Theorem 90.15.5. Let \mathcal{F} be a predeformation category. Consider the following conditions
\mathcal{F} has a minimal versal formal object satisfying (90.15.0.1),
\mathcal{F} has a minimal versal formal object satisfying (90.15.0.2),
the following conditions hold:
\mathcal{F} satisfies (S1).
\mathcal{F} satisfies (S2).
\dim _ k T\mathcal{F} is finite.
We always have
If k' \subset k is separable, then all three are equivalent.
Proof. Lemma 90.15.2 shows that (1) \Rightarrow (3). Lemmas 90.13.4 and 90.15.4 show that (3) \Rightarrow (2). If k' \subset k is separable then \text{Der}_\Lambda (k, k) = 0 and we see that (90.15.0.1) = (90.15.0.2), 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 90.13.4 to see that one can right away construct a versal object which satisfies (90.15.0.1) in this case. This avoids the use of Lemma 90.13.4 in the classical case. Details omitted. \square
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