2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221782In this paper, we examine the `derived completion' of the representation ring of a pro-p group G_p^ with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the Eilenberg-MacLane spectrum HZ, and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we define a deformation representation ring functor R[-] from groups to ring spectra, and show that the map R[G_p^] --> R[G] becomes an equivalence after completion when G is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the p-adic Heisenberg group.This is the version published by Algebraic & Geometric Topology on 26 February 2006Algebraic TopologyGroup Theory55P60, 19A22, 55P43text