Complete cohomology theories and the homology of their omega spectra

Suppose Ê is a spectrum obtained by the completion of a ring spectrum E at some ideal I of its coefficients E∗. We study the homology of the Ω spectrum representing Ê and show that the techniques of coalgebraic algebra provide a language enabling a simple description of this homology and of the homological view of the process of completion. Results are obtained for a wide range of connective theories and the main examples of complete Landweber exact theories including that of Morava E theory.