Global dimension and left derived functors of Horn

摘要

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Horn and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Horn and the right projective resolutions of right .R-modules. It is shown that rD(R) ≤ n (n ≥ 2) if and only if the gl right Proj-dim M_R≤ n- 2 if and only if Ext_(n-1)(N, M) = 0 for all right R-modules N and M if and only if every (n - 2)th Proj-cosyzygy of a right .R-module has a projective envelope with the unique mapping property. It is also proved that rD(R) ≤ n (n ≥ 1) if and only if every (n - l)th Proj-cosyzygy of a right .R-module has an epic projective envelope if and only if every nth Vroj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R) ≤ 2 are characterized.