Let R[P] be the one point extension of a k-algebra R by a projective R-module P.We prove that the extension of a complete ideal cotorsion pair in R-Mod is still a complete ideal cotorsion pair in R[P]-Mod.As an applic...Let R[P] be the one point extension of a k-algebra R by a projective R-module P.We prove that the extension of a complete ideal cotorsion pair in R-Mod is still a complete ideal cotorsion pair in R[P]-Mod.As an application,it is obtainable that the operation(-)_(m)[P]satisfies the so-called distributive law relating the operations of products and extensions of ideals under appropriate conditions.展开更多
Let C be a triangulated category. We define m-term subcategories on C induced by n-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on ...Let C be a triangulated category. We define m-term subcategories on C induced by n-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on 2-term subcategories G and support τ-tilting subcategories on an abelian quotient of G. If an m-term subcategory is induced by a co-t-structure, then we have a one-to-one correspondence between cotorsion pairs on it and cotorsion pairs on C under certain conditions.展开更多
Given a cotorsion pair （X, Y） in an abelian category A, we define cotorsion pairs （XN,dgYN） and （dgXN, YN） in the category CN（A） of N-complexes on A. We prove that if the cotorsion pair （X, Y） is complete an...Given a cotorsion pair （X, Y） in an abelian category A, we define cotorsion pairs （XN,dgYN） and （dgXN, YN） in the category CN（A） of N-complexes on A. We prove that if the cotorsion pair （X, Y） is complete and hereditary in a bicomplete abelian category, then both of the induced cotorsion pairs are complete, compatible and hereditary. We also create complete cotorsion pairs （dwXN, （dwXN）⊥）, （eXXN, （exXN）⊥） and （⊥（dwYN）, dwYN）, （X（exYN）, exYN） in a termwise manner by starting with a cotorsion pair （X,Y） that is cogenerated by a set. As applications of these results, we obtain more abelian model structures from the cotorsion pairs.展开更多
The notion of a tilting pair Miyashita in 2001. It is a useful tool in cotorsion pairs related to a fixed tilting （covariantly） finite subcategory and a tilting pair were given in this paper. over artin algebras was...The notion of a tilting pair Miyashita in 2001. It is a useful tool in cotorsion pairs related to a fixed tilting （covariantly） finite subcategory and a tilting pair were given in this paper. over artin algebras was introduced by the tilting theory. Approximations and pair were discussed. A eontravariantly eotorsion pair associated with a fixed展开更多
This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category.As an application,this result generalizes the work by W.J.Chen...This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category.As an application,this result generalizes the work by W.J.Chen,Z.K.Liu,and X.Y.Yang in a triangulated case[J.Algebra Appl.,2018,17(5):1-15].Moreover,it highlights new phenomena when it applied to an exact category.Finally,we give some applications of our main results.In particular,we obtain Krause's recollement whose proofs are both elementary and very general.展开更多
It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian.In this article,we consider a special class of hearts of twin cotorsion pairs induced by d-cluster tilting su...It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian.In this article,we consider a special class of hearts of twin cotorsion pairs induced by d-cluster tilting subcategories in extriangulated categories.We give a necessary and sufficient condition for such hearts to be abelian.In particular,we can also see that such hearts are hereditary.As an application,this generalizes the work by Liu in the exact case,thereby providing new insights into the triangulated case.展开更多
Assume that S is an almost excellent extension of R. Using functors HomR（S,-） and -×R S, we establish some connections between classes of modules lR and lS, cotorsion pairs （AR, BR） and （AS, BS）. If lS is a...Assume that S is an almost excellent extension of R. Using functors HomR（S,-） and -×R S, we establish some connections between classes of modules lR and lS, cotorsion pairs （AR, BR） and （AS, BS）. If lS is a T-extension or （and） H-extension of lR, we show that lS is a （resp., monomorphic, epimorphic, special） preenveloping class if and only if so is lR. If （AS, BS） is a TH- extension of （AR, BR）, we obtain that （AS, BS） is complete （resp., of finite type, of cofinite type, hereditary, perfect, n-tilting） if and only if so is （AR, BR）.展开更多
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
We introduce two adjoint pairs （eλ^i（）i） and （（）i,ep^i） and give a new method to construct cotorsion pairs. As applications, we characterize all projective and injective representations of a generalized path ...We introduce two adjoint pairs （eλ^i（）i） and （（）i,ep^i） and give a new method to construct cotorsion pairs. As applications, we characterize all projective and injective representations of a generalized path algebra and exhibit projective and injective objects of the category Mp which is a generalization of monomorphisms category.展开更多
In this paper,we introduce the notion of G_(C)-X-injective modules,where X denotes a class of left S-modules and C represents a faithfully semidualizing bimodule.Under the condition that X satisfies certain hypotheses...In this paper,we introduce the notion of G_(C)-X-injective modules,where X denotes a class of left S-modules and C represents a faithfully semidualizing bimodule.Under the condition that X satisfies certain hypotheses,some properties and some equivalent characterizations of G_(C)-X-injective modules are investigated,and we also show that the triple(■,cores■,■)is a weak co-AB-context.As an application,two complete cotorsion pairs and a new model structure in Mod S are given.展开更多
Let n≥1 be an integer.In this article,we study the notions of Cartan-Eilenberg FP_(n)injective and Cartan-Eilenberg FP_(n)-flat complexes.We also investigate duality pairs relative to the Cartan-Eilenberg FP_(n)-inje...Let n≥1 be an integer.In this article,we study the notions of Cartan-Eilenberg FP_(n)injective and Cartan-Eilenberg FP_(n)-flat complexes.We also investigate duality pairs relative to the Cartan-Eilenberg FP_(n)-injective and Cartan-Eilenberg FP_(n)-flat complexes.As applications,we obtain that any complex admits Cartan-Eilenberg FP_(n)-injective(resp.,Cartan-Eilenberg FP_(n)-flat)cover and preenvelope and we characterize left n-coherent rings using these two classes of complexes.展开更多
In this paper, we investigate Ding projective dimensions and Ding injective di- mensions of modules and rings. Let R be a ring with rDPD（R） = n 〈 ∞, and let YYl = {Mild（M） 〈 ∞}. We prove that （DP,W1） is a co...In this paper, we investigate Ding projective dimensions and Ding injective di- mensions of modules and rings. Let R be a ring with rDPD（R） = n 〈 ∞, and let YYl = {Mild（M） 〈 ∞}. We prove that （DP,W1） is a complete hereditary cotorsion pair such that a module M belongs to DD∩W1 if and only if M is projective, moreover, 1421 = （M[pd（M） 〈 ∞} = {MIfd（M） ≤ n} = {MIpd（M） ≤ n}. Then we introduce and inves- tigate Ding derived functor Dext^i（-, -）, and use it to characterize global Ding dimension. We show that if R is a Ding-Chen ring, or if R is a ring with rDPD（R） 〈≤ and rDID（R） 〈 ≤, then rDPD（R） 〈 n if and only if rDID（R） 〈 n if and only if Dext^n＋i（M,N） = 0 for all modules M and N and all integer i ≥ 1.展开更多
In this paper,we introduce the notions of_((m,n))-coherent rings and FP_((m,n))-projective modules for nonnegative integers m,n.We prove that(FP_((m,n))-Proj,(FPn-id)_(≤m))is a complete cotorsion pair for any m,n≥0 ...In this paper,we introduce the notions of_((m,n))-coherent rings and FP_((m,n))-projective modules for nonnegative integers m,n.We prove that(FP_((m,n))-Proj,(FPn-id)_(≤m))is a complete cotorsion pair for any m,n≥0 and it is hereditary if and only if the ring R is a left n-coherent ring for all m≥0 and n≥1.Moreover,we study the existence of FP_((m,n))-Proj covers and envelopes and obtain that if FP_((m,n))-Proj is closed under pure quotients,then FP_((m,n))-Proj is covering for any n≥2.As applications,we obtain that every R-module has an epic FP_((m,n))-Proj-envelope if and only if the left FP_((m,n))-global dimension of R is at most 1 and FP_((m,n))-Proj is closed under direct products.展开更多
Let C=(C,E,s)be an extriangulated category.In this paper,we give the notion of ideal balanced pairs in C and some equivalent characterizations of ideal balanced pairs.We show that there is a one-to-one correspondence ...Let C=(C,E,s)be an extriangulated category.In this paper,we give the notion of ideal balanced pairs in C and some equivalent characterizations of ideal balanced pairs.We show that there is a one-to-one correspondence between balanced pairs and ideal balanced pairs satisfying certain conditions when C is of a negative first extension.We also prove that there is a bijective correspondence between ideal balanced pairs(I,J)and additive subfunctors F■E with enough projective morphisms and enough injective morphisms in C.展开更多
Let(X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to(X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or ...Let(X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to(X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y(resp. Y-coresolution dimension of X)is finite, then the bounded homotopy category of Y(resp. X) is contained in that of X(resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.展开更多
Let R be an associative ring with identity.In this paper,we consider generalizations of Gorenstein FP-injective R-modules and FP-injective complexes,give the definitions and characterizations of strongly Gorenstein FP...Let R be an associative ring with identity.In this paper,we consider generalizations of Gorenstein FP-injective R-modules and FP-injective complexes,give the definitions and characterizations of strongly Gorenstein FPinjective R-modules and strongly FP-injective complexes,which are induced by strongly FP-injective modules.Then we investigate the strongly FP-injective dimension of complexes.展开更多
Let(B,E,s)be an extriangulated category and S be an extension closed subcategory of B.In this article,we prove that the Gabriel-Zisman localization B/S can be realized as an ideal quotient inside B when S satisfies so...Let(B,E,s)be an extriangulated category and S be an extension closed subcategory of B.In this article,we prove that the Gabriel-Zisman localization B/S can be realized as an ideal quotient inside B when S satisfies some mild conditions.The ideal quotient is an extriangulated category.We show that the equivalence between the ideal quotient and the localization preserves the extriangulated category structure.We also discuss the relations of our results with Hovey twin cotorsion pairs and Verdier quotients.展开更多
Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the propert...Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the property of the extension closure of some classes of objects in(T↓A),the exactness of the functor p and the detailed description of orthogonal classes of a given class p(X,Y)in(T↓A).Moreover,we characterize when special precovering classes in abelian categories A and B can induce special precovering classes in(T↓A).As an application,we prove that under suitable conditions,the class of Gorenstein projective leftΛ-modules over a triangular matrix ringΛ=(R M 0 S)is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering.Consequently,we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.展开更多
To investigate cohomology theories based on flats, Asadollahi and Salarian gave the definition of F-Gorenstein flat R-modules, and these modules are exactly Gorenstein fiat provided that R is right coherent. In this p...To investigate cohomology theories based on flats, Asadollahi and Salarian gave the definition of F-Gorenstein flat R-modules, and these modules are exactly Gorenstein fiat provided that R is right coherent. In this paper, we get some properties of F-Gorenstein flat R-modules and establish the stability of F-Gorenstein flat categories.展开更多
We introduce and study the relative lett derive functor Torn（￡,￡1） category, which unifies several related left derived functors. Then we give some criteria for computing the -resolution dimensions of modules in t...We introduce and study the relative lett derive functor Torn（￡,￡1） category, which unifies several related left derived functors. Then we give some criteria for computing the -resolution dimensions of modules in terms of the properties of Torn（￡,￡1） . We also construct a complete and hereditary cotorsion pair relative to balanced pairs. Some known results are obtained as corollaries.展开更多
基金Supported by Zhejiang Provincial Natural Science Foundation of China(LY18A010032)
文摘Let R[P] be the one point extension of a k-algebra R by a projective R-module P.We prove that the extension of a complete ideal cotorsion pair in R-Mod is still a complete ideal cotorsion pair in R[P]-Mod.As an application,it is obtainable that the operation(-)_(m)[P]satisfies the so-called distributive law relating the operations of products and extensions of ideals under appropriate conditions.
基金supported by the National Natural Science Foundation of China(Grant No.12171397)Panyue Zhou is supported by the National Natural Science Foundation of China(Grant No.12371034)by the Hunan Provincial Natural Science Foundation of China(Grant No.2023JJ30008)。
文摘Let C be a triangulated category. We define m-term subcategories on C induced by n-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on 2-term subcategories G and support τ-tilting subcategories on an abelian quotient of G. If an m-term subcategory is induced by a co-t-structure, then we have a one-to-one correspondence between cotorsion pairs on it and cotorsion pairs on C under certain conditions.
基金This research was partially supported by the National Natural Science Foundation of China （11361051, 11761060）, Program for New Century Excellent Talents in University （NCET-13-0957）, and Improvement of Young Teachers＇ Scientific Research Ability （NWNU-LKQN-16-5）.
文摘Given a cotorsion pair （X, Y） in an abelian category A, we define cotorsion pairs （XN,dgYN） and （dgXN, YN） in the category CN（A） of N-complexes on A. We prove that if the cotorsion pair （X, Y） is complete and hereditary in a bicomplete abelian category, then both of the induced cotorsion pairs are complete, compatible and hereditary. We also create complete cotorsion pairs （dwXN, （dwXN）⊥）, （eXXN, （exXN）⊥） and （⊥（dwYN）, dwYN）, （X（exYN）, exYN） in a termwise manner by starting with a cotorsion pair （X,Y） that is cogenerated by a set. As applications of these results, we obtain more abelian model structures from the cotorsion pairs.
文摘The notion of a tilting pair Miyashita in 2001. It is a useful tool in cotorsion pairs related to a fixed tilting （covariantly） finite subcategory and a tilting pair were given in this paper. over artin algebras was introduced by the tilting theory. Approximations and pair were discussed. A eontravariantly eotorsion pair associated with a fixed
基金the National Natural Science Foundation of China(Grant Nos.11901190,11671126,12071120)the Scientific Research Fund of Hunan Provincial Education Department(Grant No.19B239).
文摘This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category.As an application,this result generalizes the work by W.J.Chen,Z.K.Liu,and X.Y.Yang in a triangulated case[J.Algebra Appl.,2018,17(5):1-15].Moreover,it highlights new phenomena when it applied to an exact category.Finally,we give some applications of our main results.In particular,we obtain Krause's recollement whose proofs are both elementary and very general.
基金Panyue Zhou was supported by the Hunan Provincial Natural Science Foundation of China(Grant No.2023JJ30008).
文摘It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian.In this article,we consider a special class of hearts of twin cotorsion pairs induced by d-cluster tilting subcategories in extriangulated categories.We give a necessary and sufficient condition for such hearts to be abelian.In particular,we can also see that such hearts are hereditary.As an application,this generalizes the work by Liu in the exact case,thereby providing new insights into the triangulated case.
基金Supported by Natural Science Foundation of China (Grant No. A0324656)Natural Science Foundation of Fujian Province (Grant No. 2009J01003)+1 种基金Scientific Research Foundation of Fujian Provincial Department of Science and Technology (Grant No. 2007F5038)Foundation of Fujian Normal University (Grant Nos. 2008100209, 09A004)
文摘Assume that S is an almost excellent extension of R. Using functors HomR（S,-） and -×R S, we establish some connections between classes of modules lR and lS, cotorsion pairs （AR, BR） and （AS, BS）. If lS is a T-extension or （and） H-extension of lR, we show that lS is a （resp., monomorphic, epimorphic, special） preenveloping class if and only if so is lR. If （AS, BS） is a TH- extension of （AR, BR）, we obtain that （AS, BS） is complete （resp., of finite type, of cofinite type, hereditary, perfect, n-tilting） if and only if so is （AR, BR）.
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This work was carried out while the author was a visitor at University of California, Berkeley she thanks Prof. T. Y. Lam for the very helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China （Grant No. 11201424） and the Natural Science Foundation of Zhejiang Province （No. LY12A01026）.
文摘We introduce two adjoint pairs （eλ^i（）i） and （（）i,ep^i） and give a new method to construct cotorsion pairs. As applications, we characterize all projective and injective representations of a generalized path algebra and exhibit projective and injective objects of the category Mp which is a generalization of monomorphisms category.
文摘In this paper,we introduce the notion of G_(C)-X-injective modules,where X denotes a class of left S-modules and C represents a faithfully semidualizing bimodule.Under the condition that X satisfies certain hypotheses,some properties and some equivalent characterizations of G_(C)-X-injective modules are investigated,and we also show that the triple(■,cores■,■)is a weak co-AB-context.As an application,two complete cotorsion pairs and a new model structure in Mod S are given.
基金Supported by the National Natural Science Foundation of China(Grant No.12061061)the Science and Technology Project of Gansu Province(Grant No.22JR5RA153).
文摘Let n≥1 be an integer.In this article,we study the notions of Cartan-Eilenberg FP_(n)injective and Cartan-Eilenberg FP_(n)-flat complexes.We also investigate duality pairs relative to the Cartan-Eilenberg FP_(n)-injective and Cartan-Eilenberg FP_(n)-flat complexes.As applications,we obtain that any complex admits Cartan-Eilenberg FP_(n)-injective(resp.,Cartan-Eilenberg FP_(n)-flat)cover and preenvelope and we characterize left n-coherent rings using these two classes of complexes.
基金Supported by the National Natural Science Foundation of China(11201424)the Zhejiang Natural Science Foundation of China(LY12A01026)
文摘In this paper, we investigate Ding projective dimensions and Ding injective di- mensions of modules and rings. Let R be a ring with rDPD（R） = n 〈 ∞, and let YYl = {Mild（M） 〈 ∞}. We prove that （DP,W1） is a complete hereditary cotorsion pair such that a module M belongs to DD∩W1 if and only if M is projective, moreover, 1421 = （M[pd（M） 〈 ∞} = {MIfd（M） ≤ n} = {MIpd（M） ≤ n}. Then we introduce and inves- tigate Ding derived functor Dext^i（-, -）, and use it to characterize global Ding dimension. We show that if R is a Ding-Chen ring, or if R is a ring with rDPD（R） 〈≤ and rDID（R） 〈 ≤, then rDPD（R） 〈 n if and only if rDID（R） 〈 n if and only if Dext^n＋i（M,N） = 0 for all modules M and N and all integer i ≥ 1.
基金supported by the National Natural Science Foundation of China(No.12471036),the project of Young and Middle-Aged Talents of Hubei Province(No.Q20234405),and the Scientific Research Fund of Hunan Provincial Education Department(No.24A0221)。
文摘In this paper,we introduce the notions of_((m,n))-coherent rings and FP_((m,n))-projective modules for nonnegative integers m,n.We prove that(FP_((m,n))-Proj,(FPn-id)_(≤m))is a complete cotorsion pair for any m,n≥0 and it is hereditary if and only if the ring R is a left n-coherent ring for all m≥0 and n≥1.Moreover,we study the existence of FP_((m,n))-Proj covers and envelopes and obtain that if FP_((m,n))-Proj is closed under pure quotients,then FP_((m,n))-Proj is covering for any n≥2.As applications,we obtain that every R-module has an epic FP_((m,n))-Proj-envelope if and only if the left FP_((m,n))-global dimension of R is at most 1 and FP_((m,n))-Proj is closed under direct products.
基金supported by National Natural Science Foundation of China(Grant No.12071064)Natural Science Foundation of Jilin Province of China(Grant No.YDZJ202101ZYTS168)。
文摘Let C=(C,E,s)be an extriangulated category.In this paper,we give the notion of ideal balanced pairs in C and some equivalent characterizations of ideal balanced pairs.We show that there is a one-to-one correspondence between balanced pairs and ideal balanced pairs satisfying certain conditions when C is of a negative first extension.We also prove that there is a bijective correspondence between ideal balanced pairs(I,J)and additive subfunctors F■E with enough projective morphisms and enough injective morphisms in C.
基金supported by National Natural Science Foundation of China(Grant No.11171142)
文摘Let(X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to(X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y(resp. Y-coresolution dimension of X)is finite, then the bounded homotopy category of Y(resp. X) is contained in that of X(resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.
基金supported by the Gansu Natural Science Foundation Project(No.25JRRK002).
文摘Let R be an associative ring with identity.In this paper,we consider generalizations of Gorenstein FP-injective R-modules and FP-injective complexes,give the definitions and characterizations of strongly Gorenstein FPinjective R-modules and strongly FP-injective complexes,which are induced by strongly FP-injective modules.Then we investigate the strongly FP-injective dimension of complexes.
基金supported by National Natural Science Foundation of China(Grant No.12171397)Panyue Zhou was supported by National Natural Science Foundation of China(Grant No.12371034)the Hunan Provincial Natural Science Foundation of China(Grant No.2023JJ30008)。
文摘Let(B,E,s)be an extriangulated category and S be an extension closed subcategory of B.In this article,we prove that the Gabriel-Zisman localization B/S can be realized as an ideal quotient inside B when S satisfies some mild conditions.The ideal quotient is an extriangulated category.We show that the equivalence between the ideal quotient and the localization preserves the extriangulated category structure.We also discuss the relations of our results with Hovey twin cotorsion pairs and Verdier quotients.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671069 and 11771212)Zhejiang Provincial Natural Science Foundation of China (Grant No. LY18A010032)+1 种基金Qing Lan Project of Jiangsu Province and Jiangsu Government Scholarship for Overseas Studies (Grant No. JS2019-328)during a visit of the first author to Charles University in Prague with the support by Jiangsu Government Scholarship
文摘Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the property of the extension closure of some classes of objects in(T↓A),the exactness of the functor p and the detailed description of orthogonal classes of a given class p(X,Y)in(T↓A).Moreover,we characterize when special precovering classes in abelian categories A and B can induce special precovering classes in(T↓A).As an application,we prove that under suitable conditions,the class of Gorenstein projective leftΛ-modules over a triangular matrix ringΛ=(R M 0 S)is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering.Consequently,we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.
文摘To investigate cohomology theories based on flats, Asadollahi and Salarian gave the definition of F-Gorenstein flat R-modules, and these modules are exactly Gorenstein fiat provided that R is right coherent. In this paper, we get some properties of F-Gorenstein flat R-modules and establish the stability of F-Gorenstein flat categories.
基金Supported by NSFC(Grant Nos.11171142,11571164)a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions
文摘We introduce and study the relative lett derive functor Torn（￡,￡1） category, which unifies several related left derived functors. Then we give some criteria for computing the -resolution dimensions of modules in terms of the properties of Torn（￡,￡1） . We also construct a complete and hereditary cotorsion pair relative to balanced pairs. Some known results are obtained as corollaries.
