The Stacks project

Lemma 105.2.6. In Situation 105.2.1 let $x : U \to \mathcal{X}$ be a morphism where $U$ is a scheme locally of finite type over $S$. Let $u_0 \in U$ be a finite type point. Set $k = \kappa (u_0)$ and denote $x_0 : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ the induced map. The following are equivalent

  1. $x$ is versal at $u_0$ (Artin's Axioms, Definition 96.12.2),

  2. $\hat x : \mathcal{F}_{U, k, u_0} \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth,

  3. the formal object associated to $x|_{\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u_0}^\wedge )}$ is versal, and

  4. there is an open neighbourhood $U' \subset U$ of $x$ such that $x|_{U'} : U' \to \mathcal{X}$ is smooth.

Moreover, in this case the completion $\mathcal{O}_{U, u_0}^\wedge $ is a versal ring to $\mathcal{X}$ at $x_0$.

Proof. Since $U \to S$ is locally of finite type (as a composition of such morphisms), we see that $\mathop{\mathrm{Spec}}(k) \to S$ is of finite type (again as a composition). Thus the statement makes sense. The equivalence of (1) and (2) is the definition of $x$ being versal at $u_0$. The equivalence of (1) and (3) is Artin's Axioms, Lemma 96.12.3. Thus (1), (2), and (3) are equivalent.

If $x|_{U'}$ is smooth, then the functor $\hat x : \mathcal{F}_{U, k, u_0} \to \mathcal{F}_{\mathcal{X}, k, x_0}$ is smooth by Artin's Axioms, Lemma 96.3.2. Thus (4) implies (1), (2), and (3). For the converse, assume $x$ is versal at $u_0$. Choose a surjective smooth morphism $y : V \to \mathcal{X}$ where $V$ is a scheme. Set $Z = V \times _\mathcal {X} U$ and pick a finite type point $z_0 \in |Z|$ lying over $u_0$ (this is possible by Morphisms of Spaces, Lemma 65.25.5). By Artin's Axioms, Lemma 96.12.6 the morphism $Z \to V$ is smooth at $z_0$. By definition we can find an open neighbourhood $W \subset Z$ of $z_0$ such that $W \to V$ is smooth. Since $Z \to U$ is open, let $U' \subset U$ be the image of $W$. Then we see that $U' \to \mathcal{X}$ is smooth by our definition of smooth morphisms of stacks.

The final statement follows from the definitions as $\mathcal{O}_{U, u_0}^\wedge $ prorepresents $\mathcal{F}_{U, k, u_0}$. $\square$


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