When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that ...When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras.When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness.For a finite connected DG algebra A, we prove that Dc(A) and Dc(A op) admit Auslander-Reiten triangles if and only if A and A op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, Dc(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D lf b (A) and D lf b (A op) instead, when A is a regular DG algebra.展开更多
Let Λ be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain sub...Let Λ be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category S(Gprj-Λ) containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category Gprj. In particular, for the finite components, we show that under certain mild conditions,their cardinalities are divisible by 3. We see that this three-periodicity phenomenon reoccurs several times in the paper.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
Bocs, which is the abbreviated form of bimodule over a categary with coalgebra structure, was introduced by Kleiner and Rojter in 1975 and developed by Drozd in 1979, then formulated by Crawley-Boevey in 1988. Let k b...Bocs, which is the abbreviated form of bimodule over a categary with coalgebra structure, was introduced by Kleiner and Rojter in 1975 and developed by Drozd in 1979, then formulated by Crawley-Boevey in 1988. Let k be an algebraically closed field, A a finitely dimensional k-algebra. Then there exists a bocs B over k associated to A. From this relation Drozd proved one of the most important theorems in representation theory of algebra, namely, a finitely dimensional k-algebra is either of representation tame type or of representation wild type,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
We give sufficient conditions and necessary conditions on duality preservability of Auslander-Reiten quivers of derived categories and cluster categories over hereditary algebras. of generalized path algebras as cleft...We give sufficient conditions and necessary conditions on duality preservability of Auslander-Reiten quivers of derived categories and cluster categories over hereditary algebras. of generalized path algebras as cleft Meantime, we characterize the condition extensions of path algebras.展开更多
We single out a class of translation quivers and prove that the translation quivers in this class are generalized multicoils. These generalized multicoils form a class of special generalized multicoils. They are easie...We single out a class of translation quivers and prove that the translation quivers in this class are generalized multicoils. These generalized multicoils form a class of special generalized multicoils. They are easier to visualize, but still show all the strange behaviours of generalized multicoils, and contain quasi-stable tubes as special examples.展开更多
Let A be a finite dimensional, connected, basic algebra over an algebraically closed field. We prove that A is of finite representation type if and only if there is a natural number m such that rad^m（End（M）） = 0, ...Let A be a finite dimensional, connected, basic algebra over an algebraically closed field. We prove that A is of finite representation type if and only if there is a natural number m such that rad^m（End（M）） = 0, for any indecomposable A-modules M. This gives a partial answer to one of problems posed by Skowrofiski.展开更多
The relative transpose via Gorenstein projective modules is introduced,and some corresponding results on the Auslander-Reiten sequences and the Auslander-Reiten formula to this relative version are generalized.
The aim of the paper is to classify the indecomposable modules and describe the Auslander-Reiten sequences for the admissible algebras with formal two-ray modules.
Let T(A)=A?D(A) be a representation-finite trivial extension algebra. We haveproved: (i) every indecomposable module of T(A) is determined by its top and socle, (ii)every indecomposable module of T(A) is determined by...Let T(A)=A?D(A) be a representation-finite trivial extension algebra. We haveproved: (i) every indecomposable module of T(A) is determined by its top and socle, (ii)every indecomposable module of T(A) is determined by its Loewy factors and socal factorsrespectively.展开更多
We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis,and the corresponding irreducible el...We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis,and the corresponding irreducible elements forms a PBW type basis of the Ringel-Hall algebra.We aim to generalize this result to the derived Hall algebra DH(An)of type An.First,we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category D^b(An)using the Auslander-Reiten quiver of D^b(An),and then we prove that all possible compositions between these skew commutator relations are trivial.As an application,we give a PBW type basis of DH(An).展开更多
Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projecti...Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra展开更多
基金supported by the National Natural Science Foundation of China (Grant No. 10731070)the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)
文摘When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras.When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness.For a finite connected DG algebra A, we prove that Dc(A) and Dc(A op) admit Auslander-Reiten triangles if and only if A and A op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, Dc(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D lf b (A) and D lf b (A op) instead, when A is a regular DG algebra.
基金supported by National Natural Science Foundation of China (Grant No. 12101316)。
文摘Let Λ be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category S(Gprj-Λ) containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category Gprj. In particular, for the finite components, we show that under certain mild conditions,their cardinalities are divisible by 3. We see that this three-periodicity phenomenon reoccurs several times in the paper.
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.
文摘Bocs, which is the abbreviated form of bimodule over a categary with coalgebra structure, was introduced by Kleiner and Rojter in 1975 and developed by Drozd in 1979, then formulated by Crawley-Boevey in 1988. Let k be an algebraically closed field, A a finitely dimensional k-algebra. Then there exists a bocs B over k associated to A. From this relation Drozd proved one of the most important theorems in representation theory of algebra, namely, a finitely dimensional k-algebra is either of representation tame type or of representation wild type,
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 by the National Natural Science Foundation of China （Grant Nos. 11271318, 11571173） and the Natural Science Foundation of Zhejiang Province （No. LZ13A010001）.
文摘We give sufficient conditions and necessary conditions on duality preservability of Auslander-Reiten quivers of derived categories and cluster categories over hereditary algebras. of generalized path algebras as cleft Meantime, we characterize the condition extensions of path algebras.
文摘We single out a class of translation quivers and prove that the translation quivers in this class are generalized multicoils. These generalized multicoils form a class of special generalized multicoils. They are easier to visualize, but still show all the strange behaviours of generalized multicoils, and contain quasi-stable tubes as special examples.
基金Supported by the Education Department Foundation of Hunan Province (Grant No04C469)
文摘Let A be a finite dimensional, connected, basic algebra over an algebraically closed field. We prove that A is of finite representation type if and only if there is a natural number m such that rad^m（End（M）） = 0, for any indecomposable A-modules M. This gives a partial answer to one of problems posed by Skowrofiski.
基金supported by the National Natural Science Foundation of China (No. 10725104)the Natural Science Foundation of Shanghai (No. ZR0614049)the Shanghai Leading Academic Discipline Project(No. S30104).
文摘The relative transpose via Gorenstein projective modules is introduced,and some corresponding results on the Auslander-Reiten sequences and the Auslander-Reiten formula to this relative version are generalized.
基金The author gratefully acknowledges the support from the Schweizerischer Nationalfonds and the Polish Scientific Grant KBN No.1 P03A 018 27
文摘The aim of the paper is to classify the indecomposable modules and describe the Auslander-Reiten sequences for the admissible algebras with formal two-ray modules.
基金Project supported by the National Natural Science Foundation of China.
文摘Let T(A)=A?D(A) be a representation-finite trivial extension algebra. We haveproved: (i) every indecomposable module of T(A) is determined by its top and socle, (ii)every indecomposable module of T(A) is determined by its Loewy factors and socal factorsrespectively.
基金Supported by the Natural Science Foundation of China(Grant No.11861061)。
文摘We know that in Ringel-Hall algebra of Dynkin type,the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Grobner-Shirshov basis,and the corresponding irreducible elements forms a PBW type basis of the Ringel-Hall algebra.We aim to generalize this result to the derived Hall algebra DH(An)of type An.First,we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category D^b(An)using the Auslander-Reiten quiver of D^b(An),and then we prove that all possible compositions between these skew commutator relations are trivial.As an application,we give a PBW type basis of DH(An).
基金supported by National Natural Science Foundation of China(Grant No.11271257)National Science Foundation of Shanghai Municiple(Granted No.13ZR1422500)
文摘Let A be a finite-dimensional algebra over an algebraically closed field k,ε the category of all exact sequences in A-rood, Mp （respectively, Ml） the full subcategory of C consisting of those objects with projective （respectively, injective） middle terms. It is proved that Mp （respectively, MI） is contravariantly finite （respectively, covariantly finite） in ε. As an application, it is shown that Mp = MI is functorially finite and has Auslander-Reiten sequences provided A is selfinjective. Keywords category of exact sequences, contravariantly finite subcategory, functorially finite subcategory Auslander-Reiten sequences, selfinjective algebra
