Koszulity and Koszul modules of dual extension algebras

doi:10.1007/s11464-016-0601-4

Front. Math. China

2017, Vol. 12

Issue (3) : 583-596 https://doi.org/10.1007/s11464-016-0601-4

RESEARCH ARTICLE

Koszulity and Koszul modules of dual extension algebras

Huanhuan LI¹, Yunge XU²(

)

¹. School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China
². Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, China

Download: PDF(177 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Let Aand Bbe algebras, and let T be the dual extension algebra of A and B. We provide a different method to prove that Tis Koszul if and only if both A and B are Koszul. Furthermore, we prove that an algebra is Koszul if and only if one of its iterated dual extension algebras is Koszul, if and only if all its iterated dual extension algebras are Koszul. Finally, we give a necessary and sufficient condition for a dual extension algebra to have the property that all linearly presented modules are Koszul modules, which provides an effective way to construct algebras with such a property.

Keywords Dual extension linearly presented Koszul algebra Koszul module

Corresponding Author(s): Yunge XU

Issue Date: 20 April 2017

Cite this article:

Huanhuan LI,Yunge XU. Koszulity and Koszul modules of dual extension algebras[J]. Front. Math. China, 2017, 12(3): 583-596.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0601-4
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I3/583

1	AquinoR M, GreenE L. On modules with linear presentations over Koszul algebras. Comm Algebra, 2005, 33(1): 19–36 https://doi.org/10.1081/AGB-200036765
2	AuslanderM, ReitenI, SmaløS O. Representation Theory of Artin Algebras. Cambridge Stud Adv Math, Vol 36. Cambridge: Cambridge Univ Press, 1995 https://doi.org/10.1017/CBO9780511623608
3	BackelinJ, FrobergR. Veronese subrings, Koszul algebras and rings with linear resolutions. Rev Roumaine Math Pures Appl, 1985, 30: 85–97
4	BeilinsonA, GinszburgV, SoergelW. Koszul duality patterns in representation theory. J Amer Math Soc, 1996, 9: 473–525 https://doi.org/10.1090/S0894-0347-96-00192-0
5	BianN, YeY, ZhangP. Generalized d-Koszul modules. Math Res Lett, 2011, 18(2): 191–200 https://doi.org/10.4310/MRL.2011.v18.n2.a1
6	ClineE, ParshallB, ScottL. Finite dimensional algebras and highest weight categories. J Reine Angew Math, 1988, 391: 85–99
7	DengB M. A Construction of algebras with arbitrary finite global dimensions. Comm Algebra, 1998, 26(9): 2959–2965 https://doi.org/10.1080/00927879808826320
8	FrobergR. Koszul algebras. In: Advances in Commutative Ring Theory (Fez, 1997). Lecture Notes in Pure and Appl Math, Vol 205. New York: Dekker, 1999, 337–350
9	GreenE L. Remarks on projective resolutions. In: Dlab V, Gabriel P, eds. Representation Theory II: Proceedings of the Second International Conference on Representations of Algebras, Ottawa, Carleton University, August 13–25, 1979. Lecture Notes in Math, Vol 832. Berlin: Springer, 1980, 259–279 https://doi.org/10.1007/bfb0088468
10	GreenE L, MarcosE, ZhangP. Koszul modules and modules with linear presentations. Comm Algebra, 2003, 31(6): 2745–2770 https://doi.org/10.1081/AGB-120021891
11	GreenE L, Martinez-VillaR. Koszul and Yoneda algebras. In: Representation Theory of Algebras (Cocoyoc, 1994). CMS Conference Proceedings, Vol 18. Providence: Amer Math Soc, 1996, 247–297
12	GreenE L, Martinez-VillaR, ReitenI, SolbergF, ZachariaD. On modules with linear presentations. J Algebra, 1998, 205: 578–604 https://doi.org/10.1006/jabr.1997.7402
13	HappelD. A family of algebras with two simple modules and Fibonacci numbers. Arch Math, 1991, 57: 133–139 https://doi.org/10.1007/BF01189999
14	ManinY I. Some remarks on Koszul algebras and quantum groups. Ann Inst Fourier, 1987, 37: 191–205 https://doi.org/10.5802/aif.1117
15	Martinez-VillaR, Montano-BermudezG. Triangular matrix and Koszul algebras. Int J Algebra, 2007, 1(10): 441–467 https://doi.org/10.12988/ija.2007.07049
16	Membrillo-Hern�andezF H. Quasi-hereditary algebras with two simple modules and Fibonacci numbers. Comm Algebra, 1994, 22(11): 4499–4509 https://doi.org/10.1080/00927879408825083
17	PolishchukA, PositselskiL. Quadratic Algebras. Univ Lecture Ser, Vol 37. Providence: Amer Math Soc, 2006
18	PriddyS B. Koszul resolutions. Trans Amer Math Soc, 1970, 152: 39–60 https://doi.org/10.1090/S0002-9947-1970-0265437-8
19	XiC C. Quasi-hereditary algebras with a duality. J Reine Angew Math, 1994, 449: 201–215
20	XiC C. Global dimensions of dual extension algebras. Manuscripta Math, 1995, 88(1): 25–31 https://doi.org/10.1007/BF02567802
21	ZhaoD, HanY. Koszul algebras and finite Galois coverings. Sci China, Ser A, 2009, 52(10): 2145–2153 https://doi.org/10.1007/s11425-009-0082-y

[1]	Zhi CHENG. Trivial extension of Koszul algebras[J]. Front. Math. China, 2017, 12(5): 1045-1056.
[2]	Jiafeng LÜ,Junling ZHENG. Koszul property of a class of graded algebras with nonpure resolutions[J]. Front. Math. China, 2016, 11(4): 985-1002.
[3]	Bo HOU, Jinmei FAN. Hochschild cohomology of ?n-Galois coverings of an algebra[J]. Front Math Chin, 2012, 7(6): 1113-1128.

Viewed

Full text

Abstract

Cited

Shared

Discussed