On unstable modules over the Dickson algebras, the Singer functors Rs and the functors Fixs

Powell, Geoffrey M L

doi:10.2140/agt.2012.12.2451

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On unstable modules over the Dickson algebras, the Singer functors $R_s$ and the functors $\mathrm{Fix}_s$

Geoffrey M L Powell

Algebraic & Geometric Topology 12 (2012) 2451–2491

DOI: 10.2140/agt.2012.12.2451

Abstract

The category $D_{s}$ – $U$ of unstable modules over the Steenrod algebra equipped with a compatible module structure over the Dickson algebra $D_{s}$ is studied at the prime $2$ , with applications to the Singer functor $R_{s}$ , considered as a functor from unstable modules $U$ to $D_{s}$ – $U$ . An explicit copresentation of $R_{s} M$ is given using Lannes’ $T$ –functor when $M$ is a reduced unstable module; applying Lannes’ functor ${Fix}_{s}$ , this is used to show that $R_{s}$ gives a fully-faithful embedding of $U$ in $D_{s}$ – $U$ . In addition, the right adjoint $ℨ_{s}$ to $R_{s}$ is introduced and is related to the indecomposables functor and the functor ${Fix}_{s}$ .

Keywords

Dickson algebra, Singer functor, unstable module, Lannes' $T$–functor, Fix

Mathematical Subject Classification 2010

Primary: 55S10

Secondary: 18E10

References

Publication

Received: 6 December 2010

Revised: 5 July 2012

Accepted: 2 August 2012

Published: 28 January 2012

Authors

Geoffrey M L Powell
Laboratoire Angevin de Recherche en Mathématiques UMR 6093 Faculté des Sciences Université d’Angers 2 Boulevard Lavoisier 49045 Angers France

Volume 12, issue 4 (2012)