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.

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关于自射代数的简单系统和τ-周期模

设 $A$ 是代数闭域上的有限维自射代数，$\mathcal{C}$ 是 Auslander- $A$ 的 Reiten quiver 和 $\mathcal{S}$ 是稳定模块类别 $\stmod{A}$ 的简单系统。我们表明交集 $\mathcal{S}\cap\mathcal{C}$ 的大小严格小于 $n$，并且仅由准长度严格小于 $n$ 的模块组成。特别是，$A$ 的 Auslander-Reiten 箭袋的同质管中的所有模块都不能在任何头脑简单的系统中。