The Stacks project

Lemma 98.12.3. With notation as in Definition 98.12.2. Let $R = \mathcal{O}_{U, u_0}^\wedge $. Let $\xi $ be the formal object of $\mathcal{X}$ over $R$ associated to $x|_{\mathop{\mathrm{Spec}}(R)}$, see ( Then

\[ x\text{ is versal at }u_0 \Leftrightarrow \xi \text{ is versal} \]

Proof. Observe that $\mathcal{O}_{U, u_0}$ is a Noetherian local $S$-algebra with residue field $k$. Hence $R = \mathcal{O}_{U, u_0}^\wedge $ is an object of $\mathcal{C}_\Lambda ^\wedge $, see Formal Deformation Theory, Definition 90.4.1. Recall that $\xi $ is versal if $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth and $x$ is versal at $u_0$ if $\hat x : \mathcal{F}_{(\mathit{Sch}/U)_{fppf}, k, u_0} \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth. There is an identification of predeformation categories

\[ \underline{R}|_{\mathcal{C}_\Lambda } = \mathcal{F}_{(\mathit{Sch}/U)_{fppf}, k, u_0}, \]

see Formal Deformation Theory, Remark 90.7.12 for notation. Namely, given an Artinian local $S$-algebra $A$ with residue field identified with $k$ we have

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda ^\wedge }(R, A) = \{ \varphi \in \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(A), U) \mid \varphi |_{\mathop{\mathrm{Spec}}(k)} = u_0\} \]

Unwinding the definitions the reader verifies that the resulting map

\[ \underline{R}|_{\mathcal{C}_\Lambda } = \mathcal{F}_{(\mathit{Sch}/U)_{fppf}, k, u_0} \xrightarrow {\hat x} \mathcal{F}_{\mathcal{X}, k, x_0}, \]

is equal to $\underline{\xi }$ and we see that the lemma is true. $\square$

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