DC-projective dimensions, Foxby equivalence and SDC-projective modules

This is a study of Ding projective modules relative to a semidualizing module and related topics. Firstly, we study [Formula: see text]-projective dimensions and [Formula: see text]-projective modules under change of rings. Secondly, we establish a new version of the Foxby equivalence with respect to [Formula: see text]-projective modules and [Formula: see text]-injective modules. Thirdly, we characterize Ding projective modules in [Formula: see text] and Ding injective modules in [Formula: see text]. At last, as applications, some new characterizations of perfect rings and quasi-Frobenius rings are given.

Foxby Equivalence and Cotorsion Theories relative to Semi-dualizing Modules

Foxby duality has proven to be an important tool in studying the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. Recently the notion of a semi-dualizing module has been given [2]. Given a semi-dualizing module the relative Foxby classes can be defined and there is still an associated Foxby duality. We consider these classes (separately called the Auslander and Bass classes) and two naturally defined subclasses which are equivalent to the full subcategories of injective and flat modules. We consider the question of when these subclasses form part of one of the two classes of a cotorsion theory. We show that when this is the case, the associated cotorsion theory is not only complete but in fact is perfect. We show by examples that even when the semi-dualizing module is in fact dualizing over a local Cohen-Macaulay ring it both may or may not occur that we get this associated cotorsion theory.