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Tag 032O

Chapter 10: Commutative Algebra > Section 10.155: Japanese rings

Lemma 10.155.13. Let $R$ be a Noetherian domain. If $R$ is N-1 then $R[x]$ is N-1. If $R$ is N-2 then $R[x]$ is N-2.

Proof. Assume $R$ is N-1. Let $R'$ be the integral closure of $R$ which is finite over $R$. Hence also $R'[x]$ is finite over $R[x]$. The ring $R'[x]$ is normal (see Lemma 10.36.8), hence N-1. This proves the first assertion.

For the second assertion, by Lemma 10.155.7 it suffices to show that $R'[x]$ is N-2. In other words we may and do assume that $R$ is a normal N-2 domain. In characteristic zero we are done by Lemma 10.155.11. In characteristic $p > 0$ we have to show that the integral closure of $R[x]$ is finite in any finite purely inseparable extension of $f.f.(R[x]) = K(x) \subset L$ with $K = f.f.(R)$. Clearly there exists a finite purely inseparable field extension $K \subset L'$ and $q = p^e$ such that $L \subset L'(x^{1/q})$. As $R[x]$ is Noetherian it suffices to show that the integral closure of $R[x]$ in $L'(x^{1/q})$ is finite over $R[x]$. And this integral closure is equal to $R'[x^{1/q}]$ with $R \subset R' \subset L'$ the integral closure of $R$ in $L'$. Since $R$ is N-2 we see that $R'$ is finite over $R$ and hence $R'[x^{1/q}]$ is finite over $R[x]$. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 42776–42781 (see updates for more information).

    \begin{lemma}
    \label{lemma-polynomial-ring-N-2}
    Let $R$ be a Noetherian domain.
    If $R$ is N-1 then $R[x]$ is N-1.
    If $R$ is N-2 then $R[x]$ is N-2.
    \end{lemma}
    
    \begin{proof}
    Assume $R$ is N-1. Let $R'$ be the integral closure of $R$
    which is finite over $R$. Hence also $R'[x]$ is finite over
    $R[x]$. The ring $R'[x]$ is normal (see
    Lemma \ref{lemma-polynomial-domain-normal}), hence N-1.
    This proves the first assertion.
    
    \medskip\noindent
    For the second assertion, by Lemma \ref{lemma-finite-extension-N-2}
    it suffices to show that $R'[x]$ is N-2. In other words we may
    and do assume that $R$ is a normal N-2 domain. In characteristic zero
    we are done by Lemma \ref{lemma-domain-char-zero-N-1-2}.
    In characteristic $p > 0$ we have to show that the integral
    closure of $R[x]$ is finite in any finite purely inseparable extension
    of $f.f.(R[x]) = K(x) \subset L$ with $K = f.f.(R)$. Clearly there
    exists a finite purely inseparable field extension $K \subset L'$
    and $q = p^e$ such that $L \subset L'(x^{1/q})$. As $R[x]$ is
    Noetherian it suffices to show that the integral closure of $R[x]$
    in $L'(x^{1/q})$ is finite over $R[x]$. And this integral closure
    is equal to $R'[x^{1/q}]$ with $R \subset R' \subset L'$ the integral
    closure of $R$ in $L'$.
    Since $R$ is N-2 we see that $R'$ is finite over $R$ and hence
    $R'[x^{1/q}]$ is finite over $R[x]$.
    \end{proof}

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