Exercise 7, § 4

Let A be a Krull ring and P the set of its prime ideals of height 1. Suppose that for every prime ideal $p \in P$, the ring $A/p$ is noetherian, and denote by K the field of fractions of A. Prove that A is noetherian. (Let $p \in P$ and $x \in K$ be such that $v_p(x) = 1$ and $v_q(x) \leq 0$ for $q \in P - \{p\}$. Put $B = A[x]$. Then one has the following properties: (i) $p = xB \cap A$; (ii) the inclusion of A in B induces, by passing to quotients, an isomorphism of $A/p$ onto $B/xB$; (iii) for every integer $n \geq 0$, the ring $B/x^nB$ is noetherian ; (iv) for every integer $n \geq 0$, the ring $A/(x^nB \cap A)$ is noetherian (use Exercise 6); (v) for every integer $n \geq 0$, the A-module $A/p^{(n)}$ is noetherian. Recall that one puts $p^{(n)} = A \cap p^nA_p$, cf. IV, § 2, Exercise 18.)