Let G be a closed subgroup of the extended nth Morava stabilizer group and let E_n^{hG} denote the continuous homotopy fixed point spectrum of Devinatz and Hopkins. I will define the notion of "essentially finite rank" for certain kinds of modules and observe that this is relevant to the E_n^{hG} homology of K(n-2) acyclic finite spectra annihilated by p. I will outline a program for proving that these modules are of essentially finite rank; this is relevant to the title of this talk because the K(n)-local sphere is in fact E_n^{hG}, where G is the entire extended nth Morava stabilizer group. |