# The Stacks Project

## Tag 033L

Lemma 27.7.4. Let $X$ be an integral scheme. Then $X$ is normal if and only if for every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is a normal domain.

Proof. This follows from Algebra, Lemma 10.36.10. $\square$

The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 801–806 (see updates for more information).

\begin{lemma}
\label{lemma-integral-normal}
Let $X$ be an integral scheme.
Then $X$ is normal if and only if for every affine open
$U \subset X$ the ring $\mathcal{O}_X(U)$ is a normal domain.
\end{lemma}

\begin{proof}
This follows from
Algebra, Lemma \ref{algebra-lemma-normality-is-local}.
\end{proof}

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