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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

Polynomial Algebra Integral Closure Weyl Algebra Irreducible Element Transcendence Degree 
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.

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Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

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

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