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Degrees of Irreducible Morphisms over Perfect Fields

Authors
  • Chaio, Claudia1
  • Le Meur, Patrick2, 3
  • Trepode, Sonia1
  • 1 Universidad Nacional de Mar del Plata, Centro Marplatense de Investigaciones Matemáticas, CONICET. Funes 3350, Mar del Plata, 7600, Argentina , Mar del Plata (Argentina)
  • 2 Université Blaise Pascal & CNRS, Laboratoire de Mathématiques, Complexe Scientifique Les Cézeaux, BP 80026, Aubière cedex, 63171, France , Aubière cedex (France)
  • 3 Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, Université Paris Diderot, Sorbonne Université, CNRS, Paris, F-75013, France , Paris (France)
Type
Published Article
Journal
Algebras and Representation Theory
Publisher
Springer Netherlands
Publication Date
May 16, 2018
Volume
22
Issue
2
Pages
495–515
Identifiers
DOI: 10.1007/s10468-018-9781-9
Source
Springer Nature
Keywords
License
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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.

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