Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 6, Number 1 (2006), 253-285.
Completed representation ring spectra of nilpotent groups
In this paper, we examine the “derived completion” of the representation ring of a pro- group 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 from groups to ring spectra, and show that the map becomes an equivalence after completion when is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the –adic Heisenberg group.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.agt/1513796512