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

Polynomial Algebra Integral Closure Weyl Algebra Irreducible Element Transcendence Degree## Preview

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