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Krull-Gabriel dimension of Skew group algebras

Shantanu Sardar

TL;DR

The paper develops Galois semi-covering functors to compare the Krull–Gabriel dimension of algebras with their skew group algebras $\bar{\Lambda}$ under a finite abelian group action. It proves $KG(\bar{\Lambda})=KG(\Lambda)$ by constructing exact, faithful functors between module, morphism, and functor categories, and establishing explicit Hom-space isomorphisms under the group action. The results yield concrete consequences for skew-gentle and (skew) Brauer graph algebras, including confirmations of Prest's finiteness conjecture and Schröer's stable-rank conjecture in these classes, and determine all possible stable ranks for these skew algebras. Overall, the work provides a unified framework to transfer KG-dimension across skew constructions and enables precise KG-dimension calculations in key tame and special biserial families.

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.

Krull-Gabriel dimension of Skew group algebras

TL;DR

The paper develops Galois semi-covering functors to compare the Krull–Gabriel dimension of algebras with their skew group algebras under a finite abelian group action. It proves by constructing exact, faithful functors between module, morphism, and functor categories, and establishing explicit Hom-space isomorphisms under the group action. The results yield concrete consequences for skew-gentle and (skew) Brauer graph algebras, including confirmations of Prest's finiteness conjecture and Schröer's stable-rank conjecture in these classes, and determine all possible stable ranks for these skew algebras. Overall, the work provides a unified framework to transfer KG-dimension across skew constructions and enables precise KG-dimension calculations in key tame and special biserial families.

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.
Paper Structure (7 sections, 28 theorems, 19 equations, 1 figure)

This paper contains 7 sections, 28 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Galois semi-covering Functor over the module category
  3. Galois semi-covering functor over morphism category
  4. Galois semi-covering functor over functor categories
  5. KG-dimension of the skew group algebras
  6. KG-dimension of skew gentle algebras
  7. KG-dimension of (skew) Brauer graph algebras

Key Result

Theorem 1.3

Assume that a group $G$ acts on an algebra $\Lambda$ and $\bar{\Lambda}$ is the associated skew group algebra. Then for any $f, h\in \mathrm{H}(\mathrm{mod}\hbox{-}\Lambda)$, the functor $\mathrm{H}F_\lambda: \mathrm{H}(\mathrm{mod}\hbox{-}\Lambda) \to \mathrm{H}(\mathrm{mod}\hbox{-}\bar{\Lambda})$

Figures (1)

  • Figure :

Theorems & Definitions (39)

  • Definition 1.2: Skew group algebras
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • ...and 29 more