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Tilted algebras and short chains of modules

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Abstract

We provide an affirmative answer for the question raised almost 20 years ago in Reiten et al. (Proc. Am. Math. Soc. 117:343–354, 1993) concerning the characterization of tilted artin algebras by the existence of a sincere finitely generated module which is not the middle of a short chain.

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References

  1. Assem, I., Simson, D., Skowroński, 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)

  2. Assem I., Skowroński A.: Minimal representation-infinite coil algebras. Manuscr. Math. 67, 305–331 (1990)

    Article  MATH  Google Scholar 

  3. Auslander M., Reiten I.: Modules determined by their composition factors. Ill. J. Math. 29, 280–301 (1985)

    MathSciNet  MATH  Google Scholar 

  4. Auslander M., Reiten I., Smalø S.O.: Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, vol 36. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  5. Baer D.: Wild hereditary artin algebras and linear methods. Manuscr. Math. 55, 69–82 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bakke Ø.: Some characterizations of tilted algebras. Math. Scand. 63, 43–50 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Bautista R., Smalø S.O.: Nonexistent cycles. Comm. Algebra 11, 1755–1767 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bongartz, K.: Tilted algebras. In: Representations of Algebras, Lecture Notes in Mathematics, vol. 903, pp. 26–38. Springer, Berlin (1981)

  9. Brenner, S., Butler, M.C.R.: Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors, In: Representation Theory II, Lecture Notes in Mathematics, vol. 832, pp. 103–169. Springer, Berlin (1980)

  10. Happel D., Liu S.: Module categories without short cycles are of finite type. Proc. Amer. Math. Soc. 120, 371–375 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Happel D., Reiten I.: Hereditary abelian categories with tilting object over arbitrary basic fields. J. Algebra 256, 414–432 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Happel, D., Reiten, I., Smalø, S. O.: Tilting in abelian categories and quasitilted algebras. Memoires Amer. Math. Soc. 120(575) (1996)

  13. Happel D., Ringel C.M.: Tilted algebras. Trans. Amer. Math. Soc. 274, 399–443 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jaworska A., Malicki P., Skowroński A.: On Auslander-Reiten components of algebras without external short paths, J. Lond. Math. Soc 85, 245–268 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lenzing H., de la Peña J.A.: Concealed-canonical algebras and separating tubular families. Proc. Lond. Math. Soc. 78, 513–540 (1999)

    Article  MATH  Google Scholar 

  16. Lenzing H., Skowroński A.: Quasi-tilted algebras of canonical type. Colloq. Math. 71, 161–181 (1996)

    MathSciNet  MATH  Google Scholar 

  17. Liu S.: Degrees of irreducible maps and the shapes of the Auslander-Reiten quivers. J. Lond. Math. Soc. 45, 32–54 (1992)

    Article  MATH  Google Scholar 

  18. Liu S.: Semi-stable components of an Auslander-Reiten quiver. J. Lond. Math. Soc. 47, 405–416 (1993)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  20. Malicki, P., Skowroński, A.: Algebras with separating Auslander-Reiten components, In: Representations of Algebras and Related Topics, European Mathematical Society Series of Congress Reports, European Mathematical Society Publishing House, Zürich, pp. 251–353 (2011)

  21. Meltzer H.: Auslander-Reiten components of concealed-canonical algebras. Colloq. Math. 71, 183–202 (1996)

    MathSciNet  MATH  Google Scholar 

  22. Reiten I., Skowroński A., Smalø S.O.: Short chains and short cycles of modules. Proc. Am. Math. Soc. 117, 343–354 (1993)

    Article  MATH  Google Scholar 

  23. Ringel C.M.: Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., vol. 1099. Springer, Berlin (1984)

    Google Scholar 

  24. Ringel, C.M.: Representation theory of finite dimensional algebras. In: Representations of Algebras, London Mathematical Society Lecture Note Series, vol. 116, pp. 7–79. Cambridge University Press, Cambridge (1986)

  25. Ringel, C. M.: The canonical algebras, with an appendix by W. Crawley-Boevey. In: Topics in Algebra, Banach Center Publications, vol. 26, Part 1, pp. 407–432, PWN Warsaw (1990)

  26. Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras 2: Tubes and Concealed Algebras of Euclidean Type, London Mathematical Society Student Texts, vol. 71, Cambridge University Press, Cambridge (2007)

  27. Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras 3: Representation-Infinite Tilted Algebras, London Mathematical Society Student Texts, vol. 72, Cambridge University Press, Cambridge (2007)

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

    MathSciNet  MATH  Google Scholar 

  29. Skowroński A.: Generalized standard Auslander-Reiten components. J. Math. Soc. Jpn. 46, 517–543 (1994)

    Article  MATH  Google Scholar 

  30. Skowroński A.: Regular Auslander-Reiten components containing directing modules. Proc. Am. Math. Soc. 120, 19–26 (1994)

    MATH  Google Scholar 

  31. Skowroński A.: Directing modules and double tilted algebras. Bull. Pol. Acad. Sci. Math. 50, 77–87 (2002)

    MATH  Google Scholar 

  32. Skowroński, A.: Selfinjective algebras: finite and tame type, In: Trends in Representation Theory of Algebras and Related Topics, Contemporary Mathematics, vol. 406, American Mathematical Society, Providence, pp. 169–238 (2006)

  33. Skowroński A., Yamagata K.: Galois coverings of selfinjective algebras by repetitive algebras. Trans. Am. Math. Soc. 351, 715–734 (1999)

    Article  MATH  Google Scholar 

  34. Skowroński A., Yamagata K.: On selfinjective artin algebras having nonperiodic generalized standard Auslander-Reiten components. Colloq. Math. 96, 235–244 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Skowroński, A., Yamagata K.: Selfinjective algebras of quasitilted type, In: Trends in Representation Theory of Algebras and Related Topics, European Mathematical Society Series of Congress Reports, pp. 639–708. European Mathematical Society Publishing House, Zurich (2008)

  36. Zhang Y.: The structure of stable components. Can. J. Math. 43, 652–672 (1991)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100, Toruń, Poland

    Alicja Jaworska, Piotr Malicki & Andrzej Skowroński

Authors
  1. Alicja Jaworska
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  2. Piotr Malicki
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  3. Andrzej Skowroński
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Correspondence to Andrzej Skowroński.

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Dedicated to Idun Reiten on the occasion of her 70th birthday.

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Jaworska, A., Malicki, P. & Skowroński, A. Tilted algebras and short chains of modules. Math. Z. 273, 19–27 (2013). https://doi.org/10.1007/s00209-012-0993-0

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