The Stacks project

Lemma 98.12.6. Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition 98.12.2. Assume

  1. $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces,

  2. $\Delta $ is locally of finite type (for example if $\mathcal{X}$ 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

\[ \xymatrix{ \mathcal{Z} \ar[r] \ar[d] & (\mathit{Sch}/U)_{fppf} \ar[d]^ x \\ (\mathit{Sch}/V)_{fppf} \ar[r]^ y & \mathcal{X} } \]

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 ├ętale 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. $\square$

Comments (2)

Comment #2985 by Tanya Kaushal Srivastava on

Typo in the statement of the lemma: Let Z be the algebraic space representing ---> Let be the algebraic space representing . (There is no around). $$

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