Bounded submodules of modules

Abstract

Let m , n be positive integers such that m ⩽ n and k be a field. 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 paper 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.