Frontiers of Mathematics in China >
Koszulity and Koszul modules of dual extension algebras
Received date: 02 Mar 2016
Accepted date: 01 Jun 2016
Published date: 20 Apr 2017
Copyright
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.
Key words: Dual extension; linearly presented; Koszul algebra; Koszul module
Huanhuan LI , Yunge XU . Koszulity and Koszul modules of dual extension algebras[J]. Frontiers of Mathematics in China, 2017 , 12(3) : 583 -596 . DOI: 10.1007/s11464-016-0601-4
1 |
AquinoR M, GreenE L. On modules with linear presentations over Koszul algebras. Comm Algebra, 2005, 33(1): 19–36
|
2 |
AuslanderM, ReitenI, SmaløS O. Representation Theory of Artin Algebras. Cambridge Stud Adv Math, Vol 36. Cambridge: Cambridge Univ Press, 1995
|
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
|
5 |
BianN, YeY, ZhangP. Generalized d-Koszul modules. Math Res Lett, 2011, 18(2): 191–200
|
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
|
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
|
10 |
GreenE L, MarcosE, ZhangP. Koszul modules and modules with linear presentations. Comm Algebra, 2003, 31(6): 2745–2770
|
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
|
13 |
HappelD. A family of algebras with two simple modules and Fibonacci numbers. Arch Math, 1991, 57: 133–139
|
14 |
ManinY I. Some remarks on Koszul algebras and quantum groups. Ann Inst Fourier, 1987, 37: 191–205
|
15 |
Martinez-VillaR, Montano-BermudezG. Triangular matrix and Koszul algebras. Int J Algebra, 2007, 1(10): 441–467
|
16 |
Membrillo-Hern�andezF H. Quasi-hereditary algebras with two simple modules and Fibonacci numbers. Comm Algebra, 1994, 22(11): 4499–4509
|
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
|
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
|
21 |
ZhaoD, HanY. Koszul algebras and finite Galois coverings. Sci China, Ser A, 2009, 52(10): 2145–2153
|
/
〈 | 〉 |