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).

## 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.

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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