The Stacks project

Lemma 98.11.4. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces and $\mathcal{X}$ is limit preserving. Then $\Delta $ is locally of finite type.

Proof. We apply Criteria for Representability, Lemma 97.5.6. Let $V$ be an affine scheme $V$ locally of finite presentation over $S$ and let $\theta $ be an object of $\mathcal{X} \times \mathcal{X}$ over $V$. Let $F_\theta $ be an algebraic space representing $\mathcal{X} \times _{\Delta , \mathcal{X} \times \mathcal{X}, \theta } (\mathit{Sch}/V)_{fppf}$ and let $f_\theta : F_\theta \to V$ be the canonical morphism (see Algebraic Stacks, Section 94.9). It suffices to show that $F_\theta \to V$ has the corresponding properties. By Lemmas 98.11.2 and 98.11.3 we see that $F_\theta \to S$ is locally of finite presentation. It follows that $F_\theta \to V$ is locally of finite type by Morphisms of Spaces, Lemma 67.23.6. $\square$


Comments (2)

Comment #2644 by Xiaowen on

in the proof should be .

There are also:

  • 2 comment(s) on Section 98.11: Limit preserving

Post a comment

Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CXI. Beware of the difference between the letter 'O' and the digit '0'.