Cyclic extensions of free pro-\(p\) groups and \(p\)-adic modules. (Q2448555)

A classical result in the theory of integral representations states that a finitely generated \(\mathbb Z_p\)-torsion free \(\mathbb Z_pC_p\) module decomposes as a finite direct sum of cyclic modules of the form \(\mathbb Z_pC_p\), \(\mathbb Z_p\) or \(J(\mathbb Z_pC_p)\). Here \(p\) is a prime, \(C_p\) is a cyclic group of order \(p\), \(\mathbb Z_pC_p\) is the group ring over the \(p\)-adic integers \(\mathbb Z_p\), and \(J(\mathbb Z_pC_p)\) is the augmentation ideal. The main goal of the paper under review is to show the very far from obvious fact that this result extends to infinitely generated \(\mathbb Z_p\)-torsion free pro-\(p\) \(\mathbb Z_pC_p\) modules. The \(\mathbb Z_p\)-torsion free pro-\(p\) \(\mathbb Z_{p^n}C_p\) modules are also described that arise from the action of \(C_{p^n}\) on the Abelianization \(F/[F,F]\), provided that the centralizers of the non-identity elements of finite order in the pro-\(p\) semidirect product \(F\rtimes C_{p^n}\) are all finite.