Exercise 17, § 1

Let $A = \bigoplus_{n \in \mathbf{Z}} A_n$ be a graded Krull domain and $p \in A$, be such that $Ap$ is prime and $\neq 0$.

(a) Show that, if B is the ring $A_{(p)}$ defined in Chapter 111, § 1, Exercise 1 (a), B is a Krull domain and the groups $C(A)$ and $C(B)$ are canonically isomorphic. (Show first that $C(B)$ is isomorphic to $C(B[p, p^{-1}])$ and observe that $B[p, p^{-1}] = A[p^{-1}]$.)

(b) Under the hypotheses of Exercise 15 (b), show that the groups C(A) and C(A[X, Y]/(aX + bY)) are isomorphic.