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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

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

Dickson algebra, Singer functor, unstable module, Lannes' $T$–functor, Fix
Mathematical Subject Classification 2010
Primary: 55S10
Secondary: 18E10
Received: 6 December 2010
Revised: 5 July 2012
Accepted: 2 August 2012
Published: 28 January 2012
Geoffrey M L Powell
Laboratoire Angevin de Recherche en Mathématiques
UMR 6093
Faculté des Sciences
Université d’Angers
2 Boulevard Lavoisier
49045 Angers