Completions of Extremely Noncatenary Noetherian UFDs
Let $T$ be a complete local ring. We present necessary and sufficient conditions for $T$ to be the completion of a local (Noetherian) unique factorization domain $A$ such that there exist height one prime ideals $\{J_k\}_{k = 1}^{\infty}$ of $A$ satisfying the following conditions: (1) $J_k = J_{\ell}$ if and only if $k = \ell$, (2) there exist positive integers $n \neq m$ such that for each $k \in \mathbb{N}$, there are two saturated chains of prime ideals of $A$ of the form $J_k \subsetneq J^{(1)}_{k,2} \subsetneq \cdots \subsetneq J^{(1)}_{k,n - 1} \subsetneq M$ and $J_k \subsetneq J^{(2)}_{k,2} \subsetneq \cdots \subsetneq J^{(2)}_{k,m - 1} \subsetneq M,$ where $M$ is the maximal ideal of $A$, and (3) the prime ideals from condition (2) satisfy $J^{(i)}_{k,a} = J^{(j)}_{\ell,b}$ if and only if $i = j$, $k = \ell$, and $a = b$. We also find sufficient conditions for $T$ to be the completion of a local (Noetherian) unique factorization domain $B$ such that $B/J$ is not catenary for all height one prime ideals $J$ of $B$.