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.