Theorem 89.15.5. Let $\mathcal{F}$ be a predeformation category. Consider the following conditions

$\mathcal{F}$ has a minimal versal formal object satisfying (89.15.0.1),

$\mathcal{F}$ has a minimal versal formal object satisfying (89.15.0.2),

the following conditions hold:

$\mathcal{F}$ satisfies (S1).

$\mathcal{F}$ satisfies (S2).

$\dim _ k T\mathcal{F}$ is finite.

We always have

\[ (1) \Rightarrow (3) \Rightarrow (2). \]

If $k' \subset k$ is separable, then all three are equivalent.

**Proof.**
Lemma 89.15.2 shows that (1) $\Rightarrow $ (3). Lemmas 89.13.4 and 89.15.4 show that (3) $\Rightarrow $ (2). If $k' \subset k$ is separable then $\text{Der}_\Lambda (k, k) = 0$ and we see that (89.15.0.1) $=$ (89.15.0.2), i.e., (1) is the same as (2).

An alternative proof of (3) $\Rightarrow $ (1) in the classical case is to add a few words to the proof of Lemma 89.13.4 to see that one can right away construct a versal object which satisfies (89.15.0.1) in this case. This avoids the use of Lemma 89.13.4 in the classical case. Details omitted.
$\square$

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