Lemma 101.27.5. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
If $\mathcal{Y}$ is locally Noetherian and $f$ locally of finite type then $f$ is locally of finite presentation.
If $\mathcal{Y}$ is locally Noetherian and $f$ of finite type and quasi-separated then $f$ is of finite presentation.
Proof. Assume $f : \mathcal{X} \to \mathcal{Y}$ locally of finite type and $\mathcal{Y}$ locally Noetherian. This means there exists a diagram as in Lemma 101.16.1 with $h$ locally of finite type and surjective vertical arrow $a$. By Morphisms of Spaces, Lemma 67.28.7 $h$ is locally of finite presentation. Hence $\mathcal{X} \to \mathcal{Y}$ is locally of finite presentation by definition. This proves (1). If $f$ is of finite type and quasi-separated then it is also quasi-compact and quasi-separated and (2) follows immediately. $\square$
Your email address will not be published.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)
There are also: