Lemma 105.14.2. Let $\mathcal{X}$ be an algebraic stack. Assume $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is finite. Let $f : \mathcal{X} \to M$ be the moduli space constructed in Theorem 105.13.9.

If $\mathcal{X}$ is quasi-separated, then $M$ is quasi-separated.

If $\mathcal{X}$ is separated, then $M$ is separated.

Add more here, for example relative versions of the above.

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