Lemma 105.2.4. In Situation 105.2.1 let $x_0 : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Then a versal ring to $\mathcal{X}$ at $x_0$ exists. Given a pair $A$, $A'$ of these, then $A \cong A'[[t_1, \ldots , t_ r]]$ or $A' \cong A[[t_1, \ldots , t_ r]]$ as $S$-algebras for some $r$.

**Proof.**
By Lemma 105.2.3 and Formal Deformation Theory, Lemma 88.13.4 (note that the assumptions of this lemma hold by Formal Deformation Theory, Lemmas 88.16.6 and Definition 88.16.8). By the uniquness result of Formal Deformation Theory, Lemma 88.14.5 there exists a “minimal” versal ring $A$ of $\mathcal{X}$ at $x_0$ such that any other versal ring of $\mathcal{X}$ at $x_0$ is isomorphic to $A[[t_1, \ldots , t_ r]]$ for some $r$. This clearly implies the second statement.
$\square$

## 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Your email address will not be published.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)