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 (T,𝔪) be a complete local (Noetherian) domain such that depthT > 1. In addition, suppose T contains the rationals, |T| = |T𝔪|, 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 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