The Stacks project

Lemma 97.13.8. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $k$ be a finite type field over $S$ and let $x_0$ be an object of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(k)$ with image $s \in S$. Assume

  1. $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces,

  2. $\mathcal{X}$ satisfies axioms [1], [2], [3] (see Section 97.14),

  3. every formal object of $\mathcal{X}$ is effective,

  4. openness of versality holds for $\mathcal{X}$, and

  5. $\mathcal{O}_{S, s}$ is a G-ring.

Then there exist a morphism of finite type $U \to S$ and an object $x$ of $\mathcal{X}$ over $U$ such that

\[ x : (\mathit{Sch}/U)_{fppf} \longrightarrow \mathcal{X} \]

is smooth and such that there exists a finite type point $u_0 \in U$ whose residue field is $k$ and such that $x|_{u_0} \cong x_0$.

Proof. By axiom [2], Lemma 97.6.1, and Remark 97.6.2 we see that $\mathcal{F}_{\mathcal{X}, k, x_0}$ satisfies (S1) and (S2). Since also the tangent space has finite dimension by axiom [3] we deduce from Formal Deformation Theory, Lemma 89.13.4 that $\mathcal{F}_{\mathcal{X}, k, x_0}$ has a versal formal object $\xi $. Assumption (3) says $\xi $ is effective. By axiom [1] and Lemma 97.12.7 there exists a morphism of finite type $U \to S$, an object $x$ of $\mathcal{X}$ over $U$, and a finite type point $u_0$ of $U$ with residue field $k$ such that $x$ is versal at $u_0$ and such that $x|_{\mathop{\mathrm{Spec}}(k)} \cong x_0$. By openness of versality we may shrink $U$ and assume that $x$ is versal at every finite type point of $U$. We claim that

\[ x : (\mathit{Sch}/U)_{fppf} \longrightarrow \mathcal{X} \]

is smooth which proves the lemma. Namely, by Lemma 97.13.3 $x$ satisfies ( whereupon Lemma 97.13.7 finishes the proof. $\square$

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