The role of w-tilting modules in relative Gorenstein (co)homology

Abstract

Let $R$ be a ring, $C$ be a left $R$-module and $S={\mathrm{End}}_R\left(C\right)$. When $C$ is semidualizing, the Auslander class ${\mathcal{A}}_C\left(S\right)$ and the Bass class ${\mathcal{ℬ}}_C\left(R\right)$ associated with $C$ have been the subject of extensive investigations. It has been proved that these classes, also known as Foxby classes, are one of the central concepts of (relative) Gorenstein homological algebra. In this paper, we answer several natural questions which arise when we weaken the condition of $C$ being semidualizing: if we let $C$ be w-tilting (see Definition 2.1), we establish the conditions for the pair $\left({\mathcal{A}}_C\left(S\right),{\mathcal{A}}_C{\left(S\right)}^{{\perp }_1}\right)$ to be a perfect cotorsion theory and for the pair $\left({}^{{\perp }_1}B_C\left(R\right),B_C\left(R\right)\right)$ to be a complete hereditary cotorsion theory. This tells us when the classes of Auslander and Bass are preenveloping and precovering, which generalizes a number of results disseminated in the literature. We investigate Gorenstein flat modules relative to a not necessarily semidualizing module $C$ and we find conditions for the class of $G_C$-projective modules to be special precovering, the class of $G_C$-flat modules to be covering, the one of Gorenstein $C$-projective modules to be precovering and that of Gorenstein $C$-injective modules to be preenveloping. We also find how to recover Foxby classes from ${\mathrm{Add}}_R\left(C\right)$-resolutions of $R$.