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Tag 033O

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

Remark 27.7.8. Let $X$ be a normal scheme. If $X$ is locally Noetherian then we see that $X$ is integral if and only if $X$ is connected, see Lemma 27.7.7. But there exists a connected affine scheme $X$ such that $\mathcal{O}_{X, x}$ is a domain for all $x \in X$, but $X$ is not irreducible, see Examples, Section 99.5. This example is even a normal scheme (proof omitted), so beware!

    The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 875–885 (see updates for more information).

    \begin{remark}
    \label{remark-normal-connected-irreducible}
    Let $X$ be a normal scheme. If $X$ is locally Noetherian then we see that
    $X$ is integral if and only if $X$ is connected, see
    Lemma \ref{lemma-normal-locally-Noetherian}.
    But there exists a connected affine scheme $X$ such that
    $\mathcal{O}_{X, x}$ is a domain for all $x \in X$, but $X$ is not
    irreducible, see Examples, Section
    \ref{examples-section-connected-locally-integral-not-integral}.
    This example is even a normal scheme (proof omitted), so beware!
    \end{remark}

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