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