This paper proved that graded modules for greded division ring R are graded free modules and R is a IBN ring,and if R is a graded commutative ring,RM is a graded module andRM is a finite semisimple module,then gr.inj....This paper proved that graded modules for greded division ring R are graded free modules and R is a IBN ring,and if R is a graded commutative ring,RM is a graded module andRM is a finite semisimple module,then gr.inj.dimRM=inj.dimRM.展开更多
Let H be a finite dimensional cocommutative Hopf algebra and A an H-module algebra. In this paper, we characterize the projectivity (injectivity) of M as a left A#σH-module when it is projective (injective) as a left...Let H be a finite dimensional cocommutative Hopf algebra and A an H-module algebra. In this paper, we characterize the projectivity (injectivity) of M as a left A#σH-module when it is projective (injective) as a left A-module. The sufficient and necessary condition for A#σH, the crossed product, to have finite global homological dimension is given, in terms of the global homological dimension of A and the surjectivity of trace maps, provided that H is cocommutative and A is commutative.展开更多
Let H be a Hopf algebra over a field with bijective antipode, and A a commutative cleft right H-comodule algebra. In this paper, we investigate the ho-mological dimensions and the semisimplicity of the category of rel...Let H be a Hopf algebra over a field with bijective antipode, and A a commutative cleft right H-comodule algebra. In this paper, we investigate the ho-mological dimensions and the semisimplicity of the category of relative Hopf modulesAMH.展开更多
Let H be a finite dimensional cosemisimple Hopf algebra, C a left H-comodule coalgebra and let C = C/C(H^*)^+ be the quotient coalgebra and the smash coproduct of C and H. It is shown that if C/C is a eosemisimple...Let H be a finite dimensional cosemisimple Hopf algebra, C a left H-comodule coalgebra and let C = C/C(H^*)^+ be the quotient coalgebra and the smash coproduct of C and H. It is shown that if C/C is a eosemisimple coextension and C is an injective right C-comodule, then gl. dim(the smash coproduct of C and H) = gl. dim(C) = gl. dim(C), where gl. dim(C) denotes the global dimension of coalgebra C.展开更多
We introduce and study (pre)resolving subcategories of a triangulated category and the homological dimension relative to these subcategories. We apply the obtained properties to relative Gorenstein categories.
In Enochs'relative homological dimension theory occur the(co)resolvent and(co)proper dimensions,which are defined by proper and coproper resolutions constructed by precovers and preenvelopes,respectively.Recently,...In Enochs'relative homological dimension theory occur the(co)resolvent and(co)proper dimensions,which are defined by proper and coproper resolutions constructed by precovers and preenvelopes,respectively.Recently,some authors have been interested in relative homological dimensions defined by just exact sequences.In this paper,we contribute to the investigation of these relative homological dimensions.First we study the relation between these two kinds of relative homological dimensions and establish some transfer results under adjoint pairs.Then relative global dimensions are studied,which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories.At the end of this paper,relative derived functors are studied and generalizations of some known results of balance for relative homology are established.展开更多
Let A be a small abelian category.For a closed subbifunctor F of Ext_A^1(-,-),Buan has generalized the construction of Verdier’s quotient category to get a relative derived category,where he localized with respect ...Let A be a small abelian category.For a closed subbifunctor F of Ext_A^1(-,-),Buan has generalized the construction of Verdier’s quotient category to get a relative derived category,where he localized with respect to F-acyclic complexes.In this paper,the homological properties of relative derived categories are discussed,and the relation with derived categories is given.For Artin algebras,using relative derived categories,we give a relative version on derived equivalences induced by F-tilting complexes.We discuss the relationships between relative homological dimensions and relative derived equivalences.展开更多
Let A be an abelian category,C an additive,full and self-orthogonal subcategory of A closed under direct summands,rG(C)the right Gorenstein subcategory of A relative to C,and⊥C the left orthogonal class of C.For an o...Let A be an abelian category,C an additive,full and self-orthogonal subcategory of A closed under direct summands,rG(C)the right Gorenstein subcategory of A relative to C,and⊥C the left orthogonal class of C.For an object A in A,we prove that if A is in the right 1-orthogonal class of rG(C),then the C-projective and rG(C)-projective dimensions of A are identical;if the rG(C)-projective dimension of A is finite,then the rG(C)-projective and⊥C-projective dimensions of A are identical.We also prove that the supremum of the C-projective dimensions of objects with finite C-projective dimension and that of the rG(C)-projective dimensions of objects with finite rG(C)-projective dimension coincide.Then we apply these results to the category of modules.展开更多
In this paper we are concerned with absolute,relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory,and o...In this paper we are concerned with absolute,relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory,and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules.In the second part of the paper we consider a depth equality,called the depth formula,which has been initially introduced by Auslander and developed further by Huneke and Wiegand.As an application of our main result,we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension.展开更多
We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes. Let X and Y be two classes of R-modules with Ext〉I(X,Y) = 0 for each objec...We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes. Let X and Y be two classes of R-modules with Ext〉I(X,Y) = 0 for each object X ∈ X and each object Y ∈ Y. We show that if A,B ∈ C^(R) are X-complexes and U, V ∈ Cr(R) are Y-complexes, then U V Hom(A, U) Hom(A, Y); A B Hom(B, U) Hom(A, U). As an application, we give a sufficient condition for the Hom evaluation morphism being invertible.展开更多
文摘This paper proved that graded modules for greded division ring R are graded free modules and R is a IBN ring,and if R is a graded commutative ring,RM is a graded module andRM is a finite semisimple module,then gr.inj.dimRM=inj.dimRM.
基金Foundationitem:The NSF(10271081)of Chinathe NSF(1042004)of Beijing City
文摘Let H be a finite dimensional cocommutative Hopf algebra and A an H-module algebra. In this paper, we characterize the projectivity (injectivity) of M as a left A#σH-module when it is projective (injective) as a left A-module. The sufficient and necessary condition for A#σH, the crossed product, to have finite global homological dimension is given, in terms of the global homological dimension of A and the surjectivity of trace maps, provided that H is cocommutative and A is commutative.
基金The Fundation of Key Research Program (02021029) and the NSF (2004kj352) of Anhui Province, China.
文摘Let H be a Hopf algebra over a field with bijective antipode, and A a commutative cleft right H-comodule algebra. In this paper, we investigate the ho-mological dimensions and the semisimplicity of the category of relative Hopf modulesAMH.
基金Supported by the Foundation of Key Research Program (No. 02021029)the NSF (No. 2004kj352) of Anhui Province, China
文摘Let H be a finite dimensional cosemisimple Hopf algebra, C a left H-comodule coalgebra and let C = C/C(H^*)^+ be the quotient coalgebra and the smash coproduct of C and H. It is shown that if C/C is a eosemisimple coextension and C is an injective right C-comodule, then gl. dim(the smash coproduct of C and H) = gl. dim(C) = gl. dim(C), where gl. dim(C) denotes the global dimension of coalgebra C.
基金Supported by the National Natural Science Foundation of China(Grant No.11571164)a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions,Postgraduate Research and Practice Innovation Program of Jiangsu Province(Grant No.KYZZ16 0034)Nanjing University Innovation and Creative Program for PhD candidate(Grant No.2016011)
文摘We introduce and study (pre)resolving subcategories of a triangulated category and the homological dimension relative to these subcategories. We apply the obtained properties to relative Gorenstein categories.
基金The second and fourth authors were partially supported by the grant MTM2014-54439-P from Ministerio de Economia y CompetitividadThe third author was partially supported by NSFC(11771202).
文摘In Enochs'relative homological dimension theory occur the(co)resolvent and(co)proper dimensions,which are defined by proper and coproper resolutions constructed by precovers and preenvelopes,respectively.Recently,some authors have been interested in relative homological dimensions defined by just exact sequences.In this paper,we contribute to the investigation of these relative homological dimensions.First we study the relation between these two kinds of relative homological dimensions and establish some transfer results under adjoint pairs.Then relative global dimensions are studied,which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories.At the end of this paper,relative derived functors are studied and generalizations of some known results of balance for relative homology are established.
基金Supported by National Natural Science Foundation of China(Grant No.11201022)the Fundamental Research Funds for the Central Universities(Grant No.2015JBM101)
文摘Let A be a small abelian category.For a closed subbifunctor F of Ext_A^1(-,-),Buan has generalized the construction of Verdier’s quotient category to get a relative derived category,where he localized with respect to F-acyclic complexes.In this paper,the homological properties of relative derived categories are discussed,and the relation with derived categories is given.For Artin algebras,using relative derived categories,we give a relative version on derived equivalences induced by F-tilting complexes.We discuss the relationships between relative homological dimensions and relative derived equivalences.
基金This research was partially supported by NSFC(Grant Nos.11571164,11971225,11901341)the NSF of Shandong Province(Grant No.ZR2019QA015)。
文摘Let A be an abelian category,C an additive,full and self-orthogonal subcategory of A closed under direct summands,rG(C)the right Gorenstein subcategory of A relative to C,and⊥C the left orthogonal class of C.For an object A in A,we prove that if A is in the right 1-orthogonal class of rG(C),then the C-projective and rG(C)-projective dimensions of A are identical;if the rG(C)-projective dimension of A is finite,then the rG(C)-projective and⊥C-projective dimensions of A are identical.We also prove that the supremum of the C-projective dimensions of objects with finite C-projective dimension and that of the rG(C)-projective dimensions of objects with finite rG(C)-projective dimension coincide.Then we apply these results to the category of modules.
基金partly supported by the National Natural Science Foundation of China(Grant Nos.12271230,11761045 and 11971388)the Natural Science Foundation of Gansu Province(Grant No.21JR7RA297)。
文摘In this paper we are concerned with absolute,relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander-Buchweitz approximation theory,and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules.In the second part of the paper we consider a depth equality,called the depth formula,which has been initially introduced by Auslander and developed further by Huneke and Wiegand.As an application of our main result,we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension.
基金Supported by National Natural Science Foundation of China (Grant No. 10961021)Program of Science and Technique of Gansu Province (Grant No. 1010RJZA025)
文摘We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes. Let X and Y be two classes of R-modules with Ext〉I(X,Y) = 0 for each object X ∈ X and each object Y ∈ Y. We show that if A,B ∈ C^(R) are X-complexes and U, V ∈ Cr(R) are Y-complexes, then U V Hom(A, U) Hom(A, Y); A B Hom(B, U) Hom(A, U). As an application, we give a sufficient condition for the Hom evaluation morphism being invertible.