Relations between Corenstein derived categories, Gorenstein defect categories and Gorenstein stable categories are established. Using these, the Gorensteinness of an algebra A and invariants with respect to recolle- m...Relations between Corenstein derived categories, Gorenstein defect categories and Gorenstein stable categories are established. Using these, the Gorensteinness of an algebra A and invariants with respect to recolle- ments of the bounded Gorenstein derived category Dgbp（A-mod） of A are investigated. Specifically, the Goren- steinness of A is characterized in terms of recollements of Dbp（A-mod） and Corenstein derived equivalences. It is also shown that Cohen-Macaulay-finiteness is invariant with respect to the recollements of D^p（A-mod）.展开更多
The notions of quasi k-Gorenstein algebras and W^t-approximation representations are introduced. The existence and uniqueness (up to projective equivalences) of W^t-approximation representations over quasi k-Gorenstei...The notions of quasi k-Gorenstein algebras and W^t-approximation representations are introduced. The existence and uniqueness (up to projective equivalences) of W^t-approximation representations over quasi k-Gorenstein algebras are established. Some applications of W^t-approximation representations to homologically finite subcategories are given.展开更多
Let H be a semisimple weak Hopf algebra, A a left H-module algebra. We prove that the injective dimension of the regular A-module is equal to the one of A as the module over the smash product A#H,nd equal to the one o...Let H be a semisimple weak Hopf algebra, A a left H-module algebra. We prove that the injective dimension of the regular A-module is equal to the one of A as the module over the smash product A#H,nd equal to the one of the regular A#H-module. Also, we give a necessary and sufficient condition for A being a Gorenstein algebra, in terms of the fixed subalgebra of A under the action of H on A.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
We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian ...We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause＇s work [Math. Ann., 2012, 353： 765-781]. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived category of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.展开更多
Let R be an Artin algebra and e be an idempotent of R. Assume that Tor_(i)^(eRe)(Re, G) = 0 for any G ∈ GprojeRe and i sufficiently large. Necessary and sufficient conditions are given for the Schur functor S_(e) to ...Let R be an Artin algebra and e be an idempotent of R. Assume that Tor_(i)^(eRe)(Re, G) = 0 for any G ∈ GprojeRe and i sufficiently large. Necessary and sufficient conditions are given for the Schur functor S_(e) to induce a triangle-equivalence ■. Combining this with a result of Psaroudakis et al.(2014),we provide necessary and sufficient conditions for the singular equivalence ■ to restrict to a triangle-equivalence ■. Applying these to the triangular matrix algebra ■,corresponding results between candidate categories of T and A(resp. B) are obtained. As a consequence,we infer Gorensteinness and CM(Cohen-Macaulay)-freeness of T from those of A(resp. B). Some concrete examples are given to indicate that one can realize the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algebras.展开更多
Let A be an algebra of finite Cohen-Macaulay type and F its Cohen-Macaulay Auslander algebra. We are going to characterize the morphism category Mor（∧-Gproj） of Gorenstein-projective ∧-modules in terms of the modu...Let A be an algebra of finite Cohen-Macaulay type and F its Cohen-Macaulay Auslander algebra. We are going to characterize the morphism category Mor（∧-Gproj） of Gorenstein-projective ∧-modules in terms of the module category F-mod by a categorical equivalence. Based on this, we obtain that some factor category of the epimorphism category Epi（∧-Gproj） is a Frobenius category, and also, we clarify the relations among Mor（∧-Gproj）, Mor（T2（∧）-Gproj） and Mor（△-Gproj）, where T2（∧） and △ are respectively the lower triangular matrix algebra and the Morita ring closely related to ∧.展开更多
Let A be an Artinian algebra and F an additive subbifunctor of Ext,（-, -） having enough projectives and injectives. We prove that the dualizing subvarieties of mod A closed under F-extensions have F-almost split seq...Let A be an Artinian algebra and F an additive subbifunctor of Ext,（-, -） having enough projectives and injectives. We prove that the dualizing subvarieties of mod A closed under F-extensions have F-almost split sequences. Let T be an F-cotilting module in mod A and S a cotilting module over F = End（T）. Then Horn（-, T） induces a duality between F-almost split sequences in ⊥FT and almost sl31it sequences in ⊥S, where addrS = Hom∧（f（F）, T）. Let A be an F-Gorenstein algebra, T a strong F-cotilting module and 0→A→B→C→0 and F-almost split sequence in ⊥FT.If the injective dimension of S as a Г-module is equal to d, then C≌（ΩCM^-dΩ^dDTrA^＊）^＊,where（-）^＊=Hom（g,T）.In addition, if the F-injective dimension of A is equal to d, then A≌ΩMF^-dDΩFop^-d TrC≌ΩCMF^-d ≌F^d DTrC.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11101259)
文摘Relations between Corenstein derived categories, Gorenstein defect categories and Gorenstein stable categories are established. Using these, the Gorensteinness of an algebra A and invariants with respect to recolle- ments of the bounded Gorenstein derived category Dgbp（A-mod） of A are investigated. Specifically, the Goren- steinness of A is characterized in terms of recollements of Dbp（A-mod） and Corenstein derived equivalences. It is also shown that Cohen-Macaulay-finiteness is invariant with respect to the recollements of D^p（A-mod）.
文摘The notions of quasi k-Gorenstein algebras and W^t-approximation representations are introduced. The existence and uniqueness (up to projective equivalences) of W^t-approximation representations over quasi k-Gorenstein algebras are established. Some applications of W^t-approximation representations to homologically finite subcategories are given.
基金Acknowledgement I should like to thank Professor Zhang Pu＇s remarks and many helpful suggestions which improve the writing in English. the National Natural Science Foundation of China （No. 10301033 10271113）.
文摘Let H be a semisimple weak Hopf algebra, A a left H-module algebra. We prove that the injective dimension of the regular A-module is equal to the one of A as the module over the smash product A#H,nd equal to the one of the regular A#H-module. Also, we give a necessary and sufficient condition for A being a Gorenstein algebra, in terms of the fixed subalgebra of A under the action of H on A.
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.
基金Acknowledgements The author would like to thank his supervisor Professor Zhaoyong Huang, for the valuable help, suggestions, guidance, and encouragement during his studies and preparation of this paper. He also thanks the referees for their careful reading and for pointing out related references. This work was partially supported by the National Natural Science Foundation of China （Grant No. 11571164）.
文摘We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause＇s work [Math. Ann., 2012, 353： 765-781]. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived category of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.
基金supported by National Natural Science Foundation of China (Grant Nos. 11626179, 12101474, 12171206 and 11701455)Natural Science Foundation of Jiangsu Province (Grant No. BK20211358)+1 种基金Natural Science Basic Research Plan in Shaanxi Province of China (Grant Nos. 2017JQ1012 and 2020JM-178)Fundamental Research Funds for the Central Universities (Grant Nos. JB160703 and 2452020182)。
文摘Let R be an Artin algebra and e be an idempotent of R. Assume that Tor_(i)^(eRe)(Re, G) = 0 for any G ∈ GprojeRe and i sufficiently large. Necessary and sufficient conditions are given for the Schur functor S_(e) to induce a triangle-equivalence ■. Combining this with a result of Psaroudakis et al.(2014),we provide necessary and sufficient conditions for the singular equivalence ■ to restrict to a triangle-equivalence ■. Applying these to the triangular matrix algebra ■,corresponding results between candidate categories of T and A(resp. B) are obtained. As a consequence,we infer Gorensteinness and CM(Cohen-Macaulay)-freeness of T from those of A(resp. B). Some concrete examples are given to indicate that one can realize the Gorenstein defect category of a triangular matrix algebra as the singularity category of one of its corner algebras.
基金Supported by the National Natural Science Foundation of China （Grant No. 11771272）
文摘Let A be an algebra of finite Cohen-Macaulay type and F its Cohen-Macaulay Auslander algebra. We are going to characterize the morphism category Mor（∧-Gproj） of Gorenstein-projective ∧-modules in terms of the module category F-mod by a categorical equivalence. Based on this, we obtain that some factor category of the epimorphism category Epi（∧-Gproj） is a Frobenius category, and also, we clarify the relations among Mor（∧-Gproj）, Mor（T2（∧）-Gproj） and Mor（△-Gproj）, where T2（∧） and △ are respectively the lower triangular matrix algebra and the Morita ring closely related to ∧.
基金Partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20060284002)National Natural Science Foundation of China (Grant No. 10771095)National Natural Science Foundation of Jiangsu Province of China (Grant No. BK2007517)
文摘Let A be an Artinian algebra and F an additive subbifunctor of Ext,（-, -） having enough projectives and injectives. We prove that the dualizing subvarieties of mod A closed under F-extensions have F-almost split sequences. Let T be an F-cotilting module in mod A and S a cotilting module over F = End（T）. Then Horn（-, T） induces a duality between F-almost split sequences in ⊥FT and almost sl31it sequences in ⊥S, where addrS = Hom∧（f（F）, T）. Let A be an F-Gorenstein algebra, T a strong F-cotilting module and 0→A→B→C→0 and F-almost split sequence in ⊥FT.If the injective dimension of S as a Г-module is equal to d, then C≌（ΩCM^-dΩ^dDTrA^＊）^＊,where（-）^＊=Hom（g,T）.In addition, if the F-injective dimension of A is equal to d, then A≌ΩMF^-dDΩFop^-d TrC≌ΩCMF^-d ≌F^d DTrC.
