Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

Abstract

Let \({(R, \mathfrak{m})}\) be a commutative Noetherian local ring of Krull dimension d, and let C be a semidualizing R-module. In this paper, it is shown that if R is complete, then C is a dualizing module if and only if the top local cohomology module of \({R, H _{\mathfrak{m}} ^{d} (R)}\), has finite GC-injective dimension. This generalizes a recent result due to Yoshizawa, where the ring is assumed to be complete Cohen-Macaulay.