Question about global quotient stacks

2010-08-13T10:00:10Z

In "Brauer groups and quotient stacks", Edidin et. al prove the following theorem:

Theorem 2.7. Let $\mathcal{X}$ be an algebraic stack over a Noetherian base (of finite type). Then the diagonal $\mathcal{X}\to \mathcal{X}\times \mathcal{X}$ is quasi-finite if and only if there is a finite surjective morphism $X\to \mathcal{X}$ for a scheme $X$.

On the other hand, Kresch in "Cycle groups for Artin stacks" proves the following:

Proposition 3.5.7. Let $\mathcal{X}$ be a stack of finite type over a field. The the following are equivalent: 1) The diagonal is quasi-finite; 2) The stabilizer $\mathcal{X}\times_{\mathcal{X}\times\mathcal{X}}\mathcal{X}\to \mathcal{X}$ is quasi-finite. Further, if $\mathcal{X}$ has quasi-finite diagonal $\mathcal{X}$ admits a stratification by quotient stacks.

Now, suppose that $\mathcal{X}$ is already a quotient stack $[Y/G]$, say with $Y$ an affine scheme and $G$ some group scheme (both of finite type over a field). Then $\mathrm{id}: Y\to\mathcal{X}$ is a finite surjective morphism, so by 2.7 above have quasi-finite diagonal. Then by 3.5.7 the stabilizer is quasi-finite, but this seems false in general. For instance, take $G=GL(n)$ and then you are almost guaranteed to have non-finite stabilizers.

What am I missing here? It's obvious that there's something here that I've gotten wrong.

Answer by Angelo for Question about global quotient stacks

2010-08-13T11:23:53Z

The morphism $Y \to [Y/G]$ is a $G$-torsor, so it is finite only if $G$ is finite.

Question about global quotient stacks - MathOverflow

Question about global quotient stacks

Answer by Angelo for Question about global quotient stacks