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Tag 01OZ

Chapter 27: Properties of Schemes > Section 27.5: Noetherian schemes

Lemma 27.5.5. A (locally) Noetherian scheme has a (locally) Noetherian underlying topological space, see Topology, Definition 5.9.1.

Proof. This is because a Noetherian scheme is a finite union of spectra of Noetherian rings and Algebra, Lemma 10.30.5 and Topology, Lemma 5.9.4. $\square$

    The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 460–465 (see updates for more information).

    \begin{lemma}
    \label{lemma-Noetherian-topology}
    A (locally) Noetherian scheme has a (locally)
    Noetherian underlying topological space,
    see Topology, Definition \ref{topology-definition-noetherian}.
    \end{lemma}
    
    \begin{proof}
    This is because a Noetherian scheme is a finite union of spectra
    of Noetherian rings and
    Algebra, Lemma \ref{algebra-lemma-Noetherian-topology} and
    Topology, Lemma \ref{topology-lemma-finite-union-Noetherian}.
    \end{proof}

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