Degrees of Irreducible Morphisms over Perfect Fields
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 in terms of the induced natural transformations between representable functors on this graded 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 associated natural transformation has a representable kernel. 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.
Cite
@article{arxiv.1704.03933,
title = {Degrees of Irreducible Morphisms over Perfect Fields},
author = {Claudia Chaio and Patrick Le Meur and Sonia Trepode},
journal= {arXiv preprint arXiv:1704.03933},
year = {2018}
}
Comments
Final version to appear in Algebras and Representation Theory. Due to changes in arXiv:1412.0188, the theorem C and the proposition 5.02 (4.0.2 in v.1) are slightly reformulated and the morphisms called <<strongly irreducible>> in v.1 are now called <<freely irreducible>>. Sections renumbered to stick to the published version. Minor changes and corrections