## Tag `032N`

Chapter 10: Commutative Algebra > Section 10.155: Japanese rings

Lemma 10.155.12. Let $R$ be a Noetherian domain with fraction field $K$ of characteristic $p > 0$. Then $R$ is N-2 if and only if for every finite purely inseparable extension $K \subset L$ the integral closure of $R$ in $L$ is finite over $R$.

Proof.Assume the integral closure of $R$ in every finite purely inseparable field extension of $K$ is finite. Let $K \subset L$ be any finite extension. We have to show the integral closure of $R$ in $L$ is finite over $R$. Choose a finite normal field extension $K \subset M$ containing $L$. As $R$ is Noetherian it suffices to show that the integral closure of $R$ in $M$ is finite over $R$. By Fields, Lemma 9.27.3 there exists a subextension $K \subset M_{insep} \subset M$ such that $M_{insep}/K$ is purely inseparable, and $M/M_{insep}$ is separable. By assumption the integral closure $R'$ of $R$ in $M_{insep}$ is finite over $R$. By Lemma 10.155.8 the integral closure $R''$ of $R'$ in $M$ is finite over $R'$. Then $R''$ is finite over $R$ by Lemma 10.7.3. Since $R''$ is also the integral closure of $R$ in $M$ (see Lemma 10.35.16) we win. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 42747–42753 (see updates for more information).

```
\begin{lemma}
\label{lemma-domain-char-p-N-1-2}
Let $R$ be a Noetherian domain with fraction field $K$ of
characteristic $p > 0$. Then $R$ is N-2 if and only if
for every finite purely inseparable extension $K \subset L$ the integral
closure of $R$ in $L$ is finite over $R$.
\end{lemma}
\begin{proof}
Assume the integral closure of $R$ in every finite purely inseparable
field extension of $K$ is finite.
Let $K \subset L$ be any finite extension. We have to show the
integral closure of $R$ in $L$ is finite over $R$.
Choose a finite normal field extension $K \subset M$
containing $L$. As $R$ is Noetherian it suffices to show that
the integral closure of $R$ in $M$ is finite over $R$.
By Fields, Lemma \ref{fields-lemma-normal-case}
there exists a subextension $K \subset M_{insep} \subset M$
such that $M_{insep}/K$ is purely inseparable, and $M/M_{insep}$
is separable. By assumption the integral closure $R'$ of $R$ in
$M_{insep}$ is finite over $R$. By
Lemma \ref{lemma-Noetherian-normal-domain-finite-separable-extension}
the integral
closure $R''$ of $R'$ in $M$ is finite over $R'$. Then $R''$ is finite
over $R$ by Lemma \ref{lemma-finite-transitive}.
Since $R''$ is also the integral closure
of $R$ in $M$ (see Lemma \ref{lemma-integral-closure-transitive}) we win.
\end{proof}
```

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