In this article,we obtain an upper bound for the regularity of the binomial edge ideal of a graph whose every block is either a cycle or a clique.As a consequence,we obtain an upper bound for the regularity of binomia...In this article,we obtain an upper bound for the regularity of the binomial edge ideal of a graph whose every block is either a cycle or a clique.As a consequence,we obtain an upper bound for the regularity of binomial edge ideal of a cactus graph.We also identify a certain subclass attaining the upper bound.展开更多
In this note,we will generalize some results about the Cohen-Macaulaymess and Goronsteinness of Rees rings and associated graded rings of ideals having higher analytic deviation.
A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z_(2)-graded algebra E(θ),which corresponds to a noncommutative quadric hypersurface E^(!)/(z)for some central regular element z∈E_(2)^(...A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z_(2)-graded algebra E(θ),which corresponds to a noncommutative quadric hypersurface E^(!)/(z)for some central regular element z∈E_(2)^(!).It turns out that the bounded derived category D^(b)(gr_(Z_(2))E(θ))is equivalent to the stable category of the maximal Cohen-Macaulay modules over E^(!)/(z)provided that E!is noetherian.As a consequence,E^(!)/(z)is a noncommutative isolated singularity if and only if the corresponding Clifford deformation E(θ)is a semisimple Z_(2)-graded algebra.The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces.As an application,we recover Knörrer's periodicity theorem without using matrix factorizations.展开更多
Let R and S be Artin algebras and F be their triangular matrix extension via a bimodule sMR. We study totally acyclic complexes of projective F-modules and obtain a complete description of Gorenstein projective F-modu...Let R and S be Artin algebras and F be their triangular matrix extension via a bimodule sMR. We study totally acyclic complexes of projective F-modules and obtain a complete description of Gorenstein projective F-modules. We then construct some examples of Cohen-Macaulay finite and virtually Gorenstein triangular matrix algebras.展开更多
Let R=K[x1,…,xn]be the polynomial ring in n variables over a field K and I be a matroidal ideal of R.We show that I is sequentially Cohen-Macaulay if and only if the Alexander dual Iy has linear quotients.As a conseq...Let R=K[x1,…,xn]be the polynomial ring in n variables over a field K and I be a matroidal ideal of R.We show that I is sequentially Cohen-Macaulay if and only if the Alexander dual Iy has linear quotients.As a consequence,I is sequentially Cohen-Macaulay if and only if I is shellable.展开更多
Let A and B be Artin R-algebras of finite Cohen-Macaulay type. Then we prove that, if A and B are standard derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived equivalent. And we show that...Let A and B be Artin R-algebras of finite Cohen-Macaulay type. Then we prove that, if A and B are standard derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived equivalent. And we show that Gorenstein projective conjecture is an invariant under standard derived equivalence between Artin R-algebras.展开更多
We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen- Macaulay. As an application, we give a complete charac...We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen- Macaulay. As an application, we give a complete characterization of the completely square- free lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.展开更多
Let （R,m） be a Cohen-Macaulay local ring of dimension d with infinite residue field, I an m-primary ideal and K an ideal containing I. Let J be a minimal reduction of I such that, for some positive integer k, KIn ∩...Let （R,m） be a Cohen-Macaulay local ring of dimension d with infinite residue field, I an m-primary ideal and K an ideal containing I. Let J be a minimal reduction of I such that, for some positive integer k, KIn ∩ J = JKIn-1 for n ≤ k ？ 1 and λ（ JKKIIkk-1 ） = 1. We show that if depth G（I） ≥ d-2, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of FK（I） assuming that depth G（I） ≥ d - 1.展开更多
Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R i...Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.展开更多
The standard Podle′s quantum sphere is Artin-Schelter regular as showed by Kra¨hmer(2012).The non-standard Podle′s quantum spheres are proved to be Auslander-regular,Cohen-Macaulay and Artin-Schelter regular in...The standard Podle′s quantum sphere is Artin-Schelter regular as showed by Kra¨hmer(2012).The non-standard Podle′s quantum spheres are proved to be Auslander-regular,Cohen-Macaulay and Artin-Schelter regular in this paper.展开更多
Let （R, m） be a Cohen-Macaulay local ring of dimension d, C a canonical R-module and M an almost Cohen-Macaulay R-module of dimension n and of depth t. We prove that dim Extd-n R（M,C） = n and if n ≤ 3 then Extd-n...Let （R, m） be a Cohen-Macaulay local ring of dimension d, C a canonical R-module and M an almost Cohen-Macaulay R-module of dimension n and of depth t. We prove that dim Extd-n R（M,C） = n and if n ≤ 3 then Extd-n（M,C） is an almost Cohen-Macaulay R-module. In particular, if n = d ≤ 3 then HomR（M, C） is an almost Cohen-Macaulay R-module. In addition, with some conditions, we show that Ext1R（M, C） is also almost Cohen-Macaulay. Finally, we study the vanishing Ext2R （Extd-n （M, C）, C） and Ext2R （Extd-n（M, C）, C）.展开更多
Let R = k[x1,...,xn], where k is a field. The path ideal （of length t ≥ 2） of a directed graph G is the monomial ideal], denoted by It （G）, whose generators correspond to the directed paths of length t in G. Let ...Let R = k[x1,...,xn], where k is a field. The path ideal （of length t ≥ 2） of a directed graph G is the monomial ideal], denoted by It （G）, whose generators correspond to the directed paths of length t in G. Let F be a directed rooted tree. We characterize all such trees whose path ideals are unmixed and Cohen-Macaulay. Moreover, we show that R/It（F） is Corenstein if and only if the Stanley-Reisner simplicial complex of It（Г） is a matroid.展开更多
We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen- Macaulay if and only if it is pure and its 1-dimensional link...We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen- Macaulay if and only if it is pure and its 1-dimensional links are connected, and that a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For t≥2, a lexsegment complex is strictly Cohen-Macaulay in codimension t if and only if it is the join of a lexsegment pure discon- nected flag complex with a （t - 2）-dimensional simplex. When the Stanley-Reisner ideal of a pure lexsegment complex is not quadratic, the complex is Cohen-Macaulay if and only if it is Cohen-Macaulay in some codimension. Our results are based on a characterization of Cohen-Macaulay and Buchsbaum lexsegment complexes by Bonanzinga, Sorrenti and Terai.展开更多
Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is...Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.展开更多
We study properties of graded maximal Cohen-Macaulay modules over an N-graded locally finite,Auslander Gorenstein,and Cohen-Macaulay algebra of dimension two.As a consequence,we extend a part of the McKay corresponden...We study properties of graded maximal Cohen-Macaulay modules over an N-graded locally finite,Auslander Gorenstein,and Cohen-Macaulay algebra of dimension two.As a consequence,we extend a part of the McKay correspondence in dimension two to a more general setting.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.
文摘In this article,we obtain an upper bound for the regularity of the binomial edge ideal of a graph whose every block is either a cycle or a clique.As a consequence,we obtain an upper bound for the regularity of binomial edge ideal of a cactus graph.We also identify a certain subclass attaining the upper bound.
文摘In this note,we will generalize some results about the Cohen-Macaulaymess and Goronsteinness of Rees rings and associated graded rings of ideals having higher analytic deviation.
基金supported by ZJNSF(LY19A010011)NSFC(11971141,12371017)supported by NSFC(11971449,12131015,12371042).
文摘A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z_(2)-graded algebra E(θ),which corresponds to a noncommutative quadric hypersurface E^(!)/(z)for some central regular element z∈E_(2)^(!).It turns out that the bounded derived category D^(b)(gr_(Z_(2))E(θ))is equivalent to the stable category of the maximal Cohen-Macaulay modules over E^(!)/(z)provided that E!is noetherian.As a consequence,E^(!)/(z)is a noncommutative isolated singularity if and only if the corresponding Clifford deformation E(θ)is a semisimple Z_(2)-graded algebra.The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces.As an application,we recover Knörrer's periodicity theorem without using matrix factorizations.
文摘Let R and S be Artin algebras and F be their triangular matrix extension via a bimodule sMR. We study totally acyclic complexes of projective F-modules and obtain a complete description of Gorenstein projective F-modules. We then construct some examples of Cohen-Macaulay finite and virtually Gorenstein triangular matrix algebras.
文摘Let R=K[x1,…,xn]be the polynomial ring in n variables over a field K and I be a matroidal ideal of R.We show that I is sequentially Cohen-Macaulay if and only if the Alexander dual Iy has linear quotients.As a consequence,I is sequentially Cohen-Macaulay if and only if I is shellable.
基金Acknowledgements S.Y. Pan was supported by the National Natural Science Foundation of China （Grant No. 11201022）, the Fundamental Research Funds for the Central Universities （2013JBM096, 2013RC027）, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Ministry of Education. This revision of the first draft was done when S. Y. Pan was a postdoctor of Bishop＇s University, he would like to thank Professor Thomas Briistle for his warm hospitality. X. J. Zhang was supported by National Natural Science Foundation of China （Grant No. 11101217）.
文摘Let A and B be Artin R-algebras of finite Cohen-Macaulay type. Then we prove that, if A and B are standard derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived equivalent. And we show that Gorenstein projective conjecture is an invariant under standard derived equivalence between Artin R-algebras.
文摘We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen- Macaulay. As an application, we give a complete characterization of the completely square- free lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.
基金Supported by the National Natural Science Foundation of China (Grant No.10771152)
文摘Let （R,m） be a Cohen-Macaulay local ring of dimension d with infinite residue field, I an m-primary ideal and K an ideal containing I. Let J be a minimal reduction of I such that, for some positive integer k, KIn ∩ J = JKIn-1 for n ≤ k ？ 1 and λ（ JKKIIkk-1 ） = 1. We show that if depth G（I） ≥ d-2, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of FK（I） assuming that depth G（I） ≥ d - 1.
文摘Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.
基金supported by National Natural Science Foundation of China (Grant Nos. 10731070 and 11171067)Science and Technology Committee,Shanghai Municipality (Grant No. 11XD1400500)a training program for innovative talents of key disciplines,Fudan University
文摘The standard Podle′s quantum sphere is Artin-Schelter regular as showed by Kra¨hmer(2012).The non-standard Podle′s quantum spheres are proved to be Auslander-regular,Cohen-Macaulay and Artin-Schelter regular in this paper.
文摘Let （R, m） be a Cohen-Macaulay local ring of dimension d, C a canonical R-module and M an almost Cohen-Macaulay R-module of dimension n and of depth t. We prove that dim Extd-n R（M,C） = n and if n ≤ 3 then Extd-n（M,C） is an almost Cohen-Macaulay R-module. In particular, if n = d ≤ 3 then HomR（M, C） is an almost Cohen-Macaulay R-module. In addition, with some conditions, we show that Ext1R（M, C） is also almost Cohen-Macaulay. Finally, we study the vanishing Ext2R （Extd-n （M, C）, C） and Ext2R （Extd-n（M, C）, C）.
文摘Let R = k[x1,...,xn], where k is a field. The path ideal （of length t ≥ 2） of a directed graph G is the monomial ideal], denoted by It （G）, whose generators correspond to the directed paths of length t in G. Let F be a directed rooted tree. We characterize all such trees whose path ideals are unmixed and Cohen-Macaulay. Moreover, we show that R/It（F） is Corenstein if and only if the Stanley-Reisner simplicial complex of It（Г） is a matroid.
文摘We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen- Macaulay if and only if it is pure and its 1-dimensional links are connected, and that a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For t≥2, a lexsegment complex is strictly Cohen-Macaulay in codimension t if and only if it is the join of a lexsegment pure discon- nected flag complex with a （t - 2）-dimensional simplex. When the Stanley-Reisner ideal of a pure lexsegment complex is not quadratic, the complex is Cohen-Macaulay if and only if it is Cohen-Macaulay in some codimension. Our results are based on a characterization of Cohen-Macaulay and Buchsbaum lexsegment complexes by Bonanzinga, Sorrenti and Terai.
基金supported by National Natural Science Foundation of China (Grant No.11171296)the Zhejiang Provincial Natural Science Foundation of China (Grant No. D7080064)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110101110010)
文摘Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.
基金The authors thank the referees for the careful reading and very useful suggestions and thank Ken Brown,Daniel Rogalski,Robert Won,and Quanshui Wu for many useful conversations and valuable comments on the subject.Y.-H.Wang and X.-S.Qin thank the Department of Mathematics,University of Washington for its very supportive hospitality during their visitsX.-S.Qin was partially supported by the Foundation of China Scholarship Council(Grant No.[2016]3100)+3 种基金Y.-H.Wang was partially supported by the National Natural Science Foundation of China(Grant Nos.11971289,11871071)the Foundation of Shanghai Science and Technology Committee(Grant No.15511107300)the Foundation of China Scholarship Council(Grant No.[2016]3009)J.J.Zhang was partially supported by the US National Science Foundation(Grant No.DMS-1700825).
文摘We study properties of graded maximal Cohen-Macaulay modules over an N-graded locally finite,Auslander Gorenstein,and Cohen-Macaulay algebra of dimension two.As a consequence,we extend a part of the McKay correspondence in dimension two to a more general setting.
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.
