Modules over infinite nilpotent groups (Q2765552)

Let \(G\) be a free nilpotent group of finite rank, \(Z(G)\) the center of \(G\), \(F\) a field, \(M\) a finitely generated \(FG\)-module. The module \(M\) is called reduced if there is a prime ideal \(P\) of the group ring \(FZ(G)\) such that \(M\) is annihilated by \(P\) and is torsion-free as \(FZ(G)/P\)-module. The author discusses the question: how can one construct a faithful module for \(FG\)? He proposes a way of construction, which is based on the following main results of the paper.NEWLINENEWLINENEWLINETheorem 1. Let \(G\) be a free nilpotent group of class 2 and let \(M\) be a cyclic reduced \(FG\)-module which is faithful for \(G\). Let \(P\) denote the annihilator of \(M\) in \(FZ(G)\). Then either: (1) \(M\cong FG/PFG\); or (2) there is a normal subgroup \(H\) of \(G\) such that \(G/H\) is infinite cyclic and \(M\) is torsion-free as \(FH/PFH\)-module; furthermore, \(M\) contains a finitely generated \(FH\)-submodule \(U\) such that \(M/U\) is \(FZ(G)/P\)-torsion.NEWLINENEWLINENEWLINETheorem 2. Let \(G\) be a free nilpotent group of finite rank and let \(M\) be a finitely generated reduced \(FG\)-module which is faithful for \(G\). Let \(P\) denote the annihilator of \(M\) in \(FZ(G)\). Then either: (1) \(M\) has a non-zero submodule of the form \(V\otimes_{F[G,G]}FG\) for some \(F[G,G]\)-submodule \(V\) of \(M\); or (2) there is a normal subgroup \(H\) of \(G\) such that \(G/H\) is infinite cyclic and \(M\) is torsion-free as \(FH/PFH\)-module; furthermore, \(M\) contains a finitely generated \(FH\)-submodule \(U\) such that \(M/U\) is \(F[G,G]/PF[G,G]\)-torsion.