The Stacks project

101.24 Quasi-finite morphisms

We have defined “locally quasi-finite” morphisms of algebraic stacks in Section 101.23 and “quasi-compact” morphisms of algebraic stacks in Section 101.7. 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 67.27.1), 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 100.3 when the morphism is representable by algebraic spaces.

reference

Definition 101.24.1. 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 101.23.2) and quasi-compact (Definition 101.7.2).

Lemma 101.24.2. The composition of quasi-finite morphisms is quasi-finite.

Proof. Combine Lemmas 101.23.5 and 101.7.4. $\square$

Lemma 101.24.3. A base change of a quasi-finite morphism is quasi-finite.

Proof. Combine Lemmas 101.23.3 and 101.7.3. $\square$

Lemma 101.24.4. 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.

Proof. Combine Lemmas 101.23.8 and 101.7.7. $\square$