Minimal injective resolutions of Cohen-Macaulay isolated singularities

Abstract

Let $(R, \frak {m})$ be a commutative Gorenstein complete local ring with dim R = d and let $\Lambda $ be an R-algebra which is not necessarily commutative but finitely generated as an R-module. In this paper the structure of minimal injective resolutions $E^\bullet _\Lambda (M)$ for the $\Lambda $ -lattices M is explored, in terms of the Cousin complexes $C^{\bullet }_R(M)$ for M and the minimal projective resolutions of the $\Lambda ^{op}$ -modules $M^* = \hbox {Hom}_R(M,R)$ as well, under the assumption that $\Lambda $ is a Cohen-Macaulay isolated singularity. As a consequence we get the following. Assume that R is a regular local ring and let $k \in \Bbb Z$ . Then the ring $\Lambda $ is k-Gorenstein if and only if the ring $\Delta = (R/\frak {m})\otimes _R \Lambda $ is (k - d)-Gorenstein.