Lemma 101.7.6 (050X)—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 101.7.6 (cite)

numberstags

$\xymatrix{ \mathcal{X} \ar[rr]_ f \ar[rd]_ p & & \mathcal{Y} \ar[dl]^ q \\ & \mathcal{Z} }$

be a $2$ -commutative diagram of morphisms of algebraic stacks. If $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact.

Proof. Let $\mathcal{T}$ be a quasi-compact algebraic stack, and let $\mathcal{T} \to \mathcal{Z}$ be a morphism. By Properties of Stacks, Lemma 100.5.3 the morphism $\mathcal{T} \times _\mathcal {Z} \mathcal{X} \to \mathcal{T} \times _\mathcal {Z} \mathcal{Y}$ is surjective and by assumption $\mathcal{T} \times _\mathcal {Z} \mathcal{X}$ is quasi-compact. Hence $\mathcal{T} \times _\mathcal {Z} \mathcal{Y}$ is quasi-compact by Properties of Stacks, Lemma 100.6.2. $\square$

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