Completions of Extremely Noncatenary Noetherian UFDs

Abstract

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 $n\ne m$ such that for each $k\in \mathbb{N}$, there are two saturated chains of prime ideals of $A$ of the form $J_k⊊J_{k,2}^{(1)}⊊\cdots ⊊J_{k,n-1}^{(1)}⊊M$ and $J_k⊊J_{k,2}^{(2)}⊊\cdots ⊊J_{k,m-1}^{(2)}⊊M,$ where $M$ is the maximal ideal of $A$, and (3) the prime ideals from condition (2) satisfy $J_{k,a}^{(i)}=J_{\ell ,b}^{(j)}$ 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 $B$.

arXiv comment18 pages