The algebraic duality resolution at $p=2$

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 ${\mathbb{Z}}_2$ whose kernel we denote by ${\mathbb{S}}_2^1$. The algebraic duality resolution is a finite resolution of the trivial ${\mathbb{Z}}_2\left[\left[{\mathbb{S}}_2^1\right]\right]$–module ${\mathbb{Z}}_2$ 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 ${\mathbb{Z}}_3\left[\left[{\mathbb{G}}_2^1\right]\right]$–module ${\mathbb{Z}}_3$ 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.