Nebraska Cooperative Fish & Wildlife Research Unit
Date of this Version
Spring 2014
Citation
2014 Rocky Mountain Mathematics Consortium
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.
Included in
Aquaculture and Fisheries Commons, Environmental Indicators and Impact Assessment Commons, Environmental Monitoring Commons, Natural Resource Economics Commons, Natural Resources and Conservation Commons, Water Resource Management Commons
Comments
JOURNAL OF COMMUTATIVE ALGEBRA Volume 6, Number 1, Spring 2014