The Cotorsion Pair Cogenerated by the Subcategory of Stable Functors in ((R-mod)^op, Ab)

Let R be an associative ring with identity. Denote by ((R-mod)^op, Ab) the category consisting of contravariant functors from the category of finitely presented left R-modules R-mod to the category of abelian groups Ab. An object in ((R-mod)^op, Ab) is said to be a stable functor if it vanishes on the regular module R. Let T be the subcategory of stable functors. There are two torsion pairs t1=(Gen(-,R),T)and t2=(T,F1), where 1 is the subcategory of ((R-mod)^op, Ab) consisting of functors with flat dimension at most 1. In this article, let R be a ring of weakly global dimension at most 1, and assume R satisfies that for any exact sequence 0 → M → N → K → 0, if M and N are pure injective, then K is also pure injective. We calculate the cotorsion pair (⊥T,(⊥T)⊥)cogenerated by T clearly. It is shown that G∈⊥T if and only if G/t1(G) is a projective object in T, i.e., G/t1(G)=(-,M) for some R-module M;and G∈(⊥T)⊥ if and only if G/t2(G) is of the form (-, E), where E is an injective R-module.

Balanced Pairs on Triangulated Categories

Let C be a triangulated category.We first introduce the notion of balanced pairs in C,and then establish the bijective correspondence between balanced pairs and proper classesξwith enoughξ-projectives andξ-injectives.Assume thatξ:=ξX=ξ^(Y) is the proper class induced by a balanced pair(X,Y).We prove that(C,Eξ,sξ)is an extriangulated category.Moreover,it is proved that(C,Eξ,sξ)is a triangulated category if and only if X=Y=0,and that(C,Eξ,sξ)is an exact category if and only if X=Y=C.As an application,we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.