The Stacks project

Lemma 97.12.5. Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition 97.12.2. 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 97.5.2. 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 97.7.1. Hence the lemma follows from Formal Deformation Theory, Lemma 89.29.5. $\square$

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