Lemma 15.116.2. Let $A \subset B$ be an extension of discrete valuation rings. If $B$ is Nagata and the extension $L/K$ of fraction fields is separable, then $A$ is Nagata.

Proof. A discrete valuation ring is Nagata if and only if it is N-2. Let $K_1/K$ be a finite purely inseparable field extension. We have to show that the integral closure $A_1$ of $A$ in $K_1$ is finite over $A$, see Algebra, Lemma 10.161.12. Since $L/K$ is separable and $K_1/K$ is purely inseparable, the algebra $L \otimes _ K K_1$ is a field (by Algebra, Lemmas 10.43.6 and 10.46.10). Let $B_1$ be the integral closure of $B$ in $L \otimes _ K K_1$. Since $B$ is Nagata, $B_1$ is finite over $B$. Since $B \otimes _ A A_1 \subset B_1$ and $B$ is Noetherian, we see that $B \otimes _ A A_1$ is finite over $B$. As $A \to B$ is faithfully flat, this implies $A_1$ is finite over $A$, see Algebra, Lemma 10.83.2. $\square$

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