Graded components of local cohomology modules of invariant rings

Abstract

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 $S=A[X_1,\dots ,X_m]$ and $R=B[X_1,\dots ,X_m]$ be standard graded with $\mathrm{deg} A=0,\mathrm{deg} B=0$ and $\mathrm{deg} X_i=1$ for all i. Extend the action of G on A to S by fixing X_i. Note $S^G=R$. Let I be an arbitrary homogeneous ideal in R. The main goal of this paper is to establish a comparative study of graded components of local cohomology modules $H_I^i(R)$ that would be analogs to those proven in the paper [5] for $H_J^i(S)$ where J is an arbitrary homogeneous ideal in S.

Keywords:

• Graded local cohomology
• invariant rings
• local comohology

2010 Mathematics Subject Classification:

• Primary 13D45
• 14B15
• Secondary 13A50
• 32C36

Acknowledgments

We thank the referee for the useful comments and suggestions to improve the paper. The second author would like to thank UGC, Govt. of India for providing financial support for this study.