Lemma 93.10.5. In Example 93.10.1 assume X \to Y is a morphism of proper k-schemes. 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 \to Y}(A))/\cong
of isomorphism classes of objects has a hull. If \text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X) = \text{Der}_ k(\mathcal{O}_ Y, \mathcal{O}_ Y) = 0, then F is prorepresentable.
Proof.
The existence of a hull follows immediately from Lemmas 93.10.2 and 93.10.4 and Formal Deformation Theory, Lemma 90.16.6 and Remark 90.15.7.
Assume \text{Der}_ k(\mathcal{O}_ X, \mathcal{O}_ X) = \text{Der}_ k(\mathcal{O}_ Y, \mathcal{O}_ Y) = 0. Then the exact sequence of Lemma 93.10.3 combined with Lemma 93.9.3 shows that \text{Inf}(\mathcal{D}\! \mathit{ef}_{X \to Y}) = 0. Then \mathcal{D}\! \mathit{ef}_{X \to Y} and F are equivalent by Formal Deformation Theory, Lemma 90.19.13. Hence F is a deformation functor (because \mathcal{D}\! \mathit{ef}_{X \to Y} is a deformation category) with finite tangent space and we can apply Formal Deformation Theory, Theorem 90.18.2.
\square
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