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Short proof of a conjecture concerning split-by-nilpotent extensions

Abstract

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E. We provide a short proof to a conjecture by Assem and Zacharia concerning properties of \(\mathop {\text {mod}}B\) inherited by \(\mathop {\text {mod}}C\). We show if B is a tilted algebra, then C is a tilted algebra.

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References

  1. 1.

    Assem, I., Marmaridis, N.: Tilting modules over split-by-nilpotent extensions. Commun. Algebra 26, 1547–1555 (1998)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Assem, I., Simson, D., Skowronski, A.: Elements of the Representation Theory of Associative Algebras, 1: Techniques of Representation Theory, London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  3. 3.

    Assem, I., Zacharia, D.: On split-by-nilpotent extensions. Colloq. Math. 98, 259–275 (2003)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Jaworska, A., Malicki, P., Skowroński, A.: Tilted algebras and short chains of modules. Math. Z. 273, 19–27 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Liu, S.: Tilted algebras and generalized standard Auslander-Reiten components. Arch. Math. 61, 12–19 (1993)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Skowroński, A.: Generalized standard Auslander-Reiten components without oriented cycles. Osaka J. Math. 30, 515–527 (1993)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Stephen Zito.

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The author was supported by the University of Connecticut-Waterbury.

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Zito, S. Short proof of a conjecture concerning split-by-nilpotent extensions. Arch. Math. 111, 479–483 (2018). https://doi.org/10.1007/s00013-018-1208-7

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Mathematics Subject Classification

  • 16G20
  • 16G70

Keywords

  • Split-by-nilpotent extensions
  • Tilted algebras