Degrees of Irreducible Morphisms over Perfect Fields
- 13 Downloads
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.
KeywordsRepresentation 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.
- 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
- 6.Chaio, C., Le Meur, P., Trepode, S.: Covering techniques in Auslander-Reiten theory. J Pure Appl. Algebra. (2018). https://doi.org/10.1016/j.jpaa.2018.04.013
- 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