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

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

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). $\square$

## 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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)