Action of finite groups on Rees algebras and Gorensteinness in invariant subrings

Abstract

Let G be a finite group of order N and assume that G acts on a Cohen-Macaulay local ring A as automorphisms of rings. Let N be a unit in A. For a given G-stable ideal I in A we denote by R(I) = ⊕n≥0In and = G(I) = ⊕n≥0In/In+1 the Rees algebra and the associated graded ring of I, respectively. Then G naturally acts on R(I) and G(I) too. In this paper the conditions under which the invariant subrings R(I)G of R(I) are Cohen-Macaulay and/or Gorenstein rings are described in connection with the corresponding ring-theoretic properties of G(I)G and the a-invariants a(G(I)G of G(I)G. Consequences and some applications are discussed.