Extensions of Valuations to the Henselization and Completion
Steven Dale Cutkosky

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 is a local domain which is dominated by a valuation , then there does not always exist a regular local ring which birationally dominates R and is dominated by and an extension of to the Henselization of such that the associated graded rings of and along the valuations are equal. We also show that there does not always exist , a prime ideal of the completion such that and an extension of to such that the associated graded rings of and along the valuation are equal.
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