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Tag 094L

Chapter 27: Properties of Schemes > Section 27.2: Constructible sets

Lemma 27.2.4. Let $X$ be a quasi-compact and quasi-separated scheme. Then the underlying topological space of $X$ is a spectral space.

Proof. By Topology, Definition 5.23.1 we have to check that $X$ is sober, quasi-compact, has a basis of quasi-compact opens, and the intersection of any two quasi-compact opens is quasi-compact. This follows from Schemes, Lemma 25.11.1 and 25.11.2 and Lemma 27.2.3 above. $\square$

    The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 101–105 (see updates for more information).

    \begin{lemma}
    \label{lemma-quasi-compact-quasi-separated-spectral}
    Let $X$ be a quasi-compact and quasi-separated scheme.
    Then the underlying topological space of $X$ is a spectral space.
    \end{lemma}
    
    \begin{proof}
    By Topology, Definition \ref{topology-definition-spectral-space}
    we have to check that $X$ is sober, quasi-compact, has a basis
    of quasi-compact opens, and the intersection of any two
    quasi-compact opens is quasi-compact. This follows from
    Schemes, Lemma \ref{schemes-lemma-scheme-sober} and
    \ref{schemes-lemma-basis-affine-opens}
    and
    Lemma \ref{lemma-quasi-separated-quasi-compact-open-retrocompact}
    above.
    \end{proof}

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