Exercise 16, § 5

a) Let A be a ring (commutative), $A'$ a commutative a-algebra which is a flat a-module, B an a-algebra. Prove that, if B is an Azumaya algebra, $B' = A' \otimes_A B$ is an $A'$-Azumaya algebra; the converse is true if one assumes that $A'$ is a faithfully flat a-module (cf. Alg., chap. II, 3rd ed., § 5, No. 1, cor. of Prop. 4 and Alg. comm., chap. I, § 2, No. 9, Prop. 10 and § 3, No. 6, Prop. 12). b) Prove that the tensor product $B \otimes_A C$ of two Azumaya a-algebras B, C is an Azumaya a-algebra (cf. Alg., chap. VIII, § 7, No. 4, cor. 2 of Th. 2).