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Tag 01J4

Chapter 25: Schemes > Section 25.12: Reduced schemes

Definition 25.12.5. Let $X$ be a scheme. Let $Z \subset X$ be a closed subset. A scheme structure on $Z$ is given by a closed subscheme $Z'$ of $X$ whose underlying set is equal to $Z$. We often say ''let $(Z, \mathcal{O}_Z)$ be a scheme structure on $Z$'' to indicate this. The reduced induced scheme structure on $Z$ is the one constructed in Lemma 25.12.4. The reduction $X_{red}$ of $X$ is the reduced induced scheme structure on $X$ itself.

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 2095–2105 (see updates for more information).

    \begin{definition}
    \label{definition-reduced-induced-scheme}
    Let $X$ be a scheme. Let $Z \subset X$ be a closed subset.
    A {\it scheme structure on $Z$} is given by a closed subscheme $Z'$ of
    $X$ whose underlying set is equal to $Z$. We often say
    ``let $(Z, \mathcal{O}_Z)$ be a scheme structure on $Z$'' to
    indicate this. The {\it reduced induced scheme structure}
    on $Z$ is the one constructed in Lemma \ref{lemma-reduced-closed-subscheme}.
    The {\it reduction $X_{red}$ of $X$} is the reduced induced scheme
    structure on $X$ itself.
    \end{definition}

    Comments (2)

    Comment #2616 by Harry on July 4, 2017 a 3:00 pm UTC

    I guess "underlying closed" should be "underlying set". Another cosmetic suggestion: Change the second sentence to "Let $Z\subset X$ be a closed subset.", which is the form that's used in the subsequent paragraphs.

    Comment #2636 by Johan (site) on July 7, 2017 a 12:59 pm UTC

    Thanks Harry, fixed here.

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