Cohomologia Local de Módulos Sobre Anéis Invariantes

Abstract

In this work we study local cohomology of modules on invariant rings inspired by the results of Sharp, Huneke and Lyubeznik. In fact the main result was demonstrated by Tony J. Puthenpurakal: 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 Rtt be the ring of invariants of G. Let I be an ideal in Rtt. Fix i ≥ 0, if Rtt is Gorenstein then: (I) injdimRG HI (Rtt) ≤ dimsupp Hi(Rtt); (II) Hj (Hi(Rtt)) is injective, where m is any maximal ideal of Rtt; m I (III) µj(P, Hi(Rtt)) = µj(P j, Hi (R)) where P j is any prime in R lying above. I IR We also prove that if P is a prime ideal in Rtt with Rtt not Gorenstein then either the Bass number µj(P, Hi(Rtt))is zero for all j or there exists c such that µj(P, Hi(Rtt)) = 0 I I for j < c and µj(P, Hi(Rtt)) > 0 for all j ≥ c.