Moduli Spaces of Commutative Ring Spectra

Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E∗E is flat over E∗. We wish to address the following question: given a commutative E∗-algebra A in E∗E-comodules, is there an E∞-ring spectrum X with E∗X ∼= A as comodule algebras? We will formulate this as a moduli problem, and give a way – suggested by work of Dwyer, Kan, and Stover – of dissecting the resulting moduli space as a tower with layers governed by appropriate André-Quillen cohomology groups. A special case is A = E∗E itself. The final section applies this to discuss the Lubin-Tate or Morava spectra En. Some years ago, Alan Robinson developed an obstruction theory based on Hochschild cohomology to decide whether or not a homotopy associative ring spectrum actually has the homotopy type of an A∞-ring spectrum. In his original paper on the subject [35] he used this technique to show that the Morava K-theory spectra K(n) can be realized as an A∞-ring spectrum; subsequently, in [3], Andrew Baker used these techniques to show that a completed version of the Johnson-Wilson spectrum E(n) can also be given such a structure. Then, in the mid-90s, the second author and Haynes Miller showed that the entire theory of universal deformations of finite height formal group laws over fields of non-zero characteristic can be lifted to A∞-ring spectra in an essentially unique way. This implied, in particular, that the Morava E-theory (or Lubin-Tate) spectra En were A∞ (which could have been deduced from Baker’s work), but it also showed much more. Indeed, the theory of Lubin and Tate [25] gives a functor from a category of finite height formal group laws to the category of complete local rings, and one way to state the results of [34] is that this functor factors in an essentially unique way through A∞-ring spectra. It was the solution of the diagram lifting problem that gave this result its additional heft; for example, it implied that the Morava stabilizer group acted on En – simply because Lubin-Tate theory implied that this group acted on (En)∗. In this paper, we would like to carry this program several steps further. One step forward would be to address E∞-ring spectra rather than A∞-ring spectra. There is an existing literature on this topic developed by Robinson and others, some based on Γ-homology. See [36], [37], and [4]. This can be used, to prove, among other things, that the spectra En are E∞, and we guess that the obstruction theory we uncover here reduces to that theory. Another ∗The authors were partially supported by the National Science Foundation.