#### Volume 6, issue 1 (2006)

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Completed representation ring spectra of nilpotent groups

### Tyler Lawson

Algebraic & Geometric Topology 6 (2006) 253–285
 arXiv: 0902.4867
##### Abstract

In this paper, we examine the “derived completion” of the representation ring of a pro-$p$ group ${\mathsc{G}}_{p}^{\wedge }$ 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 $R\left[-\right]$ from groups to ring spectra, and show that the map $R\left[{\mathsc{G}}_{p}^{\wedge }\right]\to R\left[\mathsc{G}\right]$ becomes an equivalence after completion when $\mathsc{G}$ 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.

##### Keywords
S-algebra, R-module, completion, Bousfield localization, representation ring
##### Mathematical Subject Classification 2000
Primary: 55P60
Secondary: 55P43, 19A22