Homological Aspects of the Dual Auslander Transpose II

TL;DR

This paper explores homological properties related to semidualizing bimodules, establishing equivalences and characterizations that deepen understanding of Gorenstein and tilting modules, and extends classical theorems in homological algebra.

Contribution

It introduces a Morita equivalence between certain cotorsionfree and adstatic modules, relates relative and standard homological dimensions, and provides dual characterizations of Gorenstein and Auslander algebras.

Findings

Established Morita equivalence between $ abla$-cotorsionfree and $ abla$-adstatic modules.

Connected relative homological dimensions with classical homological dimensions.

Provided dual versions of key approximation theorems and characterizations of Gorenstein artin algebras.

Abstract

Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of $\infty$-$\omega$-cotorsionfree 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 ${\rm Hom}(\omega,M)$. By investigating the properties of the Bass injective dimension of modules (resp. complexes), we get some equivalent characterizations of semi-tilting modules (resp. Gorenstein artin algebras). Finally we obtain a dual version of the Auslander-Bridger's approximation theorem. As a consequence, we get some equivalent characterizations of Auslander $n$-Gorenstein artin algebras.