The Adjoint Cotranspose Of Modules With Respect To Subcategories

This paper aims to establish the adjoint counterparts of Gorenstein cotransposes,which are facilitated by the consideration of a more general situation.For a subcategory of modules X,we will define and study the adjoint X-cotranspose of modules relative to an arbitrary bimodule.Let X be a subcategory of left S-modules and RUS an(R,S)-bimodule.As a generalization of the adjoint cotranspose,we introduce the adjoint X-cotranspose of a left S-module relative to RUS and study its homological properties.Let V be a subcategory of X.The relations between adjoint X-cotransposes and adjoint V-cotransposes are investigated under the condition that V is a generator or cogenerator for X.Then we give some applications of these results to the categories of interest.In particular,the adjoint counterparts of Gorenstein cotransposes are established.This paper is organized as follows.In Section 2,we give some terminology and some preliminary results.In Section 3,for a subcategory X of left S-modules and a left S-module A,we introduce the notion of the adjoint X-cotranspose of A relative to an(R,S)-bimodule RUS,which extends the notion of adjoint cotransposes.Let V be a subcategory of X.The relations between adjoint X-cotransposes and adjoint V-cotransposes are established,whenever V is a generator or cogenerator for X.Suppose that X is closed under extensions with a generator V and Tor1S(U,X)=0.We prove that,if a left R-module M is an adjoint X-cotranspose of A relative to U,then there exists an exact sequence 0→U(?)SX→ acTrVUA→ M→ 0 with acTrXUA an adjoint V-cotransposes of A relative to U and X∈X.We also investigate when the converse statement holds(see Propositions 3.4 and 3.6,Theorem 3.7).If X is closed under extensions with a cogenerator W and Tor1S(U,X)=0,then any adjoint X-cotranspose of A is exactly an adjoint W-cotranspose of another left S-module B,where B E A*X,which means that there is an exact sequence 0→A→B→X→0 with X ∈ X(see Theorem 3.10).Then we give some applications of these results to some categories,such as the Auslander class relative to C and its intersection with Gorenstein flat modules.In particular,the adjoint counterparts of Gorenstein cotransposes are established.We prove that any adjoint HC-Gorenstein cotranspose of A is an image of an adjoint cotranspose of A with the kernel WF-Gorenstein flat.