Lemma 100.7.8. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
If $\mathcal{X}$ is quasi-compact and $\mathcal{Y}$ is quasi-separated, then $f$ is quasi-compact.
If $\mathcal{X}$ is quasi-compact and quasi-separated and $\mathcal{Y}$ is quasi-separated, then $f$ is quasi-compact and quasi-separated.
A fibre product of quasi-compact and quasi-separated algebraic stacks is quasi-compact and quasi-separated.
Proof. Part (1) follows from Lemma 100.7.7. Part (2) follows from (1) and Lemma 100.4.12. For (3) let $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Z} \to \mathcal{Y}$ be morphisms of quasi-compact and quasi-separated algebraic stacks. Then $\mathcal{X} \times _\mathcal {Y} \mathcal{Z} \to \mathcal{Z}$ is quasi-compact and quasi-separated as a base change of $\mathcal{X} \to \mathcal{Y}$ using (2) and Lemmas 100.7.3 and 100.4.4. Hence $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ is quasi-compact and quasi-separated as an algebraic stack quasi-compact and quasi-separated over $\mathcal{Z}$, see Lemmas 100.4.11 and 100.7.4. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)