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Tag 01OX

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

Lemma 27.5.3. Any immersion $Z \to X$ with $X$ locally Noetherian is quasi-compact.

Proof. A closed immersion is clearly quasi-compact. A composition of quasi-compact morphisms is quasi-compact, see Topology, Lemma 5.12.2. Hence it suffices to show that an open immersion into a locally Noetherian scheme is quasi-compact. Using Schemes, Lemma 25.19.2 we reduce to the case where $X$ is affine. Any open subset of the spectrum of a Noetherian ring is quasi-compact (for example combine Algebra, Lemma 10.30.5 and Topology, Lemmas 5.9.2 and 5.12.13). $\square$

    The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 425–428 (see updates for more information).

    \begin{lemma}
    \label{lemma-immersion-into-noetherian}
    Any immersion $Z \to X$ with $X$ locally Noetherian is quasi-compact.
    \end{lemma}
    
    \begin{proof}
    A closed immersion is clearly quasi-compact.
    A composition of quasi-compact morphisms is quasi-compact,
    see Topology, Lemma \ref{topology-lemma-composition-quasi-compact}.
    Hence it suffices to show that an open immersion into
    a locally Noetherian scheme is quasi-compact.
    Using Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine}
    we reduce to the case where $X$ is affine.
    Any open subset of the spectrum of a Noetherian ring
    is quasi-compact (for example
    combine Algebra, Lemma \ref{algebra-lemma-Noetherian-topology} and
    Topology, Lemmas \ref{topology-lemma-Noetherian} and
    \ref{topology-lemma-Noetherian-quasi-compact}).
    \end{proof}

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