The Stacks project

Lemma 106.2.4. In Situation 106.2.1 let $x_0 : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Then a versal ring to $\mathcal{X}$ at $x_0$ exists. Given a pair $A$, $A'$ of these, then $A \cong A'[[t_1, \ldots , t_ r]]$ or $A' \cong A[[t_1, \ldots , t_ r]]$ as $S$-algebras for some $r$.

Proof. By Lemma 106.2.3 and Formal Deformation Theory, Lemma 89.13.4 (note that the assumptions of this lemma hold by Formal Deformation Theory, Lemmas 89.16.6 and Definition 89.16.8). By the uniquness result of Formal Deformation Theory, Lemma 89.14.5 there exists a “minimal” versal ring $A$ of $\mathcal{X}$ at $x_0$ such that any other versal ring of $\mathcal{X}$ at $x_0$ is isomorphic to $A[[t_1, \ldots , t_ r]]$ for some $r$. This clearly implies the second statement. $\square$


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