Lemma 90.8.11. 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 90.7.12. By Lemma 90.8.8 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 $f$. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: