Algebras with periodic shifts of Ext degrees. (English. Russian original) Zbl 1203.16009
Math. Notes 86, No. 5, 665-681 (2009); translation from Mat. Zametki 86, No. 5, 705-724 (2009).
Summary: The so-called \(\lambda\)-Koszul algebra and \(\lambda\)-Koszul module are introduced. We give different equivalent descriptions of the \(\lambda\)-Koszul algebra in terms of its minimal graded projective resolution and the Yoneda Ext-algebra \(E(A)=\bigoplus_{i\geq 0}\text{Ext}_A^i(\mathbb{F},\mathbb{F})\). The “\(\lambda\)-Koszulity” of a finitely generated graded module is discussed and the concepts of (strongly) weakly \(\lambda\)-Koszul module are introduced. Finally, we discuss the \(A_\infty\)-structure on the Yoneda Ext-algebra of a \(\lambda\)-Koszul algebra.
MSC:
16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
16S37 | Quadratic and Koszul algebras |
16E05 | Syzygies, resolutions, complexes in associative algebras |
Keywords:
Koszul algebras; Koszul modules; Yoneda Ext-algebras; \(A_\infty\)-algebras; Lie algebras; modules over graded algebras; projective resolutionsReferences:
[1] | S. B. Priddy, ”Koszul resolutions,” Trans. Amer.Math. Soc. 152(1), 39–60 (1970). · Zbl 0261.18016 · doi:10.1090/S0002-9947-1970-0265437-8 |
[2] | A. Beilinson, V. Ginzburg, and W. Soergel, ”Koszul duality patterns in representation theory,” J. Amer.Math. Soc. 9(2), 473–527 (1996). · Zbl 0864.17006 · doi:10.1090/S0894-0347-96-00192-0 |
[3] | M. Artin and W. F. Schelter, ”Graded algebras of global dimension 3,” Adv. in Math. 66(2), 171–216 (1987). · Zbl 0633.16001 · doi:10.1016/0001-8708(87)90034-X |
[4] | R. Berger, ”Koszulity for nonquadratic algebras,” J. Algebra 239(2), 705–734 (2001). · Zbl 1035.16023 · doi:10.1006/jabr.2000.8703 |
[5] | E. L. Green, E. N. Marcos, R. Martínez-Villa, and P. Zhang, ”D-Koszul algebras,” J. Pure Appl. Algebra 193(1–3), 141–162 (2004). · Zbl 1075.16013 · doi:10.1016/j.jpaa.2004.03.012 |
[6] | J.-W. He and D.-M. Lu, ”Higher Koszul algebras and A-infinity algebras,” J. Algebra 293(2), 335–362 (2005). · Zbl 1143.16027 · doi:10.1016/j.jalgebra.2005.05.025 |
[7] | E. L. Green and R. Martínez-Villa, ”Koszul and Yoneda algebras,” in Representation Theory of Algebras, CMS Conf. Proc., Cocoyoc, 1994 (Amer.Math. Soc., Providence, RI, 1996), Vol. 18, pp. 247–297. · Zbl 0860.16009 |
[8] | R. Martínez-Villa and D. Zacharia, ”Approximations with modules having linear resolutions,” J. Algebra 266(2), 671–697 (2003). · Zbl 1061.16035 · doi:10.1016/S0021-8693(03)00261-8 |
[9] | J. D. Stasheff, ”Homotopy associativity of H-spaces. I,” Trans. Amer.Math. Soc. 108(2), 275–292 (1963); ”Homotopy associativity of H-spaces. II,” Trans. Amer.Math. Soc. 108 (2), 293–312 (1963). · Zbl 0114.39402 · doi:10.2307/1993608 |
[10] | B. Keller, ”Introduction to A-infinity algebras and modules,” in Homology Homotopy Appl. (2001), Vol. 3, pp. 1–35. · Zbl 0989.18009 · doi:10.4310/HHA.2001.v3.n1.a1 |
[11] | B. Keller, ”Deriving DG categories,” Ann. Sci. École Norm. Sup. (4) 27(1), 63–102 (1994). · Zbl 0799.18007 · doi:10.24033/asens.1689 |
[12] | B. Keller, ”Bimodule complexes via strong homotopy actions,” Algebr. Represent. Theory 3(4), 357–376 (2000). · Zbl 0968.18005 · doi:10.1023/A:1009954126727 |
[13] | B. Keller, A-Infinity Algebras in Representation Theory, Contribution to the Proceedings of ICRA IX (Beijing, 2000). |
[14] | D.M. Lu, J. H. Palmieri, Q. S. Wu, and J. J. Zhang, ”A -algebras for ring theorists,” Algebra Colloq. 11(1), 91–128 (2004). · Zbl 1066.16049 |
[15] | E. L. Green and E. N. Marcos, ”{\(\delta\)}-Koszul algebras,” Comm. Algebra 33(6), 1753–1764 (2005). · Zbl 1096.16012 · doi:10.1081/AGB-200061501 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.