A Note on Cartan-Eilenberg Gorenstein categories

Abstract

In this article, we investigate the stability of Cartan-Eilenberg Gorenstein categories. To this end, we introduce and study the concept of two-degree Cartan-Eilenberg W -Gorenstein complexes. We prove that a complex C is two-degree Cartan-Eilenberg W -Gorenstein if and only if C is Cartan-Eilenberg W -Gorenstein. As applications, we show that a complex C is two-degree C-E Gorenstein projective if and only if C is C-E Gorenstein projective. Moreover, we obtain similar results for some known modules such as W-Gorenstein projective modules and V-Gorenstein projective modules.

Cartan–eilenberg Fp-Injective Complexes

“…Therefore, Cartan-Eilenberg complexes play an important role in the category of complexes and the homotopy category. In [9,13,14,26], the authors also considered Cartan-Eilenberg complexes and obtained some important results. For instance, Enochs proved that every complex has a Cartan-Eilenberg injective envelope, every complex has a Cartan-Eilenberg projective precover and a complex is Cartan-Eilenberg flat if and only if it is the direct limit of finitely generated Cartan-Eilenberg projective complexes [9].…”

“…For instance, Enochs proved that every complex has a Cartan-Eilenberg injective envelope, every complex has a Cartan-Eilenberg projective precover and a complex is Cartan-Eilenberg flat if and only if it is the direct limit of finitely generated Cartan-Eilenberg projective complexes [9]. In [13,14,26], the authors investigated the Cartan-Eilenberg Gorenstein complexes, the stability of Cartan-Eilenberg Gorenstein categories and established some relationships between Cartan-Eilenberg complexes and DG complexes.…”