Completions of Extremely Noncatenary Noetherian UFDs

TL;DR

This paper characterizes when a complete local ring can be realized as the completion of a Noetherian UFD with specific prime ideal chain properties, and explores conditions for non-catenary completions.

Contribution

It provides necessary and sufficient conditions for constructing noncatenary and extremely noncatenary Noetherian UFDs with prescribed prime ideal chains.

Findings

Characterization of completions of certain Noetherian UFDs.

Conditions for noncatenary prime ideal chains.

Existence of UFDs with noncatenary prime ideal quotients.

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 $n \neq m$ 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 $B$.