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Tag 0B7L

Chapter 31: Limits of Schemes > Section 31.11: Characterizing affine schemes

Lemma 31.11.5. Let $i : Z \to X$ be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Then $X$ is quasi-affine if and only if $Z$ is quasi-affine.

Proof. Recall that a scheme is quasi-affine if and only if the structure sheaf is ample, see Properties, Lemma 27.27.1. Hence if $Z$ is quasi-affine, then $\mathcal{O}_Z$ is ample, hence $\mathcal{O}_X$ is ample by Lemma 31.11.4, hence $X$ is quasi-affine. A proof of the converse, which can also be seen in an elementary way, is gotten by reading the argument just given backwards. $\square$

    The code snippet corresponding to this tag is a part of the file limits.tex and is located in lines 2645–2650 (see updates for more information).

    \begin{lemma}
    \label{lemma-thickening-quasi-affine}
    Let $i : Z \to X$ be a closed immersion of schemes
    inducing a homeomorphism of underlying topological spaces.
    Then $X$ is quasi-affine if and only if $Z$ is quasi-affine.
    \end{lemma}
    
    \begin{proof}
    Recall that a scheme is quasi-affine
    if and only if the structure sheaf is ample, see
    Properties, Lemma \ref{properties-lemma-quasi-affine-O-ample}.
    Hence if $Z$ is quasi-affine, then $\mathcal{O}_Z$ is ample,
    hence $\mathcal{O}_X$ is ample by
    Lemma \ref{lemma-ample-on-reduction}, hence
    $X$ is quasi-affine. A proof of the converse, which
    can also be seen in an elementary way, is gotten by
    reading the argument just given backwards.
    \end{proof}

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