# The Stacks Project

## Tag 06IU

Lemma 80.13.2. 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 80.7.12. 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 $$\xymatrix{ & B \ar[d]^{f} \\ R \ar[ur] \ar[r] & A }$$ commutes.

Proof. If $\xi$ is versal then (1) holds by Lemma 80.8.8 and (2) holds by Remark 80.8.10. Assume (1) and (2) hold. By Remark 80.8.10 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 $I = \mathop{\rm Ker}(k)$. Let $\overline{g} : B \times_A B \to k[I]$ be the ring map $(u, v) \mapsto \overline{u} \oplus (v - u)$, cf. Lemma 80.10.8. 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 80.10.8 (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 80.10.8 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 $\xi \to y$. $\square$

The code snippet corresponding to this tag is a part of the file formal-defos.tex and is located in lines 3715–3752 (see updates for more information).

\begin{lemma}
\label{lemma-versal-criterion}
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 \ref{remark-formal-objects-yoneda}.
Then $\xi$ is versal if and only if the following two conditions hold:
\begin{enumerate}
\item the map
$d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to T\mathcal{F}$
on tangent spaces is surjective, and
\item 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
$$\xymatrix{ & B \ar[d]^{f} \\ R \ar[ur] \ar[r] & A }$$
commutes.
\end{enumerate}
\end{lemma}

\begin{proof}
If $\xi$ is versal then (1) holds by
Lemma \ref{lemma-smooth-morphism-essentially-surjective}
and (2) holds by
Remark \ref{remark-versal-object}.
Assume (1) and (2) hold. By
Remark \ref{remark-versal-object}
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 $I = \Ker(k)$. Let $\overline{g} : B \times_A B \to k[I]$
be the ring map $(u, v) \mapsto \overline{u} \oplus (v - u)$,
cf.\ Lemma \ref{lemma-lifting-along-small-extension}.
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 \ref{lemma-lifting-along-small-extension}
(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 \ref{lemma-lifting-along-small-extension}
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 $\xi \to y$.
\end{proof}

Comment #2639 by Xiaowen Hu on July 10, 2017 a 10:27 am UTC

Set $I=\ker(k)$ should be $I=\ker(f)$.I=Ker(k)

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