Bounded Submodules of Modules

Let $m$, $n$ be positive integers such that $m\leq n$. We consider all pairs $(B,A)$ where $B$ is a finite dimensional $T^n$-bounded $k[T]$-module and $A$ is a submodule of $B$ which is $T^m$-bounded. They form the objects of the submodule category $S_m(k[T]/T^n)$ which is a Krull-Schmidt category with Auslander-Reiten sequences. The case $m=n$ deals with submodules of $k[T]/T^n$-modules and has been studied well. In this manuscript we determine the representation type of the categories $S_m(k[T]/T^n)$ also for the cases where $m<n$: It turns out that there are only finitely many indecomposables in $S_m(k[T]/T^n)$ if either $m<3$, $n<6$, or $(m,n)=(3,6)$; the category is tame if $(m,n)$ is one of the pairs $(3,7)$, $(4,6)$, $(5,6)$, or $(6,6)$; otherwise, $S_m(k[T]/T^n)$ has wild representation type. Moreover, in each of the finite or tame cases we describe the indecomposables and picture the Auslander-Reiten quiver.