Degrees of Irreducible Morphisms over Perfect Fields

  • Claudia Chaio
  • Patrick Le Meur
  • Sonia Trepode


The module category of any artin algebra is filtered by the powers of its radical, thus defining an associated graded category. As an extension of the degree of irreducible morphisms, this text introduces the degree of morphisms in the module category. When the ground ring is a perfect field, and the given morphism behaves nicely with respect to covering theory (as do irreducible morphisms with indecomposable domain or indecomposable codomain), it is shown that the degree of the morphism is finite if and only if its induced functor has a representable kernel. This gives a generalisation of Igusa and Todorov result, about irreducible morphisms with finite left degree and over an algebraically closed field. As a corollary, generalisations of known results on the degrees of irreducible morphisms over perfect fields are given. Finally, this study is applied to the composition of paths of irreducible morphisms in relationship to the powers of the radical.


Representation theory Finite-dimensional algebras Auslander-Reiten theory Irreducible morphisms Degrees of morphisms Covering theory 

Mathematics Subject Classification (2010)

16G10 16G60 16G70 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Auslander, M., Reiten, I., Smalø, S.O.: Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995). MR 1314422CrossRefzbMATHGoogle Scholar
  2. 2.
    Brenner, S.: On the kernel of an irreducible map. Linear Algebra Appl. 365, 91–97 (2003). Special issue on linear algebra methods in representation theory. MR 1987328 (2004f:16024)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Coelho, F.U., da Silva, D.D.: Relative degrees of irreducible morphisms. J. Algebra 428, 471–489 (2015). MR 3314301MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chaio, C.: Problems solved by using degrees of irreducible morphisms, Expository lectures on representation theory, Contemp. Math., vol. 607, Amer. Math. Soc., Providence, RI, pp. 179–203. MR 3204871 (2014)Google Scholar
  5. 5.
    Chaio, C., Le Meur, P., Trepode, S.: Degrees of irreducible morphisms and finite-representation type. J. Lond. Math. Soc. 84(1), 35–57 (2011). MR 2819689MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chaio, C., Le Meur, P., Trepode, S.: Covering techniques in Auslander-Reiten theory. J Pure Appl. Algebra. (2018).
  7. 7.
    Igusa, K., Todorov, G.: Radical layers of representable functors. J. Algebra 89(1), 105–147 (1984). MR MR748231 (86f:16029a)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Igusa, K., Todorov, G.: A characterization of finite Auslander-Reiten quivers. J. Algebra 89(1), 148–177 (1984). MR 748232MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Krause, H.: The kernel of an irreducible map. Proc. Amer. Math. Soc. 121(1), 57–66 (1994). MR 1181169MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Liu, S.: Degrees of irreducible maps and the shapes of Auslander-Reiten quivers. J. London Math. Soc. 45(1), 32–54 (1992). MR 1157550MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liu, S.: Shapes of connected components of the Auslander-Reiten quivers of Artin algebras, Representation theory of algebras and related topics (Mexico City, 1994), CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 109–137. MR 1388561Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Claudia Chaio
    • 1
  • Patrick Le Meur
    • 2
    • 3
  • Sonia Trepode
    • 1
  1. 1.Centro Marplatense de Investigaciones MatemáticasUniversidad Nacional de Mar del PlataMar del PlataArgentina
  2. 2.Laboratoire de MathématiquesUniversité Blaise Pascal & CNRSAubière cedexFrance
  3. 3.Université Paris Diderot, Sorbonne Université, CNRSInstitut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRGParisFrance

Personalised recommendations