The Stacks project

Theorem 90.18.2. Let $F: \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. Then $F$ is prorepresentable if and only if (a) $F$ is a deformation functor, (b) $\dim _ k TF$ is finite, and (c) $\gamma : \text{Der}_\Lambda (k, k) \to TF$ is injective.

Proof. Assume $F$ is prorepresentable by $R \in \widehat{\mathcal{C}}_\Lambda $. We see $F$ is a deformation functor by Example 90.16.10. We see $\dim _ k TF$ is finite by Example 90.11.11. Finally, $\text{Der}_\Lambda (k, k) \to TF$ is identified with $\text{Der}_\Lambda (k, k) \to \text{Der}_\Lambda (R, k)$ by Example 90.11.14 which is injective because $R \to k$ is surjective.

Conversely, assume (a), (b), and (c) hold. By Lemma 90.16.6 we see that (S1) and (S2) hold. Hence by Theorem 90.15.5 there exists a minimal versal formal object $\xi $ of $F$ such that ( holds. Say $\xi $ lies over $R$. The map

\[ d\underline{\xi } : \text{Der}_\Lambda (R, k) \to T\mathcal{F} \]

is bijective on $\text{Der}_\Lambda (k, k)$-orbits. Since the action of $\text{Der}_\Lambda (k, k)$ on the left hand side is free by (c) and Lemma 90.12.6 we see that the map is bijective. Thus we see that $\underline{\xi }$ is an isomorphism by Lemma 90.18.1. $\square$

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