Noncatenary Unique Factorization Domains

Abstract

We demonstrate a class of local (Noetherian) unique factorization domains (UFDs) that are noncatenary at infinitely many places. In particular, if $A$ is in our class of UFDs, then the prime spectrum of $A$ contains infinitely many disjoint (except at the maximal ideal) noncatenary subsets. As a consequence of our result, there are infinitely many height one prime ideals $P$ of $A$ such that $A/P$ is not catenary. We also construct a countable local UFD $A$ satisfying the property that for every height one prime ideal $P$ of $A$, $A/P$ is not catenary.