The Stacks project

Lemma 101.19.1. 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 101.6.1. 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 68.6 and Lemma 68.15.2). Then $G_ x$ is separated by More on Groupoids in Spaces, Lemma 79.9.4 whereupon we conclude that $G_ x$ is a scheme by More on Groupoids in Spaces, Proposition 79.10.3. 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 72.10.8. $\square$

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