On certain classes of modules over group algebras
Ioannis Emmanouil, Wei Ren
TL;DR
This work develops a cohesive framework for three subfamilies of Gorenstein modules over group algebras: cofibrant, cofibrant-flat, and fibrant modules, tying them to the familiar Gorenstein projective/flat/injective classes through diagonal tensor and Hom constructions with the module $B$. It shows these subclasses form complete hereditary cotorsion pairs and admit exact model structures, enabling robust approximations and dualities. The paper then implements a systematic program of approximations for cofibrant-flat modules by cofibrant or flat modules, and establishes two Hovey triples in the cofibrant-flat setting, yielding triangulated equivalences between corresponding homotopy and stable categories. Finally, it analyzes fibrant modules, linking them to Gorenstein injectives, proving a cogenerated, set-based cotorsion pair, and demonstrating a duality with cofibrant-flat via Pontryagin duality; under suitable group-class conditions, the Gorenstein and cofibrant-flat (and fibrant) theories coincide, enriching the structure of Gorenstein homological algebra for group algebras.
Abstract
In this paper, we examine the relation between certain subclasses of the classes of Gorenstein projective, Gorenstein flat and Gorenstein injective modules over a group algebra, which consist of the cofibrant, cofibrant-flat and fibrant modules respectively. These subclasses have all structural properties that the Gorenstein classes are known to have, regarding the existence of complete cotorsion pairs and various approximations between them. Furthermore, these subclasses have certain properties, which are conjecturally anticipated to hold for the Gorenstein classes as well. Since the Gorenstein classes are actually equal to the corresponding subclasses for large families of groups, we thus obtain (indirectly) some information about the Gorenstein classes over these groups.