The Stacks project

Lemma 101.48.2. A quasi-separated algebraic stack $\mathcal{X}$ is decent. More generally, if $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is quasi-compact, then $\mathcal{X}$ is decent.

Proof. Namely, if $\mathcal{X}$ is quasi-separated, then any morphism $f : T \to \mathcal{X}$ whose source is a quasi-compact scheme $T$, is quasi-compact, see Lemma 101.7.7. If $\Delta $ is on known to be quasi-compact, then one uses the description

\[ T \times _{f, \mathcal{X}, f'} T' = (T \times T') \times _{(f, f'), \mathcal{X} \times \mathcal{X}, \Delta } \mathcal{X} \]

to prove this. Details omitted. $\square$

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