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SPRING 2014 Examples of non-Noetherian domains inside power series rings
William Heinzer, Christel Rotthaus, Sylvia Wiegand
J. Commut. Algebra 6(1): 53-93 (SPRING 2014). DOI: 10.1216/JCA-2014-6-1-53

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.

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William Heinzer. Christel Rotthaus. Sylvia Wiegand. "Examples of non-Noetherian domains inside power series rings." J. Commut. Algebra 6 (1) 53 - 93, SPRING 2014. https://doi.org/10.1216/JCA-2014-6-1-53

Information

Published: SPRING 2014
First available in Project Euclid: 2 June 2014

zbMATH: 1304.13003
MathSciNet: MR3215561
Digital Object Identifier: 10.1216/JCA-2014-6-1-53

Subjects:
Primary: 13A15 , 13B35 , 13J10

Keywords: Noetherian and non-Noetherian integral domains , Power series

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

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Vol.6 • No. 1 • SPRING 2014
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