Let A be a finite dimensional k-algebra and T be a supportτ-tilting right A-module.In this note,we give lower and upper bounds for the global dimension of the endomorphism algebra End_(A)(T)under some mild conditions...Let A be a finite dimensional k-algebra and T be a supportτ-tilting right A-module.In this note,we give lower and upper bounds for the global dimension of the endomorphism algebra End_(A)(T)under some mild conditions.Finally,we give some examples to illustrate that both the upper and lower bounds can be reached.展开更多
It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian.In this article,we consider a special class of hearts of twin cotorsion pairs induced by d-cluster tilting su...It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian.In this article,we consider a special class of hearts of twin cotorsion pairs induced by d-cluster tilting subcategories in extriangulated categories.We give a necessary and sufficient condition for such hearts to be abelian.In particular,we can also see that such hearts are hereditary.As an application,this generalizes the work by Liu in the exact case,thereby providing new insights into the triangulated case.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
This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category.As an application,this result generalizes the work by W.J.Chen...This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category.As an application,this result generalizes the work by W.J.Chen,Z.K.Liu,and X.Y.Yang in a triangulated case[J.Algebra Appl.,2018,17(5):1-15].Moreover,it highlights new phenomena when it applied to an exact category.Finally,we give some applications of our main results.In particular,we obtain Krause's recollement whose proofs are both elementary and very general.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 the National Natural Science Foundation of China(11971255,11901567,12071120)supported by the Hunan Provincial Natural Science Foundation of China(2023JJ30008)the National Natural Science Foundation of China(12371034).
文摘Let A be a finite dimensional k-algebra and T be a supportτ-tilting right A-module.In this note,we give lower and upper bounds for the global dimension of the endomorphism algebra End_(A)(T)under some mild conditions.Finally,we give some examples to illustrate that both the upper and lower bounds can be reached.
基金Panyue Zhou was supported by the Hunan Provincial Natural Science Foundation of China(Grant No.2023JJ30008).
文摘It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian.In this article,we consider a special class of hearts of twin cotorsion pairs induced by d-cluster tilting subcategories in extriangulated categories.We give a necessary and sufficient condition for such hearts to be abelian.In particular,we can also see that such hearts are hereditary.As an application,this generalizes the work by Liu in the exact case,thereby providing new insights into the triangulated case.
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 National Natural Science Foundation of China(Grant Nos.11901190,11671126,12071120)the Scientific Research Fund of Hunan Provincial Education Department(Grant No.19B239).
文摘This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category.As an application,this result generalizes the work by W.J.Chen,Z.K.Liu,and X.Y.Yang in a triangulated case[J.Algebra Appl.,2018,17(5):1-15].Moreover,it highlights new phenomena when it applied to an exact category.Finally,we give some applications of our main results.In particular,we obtain Krause's recollement whose proofs are both elementary and very general.
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.
