Abstract
The so-called λ-Koszul algebra and λ-Koszul module are introduced. We give different equivalent descriptions of the λ-Koszul algebra in terms of its minimal graded projective resolution and the Yoneda Ext-algebra E(A) = ⊕i≥0 Ext i A \( \left( {\mathbb{F},\mathbb{F}} \right) \). The {nt“}λ-Koszulity” of a finitely generated graded module is discussed and the concepts of (strongly) weakly λ-Koszul module are introduced. Finally, we discuss the A ∞-structure on the Yoneda Ext-algebra of a λ-Koszul algebra.
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References
S. B. Priddy, “Koszul resolutions,” Trans. Amer.Math. Soc. 152(1), 39–60 (1970).
A. Beilinson, V. Ginzburg, and W. Soergel, “Koszul duality patterns in representation theory,” J. Amer.Math. Soc. 9(2), 473–527 (1996).
M. Artin and W. F. Schelter, “Graded algebras of global dimension 3,” Adv. in Math. 66(2), 171–216 (1987).
R. Berger, “Koszulity for nonquadratic algebras,” J. Algebra 239(2), 705–734 (2001).
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).
J.-W. He and D.-M. Lu, “Higher Koszul algebras and A-infinity algebras,” J. Algebra 293(2), 335–362 (2005).
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.
R. Martínez-Villa and D. Zacharia, “Approximations with modules having linear resolutions,” J. Algebra 266(2), 671–697 (2003).
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).
B. Keller, “Introduction to A-infinity algebras and modules,” in Homology Homotopy Appl. (2001), Vol. 3, pp. 1–35.
B. Keller, “Deriving DG categories,” Ann. Sci. École Norm. Sup. (4) 27(1), 63–102 (1994).
B. Keller, “Bimodule complexes via strong homotopy actions,” Algebr. Represent. Theory 3(4), 357–376 (2000).
B. Keller, A-Infinity Algebras in Representation Theory, Contribution to the Proceedings of ICRA IX (Beijing, 2000).
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).
E. L. Green and E. N. Marcos, “δ-Koszul algebras,” Comm. Algebra 33(6), 1753–1764 (2005).
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Published in Russian in Matematicheskie Zametki, 2009, Vol. 86, No. 5, pp. 705–724.
The text was submitted by the author in English.
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Lü, JF. Algebras with periodic shifts of ext degrees. Math Notes 86, 665–681 (2009). https://doi.org/10.1134/S0001434609110091
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DOI: https://doi.org/10.1134/S0001434609110091