On twists of modules over noncommutative Iwasawa algebras (Q299146)

Let \(G\) be a compact \(p\)-adic Lie group with a closed normal subgroup \(H\) such that \(G/H:=\Gamma\simeq \mathbb{Z}_p\) (the ring of \(p\)-adic integers, \(p\) an odd prime). Let \(\mathcal{O}\) be the ring of integers of a finite extension of \(\mathbb{Q}_p\) and let \(\mathcal{O}[[G]]\) be the associated Iwasawa algebra. Noncommutative Iwasawa theory, as presented for example in [\textit{J. Coates} et al., Publ. Math., Inst. Hautes Étud. Sci. 101, 163--208 (2005; Zbl 1108.11081)] deals with \(\mathcal{O}[[G]]\)-modules which are finitely generated over \(\mathcal{O}[[H]]\), i.e., modules in the category \(\mathcal{M}_H(G)\). The paper under review provides a noncommutative version of the \textit{twisting lemma}: for any \(M\in \mathcal{M}_H(G)\) there exists a continuous character \(\rho:\Gamma \rightarrow \mathbb{Z}_p^*\) such that, for any open normal subgroup \(U\) of \(G\), the group of \(U\) coinvariants of \(M(\rho):=M\otimes_{\mathbb{Z}_p}\mathbb{Z}_p(\rho)\) is finite (where \(\mathbb{Z}_p(\rho)\) is a free rank one \(\mathbb{Z}_ p\)-module on which \(\Gamma\) acts via \(\rho\)).NEWLINENEWLINEThe authors use linear algebra to prove the statement for a module \(N\) such that \(N\otimes_{\mathbb{Z}_p}\mathbb{Q}_p\) is isomorphic to \((\mathbb{Z}_p[[H]]\otimes_{\mathbb{Z}_p}\mathbb{Q}_p)^{\oplus d}\). Then, for any \(M\in \mathcal{M}_H(G)\), they provide a surjection \(N \twoheadrightarrow M\) (for some \(N\) as above) to prove the general case.