Many of the standard sources which discuss the Hochschild Kostant Rosenberg theorem and cyclic homology for smooth varieties such as Loday and Weibel's paper "The Hodge Filtration and Cyclic Homology" ignore the positive characteristic case. Based upon these sources, I wasn't really sure if this was from a lack of knowledge or because the theorems are just not really that good in characteristic p. Here are a few rather simple(and hopefully correct!) observations about Hochschild homology and cyclic homology of smooth varieties over a field of characteristic p>0 which hopefully get the ball rolling. These are all trivial observations(as long as they are right) but they seem to suggest that there is some interesting math in the characteristic p>0 case, and I was wondering whether I had made a mistake or misunderstood something in the literature/what the opinion of experts was on these questions.

1) If $X$ is a smooth scheme over a field k of characteristic $p>0$, then we can prove the Hochschild Kostant Rosenberg theorem as follows. The basic observation is that to compute $HH_*(X)=Tor_{O_{X\times X}}(\Delta_{*}O_X,\Delta_{*}OX)$ by the adjunction this is the same as $Tor_{O_X}(\Delta^*\Delta_*O_X,O_X)$ but the complex in the first argument is canonically isomorphic to the tangent complex $\bigoplus\Lambda^iT(X)[-i]$ as proven on page 247 in Huybrecht's book on the Fourier Mukai transform.

2)If $A$ is a smooth commutative ring, again over a field k of characteristic $p>d$, the Krull dimension of the ring, then all the arguments that are given in Loday's book regarding the relationship between de Rham cohomology and cyclic homology seem to work exactly the same when the characteristic $p>d$. In particular, the spectral sequence converging to cyclic homology still degenerates on the second page. This should lead to the following scheme theoretic theorem as well. The periodic cyclic homology is isomorphic to $\prod H^*_{dr}$ if the characteristic $p>d$.

3) The above theorems seems to suggest that maybe the above degenerates for smooth algebras over a field, independent of the characteristic. Somewhat independent of that one could wonder if the de Rham cohomology and periodic cyclic homology always agree for smooth varieties over a field. Does anyone know of any counterexamples to this? Again, in the affine case, this might for example follow from a sort of Cartier isomorphism, optimistically a quasi-iso from $(C^*(A,A)((u)),d+uB) \mapsto C^*(A,A)((u)),d)$. In the case of ordinary de Rham theory, for general schemes there are obstructions to realizing the Cartier isomorphism at the chain level like this--- but I think these obstructions all vanish for affine schemes, hence this guess. Anyways, I have the impression that Kaledin proved something like this, but I haven't had a chance to study it yet, so I thought I'd just ask the MO community.