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. $\square$

Comment #2644 by Xiaowen on

$Y$ in the proof should be $F_\theta$.

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