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Tag 033N

Chapter 27: Properties of Schemes > Section 27.7: Normal schemes

Lemma 27.7.7. Let $X$ be a locally Noetherian scheme. The following are equivalent:

  1. $X$ is normal, and
  2. $X$ is a disjoint union of integral normal schemes.

Proof. Omitted. Hint: This is purely topological from Lemma 27.7.6. $\square$

    The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 860–868 (see updates for more information).

    \begin{lemma}
    \label{lemma-normal-locally-Noetherian}
    Let $X$ be a locally Noetherian scheme.
    The following are equivalent:
    \begin{enumerate}
    \item $X$ is normal, and
    \item $X$ is a disjoint union of integral normal schemes.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Omitted. Hint: This is purely topological from
    Lemma \ref{lemma-normal-Noetherian}.
    \end{proof}

    Comments (2)

    Comment #276 by BB on August 8, 2013 a 8:26 pm UTC

    Maybe I'm missing some definition, but probably "normal" should not be an assumption on the locally noetherian scheme X in the statement.

    Comment #277 by Aise Johan de Jong (site) on August 9, 2013 a 12:34 am UTC

    Fixed here. Thanks!

    There is also 1 comment on Section 27.7: Properties of Schemes.

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