# Abstract Goerss-Hopkins theory

## TL;DR

This paper develops an abstract framework for Goerss-Hopkins theory within prestable $ty$-categories, providing new tools to analyze realizability problems in stable homotopy theory.

## Contribution

It introduces an abstract setting for Goerss-Hopkins theory applicable to prestable $ty$-categories with periodicity, extending classical results to synthetic spectra.

## Key findings

- Obstructions to realizing comodule algebras as homology of ring spectra
- Recovers classical Goerss-Hopkins results in synthetic spectra
- Provides a new categorical framework for homotopical realization problems

## Abstract

We present an abstract version of Goerss-Hopkins theory in the setting of a prestable $\infty$-category equipped with a suitable periodicity operator. In the case of the $\infty$-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.