Let $\mathcal{X}$ be a separated Deligne-Mumford stack over an algebraically closed field $k$ and let $X$ be the corresponding coarse moduli space, which we assume to exist. There is a map $p:\mathcal{X}\to X$ of stacks. Is it true that every point of $X$ has an etale neighborhood $U\to X$ such that its pullback under $p$ is the map $[V/G]\to V/G=U$ where $V$ is an affine variety over $k$ on which a finite group $G$ acts. While reading some papers I've got the impression that the authors are implicitly using this statement or a similar one, but I wasn't able to locate a precise statement or reference in the literature. So I would be grateful if someone points me to one.

In the example I'm interested in $\mathcal{X}$ is in fact a quotient stack, but I do not want to assume that $char(k)=0$ or that the orders of the stabilizers are coprime with $char(k)$ (unless this follows from the previous conditions?).


1 Answer 1


This is Lemma 2.2.3 of the paper

Abramovich-Vistoli: Compactifying the space of stable maps;

see also section 5.4 of the "Guide to the stacks literature" by Jarod Alper.


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