#### Volume 12, issue 4 (2012)

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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
##### 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 ${\mathfrak{ℨ}}_{s}$ 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
Primary: 55S10
Secondary: 18E10