The Stacks project

Lemma 106.14.3. Let $p : \mathcal{X} \to Y$ be a morphism from an algebraic stack to an algebraic space. Assume

  1. $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is finite,

  2. $p$ is proper, and

  3. $Y$ is locally Noetherian.

Let $f : \mathcal{X} \to M$ be the moduli space constructed in Theorem 106.13.9. Then $M \to Y$ is proper.

Proof. By Lemma 106.14.1 we see that $M \to Y$ is locally of finite type. By Lemma 106.14.2 we see that $M \to Y$ is separated. Of course $M \to Y$ is quasi-compact and universally closed as these are topological properties and $\mathcal{X} \to Y$ has these properties and $\mathcal{X} \to M$ is a universal homeomorphism. $\square$

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