Lemma 97.12.6. Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition 97.12.2. 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 97.11.4.) Observe that $Z$ exists by assumption (1) and Algebraic Stacks, Lemma 93.10.11. 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 66.25.2. 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 97.3.3 the square is a fibre product of predeformation categories. By Lemma 97.12.5 we see that the right vertical arrow is smooth. By Formal Deformation Theory, Lemma 89.8.7 the left vertical arrow is smooth. By Lemma 97.3.2 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 89.8.7. 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.
$\square$

## Comments (2)

Comment #2985 by Tanya Kaushal Srivastava on

Comment #3109 by Johan on