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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 42809–42815 (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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