Lemma 98.12.5 (07XG)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 98: Artin's Axioms
Section 98.12: Versality
Lemma 98.12.5 (cite)

Lemma 98.12.5. Let $S$ , $\mathcal{X}$ , $U$ , $x$ , $u_0$ be as in Definition 98.12.2. Let $l$ be a field and let $u_{l, 0} : \mathop{\mathrm{Spec}}(l) \to U$ be a morphism with image $u_0$ such that $l/k = \kappa (u_0)$ is finite. Set $x_{l, 0} = x_0|_{\mathop{\mathrm{Spec}}(l)}$ . If $\mathcal{X}$ satisfies (RS) and $x$ is versal at $u_0$ , then

$\mathcal{F}_{(\mathit{Sch}/U)_{fppf}, l, u_{l, 0}} \longrightarrow \mathcal{F}_{\mathcal{X}, l, x_{l, 0}}$

is smooth.

Proof. Note that $(\mathit{Sch}/U)_{fppf}$ satisfies (RS) by Lemma 98.5.2. Hence the functor of the lemma is the functor

$(\mathcal{F}_{(\mathit{Sch}/U)_{fppf}, k , u_0})_{l/k} \longrightarrow (\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k}$

associated to $\hat x$ , see Lemma 98.7.1. Hence the lemma follows from Formal Deformation Theory, Lemma 90.29.5. $\square$

The Stacks project

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