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

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 ${Fix}_s$.