Abstract. 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  he used this technique to show that the Morava K-theory spectra K(n) can be realized as an A∞-ring spectrum; subsequently, in , 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.
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