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Let $K$ be a perfect field with characteristic $>0$ and $\mathcal{H}$ the category of graded connected abelian hopf algebras over $K$. By a theorem of Schoeller there is a canonical equivalence between $\mathcal{H}$ and the category of Dieudonne modules over $K.$

Question. Is there a homotopy-coherent version of Dieudonné modules and a homotopy-coherent version of Schoeller's theorem that identifies a derived version of Dieudonné module with a connected $E_\infty$-hopf algebra over $K$?

By definition, an $E_\infty$-hopf algebra over $K$ is an abelian group object (in the derived sense) in the $\infty$-category of $E_\infty$-coalgebras in the derived $\infty$-category of $K,$ which is connected if its zeroth homology is canonically $K.$

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    $\begingroup$ There's a connection between $THH$ and Witt rings and a connection between Witt rings and Dieudonné modules, so that seems like a good place to start. There is also a theorem of Goerss relating the Dieudonné module of $H_*(\Omega^\infty E;\mathbb{F}_p)$ to $E_*(\Omega^2S^3)$. It would be very valuable to understand that in a more structured way, preferably without recourse to Brown-Gitler technology and apparently-unnatural changes of grading. $\endgroup$
    – Neil Strickland
    Commented May 11, 2021 at 18:09
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    $\begingroup$ I tried to work with this some years ago but did not get very far. I struggled with some foundational problems, but they can probably be avoided using $\infty$-categories, which were not available at that point. $\endgroup$
    – Neil Strickland
    Commented May 11, 2021 at 18:11
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    $\begingroup$ @NeilStrickland I would also like to mention that there is a theory of topological Cartier modules due to Antieau and Nikolaus. $\endgroup$
    – Z. M
    Commented May 12, 2021 at 14:12
  • $\begingroup$ Thanks for your comments! $\endgroup$
    – Hadrian Heine
    Commented May 14, 2021 at 23:31

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