Unramified extensions of DVRs are ind-smooth AKA Néron desingularization

Lemma 16.4.6. Let $R \subset \Lambda$ be an extension of discrete valuation rings which has ramification index $1$ and induces a separable extension of residue fields and of fraction fields. Then $\Lambda$ is a filtered colimit of smooth $R$-algebras.

Proof. By Algebra, Lemma 10.126.4 it suffices to show that any $R \to A \to \Lambda$ as in Situation 16.4.1 can be factored as $A \to B \to \Lambda$ with $B$ a smooth $R$-algebra. After replacing $A$ by its image in $\Lambda$ we may assume that $A$ is a domain whose fraction field $K$ is a subfield of the fraction field of $\Lambda$. In particular, $A$ is separable over the fraction field of $R$ by our assumptions. Then $R \to A$ is smooth at $\mathfrak q = (0)$ by Algebra, Lemma 10.139.9. After a finite number of Néron blowups, we may assume $R \to A$ is smooth at $\mathfrak p$, see Lemma 16.4.5. Then, after replacing $A$ by a localization at an element $a \in A$, $a \not\in \mathfrak p$ it becomes smooth over $R$ and the lemma is proved. $\square$

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