Definition 107.2.2. 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 $S$. A versal ring to $\mathcal{X}$ at $x_0$ is a complete Noetherian local $S$-algebra $A$ with residue field $k$ such that there exists a versal formal object $(A, \xi _ n, f_ n)$ as in Artin's Axioms, Definition 98.12.1 with $\xi _1 \cong x_0$ (a $2$-isomorphism).
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