I was studying Jarod Alper's book on stacks (https://sites.math.washington.edu/~jarod/moduli.pdf) and after the definition of stacks and some basic discussion about them, he proposes the following exercise:
If $\mathcal{X}$ is DM stack, $k$ a field and $x \in \mathcal{X}(k)$, then the group scheme $G_x$ is étale over $k$.
Here $G_x$ is the group stabilizer of $x$, and this is exercise 3.2.8 at page 102 of the current version.
My problem is that, a priori, $G_x$ is just a group algebraic space, not a scheme, and i am having trouble proving its schemeness. Just above the exercise, Alper mentions a result to be proved later that says that, if the diagonal of $\mathcal{X}$ is quasi-separated, then $G_x$ is a group scheme locally of finite type over $k$. Assuming this i think i have managed to solve the exercise, but this requires the diagonal of $\mathcal{X}$ to be quasi-separated (i will write qs in the following), but i have not managed to show this.
I have seen that some authors define DM stacks by asking that the diagonal be qs, but Alper does not, and neither does Olsson in his book on stacks. For completion, i recall that Alper defines a DM stack $\mathcal{X}$ to be a stack with an étale surjective representable map $U \to \mathcal{X}$ from a scheme $U$ (he then shows representability of the diagonal).
Is it true that Alper's definition implies that $\Delta_\mathcal{X}$ is qs? If this is not true, how could i solve the exercise?
Best Answer
The diagonal of a Deligne-Mumford stack is not necessarily quasi-separated; take the classifying stack $BG$ of the non-quasi-separated scheme $G = \mathbb{A}^{\infty} \cup_{\mathbb{A}^{\infty} \setminus 0} \mathbb{A}^{\infty}$ viewed as a group scheme over $\mathbb{A}^{\infty} = {\rm Spec} k[x_1, x_2, \ldots, x_n]$.
To solve the exercise, first show that $G_x$ is an etale group algebraic space over $k$. If $k= \bar{k}$, use that any section of the structure morphism $G_x \to {\rm Spec} k$ is an open immersion to give an open covering of $G_x$ by schemes, and then use that any group scheme over a field is separated (see Tag 047L). Over a general field, apply effective descent for separated and locally quasi-finite morphisms to conclude that $G_x$ is a scheme. This shows that the stabilizer of any field-valued point of a Deligne-Mumford stack (w/out any conditions on the diagonal) is a separated etale group scheme.