Bounded Submodules of Modules

Abstract

Let $m$, $n$ be positive integers such that $m\le n$. 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.

References in corpus (2)

• The Auslander-Reiten Translation in Submodule Categories
• Submodule Categories of Wild Representation Type

Cited by in corpus (11)

• Invariant Subspaces of Nilpotent Linear Operators. I
• A note on Bruhat decomposition of GL(n) over local principal ideal rings
• Operations on Arc Diagrams and Degenerations for Invariant Subspaces of Linear Operators
• The entries in the LR-tableau
• Operations on Arc Diagrams and Degenerations for Invariant Subspaces of Linear Operators. Part II
• Hall polynomials via automorphisms of short exact sequences
• A Construction of Metabelian Groups
• Abelian p-groups with a fixed elementary subgroup or with a fixed elementary quotient
• Systems of submodules and a remark by M.C.R. Butler
• Invariant Subspaces of Nilpotent Operators. Level, Mean, and Colevel: The Triangle $\mathbb{T}(n)$
• Geometric representations of GL(n,R), cellular Hecke algebras and the embedding problem