Lemma 15.116.9 (0BRP)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.116: Eliminating ramification, II
Lemma 15.116.9 (cite)

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

$B$ is essentially of finite type over $A$ ,
either $A$ or $B$ is a Nagata ring, and
$L/K$ 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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