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
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
In this article,we investigate the representations of the Drinfeld doubles D(Rmn(q))of the Radford Hopf algebras Rmn(q)over an algebraically closed field k,where m>1 and n>1 are integers and q∈k is a root of un...In this article,we investigate the representations of the Drinfeld doubles D(Rmn(q))of the Radford Hopf algebras Rmn(q)over an algebraically closed field k,where m>1 and n>1 are integers and q∈k is a root of unity of order n.Under the assumption char(k)■mn,all the finite-dimensional indecomposable modules over D(Rmn(q))are displayed and classified up to isomorphism.The Auslander-Reiten sequences in the category of finite-dimensional D(Rmn(q))-modules are also all displayed.It is shown that D(Rmn(q))is of tame representation type.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
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.
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
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
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
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.
基金supported by National Natural Science Foundation of China(Grant Nos.12201545 and 12071412)。
文摘In this article,we investigate the representations of the Drinfeld doubles D(Rmn(q))of the Radford Hopf algebras Rmn(q)over an algebraically closed field k,where m>1 and n>1 are integers and q∈k is a root of unity of order n.Under the assumption char(k)■mn,all the finite-dimensional indecomposable modules over D(Rmn(q))are displayed and classified up to isomorphism.The Auslander-Reiten sequences in the category of finite-dimensional D(Rmn(q))-modules are also all displayed.It is shown that D(Rmn(q))is of tame representation type.
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.
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,
