Degrees of irreducible morphism over perfect fields

CHAIO, LE MEUR, TREPODE

Algebras and Representation Theory

SPRINGER

Año: 2019 vol. 22 p. 495 - 515

1386-923X

The module category of any artin algebra is filtered by the powers ofits radical, thus defining an associated graded category. As anextension of the degree of irreducible morphisms, this textintroduces the degree of morphisms in the module category.When the ground ring is a perfect field, and thegiven morphism behaves nicely with respect to covering theory (as doirreducible morphisms with indecomposable domain or indecomposablecodomain), it is shown that the degree of the morphism is finiteif and only if its induced functor has arepresentable 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 ofknown results on the degrees of irreducible morphisms over perfectfields are given. Finally, this study is applied to thecomposition of paths of irreducible morphisms in relationship to thepowers of the radical.