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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
The algebraic duality resolution at $p=2$

Agnès Beaudry

Algebraic & Geometric Topology 15 (2015) 3653–3705

The goal of this paper is to develop some of the machinery necessary for doing K(2)–local computations in the stable homotopy category using duality resolutions at the prime p = 2. The Morava stabilizer group S2 admits a surjective homomorphism to 2 whose kernel we denote by S21. The algebraic duality resolution is a finite resolution of the trivial 2[[S21]]–module 2 by modules induced from representations of finite subgroups of S21. Its construction is due to Goerss, Henn, Mahowald and Rezk. It is an analogue of their finite resolution of the trivial 3[[G21]]–module 3 at the prime p = 3. The construction was never published and it is the main result in this paper. In the process, we give a detailed description of the structure of Morava stabilizer group S2 at the prime 2. We also describe the maps in the algebraic duality resolution with the precision necessary for explicit computations.

finite resolution, K(2)-local, chromatic homotopy theory
Mathematical Subject Classification 2010
Primary: 55Q45
Secondary: 55T99, 55P60
Received: 19 December 2014
Revised: 30 March 2015
Accepted: 14 April 2015
Published: 12 January 2016
Agnès Beaudry
Department of Mathematics
University of Chicago
1118 East 58th Street
Chicago, IL 60637