# The Stacks Project

## Tag 07XD

### 88.12. Versality

In the previous section we explained how to approximate objects over complete local rings by algebraic objects. But in order to show that a stack $\mathcal{X}$ is an algebraic stack, we need to find smooth $1$-morphisms from schemes towards $\mathcal{X}$. Since we are not going to assume a priori that $\mathcal{X}$ has a representable diagonal, we cannot even speak about smooth morphisms towards $\mathcal{X}$. Instead, borrowing terminology from deformation theory, we will introduce versal objects.

Definition 88.12.1. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\xi = (R, \xi_n, f_n)$ be a formal object. Set $k = R/\mathfrak m$ and $x_0 = \xi_1$. We will say that $\xi$ is versal if $\xi$ as a formal object of $\mathcal{F}_{\mathcal{X}, k, x_0}$ (Remark 88.9.2) is versal in the sense of Formal Deformation Theory, Definition 80.8.9.

We briefly spell out what this means. With notation as in the definition, suppose given morphisms $\xi_1 = x_0 \to y \to z$ of $\mathcal{X}$ lying over closed immersions $\mathop{\rm Spec}(k) \to \mathop{\rm Spec}(A) \to \mathop{\rm Spec}(B)$ where $A, B$ are Artinian local rings with residue field $k$. Suppose given an $n \geq 1$ and a commutative diagram $$\vcenter{ \xymatrix{ & y \ar[ld] \\ \xi_n & \xi_1 \ar[u] \ar[l] } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & \mathop{\rm Spec}(A) \ar[ld] \\ \mathop{\rm Spec}(R/\mathfrak m^n) & \mathop{\rm Spec}(k) \ar[u] \ar[l] } }$$ Versality means that for any data as above there exists an $m \geq n$ and a commutative diagram $$\vcenter{ \xymatrix{ & & z \ar[lldd] \\ & & y \ar[ld] \ar[u] \\ \xi_m & \xi_n \ar[l] & \xi_1 \ar[u] \ar[l] } } \quad\text{lying over} \vcenter{ \xymatrix{ & & \mathop{\rm Spec}(B) \ar[lldd] \\ & & \mathop{\rm Spec}(A) \ar[ld] \ar[u] \\ \mathop{\rm Spec}(R/\mathfrak m^m) & \mathop{\rm Spec}(R/\mathfrak m^n) \ar[l] & \mathop{\rm Spec}(k) \ar[u] \ar[l] } }$$ Please compare with Formal Deformation Theory, Remark 80.8.10.

Let $S$ be a locally Noetherian scheme. Let $U$ be a scheme over $S$ with structure morphism $U \to S$ locally of finite type. Let $u_0 \in U$ be a finite type point of $U$, see Morphisms, Definition 28.15.3. Set $k = \kappa(u_0)$. Note that the composition $\mathop{\rm Spec}(k) \to S$ is also of finite type, see Morphisms, Lemma 28.14.3. Let $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $x$ be an object of $\mathcal{X}$ which lies over $U$. Denote $x_0$ the pullback of $x$ by $u_0$. By the $2$-Yoneda lemma $x$ corresponds to a $1$-morphism $$x : (\textit{Sch}/U)_{fppf} \longrightarrow \mathcal{X},$$ see Algebraic Stacks, Section 84.5. We obtain a morphism of predeformation categories $$\tag{88.12.1.1} \hat x : \mathcal{F}_{(\textit{Sch}/U)_{fppf}, k, u_0} \longrightarrow \mathcal{F}_{\mathcal{X}, k, x_0},$$ over $\mathcal{C}_\Lambda$ see (88.3.1.1).

Definition 88.12.2. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be fibred in groupoids over $(\textit{Sch}/S)_{fppf}$. Let $U$ be a scheme locally of finite type over $S$. Let $x$ be an object of $\mathcal{X}$ lying over $U$. Let $u_0$ be finite type point of $U$. We say $x$ is versal at $u_0$ if the morphism $\hat x$ (88.12.1.1) is smooth, see Formal Deformation Theory, Definition 80.8.1.

This definition matches our notion of versality for formal objects of $\mathcal{X}$.

Lemma 88.12.3. With notation as in Definition 88.12.2. Let $R = \mathcal{O}_{U, u_0}^\wedge$. Let $\xi$ be the formal object of $\mathcal{X}$ over $R$ associated to $x|_{\mathop{\rm Spec}(R)}$, see (88.9.3.1). Then $$x\text{ is versal at }u_0 \Leftrightarrow \xi\text{ is versal}$$

Proof. Observe that $\mathcal{O}_{U, u_0}$ is a Noetherian local $S$-algebra with residue field $k$. Hence $R = \mathcal{O}_{U, u_0}^\wedge$ is an object of $\mathcal{C}_\Lambda^\wedge$, see Formal Deformation Theory, Definition 80.4.1. Recall that $\xi$ is versal if $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth and $x$ is versal at $u_0$ if $\hat x : \mathcal{F}_{(\textit{Sch}/U)_{fppf}, k, u_0} \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth. There is an identification of predeformation categories $$\underline{R}|_{\mathcal{C}_\Lambda} = \mathcal{F}_{(\textit{Sch}/U)_{fppf}, k, u_0},$$ see Formal Deformation Theory, Remark 80.7.12 for notation. Namely, given an Artinian local $S$-algebra $A$ with residue field identified with $k$ we have $$\mathop{\rm Mor}\nolimits_{\mathcal{C}_\Lambda^\wedge}(R, A) = \{\varphi \in \mathop{\rm Mor}\nolimits_S(\mathop{\rm Spec}(A), U) \mid \varphi|_{\mathop{\rm Spec}(k)} = u_0\}$$ Unwinding the definitions the reader verifies that the resulting map $$\underline{R}|_{\mathcal{C}_\Lambda} = \mathcal{F}_{(\textit{Sch}/U)_{fppf}, k, u_0} \xrightarrow{\hat x} \mathcal{F}_{\mathcal{X}, k, x_0},$$ is equal to $\underline{\xi}$ and we see that the lemma is true. $\square$

Here is a sanity check.

Lemma 88.12.4. Let $S$ be a locally Noetherian scheme. Let $f : U \to V$ be a morphism of schemes locally of finite type over $S$. Let $u_0 \in U$ be a finite type point. The following are equivalent

1. $f$ is smooth at $u_0$,
2. $f$ viewed as an object of $(\textit{Sch}/V)_{fppf}$ over $U$ is versal at $u_0$.

Proof. This is a restatement of More on Morphisms, Lemma 36.12.1. $\square$

It turns out that this notion is well behaved with respect to field extensions.

Lemma 88.12.5. Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition 88.12.2. Let $l$ be a field and let $u_{l, 0} : \mathop{\rm Spec}(l) \to U$ be a morphism with image $u_0$ such that $l/k = \kappa(u_0)$ is finite. Set $x_{l, 0} = x_0|_{\mathop{\rm Spec}(l)}$. If $\mathcal{X}$ satisfies (RS) and $x$ is versal at $u_0$, then $$\mathcal{F}_{(\textit{Sch}/U)_{fppf}, l, u_{l, 0}} \longrightarrow \mathcal{F}_{\mathcal{X}, l, x_{l, 0}}$$ is smooth.

Proof. Note that $(\textit{Sch}/U)_{fppf}$ satisfies (RS) by Lemma 88.5.2. Hence the functor of the lemma is the functor $$(\mathcal{F}_{(\textit{Sch}/U)_{fppf}, k , u_0})_{l/k} \longrightarrow (\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k}$$ associated to $\hat x$, see Lemma 88.7.1. Hence the lemma follows from Formal Deformation Theory, Lemma 80.29.5. $\square$

The following lemma is another sanity check. It more or less signifies that if $x$ is versal at $u_0$ as in Definition 88.12.2, then $x$ viewed as a morphism from $U$ to $\mathcal{X}$ is smooth whenever we make a base change by a scheme.

Lemma 88.12.6. Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition 88.12.2. Assume

1. $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces,
2. $\mathcal{X}$ is limit preserving, and
3. $\mathcal{X}$ has (RS).

Let $V$ be a scheme locally of finite type over $S$ and let $y$ be an object of $\mathcal{X}$ over $V$. Form the $2$-fibre product $$\xymatrix{ \mathcal{Z} \ar[r] \ar[d] & (\textit{Sch}/U)_{fppf} \ar[d]^x \\ (\textit{Sch}/V)_{fppf} \ar[r]^y & \mathcal{X} }$$ Let $Z$ be the algebraic space representing $\mathcal{W}$ and let $z_0 \in |Z|$ be a finite type point lying over $u_0$. If $x$ is versal at $u_0$, then the morphism $Z \to V$ is smooth at $z_0$.

Proof. Observe that $Z$ exists by Algebraic Stacks, Lemma 84.10.11. By Lemma 88.11.4 we see that $Z \to V \times_S U$ is locally of finite type. Choose a scheme $W$, a closed point $w_0 \in W$, and an étale morphism $W \to Z$ mapping $w_0$ to $z_0$, see Morphisms of Spaces, Definition 58.25.2. Then $W$ is locally of finite type over $S$ and $w_0$ is a finite type point of $W$. Let $l = \kappa(z_0)$. Denote $z_{l, 0}$, $v_{l, 0}$, $u_{l, 0}$, and $x_{l, 0}$ the objects of $\mathcal{Z}$, $(\textit{Sch}/V)_{fppf}$, $(\textit{Sch}/U)_{fppf}$, and $\mathcal{X}$ over $\mathop{\rm Spec}(l)$ obtained by pullback to $\mathop{\rm Spec}(l) = w_0$. Consider $$\xymatrix{ \mathcal{F}_{(\textit{Sch}/W)_{fppf}, l, w_0} \ar[r] & \mathcal{F}_{\mathcal{Z}, l, z_{l, 0}} \ar[d] \ar[r] & \mathcal{F}_{(\textit{Sch}/U)_{fppf}, l, u_{l, 0}} \ar[d] \\ & \mathcal{F}_{(\textit{Sch}/V)_{fppf}, l, v_{l, 0}} \ar[r] & \mathcal{F}_{\mathcal{X}, l, x_{l, 0}} }$$ By Lemma 88.3.3 the square is a fibre product of predeformation categories. By Lemma 88.12.5 we see that the right vertical arrow is smooth. By Formal Deformation Theory, Lemma 80.8.7 the left vertical arrow is smooth. By Lemma 88.3.2 we see that the left horizontal arrow is smooth. We conclude that the map $$\mathcal{F}_{(\textit{Sch}/W)_{fppf}, l, w_0} \to \mathcal{F}_{(\textit{Sch}/V)_{fppf}, l, v_{l, 0}}$$ is smooth by Formal Deformation Theory, Lemma 80.8.7. Thus we conclude that $W \to V$ is smooth at $w_0$ by More on Morphisms, Lemma 36.12.1. This exactly means that $Z \to V$ is smooth at $z_0$ and the proof is complete. $\square$

We restate the approximation result in terms of versal objects.

Lemma 88.12.7. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\xi = (R, \xi_n, f_n)$ be a formal object of $\mathcal{X}$ with $\xi_1$ lying over $\mathop{\rm Spec}(k) \to S$ with image $s \in S$. Assume

1. $\xi$ is versal,
2. $\xi$ is effective,
3. $\mathcal{O}_{S, s}$ is a G-ring, and
4. $p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}$ is limit preserving on objects.

Then there exist a morphism of finite type $U \to S$, a finite type point $u_0 \in U$ with residue field $k$, and an object $x$ of $\mathcal{X}$ over $U$ such that $x$ is versal at $u_0$ and such that $x|_{\mathop{\rm Spec}(\mathcal{O}_{U, u_0}/\mathfrak m_{u_0}^n)} \cong \xi_n$.

Proof. Choose an object $x_R$ of $\mathcal{X}$ lying over $\mathop{\rm Spec}(R)$ whose associated formal object is $\xi$. Let $N = 2$ and apply Lemma 88.10.1. We obtain $A, \mathfrak m_A, x_A, \ldots$. Let $\eta = (A^\wedge, \eta_n, g_n)$ be the formal object associated to $x_A|_{\mathop{\rm Spec}(A^\wedge)}$. We have a diagram $$\vcenter{ \xymatrix{ & \eta \ar[d] \\ \xi \ar[r] \ar@{..>}[ru] & \xi_2 = \eta_2 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & A^\wedge \ar[d] \\ R \ar[r] \ar@{..>}[ru] & R/\mathfrak m_R^2 = A/\mathfrak m_A^2 } }$$ The versality of $\xi$ means exactly that we can find the dotted arrows in the diagrams, because we can successively find morphisms $\xi \to \eta_3$, $\xi \to \eta_4$, and so on by Formal Deformation Theory, Remark 80.8.10. The corresponding ring map $R \to A^\wedge$ is surjective by Formal Deformation Theory, Lemma 80.4.2. On the other hand, we have $\dim_k \mathfrak m_R^n/\mathfrak m_R^{n + 1} = \dim_k \mathfrak m_A^n/\mathfrak m_A^{n + 1}$ for all $n$ by construction. Hence $R/\mathfrak m_R^n$ and $A/\mathfrak m_A^n$ have the same (finite) length as $\Lambda$-modules by additivity of length and Formal Deformation Theory, Lemma 80.3.4. It follows that $R/\mathfrak m_R^n \to A/\mathfrak m_A^n$ is an isomorphism for all $n$, hence $R \to A^\wedge$ is an isomorphism. Thus $\eta$ is isomorphic to a versal object, hence versal itself. By Lemma 88.12.3 we conclude that $x_A$ is versal at the point $u_0$ of $U = \mathop{\rm Spec}(A)$ corresponding to $\mathfrak m_A$. $\square$

Example 88.12.8. In this example we show that the local ring $\mathcal{O}_{S, s}$ has to be a G-ring in order for the result of Lemma 88.12.7 to be true. Namely, let $\Lambda$ be a Noetherian ring and let $\mathfrak m$ be a maximal ideal of $\Lambda$. Set $R = \Lambda_\mathfrak m^\wedge$. Let $\Lambda \to C \to R$ be a factorization with $C$ of finite type over $\Lambda$. Set $S = \mathop{\rm Spec}(\Lambda)$, $U = S \setminus \{\mathfrak m\}$, and $S' = U \amalg \mathop{\rm Spec}(C)$. Consider the functor $F : (\textit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ defined by the rule $$F(T) = \left\{ \begin{matrix} * & \text{if }T \to S\text{ factors through }S' \\ \emptyset & \text{else} \end{matrix} \right.$$ Let $\mathcal{X} = \mathcal{S}_F$ is the category fibred in sets associated to $F$, see Algebraic Stacks, Section 84.7. Then $\mathcal{X} \to (\textit{Sch}/S)_{fppf}$ is limit preserving on objects and there exists an effective, versal formal object $\xi$ over $R$. Hence if the conclusion of Lemma 88.12.7 holds for $\mathcal{X}$, then there exists a finite type ring map $\Lambda \to A$ and a maximal ideal $\mathfrak m_A$ lying over $\mathfrak m$ such that

1. $\kappa(\mathfrak m) = \kappa(\mathfrak m_A)$,
2. $\Lambda \to A$ and $\mathfrak m_A$ satisfy condition (4) of Algebra, Lemma 10.139.2, and
3. there exists a $\Lambda$-algebra map $C \to A$.

Thus $\Lambda \to A$ is smooth at $\mathfrak m_A$ by the lemma cited. Slicing $A$ we may assume that $\Lambda \to A$ is étale at $\mathfrak m_A$, see for example More on Morphisms, Lemma 36.34.5 or argue directly. Write $C = \Lambda[y_1, \ldots, y_n]/(f_1, \ldots, f_m)$. Then $C \to R$ corresponds to a solution in $R$ of the system of equations $f_1 = \ldots = f_m = 0$, see Smoothing Ring Maps, Section 16.13. Thus if the conclusion of Lemma 88.12.7 holds for every $\mathcal{X}$ as above, then a system of equations which has a solution in $R$ has a solution in the henselization of $\Lambda_{\mathfrak m}$. In other words, the approximation property holds for $\Lambda_{\mathfrak m}^h$. This implies that $\Lambda_{\mathfrak m}^h$ is a G-ring (insert future reference here; see also discussion in Smoothing Ring Maps, Section 16.1) which in turn implies that $\Lambda_{\mathfrak m}$ is a G-ring.

The code snippet corresponding to this tag is a part of the file artin.tex and is located in lines 1432–1846 (see updates for more information).

\section{Versality}
\label{section-versality}

\noindent
In the previous section we explained how to approximate objects over
complete local rings by algebraic objects. But in order to show that
a stack $\mathcal{X}$ is an algebraic stack, we need to find smooth
$1$-morphisms from schemes towards $\mathcal{X}$. Since we are not going
to assume a priori that $\mathcal{X}$ has a representable diagonal, we
cannot even speak about smooth morphisms towards $\mathcal{X}$. Instead,
borrowing terminology from deformation theory, we will introduce versal
objects.

\begin{definition}
\label{definition-versal-formal-object}
Let $S$ be a locally Noetherian scheme. Let
$p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids.
Let $\xi = (R, \xi_n, f_n)$ be a formal object. Set $k = R/\mathfrak m$ and
$x_0 = \xi_1$. We will say that $\xi$ is {\it versal} if $\xi$
as a formal object of $\mathcal{F}_{\mathcal{X}, k, x_0}$
(Remark \ref{remark-formal-objects-match}) is versal in the sense
of Formal Deformation Theory, Definition \ref{formal-defos-definition-versal}.
\end{definition}

\noindent
We briefly spell out what this means. With notation as in the definition,
suppose given morphisms $\xi_1 = x_0 \to y \to z$ of $\mathcal{X}$ lying over
closed immersions
$\Spec(k) \to \Spec(A) \to \Spec(B)$
where $A, B$ are Artinian local rings with residue field $k$.
Suppose given an $n \geq 1$ and a commutative diagram
$$\vcenter{ \xymatrix{ & y \ar[ld] \\ \xi_n & \xi_1 \ar[u] \ar[l] } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & \Spec(A) \ar[ld] \\ \Spec(R/\mathfrak m^n) & \Spec(k) \ar[u] \ar[l] } }$$
Versality means that for any data as above
there exists an $m \geq n$ and a commutative diagram
$$\vcenter{ \xymatrix{ & & z \ar[lldd] \\ & & y \ar[ld] \ar[u] \\ \xi_m & \xi_n \ar[l] & \xi_1 \ar[u] \ar[l] } } \quad\text{lying over} \vcenter{ \xymatrix{ & & \Spec(B) \ar[lldd] \\ & & \Spec(A) \ar[ld] \ar[u] \\ \Spec(R/\mathfrak m^m) & \Spec(R/\mathfrak m^n) \ar[l] & \Spec(k) \ar[u] \ar[l] } }$$
Please compare with Formal Deformation Theory, Remark
\ref{formal-defos-remark-versal-object}.

\medskip\noindent
Let $S$ be a locally Noetherian scheme. Let $U$ be a scheme over $S$
with structure morphism $U \to S$ locally of finite type. Let
$u_0 \in U$ be a finite type point of $U$, see
Morphisms, Definition \ref{morphisms-definition-finite-type-point}.
Set $k = \kappa(u_0)$.
Note that the composition $\Spec(k) \to S$ is also of finite type,
see Morphisms, Lemma \ref{morphisms-lemma-composition-finite-type}.
Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids.
Let $x$ be an object of $\mathcal{X}$ which lies over $U$. Denote $x_0$
the pullback of $x$ by $u_0$. By the $2$-Yoneda lemma $x$ corresponds
to a $1$-morphism
$$x : (\Sch/U)_{fppf} \longrightarrow \mathcal{X},$$
see Algebraic Stacks, Section \ref{algebraic-section-2-yoneda}. We obtain a
morphism of predeformation categories

\label{equation-hat-x}
\hat x :
\mathcal{F}_{(\Sch/U)_{fppf}, k, u_0}
\longrightarrow
\mathcal{F}_{\mathcal{X}, k, x_0},

over $\mathcal{C}_\Lambda$ see (\ref{equation-functoriality}).

\begin{definition}
\label{definition-versal}
Let $S$ be a locally Noetherian scheme.
Let $\mathcal{X}$ be fibred in groupoids over $(\Sch/S)_{fppf}$.
Let $U$ be a scheme locally of finite type over $S$.
Let $x$ be an object of $\mathcal{X}$ lying over $U$.
Let $u_0$ be finite type point of $U$.
We say $x$ is {\it versal} at $u_0$ if the morphism $\hat x$
(\ref{equation-hat-x}) is smooth, see Formal Deformation Theory, Definition
\ref{formal-defos-definition-smooth-morphism}.
\end{definition}

\noindent
This definition matches our notion of versality for formal objects of
$\mathcal{X}$.

\begin{lemma}
\label{lemma-versality-matches}
With notation as in Definition \ref{definition-versal}.
Let $R = \mathcal{O}_{U, u_0}^\wedge$.
Let $\xi$ be the formal object of $\mathcal{X}$
over $R$ associated to $x|_{\Spec(R)}$, see (\ref{equation-approximation}).
Then
$$x\text{ is versal at }u_0 \Leftrightarrow \xi\text{ is versal}$$
\end{lemma}

\begin{proof}
Observe that $\mathcal{O}_{U, u_0}$ is a Noetherian local $S$-algebra
with residue field $k$. Hence $R = \mathcal{O}_{U, u_0}^\wedge$ is an object of
$\mathcal{C}_\Lambda^\wedge$, see Formal Deformation Theory, Definition
\ref{formal-defos-definition-completion-CLambda}.
Recall that $\xi$ is versal if
$\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}_{\mathcal{X}, k, x_0}$
is smooth and $x$ is versal at $u_0$ if
$\hat x : \mathcal{F}_{(\Sch/U)_{fppf}, k, u_0} \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth.
There is an identification of predeformation categories
$$\underline{R}|_{\mathcal{C}_\Lambda} = \mathcal{F}_{(\Sch/U)_{fppf}, k, u_0},$$
see Formal Deformation Theory, Remark
\ref{formal-defos-remark-formal-objects-yoneda} for notation.
Namely, given an Artinian local $S$-algebra $A$ with residue field
identified with $k$ we have
$$\Mor_{\mathcal{C}_\Lambda^\wedge}(R, A) = \{\varphi \in \Mor_S(\Spec(A), U) \mid \varphi|_{\Spec(k)} = u_0\}$$
Unwinding the definitions the reader verifies that the resulting map
$$\underline{R}|_{\mathcal{C}_\Lambda} = \mathcal{F}_{(\Sch/U)_{fppf}, k, u_0} \xrightarrow{\hat x} \mathcal{F}_{\mathcal{X}, k, x_0},$$
is equal to $\underline{\xi}$ and we see that the lemma is true.
\end{proof}

\noindent
Here is a sanity check.

\begin{lemma}
\label{lemma-versal-implies-smooth}
Let $S$ be a locally Noetherian scheme. Let $f : U \to V$
be a morphism of schemes locally of finite type over $S$.
Let $u_0 \in U$ be a finite type point. The following are equivalent
\begin{enumerate}
\item $f$ is smooth at $u_0$,
\item $f$ viewed as an object of $(\Sch/V)_{fppf}$ over $U$ is
versal at $u_0$.
\end{enumerate}
\end{lemma}

\begin{proof}
This is a restatement of More on Morphisms, Lemma
\ref{more-morphisms-lemma-lifting-along-artinian-at-point}.
\end{proof}

\noindent
It turns out that this notion is well behaved with respect to field
extensions.

\begin{lemma}
\label{lemma-versal-change-of-field}
Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in
Definition \ref{definition-versal}. Let $l$ be a field and let
$u_{l, 0} : \Spec(l) \to U$ be a morphism with image $u_0$ such that
$l/k = \kappa(u_0)$ is finite. Set $x_{l, 0} = x_0|_{\Spec(l)}$.
If $\mathcal{X}$ satisfies (RS) and $x$ is versal at $u_0$, then
$$\mathcal{F}_{(\Sch/U)_{fppf}, l, u_{l, 0}} \longrightarrow \mathcal{F}_{\mathcal{X}, l, x_{l, 0}}$$
is smooth.
\end{lemma}

\begin{proof}
Note that $(\Sch/U)_{fppf}$ satisfies (RS) by
Lemma \ref{lemma-algebraic-stack-RS}.
Hence the functor of the lemma is the functor
$$(\mathcal{F}_{(\Sch/U)_{fppf}, k , u_0})_{l/k} \longrightarrow (\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k}$$
associated to $\hat x$, see Lemma \ref{lemma-change-of-field}.
Hence the lemma follows from
Formal Deformation Theory, Lemma
\ref{formal-defos-lemma-change-of-fields-smooth}.
\end{proof}

\noindent
The following lemma is another sanity check. It more or less
signifies that if $x$ is versal at $u_0$ as in
Definition \ref{definition-versal},
then $x$ viewed as a morphism from $U$ to $\mathcal{X}$ is
smooth whenever we make a base change by a scheme.

\begin{lemma}
\label{lemma-base-change-versal}
Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in
Definition \ref{definition-versal}. Assume
\begin{enumerate}
\item $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$
is representable by algebraic spaces,
\item $\mathcal{X}$ is limit preserving, and
\item $\mathcal{X}$ has (RS).
\end{enumerate}
Let $V$ be a scheme locally of finite type over $S$
and let $y$ be an object of $\mathcal{X}$ over $V$.
Form the $2$-fibre product
$$\xymatrix{ \mathcal{Z} \ar[r] \ar[d] & (\Sch/U)_{fppf} \ar[d]^x \\ (\Sch/V)_{fppf} \ar[r]^y & \mathcal{X} }$$
Let $Z$ be the algebraic space representing $\mathcal{W}$
and let $z_0 \in |Z|$ be a finite type point lying over $u_0$.
If $x$ is versal at $u_0$, then
the morphism $Z \to V$ is smooth at $z_0$.
\end{lemma}

\begin{proof}
Observe that $Z$ exists by Algebraic Stacks, Lemma
\ref{algebraic-lemma-representable-diagonal}.
By Lemma \ref{lemma-diagonal} we see that
$Z \to V \times_S U$ is locally of finite type.
Choose a scheme $W$, a closed point $w_0 \in W$, and
an \'etale morphism $W \to Z$ mapping $w_0$ to $z_0$, see
Morphisms of Spaces, Definition
\ref{spaces-morphisms-definition-finite-type-point}.
Then $W$ is locally of finite type over $S$ and
$w_0$ is a finite type point of $W$.
Let $l = \kappa(z_0)$. Denote $z_{l, 0}$, $v_{l, 0}$,
$u_{l, 0}$, and $x_{l, 0}$ the objects of
$\mathcal{Z}$, $(\Sch/V)_{fppf}$, $(\Sch/U)_{fppf}$,
and $\mathcal{X}$ over $\Spec(l)$ obtained by pullback to $\Spec(l) = w_0$.
Consider
$$\xymatrix{ \mathcal{F}_{(\Sch/W)_{fppf}, l, w_0} \ar[r] & \mathcal{F}_{\mathcal{Z}, l, z_{l, 0}} \ar[d] \ar[r] & \mathcal{F}_{(\Sch/U)_{fppf}, l, u_{l, 0}} \ar[d] \\ & \mathcal{F}_{(\Sch/V)_{fppf}, l, v_{l, 0}} \ar[r] & \mathcal{F}_{\mathcal{X}, l, x_{l, 0}} }$$
By Lemma \ref{lemma-fibre-product-deformation-categories}
the square is a fibre product of predeformation categories.
By Lemma \ref{lemma-versal-change-of-field}
we see that the right vertical arrow is smooth.
By Formal Deformation Theory, Lemma
\ref{formal-defos-lemma-smooth-properties}
the left vertical arrow is smooth.
By Lemma \ref{lemma-formally-smooth-on-deformation-categories}
we see that the left horizontal arrow is smooth.
We conclude that the map
$$\mathcal{F}_{(\Sch/W)_{fppf}, l, w_0} \to \mathcal{F}_{(\Sch/V)_{fppf}, l, v_{l, 0}}$$
is smooth by Formal Deformation Theory, Lemma
\ref{formal-defos-lemma-smooth-properties}.
Thus we conclude that $W \to V$ is smooth at $w_0$ by
More on Morphisms, Lemma
\ref{more-morphisms-lemma-lifting-along-artinian-at-point}.
This exactly means that $Z \to V$ is smooth at $z_0$
and the proof is complete.
\end{proof}

\noindent
We restate the approximation result in terms of
versal objects.

\begin{lemma}
\label{lemma-approximate-versal}
Let $S$ be a locally Noetherian scheme. Let
$p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids.
Let $\xi = (R, \xi_n, f_n)$ be a formal object of $\mathcal{X}$ with
$\xi_1$ lying over $\Spec(k) \to S$ with image $s \in S$. Assume
\begin{enumerate}
\item $\xi$ is versal,
\item $\xi$ is effective,
\item $\mathcal{O}_{S, s}$ is a G-ring, and
\item $p : \mathcal{X} \to (\Sch/S)_{fppf}$ is limit preserving on objects.
\end{enumerate}
Then there exist a morphism of finite type $U \to S$, a finite type
point $u_0 \in U$ with residue field $k$, and an object $x$ of $\mathcal{X}$
over $U$ such that $x$ is versal at $u_0$ and such that
$x|_{\Spec(\mathcal{O}_{U, u_0}/\mathfrak m_{u_0}^n)} \cong \xi_n$.
\end{lemma}

\begin{proof}
Choose an object $x_R$ of $\mathcal{X}$ lying over $\Spec(R)$ whose associated
formal object is $\xi$. Let $N = 2$ and apply Lemma \ref{lemma-approximate}.
We obtain $A, \mathfrak m_A, x_A, \ldots$.
Let $\eta = (A^\wedge, \eta_n, g_n)$ be the formal object associated to
$x_A|_{\Spec(A^\wedge)}$. We have a diagram
$$\vcenter{ \xymatrix{ & \eta \ar[d] \\ \xi \ar[r] \ar@{..>}[ru] & \xi_2 = \eta_2 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & A^\wedge \ar[d] \\ R \ar[r] \ar@{..>}[ru] & R/\mathfrak m_R^2 = A/\mathfrak m_A^2 } }$$
The versality of $\xi$ means exactly that we can find the
dotted arrows in the diagrams, because we can successively find
morphisms $\xi \to \eta_3$, $\xi \to \eta_4$, and so on by
Formal Deformation Theory, Remark \ref{formal-defos-remark-versal-object}.
The corresponding ring map $R \to A^\wedge$ is surjective by
Formal Deformation Theory, Lemma
\ref{formal-defos-lemma-surjective-cotangent-space}.
On the other hand, we have
$\dim_k \mathfrak m_R^n/\mathfrak m_R^{n + 1} = \dim_k \mathfrak m_A^n/\mathfrak m_A^{n + 1}$ for all $n$ by construction.
Hence $R/\mathfrak m_R^n$ and $A/\mathfrak m_A^n$ have the same (finite)
length as $\Lambda$-modules by additivity of length and
Formal Deformation Theory, Lemma \ref{formal-defos-lemma-length}.
It follows that $R/\mathfrak m_R^n \to A/\mathfrak m_A^n$ is an isomorphism
for all $n$, hence $R \to A^\wedge$ is an isomorphism. Thus $\eta$ is
isomorphic to a versal object, hence versal itself. By
Lemma \ref{lemma-versality-matches}
we conclude that $x_A$ is versal at the point $u_0$ of
$U = \Spec(A)$ corresponding to $\mathfrak m_A$.
\end{proof}

\begin{example}
\label{example-approximate-versal-implies}
In this example we show that the local ring $\mathcal{O}_{S, s}$ has to be
a G-ring in order for the result of Lemma \ref{lemma-approximate-versal} to
be true. Namely, let $\Lambda$ be a Noetherian ring and let $\mathfrak m$
be a maximal ideal of $\Lambda$. Set $R = \Lambda_\mathfrak m^\wedge$. Let
$\Lambda \to C \to R$ be a factorization with $C$ of finite type over
$\Lambda$. Set $S = \Spec(\Lambda)$, $U = S \setminus \{\mathfrak m\}$, and
$S' = U \amalg \Spec(C)$. Consider the functor
$F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ defined by the rule
$$F(T) = \left\{ \begin{matrix} * & \text{if }T \to S\text{ factors through }S' \\ \emptyset & \text{else} \end{matrix} \right.$$
Let $\mathcal{X} = \mathcal{S}_F$ is the category fibred in sets associated
to $F$, see Algebraic Stacks, Section \ref{algebraic-section-split}.
Then $\mathcal{X} \to (\Sch/S)_{fppf}$ is limit preserving on objects and
there exists an effective, versal formal object $\xi$ over $R$.
Hence if the conclusion of Lemma \ref{lemma-approximate-versal} holds
for $\mathcal{X}$, then there exists a finite type ring map $\Lambda \to A$
and a maximal ideal $\mathfrak m_A$ lying over $\mathfrak m$ such that
\begin{enumerate}
\item $\kappa(\mathfrak m) = \kappa(\mathfrak m_A)$,
\item $\Lambda \to A$ and $\mathfrak m_A$ satisfy condition (4) of
Algebra, Lemma \ref{algebra-lemma-smooth-test-artinian}, and
\item there exists a $\Lambda$-algebra map $C \to A$.
\end{enumerate}
Thus $\Lambda \to A$ is smooth at $\mathfrak m_A$ by the lemma cited.
Slicing $A$ we may assume that $\Lambda \to A$ is \'etale at
$\mathfrak m_A$, see for example
More on Morphisms, Lemma \ref{more-morphisms-lemma-slice-smooth}
or argue directly. Write $C = \Lambda[y_1, \ldots, y_n]/(f_1, \ldots, f_m)$.
Then $C \to R$ corresponds to a solution in $R$ of the system of equations
$f_1 = \ldots = f_m = 0$, see Smoothing Ring Maps, Section
\ref{smoothing-section-approximation-G-rings}.
Thus if the conclusion of
Lemma \ref{lemma-approximate-versal} holds for every $\mathcal{X}$ as
above, then a system of equations which has a solution in $R$ has a
solution in the henselization of $\Lambda_{\mathfrak m}$.
In other words, the approximation property holds for
$\Lambda_{\mathfrak m}^h$. This implies that $\Lambda_{\mathfrak m}^h$
Smoothing Ring Maps, Section \ref{smoothing-section-introduction})
which in turn implies that $\Lambda_{\mathfrak m}$ is a G-ring.
\end{example}

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