The Stacks project

Remark 89.15.7. Let $\mathcal{F}$ be a predeformation category. Recall that $\mathcal{F} \to \overline{\mathcal{F}}$ is smooth, see Remark 89.8.5. Hence if $\xi \in \widehat{\mathcal{F}}(R)$ is a versal formal object, then the composition

\[ \underline{R}|_{\mathcal{C}_\Lambda } \longrightarrow \mathcal{F} \longrightarrow \overline{\mathcal{F}} \]

is smooth (Lemma 89.8.7) and we conclude that the image $\overline{\xi }$ of $\xi $ in $\overline{\mathcal{F}}$ is a versal formal object. If (89.15.0.1) holds, then $\overline{\xi }$ induces an isomorphism $T\underline{R}|_{\mathcal{C}_\Lambda } \to T\overline{\mathcal{F}}$ because $\mathcal{F} \to \overline{\mathcal{F}}$ identifies tangent spaces. Hence in this case $\overline{\xi }$ is a hull for $\overline{\mathcal{F}}$, see Remark 89.15.6. By Theorem 89.15.5 we can always find such a $\xi $ if $k' \subset k$ is separable and $\mathcal{F}$ is a predeformation category satisfying (S1), (S2), and $\dim _ k T\mathcal{F} < \infty $.


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