On a conjecture of O. ~I. ~Tavgen'
This article is devoted to a conjecture of \textit{O.~I.~Tavgen'} about a group-theoretic characterization of linearity for an amalgamated product of two groups [Dokl. Akad. Nauk, Ross. Akad. Nauk 328, No. 2, 153-156 (1993; Zbl 0830.20071)]. The authors give a negative answer to a more general conjecture of O.~Tavgen' about Artin-Rees properties: Let \(G\) be an analytic pro-\(p\)-group and let \(M\) be a finitely generated (in the topological sense) \(\mathbb{Z}_p\)-torsion free \(\mathbb{Z}_pG\)-module (i.e., \(M\) is free as an Abelian pro-\(p\)-group). Let \(U\) be a closed submodule and let \(\dim_{\mathbb{Z}_p}U<\infty\). Then there is a closed submodule \(N\subseteq M\) such that \(N\cap U=0\) and \(\dim_{\mathbb{Z}_p}M/N<\infty\).
scientific article; zbMATH DE number 1431904