Action of certain groups on local cohomology modules and Cousin complexes (Q2773014)

Let \(R\) denote a commutative Noetherian ring. For a group of automorphisms \(G\) of \(R\) let \(R^G\) denote the ring of invariants. For an \(R\)-module \(M\) and a group \(H\) of \(R^G\)-automorphisms of \(M\) let \(M^H\) denote the corresponding fixed submodule of \(M.\) For covariant and contravariant functors and a finite \(R^G\)-automorphism group \(H\) whose order is invertible the authors show that forming the fixed submodules commutes with taking the functors. This basic result is applied to complexes. In this way the authors study group actions on local cohomology modules and Cousin complexes. In a final section they investigate the following question: NEWLINENEWLINENEWLINEWhen does a `good property' of \(R\) pass to \(R^G\)? This is answered affirmatively for the Cohen-Macaulay, Buchsbaum, generalized Cohen-Macaulay and Serre's \(S_k\)-property. In this way the authors generalize and make more precise previously known statements.