On the exactness of flat resolvents

Abstract

Let R be a left coherent ring, FP — idRR the FP — injective dimension of RR and wD(R) the weak global dimension of R. It is shown that 1) FP -idRR 0) if and only if every flat resolvent 0 → M → F° → F1... of a finitely presented right R—module M is exact at F'(i > n−1) if and only if every nth F -cosyzygy of a finitely presented right R — module has a flat preenvelope which is a monomorphism; 2) wD(R) 1) if and only if every (n−l)th F-cosyzygy of a finitely presented right R—module has a flat preenvelope which is an epimorphism; 3) wD(R) 0) if and only if every nth F — cosyzygy of a finitely presented right R — module is flat. In particular, left FC rings and left semihereditary rings are characterized