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.



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


