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The Stacks project

Lemma 101.28.14. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks such that \mathcal{X} is a gerbe over \mathcal{Y}. If \Delta _\mathcal {X} is quasi-compact, so is \Delta _\mathcal {Y}.

Proof. Consider the diagram

\xymatrix{ \mathcal{X} \ar[r] & \mathcal{X} \times _\mathcal {Y} \mathcal{X} \ar[r] \ar[d] & \mathcal{X} \times \mathcal{X} \ar[d] \\ & \mathcal{Y} \ar[r] & \mathcal{Y} \times \mathcal{Y} }

By Proposition 101.28.11 we find that the arrow on the top left is surjective. Since the composition of the top horizontal arrows is quasi-compact, we conclude that the top right arrow is quasi-compact by Lemma 101.7.6. The square is cartesian and the right vertical arrow is surjective, flat, and locally of finite presentation. Thus we conclude by Lemma 101.27.16. \square


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