Exercise 16, § 1

The notation of the preceding exercise is retained, and $B$ is identified with the quotient of the polynomial ring $A[U]$ by the prime ideal $p$ generated by $U^2 + U - \pi T(\pi T - 1)$. Let $q_i$ be the ideal of $A[U]$ such that $n_i = q_i/p$. Then $\mathrm{ht}(q_1) = 3$, $\mathrm{ht}(p) = 1$ and the chain $p \subset q_1$ is saturated. In particular, $A[U]$ and $A[U]_{q_1}$ are not catenary.