The Stacks project

Remark 90.15.6. Let $F : \mathcal{C}_\Lambda \to \textit{Sets}$ be a predeformation functor satisfying (S1) and (S2) and $\dim _ k TF < \infty $. Recall that these conditions correspond to the conditions (H1), (H2), and (H3) from Schlessinger's paper, see Remark 90.13.5. Now, in the classical case (or if $k' \subset k$ is separable) following Schlessinger we introduce the notion of a hull: a hull is a versal formal object $\xi \in \widehat{F}(R)$ such that $d\underline{\xi } : T\underline{R}|_{\mathcal{C}_\Lambda } \to TF$ is an isomorphism, i.e., ( holds. Thus Theorem 90.15.5 tells us

\[ (H1) + (H2) + (H3) \Rightarrow \text{ there exists a hull} \]

in the classical case. In other words, our theorem recovers Schlessinger's theorem on the existence of hulls.

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