Integral closures of ideals in completions of regular local domains

Details Integral closures of ideals in completions of regular local domains Integral closures of ideals in completions of regular local domains

2004-05-16

arxiv.org/abs/math/0405315v1

Mathematics

Commutative Algebra

9 pages

Scientific paper

In this paper we exhibit an example of a three-dimensional regular local domain (A, n) having a height-two prime ideal P with the property that the extension PA^ of P to the n-adic completion A^ of A is not integrally closed. We use a construction we have studied in earlier papers: For R=k[x,y,z], where k is a field of characteristic zero and x,y,z are indeterminates over k, the example A is an intersection of the localization of the power series ring k[y,z][[x]] at the maximal ideal (x,y,z) with the field k(x,y,z,f, g) where f, g are elements of (x,y,z)k[y,z][[x]] that are algebraically independent over k(x,y,z). The elements f, g are chosen in such a way that using results from our earlier papers A is Noetherian and it is possible to describe A as a nested union of rings associated to A that are localized polynomial rings over k in five variables.

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