Algebraic & Geometric Topology

Completed representation ring spectra of nilpotent groups

Tyler Lawson

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In this paper, we examine the “derived completion” of the representation ring of a pro-p group Gp 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 , 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[Gp]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.

Article information

Algebr. Geom. Topol., Volume 6, Number 1 (2006), 253-285.

Received: 11 April 2005
Revised: 31 October 2005
Accepted: 5 January 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P60: Localization and completion
Secondary: 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 19A22: Frobenius induction, Burnside and representation rings

S-algebra R-module completion Bousfield localization representation ring


Lawson, Tyler. Completed representation ring spectra of nilpotent groups. Algebr. Geom. Topol. 6 (2006), no. 1, 253--285. doi:10.2140/agt.2006.6.253.

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