Extensions of Valuations to the Henselization and Completion

Abstract

We show that if R is a local domain which is dominated by a valuation $\nu $ , then there does not always exist a regular local ring $R^{\prime }$ which birationally dominates R and is dominated by v and an extension of $\nu $ to the Henselization $(R^{\prime })^{h}$ of $R^{\prime }$ such that the associated graded rings of $R^{\prime }$ and $(R^{\prime })^{h}$ along the valuations are equal. We also show that there does not always exist $R^{\prime }$ , a prime ideal p of the completion of $\widehat R^{\prime }$ such that $p^{}\cap R^{\prime }=(0)$ and an extension of $\nu $ to $\widehat R^{\prime }$ such that the associated graded rings of $R^{\prime }$ and $R^{\prime }/p$ along the valuation are equal.