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Koszul property of a class of graded algebras with nonpure resolutions

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Abstract

Given any integers a, b, c, and d with a > 1, c ≥ 0, b ≥ a + c, and d ≥ b + c, the notion of (a, b, c, d)-Koszul algebra is introduced, which is another class of standard graded algebras with “nonpure” resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of (a, b, c, d)-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be (a, b, c, d)-Koszul are provided.

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References

  1. Artin M, Schelter W F. Graded algebras of global dimension 3. Adv Math, 1987, 66: 171–216

    Article  MathSciNet  MATH  Google Scholar 

  2. Beilinson A, Ginzburg V, Soergel W. Koszul duality patterns in representation theory. J Amer Math Soc, 1996, 9(2): 473–527

    Article  MathSciNet  MATH  Google Scholar 

  3. Berger R. Koszulity for nonquadratic algebras. J Algebra, 2001, 239: 705–734

    Article  MathSciNet  MATH  Google Scholar 

  4. Bian N, Ye Y, Zhang P. Generalized d-Koszul modules. Math Res Lett, 2010, 18(2): 191–200

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenner S, Butler M C R, King A D. Periodic algebras which are almost Koszul. Algebr Represent Theory, 2002, 5: 331–367

    Article  MathSciNet  MATH  Google Scholar 

  6. Fløystad G, Vatne J E. Artin-Schelter regular algebras of dimension five. Algebra, Geometry, and Mathematical Physics, Banach Center Publications, 2011, 93: 19–39

    Article  MathSciNet  MATH  Google Scholar 

  7. Green E L, Marcos E N. δ-Koszul algebras. Comm Algebra, 2005, 33(6): 1753–1764

    Article  MathSciNet  MATH  Google Scholar 

  8. Green E L, Marcos E N, Martínez-Villa R, Zhang P. D-Koszul algebras. J Pure Appl Algebra, 2004, 193(1): 141–162

    Article  MathSciNet  MATH  Google Scholar 

  9. Green E L, Snashall N. Finite generation of Ext for a generalization of D-Koszul algebras. J Algebra, 2006, 295: 458–472

    Article  MathSciNet  MATH  Google Scholar 

  10. Lu D-M, Palmieri J H, Wu Q-S, Zhang J J. Regular algebras of dimension 4 and their A8-Ext-algebras. Duke Math J, 2007, 137(3): 537–584

    Article  MathSciNet  MATH  Google Scholar 

  11. Lu D-M, Si J-R. Koszulity of algebras with nonpure resolutions. Comm Algebra, 2009, 38(1): 68–85

    Article  MathSciNet  MATH  Google Scholar 

  12. Lü J-F. Nonpure piecewise-Koszul algebras. Indian J Pure Appl Math, 2012, 43(1): 1–23

    Article  MathSciNet  MATH  Google Scholar 

  13. Lü J-F. (a, b, c)-Koszul algebras. Bull Malays Math Sci Soc, 2013, 36(2): 447–463

    MathSciNet  MATH  Google Scholar 

  14. Lü J-F, Chen M-S. Discrete Koszul algebras. Algebr Represent Theory, 2012, 15(2): 273–293

    Article  MathSciNet  MATH  Google Scholar 

  15. Lü J-F, He J-W, Lu D-M. Piecewise-Koszul algebras. Sci China Ser A, 2007, 50(12): 1795–1804

    Article  MathSciNet  MATH  Google Scholar 

  16. Priddy S. Koszul resolutions. Trans Amer Math Soc, 1970, 152(1): 39–60

    Article  MathSciNet  MATH  Google Scholar 

  17. Volodymyr M, Serge O, Catharina S. Quadratic duals, Koszul dual functors, and applications. Trans Amer Math Soc, 2009, 361(3): 1129–1172

    MathSciNet  MATH  Google Scholar 

  18. Zhou G-S, Lu D-M. Artin-Schelter regular algebras of dimension five with two generators. J Pure Appl Algebra, 2014, 218(5): 937–961

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Jiafeng Lü & Junling Zheng

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  1. Jiafeng Lü
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  2. Junling Zheng
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Correspondence to Jiafeng Lü.

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Lü, J., Zheng, J. Koszul property of a class of graded algebras with nonpure resolutions. Front. Math. China 11, 985–1002 (2016). https://doi.org/10.1007/s11464-016-0566-3

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  • DOI: https://doi.org/10.1007/s11464-016-0566-3

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