Derived Completions in Stable Homotopy theory

It has long been recognized that the development of a theory of ring and module spectra, which bears the same relationship to the category of spectra as the ordinary theory of rings and modules does to the category of abelian groups, is a very desirable thing. A number of such theories exist. The different approaches [12] and [15] solve the problem in a satisfactory way, and more recently (see [18]) versions using orthogonal spectra and Γ-spaces are also available. The goal of transporting constructions which are available for ordinary rings and modules to this new category of ring spectra is also a very worthwhile one. Some of these constructions have already been made by the authors of [12] and [15]. In this paper, we will use the notion of an S-algebra (as in [12]) as the spectrum version of a ring. Given an S-algebra A and module spectra M and N , one can construct spectra M ∧ A N and HomA(M,N), analogous to the constructions M ⊗ A N and HomA(M,N) for