Krull-Gabriel dimension of Skew group algebras

TL;DR

This paper investigates the Krull-Gabriel dimension of skew group algebras, establishing a Galois semi-covering functor that confirms conjectures on its finiteness and relation to stable rank, with applications to various algebra classes.

Contribution

It introduces a Galois semi-covering functor linking morphism categories of A and its skew group algebra AG, proving their Krull-Gabriel dimensions are equal and confirming related conjectures.

Findings

Krull-Gabriel dimension of A and AG are equal

Confirmed Prests conjecture on finiteness of Krull-Gabriel dimension

Determined possible stable ranks for skew Brauer graph algebras

Abstract

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional algebra A and AG is the associated skew group algebra. The author with S. Trepode and A. G. Chaio introduced the notion of a Galois semi-covering functor to study the irreducible morphisms over skew group algebras. In this paper, we establish a Galois semi-covering functor between the morphism categories as well as the functor categories over the algebras A and AG and prove that their Krull-Gabriel dimension are equal. This computation confirms Prests conjecture on the finiteness of Krull-Gabriel dimension and Schroers conjecture on its connection with the stable rank (the least stabilized radical power) over skew gentle algebras. Moreover, we determine all posible stable ranks for (skew) Brauer graph algebras.