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
Comments (0)
There are also: