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On Sectional Paths in a Category of Complexes of Fixed Size

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Abstract

We show how to build the Auslander-Reiten quiver of the category C n (proj Λ) of complexes of size n ≥ 2, for any artin algebra Λ. We also give conditions over the complexes in C n (proj Λ) under which the composition of irreducible morphisms in sectional paths vanishes.

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References

  1. Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras. London Math. Soc. Student Texts 65. Cambridge University Press (2006)

  2. Auslander, M., Reiten, I., Smalø, S.: Representation theory of artin algebras. vol 36 of Cambridge Studies in Advanced Mathematics. Cambridge University Press (1995)

  3. Bautista, R.: Sections in Auslander-Reiten quivers. Representation theory II. In: Proceedings of the Second International Conference, Carleton University, Ottawa, Ont., 1979, Lecture Notes in Math. 832, pp 74–96. Springer, Berlin-New York (1980)

    Google Scholar 

  4. Bautista, R.: The category of morphisms between projectives modules. Communications in Algebra 32(11), 4303–4331 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bautista, R., Smalø, S.O.: Non-existent cycles. Communications in Algebra 11, 1755–1767 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bautista, R., Souto Salorio, M.J.: Irreducible morphisms in the bounded derived category. Journal Pure and Applied 215, 866–884 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bautista, R., Souto Salorio, M.J., Zuazua, R.: Almost split sequences for complexes of fixed size. Journal of Algebra 287, 140–168 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chaio, C., Platzeck, M.I., Trepode, S.: On the degree of irreducible morphisms. Journal of Algebra 281 1, 200–224 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chaio, C., Souto Salorio, M.J., Trepode, S.: Composite of irreducible morphisms in the bounded derived category. Journal of Pure and Applied Algebra 215 (12), 2957–2968 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Girardo, H., Merklen, H.: Irreducible morphism of the category of complexes. Journal of Algebra 321, 2716–2736 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras. London Math. Soc. Lecture Note Ser., vol. 119, Cambridge (1998)

  12. Igusa, K., Todorov, G.: A characterization of finite Auslander-Reiten quivers. Journal Algebra 89, 148–177 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, S.: Auslander-reiten theory in a Krull-Schmidt category. Sao Paulo J. Math. Sci. 4, 425–472 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wheeler, W.: The triangulated structure of the stable derived category. Journal of Algebra 165(1), 23–40 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Funes 3350, Universidad Nacional de Mar del Plata, 7600, Mar del Plata, Argentina

    Claudia Chaio & Isabel Pratti

  2. Departamento de Computación, Facultade de Informática, Universidade da Coruña, Campus de A Coruña, 15071 A, Coruña, España

    María José Souto-Salorio

Authors
  1. Claudia Chaio
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  2. Isabel Pratti
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  3. María José Souto-Salorio
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Corresponding author

Correspondence to María José Souto-Salorio.

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Presented by Michel Van den Bergh.

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Chaio, C., Pratti, I. & Souto-Salorio, M.J. On Sectional Paths in a Category of Complexes of Fixed Size. Algebr Represent Theor 20, 289–311 (2017). https://doi.org/10.1007/s10468-016-9643-2

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  • DOI: https://doi.org/10.1007/s10468-016-9643-2

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