Lemma 106.14.3. Let p : \mathcal{X} \to Y be a morphism from an algebraic stack to an algebraic space. Assume
\mathcal{I}_\mathcal {X} \to \mathcal{X} is finite,
p is proper, and
Y is locally Noetherian.
Let f : \mathcal{X} \to M be the moduli space constructed in Theorem 106.13.9. Then M \to Y is proper.
Proof. By Lemma 106.14.1 we see that M \to Y is locally of finite type. By Lemma 106.14.2 we see that M \to Y is separated. Of course M \to Y is quasi-compact and universally closed as these are topological properties and \mathcal{X} \to Y has these properties and \mathcal{X} \to M is a universal homeomorphism. \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Your email address will not be published.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)