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This a alternative form of the question I posted some time ago.

We, the people who don't know topology, are told that in characteristic p the formalizm of DG algebras is not quite adequate for applications, that one needs to work with spectra. Yet it seems that in some cases you can still get away with working on a DG algebra level, e.g. when the algebra lifts $mod\ p^2$ and some cohomological vanishing happens in degrees $\geq 2p$ (as in Kaledin's Proposition 5.10 of https://arxiv.org/pdf/math/0611623.pdf or the result of Deligne-Illusie on the de Rham complex in characterstic p).

I am wondering if there is a standard result somewhere that says: "the cohomology of little disk spaces (or configuration spaces) with coefficients in $\mathbb{F}_p$ are such and such, so in full generality DG algebras would be plain wrong but if some vanishing happends then working with a DG algebra equipped with some Steenrod operations is a good idea".

I apologize for the sloppy question, and will be grateful for suggesting a better version of the same question.

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    $\begingroup$ Associative dg algebras do behave in all characteristics. Problems arise with any other type of algebra, not just a specific class of operads, essentially because homology of symmetric groups is non-trivial in finite characteristic. $\endgroup$
    – Jon Pridham
    Dec 21, 2018 at 8:22
  • $\begingroup$ Where can I find computations over $\mathbb{F}_p$? $\endgroup$
    – Vladimir Baranovsky
    Dec 21, 2018 at 20:22
  • $\begingroup$ For one example, in the commutative setting, Avramov's papers on the cotangent complex do a lot of computations for simplicial algebras by DG methods. $\endgroup$
    – Jon Pridham
    Dec 23, 2018 at 8:10

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