Completions of Quasi-excellent Domains

TL;DR

This paper characterizes when a complete local Noetherian ring of characteristic zero can be realized as the completion of a quasi-excellent local domain, providing conditions and exploring chain lengths of prime ideals.

Contribution

It establishes necessary and sufficient conditions for such completions, including the case with rational content, and investigates the structure of prime ideal chains in quasi-excellent domains.

Findings

Conditions for T to be the completion of a quasi-excellent local domain

Characterization of completions when T contains the rationals

Examples of noncatenary quasi-excellent local domains

Abstract

Let $T$ be a complete local (Noetherian) ring of characteristic zero. We find necessary and sufficient conditions for $T$ to be the completion of a quasi-excellent local domain. In the case that $T$ contains the rationals, we provide necessary and sufficient conditions for $T$ to be the completion of a countable quasi-excellent local domain. We also prove results regarding the possible lengths of maximal saturated chains of prime ideals of these quasi-excellent local domains, and we show that these results lead to interesting examples of noncatenary quasi-excellent local domains.