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Selfinjective Koszul algebras of finite complexity

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Abstract

In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category

of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ l ≤ m there exist a family of modules of complexity l parameterized by G(l,m), the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL(2, C) over the exterior algebra of a 2-dimensional vector space.

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References

  1. Webb, P. J.: The Auslander-Reiten quiver of a finite group. Math. Z., 179, 97–121 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. Guo, J. Y., Wu Q.: Loewy matrix, Koszul cone and applications. Comm. Algebra, 28(2), 925–941 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Auslander, M.: Rational singularities and almost splitting sequences. Trans. Amer. Math. Soc., 293, 511–531 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  4. Auslander, M., Reiten, I.: McKay quivers and extended Dynkin diagrams. Trans. Amer. Math. Soc., 293, 293–301 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  5. Crawley-Boevey, W., Holland M. P.: Noncommutative deformations of Kleinian singularities. Duke J. Math., 92, 251–289 (1998)

    Article  MathSciNet  Google Scholar 

  6. Guo, J. Y., Martínez-Villa, R.: Algebra pairs associated to McKay quivers. Comm. Algebra, 30(2), 1017–1032 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Tang, A.: A note on the stable equivalence conjecture. Comm. Algebra, 24(9), 2793–2809 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Auslander, M., Reiten, I.: k-Gorenstein algebras and syzygy modules. J. Pure Appl. Algebras, 92, 1–27 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math., 86 111–152 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Martínez-Villa, R.: Graded, selfinjective and Koszul algebras. J. Algebra, 215(1), 34–72 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Smith, P.: Some finite-dimensional algebras related to elliptic curves, Representation theory of algebras and related topics. CMS Conference Proceedings, 19, 315–348 (1996)

    Google Scholar 

  12. Green, E. L., Martínez-Villa, R.: Koszul and Yoneda algebras, representation theory of algebras. CMS Conference Proceedings, 19, 247–298 (1996)

    Google Scholar 

  13. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry, GTM. 150, Springer-Verlag, New York, 1994

    Google Scholar 

  14. Guo, J. Y., Wan, Q. H., Wu, Q. X.: On the Koszul Modules of Exterior Algebras. Acta Mathematica Sinica, English Series, 23(11), 1967–1984 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Auslander, M., Reiten, I., Smalø, S.: Representation Theory of Artin Algebras, Cambridge Studies in Advanced Math. 36, Cambridge Univ. Press, Cambridge, 1995

    MATH  Google Scholar 

  16. McConell, J. C., Robson, J. C.: Nocommutative Noetherian Rings, A Piley Interscience Publication, Chichester, 1987

    Google Scholar 

  17. Ringel, C. M.: Cones, representation theory of algebras. CMS Conference Proceedings, 18, 587–601 (1996)

    MathSciNet  Google Scholar 

  18. Guo, J. Y., Li, B., Wu, Q.: Equivalence of the category of Koszul modules of complexity one of an exterior algebra. Acta Mathematica Sinica, English Series, 22(3), 849–854 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Yoshino, Y.: Modules with linear resolution over a polynomial ring in two variables. Nagoya Math. J., 113, 89–98 (1989)

    MATH  MathSciNet  Google Scholar 

  20. Martínez-Villa, R., Zacharia, D.: Approximations with modules having linear resolution. J. Algebra, 266(2), 671–697 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Martínez-Villa, R.: Application of Koszul algebras: the preprojective algebra, representation theory of algebras. CMS Conference Proceedings, 18, 487–504 (1996)

    Google Scholar 

  22. Guo, J. Y., Martínez-Villa, R., Takane, M.: Koszul generalized Auslander regular algebras, in Algebras and Modules II. Canadian Mathematical Society Conference Proceedings, 24, 263–283 (1998)

    Google Scholar 

  23. Takane, M.: The Coxeter transformations of representation infinite quiver, representation theory of algebras and related topics. CMS Conference Proceedings, 19, 349–372 (1996)

    MathSciNet  Google Scholar 

  24. McKay, J.: Graph, singularities and finite groups. Proc. Symp. Pure Math., 37, 183–186 (1980)

    MathSciNet  Google Scholar 

  25. Martínez-Villa, R.: Skew group algebras and their Yoneda algebras. Math. J. Okayama Univ., 43, 1–16 (2001)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Xiangtan University, Xiangtan, 411105, P. R. China

    Jin Yun Guo

  2. Department of Mathematics, Hu’nan Normal University, Changsha, 410081, P. R. China

    Ai Hua Li & Qiu Xian Wu

Authors
  1. Jin Yun Guo
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  2. Ai Hua Li
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  3. Qiu Xian Wu
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Correspondence to Jin Yun Guo.

Additional information

Supported by NSFC #10671061, SRFDP #200505042004 and the Cultivation Fund of the Key Scientific and Technical Innovation Project #21000115 of the Ministry of Education of China

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Guo, J.Y., Li, A.H. & Wu, Q.X. Selfinjective Koszul algebras of finite complexity. Acta. Math. Sin.-English Ser. 25, 2179–2198 (2009). https://doi.org/10.1007/s10114-009-6703-0

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  • DOI: https://doi.org/10.1007/s10114-009-6703-0

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