Bass numbers and endomorphism rings of Gorenstein injective modules

Let R be a commutative noetherian ring admitting a dualizing complex and let p be a prime ideal of R. In this paper we investigate when G(R/ p) is an Rₚ-module. We give some necessary and sufficient conditions under which G(R/ p) is an Rₚ-module. We also study the Bass numbers of G(R/ p) and we show that if GidRR/ p is finite, then μⁱ( q,G(R/ p)) is finite for all i≥ 0 and all q∈ Spec R. If GpdRR/ p is finite, then μⁱ( p,G(R/ p)) is finite for all i≥ 0. We define a subring S( p)ₚ of End_Rₚ(G(Rₚ/ pRₚ)) and we show that it is noetherian and contains a subring which is a quotient of Rₚ.