Extensions of valuations to the completion of a local domain

Abstract

Let (R, m) be a normal local domain of dimension n > 1. Suppose that R is analytically irreducible, i.e. that ii, the m-adic completion of R, is a domain. Given a valuation domain V birationally dominating R, what can be said about valuation domains W which extend V and birationally dominate Ai? In [l, Lemma 131, Abhyankar shows that V has an extension W which birationally dominates k. Here we discuss the uniqueness of such an extension. For this, we will assume, in addition, that i is normal, i.e., that R is analytically normal and that R satisfies a property, defined below, which is weaker than excellence. Matsumura’s work on dimensions of formal fibers was a motivating influence for the questions we consider on extending valuations from R to ff. In [lo], Matsumura asks if the dimension of the generic formal fiber, i.e., the fiber over 0 in the embedding R+ fi’, can be a positive integer other than 0, y1 2 or n 1. Since extensions of rank 1 valuations to i which grow in rank produce nonzero primes in the generic formal fiber, we initiated a study of these extensions hoping to shed some light on Matsumura’s question.’