Lemma 101.27.6 (06Q6)—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 101.27.6 (cite)

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Lemma 101.27.6. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks If $g \circ f$ is locally of finite presentation and $g$ is locally of finite type, then $f$ is locally of finite presentation.

Proof. Choose an algebraic space $W$ and a surjective smooth morphism $W \to \mathcal{Z}$ . Choose an algebraic space $V$ and a surjective smooth morphism $V \to \mathcal{Y} \times _\mathcal {Z} W$ . Choose an algebraic space $U$ and a surjective smooth morphism $U \to \mathcal{X} \times _\mathcal {Y} V$ . The lemma follows upon applying Morphisms of Spaces, Lemma 67.28.9 to the morphisms $U \to V \to W$ . $\square$

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