Lemma 101.7.10. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. Let W \to \mathcal{Y} be surjective, flat, and locally of finite presentation where W is an algebraic space. If the base change W \times _\mathcal {Y} \mathcal{X} \to W is quasi-compact, then f is quasi-compact.
Proof. Assume W \times _\mathcal {Y} \mathcal{X} \to W is quasi-compact. Let \mathcal{Z} \to \mathcal{Y} be a morphism with \mathcal{Z} a quasi-compact algebraic stack. Choose a scheme U and a surjective smooth morphism U \to W \times _\mathcal {Y} \mathcal{Z}. Since U \to \mathcal{Z} is flat, surjective, and locally of finite presentation and \mathcal{Z} is quasi-compact, we can find a quasi-compact open subscheme U' \subset U such that U' \to \mathcal{Z} is surjective. Then U' \times _\mathcal {Y} \mathcal{X} = U' \times _ W (W \times _\mathcal {Y} \mathcal{X}) is quasi-compact by assumption and surjects onto \mathcal{Z} \times _\mathcal {Y} \mathcal{X}. Hence \mathcal{Z} \times _\mathcal {Y} \mathcal{X} is quasi-compact as desired. \square
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