Definition 98.12.1. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ 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 98.9.2) is versal in the sense of Formal Deformation Theory, Definition 90.8.9.
98.12 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.
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 $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(B)$ where $A, B$ are Artinian local rings with residue field $k$. Suppose given an $n \geq 1$ and a commutative diagram
\[ \vcenter { \xymatrix{ & y \ar[ld] \\ \xi _ n & \xi _1 \ar[u] \ar[l] } } \quad \text{lying over}\quad \vcenter { \xymatrix{ & \mathop{\mathrm{Spec}}(A) \ar[ld] \\ \mathop{\mathrm{Spec}}(R/\mathfrak m^ n) & \mathop{\mathrm{Spec}}(k) \ar[u] \ar[l] } } \]
Versality means that for any data as above there exists an $m \geq n$ and a commutative diagram
\[ \vcenter { \xymatrix{ & & z \ar[lldd] \\ & & y \ar[ld] \ar[u] \\ \xi _ m & \xi _ n \ar[l] & \xi _1 \ar[u] \ar[l] } } \quad \text{lying over} \vcenter { \xymatrix{ & & \mathop{\mathrm{Spec}}(B) \ar[lldd] \\ & & \mathop{\mathrm{Spec}}(A) \ar[ld] \ar[u] \\ \mathop{\mathrm{Spec}}(R/\mathfrak m^ m) & \mathop{\mathrm{Spec}}(R/\mathfrak m^ n) \ar[l] & \mathop{\mathrm{Spec}}(k) \ar[u] \ar[l] } } \]
Please compare with Formal Deformation Theory, Remark 90.8.10.
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 29.16.3. Set $k = \kappa (u_0)$. Note that the composition $\mathop{\mathrm{Spec}}(k) \to S$ is also of finite type, see Morphisms, Lemma 29.15.3. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ 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
\[ x : (\mathit{Sch}/U)_{fppf} \longrightarrow \mathcal{X}, \]
see Algebraic Stacks, Section 94.5. We obtain a morphism of predeformation categories
98.12.1.1
\begin{equation} \label{artin-equation-hat-x} \hat x : \mathcal{F}_{(\mathit{Sch}/U)_{fppf}, k, u_0} \longrightarrow \mathcal{F}_{\mathcal{X}, k, x_0}, \end{equation}
over $\mathcal{C}_\Lambda $ see (98.3.1.1).
Definition 98.12.2. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. 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 $\hat x$ (98.12.1.1) is smooth, see Formal Deformation Theory, Definition 90.8.1.
This definition matches our notion of versality for formal objects of $\mathcal{X}$.
Lemma 98.12.3. With notation as in Definition 98.12.2. Let $R = \mathcal{O}_{U, u_0}^\wedge $. Let $\xi $ be the formal object of $\mathcal{X}$ over $R$ associated to $x|_{\mathop{\mathrm{Spec}}(R)}$, see (98.9.3.1). 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 90.4.1. 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 $\hat x : \mathcal{F}_{(\mathit{Sch}/U)_{fppf}, k, u_0} \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth. There is an identification of predeformation categories
\[ \underline{R}|_{\mathcal{C}_\Lambda } = \mathcal{F}_{(\mathit{Sch}/U)_{fppf}, k, u_0}, \]
see Formal Deformation Theory, Remark 90.7.12 for notation. Namely, given an Artinian local $S$-algebra $A$ with residue field identified with $k$ we have
\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda ^\wedge }(R, A) = \{ \varphi \in \mathop{\mathrm{Mor}}\nolimits _ S(\mathop{\mathrm{Spec}}(A), U) \mid \varphi |_{\mathop{\mathrm{Spec}}(k)} = u_0\} \]
Unwinding the definitions the reader verifies that the resulting map
\[ \underline{R}|_{\mathcal{C}_\Lambda } = \mathcal{F}_{(\mathit{Sch}/U)_{fppf}, k, u_0} \xrightarrow {\hat x} \mathcal{F}_{\mathcal{X}, k, x_0}, \]
is equal to $\underline{\xi }$ and we see that the lemma is true. $\square$
Here is a sanity check.
Lemma 98.12.4. 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
$f$ is smooth at $u_0$,
$f$ viewed as an object of $(\mathit{Sch}/V)_{fppf}$ over $U$ is versal at $u_0$.
Proof. This is a restatement of More on Morphisms, Lemma 37.12.1. $\square$
It turns out that this notion is well behaved with respect to field extensions.
Lemma 98.12.5. Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition 98.12.2. Let $l$ be a field and let $u_{l, 0} : \mathop{\mathrm{Spec}}(l) \to U$ be a morphism with image $u_0$ such that $l/k = \kappa (u_0)$ is finite. Set $x_{l, 0} = x_0|_{\mathop{\mathrm{Spec}}(l)}$. If $\mathcal{X}$ satisfies (RS) and $x$ is versal at $u_0$, then is smooth.
\[ \mathcal{F}_{(\mathit{Sch}/U)_{fppf}, l, u_{l, 0}} \longrightarrow \mathcal{F}_{\mathcal{X}, l, x_{l, 0}} \]
Proof. Note that $(\mathit{Sch}/U)_{fppf}$ satisfies (RS) by Lemma 98.5.2. Hence the functor of the lemma is the functor
\[ (\mathcal{F}_{(\mathit{Sch}/U)_{fppf}, k , u_0})_{l/k} \longrightarrow (\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k} \]
associated to $\hat x$, see Lemma 98.7.1. Hence the lemma follows from Formal Deformation Theory, Lemma 90.29.5. $\square$
The following lemma is another sanity check. It more or less signifies that if $x$ is versal at $u_0$ as in Definition 98.12.2, then $x$ viewed as a morphism from $U$ to $\mathcal{X}$ is smooth whenever we make a base change by a scheme.
Lemma 98.12.6. Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition 98.12.2. Assume
$\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces,
$\Delta $ is locally of finite type (for example if $\mathcal{X}$ is limit preserving), and
$\mathcal{X}$ 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
\[ \xymatrix{ \mathcal{Z} \ar[r] \ar[d] & (\mathit{Sch}/U)_{fppf} \ar[d]^ x \\ (\mathit{Sch}/V)_{fppf} \ar[r]^ y & \mathcal{X} } \]
Let $Z$ be the algebraic space representing $\mathcal{Z}$ 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. (The parenthetical remark in the statement holds by Lemma 98.11.4.) Observe that $Z$ exists by assumption (1) and Algebraic Stacks, Lemma 94.10.11. By assumption (2) 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 67.25.2. 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 $\mathcal{Z}$, $(\mathit{Sch}/V)_{fppf}$, $(\mathit{Sch}/U)_{fppf}$, and $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(l)$ obtained by pullback to $\mathop{\mathrm{Spec}}(l) = w_0$. Consider
\[ \xymatrix{ \mathcal{F}_{(\mathit{Sch}/W)_{fppf}, l, w_0} \ar[r] & \mathcal{F}_{\mathcal{Z}, l, z_{l, 0}} \ar[d] \ar[r] & \mathcal{F}_{(\mathit{Sch}/U)_{fppf}, l, u_{l, 0}} \ar[d] \\ & \mathcal{F}_{(\mathit{Sch}/V)_{fppf}, l, v_{l, 0}} \ar[r] & \mathcal{F}_{\mathcal{X}, l, x_{l, 0}} } \]
By Lemma 98.3.3 the square is a fibre product of predeformation categories. By Lemma 98.12.5 we see that the right vertical arrow is smooth. By Formal Deformation Theory, Lemma 90.8.7 the left vertical arrow is smooth. By Lemma 98.3.2 we see that the left horizontal arrow is smooth. We conclude that the map
\[ \mathcal{F}_{(\mathit{Sch}/W)_{fppf}, l, w_0} \to \mathcal{F}_{(\mathit{Sch}/V)_{fppf}, l, v_{l, 0}} \]
is smooth by Formal Deformation Theory, Lemma 90.8.7. Thus we conclude that $W \to V$ is smooth at $w_0$ by More on Morphisms, Lemma 37.12.1. 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 98.12.7. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ 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 $\mathop{\mathrm{Spec}}(k) \to S$ with image $s \in S$. Assume
$\xi $ is versal,
$\xi $ is effective,
$\mathcal{O}_{S, s}$ is a G-ring, and
$p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ 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 $x|_{\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u_0}/\mathfrak m_{u_0}^ n)} \cong \xi _ n$.
Proof. Choose an object $x_ R$ of $\mathcal{X}$ lying over $\mathop{\mathrm{Spec}}(R)$ whose associated formal object is $\xi $. Let $N = 2$ and apply Lemma 98.10.1. We obtain $A, \mathfrak m_ A, x_ A, \ldots $. Let $\eta = (A^\wedge , \eta _ n, g_ n)$ be the formal object associated to $x_ A|_{\mathop{\mathrm{Spec}}(A^\wedge )}$. 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 90.8.10. The corresponding ring map $R \to A^\wedge $ is surjective by Formal Deformation Theory, Lemma 90.4.2. 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 90.3.4. 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 98.12.3 we conclude that $x_ A$ is versal at the point $u_0$ of $U = \mathop{\mathrm{Spec}}(A)$ corresponding to $\mathfrak m_ A$. $\square$
Example 98.12.8. 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 98.12.7 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 $S = \mathop{\mathrm{Spec}}(\Lambda )$, $U = S \setminus \{ \mathfrak m\} $, and $S' = U \amalg \mathop{\mathrm{Spec}}(C)$. Consider the functor $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ defined by the rule Let $\mathcal{X} = \mathcal{S}_ F$ is the category fibred in sets associated to $F$, see Algebraic Stacks, Section 94.7. Then $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects and there exists an effective, versal formal object $\xi $ over $R$. Hence if the conclusion of Lemma 98.12.7 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
$\kappa (\mathfrak m) = \kappa (\mathfrak m_ A)$,
$\Lambda \to A$ and $\mathfrak m_ A$ satisfy condition (4) of Algebra, Lemma 10.141.2, and
there exists a $\Lambda $-algebra map $C \to A$.
\[ F(T) = \left\{ \begin{matrix} *
& \text{if }T \to S\text{ factors through }S'
\\ \emptyset
& \text{else}
\end{matrix} \right. \]
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 37.38.5 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 98.12.7 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.
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