Geodesic
Koszul-Like Algebras and Modules
Matematičeskie zametki, Tome 93 (2013) no. 3, pp. 413-435.

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In this paper, the notion of Koszul-like algebra is introduced; this notion generalizes the notion of Koszul algebra and includes some Artin–Schelter regular algebras of global dimension $5$ as special examples. Basic properties of Koszul-like modules are discussed. In particular, some necessary and sufficient conditions for $\mathcal{KL}(A)=\mathcal{L}(A)$ are provided, where $\mathcal{KL}(A)$ and $\mathcal{L}(A)$ denote the categories of Koszul-like modules and modules with linear presentations (see [1]–[3], etc.) respectively, and $A$ is a Koszul-like algebra. We construct new Koszul-like algebras from the known ones by the “one-point extension”. Some criteria for a graded algebra to be Koszul-like are provided. Finally, we construct many classical Koszul objects from the given Koszul-like objects.
Mots-clés : Koszul algebra, Koszul-like algebra/module
Keywords: module with linear presentations, one-point extension, Yoneda algebra. 
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Lü Jia-Feng. Koszul-Like Algebras and Modules. Matematičeskie zametki, Tome 93 (2013) no. 3, pp. 413-435. http://geodesic.mathdoc.fr/item/MZM_2013_93_3_a9/

[1] R. M. Aquino, E. L. Green, “On modules with linear presentations over Koszul algebras”, Comm. Algebra, 33:1 (2005), 19–36 | DOI | MR | Zbl

[2] E. L. Green, R. Martínez-Villa, I. Reiten, Ø. Solberg, D. Zacharia, “On modules with linear presentations”, J. Algebra, 205:2 (1998), 578–604 | DOI | MR | Zbl

[3] E. L. Green, E. N. Marcos, P. Zhang, “Koszul modules and modules with linear presentations”, Comm. Algebra, 31:6 (2003), 2745–2770 | DOI | MR | Zbl

[4] S. Priddy, “Koszul resolutions”, Trans. Amer. Math. Soc., 152 (1970), 39–60 | DOI | MR | Zbl

[5] M. Artin, W. F. Schelter, “Graded algebras of global dimension $3$”, Adv. Math., 66:2 (1987), 171–216 | DOI | MR | Zbl

[6] R. Berger, “Koszulity for nonquadratic algebras”, J. Algebra, 239:2 (2001), 705–734 | DOI | MR | Zbl

[7] E. L. Green, E. N. Marcos, R. Martínez-Villa, P. Zhang, “$D$-Koszul algebras”, J. Pure Appl. Algebra, 193:1-3 (2004), 141–162 | DOI | MR | Zbl

[8] J.-W. He, D.-M. Lu, “Higher Koszul algebras and $A$-infinity algebras”, J. Algebra, 293:2 (2005), 335–362 | DOI | MR | Zbl

[9] S. Brenner, M. C. R. Butler, A. D. King, “Periodic algebras which are almost Koszul”, Algebr. Represent. Theory, 5:4 (2002), 331–367 | DOI | MR | Zbl

[10] E. L. Green, E. N. Marcos, “$\delta$-Koszul algebras”, Comm. Algebra, 33:6 (2005), 1753–1764 | DOI | MR | Zbl

[11] E. L. Green, N. Snashall, “Finite generation of Ext for a generalization of $D$-Koszul algebras”, J. Algebra, 295:2 (2006), 458–472 | DOI | MR | Zbl

[12] J.-F. Lü, J.-W. He, D.-M Lu, “Piecewise-Koszul algebras”, Sci. China Ser. A, 50:12 (2007), 1795–1804 | MR | Zbl

[13] T. Cassidy, B. Shelton, “Generalizing the notion of a Koszul algebra”, Math. Z., 260:1 (2008), 93–114 | DOI | MR | Zbl

[14] D.-M Lu, J.-R. Si, “Koszulity of algebras with non-pure resolutions”, Comm. Algebra, 38:1 (2010), 68–85 | MR | Zbl

[15] Lyu Sia-Fen, “Algebry s periodicheskimi sdvigami stepenei Ext”, Matem. zametki, 86:5 (2009), 705–724 | DOI | MR | Zbl

[16] J.-F. Lü, Z.-B. Zhao, “$(p,\lambda)$-Koszul algebras and modules”, Indian J. Pure Appl. Math., 41:3 (2010), 443–473 | DOI | MR | Zbl

[17] Pei-Sen Chen, Sia-Fen Lyu, “Slabo koshulepodobnye moduli”, Matem. zametki, 91:1 (2012), 93–119 | DOI

[18] A. Beilinson, V. Ginszburg, W. Soergel, “Koszul duality patterns in representation theory”, J. Amer. Math. Soc., 9:2 (1996), 473–527 | DOI | MR | Zbl

[19] A. Polishchuk, L. Positselski, Quadratic Algebras, Univ. Lecture Ser., 37, Amer. Math. Soc., Providence, RI, 2005 | MR | Zbl

[20] G. Fløystad, J. E. Vatne, Artin–Schelter Regular Algebras of Dimension Five, 2010, arXiv: 0911.5129v3

[21] D. Zacharia, Ext and Koszul Algebras, Preprint, 2000

[22] E. L. Green, R. Martínez-Villa, “Koszul and Yoneda algebras”, Representation Theory of Algebras (Cocoyoc, 1994), CMS Conf. Proc., 18, Amer. Math. Soc., Providence, RI, 1996, 247–297 | MR | Zbl

[23] C. A. Weibel, “An Introduction to Homological Algebra”, Cambridge Stud. Adv. Math., 38, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl