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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like
$\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.