#### Vol. 13, No. 1, 2020

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Almost excellent unique factorization domains

### Sarah M. Fleming and Susan Loepp

Vol. 13 (2020), No. 1, 165–180
##### Abstract

Let $\left(T,\mathfrak{𝔪}\right)$ be a complete local (Noetherian) domain such that $depthT>1$. In addition, suppose $T$ contains the rationals, $|T|=|T∕\mathfrak{𝔪}|$, 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.

##### Keywords
completions of local rings, excellent rings, unique factorization domains
##### Mathematical Subject Classification 2010
Primary: 13F15, 13F40
Secondary: 13B35, 13J10