Simple-minded systems in stable module categories

TL;DR

This paper studies simple-minded systems in stable module categories, showing their invariance under stable equivalences and describing their structure across various algebra classes, with implications for the Auslander-Reiten conjecture.

Contribution

It establishes the invariance of simple-minded systems under stable equivalences and describes their structure in different algebra classes.

Findings

Simple-minded systems are invariant under stable equivalences.

Descriptions of simple-minded systems for several algebra classes.

Connections to the Auslander-Reiten conjecture.

Abstract

Simple-minded systems in stable module categories are defined by orthogonality and generating properties so that the images of the simple modules under a stable equivalence form such a system. Simple-minded systems are shown to be invariant under stable equivalences; thus the set of all simple-minded systems is an invariant of a stable module category. The simple-minded systems of several classes of algebras are described and connections to the Auslander-Reiten conjecture are pointed out.