The Algebraic Duality Resolution at $p=2$

Abstract

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.

Expository changes, with some clarifications and corrections. To appear in AGT

References in corpus (3)

• Homotopy fixed points for L_K(n)(E_n ^ X) using the continuous action
• Finite subgroups of extended Morava stabilizer groups
• The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited

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• The Chromatic Splitting Conjecture at n=p=2
• The telescope conjecture at height 2 and the tmf resolution
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• The topological modular forms of $\mathbb{R}{P}^2$ and $\mathbb{R}{P}^2\wedge \mathbb{C}{P}^2$
• Homotopy groups of $E_C^{h{G}_{24}}\wedge A_1$
• Vanishing lines in chromatic homotopy theory