The Stacks project

Lemma 107.2.4. In Situation 107.2.1 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 107.2.3 and Formal Deformation Theory, Lemma 90.13.4 (note that the assumptions of this lemma hold by Formal Deformation Theory, Lemmas 90.16.6 and Definition 90.16.8). By the uniquness result of Formal Deformation Theory, Lemma 90.14.5 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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