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 present-ed 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 mod-ule R.Let T be the subcategory of stable functors.There are two hereditary torsion pairs t1 =(Gen(-,R),T)and t2 =(T,F1),where F1 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.