#### Volume 15, issue 6 (2015)

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The algebraic duality resolution at $p=2$

### Agnès Beaudry

Algebraic & Geometric Topology 15 (2015) 3653–3705
##### Abstract

The goal of this paper is to develop some of the machinery necessary for doing $K\left(2\right)$–local computations in the stable homotopy category using duality resolutions at the prime $p=2$. The Morava stabilizer group ${\mathbb{S}}_{2}$ admits a surjective homomorphism to ${ℤ}_{2}$ whose kernel we denote by ${\mathbb{S}}_{2}^{1}$. The algebraic duality resolution is a finite resolution of the trivial ${ℤ}_{2}\left[\left[{\mathbb{S}}_{2}^{1}\right]\right]$–module ${ℤ}_{2}$ by modules induced from representations of finite subgroups of ${\mathbb{S}}_{2}^{1}$. Its construction is due to Goerss, Henn, Mahowald and Rezk. It is an analogue of their finite resolution of the trivial ${ℤ}_{3}\left[\left[{\mathbb{G}}_{2}^{1}\right]\right]$–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 ${\mathbb{S}}_{2}$ at the prime $2$. We also describe the maps in the algebraic duality resolution with the precision necessary for explicit computations.

##### Keywords
finite resolution, K(2)-local, chromatic homotopy theory
##### Mathematical Subject Classification 2010
Primary: 55Q45
Secondary: 55T99, 55P60