Exercise 15, § 16

a) Let A be a regular integral domain (AC, X, §4, n$^o2$, p. 55) and K its field of fractions. The homomorphism $r_{K/A}:$ Br(A) $\rightarrow$ Br(K) is injective (VIII, p. 271, Exercise 11)$.*$

b) Prove that the ring $A =\mathbf{Q}[X,Y]/(X^2+ Y^2)$ is an integral domain; let K be its field of fractions. Let H be the quaternion algebra of type $(-1,0,-1)$ over $\mathbf{Q}$, which is a field (cf. III, §2, No. 5, p. 445). Prove that the class of the Azumaya A-algebra $H_{(A)}$ in Br(A) is not zero but that $H_{(K)}$ is isomorphic to $\mathbf{M}_2$(K), so that the homomorphism $r_{K/A}$ is not injective (observe that K contains an element of square $-1$).