Nebraska Cooperative Fish and Wildlife Research Unit

 

Date of this Version

Spring 2014

Citation

2014 Rocky Mountain Mathematics Consortium

Comments

JOURNAL OF COMMUTATIVE ALGEBRA Volume 6, Number 1, Spring 2014

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 ∩ 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 SpecA → SpecB 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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