Moduli Problems for Structured Ring Spectra

In this document we make good on all the assertions we made in the previous paper " Moduli spaces of commutative ring spectra " [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E ∞-ring spectra. In that paper, we discussed the the Hopkins-Miller theorem on the Lubin-Tate or Morava spectra E n ; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (E n) * X ∼ = (E n) * E n as commutative (E n) * algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E * and a commutative ring A in E * E co-modules and we wish to discuss the homotopy type of the space T M(A) of all E ∞-ring spectra so that E * X ∼ = A. We do not, a priori, assume that T M(A) is non-empty, or even that there is a spectrum X so that E * X ∼ = A as comodules. For a variety of applications we are not simply interested in this absolute problem, but in a relative version as well. We fix an E ∞-ring spectrum Y and write k = E * Y for the resulting commutative algebra in E * E comodules. Then we may choose a morphism of commutative algebras k → A in E * E-comodules and write T M(A/k) for the moduli space of Y-algebras X so that E * X ∼ = A as a k-algebra. The absolute case can be recovered by setting Y = S 0 , the zero-sphere. While we are assuming the existence of Y , we are not assuming that T M(A/k) is non-empty or even that there exists a spectrum X with E * X ∼ = A.5 which together give a decomposition of T M(A/k) as the homotopy inverse limit of a tower of fibra-tions · · · → T M n (A/k) → T M n−1 (A/k) → · · · → T M 1 (A/k) where 1. T M 1 …