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Tag 0335

Chapter 10: Commutative Algebra > Section 10.156: Nagata rings

Proposition 10.156.16. The following types of rings are Nagata and in particular universally Japanese:

  1. fields,
  2. Noetherian complete local rings,
  3. $\mathbf{Z}$,
  4. Dedekind domains with fraction field of characteristic zero,
  5. finite type ring extensions of any of the above.

Proof. The Noetherian complete local ring case is Lemma 10.156.8. In the other cases you just check if $R/\mathfrak p$ is N-2 for every prime ideal $\mathfrak p$ of the ring. This is clear whenever $R/\mathfrak p$ is a field, i.e., $\mathfrak p$ is maximal. Hence for the Dedekind ring case we only need to check it when $\mathfrak p = (0)$. But since we assume the fraction field has characteristic zero Lemma 10.155.11 kicks in. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 43649–43659 (see updates for more information).

    \begin{proposition}
    \label{proposition-ubiquity-nagata}
    The following types of rings are Nagata and in particular universally Japanese:
    \begin{enumerate}
    \item fields,
    \item Noetherian complete local rings,
    \item $\mathbf{Z}$,
    \item Dedekind domains with fraction field of characteristic zero,
    \item finite type ring extensions of any of the above.
    \end{enumerate}
    \end{proposition}
    
    \begin{proof}
    The Noetherian complete local ring case is
    Lemma \ref{lemma-Noetherian-complete-local-Nagata}.
    In the other cases you just check if $R/\mathfrak p$ is N-2 for every
    prime ideal $\mathfrak p$ of the ring. This is clear whenever
    $R/\mathfrak p$ is a field, i.e., $\mathfrak p$ is maximal.
    Hence for the Dedekind ring case we only need to check it when
    $\mathfrak p = (0)$. But since we assume the fraction field has
    characteristic zero Lemma \ref{lemma-domain-char-zero-N-1-2} kicks in.
    \end{proof}

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