Lemma 88.8.11. Let $\mathcal{F}$ be a predeformation category. Let $\xi$ be a versal formal object of $\mathcal{F}$. For any formal object $\eta$ of $\widehat{\mathcal{F}}$, there exists a morphism $\xi \to \eta$.

Proof. By assumption the morphism $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ is smooth. Then $\iota (\xi ) : \underline{R} \to \widehat{\mathcal{F}}$ is the completion of $\underline{\xi }$, see Remark 88.7.12. By Lemma 88.8.8 there exists an object $f$ of $\underline{R}$ such that $\iota (\xi )(f) = \eta$. Then $f$ is a ring map $f : R \to S$ in $\widehat{\mathcal{C}}_\Lambda$. And $\iota (\xi )(f) = \eta$ means that $f_*\xi \cong \eta$ which means exactly that there is a morphism $\xi \to \eta$ lying over $f$. $\square$

There are also:

• 2 comment(s) on Section 88.8: Smooth morphisms

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).