Extensions of Valuations to the Henselization and Completion

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.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Abhyankar, S.: Local uniformization on algebraic surfaces over ground fields of characteristic \(p\neq 0\). Ann. of Math. (2) 63(2), 491–526 (1956)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Abhyankar, S.: Ramification Theoretic Methods in Algebraic Geometry. Princeton Univ. Press, Princeton (1959)

    Google Scholar 

  3. 3.

    Cutkosky, S.D.: Local factorization and monomialization of morphisms. Astérisque, 260 (1999)

  4. 4.

    Cutkosky, S.D.: Finite generation of extensions of associated graded rings along a valuation. To appear in J. Lond. Math. Soc.

  5. 5.

    Cutkosky, S.D., ElHitti, S.: Formal prime ideals of infinite value and their algebraic resolution. Ann. Fac. Sci. Toulouse Math. (6) 19(3–4), 635–649 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Cutkosky, S.D., Ghezzi, L.: Completions of valuation rings. Contemp. Math. 386, 13–34 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Cutkosky, S.D., Vinh, P.A.: Valuation semigroups of two dimensional local rings. Proc. Lond. Math. Soc. (3) 108(2), 350–384 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    ElHitti, S.: Perron transforms. Comm. Algebra 42(5), 2003–2045 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Heinzer, W., Sally, J.: Extensions of valuations to the completion of a local domain. J. Pure Appl. Algebra 71(2–3), 175–185 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Herrera Govantes, F.J., Olalla Acosta, M.A., Spivakovsky, M., Teissier, B.: Extending a valuation centered in a local domain to the formal completion. Proc. Lond. Math. Soc. (3) 105(3), 571–621 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Kashcheyeva, O.: Constructing examples of semigroups of valuations. J. Pure Appl. Algebra 220(12), 3826–3860 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Lang, S.: Algebra. Revised third edition. Springer, New York (2002)

    Google Scholar 

  13. 13.

    Moghaddam, M.: A construction for a class of valuations of the field \(k(X_{1},\ldots , X_{d},Y)\) with large value group. J. Algebra 319(7), 2803–2829 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Nagata, M.: Local Rings. Interscience publishers, New York-London (1962)

    Google Scholar 

  15. 15.

    Novacoski, J., Spivakovsky, M.: Key polynomials and pseudo-convergent sequences. J. Algebra 495, 199–219 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Spivakovsky, M.: Valuations in function fields of surfaces. Am. J. Math. 112(1), 107–156 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Teissier, B.: Valuations, deformations and toric geometry. Valuation theory and its applications II. Fields Inst. Commun. 33, 361–459 (2003). Am. Math. Soc., Providence, RI – 459

    MATH  Google Scholar 

  18. 18.

    Teissier, B.: Overweight deformations of affine toric varieties and local uniformization. Valuation Theory in Interaction, pp. 474–565. EMS Ser. Congr. Rep., Eur. Math. Soc Zürich (2014)

  19. 19.

    Zariski, O., Samuel, P.: Commutative Algebra, Volume I. D. Van Nostrand Company, Inc, Princeton (1958)

    Google Scholar 

  20. 20.

    Zariski, O., Samuel, P.: Commutative Algebra, Volume II. D. Van Nostrand Company, Inc., N.J.-Toronto-London-New York (1960)

    Google Scholar 

Download references

Funding

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Steven Dale Cutkosky.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Valuation
  • Local ring
  • Henselization
  • Comletion

Mathematics Subject Classification (2010)

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