Tangent space to Milnor K-groups of rings

Article

Abstract

We prove that the tangent space to the (n + 1)th Milnor K-group of a ring R is isomorphic to the group of nth absolute Kähler differentials of R when the ring R contains 1/2 and has sufficiently many invertible elements. More precisely, the latter condition means that R is weakly 5-fold stable in the sense of Morrow.

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References

  1. 1.
    S. Bloch, “On the tangent space to Quillen K-theory,” in Algebraic K-Theory. I: Higher K-Theories: Proc. Conf., Battelle Mem. Inst., Seattle, WA, 1972 (Springer, Berlin, 1973)Google Scholar
  2. 1.
    S. Bloch, Lect. Notes Math. 341, pp. 205–210.Google Scholar
  3. 2.
    C. Contou-CarrÈre, “Jacobienne locale, groupe de bivecteurs de Witt universel, et symbole modéré,” C. R. Acad. Sci. Paris, Sér. I, 318 (8), 743–746 (1994).Google Scholar
  4. 3.
    C. Contou-CarrÈre, “Jacobienne locale d’une courbe formelle relative,” Rend. Semin. Mat. Univ. Padova 130, 1–106 (2013).MathSciNetCrossRefGoogle Scholar
  5. 4.
    B. F. Dribus, “A Goodwillie-type theorem for Milnor K-theory,” arXiv: 1402.2222 [math.KT].Google Scholar
  6. 5.
    T. G. Goodwillie, “Relative algebraic K-theory and cyclic homology,” Ann. Math., Ser. 2, 124 (2), 347–402 (1986).Google Scholar
  7. 6.
    S. O. Gorchinskiy and D. V. Osipov, “Explicit formula for the higher-dimensional Contou-CarrÈre symbol,” Usp. Mat. Nauk 70 (1), 183–184 (2015)Google Scholar
  8. 6.
    S. O. Gorchinskiy and D. V. Osipov, Russ. Math. Surv. 70, 171–173 (2015)].CrossRefGoogle Scholar
  9. 7.
    S. O. Gorchinskiy and D. V. Osipov, “Higher Contou-CarrÈre symbol: Local theory,” Mat. Sb. 206 (9), 21–98 (2015).CrossRefGoogle Scholar
  10. 8.
    W. van der Kallen, “Le K2 des nombres duaux,” C. R. Acad. Sci. Paris, Sér. A 273, 1204–1207 (1971).Google Scholar
  11. 9.
    W. van der Kallen, “The K2 of rings with many units,” Ann. Sci. éc. Norm. Supér., Sér. 4, 10 (4), 473–515 (1977).Google Scholar
  12. 10.
    M. Kerz, “The Gersten conjecture for Milnor K-theory,” Invent. Math. 175 (1), 1–33 (2009).MathSciNetCrossRefGoogle Scholar
  13. 11.
    M. Morrow, “K2 of localisations of local rings,” J. Algebra 399, 190–204 (2014).MathSciNetCrossRefGoogle Scholar
  14. 12.
    Yu. P. Nesterenko and A. A. Suslin, “Homology of the full linear group over a local ring, and Milnor’s K-theory,” Izv. Akad. Nauk SSSR, Ser. Mat. 53 (1), 121–146 (1989)MATHGoogle Scholar
  15. 12.
    Yu. P. Nesterenko and A. A. Suslin, Math. USSR, Izv. 34 (1), 121–145 (1990)].MATHMathSciNetCrossRefGoogle Scholar
  16. 13.
    A. A. Suslin and V. A. Yarosh, “Milnor’s K3 of a discrete valuation ring,” in Algebraic K-Theory (Am. Math. Soc., Providence, RI, 1991)Google Scholar
  17. 13.
    A. A. Suslin and V. A. Yarosh, Adv. Sov. Math. 4, pp. 155–170.Google Scholar

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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

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