We present an abstract version of Goerss-Hopkins theory in the setting of a prestable ∞-category equipped with a suitable periodicity operator. In the case of the ∞category of synthetic spectra, this yields obstructions to realizing a comodule algebra as a homology of a commutative ring spectrum, recovering the results of Goerss and Hopkins.

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We study the relationship between the transchromatic localizations of Morava E-theory, L K(n−1) En, and formal groups. In particular, we show that the coefficient ring π0L K(n−1) En has a modular interpretation, representing deformations of formal groups with certain extra structure, and derive similar descriptions of the cooperations algebra and En−1-homology of this spectrum. As an application, we show that L K(1) E2 has exotic E∞ structures not obtained by K(1)-localizing the E∞ ring E2.

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