The Stacks project

Lemma 89.27.6. Let $\mathcal{F}$ be a deformation category such that $\dim _ k T\mathcal{F} <\infty $ and $\dim _ k \text{Inf}(\mathcal{F}) < \infty $. Then there exists a minimal versal formal object $\xi $ of $\mathcal{F}$. Say $\xi $ lies over $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$. Let $U = \underline{R}|_{\mathcal{C}_\Lambda }$. Let $f = \underline{\xi } : U \to \mathcal{F}$ be the associated morphism. Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}_\Lambda $ constructed from $f : U \to \mathcal{F}$ in Lemma 89.25.2. Then $(U, R, s, t, c)$ is a minimal smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $ and there is an equivalence $[U/R] \to \mathcal{F}$.

Proof. As $\mathcal{F}$ is a deformation category it satisfies (S1) and (S2), see Lemma 89.16.6. By Lemma 89.13.4 there exists a versal formal object. By Lemma 89.14.5 there exists a minimal versal formal object $\xi /R$ as in the statement of the lemma. Setting $U = \underline{R}|_{\mathcal{C}_\Lambda }$ the associated map $\underline{\xi } : U \to \mathcal{F}$ is smooth (this is the definition of a versal formal object). Let $(U, R, s, t, c)$ be the groupoid in functors constructed in Lemma 89.25.2 from the map $\underline{\xi }$. By Lemma 89.26.1 we see that $(U, R, s, t, c)$ is a smooth groupoid in functors and that $[U/R] \to \mathcal{F}$ is an equivalence. By Lemma 89.26.3 we see that $(U, R, s, t, c)$ is prorepresentable. Finally, $(U, R, s, t, c)$ is minimal because $U \to [U/R] = \mathcal{F}$ corresponds to the minimal versal formal object $\xi $. $\square$

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