Bounded submodules of modules. (Q2573481)

For a ring \(\Lambda\), the submodule category \(\mathcal S(\Lambda)\) is defined as follows: the objects in \(\mathcal S(\Lambda)\) are pairs \((A\subseteq B)\) where \(B\) is a finite length \(\Lambda\)-module and \(A\) is a submodule of \(B\), and the morphisms from \((A\subseteq B)\) to \((A'\subseteq B')\) are given by the \(\Lambda\)-linear maps \(f\colon B\to B'\) which map \(A\) to \(A'\). For a positive integer \(m\), the category \(\mathcal S_m(\Lambda)\) of bounded submodules is the full subcategory of \(\mathcal S(\Lambda)\) consisting of all pairs \((A\subseteq B)\) with \(\text{rad}^mA=0\). In this paper, the author studies the category \(\mathcal S_m(\Lambda)\) in the case \(\Lambda=k[T]/T^n\), where \(k\) is a field and \(m\leq n\). The category \(\mathcal S_m(k[T]/T^n)\) is a Krull-Schmidt category with Auslander-Reiten sequences. As the main result, the author obtains a complete characterization of the representation types of the category \(\mathcal S_m(k [T]/T^n)\), as follows: (a) \(\mathcal S_m(k[T]/T^n)\) has finite representation type if \(n<6\) or if \(m<3\), or if \((m,n)=(3,6)\); (b) \(\mathcal S_m(k[T]/T^n)\) has tame infinite representation type if \((m,n)\) is one of the pairs \((3,7)\), \((4,6)\), \((5,6)\) or \((6,6)\); (c) \(\mathcal S _m(k[T]/T^n)\) has wild representation type in all other cases. Moreover, the author obtains a description of the indecomposable modules and the Auslander-Reiten quivers in the finite and tame cases. The methods of the paper are based on homological algebra, Auslander-Reiten theory and covering theory.