Lemma 93.9.5. In Example 93.9.1 assume X is a proper k-scheme. Assume \Lambda is a complete local ring with residue field k (the classical case). Then the functor
F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}\! \mathit{ef}_ X(A))/\cong
of isomorphism classes of objects has a hull. If \text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X) = 0, then F is prorepresentable.
Proof.
The existence of a hull follows immediately from Lemmas 93.9.2 and 93.9.4 and Formal Deformation Theory, Lemma 90.16.6 and Remark 90.15.7.
Assume \text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X) = 0. Then \mathcal{D}\! \mathit{ef}_ X and F are equivalent by Formal Deformation Theory, Lemma 90.19.13. Hence F is a deformation functor (because \mathcal{D}\! \mathit{ef}_ X is a deformation category) with finite tangent space and we can apply Formal Deformation Theory, Theorem 90.18.2.
\square
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