The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M ∈ W H J^d（A） if and only if M admits a filtrat...The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M ∈ W H J^d（A） if and only if M admits a filtration of submodules： 0 belong to U0 belong to U1 belong to ... belong to Up = M such that all Ui/Ui-1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in W H J^d（A） in a special case. Let M ∈ W H J^d（A）. It is proved that the Koszul dual E（M） is Noetherian, Hopfian, of finite dimension in special cases, and E（M） ∈ gr0（E（A））. In particular, we show that M ∈ W H J^d（A） if and only if E（G（M）） ∈ gr0（E（A））, where G is the associated graded functor.展开更多
Given any integers a,b,c, and d with a 〉 1, c ≥ 0, b ≥ a＋c, and d ≥ b ＋ c, the notion of （a, b, c, d）-Koszul algebra is introduced, which is another class of standard graded algebras with ＂nonpure＂ resolutio...Given any integers a,b,c, and d with a 〉 1, c ≥ 0, b ≥ a＋c, and d ≥ b ＋ c, the notion of （a, b, c, d）-Koszul algebra is introduced, which is another class of standard graded algebras with ＂nonpure＂ resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of （a, b, c, d）-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be （a, b, c, d）- Koszul are provided.展开更多
For a graded algebra,the minimal projective resolution often reveals amounts of information.All generated degrees of modules in the minimal resolution of the trivial module form a sequence,which can be called the degr...For a graded algebra,the minimal projective resolution often reveals amounts of information.All generated degrees of modules in the minimal resolution of the trivial module form a sequence,which can be called the degree distribution of the algebra.We try to find lower and upper bounds of the degree distribution,introduce the notion of(s,t)-(homogeneous) determined algebras and construct such algebras with the aid of algebras with pure resolutions.In some cases,the Ext-algebra of an(s,t)-(homogeneous) determined algebra is finitely generated.展开更多
It is a small step toward the Koszul-type algebras.The piecewise-Koszul algebras are, in general,a new class of quadratic algebras but not the classical Koszul ones,simultaneously they agree with both the classical Ko...It is a small step toward the Koszul-type algebras.The piecewise-Koszul algebras are, in general,a new class of quadratic algebras but not the classical Koszul ones,simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases.We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A),and show an A_∞-structure on E(A).Relations between Koszul algebras and piecewise-Koszul algebras are discussed.In particular,our results are related to the third question of Green-Marcos.展开更多
基金supported by National Natural Science Foundation of China(Grant No.10571152)
文摘The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M ∈ W H J^d（A） if and only if M admits a filtration of submodules： 0 belong to U0 belong to U1 belong to ... belong to Up = M such that all Ui/Ui-1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in W H J^d（A） in a special case. Let M ∈ W H J^d（A）. It is proved that the Koszul dual E（M） is Noetherian, Hopfian, of finite dimension in special cases, and E（M） ∈ gr0（E（A））. In particular, we show that M ∈ W H J^d（A） if and only if E（G（M）） ∈ gr0（E（A））, where G is the associated graded functor.
基金This work was supported by the National Natural Science Foundation of China （Grant Nos. 11571316, 11001245） and the Natural Science Foundation of Zhejiang Province （Grant No. LY16A010003）.
文摘Given any integers a,b,c, and d with a 〉 1, c ≥ 0, b ≥ a＋c, and d ≥ b ＋ c, the notion of （a, b, c, d）-Koszul algebra is introduced, which is another class of standard graded algebras with ＂nonpure＂ resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of （a, b, c, d）-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be （a, b, c, d）- Koszul are provided.
基金supported by National Natural Science Foundation of China(Grant Nos.11026106 and 10971188)National Natural Science Foundation of Zhejiang Province of China(Grant No.LQ12A01028)
文摘For a graded algebra,the minimal projective resolution often reveals amounts of information.All generated degrees of modules in the minimal resolution of the trivial module form a sequence,which can be called the degree distribution of the algebra.We try to find lower and upper bounds of the degree distribution,introduce the notion of(s,t)-(homogeneous) determined algebras and construct such algebras with the aid of algebras with pure resolutions.In some cases,the Ext-algebra of an(s,t)-(homogeneous) determined algebra is finitely generated.
基金The work was supported by the National Natural Science Foundation of China (Grant No.10571152)
文摘It is a small step toward the Koszul-type algebras.The piecewise-Koszul algebras are, in general,a new class of quadratic algebras but not the classical Koszul ones,simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases.We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A),and show an A_∞-structure on E(A).Relations between Koszul algebras and piecewise-Koszul algebras are discussed.In particular,our results are related to the third question of Green-Marcos.
