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Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as in the affine case the theorem is still true so this seems like a reasonably naive question. Although there is no discussion of this in the paper, I guess the methods in the Swan paper roughly give the result as long the dimension of the variety is smaller than the characteristic (correct?) but don't generalize completely.

If it isn't known, any ideas about the possibility of looking for a counterexample could also be helpful. If the above statement is true it seems reasonable to look at surfaces in characteristic 2 such as Enriques surfaces or K3 surfaces. I have no idea if it could be possible to do calculations in these cases.

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See this paper of mine and Gabriele Vezzosi. We prove that HKR holds in particular for smooth proper schemes $X$ of dimension at most $p$, the characteristic prime. In particular, it holds for smooth proper surfaces in characteristic $2$.

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    $\begingroup$ Thanks Benjamin for pointing out your very interesting paper! $\endgroup$
    – Daniel Pomerleano
    Commented Nov 20, 2017 at 10:55
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Antieau, Bhatt, and Mathew have constructed examples of $2p$-dimensional smooth projective varieties where the HKR isomorphism does not hold. See arXiv:1909.11437.

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