We investigate the behavior of the extension dimension of subcategories of abelian categories under recollements.LetΛ',Λ,Λ"be art in algebras such that(modΛ',mod A,modΛ")is a recollement,and let...We investigate the behavior of the extension dimension of subcategories of abelian categories under recollements.LetΛ',Λ,Λ"be art in algebras such that(modΛ',mod A,modΛ")is a recollement,and let D'and D"be subcategories of modΛand modΛ"respectively.For any n,m≥0,under some conditions,we get dimΩ^(k)(D)≤dimΩ^(n)(D')+dimΩ^(m)(D")+1,where k=max{m,n}and D is the subcategory of modΛglued by D'and D";moreover,we give a sufficient condition such that the converse inequality holds true.As applications,some results for Igusa-Todorov subcategories and syzygy finite sub categories are obtained.展开更多
Given an additive category C and an integer n≥2.The higher differential additive category consists of objects X in C equipped with an endomorphism ϵ_(X)satisfying ϵ_(X)^(n).Let R be a,finite-dimensional basic algebra...Given an additive category C and an integer n≥2.The higher differential additive category consists of objects X in C equipped with an endomorphism ϵ_(X)satisfying ϵ_(X)^(n).Let R be a,finite-dimensional basic algebra over an algebraically closed field and T the augmenting functor from the category of finitely generated left R-modules to that of finitely generated left R/(t^(n))-modules.It is proved that a finitely generated left R-module M isτ-rigid(respectively,(support)τ-tilting,almost completeτ-tilting)if and only if so is T(M)as a left R[t]/(t^(n))-module.Moreover,R isτm-selfinjective if and only if so is R[t]/(t^(n)).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 introduce the definition of n-star subcategories, which is a generalization of n-star modules and n-C-star modules. We give some characterizations of n-star subcategories, and prove that M is an n-P-...In this paper, we introduce the definition of n-star subcategories, which is a generalization of n-star modules and n-C-star modules. We give some characterizations of n-star subcategories, and prove that M is an n-P-tilting subcategory with respect to cotorsion triple(P, R-Mod, I), if and only if M is an n-star subcategory with I ■Pres^(n)(M), where P denotes the subcategory of projective left R-modules and I denotes the subcategory of injective left R-modules.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.
Let l be a Krull-Schmidt n-exangulated category and A be an n-extension closed subcategory of l.Then A inherits the n-exangulated structure from the given n-exangulated category in a natural way.This construction give...Let l be a Krull-Schmidt n-exangulated category and A be an n-extension closed subcategory of l.Then A inherits the n-exangulated structure from the given n-exangulated category in a natural way.This construction gives n-exangulated categories which are neither n-exact categories in the sense of Jasso nor(n+2)-angulated categories in the sense of Geiss-Keller-Oppermann in general.Furthermore,we also give a sufficient condition on when an n-exangulated category A is an n-exact category.These results generalize work by Klapproth and Zhou.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
The representations of the Dynkin quivers and the corresponding Euclidean quivers are treated in many books. These notes provide three building blocks for dealing with representations of Dynkin （and Euclidean） quive...The representations of the Dynkin quivers and the corresponding Euclidean quivers are treated in many books. These notes provide three building blocks for dealing with representations of Dynkin （and Euclidean） quivers. They should be helpful as part of a direct approach to study representations of quivers, and they shed some new light on properties of Dynkin and Euclidean quivers.展开更多
目的探讨甲状腺BethesdaⅢ类(AUS/FLUS)结节的诊断原因,以及亚分类在预测结节恶性风险(risk of malignancy,ROM)中的价值。方法收集356例BethesdaⅢ结节患者,对其诊断原因,ROM及亚分类进行总结分析。结果在97例手术切除标本中,72例恶性...目的探讨甲状腺BethesdaⅢ类(AUS/FLUS)结节的诊断原因,以及亚分类在预测结节恶性风险(risk of malignancy,ROM)中的价值。方法收集356例BethesdaⅢ结节患者,对其诊断原因,ROM及亚分类进行总结分析。结果在97例手术切除标本中,72例恶性肿瘤均为甲状腺乳头状癌(papillary thyroid carcinoma,PTC),BethesdaⅢ类的ROM为74.2%。影响PTC诊断的主要原因有病灶小、穿刺细胞量稀少、缺乏乳头状结构及细胞核特征不典型;次要原因有间质显著纤维化或钙化、涂片不合格、固定不当、染色不佳及细胞学诊断经验欠缺等。BethesdaⅢ类的亚分类:132例为低风险组,其中12例手术切除,ROM为8.3%;122例为高风险组,其中70例手术切除,ROM为92.9%;102例为中风险组,其中15例手术切除,ROM为40.0%;高风险组和低/中风险组之间的差异有统计学意义(P<0.05)。结论BethesdaⅢ类的诊断具有一定的主观性和经验性,而对BethesdaⅢ类结节进行风险相关的亚分类,有助于实现更好的ROM分层并改善此类病变的临床管理。展开更多
Let M be an n-cluster tilting subcategory of mod-Λ,whereΛis an Artin algebra.Let S(M)denote the full subcategory of S(Λ),the submodule category of Λ,consisting of all the monomorphisms in M.We construct two functo...Let M be an n-cluster tilting subcategory of mod-Λ,whereΛis an Artin algebra.Let S(M)denote the full subcategory of S(Λ),the submodule category of Λ,consisting of all the monomorphisms in M.We construct two functors from S(M)to mod-Λ,the category of finitely presented additive contravariant functors on the stable category of M.We show that these functors are full,dense and objective and hence provide equivalences between the quotient categories of S(M)and mod-Λ.We also compare these two functors and show that they differ by the n-th syzygy functor,provided M is an n Z-cluster tilting subcategory.These functors can be considered as higher versions of the two functors studied by Ringel and Zhang(2014)in the case Λ=k[x]/and generalized later by Eiríksson(2017)to self-injective Artin algebras.Several applications are provided.展开更多
We build a connection between iterated tilted algebras with trivial cluster tilting subcategories and tilted algebras of finite type.Moreover,all tilted algebras with cluster tilting subcategories are determined in te...We build a connection between iterated tilted algebras with trivial cluster tilting subcategories and tilted algebras of finite type.Moreover,all tilted algebras with cluster tilting subcategories are determined in terms of quivers.As a result,we draw the quivers of Auslander's 1-Gorenstein algebras with global dimension 2 admitting trivial cluster tilting subcategories,which implies that such algebras are of finite type but not necessarily Nakayama.展开更多
The category consisting of finitely generated modules which are left orthogonal with a cotilting bimodule is shown to be functorially finite. The notion of left orthogonal dimension is introduced , and then a necessar...The category consisting of finitely generated modules which are left orthogonal with a cotilting bimodule is shown to be functorially finite. The notion of left orthogonal dimension is introduced , and then a necessary and sufficient condition of selforthogonal modules having finite injective dimension and a characterization of cotilting modules are given.展开更多
认知者往往依据陌生个体面孔所携带的性别、年龄、种族等多重社会范畴信息对其进行加工,以期快速识别和了解他人。在基于面孔识别的多重社会范畴加工过程中,亚范畴间存在着复杂的交互作用。研究者分别采用"Who Said What"范...认知者往往依据陌生个体面孔所携带的性别、年龄、种族等多重社会范畴信息对其进行加工,以期快速识别和了解他人。在基于面孔识别的多重社会范畴加工过程中,亚范畴间存在着复杂的交互作用。研究者分别采用"Who Said What"范式、重复启动范式、加纳选择注意范式、鼠标追踪范式等方法,发现亚范畴间的内隐加工具有彼此削弱的特性,外显加工存在交互影响的不对称性和偏差性。动态交互理论对此进行了进一步的理论分析与阐释。今后需更加科学地区分社会范畴加工的各个阶段,凸显内隐和外显加工的区别与联系;同时进一步整合各研究范式,克服方法异质导致的结果偏差甚至矛盾。展开更多
基金Supported by NSFC(Grant Nos.11971225,12171207,12001168)Henan University of Engineering(Grant Nos.DKJ2019010,XTYR-2021JZ001)the Key Research Project of Education Department of Henan Province(Grant No.21A110006)。
文摘We investigate the behavior of the extension dimension of subcategories of abelian categories under recollements.LetΛ',Λ,Λ"be art in algebras such that(modΛ',mod A,modΛ")is a recollement,and let D'and D"be subcategories of modΛand modΛ"respectively.For any n,m≥0,under some conditions,we get dimΩ^(k)(D)≤dimΩ^(n)(D')+dimΩ^(m)(D")+1,where k=max{m,n}and D is the subcategory of modΛglued by D'and D";moreover,we give a sufficient condition such that the converse inequality holds true.As applications,some results for Igusa-Todorov subcategories and syzygy finite sub categories are obtained.
基金Supported by NSFC(Grant Nos.12371038,11971225,12171207,12061026)NSF of Guangxi Province of China(Grant No.2020GXNSFAA159120)。
文摘Given an additive category C and an integer n≥2.The higher differential additive category consists of objects X in C equipped with an endomorphism ϵ_(X)satisfying ϵ_(X)^(n).Let R be a,finite-dimensional basic algebra over an algebraically closed field and T the augmenting functor from the category of finitely generated left R-modules to that of finitely generated left R/(t^(n))-modules.It is proved that a finitely generated left R-module M isτ-rigid(respectively,(support)τ-tilting,almost completeτ-tilting)if and only if so is T(M)as a left R[t]/(t^(n))-module.Moreover,R isτm-selfinjective if and only if so is R[t]/(t^(n)).
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Supported by the 2020 Scientific Research Projects in Universities of Gansu Province (Grant No. 2020A-277)。
文摘In this paper, we introduce the definition of n-star subcategories, which is a generalization of n-star modules and n-C-star modules. We give some characterizations of n-star subcategories, and prove that M is an n-P-tilting subcategory with respect to cotorsion triple(P, R-Mod, I), if and only if M is an n-star subcategory with I ■Pres^(n)(M), where P denotes the subcategory of projective left R-modules and I denotes the subcategory of injective left R-modules.
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supported by the National Natural Science Foundation of China(Grant No.12171230)supported by the Hunan Provincial Natural Science Foundation of China(Grant No.2023JJ30008)。
文摘Let l be a Krull-Schmidt n-exangulated category and A be an n-extension closed subcategory of l.Then A inherits the n-exangulated structure from the given n-exangulated category in a natural way.This construction gives n-exangulated categories which are neither n-exact categories in the sense of Jasso nor(n+2)-angulated categories in the sense of Geiss-Keller-Oppermann in general.Furthermore,we also give a sufficient condition on when an n-exangulated category A is an n-exact category.These results generalize work by Klapproth and Zhou.
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.
文摘The representations of the Dynkin quivers and the corresponding Euclidean quivers are treated in many books. These notes provide three building blocks for dealing with representations of Dynkin （and Euclidean） quivers. They should be helpful as part of a direct approach to study representations of quivers, and they shed some new light on properties of Dynkin and Euclidean quivers.
文摘目的探讨甲状腺BethesdaⅢ类(AUS/FLUS)结节的诊断原因,以及亚分类在预测结节恶性风险(risk of malignancy,ROM)中的价值。方法收集356例BethesdaⅢ结节患者,对其诊断原因,ROM及亚分类进行总结分析。结果在97例手术切除标本中,72例恶性肿瘤均为甲状腺乳头状癌(papillary thyroid carcinoma,PTC),BethesdaⅢ类的ROM为74.2%。影响PTC诊断的主要原因有病灶小、穿刺细胞量稀少、缺乏乳头状结构及细胞核特征不典型;次要原因有间质显著纤维化或钙化、涂片不合格、固定不当、染色不佳及细胞学诊断经验欠缺等。BethesdaⅢ类的亚分类:132例为低风险组,其中12例手术切除,ROM为8.3%;122例为高风险组,其中70例手术切除,ROM为92.9%;102例为中风险组,其中15例手术切除,ROM为40.0%;高风险组和低/中风险组之间的差异有统计学意义(P<0.05)。结论BethesdaⅢ类的诊断具有一定的主观性和经验性,而对BethesdaⅢ类结节进行风险相关的亚分类,有助于实现更好的ROM分层并改善此类病变的临床管理。
基金supported by a grant from University of Isfahan。
文摘Let M be an n-cluster tilting subcategory of mod-Λ,whereΛis an Artin algebra.Let S(M)denote the full subcategory of S(Λ),the submodule category of Λ,consisting of all the monomorphisms in M.We construct two functors from S(M)to mod-Λ,the category of finitely presented additive contravariant functors on the stable category of M.We show that these functors are full,dense and objective and hence provide equivalences between the quotient categories of S(M)and mod-Λ.We also compare these two functors and show that they differ by the n-th syzygy functor,provided M is an n Z-cluster tilting subcategory.These functors can be considered as higher versions of the two functors studied by Ringel and Zhang(2014)in the case Λ=k[x]/and generalized later by Eiríksson(2017)to self-injective Artin algebras.Several applications are provided.
文摘We build a connection between iterated tilted algebras with trivial cluster tilting subcategories and tilted algebras of finite type.Moreover,all tilted algebras with cluster tilting subcategories are determined in terms of quivers.As a result,we draw the quivers of Auslander's 1-Gorenstein algebras with global dimension 2 admitting trivial cluster tilting subcategories,which implies that such algebras are of finite type but not necessarily Nakayama.
文摘The category consisting of finitely generated modules which are left orthogonal with a cotilting bimodule is shown to be functorially finite. The notion of left orthogonal dimension is introduced , and then a necessary and sufficient condition of selforthogonal modules having finite injective dimension and a characterization of cotilting modules are given.
文摘认知者往往依据陌生个体面孔所携带的性别、年龄、种族等多重社会范畴信息对其进行加工,以期快速识别和了解他人。在基于面孔识别的多重社会范畴加工过程中,亚范畴间存在着复杂的交互作用。研究者分别采用"Who Said What"范式、重复启动范式、加纳选择注意范式、鼠标追踪范式等方法,发现亚范畴间的内隐加工具有彼此削弱的特性,外显加工存在交互影响的不对称性和偏差性。动态交互理论对此进行了进一步的理论分析与阐释。今后需更加科学地区分社会范畴加工的各个阶段,凸显内隐和外显加工的区别与联系;同时进一步整合各研究范式,克服方法异质导致的结果偏差甚至矛盾。
