# Hochschild and cyclic homology of smooth varieties

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.

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Oh boy, I haven't thought of this in years. In the early days of cyclic homology I remember trying to get to exactly that conclusion -- that the higher differentials in the spectral sequence for periodic cyclic homology vanish in the smooth char p case -- using some Cartier isomorphism idea. I was using some chain-level universal-example argument. It seemed to work, but I never really finished the job or wrote it down or anything. I remember also wondering, what if you get so to speak a different reason for degenerating in every characteristic. What does that do for you over $\mathbb Z$? –  Tom Goodwillie Aug 16 '10 at 21:39
Or rather, it's easy, using the Cartier isomorphism, to see that the higher differentials are zero. What I was having fun with was seeing that the extensions are trivial. I can't remember how far I got. –  Tom Goodwillie Aug 16 '10 at 23:44
Hum... that's interesting, you seem to be saying that there is a Cartier isomorphism from `$HH_*(A)((u)) \mapsto HP_*(A)$' for all smooth affine A over a field of characteristic p. How does that work? Kaledin appears to have a construction... but without going deeper into the argument it is stated only for $dimA<2p$ (Kaledin considers a much wider range of A than we consider here). For our specific case, this gives a slightly better result than other authors but not the full thing. I've thought a little bit about the Z stuff, but I guess it would depend on the answer over all the primes :) –  Daniel Pomerleano Aug 17 '10 at 18:04
Nevermind, I think this is also explained in section 4 of Kaledin's arxiv.org/PS_cache/arxiv/pdf/0708/0708.1574v2.pdf –  Daniel Pomerleano Aug 17 '10 at 19:37

HKR is true in characteristic $p>0$ as soon as you assume that $p$ is greater than the dimension of $X$.

You can find a very nice proof of that in a recent preprint of Arinkin-Caldararu: http://arxiv.org/abs/1007.1671

I am sure this is not new, but the reason I am quoting this paper is that you will find in it a very nice generalization of HKR for closed embeddings $X\subset Y$ (where the result works in positive characteristic whenever it is greater than the codimension). The Hochschild case is the diagonal inclusion of $X$ into $X\times X$.

EDIT: I am not an expert in algebraic geometry, but I believe your question is related to Deligne-Illusie proof of Hodge-to-de Rham degeneration, which has the same kind of restriction (p greater than the dimension).

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Typo: Caldararu. Besides, the classical HKR is just the beginning of the story, the richer facts are coming from the analysis at the chain level and are related to the Hodge-to-de Rham degeneration. Look at the various lecture notes of Dima Kaledin, some links are at ncatlab.org/nlab/show/Dmitry+Kaledin –  Zoran Skoda Apr 26 '11 at 9:10
Damien: Daniel already points out in his post that when characteristic > dimension, it works, see his item #2. –  Kevin H. Lin Apr 26 '11 at 9:17
Kevin: agreed. But it seems that at the same time he says in item #1 that HKR works without any restriction. –  DamienC Apr 26 '11 at 9:46
Ah, ok. Then that's my bad. –  Kevin H. Lin Apr 26 '11 at 10:03
Hi guys, this was based on a false assumption. The proof I had in mind when writing that only works when the diagonal subsheaf $\Delta \to X \times X$ was the zero section of a vector bundle and that one could build a Koszul resolution for $\Delta$. I later realized this is wildly false. The fact about char(p)> dim(X) should be because when writing the HKR map to differential operators you need to divide by n! for n<dim(X) Thanks for the neat reference to Arinkin-Caldararu. –  Daniel Pomerleano Apr 26 '11 at 23:54