## 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 $D(x)$.

- Show that every global section $s$ of ${\mathcal F}$ is killed by some power of $x$, i.e., $x^ns = 0$ for some $n\in {\mathbf N}$.
- 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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