

(i) The $(E$, A)-bimodule S is invertible.

(ii) The A-algebra S is central, and the E-module S is projective.

(iii) The canonical homomorphism $E\longrightarrow$ End$_A(S)$ is bijective.

(iv) For every maximal ideal $\mathfrak{m}$ of A, the $A/\mathfrak{m}$-algebra $S/\mathfrak{m}S$ is central simple. (Deduce the equivalence of (i), (ii), and (iii) from Theorem 1 of VIII, p. 101 and Corollary 2 of VIII, p. 100 and that of (iii) and (iv) from Theorem 1 of VIII, p. 252 and Exercise 7).

When these conditions are satisfied, we say that S is an Azumaya algebra over A.

b) If S and T are Azumaya algebras over A, then so is the A-algebra $S\otimes_AT$.

c) Let B be a commutative A-algebra; if S is an Azumaya algebra over A, then the B-algebra $S_{(B)}$ is an Azumaya algebra. If the B-algebra $S_{(B)}$ is an Azumaya algebra and B is a faithful and projective $(^*$or faithfully flat$_*)$ A-algebra, then S is an Azumaya A-algebra (apply Exercise 7, d)).