On relative projectivity of lattices in Auslander-Reiten components for group rings

Abstract

Let G be a finite group, Q a p-subgroup of G and P a Sylow p-subgroup of $N_G(Q).$ Let $\mathcal{O}$ be a complete discrete valuation ring of characteristic zero with residue class field $k:=\mathcal{O}/\pi \mathcal{O}$ of characteristic p > 0. Suppose that the group ring $\mathcal{O}G$ is of infinite representation type and $\mathcal{O}$ is sufficiently large. Let ${\mathbf{\Theta }}_s$ be a stable Auslander-Reiten component containing a Scott $\mathcal{O}G$-lattice S(Q) with vertex Q. Then all the $\mathcal{O}G$-lattices in ${\Theta }_s\setminus \{{\tau }^{i}S(Q)\}$ have P as their vertices. Also, we show that there exists an indecomposable $\mathcal{O}G$-lattice L with vertex P such that a kG-module $L/\pi L$ is Q-projective.

Communicated by J. Zhang

Keywords:

• Auslander-Reiten quivers
• representations of finite groups
• vertices

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 16G70
• 20C11
• 20C20

Acknowledgment

The author would like to thank the referee for reading the manuscript carefully.

Additional information

Funding

The author is supported partially by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 19K03451.