The discrete module category for the ring of -theory operations

We study the category of discrete modules over the ring of degree-zero stable operations in p -local complex K -theory, where p is an odd prime. We show that the K ( p ) -homology of any space or spectrum is such a module, and that this category is isomorphic to a category defined by Bousfield and used in his work on the K ( p ) -local stable homotopy category. We give a simple construction of cofree discrete modules and construct the analogue in the category of discrete modules of a four-term exact sequence due to Bousfield.