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 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
Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories. In this paper, we develop the relative homology aspect of Ausl...Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories. In this paper, we develop the relative homology aspect of Auslander bijection in extriangulated categories. Namely, we introduce the notion of generalized ARS-duality relative to an additive subfunctor, and prove that there is a bijective triangle which involves the generalized ARS-duality and the restricted Auslander bijection relative to the subfunctor. We give the Auslander’s defect formula in terms of the generalized ARS-duality, show the interplay of morphisms being determined by objects and a kind of extriangles, and characterize a class of objects which are somehow controlled by this kind of extriangles. We also give a realization for a functor relating the Auslander bijection and the generalized ARS-duality.展开更多
In this paper, the authors introduce a new definition of ∞-tilting(resp. cotilting) subcategories with infinite projective dimensions(resp. injective dimensions) in an extriangulated category. They give a Bazzoni cha...In this paper, the authors introduce a new definition of ∞-tilting(resp. cotilting) subcategories with infinite projective dimensions(resp. injective dimensions) in an extriangulated category. They give a Bazzoni characterization of ∞-tilting(resp. cotilting)subcategories. Also, they obtain a partial Auslander-Reiten correspondence between ∞-tilting(resp. cotilting) subcategories and coresolving(resp. resolving) subcategories with an E-projective generator(resp. E-injective cogenerator) in an extriangulated category.展开更多
Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulate...Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulated category is defined in this paper.We give a Bazzoni characterization of tilting(resp.,cotilting)subcategories and obtain an Auslander-Reiten correspondence between tilting(resp.,cotilting)subcategories and coresolving covariantly(resp.,resolving contravariantly)finite subcatgories which are closed under direct summands and satisfy some cogenerating(resp.,generating)conditions.Applications of the results are given:we show that tilting(resp.,cotilting)subcategories defined here unify many previous works about tilting modules(subcategories)in module categories of Artin algebras and in abelian categories admitting a cotorsion triples;we also show that the results work for the triangulated categories with a proper class of triangles introduced by A.Beligiannis.展开更多
Let(C,E,s)be an extriangulated category with a proper classξof E-triangles,and W an additive full subcategory of(C,E,s).We provide a method for constructing a proper W(ξ)-resolution(resp.,coproper W(ξ)-coresolution...Let(C,E,s)be an extriangulated category with a proper classξof E-triangles,and W an additive full subcategory of(C,E,s).We provide a method for constructing a proper W(ξ)-resolution(resp.,coproper W(ξ)-coresolution)of one term in an E-triangle inξfrom that of the other two terms.By using this way,we establish the stability of the Gorenstein category GW(ξ)in extriangulated categories.These results generalize the work of Z.Y.Huang[J.Algebra,2013,393:142–169]and X.Y.Yang and Z.C.Wang[Rocky Mountain J.Math.,2017,47:1013–1053],but the proof is not too far from their case.Finally,we give some applications about our main results.展开更多
Let C be an extriangulated category and T be any n-cluster tilting subcategory of C.We consider the index with respect to T and introduce the index Grothendieck group of T.Using the index,we prove that the index Groth...Let C be an extriangulated category and T be any n-cluster tilting subcategory of C.We consider the index with respect to T and introduce the index Grothendieck group of T.Using the index,we prove that the index Grothendieck group of T is isomorphic to the Grothendieck group of C,which implies that the index Grothendieck groups of any two n-cluster tilting subcategories are isomorphic.In particular,we show that the split Grothendieck groups of any two 2-cluster tilting subcategories are isomorphic.Then we develop a general framework for c-vectors of 2-Calabi-Yau extriangulated categories with respect to arbitrary 2-cluster tilting subcategories.We show that the c-vectors have the sign-coherence property and provide some formulas for calculating c-vectors.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
Extriangulated categories were introduced by Nakaoka and Palu via extracting the similarities between exact categories and triangulated categories.In this article we introduce the notion of ζ-tilting objects in an ex...Extriangulated categories were introduced by Nakaoka and Palu via extracting the similarities between exact categories and triangulated categories.In this article we introduce the notion of ζ-tilting objects in an extriangulated category,where ζ is a proper class of E-triangles.Our results extend the relative tilting theory in extriangulated categories.展开更多
Let(A,B,C)be a recollement of extriangulated categories.In this paper we introduce the global dimension and extension dimension of extriangulated categories,and give some upper bounds of global dimensions and extensio...Let(A,B,C)be a recollement of extriangulated categories.In this paper we introduce the global dimension and extension dimension of extriangulated categories,and give some upper bounds of global dimensions and extension dimensions of the categories involved in(A,B,C),which give a simultaneous generalization of some results in recollements of abelian categories and triangulated categories.展开更多
Let(C,E,s)be an extriangulated category with a proper classξof E-triangles.We study complete cohomology of objects in(C,E,s)by applyingξ-projective resolutions andξ-injective coresolutions constructed in(C,E,s).Van...Let(C,E,s)be an extriangulated category with a proper classξof E-triangles.We study complete cohomology of objects in(C,E,s)by applyingξ-projective resolutions andξ-injective coresolutions constructed in(C,E,s).Vanishing of complete cohomology detects objects with finiteξ-projective dimension and finiteξ-injective dimension.As a consequence,we obtain some criteria for the validity of the Wakamatsu tilting conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein.Moreover,we give a general technique for computing complete cohomology of objects with finiteξ-Gprojective dimension.As an application,the relations betweenξ-projective dimension andξ-Gprojective dimension for objects in(C,E,s)are given.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
Let C be a triangulated category.We first introduce the notion of balanced pairs in C,and then establish the bijective correspondence between balanced pairs and proper classesξwith enoughξ-projectives andξ-injectiv...Let C be a triangulated category.We first introduce the notion of balanced pairs in C,and then establish the bijective correspondence between balanced pairs and proper classesξwith enoughξ-projectives andξ-injectives.Assume thatξ:=ξX=ξ^(Y) is the proper class induced by a balanced pair(X,Y).We prove that(C,Eξ,sξ)is an extriangulated category.Moreover,it is proved that(C,Eξ,sξ)is a triangulated category if and only if X=Y=0,and that(C,Eξ,sξ)is an exact category if and only if X=Y=C.As an application,we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.展开更多
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 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 NSFC(Grant No.11901341)Shandong Provincial Natural Science Foundation(Grant No.ZR2021QA001)Youth Innovation Team of Universities of Shandong Province(Grant No.2022KJ314)。
文摘Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories. In this paper, we develop the relative homology aspect of Auslander bijection in extriangulated categories. Namely, we introduce the notion of generalized ARS-duality relative to an additive subfunctor, and prove that there is a bijective triangle which involves the generalized ARS-duality and the restricted Auslander bijection relative to the subfunctor. We give the Auslander’s defect formula in terms of the generalized ARS-duality, show the interplay of morphisms being determined by objects and a kind of extriangles, and characterize a class of objects which are somehow controlled by this kind of extriangles. We also give a realization for a functor relating the Auslander bijection and the generalized ARS-duality.
基金supported by the National Natural Science Foundation of China(Nos.12101344,11371196)the Shan Dong Provincial Natural Science Foundation of China(No.ZR2015PA001).
文摘In this paper, the authors introduce a new definition of ∞-tilting(resp. cotilting) subcategories with infinite projective dimensions(resp. injective dimensions) in an extriangulated category. They give a Bazzoni characterization of ∞-tilting(resp. cotilting)subcategories. Also, they obtain a partial Auslander-Reiten correspondence between ∞-tilting(resp. cotilting) subcategories and coresolving(resp. resolving) subcategories with an E-projective generator(resp. E-injective cogenerator) in an extriangulated category.
基金the National Natural Science Foundation of China(Grant No.11671221).
文摘Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulated category is defined in this paper.We give a Bazzoni characterization of tilting(resp.,cotilting)subcategories and obtain an Auslander-Reiten correspondence between tilting(resp.,cotilting)subcategories and coresolving covariantly(resp.,resolving contravariantly)finite subcatgories which are closed under direct summands and satisfy some cogenerating(resp.,generating)conditions.Applications of the results are given:we show that tilting(resp.,cotilting)subcategories defined here unify many previous works about tilting modules(subcategories)in module categories of Artin algebras and in abelian categories admitting a cotorsion triples;we also show that the results work for the triangulated categories with a proper class of triangles introduced by A.Beligiannis.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11771212,11901190,11671221)Qing Lan Project of Jiangsu Province,Jiangsu Government Scholarship for Overseas Studies(Grant No.JS-2019-328)+1 种基金Hunan Provincial Natural Science Foundation of China(Grant No.2018JJ3205)Scientific Research Fund of Hunan Provincial Education Department(Grant No.19B239).
文摘Let(C,E,s)be an extriangulated category with a proper classξof E-triangles,and W an additive full subcategory of(C,E,s).We provide a method for constructing a proper W(ξ)-resolution(resp.,coproper W(ξ)-coresolution)of one term in an E-triangle inξfrom that of the other two terms.By using this way,we establish the stability of the Gorenstein category GW(ξ)in extriangulated categories.These results generalize the work of Z.Y.Huang[J.Algebra,2013,393:142–169]and X.Y.Yang and Z.C.Wang[Rocky Mountain J.Math.,2017,47:1013–1053],but the proof is not too far from their case.Finally,we give some applications about our main results.
基金supported by National Natural Science Foundation of China(Grant No.12271257)。
文摘Let C be an extriangulated category and T be any n-cluster tilting subcategory of C.We consider the index with respect to T and introduce the index Grothendieck group of T.Using the index,we prove that the index Grothendieck group of T is isomorphic to the Grothendieck group of C,which implies that the index Grothendieck groups of any two n-cluster tilting subcategories are isomorphic.In particular,we show that the split Grothendieck groups of any two 2-cluster tilting subcategories are isomorphic.Then we develop a general framework for c-vectors of 2-Calabi-Yau extriangulated categories with respect to arbitrary 2-cluster tilting subcategories.We show that the c-vectors have the sign-coherence property and provide some formulas for calculating c-vectors.
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Supported by the Natural Science Foundation of China(Grant No.11771212)a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘Extriangulated categories were introduced by Nakaoka and Palu via extracting the similarities between exact categories and triangulated categories.In this article we introduce the notion of ζ-tilting objects in an extriangulated category,where ζ is a proper class of E-triangles.Our results extend the relative tilting theory in extriangulated categories.
基金This work was supported by the NSFC(12001168)Henan University of Engineering(DKJ2019010)+1 种基金the Key Research Project of Education Department of Henan Province(21A110006)the project ZR2021QA001 of Shandong Provincial Natural Science Foundation.
文摘Let(A,B,C)be a recollement of extriangulated categories.In this paper we introduce the global dimension and extension dimension of extriangulated categories,and give some upper bounds of global dimensions and extension dimensions of the categories involved in(A,B,C),which give a simultaneous generalization of some results in recollements of abelian categories and triangulated categories.
基金supported by the NSF of China(11671069,11771212)Qing Lan Project of Jiangsu Province and Natural Science Foundation of Jiangsu Province(BK20211358)+4 种基金supported by the NSF of China(11971225,11901341)Shandong Provincial Natural Science Foundation(ZR2019QA015)supported by the National Natural Science Foundation of China(11901190,11671221)the Hunan Provincial Natural Science Foundation of China(2018JJ3205)the Scientific Research Fund of Hunan Provincial Education Department(19B239).
文摘Let(C,E,s)be an extriangulated category with a proper classξof E-triangles.We study complete cohomology of objects in(C,E,s)by applyingξ-projective resolutions andξ-injective coresolutions constructed in(C,E,s).Vanishing of complete cohomology detects objects with finiteξ-projective dimension and finiteξ-injective dimension.As a consequence,we obtain some criteria for the validity of the Wakamatsu tilting conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein.Moreover,we give a general technique for computing complete cohomology of objects with finiteξ-Gprojective dimension.As an application,the relations betweenξ-projective dimension andξ-Gprojective dimension for objects in(C,E,s)are given.
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Xianhui Fu was supported by YDZJ202101ZYTS168 and the NSF of China(12071064)Jiangsheng Hu was supported by the NSF of China(12171206)+2 种基金the Natural Science Foundation of Jiangsu Province(BK20211358)Haiyan Zhu was supported by Zhejiang Provincial Natural Science Foundation of China(LY18A010032)the NSF of China(12271481).
文摘Let C be a triangulated category.We first introduce the notion of balanced pairs in C,and then establish the bijective correspondence between balanced pairs and proper classesξwith enoughξ-projectives andξ-injectives.Assume thatξ:=ξX=ξ^(Y) is the proper class induced by a balanced pair(X,Y).We prove that(C,Eξ,sξ)is an extriangulated category.Moreover,it is proved that(C,Eξ,sξ)is a triangulated category if and only if X=Y=0,and that(C,Eξ,sξ)is an exact category if and only if X=Y=C.As an application,we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.
