This paper focuses on a question raised by Holm and Jorgensen,who asked if the induced cotorsion pairs(Φ(X),Φ(X)^(⊥))and(^(⊥)Ψ(Y),Ψ(Y))in Rep(Q,A)—the category of all A-valued representations of a quiver Q—are...This paper focuses on a question raised by Holm and Jorgensen,who asked if the induced cotorsion pairs(Φ(X),Φ(X)^(⊥))and(^(⊥)Ψ(Y),Ψ(Y))in Rep(Q,A)—the category of all A-valued representations of a quiver Q—are complete whenever(X,Y)is a complete cotorsion pair in an abelian category A satisfying some mild conditions.We give an affirmative answer if the quiver Q is rooted.As an application,we show under certain mild conditions that if a subcategory L,which is not necessarily closed under direct summands,of A is special precovering(resp.,preenveloping),thenΦ(L)(resp.,Ψ(L))is special precovering(resp.,preenveloping)in Rep(Q,A).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
基金partly supported by NSF of China(Grant No.11971388)partly supported by NSF of China(Grant No.12171146)+4 种基金partly supported by NSF of China(Grant No.12271230)partly supported by NSF of China(Grant No.12171297)the Scientific Research Funds of Huaqiao University(Grant No.605-50Y22050)the Fujian Alliance Of Mathematics(Grant No.2024SXLMMS04)the Foundation for Innovative Fundamental Research Group Project of Gansu Province(Grant No.23JRRA684)。
文摘This paper focuses on a question raised by Holm and Jorgensen,who asked if the induced cotorsion pairs(Φ(X),Φ(X)^(⊥))and(^(⊥)Ψ(Y),Ψ(Y))in Rep(Q,A)—the category of all A-valued representations of a quiver Q—are complete whenever(X,Y)is a complete cotorsion pair in an abelian category A satisfying some mild conditions.We give an affirmative answer if the quiver Q is rooted.As an application,we show under certain mild conditions that if a subcategory L,which is not necessarily closed under direct summands,of A is special precovering(resp.,preenveloping),thenΦ(L)(resp.,Ψ(L))is special precovering(resp.,preenveloping)in Rep(Q,A).
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.
