Bass’ Nilpotent Unitary K1-Group of Unitary Ring

  • V. I. KopeikoEmail author

Bass’ nilpotent unitary K1-group of the unitary ring is introduced and studied. A set of unitary representatives of this group is found and a complete description of its unipotent representatives is given.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Bass, Algebraic K-theory [Russian translation], Mir, Moscow (1973).Google Scholar
  2. 2.
    H. Bass, “Some problems in classical algebraic K-theory,” Lecture Notes Math., 342, 3–73 (1973).MathSciNetzbMATHGoogle Scholar
  3. 3.
    H. Bass, “Unitary algebraic K-theory,” Lecture Notes Math., 343, 57–265 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    H. Bass, J. Milnor, and J.-P. Serre, “Solution of the congruence subgroup problem for the groups of type SLn (n ≥ 3) and Sp2n, (n ≥ 2),” Publ. Inst. Hautes Et. Sci., 33, 59–137 (1967).Google Scholar
  5. 5.
    S. E. Cappell, “Unitary nilpotent groups and hermitian K-theory,” Bull. AMS, 80, 1117–1122 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    F. Connolly and A. Ranicki, “On the calculation of UNil,” Adv. Math., 195, 205–258 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    G. Cortinas, C. Haesemeyer, M. E. Walker, and C. Weibel, “Bass’ NK groups and cdh-fibrant Hochschild homology,” Invent. Math., 89, 421–448 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. F. Davis, “Some remarks on Nil groups in algebraic K-theory,” arXiv:0803.1641v2 (2008).Google Scholar
  9. 9.
    F. T. Farrell, “The exponent of UNil,” Topology, 18, 305–312 (1979).MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. J. Hahn and O. T. O’Meara, The Classical Groups and K-Theory, Springer, Berlin (1989).Google Scholar
  11. 11.
    M. Karoubi, “Périodicité de la K-théorie hermitienne,” Lecture Notes Math., 343, 301–411 (1973).CrossRefzbMATHGoogle Scholar
  12. 12.
    V. I. Kopeiko, “The stabilization of symplectic groups over a polynomial ring,” Mat. Sb., 106, No. 1, 94–107 (1978).MathSciNetGoogle Scholar
  13. 13.
    V. I. Kopeiko, “On homotopization of the unitary K1-functor,” Algebra Analiz, 20, No. 5, 99–108 (2008).MathSciNetGoogle Scholar
  14. 14.
    V. I. Kopeiko, “Transfer unitary K1-functor under polynomial extension of rings,” Algebra Analiz, 29, No. 3, 34–60 (2017).MathSciNetGoogle Scholar
  15. 15.
    L. N. Vaserstein, “Stabilization for unitary and orthogonal groups over a ring with involution,” Mat. Sb., 81, No. 3, 328–351 (1970).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Kalmyk State University named after B. B. GorodovikovElistaRussia

Personalised recommendations