Skip to main content
SpringerLink
Log in

SK1 of graded division algebras

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

Abstract

The reduced Whitehead group SK1 of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that SK1 of a tame valued division algebra over a henselian field coincides with SK1 of its associated graded division algebra. Furthermore, it is shown that SK1 of a graded division algebra is isomorphic to SK1 of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes SK1 for generic abelian crossed products.

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. S. A. Amitsur and D. J. Saltman, Generic Abelian crossed products and p-algebras, Journal of Algebra 51 (1978), 76–87.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Boulagouaz, Le gradué d’une algèbre à division valuée, Communications in Algebra 23 (1995), 4275–4300.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Boulagouaz and K. Mounirh, Generic abelian crossed products and graded division algebras, in Algebra and Number Theory, (M. Boulagouaz and J.-P. Tignol eds.), Lecture Notes in Pure and Applied Mathematics, Vol. 208, Dekker, New York, 2000, pp. 33–47.

    Google Scholar 

  4. P. Draxl, Skew Fields, London Mathematical Society Lecture Note Series, Vol. 81, Cambridge University Press, Cambridge, 1983.

    Book  MATH  Google Scholar 

  5. A. J. Engler and A. Prestel, Valued Fields, Springer-Verlag, Berlin, 2005.

    MATH  Google Scholar 

  6. Yu. Ershov, Henselian valuations of division rings and the group SK 1, Rossiĭskaya Akademiya Nauk. Matematicheskiĭ Sbornik (N.S.) 117 (1982), 60–68 (in Russian); English transl.: Math. USSR-Sb. 45 (1983), 63–71.

    MathSciNet  Google Scholar 

  7. P. Gille, Le problème de Kneser-Tits, Astérisque, Bourbaki, Exposé no. 983, in Séminaire Bourbaki, Vol. 2007/2008, Astérisque 326, 2009, pp. 39–82.

  8. [H1]_R. Hazrat, SK1 -like functors for division algebras, Journal of Algebra 239 (2001), 573–588.

    Article  MathSciNet  MATH  Google Scholar 

  9. [H2]_R. Hazrat, Wedderburn’s factorization theorem, application to reduced K-theory, Proceedings of the American Mathematical Society 130 (2002), 311–314.

    Article  MathSciNet  MATH  Google Scholar 

  10. [H3]_R. Hazrat, On central series of the multiplicative group of division rings, Algebra Colloquium 9 (2002), 99–106.

    MathSciNet  MATH  Google Scholar 

  11. [H4]_R. Hazrat, SK1-of Azumaya algebras over Hensel pairs, Mathematische Zeitschrift 264 (2010), 295–299.

    Article  MathSciNet  MATH  Google Scholar 

  12. [HW1]_R. Hazrat and A. R. Wadsworth, On maximal subgroups of the multiplicative group of a division algebra, Journal of Algebra 322 (2009), 2528–2543.

    Article  MathSciNet  MATH  Google Scholar 

  13. [HwW1]_Y.-S. Hwang and A. R. Wadsworth, Algebraic extensions of graded and valued fields, Communications in Algebra 27 (1999), 821–840.

    Article  MathSciNet  MATH  Google Scholar 

  14. [HwW2]_Y.-S. Hwang and A. R. Wadsworth, Correspondences between valued division algebras and graded division algebras, Journal of Algebra 220 (1999), 73–114.

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Jacob and A. R. Wadsworth, Division algebras over Henselian fields, Journal of Algebra 128 (1990), 126–179.

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Jacobson, Finite-Dimensional Division Algebras over Fields, Springer-Verlag, Berlin, 1996.

    Book  MATH  Google Scholar 

  17. M.-A. Knus, Quadratic and Hermitian Forms over Rings, Springer-Verlag, Berlin, 1991.

    MATH  Google Scholar 

  18. M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The Book of Involutions, AMS Coll. Pub., Vol. 44, American Mathematical Society, Providence, RI, 1998.

    Google Scholar 

  19. T.-Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, Vol. 131, Springer-Verlag, New York, 1991.

    Google Scholar 

  20. D. W. Lewis and J.-P. Tignol, Square class groups and Witt rings of central simple algebras, Journal of Algebra 154 (1993), 360–376.

    Article  MathSciNet  MATH  Google Scholar 

  21. [Mc1]_K. McKinnie, Prime to p extensions of the generic abelian crossed product, Journal of Algebra 317 (2007), 813–832.

    Article  MathSciNet  MATH  Google Scholar 

  22. [Mc2]_K. McKinnie, Indecomposable p-algebras and Galois subfields in generic abelian crossed products, Journal of Algebra, 320 (2008), 1887–1907.

    Article  MathSciNet  MATH  Google Scholar 

  23. K. McKinnie, Degeneracy and decomposability in abelian crossed products, preprint, arXiv: 0809.1395.

  24. A. S. Merkurjev, K-theory of simple algebras, in K-theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, (B. Jacob and A. Rosenberg, eds.), Proceedings of Symposia in Pure Mathematics Vol. 58, Part 1, American Mathematical Society, Providence, RI, 1995, pp. 65–83.

    Google Scholar 

  25. P. Morandi, The Henselization of a valued division algebra, Journal of Algebra 122 (1989), 232–243.

    Article  MathSciNet  MATH  Google Scholar 

  26. [M1]_K. Mounirh, Nicely semiramified division algebras over Henselian fields, International Journal of Mathematics and Mathematical Sciences 2005 (2005), 571–577.

    Article  MathSciNet  MATH  Google Scholar 

  27. [M2]_K. Mounirh, Nondegenerate semiramified valued and graded division algebras, Communications in Algebra 36 (2008), 4386–4406.

    Article  MathSciNet  MATH  Google Scholar 

  28. K. Mounirh and A. R. Wadsworth, Subfields of nondegenerate tame semiramified division algebras, Communications in Algebra, to appear.

  29. I. A. Panin and A. A. Suslin, On a conjecture of Grothendieck concerning Azumaya algebras, St. Petersburg Mathematical Journal 9 (1998), 851–858.

    MathSciNet  Google Scholar 

  30. [P1]_V. P. Platonov, The Tannaka-Artin problem and reduced K-theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), 227–261 (in Russian); English transl.: Math. USSR-Izv. 10 (1976), 211–243.

    MathSciNet  MATH  Google Scholar 

  31. [P2]_V.P. Platonov, Infiniteness of the reduced Whitehead group in the Tannaka-Artin problem, Rossiĭskaya Akademiya Nauk 100(142) (1976), 191–200, 335 (in Russian); English transl.: Math. USSR-Sb. 29 (1976), 167–176.

    MathSciNet  Google Scholar 

  32. [P3]_V. P. Platonov, Algebraic groups and reduced K-theory, in Proceedings of the ICM (Helsinki 1978), (O. Lehto, ed.) Acad. Sci. Fennica, Helsinki, 1980, pp. 311–317.

    Google Scholar 

  33. V. P. Platonov and V. I. Yanchevskiĭ, SK1-for division rings of noncommutative rational functions. Rossiĭskaya Akademiya Nauk SSSR 249 (1979), 1064–1068 (in Russian); English transl.: Soviet Math. Doklady 20 (1979), 1393–1397.

    Google Scholar 

  34. I. Reiner, Maximal Orders, Academic Press, New York, 1975.

    MATH  Google Scholar 

  35. J.-F. Renard, J.-P. Tignol. and A. R. Wadsworth, Graded Hermitian forms and Springer’s theorem, Koninklijke Nederlandse Akademie van Wetenschappen 18 (2007), 97–134.

    MathSciNet  MATH  Google Scholar 

  36. [S1]_D. J. Saltman, Noncrossed product p-algebras and Galois p-extensions, Journal of Algebra 52 (1978), 302–314.

    Article  MathSciNet  MATH  Google Scholar 

  37. [S2]_D. J. Saltman, Lectures on Division Algebras, Regional Conference Series in Mathematics, no. 94, AMS, Providence, RI, 1999.

    MATH  Google Scholar 

  38. C. Stuth, A generalization of the Cartan-Brauer-Hua theorem, Proceedings of the American Mathematical Society 15 (1964), 211–217.

    Article  MathSciNet  MATH  Google Scholar 

  39. A. A. Suslin, SK1-of division algebras and Galois cohomology, in Algebraic K-theory, (A. A. Suslin, ed.) Adv. Soviet Math., Vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 75–99.

    Google Scholar 

  40. J.-P. Tignol, Produits croisés abéliens, Journal of Algebra 70 (1981), 420–436.

    Article  MathSciNet  MATH  Google Scholar 

  41. J.-P. Tignol and A. R. Wadsworth, Value functions and associated graded rings for semisimple algebras, Transactions of the American Mathematical Society 362 (2010), 687–726.

    Article  MathSciNet  MATH  Google Scholar 

  42. [W1]_A. R. Wadsworth, Extending valuations to finite dimensional division algebras, Proceedings of the American Mathematical Society 98 (1986), 20–22.

    MathSciNet  MATH  Google Scholar 

  43. [W2]_A. R. Wadsworth, Valuation theory on finite dimensional division algebras, in Valuation Theory and Its Applications, Vol. I, (F.-V. Kuhlmann et al. eds.), Fields Institute Commun., Vol. 32, American Mathematical Society, Providence, RI, 2002, pp. 385–449.

    Google Scholar 

  44. A. R. Wadsworth, Unitary SK 1 for graded and valued division algebras II, preprint in preparation.

  45. J. H. M. Wedderburn, On division algebras, Transactions of the American Mathematical Society 15 (1921), 162–166.

    Article  MathSciNet  Google Scholar 

  46. V. I. Yanchevskiĭ, Reduced unitary K-Theory and division rings over discretely valued Hensel fields, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 879–918 (in Russian); English transl.: Math. USSR Izvestiya 13 (1979), 175–213.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, Queen’s University, Belfast, BT7 1NN, UK

    R. Hazrat

  2. Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093-0112, USA

    A. R. Wadsworth

Authors
  1. R. Hazrat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. R. Wadsworth
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Hazrat.

Additional information

The first author acknowledges the support of EPSRC first grant scheme EP/D03695X/1.

The second author would like to thank the first author and Queen’s University, Belfast for their hospitality while the research for this paper was carried out. Both authors thank the referee for his or her careful reading of the paper and constructive suggestions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hazrat, R., Wadsworth, A.R. SK1 of graded division algebras. Isr. J. Math. 183, 117–163 (2011). https://doi.org/10.1007/s11856-011-0045-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-011-0045-1

Keywords

Search

Navigation