A remark on two-dimensional local rings with the property of approximation

Abstract

Let A be a two dimensional regular local ring such that (,) Q(A)~-+Q(A) is an extension of infinite transcendence degree ("A" denotes the completion of A and Q(A) is the fraction field of A). A result of Pfister (see [4] or Proposition 6.6 [2]) says that: A is henselian universally japanese iff A has the property of approximation ("A has the property of approximation" means that every system of polynomial equations with coefficients in A has a solution in /1 iff it has one in A). The aim of this work is to show that (.) is superfluous (Theorem 6). Let A' be a noetherian normal domain and P the set of height one prime ideals of A'. If q~P then A'q is a discrete valuation ring, Let ._acA' be a non-zero ! ! ideal. Then aA'q is a power of qAq, i.e. we have _aA'q= q~,(a) Aq. In general, we define e(a)= ~ eq(a). This is a natural number because eq(a)>0 just for a finite