Lemma 101.7.8 (075S)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 101: Morphisms of Algebraic Stacks
Section 101.7: Quasi-compact morphisms
Lemma 101.7.8 (cite)

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Lemma 101.7.8. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.

If $\mathcal{X}$ is quasi-compact and $\mathcal{Y}$ is quasi-separated, then $f$ is quasi-compact.
If $\mathcal{X}$ is quasi-compact and quasi-separated and $\mathcal{Y}$ is quasi-separated, then $f$ is quasi-compact and quasi-separated.
A fibre product of quasi-compact and quasi-separated algebraic stacks is quasi-compact and quasi-separated.

Proof. Part (1) follows from Lemma 101.7.7. Part (2) follows from (1) and Lemma 101.4.12. For (3) let $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Z} \to \mathcal{Y}$ be morphisms of quasi-compact and quasi-separated algebraic stacks. Then $\mathcal{X} \times _\mathcal {Y} \mathcal{Z} \to \mathcal{Z}$ is quasi-compact and quasi-separated as a base change of $\mathcal{X} \to \mathcal{Y}$ using (2) and Lemmas 101.7.3 and 101.4.4. Hence $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ is quasi-compact and quasi-separated as an algebraic stack quasi-compact and quasi-separated over $\mathcal{Z}$ , see Lemmas 101.4.11 and 101.7.4. $\square$

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