The Stacks project

Theorem 90.15.5. Let $\mathcal{F}$ be a predeformation category. Consider the following conditions

  1. $\mathcal{F}$ has a minimal versal formal object satisfying (,

  2. $\mathcal{F}$ has a minimal versal formal object satisfying (,

  3. the following conditions hold:

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

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

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

We always have

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

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

Proof. Lemma 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 ( $=$ (, 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 ( in this case. This avoids the use of Lemma 90.13.4 in the classical case. Details omitted. $\square$

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