Moduli spaces of commutative ring spectra

Abstract

A bstract . 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 Andre-Quillen cohomology groups. A special case is A = E∗E itself. The final section applies this to discuss the Lubin-Tate or Morava spectra E n . 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 offinite height formal group laws over fields of non-zero characteristic can be lifted to A ∞-ring spectra in an essentially unique way.