Algebras and Representation Theory

, Volume 20, Issue 2, pp 289–311 | Cite as

On Sectional Paths in a Category of Complexes of Fixed Size

  • Claudia Chaio
  • Isabel Pratti
  • María José Souto-Salorio


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.


Irreducible morphisms Complexes of fixed size Auslander-Reiten quiver Sectional paths 

Mathematics Subject Classification (2010)

16G70 18G35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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)Google Scholar
  2. 2.
    Auslander, M., Reiten, I., Smalø, S.: Representation theory of artin algebras. vol 36 of Cambridge Studies in Advanced Mathematics. Cambridge University Press (1995)Google Scholar
  3. 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. 4.
    Bautista, R.: The category of morphisms between projectives modules. Communications in Algebra 32(11), 4303–4331 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bautista, R., Smalø, S.O.: Non-existent cycles. Communications in Algebra 11, 1755–1767 (1983)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bautista, R., Souto Salorio, M.J.: Irreducible morphisms in the bounded derived category. Journal Pure and Applied 215, 866–884 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bautista, R., Souto Salorio, M.J., Zuazua, R.: Almost split sequences for complexes of fixed size. Journal of Algebra 287, 140–168 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chaio, C., Platzeck, M.I., Trepode, S.: On the degree of irreducible morphisms. Journal of Algebra 281 1, 200–224 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 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)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Girardo, H., Merklen, H.: Irreducible morphism of the category of complexes. Journal of Algebra 321, 2716–2736 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Happel, D.: Triangulated categories in the representation theory of finite dimensional algebras. London Math. Soc. Lecture Note Ser., vol. 119, Cambridge (1998)Google Scholar
  12. 12.
    Igusa, K., Todorov, G.: A characterization of finite Auslander-Reiten quivers. Journal Algebra 89, 148–177 (1984)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Liu, S.: Auslander-reiten theory in a Krull-Schmidt category. Sao Paulo J. Math. Sci. 4, 425–472 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Wheeler, W.: The triangulated structure of the stable derived category. Journal of Algebra 165(1), 23–40 (1994)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Claudia Chaio
    • 1
  • Isabel Pratti
    • 1
  • María José Souto-Salorio
    • 2
  1. 1.Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Funes 3350Universidad Nacional de Mar del PlataMar del PlataArgentina
  2. 2.Departamento de Computación, Facultade de InformáticaUniversidade da CoruñaCoruñaEspaña

Personalised recommendations