Commutative Algebra pp 69-109 | Cite as

# Differential Structure of Étale Extensions of Polynomial Algebras

## 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.

### References

- [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 - [AK]A. Altman and S. Kleiman,
*Introduction to Grothendieck Duality*, Springer Lecture Notes in Math. 146 (1970).Google Scholar - [B]H. Bass, “Algebraic JC-theory,” W. A. Benjamin, New York, 1968.Google Scholar
- [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 - [BH]H. Bass and W. Haboush,
*Linearizing certain reductive group actions*, Trans. AMS 292 (1985), 463–482.MathSciNetMATHCrossRefGoogle Scholar - [Bj]J. E. Bjork, “Rings of Differential Operators,” North Holland Pubi. Co., Amsterdam, 1979.Google Scholar
- [Bo]A. Borei, “Linear Algebraic Groups,” W. A. Benjamin, New York, 1969.Google Scholar
- [BoD]A. Borei, et al., “Algebraic
*D*-modules,” Perspectives in Mathematics, Academic Press, Boston, 1987.Google Scholar - [NB]N. Bourbaki, “Algèbre Commutative,” Chs. 8 and 9, Masson, Paris, 1983.Google Scholar
- [D]P. Dienes, “The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable,” Clarendon Press, Oxford, 1931.Google Scholar
- [F]E. Formanek,
*Two notes on the Jacobian Conjecture*, preprint (1987).Google Scholar - [H]E. Hille, “Analytic Function Theory,” Ginn and Co., Boston, 1959.Google Scholar
- [L]S. Lang, “Fundamentals of Diophantine Geometry,” Springer-Verlag, New York, 1983.Google Scholar
- [Li]J. Lipman,
*Free derivation modules on algebraic varieties*, Amer. Jour. Math.**87**(1965), 874–898.MathSciNetMATHCrossRefGoogle Scholar - [M]H. Matsumura, “Commutative Algebra,” W. A. Benjamin, New York, 1970.Google Scholar
- [N]M. Nagata, “Field Theory,” M. Dekker, New York, 1977.Google Scholar
- [S]J.-P. Serre, “Lie Algebras and Lie Groups,” Lectures at Harvard, W. A. Benjamin, New York, 1964.Google Scholar
- [W]