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

Abstract

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 [Formula: see text] be the subcategory of stable functors. There are two torsion pairs [Formula: see text] and [Formula: see text], 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 [Formula: see text] cogenerated by [Formula: see text] clearly. It is shown that [Formula: see text] if and only if G/t1(G) is a projective object in [Formula: see text], i.e., G/t1(G) = (−,M) for some R-module M; and [Formula: see text] if and only if G/t2(G) is of the form (−, E), where E is an injective R-module.