Let $S=\bigoplus_{\underline n\in\mathbb N^r } S_{\underline n}$ be a standard multigraded ring over a local ring and M be a finitely generated $\mathbb N^r $-graded $S$-module. Let $M_{\Delta}=\bigoplus_{n\in\mathbb N } M_{ne}$ and $S_{++}=\bigoplus_{\underline n\geq e } S_{\underline n}.$
Is there any procedure to relate local cohomology modules of $M$ and $M_{\Delta}$ with respect to $S_{++}$ using commutative algebra? Even in any special case?
If we consider $S$ is multi- Rees algebra of ideals , then S is Cohen-Macaulay implies $S_{\Delta}$ is Cohen-Macaulay. Proof of this involves algebraic geometry. Is there any way to prove it using commutative algebra techniques.
I need reference of homological relations of $M$ with $M_{\Delta}.$
