The Stacks project

Lemma 105.2.5. In Situation 105.2.1 let $x_0 : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Let $l/k$ be a finite extension of fields and denote $x_{l, 0} : \mathop{\mathrm{Spec}}(l) \to \mathcal{X}$ the induced morphism. Given a versal ring $A$ to $\mathcal{X}$ at $x_0$ there exists a versal ring $A'$ to $\mathcal{X}$ at $x_{l, 0}$ such that there is a $S$-algebra map $A \to A'$ which induces the given field extension $l/k$ and is formally smooth in the $\mathfrak m_{A'}$-adic topology.

Proof. Follows immediately from Artin's Axioms, Lemma 96.7.1 and Formal Deformation Theory, Lemma 88.29.6. (We also use that $\mathcal{X}$ satisfies (RS) by Artin's Axioms, Lemma 96.5.2.) $\square$


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