The Stacks project

Lemma 101.7.7. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $g \circ f$ is quasi-compact and $g$ is quasi-separated then $f$ is quasi-compact.

Proof. This is true because $f$ equals the composition $(1, f) : \mathcal{X} \to \mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{Y}$. The first map is quasi-compact by Lemma 101.4.9 because it is a section of the quasi-separated morphism $\mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{X}$ (a base change of $g$, see Lemma 101.4.4). The second map is quasi-compact as it is the base change of $f$, see Lemma 101.7.3. And compositions of quasi-compact morphisms are quasi-compact, see Lemma 101.7.4. $\square$

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