The Stacks project

Lemma 88.14.5. Let $\mathcal{F}$ be a predeformation category which has a versal formal object. Then

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

  2. minimal versal objects are unique up to isomorphism, and

  3. any versal object is the pushforward of a minimal versal object along a power series ring extension.

Proof. Suppose $\mathcal{F}$ has a versal formal object $\xi $ over $R$. Then it satisfies (S1), see Lemma 88.13.1. Let $\xi ' \to \xi $ over $R' \subset R$ be any of the morphisms constructed in Lemma 88.14.2. By Lemma 88.14.3 we see that $\xi '$ is versal, hence it is a minimal versal formal object (by construction). This proves (1). Also, $R \cong R'[[x_1, \ldots , x_ n]]$ which proves (3).

Suppose that $\xi _ i/R_ i$ are two minimal versal formal objects. By Lemma 88.8.11 there exist morphisms $\xi _1 \to \xi _2$ and $\xi _2 \to \xi _1$. The corresponding ring maps $f : R_1 \to R_2$ and $g : R_2 \to R_1$ are surjective by minimality. Hence the compositions $g \circ f : R_1 \to R_1$ and $f \circ g : R_2 \to R_2$ are isomorphisms by Algebra, Lemma 10.31.10. Thus $f$ and $g$ are isomorphisms whence the maps $\xi _1 \to \xi _2$ and $\xi _2 \to \xi _1$ are isomorphisms (because $\widehat{\mathcal{F}}$ is cofibred in groupoids by Lemma 88.7.2). This proves (2) and finishes the proof of the lemma. $\square$


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