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Degrees of Irreducible Morphisms over Perfect Fields

  • Claudia Chaio
  • Patrick Le Meur
  • Sonia Trepode
Article
  • 13 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification (2010)

16G10 16G60 16G70 

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© 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

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