The Stacks project

Lemma 15.116.9. Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. Assume

  1. $B$ is essentially of finite type over $A$,

  2. either $A$ or $B$ is a Nagata ring, and

  3. $L/K$ is separable.

Then there exists a separable solution for $A \to B$ (Definition 15.115.1).

Proof. Observe that if $A$ is Nagata, then so is $B$ (Algebra, Lemma 10.162.6 and Proposition 10.162.15). Thus the lemma follows on combining Proposition 15.116.8 and Lemma 15.116.6. $\square$

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