Differential Structure of Étale Extensions of Polynomial Algebras

  • Hyman Bass
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 15)

Abstract

The Jacobian Conjecture (see [BCW]) asserts that an étale polynomial map F: ℂ n → ℂ n is an isomorphism. In the course of some work with Haboush on group actions [BH] we observed that this is easily proved if F is equivariant for a reductive group acting on the two ℂ n ’s and having a dense orbit and a fixed point. One is thus tempted to try to prove the Jacobian conjecture by “feeding” such group actions into the picture. For example, let G = SL n (ℂ) act linearly on the target ℂ n . We may assume that F(0) = 0. In order to make F equivariant for a G-action, we must make G act on the source ℂ n by “F º G º F −1”. Of course this begs the issue, since we don’t know that F −1 exists. However F−1 exists analytically near 0. In particular we can pull back the action of the Lie algebra sl n(ℂ) as vector fields on ℂ n . Moreover the fact that the Jacobian matrix of F has a polynomial inverse implies that sl n (ℂ) pulls back to polynomial vector fields on the source ℂ n .

Keywords

Filtration Diene 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AM]
    S. S. Abhyankar and T.-T. Moh, Embeddings of the line in the plane, J. Reine angew. Math. 276 (1975), 149–166.MathSciNetGoogle Scholar
  2. [AK]
    A. Altman and S. Kleiman, Introduction to Grothendieck Duality, Springer Lecture Notes in Math. 146 (1970).Google Scholar
  3. [B]
    H. Bass, “Algebraic JC-theory,” W. A. Benjamin, New York, 1968.Google Scholar
  4. [BCW]
    H. Bass, E. H. Connell and D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. AMS 7 (1982), 287–330.MathSciNetMATHCrossRefGoogle Scholar
  5. [BH]
    H. Bass and W. Haboush, Linearizing certain reductive group actions, Trans. AMS 292 (1985), 463–482.MathSciNetMATHCrossRefGoogle Scholar
  6. [Bj]
    J. E. Bjork, “Rings of Differential Operators,” North Holland Pubi. Co., Amsterdam, 1979.Google Scholar
  7. [Bo]
    A. Borei, “Linear Algebraic Groups,” W. A. Benjamin, New York, 1969.Google Scholar
  8. [BoD]
    A. Borei, et al., “Algebraic D-modules,” Perspectives in Mathematics, Academic Press, Boston, 1987.Google Scholar
  9. [NB]
    N. Bourbaki, “Algèbre Commutative,” Chs. 8 and 9, Masson, Paris, 1983.Google Scholar
  10. [D]
    P. Dienes, “The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable,” Clarendon Press, Oxford, 1931.Google Scholar
  11. [F]
    E. Formanek, Two notes on the Jacobian Conjecture, preprint (1987).Google Scholar
  12. [H]
    E. Hille, “Analytic Function Theory,” Ginn and Co., Boston, 1959.Google Scholar
  13. [L]
    S. Lang, “Fundamentals of Diophantine Geometry,” Springer-Verlag, New York, 1983.Google Scholar
  14. [Li]
    J. Lipman, Free derivation modules on algebraic varieties, Amer. Jour. Math. 87 (1965), 874–898.MathSciNetMATHCrossRefGoogle Scholar
  15. [M]
    H. Matsumura, “Commutative Algebra,” W. A. Benjamin, New York, 1970.Google Scholar
  16. [N]
    M. Nagata, “Field Theory,” M. Dekker, New York, 1977.Google Scholar
  17. [S]
    J.-P. Serre, “Lie Algebras and Lie Groups,” Lectures at Harvard, W. A. Benjamin, New York, 1964.Google Scholar
  18. [W]
    D. Wright, On the Jacobian Conjecture, Illinois Jour. Math. 25 (1981), 423–440.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Hyman Bass
    • 1
  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA

Personalised recommendations