Skip to main content
SpringerLink
Log in

Local-Global Principle for the General Quadratic and General Hermitian Groups and the Nilpotency of KH1

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

In this article, an analog of the Quillen–Suslin’s local-global principle was established for the elementary subgroup of the general quadratic group and the general Hermitian group. It is shown that unstable K1-groups of the general Hermitian groups over module finite rings are nilpotent-by-Abelian. This generalizes earlier results by A. Bak, R. Hazrat, and N. Vavilov.

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. E. Abe, “Chevalley groups over local rings,” T ô hoku Math. J., (2) 21, 474–494 (1969).

  2. A. Bak, K-Theory of Forms, Annals of Mathematics Studies, 98. Princeton University Press, Princeton, New Jersey (1981).

  3. A. Bak, “Non-Abelian K-theory: the nilpotent class of K1 and general stability,” K-Theory 4, No. 4, 363–397 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Bak and G. Tang, “Stability for Hermitian K1,” J. Pure Appl. Algebra, 150, 107–121 (2000).

    Article  MATH  Google Scholar 

  5. A. Bak, V. Petrov, and G. Tang, “Stability for Quadratic K1,” K-Theory, 29, 1–11 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Bak, R. Basu, and R. A. Rao, “Local-global principle for transvection groups,” Proc. Amer. Math. Soc., 138, No. 4, 1191–1204 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Bak, R. Hazrat, and N. Vavilov, “Localization-completion strikes again: relative K1 is nilpotent-by-Abelian,” J. Pure Appl. Algebra, 213, 1075–1085 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups, I. Elementary subgroups,” Alg. Colloquium, 7 (2), 159–196 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Bass, Algebraic K-Theory, Benjamin, New York-Amsterdam (1968).

  10. H. Bass, “Unitary algebraic K-theory,” Lecture Notes Math., 343, 57–265 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Bass, “Quadratic modules over polynomial rings,” in: Contribution to Algebra (Collection of papers dedicated to Ellis Kolchin), Academic Press, New York (1977), pp. 1–23.

    Google Scholar 

  12. R. Basu, R. A. Rao, and R. Khanna, “On Quillen’s local-global principle,” Algebra and Algebraic Geometry (Bangalore, India, 2003), Contemp. Math., 390, AMS, Providence, Rhode Island, 17–30 (2005).

  13. A. J. Berrick and M. E. Keating, An Introduction to Rings and Modules with K-theory in View, Cambridge Studies Adv. Math., 65 (2000).

  14. P. Chattopadhyay and R. A. Rao, “Elementary symplectic orbits and improved K1-stability,” J. K-Theory, 7, No. 2, 389–403 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Fasel, R. A. Rao, and R. G. Swan, “On stably free modules over affine algebras,” Publ. Math. Inst. Hautes É tudes Sci., 116, 223–243 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  16. Fu An Li, “The structure of orthogonal groups over arbitrary commutative rings,” Chinese Ann. Math., Ser. B, 10, 341–350 (1989).

  17. R. Hazrat, “Dimension theory and nonstable K1 of quadratic modules,” K-Theory, 27, No. 4, 293–328 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Hazrat and N. Vavilov, “K1 of Chevalley groups are nilpotent,” J. Pure Appl. Algebra, 179, No. 1-2, 99–116 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Hazrat and N. Vavilov, “Bak’s work on K-theory of rings. On the occasion of his 65th birthday,” J. K-Theory, 4, No. 1, 1–65 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  20. R. Hazrat, N. Vavilov, and Z. Zhang, “Relative unitary commutator calculus, and applications,” J. Algebra, 343, 107–137 (2011).

  21. R. Hazrat, A. Stepanov, N. Vavilov, and Z. Zhang, “The yoga of commutators,” Zap. Nauchn. Semin. POMI, 387 53–82, 189 (2011).

  22. A. J. Hahn and O. T. O’Meara, The Classical Groups and K-Theory, With a foreword by J. Dieudonné, Springer-Verlag, Berlin (1989).

  23. N. Jacobson, Lectures on Quadratic Jordan Algebras, Tata Istitute of Fundamental Research, Bombay (1969).

  24. I. S. Klein and A. V. Mikhalev, “The orthogonal Steinberg group over a ring with involution,” Algebra Logika, 9, 145–166 (1970).

    MathSciNet  Google Scholar 

  25. I. S. Klein and A. V. Mikhalev, “The unitary Steinberg group over a ring with involution,” Algebra Logika, 9 510–519 (1970).

    MathSciNet  Google Scholar 

  26. V. I. Kopeiko, “The stabilization of symplectic groups over a polynomial ring,” Math. USSR Sbornik, 34, 655–669 (1978).

    Article  Google Scholar 

  27. K. McCrimmon, “A general theory of Jordan rings,” Proc. Nat. Acad. Sci. U.S.A., 56, 1072–1079 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  28. R. Parimala, “Failure of quadratic analog of Serre’s conjecture,” Bull. Amer. Math. Soc., 82, 962–964 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  29. R. Parimala, “Failure of quadratic analog of Serre’s conjecture,” Amer. J. Math., 100, 913–924 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  30. V. A. Petrov, “Odd unitary groups,” J. Math. Sci., (New York), 130, No. 3, 4752–4766 (2005).

  31. V. A. Petrov and A. K. Stavrova, “Elementary subgroups in isotropic reductive groups,” Algebra Analiz, 20, No. 4, 160–188 (2008).

    MathSciNet  MATH  Google Scholar 

  32. R. A. Rao and W. van der Kallen, “Improved stability for SK1 and WMS d of a non-singular affine algebra,” K-theory, 226, 411–420 (1994).

    Google Scholar 

  33. R. A. Rao, R. Basu, and S. Jose, “Injective stability for K1 of the orthogonal group,” J. Algebra, 323, 393–396 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  34. Sergei Sinchuk, “Injective stability for unitary K1,” revisited (to appear).

  35. A. A. Suslin, “On the structure of special linear group over polynomial rings,” Math. USSR Izv., 11, 221–238 (1977).

    Article  MATH  Google Scholar 

  36. M. R. Stein, “Stability theorems for K1, K2, and related functors modeled on Chevalley groups,” Japan. J. Math. (N.S.), 4, No. 1, 77–108 (1978).

  37. A. A. Suslin and V. I. Kopeiko, “Quadratic modules and orthogonal groups over polynomial rings,” Zap. Nauchn. Semin. LOMI, 71, 216–250 (1978).

    MathSciNet  Google Scholar 

  38. A. A. Suslin and L. N. Vaserstein, “Serre’s problem on projective modules over polynomial rings, and algebraic K-theory,” Izv. Akad. SSSR, Ser. Mat., 40, No. 5, 937–1001 (1976).

  39. Guoping Tang, “Hermitian groups and K-theory,” K-Theory, 13, No. 3, 209–267 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  40. G. Taddei, “Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau,” in: Application of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part II (Boulder, Colo., 1983), Amer. Math. Soc., Providence, Rhode Island (1986), pp. 693–710.

  41. M. S. Tulenbaev, “Schur multiplier of a group of elementary matrices of finite order,” Zap. Nauchn. Semin. LOMI, 86, 162–169 (1979).

    MathSciNet  MATH  Google Scholar 

  42. L. N. Vaserstein, “On the stabilization of the general linear group over a ring,” Mat. Sb. (N.S.), 79, (121) 405–424 (1968).

    Google Scholar 

  43. L. N. Vaserstein, “Stabilization of unitary and orthogonal groups over a ring with involution,” Mat. Sb., 81 (123), No. 3, 307–326 (1970).

  44. L. N. Vaserstein, “Stabilization for classical groups over rings,” Mat. Sb., (N.S.), 93 (135), 268–295 (1974).

    MathSciNet  Google Scholar 

  45. L. N. Vaserstein, “On the normal subgroups of GL n over a ring,” in: Algebraic K-theory, Evanston (1980), pp. 456–465.

  46. Weibe Yu, “Stability for odd unitary K1 under the Λ-stable range condition,” J. Pure Appl. Algebra, 217, 886–891 (2013).

    Article  MathSciNet  Google Scholar 

  47. J. Wilson, “The normal and subnormal structure of general linear groups,” Proc. Camb. Phil. Soc., 71, 163–177 (1972).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Indian Institute of Science Education and Research, Pune, India

    R. Basu

Authors
  1. R. Basu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Basu.

Additional information

Dedicated to the memory of late Professor Amit Roy

Published in Zapiski Nauchnykh Seminarov POMI, Vol. 452, 2016, pp. 5–31.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Basu, R. Local-Global Principle for the General Quadratic and General Hermitian Groups and the Nilpotency of KH1. J Math Sci 232, 591–609 (2018). https://doi.org/10.1007/s10958-018-3891-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-018-3891-0

Search

Navigation