[1] Liao Yihua, Chen Jianlong,. Dual pairs of tilting pairs [J]. Journal of Southeast University (English Edition), 2011, 27 (3): 347-350. [doi:10.3969/j.issn.1003-7985.2011.03.024]

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Dual pairs of tilting pairs()

倾斜对的对偶对

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

27

Issue:

2011 3

Page:

347-350

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2011-09-30

Info

Title:

Dual pairs of tilting pairs

倾斜对的对偶对

Author(s):

Liao Yihua¹;  2;  Chen Jianlong¹

¹Department of Mathematics, Southeast University, Nanjing 211189, China
²College of Mathematics and Information Science, Guangxi University, Nanning 530004, China

廖贻华¹;  2;  陈建龙¹

¹ 东南大学数学系, 南京 211189; ² 广西大学数学与信息科学学院, 南宁 530004

Keywords:

selforthogonal module;  selfinjective;  dual module;  tilting pair

自正交模;  自内射性;  对偶模;  倾斜对

PACS:

O153.3

DOI:

10.3969/j.issn.1003-7985.2011.03.024

Abstract:

A module pair(C, T)over an Artin algebra Λ is called a tilting pair if both C and T are selforthogonal modules and the conditions T∈ad(^overd)C and C∈ad(ˇoverd)T hold. The duality on a tilting pair is investigated to discuss the condition under which the dual of a tilting pair is also a tilting pair. A necessary and sufficient condition of(D(T), D(C))being an n-tilting pair over an Artin algebra for an n-tilting pair(C, T)is given. And, a necessary and sufficient condition of(T^*, C^*)being an n-tilting pair over a selfinjective Artin algebra for an n-tilting pair(C, T)is also given.

在一个Artin代数Λ上, 如果模C和T都是自正交模且满足条件T∈ad(^overd)C和C∈ad(ˇoverd)T, 则模对(C, T)称为倾斜对. 讨论了倾斜对的对偶性, 探讨在何种条件下一个倾斜对的对偶仍然是一个倾斜对. 给出了一个Artin代数上的模对(D(T), D(C))成为一个n-倾斜对的充分必要条件. 同时也给出了一个自内射Artin代数上的模对(T^*, C^*)成为一个n-倾斜对的充分必要条件.

References:

[1] Morita K. Duality of modules and its applications to the theory of rings with minimum condition [J]. Sci Rep Tokyo Kyoiku Daigaku, Sect A, 1958, 6(1): 83-142.
[2] Azumaya G. A duality theory for injective modules[J]. Amer J Math, 1959, 81(1): 249-278.
[3] Miyashita Y. Tilting modules associated with a series of idempotent ideals [J]. J Algebra, 2001, 238(2): 485-501.
[4] Wei J Q, Xi C C. A characterization of the tilting pair [J]. J Algebra, 2007, 317(1): 376-391.
[5] Wei J Q, Xi C C. Auslander-Reiten correspondence for tilting pairs [J]. J Pure Appl Algebra, 2008, 212(2): 411-422.
[6] Auslander M, Reiten I. Applications of contravariantly finite subcategories [J]. Adv Math, 1991, 86(1): 111-152.
[7] Auslander M, Reiten I, Smalø S O. Representation theory of Artin algebras [M]. London: Cambridge University Press, 1997.
[8] Colby R R, Fuller K R. Equivalence and duality for module categories(with tilting and cotilting for rings)[M]. London: Cambridge University Press, 2004.
[9] Skowronski A, Yamagata K. Stable equivalence of selfinjective Artin algebras of Dynkin type [J]. Alg Repr Theory, 2006, 9(1): 33-45.

Memo

Memo:

Biographies: Liao Yihua(1963—), male, graduate; Chen Jianlong(corresponding author), male, doctor, professor, jlchen@seu.edu.cn.
Foundation items: The National Natural Science Foundation of China(No.10971024), the Specialized Research Fund for the Doctoral Program of Higher Education( No.200802860024), the Natural Science Foundation of Jiangsu Province(No.BK2010393), Scientific Research Foundation of Guangxi University(No.XJZ100246).
Citation: Liao Yihua, Chen Jianlong. Dual pairs of tilting pairs[J].Journal of Southeast University(English Edition), 2011, 27(3):347-350.[doi:10.3969/j.issn.1003-7985.2011.03.024]

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