Advertisement

Algebraic K-theory, morita theory, and the classical groups

  • Alexander J. Hahn
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1185)

Keywords

Exact Sequence Normal Subgroup Classical Group Unitary Group Linear Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    E. Artin, Geometric Algebra, Wiley Interscience, New York, 1957.MATHGoogle Scholar
  2. [2]
    A. Bak, On modules with quadratic forms, pp. 55–66, in Lecture Notes in Mathematics 108, Springer Verlag, Berlin 1969.MATHGoogle Scholar
  3. [3]
    A. Bak, K-Theory of Forms, Annals of Mathematics Studies 98, Princeton University Press, 1981.Google Scholar
  4. [4]
    A. Bak, Le probleme des sous-groupes de congruence et le probleme metaplectique pour les groupes classiques de rang >1, C.R. Acad. Sc. Paris, 292, 307–310 (1981).MathSciNetMATHGoogle Scholar
  5. [5]
    A. Bak and U. Rehann, The congruence subgroup and metaplectic problems for SLN>2 of division algebras, J. Algebra, 78 (1982), 475–547.MathSciNetCrossRefGoogle Scholar
  6. [6]
    H. Bass, Algebraic K-Theory, Benjamin, New York, 1968.MATHGoogle Scholar
  7. [7]
    H. Bass, Unitary Algebraic K-Theory, pp. 57–265, in Lecture Notes in Mathematics 343, Springer Verlag, Berlin 1973.MATHGoogle Scholar
  8. [8]
    H. Bass, Introduction to some Methods of Algebraic K-Theory, American Mathematical Soc., Providence, R.I., 1974.CrossRefMATHGoogle Scholar
  9. [9]
    H. Bass, Clifford algebras and spinor norms over a commutative ring, Amer. J. Math. 96 (1974), 156–206.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    H. Bass, J. Milnor, and J.P. Serre, Solution of the Congruence Subgroup Problem for SLn and Sp2n. Publ. Math. IHES 33 (1967), 59–137.CrossRefMATHGoogle Scholar
  11. [11]
    D. Callan, The isomorphisms of unitary groups over non-commutative domains, J. Algebra 52 (1978), 475–503.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    F. Connolly, Linking numbers and surgery, Topology 12, 1973, 389–409.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    V.V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math. 100 (1978), 303–386.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    J. Dieudonne, La Geometrie des Groupes Classiques, 3rd ed., Springer Verlag, Berlin-New York, 1971.MATHGoogle Scholar
  15. [15]
    P. Draxl, Skew Fields, Cambridge University Press, 1982.Google Scholar
  16. [16]
    A. Fröhlich and E.M. McEvett, Forms over rings with involution, J. Algebra, 12 (1969), 79–104.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    I. Golubchik, On the general linear group over an associative ring, Uspekhi Mat. Nauk, 28:3 (1973), 179–180 (Russian).MathSciNetGoogle Scholar
  18. [18]
    A. Hahn, Isomorphism theory for orthogonal groups over arbitrary integral domains, J. Algebra, 51 (1978), 233–287.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    A. Hahn, Category equivalences and linear groups over rings, J. Algebra, 77 (1982), 505–543.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    A. Hahn, A hermitian Morita theorem for algebras with anti-structure, J. Algebra, 93 (1985), 215–235.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    A. Hahn and Z.X. Li, Hermitian Morita theory and hyperbolic unitary groups, to appear in J. Algebra.Google Scholar
  22. [22]
    A. Hahn, D. James and B. Weisfeiler, Homomorphisms of algebraic and classical groups: a survey, in Canadian Mathematical Society Conference Proceedings, Volume 4 (1984), 249–296.MathSciNetMATHGoogle Scholar
  23. [23]
    I. Hambleton, L. Taylor and B. Williams, An introduction to maps between surgery obstruction groups, pp. 49–127, in Lecture Notes in Mathematics 1051, Springer Verlag 1982.Google Scholar
  24. [24]
    J. Humphreys, Arithmetic Groups, Lecture Notes in Mathematics 789, Springer Verlag, Berlin, 1980.MATHGoogle Scholar
  25. [25]
    D. James, W. Waterhouse and B. Weisfeiler, Abstract homomorphisms of algebraic groups: problems and bibliography, Comm. Algebra, 9 (1981), 95–114.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    W. van der Kallen, Generators and relations in Algebraic K-Theory, pp. 305–210, in Proceedings of the International Conference of Mathematicians, Helsinki, 1978.Google Scholar
  27. [27]
    W. van der Kallen, Stability for K2 in Dedekind rings of arithmetic type, pp. 217–248, in Lecture Notes in Mathematics 854, Springer-Verlag, Berlin, 1980.Google Scholar
  28. [28]
    M. Kolster, Surjective stability for unitary K-groups, preprint, 1975.Google Scholar
  29. [29]
    M. Kolster, General symbols and presentation of elementary linear groups, J. für reine u. angew. Math. 353 (1984), 132–164.MathSciNetMATHGoogle Scholar
  30. [30]
    T.Y. Lam, The Algebraic Theory of Quadratic Forms, 2nd ed., Benjamin, New York, 1980.MATHGoogle Scholar
  31. [31]
    K. Leung, The isomorphism theory of projective pseudo-orthogonal groups, J. Algebra, 61 (1979), 367–387.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    J. Mennicke, Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965), 115–129.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    H. Matsumoto, Sur les sousgroupes arithmetiques des groupes semisimple deployes, Ann. Sci. Ecole Norm. Sup. (4) 2(1969), 1–62.MATHGoogle Scholar
  34. [34]
    A. Merkurjev and A. Suslin, K-Cohomologies of Severi-Brauer varieties and norm residue homomorphism, Izv. Akad. Nauk SSSR, 16 (1982), 1011–1046.Google Scholar
  35. [35]
    J. Milnor, Introduction to algebraic K-Theory, Annals of Mathematical Studies 72, Princeton University Press, 1971.Google Scholar
  36. [36]
    N.M. Mustafa-Zade, On epimorphic stability of a unitary K2-functor, Russian Math. Surveys (1980), 99–100.Google Scholar
  37. [37]
    O.T. O'Meara, Lectures on Linear groups, Amer. Math. Society, Providence, R.I., 1974.Google Scholar
  38. [38]
    O.T. O'Meara, A general isomorphism theory for linear groups, J. Algebra, 44 (1977), 93–142.MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    O.T. O'Meara, Symplectic groups, Math. Surveys, Amer. Math. Soc. Providence, R.I., 1978.Google Scholar
  40. [40]
    O.T. O'Meara, A survey of the isomorphism theory of the classical groups, pp. 225–242, in "Ring theory and Algebra III", Dekker, New York, 1980.Google Scholar
  41. [41]
    W. Pender, Automorphisms and Isomorphisms of the indefinite modular classical groups, Ph.D. Thesis, Sydney University (1972).Google Scholar
  42. [42]
    W. Pender, Classical groups over division rings of characteristic 2, Bull. Aust. Math. Soc. 7 (1972), 191–226.MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    G. Prasad and M.S. Ragunathan, On the congruence subgroup problem: determination of the "metaplectic kernel", Invent. math. 71, (1983), 21–42.MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    H.G. Quebbemann, W. Scharlau, and M. Schulte, Quaratic and hermitian forms in additive and abelian categories, J. Algebra, 59 (1979), 264–289.MathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    A. Ranicki, The algebraic theory of surgery I, Foundations, Proc. London Math. Soc. (3) 40 (1980), 87–192.MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    A. Ranicki, Exact Sequences in the Algebraic Theory of Surgery, Princeton Mathematical Notes, Princeton University Press, 1981.Google Scholar
  47. [47]
    H.S. Ren and Z.X. Wan, Automorphisms of PSL+2 (K) over any skew field K, Acta. Math. Sinica, 25, (1982), 484–492.MathSciNetMATHGoogle Scholar
  48. [48]
    J.P. Serre, Trees, Springer-Verlag, Berlin, New York, 1980.CrossRefMATHGoogle Scholar
  49. [49]
    R. Sharpe, On the structure of the unitary Steinberg group, Ann. Math. 96 (1972), 444–479.MathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    J. Silvester, Introduction to Algebraic K-Theory, Chapman and Hall, London, 1981.MATHGoogle Scholar
  51. [51]
    G. Soule, K2 et le groupe de Brauer [d'apres A.S. Merkurjev et A.A. Suslin]. Seminare Bourbaki, 1982/83, No. 601 (1982).Google Scholar
  52. [52]
    R. Steinberg, Generateurs, relations et revetements de groups algebriques, Colloque de Bruxelles, 1962, 113–127.Google Scholar
  53. [53]
    R. Steinberg, Lecture Notes on Chevalley Groups, Yale University, 1967.Google Scholar
  54. [54]
    A. Suslin, On the structure of the special linear group over polynomial rings, Math. USSR Izvestija, Vol. II (1972), No. 2, 221–328.MATHGoogle Scholar
  55. [55]
    A. Suslin and V. Kopeiko, Quadratic modules and the orthogonal group over polynomial rings, Zap. Naucn. Sem. Leningrad. Otdel. Math. Inst. Steklov. (LOMI) 71 (1977), 216–250.MathSciNetGoogle Scholar
  56. [56]
    A. Suslin, Reciprocity laws and the stable rank of polynomial rings, Math. USSR Izvestija, Vol 15(1980), No. 3, 589–623.CrossRefMATHGoogle Scholar
  57. [57]
    R. Swan, K-Theory of Finite Groups and Orders, Lecture Notes in Mathematics 149, Springer-Verlag, Berlin, 1970.MATHGoogle Scholar
  58. [58]
    J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics 383, Springer-Verlag, Berlin, 1974.MATHGoogle Scholar
  59. [59]
    L. Vaserstein, Stabilization of unitary and orthogonal groups over a ring with involution, Math. USSR Sbornik, Vol. 10 (1970), 307–326.CrossRefMATHGoogle Scholar
  60. [60]
    L. Vaserstein, The stabilization for classical groups over rings, Math. USSR Sbornik 22, (1974), 271–303.CrossRefMATHGoogle Scholar
  61. [61]
    L. Vaserstein, Foundations of algebraic K-theory, Russian Math. Surveys, 31:4 (1976), 89–156.MathSciNetCrossRefMATHGoogle Scholar
  62. [62]
    L. Vaserstein, On the normal subgroups of GLn over a ring, pp. 456–465, in Lecture Notes in Mathematics 854, Springer-Verlag, Berlin, 1981.Google Scholar
  63. [63]
    L. Vaserstein, On full subgroups in the sense of O'Meara, J. Algebra, 75 (1982), 437–44.MathSciNetCrossRefMATHGoogle Scholar
  64. [64]
    L. Vaserstein, Classical groups over rings, in Canadian Mathematical Society Conference Proceedings, Volume 4 (1984).Google Scholar
  65. [65]
    L. Vaserstein and A. Suslin, Serre's problem on projective modules over polynomial rings and algebraic K-theory, Math. USSR Izvestija, Vol. 10 (1976), No. 5, 937–1001.MathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    A. Wadsworth, Merkurjev's elementary proof of Merkurjev's theorem, Boulder Conference in Algebraic K-theory, to appear.Google Scholar
  67. [67]
    C.T.C. Wall, Surgery on Compact Manifolds, Academic Press, 1970.Google Scholar
  68. [68]
    C.T.C. Wall, On the axiomatic foundation of the theory of Hermitian forms, Proc. Camb. Phil. Soc., 67 (1970), 243–250.MathSciNetCrossRefMATHGoogle Scholar
  69. [69]
    C.T.C. Wall, Foundations of algebraic L-Theory, pp. 266–300, in Lecture Notes in Mathematics 343, Springer Verlag, Berlin, 1973.MATHGoogle Scholar
  70. [70]
    C.T.C. Wall, On the classification of Hermitian Forms III, semisiple rings, Invent. Math., 18 (1972), 119–141.MathSciNetCrossRefMATHGoogle Scholar
  71. [71]
    G.E. Wall, The Structure of a unitary factor group, Publ. Math., IHES, No. 1, (1959), 7–23.Google Scholar
  72. [72]
    Z.X. Wan, The Classical Groups, Shanghai University Press, 1981.Google Scholar
  73. [73]
    Z.X. Wan and J.G. Yang, Automorphisms of the projective quaternion unimodular group in dimension 2, Chinese Annals of Math., 3(1982), 395–402.MATHGoogle Scholar
  74. [74]
    B. Weisfeiler, Abstract homomorphisms of big subgroups of algebraic groups, pp. 135–181, in Topics in the theory of Algebraic Groups, Notre Dame Mathematical Lectures, No. 10, University of Notre Dame Press, 1982.Google Scholar
  75. [75]
    J.S. Wilson, The normal and subnormal structure of general linear groups, Proc. Camb. Phil. Soc. 71(1972), 163–177.MathSciNetCrossRefMATHGoogle Scholar
  76. [76]
    Zalesky, Linear groups, Russian Math. Surveys, 36, No. 5, (1981), 63–128.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Alexander J. Hahn
    • 1
  1. 1.University of Notre DameNotre DameUSA

Personalised recommendations