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Tag 029U

Chapter 102: Exercises > Section 102.35: Quasi-coherent Sheaves

Exercise 102.35.7. Let $X = \mathop{\mathrm{Spec}}(R)$ be an affine scheme.

  1. Let $f \in R$. Let $\mathcal{G}$ be a quasi-coherent sheaf of $\mathcal{O}_{D(f)}$-modules on the open subscheme $D(f)$. Show that $\mathcal{G} = \mathcal{F}|_{D(f)}$ for some quasi-coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$.
  2. Let $I \subset R$ be an ideal. Let $i : Z \to X$ be the closed subscheme of $X$ corresponding to $I$. Let $\mathcal{G}$ be a quasi-coherent sheaf of $\mathcal{O}_Z$-modules on the closed subscheme $Z$. Show that $\mathcal{G} = i^*\mathcal{F}$ for some quasi-coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$. (Why is this silly?)
  3. Assume that $R$ is Noetherian. Let $f \in R$. Let $\mathcal{G}$ be a coherent sheaf of $\mathcal{O}_{D(f)}$-modules on the open subscheme $D(f)$. Show that $\mathcal{G} = \mathcal{F}|_{D(f)}$ for some coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$.

    The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 3321–3346 (see updates for more information).

    \begin{exercise}
    \label{exercise-extend-quasi-coherent}
    Let $X = \Spec(R)$ be an affine scheme.
    \begin{enumerate}
    \item Let $f \in R$. Let $\mathcal{G}$ be a
    quasi-coherent sheaf of $\mathcal{O}_{D(f)}$-modules
    on the open subscheme $D(f)$.
    Show that $\mathcal{G} = \mathcal{F}|_{D(f)}$ for some
    quasi-coherent sheaf of $\mathcal{O}_X$-modules
    $\mathcal{F}$.
    \item Let $I \subset R$ be an ideal.
    Let $i : Z \to X$ be the closed subscheme of $X$ corresponding
    to $I$. Let $\mathcal{G}$ be a
    quasi-coherent sheaf of $\mathcal{O}_Z$-modules
    on the closed subscheme $Z$.
    Show that $\mathcal{G} = i^*\mathcal{F}$ for some
    quasi-coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$.
    (Why is this silly?)
    \item Assume that $R$ is Noetherian.
    Let $f \in R$. Let $\mathcal{G}$ be a
    coherent sheaf of $\mathcal{O}_{D(f)}$-modules
    on the open subscheme $D(f)$.
    Show that $\mathcal{G} = \mathcal{F}|_{D(f)}$ for some
    coherent sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$.
    \end{enumerate}
    \end{exercise}

    Comments (2)

    Comment #2363 by Dominic Wynter on January 30, 2017 a 7:23 pm UTC

    I presume that we have set $U:=D(f)$?

    Comment #2426 by Johan (site) on February 17, 2017 a 2:23 pm UTC

    Thanks, fixed here.

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