Systems of submodules and a remark by M.C.R. Butler

Abstract

Fix a poset $P$ and a natural number $n$. For various commutative local rings $\Lambda$, each of Loewy length $n$, consider the category $\textrm{sub}_\Lambda P$ of $\Lambda$-linear submodule representations of $P$. We give a criterion for when the underlying translation quiver of a connected component of the Auslander-Reiten quiver of $\textrm{sub}_\Lambda P$ is independent of the choice of the base ring $\Lambda$. If $\mathcal P$ is the one-point poset and $\Lambda=\mathbb Z/p^n$ for $p$ a prime number, then $\textrm{sub}_\Lambda P$ consists of all pairs $(B;A)$ where $B$ is a finite abelian $p^n$-bounded group and $A\subset B$ a subgroup. We can respond to a remark by M. C. R. Butler concerning the first occurence of parametrized families of such subgroup embeddings.