Gorenstein Homological Dimensions for Extriangulated Categories

Let (C, E, s) be an extriangulated category with a proper class xi of E-triangles. In a previous work, we introduced and studied the xi-Gprojective and the xi-Ginjective dimension for any object in C. In this paper, we first give some characterizations of xi-Gprojective dimension by using derived functors on C. Consequently, we show that the following equality holds under some assumptions: sup{xi-GpdM | for any M is an element of C} = sup{xi-GidM | for any M is an element of C}, where xi-GpdM (resp., xi-GidM) denotes the xi-Gprojective dimension (resp., xi-Ginjective dimension) of M. As an application, our main results generalize the work by Bennis-Mahdou and Ren-Liu, which are newfor an exact category case. Moreover, our proof is far from the usual module or triangulated case.

Automated summary

This paper investigates the xi-Gprojective and xi-Ginjective dimensions of objects in an extriangulated category, providing characterizations of the xi-Gprojective dimension. The study shows that under certain assumptions, the equality between the two dimensions holds. These results extend the work in the exact category case and present a unique proof approach distinct from traditional methods in module or triangulated categories.