Degrees of Irreducible Morphisms over Perfect Fields

Claudia Chaio (1) , Patrick Le Meur (2, 3) , Sonia Trepode (1)

1 Departamento de Matemática [Mar del Plata]
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
3 LMBP - Laboratoire de Mathématiques Blaise Pascal

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.

Domaines: Théorie des représentations [math.RT]

Claudia Chaio, Patrick Le Meur, Sonia Trepode. Degrees of Irreducible Morphisms over Perfect Fields. Algebras and Representation Theory, 2019, 22 (2), pp.495-515. ⟨10.1007/s10468-018-9781-9⟩. ⟨hal-02361291⟩