On simple-minded systems and τ-periodic modules of self-injective algebras

Let A be a finite-dimensional self-injective algebra over an algebraically closed field, 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 S be a simple-minded system of the stable module category modA. We show that the intersection S boolean AND 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. (C) 2020 Elsevier Inc. All rights reserved.