We present an abstract version of Goerss-Hopkins theory in the setting of a prestable @math -category equipped with a suitable periodicity operator. In the case of the @math -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.
To give a different example, Goerss-Hopkins towers were also used by the first author to prove that at large primes, there is an equivalence @math of homotopy categories between @math -local spectra and differential @math -comodules @cite_0 . The proof of that result proceeds by comparing topological and algebraic towers and so crucially depends on the additional generality provided in this note.
@cite_0
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We show that if @math is a @math -local Landweber exact homology theory of height @math and @math , then there exists an equivalence @math between homotopy categories of @math -local spectra and differential @math -comodules, generalizing Bousfield's and Franke's results to heights @math .