Module structures and the derived functors of iterated loop functors on unstable modules over the Steenrod algebra

The calculation of the iterated loop functors and their left derived functors on the category of unstable modules over the Steenrod algebra is a non-trivial problem; Singer constructed an explicit and functorial chain complex to calculate these functors. The results of Singer are analysed to give information on the behaviour of these functors with respect to the nilpotent filtration of the category of unstable modules.We show that, if an unstable module M supports an action of an unstable algebra K, then the derived functors of the iterated loop functors applied to M support actions of iterated doubles of K. This allows the finiteness results of Henn on unstable modules which support actions of unstable algebras to be applied to deduce structural results on the derived functors of iterated loops on such modules.