In this paper,we study the relations between the class of A-Gorenstein projective modules and the left orthogonal class of A,also the relations between the class of A-Gorenstein injective modules and the right orthogo...In this paper,we study the relations between the class of A-Gorenstein projective modules and the left orthogonal class of A,also the relations between the class of A-Gorenstein injective modules and the right orthogonal class of A.Some functor characterizations of AGorenstein projective modules and A-Gorenstein injective modules are obtained.Using the notion of complete duality pair,we discuss the relations between A-Gorenstein projective modules and B-Gorenstein flat modules.Some known results are generalized.展开更多
In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applicati...In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applications are given.展开更多
We introduce a generalization of the Gorenstein injective modules:the Gorenstein FPn-injective modules(denoted by GI_(n)).They are the cycles of the exact complexes of injective modules that remain exact when we apply...We introduce a generalization of the Gorenstein injective modules:the Gorenstein FPn-injective modules(denoted by GI_(n)).They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor Hom(A,-),with A any FP_(n)-injective module.Thus,GL_(o)is the class of classical Gorenstein injective modules,and GI_(1)is the class of Ding injective modules.We prove that over any ring R,for any n≥2,the class GI_(n)is the right half of a perfect cotorsion pair,and therefore it is an enveloping class.For n=1 we show that GI_(1)(i.e.,the Ding injectives)forms the right half of a hereditary cotorsion pair.If moreover the ring R is coherent,then the Ding injective modules form an enveloping class.We also define the dual notion,that of Gorenstein FP_(n)-projectives(denoted by GP_(n)).They generalize the Ding projective modules,and so,the Gorenstein projective modules.We prove that for any n≥2 the class GP_(n)is the left half of a complete hereditary cotorsion pair,and therefore it is special precovering.展开更多
In the Gorenstein homological theory, Gorenstein projective and Gorenstein injective dimensions play an important and fundamental role. In this paper, we aim at studying the closely related strongly Gorenstein flat an...In the Gorenstein homological theory, Gorenstein projective and Gorenstein injective dimensions play an important and fundamental role. In this paper, we aim at studying the closely related strongly Gorenstein flat and Gorenstein FP-injective dimensions, and show that some characterizations similar to Gorenstein homological dimensions hold for these two dimensions.展开更多
In this article, we introduce and study the concept of n-Gorenstein injective （resp., n-Gorenstein flat） modules as a nontrivial generalization of Gorenstein injective （resp., Gorenstein flat） modules. We invest...In this article, we introduce and study the concept of n-Gorenstein injective （resp., n-Gorenstein flat） modules as a nontrivial generalization of Gorenstein injective （resp., Gorenstein flat） modules. We investigate the properties of these modules in various ways. For example, we show that the class of n-Gorenstein injective （resp., n -Gorenstein flat） modules is closed under direct sums and direct products for n ≥ 2. To this end, we first introduce and study the notions of n-injective modules and n-flat modules.展开更多
We consider the conditions under which the class of （m, d）-injective R-modules is （pre）covering. It is shown that every left R-module over a left （m, d）-coherent ring has an （rn, d）-injective cover. Moreover, ...We consider the conditions under which the class of （m, d）-injective R-modules is （pre）covering. It is shown that every left R-module over a left （m, d）-coherent ring has an （rn, d）-injective cover. Moreover, the classes of Gorenstein （m, d）-flat modules and Gorenstein （m, d）-injecitve modules are introduced and studied. For a right （m, d）-coherent ring R, we prove that a left R-module M is Gorenstein （m, d）-flat if and only if M＋ is Gorenstein （m, d）- injective as a right R-module. Some results on Gorenstein flat modules and Gorenstein n-flat modules are generalized.展开更多
In this article,we first establish a recollement related to projectively coresolved Gorenstein flat(PGF)complexes.Secondly,we define and study PGF dimension of complexes,we denote it PG F(X)for a complex X.It is shown...In this article,we first establish a recollement related to projectively coresolved Gorenstein flat(PGF)complexes.Secondly,we define and study PGF dimension of complexes,we denote it PG F(X)for a complex X.It is shown that the PGF(X)is equal to the infimum of the set{supA|there exists a diagram of morphisms of complexes A←G→X,such that G→X is a special PGF precover of X and G→A is a PGF almost isomorphism}.展开更多
In this paper, we shall be concerned with what happens of Gorenstein homological dimensions when certain modifications are made to a ring. The five structural operations addressed later are the formation of excellent ...In this paper, we shall be concerned with what happens of Gorenstein homological dimensions when certain modifications are made to a ring. The five structural operations addressed later are the formation of excellent extensions, localizations, Morita equivalences, polynomial extensions and power series extensions.展开更多
As we know,a complex P is projective if and only if P is exact with Z_n(P)projective in R-Mod for each n∈Z and any morphism f:P→C is null homotopic for any complex C.In this article,we study the notion of DG-Gorenst...As we know,a complex P is projective if and only if P is exact with Z_n(P)projective in R-Mod for each n∈Z and any morphism f:P→C is null homotopic for any complex C.In this article,we study the notion of DG-Gorenstein projective complexes.We show that a complex G is DG-Gorenstein projective if and only if G is exact with Z_n(G)Gorenstein projective in R-Mod for each n∈Z and any morphism f:G→Q is null homotopic whenever Q is a DG-projective complex.展开更多
LetRbeanarbitrary ring and let S be a separable and quasi-Frobenius extension of R.Then for any right S-module M,the Gorenstein flat dimensions of MS andMR areidentical.As a consequence,the Gorenstein weak global dime...LetRbeanarbitrary ring and let S be a separable and quasi-Frobenius extension of R.Then for any right S-module M,the Gorenstein flat dimensions of MS andMR areidentical.As a consequence,the Gorenstein weak global dimension of S is less than or equal to that of R.展开更多
Let R be a ring,X a class of R-modules and n≥1 an integer.We intro-duce the concepts of Gorenstein n-X-injective and n-X-flat modules via special finitely presented modules.Besides,we obtain some equivalent propertie...Let R be a ring,X a class of R-modules and n≥1 an integer.We intro-duce the concepts of Gorenstein n-X-injective and n-X-flat modules via special finitely presented modules.Besides,we obtain some equivalent properties of these modules on n-X-coherent rings.Then we investigate the relations among Gorenstein n-X-injective,n-X-flat,injective and fiat modules on X-FC-rings(i.e.,self n-X-injective and n-X-coherent rings).Several known results are generalized to this new context.展开更多
There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give hom...There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give homological descriptions of the strongly Gorenstein flat dimension (of modules and rings) over arbitrary associative rings.展开更多
We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extensi...We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extension is either left-Gorenstein or separable(e.g., the integral group ring extension ZZG).Moreover, for the Frobenius extension RA = R[x]/(x^2), we show that: a graded A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module is Gorenstein projective, if and only if its underlying R-module is Gorenstein projective. It immediately follows that an R-complex is Gorenstein projective if and only if all its items are Gorenstein projective R-modules.展开更多
Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded an...Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.展开更多
In this paper,we introduce Gorenstein weak injective and weak flat modules in terms of,respectively,weak injective and weak flat modules;the classes of Gorenstein weak injective and weak flat modules are larger than t...In this paper,we introduce Gorenstein weak injective and weak flat modules in terms of,respectively,weak injective and weak flat modules;the classes of Gorenstein weak injective and weak flat modules are larger than the classical classes of Gorenstein injective and flat modules.In this new setting,we characterize rings over which all modules are Gorenstein weak injective.Moreover,we discuss the relation between the weak cosyzygy and Gorenstein weak cosyzygy of a module,and also the stability of Gorenstein weak injective modules.展开更多
Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is...Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.展开更多
Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modu...Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modules are investigated. It is shown that weak Gorenstein cotorsion R-modules over coherent rings are indeed weaker than Gorenstein cotorsion R-modules. Weak Gorenstein cotorsion dimension for modules and rings are also studied.展开更多
Let A and B be rings and U a(B,A)-bimodule.If BU is flat and UA is finitely generated projective(resp.,BU is finitely generated projective and UA is flat),then the characterizations of level modules and Gorenstein AC-...Let A and B be rings and U a(B,A)-bimodule.If BU is flat and UA is finitely generated projective(resp.,BU is finitely generated projective and UA is flat),then the characterizations of level modules and Gorenstein AC-projective modules(resp.,absolutely clean modules and Gorenstein AC-injective modules)over the formal triangular matrix ring T=(A0 UB)are given.As applications,it is proved that every Gorenstein AC-projective left T-module is projective if and only if each Gorenstein AC-projective left A-module and B-module is projective,and every Gorenstein AC-injective left T-module is injective if and only if each Gorenstein AC-injective left A-module and B-module is injective.Moreover,Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring T are studied.展开更多
We study some homological properties of Gorenstein FP∞-injective modules,and we prove(1)a ring R is not necessarily coherent if every Gorenstein FP∞-injective R-module is injective,and(2)a ring R is not necessarily ...We study some homological properties of Gorenstein FP∞-injective modules,and we prove(1)a ring R is not necessarily coherent if every Gorenstein FP∞-injective R-module is injective,and(2)a ring R is not necessarily coherent if every Gorenstein injective R-module is injective.In addition,we characterize w-Noetherian rings in terms of Gorenstein FP∞-injective modules,and we prove that a ring R is w-Noetherian if and only if every GV-torsion-free FP∞-injective R-module is Gorenstein FP∞-injective,if and only if any direct sum of GV-torsion-free FP∞-injective R-modules is Gorenstein FP∞-injective.展开更多
We prove that a certain eventually homological isomorphism between module categories induces triangle equivalences between their singularity categories,Gorenstein defect categories and stable categories of Gorenstein ...We prove that a certain eventually homological isomorphism between module categories induces triangle equivalences between their singularity categories,Gorenstein defect categories and stable categories of Gorenstein projective modules.Furthermore,we show that the Auslander-Reiten conjecture and the Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms.Applying these results to arrow removal and vertex removal,we describe the Gorenstein projective modules over some non-monomial algebras and verify the Auslander-Reiten conjecture for certain algebras.展开更多
基金Supported by Qinglan Project of Jiangsu Province(Grant No.2022)the National Natural Science Foundation of China(Grant No.11701408)Jiangsu Provincial Government Scholarship Program。
文摘In this paper,we study the relations between the class of A-Gorenstein projective modules and the left orthogonal class of A,also the relations between the class of A-Gorenstein injective modules and the right orthogonal class of A.Some functor characterizations of AGorenstein projective modules and A-Gorenstein injective modules are obtained.Using the notion of complete duality pair,we discuss the relations between A-Gorenstein projective modules and B-Gorenstein flat modules.Some known results are generalized.
基金Supported by the National Natural Science Foundation of China（11361051） Supported by the Program for New Century Excellent the Talents in University（NCET-13-0957）
文摘In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applications are given.
文摘We introduce a generalization of the Gorenstein injective modules:the Gorenstein FPn-injective modules(denoted by GI_(n)).They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor Hom(A,-),with A any FP_(n)-injective module.Thus,GL_(o)is the class of classical Gorenstein injective modules,and GI_(1)is the class of Ding injective modules.We prove that over any ring R,for any n≥2,the class GI_(n)is the right half of a perfect cotorsion pair,and therefore it is an enveloping class.For n=1 we show that GI_(1)(i.e.,the Ding injectives)forms the right half of a hereditary cotorsion pair.If moreover the ring R is coherent,then the Ding injective modules form an enveloping class.We also define the dual notion,that of Gorenstein FP_(n)-projectives(denoted by GP_(n)).They generalize the Ding projective modules,and so,the Gorenstein projective modules.We prove that for any n≥2 the class GP_(n)is the left half of a complete hereditary cotorsion pair,and therefore it is special precovering.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1120137711261050)+1 种基金China Postdoctoral Science Foundation(Grant No.2013M541509)Program of Science and Technique of Gansu Province(Grant No.1208RJZA145)
文摘In the Gorenstein homological theory, Gorenstein projective and Gorenstein injective dimensions play an important and fundamental role. In this paper, we aim at studying the closely related strongly Gorenstein flat and Gorenstein FP-injective dimensions, and show that some characterizations similar to Gorenstein homological dimensions hold for these two dimensions.
基金The NSF(11501451)of Chinathe Fundamental Research Funds(31920150038)for the Central Universities and XBMUYJRC(201406)
文摘In this article, we introduce and study the concept of n-Gorenstein injective （resp., n-Gorenstein flat） modules as a nontrivial generalization of Gorenstein injective （resp., Gorenstein flat） modules. We investigate the properties of these modules in various ways. For example, we show that the class of n-Gorenstein injective （resp., n -Gorenstein flat） modules is closed under direct sums and direct products for n ≥ 2. To this end, we first introduce and study the notions of n-injective modules and n-flat modules.
基金Supported by the Provincial Natural Science Research Program of Higher Education Institution of Anhui Province(Grant No.KJ2012Z028)
文摘We consider the conditions under which the class of （m, d）-injective R-modules is （pre）covering. It is shown that every left R-module over a left （m, d）-coherent ring has an （rn, d）-injective cover. Moreover, the classes of Gorenstein （m, d）-flat modules and Gorenstein （m, d）-injecitve modules are introduced and studied. For a right （m, d）-coherent ring R, we prove that a left R-module M is Gorenstein （m, d）-flat if and only if M＋ is Gorenstein （m, d）- injective as a right R-module. Some results on Gorenstein flat modules and Gorenstein n-flat modules are generalized.
基金Supported by the National Natural Science Foundation of China(12061061)Young Talents Team Project of Gansu Province(2025QNTD49)+1 种基金Lanshan Talents Project of Northwest Minzu University(Xbmulsrc202412)Longyuan Young Talents of Gansu Province。
文摘In this article,we first establish a recollement related to projectively coresolved Gorenstein flat(PGF)complexes.Secondly,we define and study PGF dimension of complexes,we denote it PG F(X)for a complex X.It is shown that the PGF(X)is equal to the infimum of the set{supA|there exists a diagram of morphisms of complexes A←G→X,such that G→X is a special PGF precover of X and G→A is a PGF almost isomorphism}.
基金Supported by the National Natural Science Foundation of China (Grant No. 11001222)
文摘In this paper, we shall be concerned with what happens of Gorenstein homological dimensions when certain modifications are made to a ring. The five structural operations addressed later are the formation of excellent extensions, localizations, Morita equivalences, polynomial extensions and power series extensions.
基金Supported by the National Natural Science Foundation of China(2061061)Fundamental Research Funds for the Central Universities(31920190054)+1 种基金Funds for Talent Introduction of Northwest Minzu University(XBMUYJRC201406)First-Rate Discipline of Northwest Minzu University。
文摘As we know,a complex P is projective if and only if P is exact with Z_n(P)projective in R-Mod for each n∈Z and any morphism f:P→C is null homotopic for any complex C.In this article,we study the notion of DG-Gorenstein projective complexes.We show that a complex G is DG-Gorenstein projective if and only if G is exact with Z_n(G)Gorenstein projective in R-Mod for each n∈Z and any morphism f:G→Q is null homotopic whenever Q is a DG-projective complex.
基金Supported by NSFC(Grant No.11601304)Foundation for University Key Teacher by Henan Province(2019GGJS204)。
文摘LetRbeanarbitrary ring and let S be a separable and quasi-Frobenius extension of R.Then for any right S-module M,the Gorenstein flat dimensions of MS andMR areidentical.As a consequence,the Gorenstein weak global dimension of S is less than or equal to that of R.
基金supported by a scholarship from the Graduate Research Assistantships in Developing Countries Program of the Commission for Developing Countries of the International Mathematical Union.
文摘Let R be a ring,X a class of R-modules and n≥1 an integer.We intro-duce the concepts of Gorenstein n-X-injective and n-X-flat modules via special finitely presented modules.Besides,we obtain some equivalent properties of these modules on n-X-coherent rings.Then we investigate the relations among Gorenstein n-X-injective,n-X-flat,injective and fiat modules on X-FC-rings(i.e.,self n-X-injective and n-X-coherent rings).Several known results are generalized to this new context.
文摘There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give homological descriptions of the strongly Gorenstein flat dimension (of modules and rings) over arbitrary associative rings.
基金supported by National Natural Science Foundation of China(Grant No.11401476)China Postdoctoral Science Foundation(Grant No.2016M591592)
文摘We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extension is either left-Gorenstein or separable(e.g., the integral group ring extension ZZG).Moreover, for the Frobenius extension RA = R[x]/(x^2), we show that: a graded A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module is Gorenstein projective, if and only if its underlying R-module is Gorenstein projective. It immediately follows that an R-complex is Gorenstein projective if and only if all its items are Gorenstein projective R-modules.
基金Acknowledgements This work was Foundation of China （Grant No. 11371187） Province of China （Grant No. BK20160771） supported by the National Natural Science and the Natural Science Foundation of Jiangsu
文摘Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.
基金This work was partially supported by NSFC(Grant Nos.11571164 and 11571341).
文摘In this paper,we introduce Gorenstein weak injective and weak flat modules in terms of,respectively,weak injective and weak flat modules;the classes of Gorenstein weak injective and weak flat modules are larger than the classical classes of Gorenstein injective and flat modules.In this new setting,we characterize rings over which all modules are Gorenstein weak injective.Moreover,we discuss the relation between the weak cosyzygy and Gorenstein weak cosyzygy of a module,and also the stability of Gorenstein weak injective modules.
基金supported by National Natural Science Foundation of China (Grant No.11171296)the Zhejiang Provincial Natural Science Foundation of China (Grant No. D7080064)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110101110010)
文摘Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.
基金The authors wish to express their sincere thanks to the referees for their valuable comments and suggestions. The first author was supported by the Postdoctoral Science Foundation of China （2017M611851）, the Jiangsu Planned Projects for Postdoctoral Research Funds （1601151C） and the Provincial Natural Science Foundation of Anhui Province of China （KJ2017A040）. The second author was supported by the NSFC （11771212）, and the first two authors were supported by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The third author was supported by the NSFC （11501257, 11671069, 11771212） and the Postdoctoral Science Foundation of China （2016M600426）.
文摘Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modules are investigated. It is shown that weak Gorenstein cotorsion R-modules over coherent rings are indeed weaker than Gorenstein cotorsion R-modules. Weak Gorenstein cotorsion dimension for modules and rings are also studied.
基金partly supported by NSF of China(grants 11761047 and 11861043).
文摘Let A and B be rings and U a(B,A)-bimodule.If BU is flat and UA is finitely generated projective(resp.,BU is finitely generated projective and UA is flat),then the characterizations of level modules and Gorenstein AC-projective modules(resp.,absolutely clean modules and Gorenstein AC-injective modules)over the formal triangular matrix ring T=(A0 UB)are given.As applications,it is proved that every Gorenstein AC-projective left T-module is projective if and only if each Gorenstein AC-projective left A-module and B-module is projective,and every Gorenstein AC-injective left T-module is injective if and only if each Gorenstein AC-injective left A-module and B-module is injective.Moreover,Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring T are studied.
基金supported by the Scientific Research Foundation of Chengdu University of Information Technology(No.KYTZ202015)supported by NSFC(No.12061001).
文摘We study some homological properties of Gorenstein FP∞-injective modules,and we prove(1)a ring R is not necessarily coherent if every Gorenstein FP∞-injective R-module is injective,and(2)a ring R is not necessarily coherent if every Gorenstein injective R-module is injective.In addition,we characterize w-Noetherian rings in terms of Gorenstein FP∞-injective modules,and we prove that a ring R is w-Noetherian if and only if every GV-torsion-free FP∞-injective R-module is Gorenstein FP∞-injective,if and only if any direct sum of GV-torsion-free FP∞-injective R-modules is Gorenstein FP∞-injective.
基金supported by National Natural Science Foundation of China(Grant Nos.12061060 and 11801141)Scientific and Technological Planning Project of Yunnan Province(Grant No.202305AC160005)Scientific and Technological Innovation Team of Yunnan Province(Grant No.2020CXTD25)。
文摘We prove that a certain eventually homological isomorphism between module categories induces triangle equivalences between their singularity categories,Gorenstein defect categories and stable categories of Gorenstein projective modules.Furthermore,we show that the Auslander-Reiten conjecture and the Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms.Applying these results to arrow removal and vertex removal,we describe the Gorenstein projective modules over some non-monomial algebras and verify the Auslander-Reiten conjecture for certain algebras.
