On unstable modules over the Dickson algebras, the Singer functors R_s and the functors Fix_s

摘要

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 Rs, 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 ?_s to R_s is introduced and is related to the indecomposables functor and the functor Fix_s.