Lemma 101.18.4 (06G0)—The Stacks project

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Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.18: Points of finite type
Lemma 101.18.4 (cite)

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Lemma 101.18.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. If $f$ is locally of finite type, then $f(\mathcal{X}_{\text{ft-pts}}) \subset \mathcal{Y}_{\text{ft-pts}}$ .

Proof. Take $x \in \mathcal{X}_{\text{ft-pts}}$ . Represent $x$ by a locally finite type morphism $x : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ . Then $f \circ x$ is locally of finite type by Lemma 101.17.2. Hence $f(x) \in \mathcal{Y}_{\text{ft-pts}}$ . $\square$

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