## Algebraic & Geometric Topology

### The algebraic duality resolution at $p=2$

Agnès Beaudry

#### 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 $S2$ admits a surjective homomorphism to $ℤ2$ whose kernel we denote by $S21$. The algebraic duality resolution is a finite resolution of the trivial $ℤ2[[S21]]$–module $ℤ2$ by modules induced from representations of finite subgroups of $S21$. Its construction is due to Goerss, Henn, Mahowald and Rezk. It is an analogue of their finite resolution of the trivial $ℤ3[[G21]]$–module $ℤ3$ 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 $S2$ at the prime $2$. We also describe the maps in the algebraic duality resolution with the precision necessary for explicit computations.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3653-3705.

Dates
Received: 19 December 2014
Revised: 30 March 2015
Accepted: 14 April 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841080

Digital Object Identifier
doi:10.2140/agt.2015.15.3653

Mathematical Reviews number (MathSciNet)
MR3450774

Zentralblatt MATH identifier
1350.55019

#### Citation

Beaudry, Agnès. The algebraic duality resolution at $p=2$. Algebr. Geom. Topol. 15 (2015), no. 6, 3653--3705. doi:10.2140/agt.2015.15.3653. https://projecteuclid.org/euclid.agt/1510841080

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