The Stacks Project

Tag 033N

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 are also 2 comments on Section 27.7: Properties of Schemes.

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