Products on -modules

Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of -modules such as , and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over that are concentrated in degrees divisible by ; this guarantees that various obstruction groups are trivial. We extend these results to the cases where or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising -local -modules as -modules.