The Stacks project

Lemma 88.13.2. Let $\mathcal{F}$ be a predeformation category satisfying (S1) and (S2). Let $\xi $ be a formal object of $\mathcal{F}$ corresponding to $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$, see Remark 88.7.12. Then $\xi $ is versal if and only if the following two conditions hold:

  1. the map $d\underline{\xi } : T\underline{R}|_{\mathcal{C}_\Lambda } \to T\mathcal{F}$ on tangent spaces is surjective, and

  2. given a diagram in $\widehat{\mathcal{F}}$

    \[ \vcenter { \xymatrix{ & y \ar[d] \\ \xi \ar[r] & x } } \quad \text{lying over}\quad \vcenter { \xymatrix{ & B \ar[d]^{f} \\ R \ar[r] & A } } \]

    in $\widehat{\mathcal{C}}_\Lambda $ with $B \to A$ a small extension of Artinian rings, then there exists a ring map $R \to B$ such that

    \[ \xymatrix{ & B \ar[d]^{f} \\ R \ar[ur] \ar[r] & A } \]


Proof. If $\xi $ is versal then (1) holds by Lemma 88.8.8 and (2) holds by Remark 88.8.10. Assume (1) and (2) hold. By Remark 88.8.10 we must show that given a diagram in $\widehat{\mathcal{F}}$ as in (2), there exists $\xi \to y$ such that

\[ \xymatrix{ & y \ar[d] \\ \xi \ar[ur] \ar[r] & x } \]

commutes. Let $b : R \to B$ be the map guaranteed by (2). Denote $y' = b_*\xi $ and choose a factorization $\xi \to y' \to x$ lying over $R \to B \to A$ of the given morphism $\xi \to x$. By (S1) we obtain a commutative diagram

\[ \vcenter { \xymatrix{ z \ar[r] \ar[d] & y \ar[d] \\ y' \ar[r] & x } } \quad \text{lying over}\quad \vcenter { \xymatrix{ B \times _ A B \ar[d] \ar[r] & B \ar[d]^{f} \\ B \ar[r]^{f} & A . } } \]

Set $I = \mathop{\mathrm{Ker}}(f)$. Let $\overline{g} : B \times _ A B \to k[I]$ be the ring map $(u, v) \mapsto \overline{u} \oplus (v - u)$, cf. Lemma 88.10.8. By (1) there exists a morphism $\xi \to \overline{g}_*z$ which lies over a ring map $i : R \to k[\epsilon ]$. Choose an Artinian quotient $b_1 : R \to B_1$ such that both $b : R \to B$ and $i : R \to k[\epsilon ]$ factor through $R \to B_1$, i.e., giving $h : B_1 \to B$ and $i' : B_1 \to k[\epsilon ]$. Choose a pushforward $y_1 = b_{1, *}\xi $, a factorization $\xi \to y_1 \to y'$ lying over $R \to B_1 \to B$ of $\xi \to y'$, and a factorization $\xi \to y_1 \to \overline{g}_*z$ lying over $R \to B_1 \to k[\epsilon ]$ of $\xi \to \overline{g}_*z$. Applying (S1) once more we obtain

\[ \vcenter { \xymatrix{ z_1 \ar[r] \ar[d] & z \ar[r] \ar[d] & y \ar[d] \\ y_1 \ar[r] & y' \ar[r] & x } } \quad \text{lying over}\quad \vcenter { \xymatrix{ B_1 \times _ A B \ar[d] \ar[r] & B \times _ A B \ar[r] \ar[d] & B \ar[d]^{f} \\ B_1 \ar[r] & B \ar[r] & A . } } \]

Note that the map $g : B_1 \times _ A B \to k[I]$ of Lemma 88.10.8 (defined using $h$) is the composition of $B_1 \times _ A B \to B \times _ A B$ and the map $\overline{g}$ above. By construction there exists a morphism $y_1 \to g_*z_1 \cong \overline{g}_*z$! Hence Lemma 88.10.8 applies (to the outer rectangles in the diagrams above) to give a morphism $y_1 \to y$ and precomposing with $\xi \to y_1$ gives the desired morphism $\xi \to y$. $\square$

Comments (2)

Comment #2639 by Xiaowen Hu on

Set should be .I=Ker(k)

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 06IU. Beware of the difference between the letter 'O' and the digit '0'.