Lemma 97.11.4. 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 96.5.6. 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 93.9). It suffices to show that $F_\theta \to V$ has the corresponding properties. By Lemmas 97.11.2 and 97.11.3 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 66.23.6.
$\square$

## 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Your email address will not be published.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.

## Comments (2)

Comment #2644 by Xiaowen on

Comment #2664 by Johan on

There are also: