Let $R$ be a graded commutative ring such that $X=\operatorname{Proj}(R)$ is a smooth projective variety. Let $M$ be a finite graded module over $R$ and let $\mathcal{F}=\widetilde{M}$ be the associated coherent sheaf on $X$.
If $M$ is indecomposable (i.e. can not be presented as a direct sum of two nonzero graded modules) is it true that $\mathcal{F}$ is also indecomposable?