The Stacks project

Lemma 89.13.1. Let $\mathcal{F}$ be a predeformation category. Assume $\mathcal{F}$ has a versal formal object. Then $\mathcal{F}$ satisfies (S1).

Proof. Let $\xi $ be a versal formal object of $\mathcal{F}$. Let

\[ \xymatrix{ & x_2 \ar[d] \\ x_1 \ar[r] & x } \]

be a diagram in $\mathcal{F}$ such that $x_2 \to x$ lies over a surjective ring map. Since the natural morphism $\widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda } \xrightarrow {\sim } \mathcal{F}$ is an equivalence (see Remark 89.7.7), we can consider this diagram also as a diagram in $\widehat{\mathcal{F}}$. By Lemma 89.8.11 there exists a morphism $\xi \to x_1$, so by Remark 89.8.10 we also get a morphism $\xi \to x_2$ making the diagram

\[ \xymatrix{ \xi \ar[r] \ar[d] & x_2 \ar[d] \\ x_1 \ar[r] & x } \]

commute. If $x_1 \to x$ and $x_2 \to x$ lie above ring maps $A_1 \to A$ and $A_2 \to A$ then taking the pushforward of $\xi $ to $A_1 \times _ A A_2$ gives an object $y$ as required by (S1). $\square$

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