The Stacks project

Lemma 89.9.4. Let $\mathcal{F}$ be a predeformation category. Let $\xi $ be a versal formal object of $\mathcal{F}$ lying over $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$. The following are equivalent

  1. $\mathcal{F}$ is unobstructed, and

  2. $\Lambda \to R$ is formally smooth in the $\mathfrak m_ R$-adic topology.

In the classical case these are also equivalent to

  1. $R \cong \Lambda [[x_1, \ldots , x_ n]]$ for some $n$.

Proof. If (1) holds, i.e., if $\mathcal{F}$ is unobstructed, then the composition

\[ \underline{R}|_{\mathcal{C}_\Lambda } \xrightarrow {\underline{\xi }} \mathcal{F} \to \mathcal{C}_\Lambda \]

is smooth, see Lemma 89.8.7. Hence we see that (2) holds by Lemma 89.9.3. Conversely, if (2) holds, then the composition is smooth and moreover the first arrow is essentially surjective by Lemma 89.8.11. Hence we find that the second arrow is smooth by Lemma 89.8.7 which means that $\mathcal{F}$ is unobstructed by definition. The equivalence with (3) in the classical case follows from Lemma 89.9.3. $\square$

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