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Tag 02FH

Chapter 102: Exercises > Section 102.48: Schemes, Final Exam, Fall 2007

Exercise 102.48.2. Let $X = \mathop{\mathrm{Spec}}({\mathbf Z}[x, y])$, and let ${\mathcal F}$ be a quasi-coherent ${\mathcal O}_X$-module. Suppose that ${\mathcal F}$ is zero when restricted to the standard affine open $D(x)$.

  1. Show that every global section $s$ of ${\mathcal F}$ is killed by some power of $x$, i.e., $x^ns = 0$ for some $n\in {\mathbf N}$.
  2. Do you think the same is true if we do not assume that ${\mathcal F}$ is quasi-coherent?

    The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 4837–4850 (see updates for more information).

    \begin{exercise}
    \label{exercise-kill-global-sections}
    Let $X = \Spec({\mathbf Z}[x, y])$, and let ${\mathcal F}$ be a
    quasi-coherent
    ${\mathcal O}_X$-module. Suppose that ${\mathcal F}$ is zero when restricted to
    the
    standard affine open $D(x)$.
    \begin{enumerate}
    \item Show that every global section $s$ of ${\mathcal F}$ is killed by some
    power of $x$, i.e., $x^ns = 0$ for some $n\in {\mathbf N}$.
    \item Do you think the same is true if we do not assume that ${\mathcal F}$
    is quasi-coherent?
    \end{enumerate}
    \end{exercise}

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