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Tag 033L

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

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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