Extensions of Valuations to the Henselization and Completion

TL;DR

The paper investigates the limitations of extending valuations to Henselizations and completions of local domains, showing that certain desirable properties of associated graded rings do not always hold in these extensions.

Contribution

It demonstrates the non-existence of certain regular local rings and extensions of valuations that preserve associated graded rings in Henselizations and completions.

Findings

Existence of regular local rings dominating R with valuation extensions is not guaranteed.

Associated graded rings may differ after extending valuations to Henselizations.

Counterexamples show limitations in extending valuations while preserving graded ring structures.

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