The Related Research Of Morphism Categories Of Gorenstein Projective Modules And Morita Ring

Suppose A is a CM-finite artin algebra,let ?(A)be relative Auslander algebra of A and ?(A)-mod the category of finitely generated left ?(A)-modules.We will characterize the morphism category Mor(A-Gproj)of Gorenstein projective A-modules by ?(A)-mod.Denote by be a full subcategory of ?(A)-mod consisting of modules of projec-tive dimension ?1,we will show that monomorphism category S(A-Gproj)is equivalent to p?1(F(A)),and we also obtain that some factor category of the epimorphism category F(A-Gproj)of Gorenstein projective A-modules is a Frobenius category.Let A and B be two Artin algebras,ANB an A-B-bimodule,BMA an B-A-bimodule,let ?:=(AMNB)be the Morita ring such that M(?)A N=0 and N(?)B M=0.Denote by A-mod the catgory of finitely generated left A-modules,we will define the monomorphism cate-gory S(?-mod)be the subcategory of A-mod consisting of A-modules(X,Y,f,g)such that f:M(?)A X? Y is monic B-map and g:N(?)B Y?X is monic A-map.We show that S(?-mod)is resolving subcategory of A-mod if and only if MA,NB are projectives.We also provide a sufficient and necessary condition such that it is a Frobenius category.Moreover,we describe S(?-mod)as the left perpendicular category of a cotilting A-mod,and show that this cotlting module yields the Ringel-Schmidmeier-Simson equivalence be-tween S(?-mod)and its dual:DHom?(-,G):S(?-mod)=F(A-mod).Finally,we obtain relations among the different morphism categories.Let A be a finite dimensional algebra over a field k,e1 and e2 be idempotents of A with e2Ae1=0.Then N:=Ae1(?)ke2A has a natural structure of A-A-bimodule with N(?)A N =0.Let (?):=(ANNA)be the Morita ring,we clarify the relations between Mor(A-Gproj)and Mor((?)-Gproj)by adjiont pairs,we also investigated the recollement of S(?-mod).