This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to t...This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to the strongly Gorenstein flat dimensions of rings.Also,we investigate the strongly Gorenstein flat dimensions of direct products of rings and（almost）excellent extensions of rings.展开更多
Let R be a right coherent ring and D^b（R-Mod） the bounded derived category of left R-modules. Denote by D^b（R-Mod）[GF,C] the subcategory of D^b（R-Mod） consisting of all complexes with both finite Gorenstein flat...Let R be a right coherent ring and D^b（R-Mod） the bounded derived category of left R-modules. Denote by D^b（R-Mod）[GF,C] the subcategory of D^b（R-Mod） consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K^b（F∩C） the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category D^b（R-Mod）[GF,C]/K^b（F∩C,） is triangle-equivalent to the stable category GF∩C of the Frobenius category of all Gorenstein fiat and cotorsion left R-modules.展开更多
The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-fiat module if Tor1r（M, N） is GV-torsion for all R-modules N. In this paper, we introduc...The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-fiat module if Tor1r（M, N） is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules ... → P1 → P0 → p0 → p1 → ... such that M Im（P0 → p0） and such that the functor HomR （-,F） leaves the sequence exact whenever F is w-flat. Several well- known classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension （of modules and rings） and the existence of Ding w-flat precovers are also studied.展开更多
基金Supported by the National Natural Science Foundation of China (Grant No.10961021)
文摘This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to the strongly Gorenstein flat dimensions of rings.Also,we investigate the strongly Gorenstein flat dimensions of direct products of rings and（almost）excellent extensions of rings.
基金Supported by National Natural Science Foundation of China(Grant Nos.11601433 and 11261050)the Postdoctoral Science Foundation of China(Grant No.2106M602945XB)Northwest Normal University(Grant No.NWNU-LKQN-15-12)
文摘Let R be a right coherent ring and D^b（R-Mod） the bounded derived category of left R-modules. Denote by D^b（R-Mod）[GF,C] the subcategory of D^b（R-Mod） consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K^b（F∩C） the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category D^b（R-Mod）[GF,C]/K^b（F∩C,） is triangle-equivalent to the stable category GF∩C of the Frobenius category of all Gorenstein fiat and cotorsion left R-modules.
文摘The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-fiat module if Tor1r（M, N） is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules ... → P1 → P0 → p0 → p1 → ... such that M Im（P0 → p0） and such that the functor HomR （-,F） leaves the sequence exact whenever F is w-flat. Several well- known classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension （of modules and rings） and the existence of Ding w-flat precovers are also studied.
