The algebraic duality resolution at p = 2

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 $S_{2}$ admits a surjective homomorphism to $ℤ_{2}$ whose kernel we denote by $S_{2}^{1}$ . The algebraic duality resolution is a finite resolution of the trivial $ℤ_{2} [[S_{2}^{1}]]$ –module $ℤ_{2}$ by modules induced from representations of finite subgroups of $S_{2}^{1}$ . Its construction is due to Goerss, Henn, Mahowald and Rezk. It is an analogue of their finite resolution of the trivial $ℤ_{3} [[G_{2}^{1}]]$ –module $ℤ_{3}$ 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 $S_{2}$ at the prime $2$ . We also describe the maps in the algebraic duality resolution with the precision necessary for explicit computations.

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Keywords

finite resolution, K(2)-local, chromatic homotopy theory

Mathematical Subject Classification 2010

Primary: 55Q45

Secondary: 55T99, 55P60

References

Publication

Received: 19 December 2014

Revised: 30 March 2015

Accepted: 14 April 2015

Published: 12 January 2016

Authors

Agnès Beaudry
Department of Mathematics University of Chicago 1118 East 58th Street Chicago, IL 60637 USA

Volume 15, issue 6 (2015)

Agnès Beaudry

Abstract