Exercise 16, § 5

(a) Let A be a (commutative) ring, A' a commutative A-algebra which is a flat A-module and B an A-algebra. Show that, if B is an Azumaya algebra, $B' = A' \otimes_A B$ is an Azumaya A'-algebra; the converse is true if A' is assumed to be a faithfully flat A-module (cf. Algebra, Chapter II, §5, no. 1, Corollary to Proposition 4 and Commutative Algebra, Chapter I, §2, no. 9, Proposition 10 and § 3, no. 6, Proposition 12). (b) Show that the tensor product $B \otimes_A C$ of two Azumaya A-algebras B, C is an Azumaya A-algebra (cf. Algebra, Chapter VIII, § 7, no. 4, Corollary 2 to Theorem 2).