Systems of Submodules and an Isomorphism Problem for Auslander-Reiten Quivers

Abstract

Fix a poset $\mathcal P$ and a natural number $n$. For various commutative local rings $\Lambda$, each of Loewy length $n$, consider the category $\textrm{sub}_\Lambda\mathcal P$ of $\Lambda$-linear submodule representations of $\mathcal P$. We give a criterion for when the underlying translation quiver of a connected component of the Auslander-Reiten quiver of $\sub_\Lambda\mathcal P$ is independent of the choice of the base ring $\Lambda$. If $\mathcal P$ is the one-point poset and $\Lambda=\mathbb Z/p^n$, then $\textrm{sub}_\Lambda\mathcal P$ 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.