Homological aspects of the dual Auslander transpose, II

Abstract

Let $R$ and $S$ be rings, and let ${}_R{\omega }_S$ be a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$\omega$-cotorsion-free modules and a subclass of the class of $\omega$-adstatic modules. Also, we establish the relation between the relative homological dimensions of a module $M$ and the corresponding standard homological dimensions of $Hom(\omega ,M)$. By investigating the properties of the Bass injective dimension of modules (resp., complexes), we get some equivalent characterizations of semitilting modules (resp., Gorenstein Artin algebras). Finally, we obtain a dual version of the Auslander–Bridger approximation theorem. As a consequence, we get some equivalent characterizations of Auslander $n$-Gorenstein Artin algebras.