# Is every quasicoherent sheaf $\mathscr{F}$ on $\operatorname{Proj}S$ of the form $\widetilde{M}$?

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?

• 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. – Jesko Hüttenhain Mar 30 '17 at 19:28