## 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:

- $X$ is normal, and
- $X$ 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}
```

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