Acta Mathematica Vietnamica

, Volume 44, Issue 1, pp 159–172 | Cite as

Extensions of Valuations to the Henselization and Completion

  • Steven Dale CutkoskyEmail author


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.


Valuation Local ring Henselization Comletion 

Mathematics Subject Classification (2010)

14B05 14B22 13B10 11S15 


Funding Information

The author was partially supported by NSF grant DMS-1700046.


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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

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