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

AbstractLetRRbe a ring,CCbe a leftRR-module andS=EndR(C)S={{\rm{End}}}_{R}\left(C). WhenCCis semidualizing, the Auslander classAC(S){{\mathcal{A}}}_{C}\left(S)and the Bass classℬC(R){{\mathcal{ {\mathcal B} }}}_{C}\left(R)associated withCChave 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 ofCCbeing semidualizing: if we letCCbe w-tilting (see Definition 2.1), we establish the conditions for the pair(AC(S),AC(S)⊥1)\left({{\mathcal{A}}}_{C}\left(S),{{\mathcal{A}}}_{C}{\left(S)}^{{\perp }_{1}})to be a perfect cotorsion theory and for the pair(BC⊥1(R),BC(R))\left({}^{{\perp }_{1}}B_{C}\left(R),{B}_{C}\left(R))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 moduleCCand we find conditions for the class ofGC{G}_{C}-projective modules to be special precovering, the class ofGC{G}_{C}-flat modules to be covering, the one of GorensteinCC-projective modules to be precovering and that of GorensteinCC-injective modules to be preenveloping. We also find how to recover Foxby classes fromAddR(C){{\rm{Add}}}_{R}\left(C)-resolutions ofRR.