Almost excellent unique factorization domains

Fleming, Sarah M.; Loepp, Susan

doi:10.2140/involve.2020.13.165

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Almost excellent unique factorization domains

Sarah M. Fleming and Susan Loepp

Vol. 13 (2020), No. 1, 165–180

DOI: 10.2140/involve.2020.13.165

Abstract

Let $(T, 𝔪)$ be a complete local (Noetherian) domain such that $depth T > 1$ . In addition, suppose $T$ contains the rationals, $| T | = | T ∕ 𝔪 |$ , 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 $𝔮$ of $A$ such that ${(T ∕ (𝔮 \cap A) T)}_{𝔮}$ 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

Milestones

Received: 2 September 2019

Revised: 10 December 2019

Accepted: 10 December 2019

Published: 4 February 2020

Communicated by Scott T. Chapman

Authors

Sarah M. Fleming
Williams College Williamstown, MA United States

Susan Loepp
Department of Mathematics and Statistics Williams College Williamstown, MA United States

Vol. 13, No. 1, 2020