# The Stacks Project

## Tag 033J

Lemma 27.7.2. Let $X$ be a scheme. The following are equivalent:

1. The scheme $X$ is normal.
2. For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is normal.
3. There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is normal.
4. There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is normal.

Moreover, if $X$ is normal then every open subscheme is normal.

Proof. This is clear from the definitions. $\square$

The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 772–786 (see updates for more information).

\begin{lemma}
\label{lemma-locally-normal}
Let $X$ be a scheme. The following are equivalent:
\begin{enumerate}
\item The scheme $X$ is normal.
\item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$
is normal.
\item There exists an affine open covering $X = \bigcup U_i$ such that
each $\mathcal{O}_X(U_i)$ is normal.
\item There exists an open covering $X = \bigcup X_j$
such that each open subscheme $X_j$ is normal.
\end{enumerate}
Moreover, if $X$ is normal then every open subscheme
is normal.
\end{lemma}

\begin{proof}
This is clear from the definitions.
\end{proof}

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