The Stacks project

Lemma 89.15.2. Let $\mathcal{F}$ be a predeformation category. Let $\xi $ be a versal formal object of $\mathcal{F}$ such that (89.15.0.1) holds. Then

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

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

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

Proof. Condition (S1) holds by Lemma 89.13.1. The first part of (S2) holds since (S1) holds. Let

\[ \vcenter { \xymatrix{ y \ar[r]_ c \ar[d]_ a & x_\epsilon \ar[d]^ e \\ x \ar[r]^ d & x_0 } } \quad \text{and}\quad \vcenter { \xymatrix{ y' \ar[r]_{c'} \ar[d]_{a'} & x_\epsilon \ar[d]^ e \\ x \ar[r]^ d & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A \times _ k k[\epsilon ] \ar[r] \ar[d] & k[\epsilon ] \ar[d] \\ A \ar[r] & k } } \]

be diagrams as in the second part of (S2). As above we can find morphisms $b : \xi \to y$ and $b' : \xi \to y'$ such that

\[ \xymatrix{ \xi \ar[r]^{b'} \ar[d]_ b & y' \ar[d]^{a'} \\ y \ar[r]^{a} & x } \]

commutes. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ denote the structure morphism. Say $\widehat{p}(\xi ) = R$, i.e., $\xi $ lies over $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$. We see that the pushforward of $\xi $ via $p(c) \circ p(b)$ is $x_\epsilon $ and that the pushforward of $\xi $ via $p(c') \circ p(b')$ is $x_\epsilon $. Since $\xi $ satisfies (89.15.0.1), we see that $p(c) \circ p(b) = p(c') \circ p(b')$ as maps $R \to k[\epsilon ]$. Hence $p(b) = p(b')$ as maps from $R \to A \times _ k k[\epsilon ]$. Thus we see that $y$ and $y'$ are isomorphic to the pushforward of $\xi $ along this map and we get a unique morphism $y \to y'$ over $A \times _ k k[\epsilon ]$ compatible with $b$ and $b'$ as desired.

Finally, by Example 89.11.11 we see $\dim _ k T\mathcal{F} = \dim _ k T\underline{R}|_{\mathcal{C}_\Lambda }$ is finite. $\square$


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