Extensions of Valuations to the Henselization and Completion

Acta Mathematica Vietnamica volume 44, pages159–172(2019)Cite this article

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.

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Funding

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

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Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Steven Dale Cutkosky

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  1. Steven Dale Cutkosky
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Correspondence to Steven Dale Cutkosky.

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Cutkosky, S.D. Extensions of Valuations to the Henselization and Completion. Acta Math Vietnam 44, 159–172 (2019). https://doi.org/10.1007/s40306-018-0267-y

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Keywords

  • Valuation
  • Local ring
  • Henselization
  • Comletion

Mathematics Subject Classification (2010)

  • 14B05
  • 14B22
  • 13B10
  • 11S15