Israel Journal of Mathematics

, Volume 72, Issue 1–2, pp 167–195 | Cite as

Principal homogeneous spaces for arbitrary Hopf algebras

  • Hans-Jürgen Schneider
Article

Abstract

LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A B AAH the canonical map. ThenA is a faithfully flat (as left or rightB-module) Hopf Galois extension iffA is coflat asH-comodule and can is surjective (Theorem I). This generalizes results on affine quotients of affine schemes by Oberst and Cline, Parshall and Scott to the case of non-commutative algebras. The dual of Theorem I holds and generalizes results of Gabriel on quotients of formal schemes to the case of non-cocommutative coalgebras (Theorem II). Furthermore, in the dual situation, a normal basis theorem is proved (Theorem III) generalizing results of Oberst-Schneider, Radford and Takeuchi.

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References

  1. 1.
    R. J. Blattner,Induced and produced representations of Lie algebras, Trans. Am. Math. Soc.144 (1969), 457–474.CrossRefMathSciNetGoogle Scholar
  2. 2.
    E. Cline, B. Parshall and L. Scott,Induced modules and affine quotients, Math. Ann.230 (1977), 1–14.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    E. C. Dade,Group-graded rings and modules, Math. Z.174 (1980), 241–262.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. Demazure and P. Gabriel,Groupes algébriques, North-Holland, Amsterdam, 1970.MATHGoogle Scholar
  5. 5.
    Y. Doi,On the structure of relative Hopf modules, Commun. Algebra11 (1983), 243–255.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Y. Doi,Algebras with total integrals, Commun. Algebra13 (1985), 2137–2159.MATHMathSciNetGoogle Scholar
  7. 7.
    Y. Doi and M. Takeuchi,Hopf-Galois extensions of algebras, the Miyashita-Ulbrich action, and Azumaya algebras, J. Algebra121 (1989), 488–516.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    P. Gabriel,Etude infinitésimale des schémas en groupes — groupes formels, Exp. VIIB,Schémas en groupes I, Lecture Notes in Mathematics, No. 151, Springer, Berlin, Heidelberg, New York, 1970.Google Scholar
  9. 9.
    A. Grothendieck and J. Dieudonné,Eléments de géométrie algébrique, IV, Publ. Math.24 (1965).Google Scholar
  10. 10.
    L. Gruson and M. Raynaud,Critères de platitude et de projectivite, Invent. Math.13 (1971), 1–81.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    M. Koppinen and T. Neuvonen,An imprimitivity theorem for Hopf algebras, Math. Scand.41 (1977), 193–198.MathSciNetGoogle Scholar
  12. 12.
    H. F. Kreimer,A note on the outer Galois theory of rings, Pacific J. Math.31 (1969), 417–432.MATHMathSciNetGoogle Scholar
  13. 13.
    H. F. Kreimer and M. Takeuchi,Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J.30 (1981), 675–692.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A Masuoka,On Hopf algebras with cocommutative coradicals, preprint, 1989.Google Scholar
  15. 15.
    J. W. Milnor and J. C. Moore,On the structure of Hopf algebras, Ann. of Math.81 (1965), 211–264.CrossRefMathSciNetGoogle Scholar
  16. 16.
    D. Mumford and J. Fogarty,Geometric Invariant Theory, Springer, Berlin, Heidelberg, New York, 1982.MATHGoogle Scholar
  17. 17.
    K. Newman,A correspondence between bi-ideals and sub-Hopf algebras in cocommutative Hopf algebras, J. Algebra36 (1975), 1–15.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    U. Oberst and H.-J. Schneider,Untergruppen formeller Gruppen von endlichem Index, J. Algebra31 (1974), 10–44.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    U. Oberst,Affine Quotientenschemata nach affinen, algebraischen Gruppen und induzierte Darstellungen, J. Algebra44 (1977), 503–538.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    U. Oberst,Actions of formal groups on formal schemes. Applications to control theory and combinatorics, inSeminaire d’Algebre (P. Dubreil and M.-P. Malliavin, eds.), Lecture Notes in Mathematics, No. 1146, Springer, Berlin, Heidelberg, New York, 1985.Google Scholar
  21. 21.
    D. Radford,Pointed Hopf algebras are free over Hopf subalgebras, J. Algebra45 (1977), 266–273.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    D. Radford,Freeness (projectivity) criteria for Hopf algebras over Hopf subalgebras, J. Pure Appl. Algebra11 (1977), 15–28.CrossRefMathSciNetGoogle Scholar
  23. 23.
    N. S. Rivano,Catégories tannakiennes, Lecture Notes in Mathematics, No. 265, Springer, Berlin, Heidelberg, New York, 1972.MATHGoogle Scholar
  24. 24.
    L. Rowen,Ring Theory, Volume I, Academic Press, Boston, 1988.MATHGoogle Scholar
  25. 25.
    H.-J. Schneider,Zerlegbare Untergruppen affiner Gruppen, Math. Ann.255 (1981), 139–158.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    M. Sweedler,Hopf Algebras, Benjamin, New York, 1969.Google Scholar
  27. 27.
    M. Takeuchi,A correspondence between Hopf ideals and sub-Hopf algebras, Manuscr. Math.7 (1972), 251–270.MATHCrossRefGoogle Scholar
  28. 28.
    M. Takeuchi,A note on geometrically reductive groups, J. Fac. Sci., Univ. Tokyo, Sect. 1,20, No. 3 (1973), 387–396.MATHGoogle Scholar
  29. 29.
    M. Takeuchi,On extensions of formal groups by μ A, Commun. Algebra13 (1977), 1439–1481.CrossRefGoogle Scholar
  30. 30.
    M. Takeuchi,Formal schemes over fields, Commun. Algebra14 (1977), 1483–1528.CrossRefGoogle Scholar
  31. 31.
    M. Takeuchi,Relative Hopf modules — Equivalences and freeness criteria, J. Algebra60 (1979), 452–471.MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    K.-H. Ulbrich,Galoiserweiterungen von nicht-kommutativen Ringen, Commun. Algebra10 (1982), 655–672.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    D. Voigt,Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen, Lecture Notes in Mathematics, No. 92, Springer, Berlin, New York, Heidelberg, 1977.MATHGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1990

Authors and Affiliations

  • Hans-Jürgen Schneider
    • 1
  1. 1.Mathematisches InstitutUniversität MünchenMünchen 2FRG

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