The Stacks project

Lemma 89.26.3. Let $\mathcal{F}$ be a deformation category. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a prorepresentable functor. Let $f : U \to \mathcal{F}$ be a morphism of categories cofibered in groupoids. Let $(U, R, s, t, c)$ be the groupoid in functors on $\mathcal{C}_\Lambda $ constructed from $f : U \to \mathcal{F}$ in Lemma 89.25.2. If $\dim _ k \text{Inf}(\mathcal{F}) < \infty $, then $(U, R, s, t, c)$ is prorepresentable.

Proof. Note that $U$ is a deformation functor by Example 89.16.10. By Lemma 89.26.2 we see that $R = U \times _{f, \mathcal{F}, f} U$ is a deformation functor whose tangent space $TR = T(U \times _{f, \mathcal{F}, f} U)$ sits in an exact sequence $0 \to \text{Inf}(\mathcal{F}) \to TR \to TU \oplus TU$. Since we have assumed the first space has finite dimension and since $TU$ has finite dimension by Example 89.11.11 we see that $\dim TR < \infty $. The map $\gamma : \text{Der}_\Lambda (k, k) \to TR$ see ( is injective because its composition with $TR \to TU$ is injective by Theorem 89.18.2 for the prorepresentable functor $U$. Thus $R$ is prorepresentable by Theorem 89.18.2. It follows from Lemma 89.22.4 that $(U, R, s, t, c)$ is prorepresentable. $\square$

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