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Tag 07ZL

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

Lemma 27.2.6. Let $X$ be a scheme. A subset $Z$ of $X$ is retrocompact in $X$ if and only if $E \cap U$ is quasi-compact for every affine open $U$ of $X$.

Proof. Immediate from the fact that every quasi-compact open of $X$ is a finite union of affine opens. $\square$

    The code snippet corresponding to this tag is a part of the file properties.tex and is located in lines 137–141 (see updates for more information).

    \begin{lemma}
    \label{lemma-retrocompact}
    Let $X$ be a scheme. A subset $Z$ of $X$ is retrocompact in $X$ if and only if
    $E \cap U$ is quasi-compact for every affine open $U$ of $X$.
    \end{lemma}
    
    \begin{proof}
    Immediate from the fact that every quasi-compact open of $X$ is a finite
    union of affine opens.
    \end{proof}

    Comments (1)

    Comment #3199 by Dennis Keeler on February 11, 2018 a 6:14 pm UTC

    I believe that $Z=E$ is intended here.

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