The Stacks project

Lemma 101.23.8. Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. Assume that $\mathcal{X} \to \mathcal{Z}$ is locally quasi-finite and $\mathcal{Y} \to \mathcal{Z}$ is quasi-DM. Then $\mathcal{X} \to \mathcal{Y}$ is locally quasi-finite.

Proof. Write $\mathcal{X} \to \mathcal{Y}$ as the composition

\[ \mathcal{X} \longrightarrow \mathcal{X} \times _\mathcal {Z} \mathcal{Y} \longrightarrow \mathcal{Y} \]

The second arrow is locally quasi-finite as a base change of $\mathcal{X} \to \mathcal{Z}$, see Lemma 101.23.3. The first arrow is locally quasi-finite by Lemma 101.4.8 as $\mathcal{Y} \to \mathcal{Z}$ is quasi-DM. Hence $\mathcal{X} \to \mathcal{Y}$ is locally quasi-finite by Lemma 101.23.5. $\square$

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