Let $X$ be a quasi-projective scheme over a noetherian ring, $\mathcal F$ a coherent sheaf of $\mathcal O_X$-modules, and $U$ an arbitrary open subset of $X$. Is $\Gamma(U,\mathcal F)$ a $\Gamma(U,\mathcal O_X)$-module of finite type?
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I think the answer is no: take $X=\mathbb{P}^2_{k}$, $U=\mathbb{P}^2\setminus \{ p \}$ for a closed point $p$ and $\mathcal{F}=i_*\mathcal{O}_L$, where $i\colon L\hookrightarrow\mathbb{P}^2$ is line passing through $p$. Then $\Gamma(U,\mathcal{F})\cong k[X]$ is not finitely generated over $\Gamma(U,\mathcal{O}_X)=k$.
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$\begingroup$ Thank you. I'm now interested in the case when $\mathcal F$ is invertible, but I will open another question so that I can accept the new answer. $\endgroup$– A.GMay 30, 2016 at 17:08
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$\begingroup$ Bravo for this simple and clever counterexample: +1 $\endgroup$ Jun 2, 2016 at 11:07