Completed representation ring spectra of nilpotent groups

Lawson, Tyler

doi:10.2140/agt.2006.6.253

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Completed representation ring spectra of nilpotent groups

Tyler Lawson

Algebraic & Geometric Topology 6 (2006) 253–285

DOI: 10.2140/agt.2006.6.253

arXiv: 0902.4867

Abstract

In this paper, we examine the “derived completion” of the representation ring of a pro- $p$ group $G_{p}^{\land}$ 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 $ℍ ℤ$ , 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 $R [G_{p}^{\land}] \to R [G]$ 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.

Keywords

S-algebra, R-module, completion, Bousfield localization, representation ring

Mathematical Subject Classification 2000

Primary: 55P60

Secondary: 55P43, 19A22

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Publication

Received: 11 April 2005

Revised: 31 October 2005

Accepted: 5 January 2006

Published: 26 February 2006

Authors

Tyler Lawson
Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139 USA

Volume 6, issue 1 (2006)