Let $S$ be a graded ring, let $X=\operatorname{Proj}S$ be a sheaf, and $\mathscr{F}$ be a quasicoherent sheaf on $\mathscr{O}_X$. We know that for any graded $S$-module $M$, $\widetilde{M}$ is a quasicoherent $\mathscr{O}_X$-module, but does the converse hold?

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    $\begingroup$ This is true if $S$ is finitely generated by $S_1$ as an $S_0$-algebra. This is Proposition 5.15 in Hartshorne. The assumption is quite important for the proof, I do not know if anything more general can be said. $\endgroup$ – Jesko Hüttenhain Mar 30 '17 at 19:28

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