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?
Let $R=\mathbb{C}[x,y]$ (with $x$ and $y$ in degree one), so we're considering sheaves on the projective line.
Let $M=\frac{\mathbb{C}[x,y]}{(xy)}$. Then $M$ is indecomposable.
$M$ also has a graded submodule $xM\oplus yM$, such that the quotient $M/(xM\oplus yM)$ is one dimensional and so has zero associated coherent sheaf.
So the inclusion $xM\oplus yM\to M$ induces an isomorphism of associated coherent sheaves $\widetilde{xM}\oplus\widetilde{yM}\cong\widetilde{M}$.