On simple-minded systems and $\tau$-periodic modules of self-injective algebras

TL;DR

This paper investigates the structure of simple-minded systems within the stable module category of self-injective algebras, revealing restrictions on modules in homogeneous tubes and their quasi-lengths.

Contribution

It establishes that simple-minded systems intersect stable tubes in a limited way, specifically modules with quasi-length less than the tube's rank, excluding homogeneous tubes.

Findings

Intersection of simple-minded systems with stable tubes is smaller than the tube's rank.

Modules in homogeneous tubes cannot be part of any simple-minded system.

Modules in the intersection have quasi-length strictly less than the tube's rank.

Abstract

Let $A$ be a finite-dimensional self-injective algebra over an algebraically closed field, $\mathcal{C}$ a stably quasi-serial component (i.e. its stable part is a tube) of rank $n$ of the Auslander-Reiten quiver of $A$, and $\mathcal{S}$ be a simple-minded system of the stable module category $\stmod{A}$. We show that the intersection $\mathcal{S}\cap\mathcal{C}$ is of size strictly less than $n$, and consists only of modules with quasi-length strictly less than $n$. In particular, all modules in the homogeneous tubes of the Auslander-Reiten quiver of $A$ cannot be in any simple-minded system.