Exercise 18, § 1

Let $A = \bigoplus_{n \in \mathbf{Z}} A_n$ be a graded Krull domain with positive degrees, such that $A_0$ is a field and let $m = \bigoplus_{n \gg 1} A_n$ a maximal ideal of A. Let S be the multiplicative set of homogeneous elements $\neq 0$ of A, so that the ring $S^{-1}A$ is a principal ideal domain (Exercise 16 (b)). (a) Let $p$ be a prime ideal of A of height 1 which meets $A - m$; then $p$ is not graded (otherwise $p = A$) and the ideal $S^{-1}p$ of $S^{-1}A$ is principal, generated by an element of the form $x = 1 + x_1 + \ldots + x_r$, where $x_j \in (S^{-1}A)_j$. Writing an element of p of the form $1 + a_1 + \ldots + a_r$, ($a_i \in A_i$) as a multiple of $x$ in $S^{-1}A$ and using Chapter V, § 1, no. 3, Proposition 11, show that $x \in p$; writing finally every element of p as a multiple of $x$ in $S^{-1}A$, show that $p = Ax$. (b) Deduce from (a) that $C(A)$ and $C(A_m)$ are canonically isomorphic (use Proposition 17 of no. 10).