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 R ₏-module. We give some necessary and sufficient conditions under which G (R/ p) is an R ₏-module. We also study the Bass numbers of G (R/ p) and we show that if GidRR/ p is finite, then ⁱ (q, G (R/ p) ) is finite for all i 0 and all q Spec R. If GpdRR/ p is finite, then ⁱ (p, G (R/ p) ) is finite for all i 0. We define a subring S (p) ₏ of Endₑ_ ₏ (G (R ₏/ pR ₏) ) and we show that it is noetherian and contains a subring which is a quotient of R ₏.