Examples of non-Noetherian domains inside power series rings

Abstract

Given a power series ring $R^*$ over a Noetherian integral domain $R$ and an intermediate field $L$ between $R$ and the total quotient ring of $R^*$, the integral domain $A = L \cap R^*$ often (but not always) inherits nice properties from $R^*$ such as the Noetherian property. For certain fields $L$ it is possible to approximate $A$ using a localization $B$ of a particular nested union of polynomial rings over $R$ associated to $A$; if $B$ is Noetherian, then $B = A$. If $B$ is not Noetherian, we can sometimes identify the prime ideals of $B$ that are not finitely generated. We have obtained in this way, for each positive integer $m$, a three-dimensional local unique factorization domain $B$ such that the maximal ideal of $B$ is two-generated, $B$ has precisely $m$ prime ideals of height~2, each prime ideal of $B$ of height~2 is not finitely generated and all the other prime ideals of $B$ are finitely generated. We examine the structure of the map $\text{Spec\,} A \to \text{Spec\,} B$ for this example. We also present a generalization of this example to dimension four. This four-dimensional, non-Noetherian local unique factorization domain has exactly one prime ideal $Q$ of height three, and $Q$ is not finitely generated.