The Stacks project

Remark 105.2.11. In Situation 105.2.1 let $x_0 : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. By Lemma 105.2.3 and Formal Deformation Theory, Theorem 88.26.4 we know that $\mathcal{F}_{\mathcal{X}, k, x_0}$ has a presentation by a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $. Unwinding the definitions, this means we can choose

  1. a Noetherian complete local $\Lambda $-algebra $A$ with residue field $k$ and a versal formal object $\xi $ of $\mathcal{F}_{\mathcal{X}, k, x_0}$ over $A$,

  2. a Noetherian complete local $\Lambda $-algebra $B$ with residue field $k$ and an isomorphism

    \[ \underline{B}|_{\mathcal{C}_\Lambda } \longrightarrow \underline{A}|_{\mathcal{C}_\Lambda } \times _{\underline{\xi }, \mathcal{F}_{\mathcal{X}, k, x_0}, \underline{\xi }} \underline{A}|_{\mathcal{C}_\Lambda } \]

The projections correspond to formally smooth maps $t : A \to B$ and $s : A \to B$ (because $\xi $ is versal). There is a map $c : B \to B \widehat{\otimes }_{s, A, t} B$ which turns $(A, B, s, t, c)$ into a cogroupoid in the category of Noetherian complete local $\Lambda $-algebras with residue field $k$ (on prorepresentable functors this map is constructed in Formal Deformation Theory, Lemma 88.25.2). Finally, the cited theorem tells us that $\xi $ induces an equivalence

\[ [\underline{A}|_{\mathcal{C}_\Lambda } / \underline{B}|_{\mathcal{C}_\Lambda }] \longrightarrow \mathcal{F}_{\mathcal{X}, k, x_0} \]

of groupoids cofibred over $\mathcal{C}_\Lambda $. In fact, we also get an equivalence

\[ [\underline{A}/\underline{B}] \longrightarrow \widehat{\mathcal{F}}_{\mathcal{X}, k, x_0} \]

of groupoids cofibred over the completed category $\widehat{\mathcal{C}}_\Lambda $ (see discussion in Formal Deformation Theory, Section 88.22 as to why this works). Of course $A$ is a versal ring to $\mathcal{X}$ at $x_0$.


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