98.13 Openness of versality
Next, we come to openness of versality.
Definition 98.13.1. Let $S$ be a locally Noetherian scheme.
Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say $\mathcal{X}$ satisfies openness of versality if given a scheme $U$ locally of finite type over $S$, an object $x$ of $\mathcal{X}$ over $U$, and a finite type point $u_0 \in U$ such that $x$ is versal at $u_0$, then there exists an open neighbourhood $u_0 \in U' \subset U$ such that $x$ is versal at every finite type point of $U'$.
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say $f$ satisfies openness of versality if given a scheme $U$ locally of finite type over $S$, an object $y$ of $\mathcal{Y}$ over $U$, openness of versality holds for $(\mathit{Sch}/U)_{fppf} \times _\mathcal {Y} \mathcal{X}$.
Openness of versality is often the hardest to check. The following example shows that requiring this is necessary however.
Example 98.13.2. Let $k$ be a field and set $\Lambda = k[s, t]$. Consider the functor $F : \Lambda \text{-algebras} \longrightarrow \textit{Sets}$ defined by the rule
\[ F(A) = \left\{ \begin{matrix} *
& \text{if there exist }f_1, \ldots , f_ n \in A\text{ such that }
\\ & A = (s, t, f_1, \ldots , f_ n)\text{ and } f_ i s = 0\ \forall i
\\ \emptyset
& \text{else}
\end{matrix} \right. \]
Geometrically $F(A) = *$ means there exists a quasi-compact open neighbourhood $W$ of $V(s, t) \subset \mathop{\mathrm{Spec}}(A)$ such that $s|_ W = 0$. Let $\mathcal{X} \subset (\mathit{Sch}/\mathop{\mathrm{Spec}}(\Lambda ))_{fppf}$ be the full subcategory consisting of schemes $T$ which have an affine open covering $T = \bigcup \mathop{\mathrm{Spec}}(A_ j)$ with $F(A_ j) = *$ for all $j$. Then $\mathcal{X}$ satisfies [0], [1], [2], [3], and [4] but not [5]. Namely, over $U = \mathop{\mathrm{Spec}}(k[s, t]/(s))$ there exists an object $x$ which is versal at $u_0 = (s, t)$ but not at any other point. Details omitted.
Let $S$ be a locally Noetherian scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Consider the following property
98.13.2.1
\begin{equation} \label{artin-equation-smooth} \begin{matrix} \text{for all fields }k\text{ of finite type over }S \text{ and all }x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{\mathop{\mathrm{Spec}}(k)})\text{ the}
\\ \text{map } \mathcal{F}_{\mathcal{X}, k, x_0} \to \mathcal{F}_{\mathcal{Y}, k, f(x_0)} \text{ of predeformation categories is smooth}
\end{matrix} \end{equation}
We formulate some lemmas around this concept. First we link it with (openness of) versality.
Lemma 98.13.3. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category 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 $\mathcal{X}$ over $U$. Assume that $x$ is versal at every finite type point of $U$ and that $\mathcal{X}$ satisfies (RS). Then $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ satisfies (98.13.2.1).
Proof.
Let $\mathop{\mathrm{Spec}}(l) \to U$ be a morphism with $l$ of finite type over $S$. Then the image $u_0 \in U$ is a finite type point of $U$ and $l/\kappa (u_0)$ is a finite extension, see discussion in Morphisms, Section 29.16. Hence we see that $\mathcal{F}_{(\mathit{Sch}/U)_{fppf}, l, u_{l, 0}} \to \mathcal{F}_{\mathcal{X}, l, x_{l, 0}}$ is smooth by Lemma 98.12.5.
$\square$
Lemma 98.13.4. Let $S$ be a locally Noetherian scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be composable $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $f$ and $g$ satisfy (98.13.2.1) so does $g \circ f$.
Proof.
This follows formally from Formal Deformation Theory, Lemma 90.8.7.
$\square$
Lemma 98.13.5. Let $S$ be a locally Noetherian scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ and $\mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $f$ satisfies (98.13.2.1) so does the projection $\mathcal{X} \times _\mathcal {Y} \mathcal{Z} \to \mathcal{Z}$.
Proof.
Follows immediately from Lemma 98.3.3 and Formal Deformation Theory, Lemma 90.8.7.
$\square$
Lemma 98.13.6. Let $S$ be a locally Noetherian scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $f$ is formally smooth on objects, then $f$ satisfies (98.13.2.1). If $f$ is representable by algebraic spaces and smooth, then $f$ satisfies (98.13.2.1).
Proof.
A reformulation of Lemma 98.3.2.
$\square$
Lemma 98.13.7. Let $S$ be a locally Noetherian scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume
$f$ is representable by algebraic spaces,
$f$ satisfies (98.13.2.1),
$\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects, and
$\mathcal{Y}$ is limit preserving.
Then $f$ is smooth.
Proof.
The key ingredient of the proof is More on Morphisms, Lemma 37.12.1 which (almost) says that a morphism of schemes of finite type over $S$ satisfying (98.13.2.1) is a smooth morphism. The other arguments of the proof are essentially bookkeeping.
Let $V$ be a scheme over $S$ and let $y$ be an object of $\mathcal{Y}$ over $V$. Let $Z$ be an algebraic space representing the $2$-fibre product $\mathcal{Z} = \mathcal{X} \times _{f, \mathcal{X}, y} (\mathit{Sch}/V)_{fppf}$. We have to show that the projection morphism $Z \to V$ is smooth, see Algebraic Stacks, Definition 94.10.1. In fact, it suffices to do this when $V$ is an affine scheme locally of finite presentation over $S$, see Criteria for Representability, Lemma 97.5.6. Then $(\mathit{Sch}/V)_{fppf}$ is limit preserving by Lemma 98.11.3. Hence $Z \to S$ is locally of finite presentation by Lemmas 98.11.2 and 98.11.3. Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Then $W$ is locally of finite presentation over $S$.
Since $f$ satisfies (98.13.2.1) we see that so does $\mathcal{Z} \to (\mathit{Sch}/V)_{fppf}$, see Lemma 98.13.5. Next, we see that $(\mathit{Sch}/W)_{fppf} \to \mathcal{Z}$ satisfies (98.13.2.1) by Lemma 98.13.6. Thus the composition
\[ (\mathit{Sch}/W)_{fppf} \to \mathcal{Z} \to (\mathit{Sch}/V)_{fppf} \]
satisfies (98.13.2.1) by Lemma 98.13.4. More on Morphisms, Lemma 37.12.1 shows that the composition $W \to Z \to V$ is smooth at every finite type point $w_0$ of $W$. Since the smooth locus is open we conclude that $W \to V$ is a smooth morphism of schemes by Morphisms, Lemma 29.16.7. Thus we conclude that $Z \to V$ is a smooth morphism of algebraic spaces by definition.
$\square$
The lemma below is how we will use openness of versality.
Lemma 98.13.8. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $k$ be a finite type field over $S$ and let $x_0$ be an object of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(k)$ with image $s \in S$. Assume
$\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces,
$\mathcal{X}$ satisfies axioms [1], [2], [3] (see Section 98.14),
every formal object of $\mathcal{X}$ is effective,
openness of versality holds for $\mathcal{X}$, and
$\mathcal{O}_{S, s}$ is a G-ring.
Then there exist a morphism of finite type $U \to S$ and an object $x$ of $\mathcal{X}$ over $U$ such that
\[ x : (\mathit{Sch}/U)_{fppf} \longrightarrow \mathcal{X} \]
is smooth and such that there exists a finite type point $u_0 \in U$ whose residue field is $k$ and such that $x|_{u_0} \cong x_0$.
Proof.
By axiom [2], Lemma 98.6.1, and Remark 98.6.2 we see that $\mathcal{F}_{\mathcal{X}, k, x_0}$ satisfies (S1) and (S2). Since also the tangent space has finite dimension by axiom [3] we deduce from Formal Deformation Theory, Lemma 90.13.4 that $\mathcal{F}_{\mathcal{X}, k, x_0}$ has a versal formal object $\xi $. Assumption (3) says $\xi $ is effective. By axiom [1] and Lemma 98.12.7 there exists a morphism of finite type $U \to S$, an object $x$ of $\mathcal{X}$ over $U$, and a finite type point $u_0$ of $U$ with residue field $k$ such that $x$ is versal at $u_0$ and such that $x|_{\mathop{\mathrm{Spec}}(k)} \cong x_0$. By openness of versality we may shrink $U$ and assume that $x$ is versal at every finite type point of $U$. We claim that
\[ x : (\mathit{Sch}/U)_{fppf} \longrightarrow \mathcal{X} \]
is smooth which proves the lemma. Namely, by Lemma 98.13.3 $x$ satisfies (98.13.2.1) whereupon Lemma 98.13.7 finishes the proof.
$\square$
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