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Acta Mathematica Vietnamica

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

Extensions of Valuations to the Henselization and Completion

  • Steven Dale CutkoskyEmail author
Article
  • 25 Downloads

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.

Keywords

Valuation Local ring Henselization Comletion 

Mathematics Subject Classification (2010)

14B05 14B22 13B10 11S15 

Notes

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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