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How to prove this? Let G be a finite group acting on a connected normal affine variety.Then the stabilizer at x is trivial iff Then the orbit space morphism q : X---->X // G is E'tale at x.

flag	asked Dec 21 2011 at 17:34 unknown (google) 21●1

	I will assume that you are working over an algebraically closed field. Let $\mathcal{O}$ be the completion of $X$ at $x$, and let $\mathcal{O}'$ be the completion of $X/G$ at the image of $x$. Then the fact is that $\mathcal{O}'$ is the ring of invariants in $\mathcal{O}$ under the stabilizer of $x$. In particular, $\mathcal{O}'=\mathcal{O}$ precisely when the stabilizer is trivial. – Keerthi Madapusi Pera Dec 21 2011 at 17:45
	I suppose we should also assume the action is faithful for this to be true. – Jack Huizenga Dec 21 2011 at 18:14
	Yes, of course. – Keerthi Madapusi Pera Dec 21 2011 at 18:36

Niels · Answer 1 · 2011-12-21 20:19:37Z UTC

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see SGA1 http://arxiv.org/abs/math/0206203 Exposé V Corollaire 2.3

answered Dec 21 2011 at 20:19

Niels
1,293●4●7

How to prove Etale morphism at a point.

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