Lemma 101.27.7 (0CMG)—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 101.27.7 (cite)

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Lemma 101.27.7. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks with diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ . If $f$ is locally of finite type then $\Delta$ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\Delta$ is of finite presentation.

Proof. Note that $\Delta$ is a morphism over $\mathcal{X}$ (via the second projection). If $f$ is locally of finite type, then $\mathcal{X}$ is of finite presentation over $\mathcal{X}$ and $\text{pr}_2 : \mathcal{X} \times _\mathcal {Y} \mathcal{X} \to \mathcal{X}$ is locally of finite type by Lemma 101.17.3. Thus the first statement holds by Lemma 101.27.6. The second statement follows from the first and the definitions (because $f$ being quasi-separated means by definition that $\Delta _ f$ is quasi-compact and quasi-separated). $\square$

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