Characterization of completions of noncatenary local domains and noncatenary local UFDs

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.

Cardinalities of Prime Spectra of Precompletions

“…First, we state a result about the cardinality of spectra in general. The following lemma and remark are Lemma 2.3 and Remark 2.4 from [1], adapted from [3]. Lemma 3.8 ([1] Lemma 2.3).…”

“…Remark 3.9. It is noted in [1] that, for the T and A in Lemma 3.8, there is a bijection between the nonzero prime ideals of A and the prime ideals of T that are not in G.…”

Maximal chains of prime ideals of different lengths in unique factorization domains

“…We are now in a position to construct the desired UFD. As alluded to before, the proof of Theorem 3.4 is largely based on that of Theorem 2.6 from [1], which in turn is based on the proof of the main theorem in [5].…”

“…We will show that, in fact, almost all prime ideals which satisfy (1) also satisfy (2) and (3). We begin by showing that there are uncountably many prime ideals that satisfy (1).…”

“…This was the only example of a noncatenary Noetherian UFD in the literature until 2017, when Avery, et. al. characterized the completions of noncatenary UFDs in [1], which generated a new class of examples. As a corollary of this result ( [1], Proposition 4.2), Avery, et.…”