Lemma 101.27.11 (06Q8)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.27: Morphisms of finite presentation
Lemma 101.27.11 (cite)

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Lemma 101.27.11. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Z} \to \mathcal{Y}$ be a surjective, flat, locally finitely presented morphism of algebraic stacks. If the base change $\mathcal{Z} \times _\mathcal {Y} \mathcal{X} \to \mathcal{Z}$ is locally of finite presentation, then $f$ is locally of finite presentation.

Proof. The property “locally of finite presentation” satisfies the conditions of Lemma 101.27.10. Smooth local on the source-and-target we have seen in the introduction to this section and fppf local on the target is Descent on Spaces, Lemma 74.11.10. $\square$

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