In this article, we consider endomorphism algebras of direct sums of some local left ideals over a local algebra and give a construction of quasi-hereditary algebras.
In this paper, let A be a finite dimensional associative algebra over an algebraically closed field h, modA be the category of finite dimensional left A-module and X1,X2,... ,X2, in modA be a complete exceptional sequ...In this paper, let A be a finite dimensional associative algebra over an algebraically closed field h, modA be the category of finite dimensional left A-module and X1,X2,... ,X2, in modA be a complete exceptional sequence, and let E be the endomorphism algebra of X1, X2,..., Xn. We study the global dimension of E, and calculate the Hochschild cohomology and homology groups of E.展开更多
This paper develops a duality theory for connected cochain DG algebras,with particular emphasis on the non-commutative aspects.One of the main items is a dualizing DG module which induces a duality between the derived...This paper develops a duality theory for connected cochain DG algebras,with particular emphasis on the non-commutative aspects.One of the main items is a dualizing DG module which induces a duality between the derived categories of DG left-modules and DG right-modules with finitely generated cohomology.As an application,it is proved that if the canonical module k=A/A≥1 has a semi-free resolution where the cohomological degree of the generators is bounded above,then the same is true for each DG module with finitely generated cohomology.展开更多
This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applic...This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary(the "good" prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.展开更多
We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T T′. The main result is that T T′ turns out to be an （n ＋ m）-tilting A B-modu...We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T T′. The main result is that T T′ turns out to be an （n ＋ m）-tilting A B-module, where T is an m-tilting A-module and T′ is an n-tilting B-module.展开更多
基金Supported by the National Natural Science Foundation of China(10801117)
文摘In this article, we consider endomorphism algebras of direct sums of some local left ideals over a local algebra and give a construction of quasi-hereditary algebras.
基金the National Natural Science Foundation of China （No. 10371036 10671061）+2 种基金 the Natural Science Foundation of Beijing （1042001） the Fund of Beijing Education Committee （No. KM200610005024） Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality and the Fundamental reseaxch Fund of Beijing University of Technology.
文摘In this paper, let A be a finite dimensional associative algebra over an algebraically closed field h, modA be the category of finite dimensional left A-module and X1,X2,... ,X2, in modA be a complete exceptional sequence, and let E be the endomorphism algebra of X1, X2,..., Xn. We study the global dimension of E, and calculate the Hochschild cohomology and homology groups of E.
文摘This paper develops a duality theory for connected cochain DG algebras,with particular emphasis on the non-commutative aspects.One of the main items is a dualizing DG module which induces a duality between the derived categories of DG left-modules and DG right-modules with finitely generated cohomology.As an application,it is proved that if the canonical module k=A/A≥1 has a semi-free resolution where the cohomological degree of the generators is bounded above,then the same is true for each DG module with finitely generated cohomology.
基金supported by a 2017 University of New South Wales Science Goldstar Grant(Jie Du)the Simons Foundation(Grant Nos. #359360(Brian Parshall) and #359363 (Leonard Scott))
文摘This paper aims at developing a "local-global" approach for various types of finite dimensional algebras, especially those related to Hecke algebras. The eventual intention is to apply the methods and applications developed here to the cross-characteristic representation theory of finite groups of Lie type. We first review the notions of quasi-hereditary and stratified algebras over a Noetherian commutative ring. We prove that many global properties of these algebras hold if and only if they hold locally at every prime ideal. When the commutative ring is sufficiently good, it is often sufficient to check just the prime ideals of height at most one. These methods are applied to construct certain generalized q-Schur algebras, proving they are often quasi-hereditary(the "good" prime case) but always stratified. Finally, these results are used to prove a triangular decomposition matrix theorem for the modular representations of Hecke algebras at good primes. In the bad prime case, the generalized q-Schur algebras are at least stratified, and a block triangular analogue of the good prime case is proved, where the blocks correspond to Kazhdan-Lusztig cells.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11471269, 61373140）, the Natural Science Foundation of Fujian Province （2016J01002）, and 2016 Incubation Program for Scientific Research Talent of Distinguished Young of Colledges and Universities in Fujian Province.
文摘We consider whether the tilting properties of a tilting A-module T and a tilting B-module T′ can convey to their tensor product T T′. The main result is that T T′ turns out to be an （n ＋ m）-tilting A B-module, where T is an m-tilting A-module and T′ is an n-tilting B-module.
