Integral closures of ideals in completions of regular local domains

Abstract

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.