Characterization of Completions of Noncatenary Local Domains and Noncatenary Local UFDs

TL;DR

This paper characterizes when complete local rings can be realized as completions of noncatenary local domains and UFDs, revealing new classes of such domains and their properties.

Contribution

It provides necessary and sufficient conditions for completions of noncatenary local domains and UFDs, expanding understanding of their structure and examples.

Findings

Identifies conditions for a complete local ring to be a noncatenary local domain completion.

Demonstrates existence of large classes of non-excellent and non-universally catenary domains.

Shows there is no upper limit on how noncatenary a UFD can be.

Abstract

We find necessary and sufficient conditions for a complete local ring to be the completion of a noncatenary local (Noetherian) domain, as well as necessary and sufficient conditions for it to be the completion of a noncatenary local (Noetherian) unique factorization domain. We use our first result to demonstrate a large class of quasi-excellent domains that are not excellent, as well as a large class of catenary domains that are not universally catenary. We use our second result to find a larger class of noncatenary local UFDs than was previously known, and we show that there is no bound on how noncatenary a UFD can be.