Abstract
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.
Highlights
Foxby classes (Auslander and Bass classes) have proven to be very useful when studying Gorenstein injective and projective dimensions: over Gorenstein rings, every module has finite Gorenstein projective and finite Gorenstein injective dimensions [1]
Foxby proved that the Auslander and Bass classes behave in a duality and that finitely generated modules in the Auslander class are precisely those of finite Gorenstein projective dimension, while finitely generated modules in the Bass class are those of finite Gorenstein injective dimension [2]
We will be able to prove the last two main results of the paper: Corollary 5.7, which says that, under certain circumstances, the class of all GC-flat modules is covering in R-Mod, and Corollary 5.14, which states that GCP(R) is special precovering in R-Mod
Summary
Foxby classes (Auslander and Bass classes) have proven to be very useful when studying Gorenstein injective and projective dimensions: over Gorenstein rings, every module has finite Gorenstein projective and finite Gorenstein injective dimensions [1]. Little is yet known about the existence of precovers or preenvelopes relative to these new classes of modules: it is known that if C is weakly Wakamatsu tilting (cotilting), every module of finite GC-projective (injective) dimension has a special GC-precover (preenvelope) [11, Corollary 3.6 and Theorem 4.11]. We study when the classes AddR(C) and ProdS(HomR(C, E)) (RE is an injective cogenerator) are covering (Corollary 3.5) and enveloping (Corollary 3.4), respectively, with precovers (preenvelopes) of modules in C(R) ( C(S)) being epimorphisms (monomorphisms) as can be checked in Propositions 3.8 and 3.9, respectively This fact will lead us to prove that the Bass class C(R) can be recovered from a right AddR(C)-resolution of R, as a direct sum of C and the right orthogonal class of all cokernels in such resolution (Theorem 3.10). From this fact we deduce the existence of GC-projective precovers ( special GC-projective precovers) in Corollary 5.14
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