Almost excellent unique factorization domains

Abstract

Let $\left(T,\mathfrak{𝔪}\right)$ be a complete local (Noetherian) domain such that $depthT>1$. In addition, suppose $T$ contains the rationals, $\left|T\right|=|T∕\mathfrak{𝔪}|$, and the set of all principal height-1 prime ideals of $T$ has the same cardinality as $T$. We construct a universally catenary local unique factorization domain $A$ such that the completion of $A$ is $T$ and such that there exist uncountably many height-1 prime ideals $\mathfrak{𝔮}$ of $A$ such that ${\left(T∕\left(\mathfrak{𝔮}\cap A\right)T\right)}_{\mathfrak{𝔮}}$ is a field. Furthermore, in the case where $T$ is a normal domain, we can make $A$ “close” to excellent in the following sense: the formal fiber at every prime ideal of $A$ of height not equal to 1 is geometrically regular, and uncountably many height-1 prime ideals of $A$ have geometrically regular formal fibers.