Lemma 95.19.1. In the situation above $G_ x$ is a scheme if one of the following holds

1. $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is quasi-separated

2. $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is locally separated,

3. $\mathcal{X}$ is quasi-DM,

4. $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is quasi-separated,

5. $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is locally separated, or

6. $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is locally quasi-finite.

Proof. Observe that (1) $\Rightarrow$ (4), (2) $\Rightarrow$ (5), and (3) $\Rightarrow$ (6) by Lemma 95.6.1. In case (4) we see that $G_ x$ is a quasi-separated algebraic space and in case (5) we see that $G_ x$ is a locally separated algebraic space. In both cases $G_ x$ is a decent algebraic space (Decent Spaces, Section 62.6 and Lemma 62.15.2). Then $G_ x$ is separated by More on Groupoids in Spaces, Lemma 73.9.4 whereupon we conclude that $G_ x$ is a scheme by More on Groupoids in Spaces, Proposition 73.10.3. In case (6) we see that $G_ x \to \mathop{\mathrm{Spec}}(k)$ is locally quasi-finite and hence $G_ x$ is a scheme by Spaces over Fields, Lemma 66.10.8. $\square$

There are also:

• 1 comment(s) on Section 95.19: Automorphism groups

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).