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Some remarks on δ-Koszul algebras

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Abstract

Let A be a δ-Koszul algebra, and let Ƙ δ(A) and L(A) denote the categories of δ-Koszul modules and modules with linear presentations. Some necessary and sufficient conditions for Ƙ δ(A) = L(A) are given. Set

$$\begin{array}{*{20}c} {E(A): = \mathop \oplus \limits_{i \geqslant 0} Ext_A^i (A_0 ,A_0 )} & {and} & {\mathcal{B}(A): = \sup \left\{ {\left. {i \in \mathbb{N}} \right|Ext_A^i (A_0 ,A_0 ) \cap V \ne 0} \right\},} \\ \end{array}$$

, where V is a minimal graded generating space of E(A). In the present paper, we prove that {B(A)| A is δ - Koszul} = ℕ. Finally, the Koszulity of the graded Hopf Galois extension of δ-Koszul algebras is studied.

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References

  1. R. M. Aquino and E. L. Green, “On modules with linear presentations over Koszul algebras,” Comm. Algebra 33, 19–36 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Artin and W. F. Schelter, “Graded algebras of global dimension 3,” Adv. Math. 66, 171–216 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  3. R. Berger, “Koszulity for nonquadratic algebras,” J. Algebra 239, 705–734 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  4. E. L.Green and E. N. Marcos, “δ-Koszul algebras,” Comm. Algebra 33, 1753–1764 (2005).

    Article  MathSciNet  Google Scholar 

  5. E. L. Green, E. N. Marcos, R. Martínez-Villa, and P. Zhang, “D-Koszul algebras,” J. Pure Appl. Algebra 193, 141–162 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  6. E. L. Green, R. Martínez-Villa, and I. Reiten, “On modules with linear presentations,” J. Algebra 205, 578–604 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  7. E. L.Green, E. N. Marcos, and P. Zhang, “Koszul modules and modules with linear presentations,” Comm. Algebra 31, 2745–2770 (2003).

    Article  MathSciNet  Google Scholar 

  8. R. Martínez-Villa and D. Zacharia, “Approximations with modules having linear resolutions,” J. Algebra 266, 671–697 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  9. J. W. He, F. Van Oystaeyen, and Y. H. Zhang, “Derived H-module endomorphism rings,” Glasgow Math. Journal 52, 649–661 (2010).

    Article  MATH  Google Scholar 

  10. J. F. lü, “Quasi-Koszulity and minimal Horseshoe Lemma,” Bull. Malaysian Math. Sci. Soc. (2) 35 (4), 1017–1033 (2012).

    MATH  Google Scholar 

  11. J. F. lü, “Notes on δ-Koszul algebras,” Appl. Categor. Struc. 20, 143–159 (2012).

    Article  MATH  Google Scholar 

  12. J. F. lü, “On an example of δ-Koszul algebras,” Proc. Amer.Math. Soc. 138 (11), 3777–3781 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  13. J. F. lü and M. S. Chen, “On a special class of exact sequences,” J. Alg. Appl. 10 (5), 915–930 (2011).

    Article  MATH  Google Scholar 

  14. J. F. lü, J. W. He, and D. MLu, “Piecewise-Koszul algebras,” Sci. China Ser. A 50, 1785–1794 (2007).

    Google Scholar 

  15. A. Polishchuk and L. Positselski, Quadratic Algebras, University Lectures Series (Amer. Math. Soc., Providence, RI, 2005), Vol. 37.

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

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Zhejiang Normal University, Jinhua, Zhejiang, China

    Jiafeng Lü

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  1. Jiafeng Lü
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Correspondence to Jiafeng Lü.

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Lü, J. Some remarks on δ-Koszul algebras. Math Notes 97, 402–411 (2015). https://doi.org/10.1134/S0001434615030116

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  • DOI: https://doi.org/10.1134/S0001434615030116

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