Lemma 98.13.3 (07XT)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 98: Artin's Axioms
Section 98.13: Openness of versality
Lemma 98.13.3 (cite)

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Lemma 98.13.3. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ . Let $U$ be a scheme locally of finite type over $S$ . Let $x$ be an object of $\mathcal{X}$ over $U$ . Assume that $x$ is versal at every finite type point of $U$ and that $\mathcal{X}$ satisfies (RS). Then $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ satisfies (98.13.2.1).

Proof. Let $\mathop{\mathrm{Spec}}(l) \to U$ be a morphism with $l$ of finite type over $S$ . Then the image $u_0 \in U$ is a finite type point of $U$ and $l/\kappa (u_0)$ is a finite extension, see discussion in Morphisms, Section 29.16. Hence we see that $\mathcal{F}_{(\mathit{Sch}/U)_{fppf}, l, u_{l, 0}} \to \mathcal{F}_{\mathcal{X}, l, x_{l, 0}}$ is smooth by Lemma 98.12.5. $\square$

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