The concept of Koszulity for differential graded (DG, for short) modules is introduced. It is shown that any bounded below DG module with bounded Ext-group to the trivial module over a Koszul DG algebra has a Koszul D...The concept of Koszulity for differential graded (DG, for short) modules is introduced. It is shown that any bounded below DG module with bounded Ext-group to the trivial module over a Koszul DG algebra has a Koszul DG submodule (up to a shift and truncation), moreover such a DG module can be approximated by Koszul DG modules (Theorem 3.6). Let A be a Koszul DG algebra, and Dc(A) be the full triangulated subcategory of the derived category of DG A-modules generated by the object AA. If the trivial DG module kA lies in Dc(A), then the heart of the standard t-structure on Dc(A) is anti-equivalent to the category of finitely generated modules over some finite dimensional algebra. As a corollary, Dc(A) is equivalent to the bounded derived category of its heart as triangulated categories.展开更多
The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called(s,t,d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An(s,...The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called(s,t,d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An(s,t,d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an(s,t,d)-bi-Koszul algebra is discussed. Based on it,the notion of strongly(s,t,d)-bi-Koszul algebras is raised and their homological properties are further discussed.展开更多
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 righ...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.展开更多
基金supported by National Natural Science Foundation of China (Grant No.10801099)Doctorate Foundation of Ministry of Education of China (Grant No.20060246003)Foundation of Zhejiang Province's Educational Committee (Grant No.20070501)
文摘The concept of Koszulity for differential graded (DG, for short) modules is introduced. It is shown that any bounded below DG module with bounded Ext-group to the trivial module over a Koszul DG algebra has a Koszul DG submodule (up to a shift and truncation), moreover such a DG module can be approximated by Koszul DG modules (Theorem 3.6). Let A be a Koszul DG algebra, and Dc(A) be the full triangulated subcategory of the derived category of DG A-modules generated by the object AA. If the trivial DG module kA lies in Dc(A), then the heart of the standard t-structure on Dc(A) is anti-equivalent to the category of finitely generated modules over some finite dimensional algebra. As a corollary, Dc(A) is equivalent to the bounded derived category of its heart as triangulated categories.
基金supported by National Natural Science Foundation of China (Grant No. 10571152)the Natural Science Foundation of Zhejiang Province of China (Grant No. J20080154)
文摘The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called(s,t,d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An(s,t,d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an(s,t,d)-bi-Koszul algebra is discussed. Based on it,the notion of strongly(s,t,d)-bi-Koszul algebras is raised and their homological properties are further discussed.
基金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)+3 种基金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)Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 0601021B)
文摘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.
