Constructing Noncatenary Quasi-Excellent Precompletions

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.