The Stacks project

Lemma 90.14.5. Let $\mathcal{F}$ be a predeformation category which has a versal formal object. Then

  1. $\mathcal{F}$ has a minimal versal formal object,

  2. minimal versal objects are unique up to isomorphism, and

  3. any versal object is the pushforward of a minimal versal object along a power series ring extension.

Proof. Suppose $\mathcal{F}$ has a versal formal object $\xi $ over $R$. Then it satisfies (S1), see Lemma 90.13.1. Let $\xi ' \to \xi $ over $R' \subset R$ be any of the morphisms constructed in Lemma 90.14.2. By Lemma 90.14.3 we see that $\xi '$ is versal, hence it is a minimal versal formal object (by construction). This proves (1). Also, $R \cong R'[[x_1, \ldots , x_ n]]$ which proves (3).

Suppose that $\xi _ i/R_ i$ are two minimal versal formal objects. By Lemma 90.8.11 there exist morphisms $\xi _1 \to \xi _2$ and $\xi _2 \to \xi _1$. The corresponding ring maps $f : R_1 \to R_2$ and $g : R_2 \to R_1$ are surjective by minimality. Hence the compositions $g \circ f : R_1 \to R_1$ and $f \circ g : R_2 \to R_2$ are isomorphisms by Algebra, Lemma 10.31.10. Thus $f$ and $g$ are isomorphisms whence the maps $\xi _1 \to \xi _2$ and $\xi _2 \to \xi _1$ are isomorphisms (because $\widehat{\mathcal{F}}$ is cofibred in groupoids by Lemma 90.7.2). This proves (2) and finishes the proof of the lemma. $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06T5. Beware of the difference between the letter 'O' and the digit '0'.