查看文章

A model structure on a category is a formal way of introducing a homotopy theory on that category, and if the model structure is abelian and hereditary, its homotopy category is known to be triangulated. So a good way to both build and model a triangulated category is to build a hereditary abelian model structure. Given a ring and a (non necessarily semidualizing) left -module , we introduce and study new concepts of relative Gorenstein cotorsion and cotorsion modules: -cotorsion and (strongly) -cotorsion. As an application, we prove that there is a unique hereditary abelian model structure on the category of left -modules, in which the cofibrations are the monomorphisms with -flat cokernel and the fibrations are the epimorphisms with -cotorsion kernel belonging to the Bass class . In the second part, when is a semidualizing -bimodule, we investigate the existence of abelian model structures on the category of left (resp., right) -modules where the cofibrations are the epimorphisms (resp., monomorphisms) with kernel (resp., cokernel) belonging to the Bass (resp., Auslander) class (resp., ). We also study the class of -flat modules and the Bass class from the Auslander-Buchweitz approximation theory point of view. We show that they are part of weak AB-contexts. As the concept of weak AB-context can be dualized, we also give dual results that involve the class of -cotorsion modules and the Auslander class.