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Let $A$ be a regular ring containing a field $K$ of characteristic zero and let $R = A[X_1,ldots, X_m]$. Consider $R$ as standard graded with $deg A = 0$ and $deg X_i = 1$ for all $i$. Let $G$ be a finite subgroup of $GL_m(A)$. Let $G$ act linearly on $R$ fixing $A$. Let $S = R^G$. In this paper we present a comprehensive study of graded components of local cohomology modules $H^i_I(S)$ where $I$ is an emph{arbitrary} homogeneous ideal in $S$. We prove stronger results when $G subseteq GL_m(K)$. Some of our results are new even in the case when $A$ is a field.
Let $A$ be a regular domain containing a field $K$ of characteristic zero, $G$ be a finite subgroup of the group of automorphisms of $A$ and $B=A^G$ be the ring of invariants of $G$. Let $S= A[X_1,ldots, X_m]$ and $R= B[X_1, ldots, X_m]$ be standard
Let $A$ be a commutative Noetherian ring containing a field $K$ of characteristic zero and let $R= A[X_1, ldots, X_m]$. Consider $R$ as standard graded with $deg A=0$ and $deg X_i=1$ for all $i$. We present a few results about the behavior of the gra
Let $R=K[X_1,ldots, X_n]$ where $K$ is a field of characteristic zero, and let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. We give standard grading on $R$ and $A_n(K)$. Let $I$, $J$ be homogeneous ideals of $R$. Let $M = H^i_I(R)$ and $N = H^j_J(
Let $(A,mathfrak{m})$ be a hypersurface ring with dimension $d$, and $M$ a MCM $A-$module with reduction no.2 and $mu(M)=2$ or $3$ then we have proved that depth$G(M)geq d-mu(M)+1$. If $e(A)=3$ and $mu(M)=4$ then in this case we have proved that dept
It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}]-module associated to R. This module, denoted J(R), is the free Z[t,t