Date of this Version
2014 Rocky Mountain Mathematics Consortium
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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