Constructing Noncatenary Quasi-Excellent Precompletions

TL;DR

This paper establishes conditions for constructing noncatenary quasi-excellent local subrings within a given local ring, preserving key properties like completion and prime spectrum structure, with implications for understanding complex prime ideal configurations.

Contribution

It introduces a method to build noncatenary quasi-excellent subrings that preserve completion and prime spectrum properties, expanding the understanding of local ring structures.

Findings

Existence of quasi-excellent subrings with specified prime ideal properties

Construction of noncatenary prime spectrum subsets within local rings

Preservation of completion and prime spectrum structure in subring construction

Abstract

Let $T$ be a local (Noetherian) ring and let $Q_1$ and $Q_2$ be prime ideals of $T$. We find sufficient conditions for there to exist a quasi-excellent local subring $B$ of $T$ satisfying the following conditions: (1) the completion of $B$ at its maximal ideal is isomorphic to the completion of $T$ at its maximal ideal, (2) $B \cap Q_1 = B \cap Q_2$, (3) the set of prime ideals of $T/(Q_1 \cap Q_2)$ of positive height is the same as the set of prime ideals of $B/(B \cap Q_1)$ of positive height when viewed as partially ordered sets, and (4) for $i = 1$ and for $i = 2$, there is a coheight preserving bijection between the minimal prime ideals of $T_{Q_i}$ and the minimal prime ideals of $B_{B \cap Q_1}$. Intuitively, this means that $T$ contains a quasi-excellent local subring in which $Q_1$ and $Q_2$ are "glued together" and such that both the completion and desirable properties of the prime spectrum are preserved. We use this result to show that certain complete local rings are the completion of a quasi-excellent local ring whose prime spectrum, when viewed as a partially ordered set, contains interesting noncatenary finite subsets.