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 C be a triangulated category.We first introduce the notion of balanced pairs in C,and then establish the bijective correspondence between balanced pairs and proper classesξwith enoughξ-projectives andξ-injectiv...Let C be a triangulated category.We first introduce the notion of balanced pairs in C,and then establish the bijective correspondence between balanced pairs and proper classesξwith enoughξ-projectives andξ-injectives.Assume thatξ:=ξX=ξ^(Y) is the proper class induced by a balanced pair(X,Y).We prove that(C,Eξ,sξ)is an extriangulated category.Moreover,it is proved that(C,Eξ,sξ)is a triangulated category if and only if X=Y=0,and that(C,Eξ,sξ)is an exact category if and only if X=Y=C.As an application,we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.展开更多
To investigate cohomology theories based on flats, Asadollahi and Salarian gave the definition of F-Gorenstein flat R-modules, and these modules are exactly Gorenstein fiat provided that R is right coherent. In this p...To investigate cohomology theories based on flats, Asadollahi and Salarian gave the definition of F-Gorenstein flat R-modules, and these modules are exactly Gorenstein fiat provided that R is right coherent. In this paper, we get some properties of F-Gorenstein flat R-modules and establish the stability of F-Gorenstein flat categories.展开更多
Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the propert...Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the property of the extension closure of some classes of objects in(T↓A),the exactness of the functor p and the detailed description of orthogonal classes of a given class p(X,Y)in(T↓A).Moreover,we characterize when special precovering classes in abelian categories A and B can induce special precovering classes in(T↓A).As an application,we prove that under suitable conditions,the class of Gorenstein projective leftΛ-modules over a triangular matrix ringΛ=(R M 0 S)is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering.Consequently,we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.展开更多
Let(C,E,s)be an extriangulated category with a proper classξof E-triangles,and W an additive full subcategory of(C,E,s).We provide a method for constructing a proper W(ξ)-resolution(resp.,coproper W(ξ)-coresolution...Let(C,E,s)be an extriangulated category with a proper classξof E-triangles,and W an additive full subcategory of(C,E,s).We provide a method for constructing a proper W(ξ)-resolution(resp.,coproper W(ξ)-coresolution)of one term in an E-triangle inξfrom that of the other two terms.By using this way,we establish the stability of the Gorenstein category GW(ξ)in extriangulated categories.These results generalize the work of Z.Y.Huang[J.Algebra,2013,393:142–169]and X.Y.Yang and Z.C.Wang[Rocky Mountain J.Math.,2017,47:1013–1053],but the proof is not too far from their case.Finally,we give some applications about our main results.展开更多
Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modu...Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modules are investigated. It is shown that weak Gorenstein cotorsion R-modules over coherent rings are indeed weaker than Gorenstein cotorsion R-modules. Weak Gorenstein cotorsion dimension for modules and rings are also studied.展开更多
The influence of the amplitude ratio between the two THz pulses on two-dimension THz spectroscopy(2DTS)has been studied theoretically via a classical method in which the expressions for the second-order nonlinearity w...The influence of the amplitude ratio between the two THz pulses on two-dimension THz spectroscopy(2DTS)has been studied theoretically via a classical method in which the expressions for the second-order nonlinearity were derived using perturbation approach,and the THz pulses were not treated as a delta function.Three types of nonlinear sources i.e.,anharmonicity,nonlinear damping,and nonlinear coupling,are considered in a single mode system.The simulation results demonstrated that the amplitude ratio had a notable influence on the 2DTSs,and different sources have different influences.This study is promising for guiding future experiments.展开更多
We study complete cohomology of complexes with finite Gorenstein AC-projective dimension. We show first that the class of complexes admitting a complete level resolution is exactly the class of complexes with finite G...We study complete cohomology of complexes with finite Gorenstein AC-projective dimension. We show first that the class of complexes admitting a complete level resolution is exactly the class of complexes with finite Gorenstein AC-projective dimension. This lets us give some general techniques for computing complete cohomology of complexes with finite Gorenstein AC- projective dimension. As a consequence, the classical relative cohomology for modules of finite Gorenstein AC-projective dimension is extended. Finally, the relationships between projective dimension and Gorenstein AC-projective dimension for complexes are given.展开更多
Let(C,E,s)be an extriangulated category with a proper classξof E-triangles.We study complete cohomology of objects in(C,E,s)by applyingξ-projective resolutions andξ-injective coresolutions constructed in(C,E,s).Van...Let(C,E,s)be an extriangulated category with a proper classξof E-triangles.We study complete cohomology of objects in(C,E,s)by applyingξ-projective resolutions andξ-injective coresolutions constructed in(C,E,s).Vanishing of complete cohomology detects objects with finiteξ-projective dimension and finiteξ-injective dimension.As a consequence,we obtain some criteria for the validity of the Wakamatsu tilting conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein.Moreover,we give a general technique for computing complete cohomology of objects with finiteξ-Gprojective dimension.As an application,the relations betweenξ-projective dimension andξ-Gprojective dimension for objects in(C,E,s)are given.展开更多
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Xianhui Fu was supported by YDZJ202101ZYTS168 and the NSF of China(12071064)Jiangsheng Hu was supported by the NSF of China(12171206)+2 种基金the Natural Science Foundation of Jiangsu Province(BK20211358)Haiyan Zhu was supported by Zhejiang Provincial Natural Science Foundation of China(LY18A010032)the NSF of China(12271481).
文摘Let C be a triangulated category.We first introduce the notion of balanced pairs in C,and then establish the bijective correspondence between balanced pairs and proper classesξwith enoughξ-projectives andξ-injectives.Assume thatξ:=ξX=ξ^(Y) is the proper class induced by a balanced pair(X,Y).We prove that(C,Eξ,sξ)is an extriangulated category.Moreover,it is proved that(C,Eξ,sξ)is a triangulated category if and only if X=Y=0,and that(C,Eξ,sξ)is an exact category if and only if X=Y=C.As an application,we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.
文摘To investigate cohomology theories based on flats, Asadollahi and Salarian gave the definition of F-Gorenstein flat R-modules, and these modules are exactly Gorenstein fiat provided that R is right coherent. In this paper, we get some properties of F-Gorenstein flat R-modules and establish the stability of F-Gorenstein flat categories.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671069 and 11771212)Zhejiang Provincial Natural Science Foundation of China (Grant No. LY18A010032)+1 种基金Qing Lan Project of Jiangsu Province and Jiangsu Government Scholarship for Overseas Studies (Grant No. JS2019-328)during a visit of the first author to Charles University in Prague with the support by Jiangsu Government Scholarship
文摘Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the property of the extension closure of some classes of objects in(T↓A),the exactness of the functor p and the detailed description of orthogonal classes of a given class p(X,Y)in(T↓A).Moreover,we characterize when special precovering classes in abelian categories A and B can induce special precovering classes in(T↓A).As an application,we prove that under suitable conditions,the class of Gorenstein projective leftΛ-modules over a triangular matrix ringΛ=(R M 0 S)is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering.Consequently,we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11771212,11901190,11671221)Qing Lan Project of Jiangsu Province,Jiangsu Government Scholarship for Overseas Studies(Grant No.JS-2019-328)+1 种基金Hunan Provincial Natural Science Foundation of China(Grant No.2018JJ3205)Scientific Research Fund of Hunan Provincial Education Department(Grant No.19B239).
文摘Let(C,E,s)be an extriangulated category with a proper classξof E-triangles,and W an additive full subcategory of(C,E,s).We provide a method for constructing a proper W(ξ)-resolution(resp.,coproper W(ξ)-coresolution)of one term in an E-triangle inξfrom that of the other two terms.By using this way,we establish the stability of the Gorenstein category GW(ξ)in extriangulated categories.These results generalize the work of Z.Y.Huang[J.Algebra,2013,393:142–169]and X.Y.Yang and Z.C.Wang[Rocky Mountain J.Math.,2017,47:1013–1053],but the proof is not too far from their case.Finally,we give some applications about our main results.
基金The authors wish to express their sincere thanks to the referees for their valuable comments and suggestions. The first author was supported by the Postdoctoral Science Foundation of China （2017M611851）, the Jiangsu Planned Projects for Postdoctoral Research Funds （1601151C） and the Provincial Natural Science Foundation of Anhui Province of China （KJ2017A040）. The second author was supported by the NSFC （11771212）, and the first two authors were supported by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The third author was supported by the NSFC （11501257, 11671069, 11771212） and the Postdoctoral Science Foundation of China （2016M600426）.
文摘Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modules are investigated. It is shown that weak Gorenstein cotorsion R-modules over coherent rings are indeed weaker than Gorenstein cotorsion R-modules. Weak Gorenstein cotorsion dimension for modules and rings are also studied.
基金supported from the National Natural Science Foundation of China(10974063,60907045 and 61177095)Hubei Natural Science Foundations(2010CDA001 and2012FFA074)+1 种基金Ph.D.Programs Foundation of Ministry of Education of China(20100142110042)the Fundamental Research Fundsfor the Central Universities,HUST(2011TS001,2012QN094 and 2012QN097)
文摘The influence of the amplitude ratio between the two THz pulses on two-dimension THz spectroscopy(2DTS)has been studied theoretically via a classical method in which the expressions for the second-order nonlinearity were derived using perturbation approach,and the THz pulses were not treated as a delta function.Three types of nonlinear sources i.e.,anharmonicity,nonlinear damping,and nonlinear coupling,are considered in a single mode system.The simulation results demonstrated that the amplitude ratio had a notable influence on the 2DTSs,and different sources have different influences.This study is promising for guiding future experiments.
基金This work was partially supported by the National Natural Science Foundation of China （Grant Nos. 11371187, 11501257） and Jiangsu University of Technology of China （KYY14015, KYY14016）.
文摘We study complete cohomology of complexes with finite Gorenstein AC-projective dimension. We show first that the class of complexes admitting a complete level resolution is exactly the class of complexes with finite Gorenstein AC-projective dimension. This lets us give some general techniques for computing complete cohomology of complexes with finite Gorenstein AC- projective dimension. As a consequence, the classical relative cohomology for modules of finite Gorenstein AC-projective dimension is extended. Finally, the relationships between projective dimension and Gorenstein AC-projective dimension for complexes are given.
基金supported by the NSF of China(11671069,11771212)Qing Lan Project of Jiangsu Province and Natural Science Foundation of Jiangsu Province(BK20211358)+4 种基金supported by the NSF of China(11971225,11901341)Shandong Provincial Natural Science Foundation(ZR2019QA015)supported by the National Natural Science Foundation of China(11901190,11671221)the Hunan Provincial Natural Science Foundation of China(2018JJ3205)the Scientific Research Fund of Hunan Provincial Education Department(19B239).
文摘Let(C,E,s)be an extriangulated category with a proper classξof E-triangles.We study complete cohomology of objects in(C,E,s)by applyingξ-projective resolutions andξ-injective coresolutions constructed in(C,E,s).Vanishing of complete cohomology detects objects with finiteξ-projective dimension and finiteξ-injective dimension.As a consequence,we obtain some criteria for the validity of the Wakamatsu tilting conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein.Moreover,we give a general technique for computing complete cohomology of objects with finiteξ-Gprojective dimension.As an application,the relations betweenξ-projective dimension andξ-Gprojective dimension for objects in(C,E,s)are given.
