Skip to main content
SpringerLink
Log in

Valuations and diameters of milnor K-rings

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Given a field F and a subgroup S of F x containing −1, we define a graph on F x/S associated with the relative Milnor K-ring K M* (F)/S. We prove that if the diameter of this graph is at least 4, then there exists a valuation v on F such that S is v-open. This is done by adopting to our setting a construction in a noncommutative setting due to Rapinchuk, Segev and Seitz. We study the behavior of the diameter under important K-theoretic constructions, and relate it to the elementary type conjecture. Finally, we provide an example showing that the above bound 4 is sharp.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Kr. Arason, R. Elman and B. Jacob, Rigid elements, valuations, and realization of Witt rings, Journal of Algebra 110 (1987), 449–467.

    Article  MathSciNet  Google Scholar 

  2. V. Bergelson and D. B. Shapiro, Multiplicative subgroups of finite index in a ring, Proceedings of the American Mathematical Society 116 (1992), 885–896.

    Article  MathSciNet  Google Scholar 

  3. I. Efrat, Construction of valuations from K-theory, Mathematical Research Letters 6 (1999), 335–343.

    Article  MathSciNet  Google Scholar 

  4. I. Efrat, Demuškin fields with valuations, Mathematische Zeitschrift 243 (2003), 333–353.

    Article  MathSciNet  Google Scholar 

  5. I. Efrat, Quotients of Milnor K-rings, orderings, and valuations, Pacific Journal of Mathematics 226 (2006), 259–276.

    Article  MathSciNet  Google Scholar 

  6. I. Efrat, Valuations, Orderings, and Milnor K-Theory, Mathematical Surveys and Monographs 124, American Mathematical Society, Providence, RI, 2006.

    Google Scholar 

  7. I. Efrat, Compatible valuations and generalized Milnor K-theory, Transactions of the American Mathematical Society 359 (2007), 4695–4709.

    Article  MathSciNet  Google Scholar 

  8. Y. S. Hwang and B. Jacob, Brauer group analogues of results relating the Witt ring to valuations and Galois theory, Canadian Journal of Mathematics 47 (1995), 527–543.

    Article  MathSciNet  Google Scholar 

  9. B. Jacob, On the structure of Pythagorean fields, Journal of Algebra 68 (1981), 247–267.

    Article  MathSciNet  Google Scholar 

  10. S. Lang, Algebra, 2nd ed., Addison-Wesley Publishing Company Advanced Book Program, Reading, MA, 1984.

    MATH  Google Scholar 

  11. M. Marshall, The elementary type conjecture in quadratic form theory, in Algebraic and Arithmetic Theory of Quadratic Forms (Universidad de Talca, 2002), Contemporary Mathematics 344, American Mathematical Society, Providence, RI, 2004, pp. 275–295.

    Google Scholar 

  12. J. Milnor, Algebraic K-theory and quadratic forms, Inventiones Mathematicae 9 (1969/1970), 318–344.

    Article  MathSciNet  Google Scholar 

  13. A. S. Rapinchuk and Y. Segev, Valuation-like maps and the congruence subgroup property, Inventiones Mathematicae 144 (2001), 571–607.

    Article  MathSciNet  Google Scholar 

  14. A. S. Rapinchuk, Y. Segev and G. M. Seitz, Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable, Journal of the American Mathematical Society 15 (2002), 929–978.

    Article  MathSciNet  Google Scholar 

  15. Y. Segev, On finite homomorphic images of the multiplicative group of a division algebra, Annals of Mathematics. Second Series 149 (1999), 219–251.

    Article  MathSciNet  Google Scholar 

  16. G. Turnwald, Multiplicative subgroups of finite index in rings, Proceedings of the American Mathematical Society 120 (1994), 377–381.

    Article  MathSciNet  Google Scholar 

  17. R. Ware, Valuation rings and rigid elements in fields, Canadian Journal of Mathematics 33 (1981), 1338–1355.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Be’er-Sheva, 84105, Israel

    Ido Efrat

Authors
  1. Ido Efrat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ido Efrat.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Efrat, I. Valuations and diameters of milnor K-rings. Isr. J. Math. 172, 75–92 (2009). https://doi.org/10.1007/s11856-009-0064-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-009-0064-3

Keywords

Search

Navigation