Global dimension and left derived functors of Hom

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom 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 Hom 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 MR≤n - 2 if and only if Extn-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-1)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.

Global dimension and left derived functors of Hom

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom 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 Hom 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 Projcosyzygy 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−1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Proj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R) ⩽ 2 are characterized. This work was partially supported by the Specialized Research Fund for the Doctoral Program of High Education (Grant No. 20050284015), the National Science Foundation of China (Grant No. 10331030), China Postdoctoral Science Foundation (Grant No. 20060390926), NSF of Jiangsu Province of China (Grant No. BK 2005207), Collegial Natural Science Research Program of Education Department of Jiangsu Province (Grant No. 06KJB110033), and Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 0601021B)