Roughly speaking , the question is : when a f.g. graded module induces a trivial coherent sheave on Proj ? More precisely, let S be a (complex) graded polynomial algebra, where the variables have (any) positive degrees. I do not know how to characterize the f.g. graded S-modules whose associated coherent sheaves on X=Proj(S) are equal to zero. This should help,for instance,in computing the Grothendieck group K.(X) of coherent sheaves on X (without using Chow group of X) . The standard case, when S is generated in degree 1, is well known,e.g. by D.Quillen (Higher algebraic K-theory (I), Lect.Notes Math. 341): a f.g. graded S-module induces a trivial coherent sheaf on Proj(S) iff it has (at most) finitely many non-trivial homogeneous components.
[Math] Coherent sheaves on Proj
ag.algebraic-geometryalgebraic-k-theory
Best Answer
If we change $Proj(S)$ by $Proj(S^{(d)})$ then $\widetilde{M}$ corresponds to $\widetilde{M^{(d)}}$. Let $M$ be a f.g. graded $S$-module with $S$ also f.g. There exists $d$ such that $S^{(d)}$ is genereted in degree 1 and it is esay to see that $M^{(d)}$ is finitely generated as $S^{(d)}$-module. Then you conclude that $\widetilde{M}=0$ iff for any $d$ such that $S^{(d)}$ is generated in degree 1 then there exists $n_0$ (depending on $d$) such that $M_{nd}=0$ for all $n\geq n_0$