Graded Components of Local Cohomology Modules Over Invariant Rings-II

Let A be a regular ring containing a field K of characteristic zero and let \(R = A[X_1,\ldots , X_m]\) $R=A[X_1,\dots ,X_m]$ . Consider R as standard graded with \(\deg A = 0\) $degA=0$ and \(\deg X_i = 1\) $deg{X}_i=1$ for all i. Let G be a finite subgroup of \(\textrm{GL}_m(A)\) ${\text{GL}}_m(A)$ . Let G act linearly on R fixing A. Let \(S = R^G\) $S=R^G$ . In this paper, we present a comprehensive study of graded components of local cohomology modules \(H^i_I(S)\) $H_I^i(S)$ , where I is an arbitrary homogeneous ideal in S. We prove stronger results when \(G \subseteq \textrm{GL}_m(K)\) $G\subseteq {\text{GL}}_m(K)$ . Some of our results are new even in the case when A is a field.