## 100.24 Quasi-finite morphisms

We have defined “locally quasi-finite” morphisms of algebraic stacks in Section 100.23 and “quasi-compact” morphisms of algebraic stacks in Section 100.7. Since a morphism of algebraic spaces is by definition quasi-finite if and only if it is both locally quasi-finite and quasi-compact (Morphisms of Spaces, Definition 66.27.1), we may define what it means for a morphism of algebraic stacks to be quasi-finite as follows and it agrees with the already existing notion defined in Properties of Stacks, Section 99.3 when the morphism is representable by algebraic spaces.

Definition 100.24.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is quasi-finite if $f$ is locally quasi-finite (Definition 100.23.2) and quasi-compact (Definition 100.7.2).

Lemma 100.24.2. The composition of quasi-finite morphisms is quasi-finite.

Lemma 100.24.3. A base change of a quasi-finite morphism is quasi-finite.

Lemma 100.24.4. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $g \circ f$ is quasi-finite and $g$ is quasi-separated and quasi-DM then $f$ is quasi-finite.

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