The Stacks project

Lemma 92.10.5. In Example 92.10.1 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 92.10.2 and 92.10.4 and Formal Deformation Theory, Lemma 89.16.6 and Remark 89.15.7.

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 92.10.3 combined with Lemma 92.9.3 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 89.19.13. 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 89.18.2. $\square$


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