Lemma 100.19.1. In the situation above $G_ x$ is a scheme if one of the following holds

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

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

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

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

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

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

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