Let R be a commutative noetherian ring and Γ a finite group. In this paper,we study Gorenstein homological dimensions of modules with respect to a semi-dualizing module over the group ring  . It is shown that Gorenstein homological dimensions  of an  -RΓ module M with respect to a semi-dualizing module,  are equal over R and RΓ  .

Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full subcategories of strong and weakly topologized objects and show that each is equivalent to the chu category of the original category with respect to the dualizing obj...

The notion of toric face rings generalizes both Stanley-Reisner rings and affine semigroup rings, and has been studied by Bruns, Römer, et.al. Here, we will show that, for a toric face ring R, the “graded” Matlis dual of a Cěch complex gives a dualizing complex. In the most general setting, R is not a graded ring in the usual sense. Hence technical argument is required.

We introduce the notion of a duality pair and demonstrate how the left half of such a pair is “often” covering and preenveloping. As an application, we generalize a result by Enochs et al. on Auslander and Bass classes, and we prove that the class of Gorenstein injective modules—introduced by Enochs and Jenda—is covering when the ground ring has a dualizing complex.

Let (R, m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If GC -dimension of M/aM is finite for all ideals a generated by an R-regular sequence of length at most d − t then either GC -dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.