The Category of $H$-Modules over a Spectrum

The category of H-modules over a ring spectrum is introduced. We prove that mapping cones can be constructed in the category and that these mapping cones are unique up to equivalence. Other H-modules are constructed by induction and a limit process. We prove that each of the classical Thom spectra MO, MSO, MU, MSU, and MSp is a convergent H-module over itself. Finally we construct a tower of homology theories and natural transformations from MU.