Local cohomology modules of invariant rings

Abstract Let K be a field and let R be a regular domain containing K . Let G be a finite subgroup of the group of automorphisms of R . We assume that | G | is invertible in K . Let R G be the ring of invariants of G . Let I be an ideal in R G . Fix i ⩾ 0. If R G is Gorenstein then: (i) injdim R G H i I ( R G ) ⩽ dim Supp H i I ( R G ); (ii) $H^j_{\mathfrak{m}}$ ( H i I ( R G )) is injective, where $\mathfrak{m}$ is any maximal ideal of R G ; (iii) μ j ( P, H i I ( R G )) = μ j ( P ′, H i IR ( R )) where P ′ is any prime in R lying above P . We also prove that if P is a prime ideal in R G with R G P not Gorenstein then either the bass numbers μ j ( P, H i I ( R G )) is zero for all j or there exists c such that μ j ( P, H i I ( R G )) = 0 for j < c and μ j ( P, H i I ( R G )) > 0 for all j ⩾ c .