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In the book Lie Algebras of finite and affine type by Roger Carter, in chapter 3, the conjugacy of Cartan subalgebras of a finite-dimensional Lie algebra over $\mathbb{C}$ is established via a technical lemma in algebraic geometry (Corollary 3.11). This makes me wonder whether a slightly more general version of this corollary holds.

Question 1 Let $n$ be a positive integer, $F: \mathbb{C}^n \to \mathbb{C}^n$ be a polynomial mapping, such that the determinant of the Jacobian $\det(J(F))$ never vanishes, is it true that $F(U)$ is Zariski open for any Zariski open set $U$?

More, generally,

Question 2 Does a similar result holds if we replace $\mathbb{C}^n$ by any smooth algebraic variety over $\mathbb{C}$ and $F$ by a regular mapping?

And even more generally,

Question 3 What happens if we further generalise the above question for a general smooth scheme over an algebraically closed field, which might be of positive characteristic?

I think maybe the strategy for proving Corollary 3.11 in Carter's book can be adapted to answer the above questions in the affirmative. But I prefer a more systematic approach to the above questions, thus simplifying the proof of the conjugacy of Cartan subalgebras of finite-dimensional algebras over $\mathbb{C}$.

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  • $\begingroup$ Could you state Corollary 3.11 here? $\endgroup$
    – YCor
    Oct 29, 2018 at 11:43
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    $\begingroup$ Question 1 has a positive answer, because by plain (real) differential geometry, the image of any Zariski-open subset is open, and being Zariski-constructible and open, it has to be Zariski-open. Question 2 most likely works by the same argument. Question 3: at least the case of algebraically closed fields of characteristic zero follows formally from the complex case. $\endgroup$
    – YCor
    Oct 29, 2018 at 11:45
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    $\begingroup$ If I understand correctly the question, your $F$ is étale (EGA IV, Cor. 17.11.12), hence flat, hence open — over any field. $\endgroup$
    – abx
    Oct 29, 2018 at 13:51
  • $\begingroup$ @abx, thanks for the reference. What happens if $F$ is not étale in the positive characteristic case (and we of course drop the algebraic closedness of the field)? Is there any easy counter-example? $\endgroup$
    – Rick Sternbach
    Oct 29, 2018 at 19:06

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