The Stacks project

Lemma 90.15.2. Let $\mathcal{F}$ be a predeformation category. Let $\xi $ be a versal formal object of $\mathcal{F}$ such that ( holds. Then

  1. $\mathcal{F}$ satisfies (S1).

  2. $\mathcal{F}$ satisfies (S2).

  3. $\dim _ k T\mathcal{F}$ is finite.

Proof. Condition (S1) holds by Lemma 90.13.1. The first part of (S2) holds since (S1) holds. Let

\[ \vcenter { \xymatrix{ y \ar[r]_ c \ar[d]_ a & x_\epsilon \ar[d]^ e \\ x \ar[r]^ d & x_0 } } \quad \text{and}\quad \vcenter { \xymatrix{ y' \ar[r]_{c'} \ar[d]_{a'} & x_\epsilon \ar[d]^ e \\ x \ar[r]^ d & x_0 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ A \times _ k k[\epsilon ] \ar[r] \ar[d] & k[\epsilon ] \ar[d] \\ A \ar[r] & k } } \]

be diagrams as in the second part of (S2). As above we can find morphisms $b : \xi \to y$ and $b' : \xi \to y'$ such that

\[ \xymatrix{ \xi \ar[r]^{b'} \ar[d]_ b & y' \ar[d]^{a'} \\ y \ar[r]^{a} & x } \]

commutes. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ denote the structure morphism. Say $\widehat{p}(\xi ) = R$, i.e., $\xi $ lies over $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$. We see that the pushforward of $\xi $ via $p(c) \circ p(b)$ is $x_\epsilon $ and that the pushforward of $\xi $ via $p(c') \circ p(b')$ is $x_\epsilon $. Since $\xi $ satisfies (, we see that $p(c) \circ p(b) = p(c') \circ p(b')$ as maps $R \to k[\epsilon ]$. Hence $p(b) = p(b')$ as maps from $R \to A \times _ k k[\epsilon ]$. Thus we see that $y$ and $y'$ are isomorphic to the pushforward of $\xi $ along this map and we get a unique morphism $y \to y'$ over $A \times _ k k[\epsilon ]$ compatible with $b$ and $b'$ as desired.

Finally, by Example 90.11.11 we see $\dim _ k T\mathcal{F} = \dim _ k T\underline{R}|_{\mathcal{C}_\Lambda }$ is finite. $\square$

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Your email address will not be published.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 06IV. Beware of the difference between the letter 'O' and the digit '0'.