Exercise 16, § 1

Let $A = \bigoplus_{n \in \mathbf{Z}} A_n$ be a graded Krull domain. Let $D_g(A)$ (resp. $F_g(A)$) denote the group generated by the divisors of the graded integral divisorial ideals (resp. by the divisors $\operatorname{div}(a)$, where $a$ is homogeneous in A); write $C_g(A) = D_g(A)/F_g(A)$. Let S be the multiplicative subset of homogeneous elements $\neq 0$ of A.

(a) Show that every prime ideal of A of height 1 which meets S is graded (use Chapter III, § 1, no. 4, Proposition 5).

(b) Let $B = S^{-1}A$, which is a graded ring (Chapter II, § 2, no. 9); write $B = \bigoplus_{n \in \mathbf{Z}} B_n$; show that $B_0$ is a field and that, if $A \neq A_n$, B is isomorphic to the ring $B_0[X, X^{-1}]$ of rational functions $P(X)/X^k$, where $P(X) \in B_0[X]$.

(c) Show that $C_g(A)$ and $C(A)$ are canonically isomorphic (use (a), (b) and equation (5) of no. 10).