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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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