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Tag 033M

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

Lemma 27.7.6. Let $X$ be a Noetherian scheme. The following are equivalent:

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

Proof. This is a special case of Lemma 27.7.5 because a Noetherian scheme has a Noetherian underlying topological space (Lemma 27.5.5 and Topology, Lemma 5.9.2. $\square$

    The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 841–849 (see updates for more information).

    \begin{lemma}
    \label{lemma-normal-Noetherian}
    Let $X$ be a Noetherian scheme.
    The following are equivalent:
    \begin{enumerate}
    \item $X$ is normal, and
    \item $X$ is a finite disjoint union of normal integral schemes.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    This is a special case of
    Lemma \ref{lemma-normal-locally-finite-nr-irreducibles} because a Noetherian
    scheme has a Noetherian underlying topological space
    (Lemma \ref{lemma-Noetherian-topology}
    and
    Topology, Lemma \ref{topology-lemma-Noetherian}.
    \end{proof}

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