Exercise 12, § 16

Let A be a commutative ring. Endow the set of isomorphism classes of Azumaya A-algebras (VIII, p. 270, Exercise 8) with the structure of a monoid by setting cl(S) cl(T) = cl(S $\otimes_AT$).

a) Prove that the quotient monoid for Morita equivalence is an abelian group; we denote it by Br(A) and call it the Brauer group of A (when A is a field, this definition coincides with that given in VIII, p. 279). Denote by [S] the class of an Azumaya A-algebra S in Br(A). The identity element of Br(A) is the class of A, and the inverse of an element [S] is $[S^o]$.

b) Let S be an Azumaya A-algebra. We have [S] = [A] if and only if there exists a finitely generated, projective, faithful A-module P such that S is isomorphic to End$_A(P)$.

c) Let B be a commutative A-algebra. Define a group homomorphism $r_{B/A}:$ Br(A) $\rightarrow$ Br(B) such that $r_{B/A}([S]) = [S_{(B)}]$ for every Azumaya A-algebra S. We denote the kernel of this homomorphism by Br(B$/A$).