At the end of a post on crystalline cohomology I asked a question which was answered the same day by Bhargav Bhatt. It turns out that all cohomology groups of the sheaf Ω^1 (differentials compatible with divided powers) on the crystalline site of a scheme in characteristic p are zero! As a consequence Bhargav and I get a short proof of Berthelot’s comparison theorem relating crystalline and de Rham cohomology. If you think you’re confused, note that the de Rham cohomogy is computed on the scheme and not on the crystalline site. Here is a link to a recent version of the write-up — it should appear on the arxiv soon.

As an example, let’s consider an algebraically closed field k and the power series ring A = k[[t]]. It turns out that A has a p-basis, namely {t}. This simply means that every element a of A can be uniquely written as ∑_{i = 0,1,…,p-1} a_i^pt^i. Let W = W(k) be a Cohen ring for k (i.e., the Witt ring). By a result of Berthelot and Messing the category of crystals in quasi-coherent modules on (Spec(A)/Z_p)_{cris} is equivalent to the category of pairs (M, ∇) where M is a p-adically complete W[[t]]-module and ∇ : M —> Mdt is a topologically quasi-nilpotent connection. Given F corresponding to (M, ∇) the comparison theorem (in this special case) states

the complex ∇ : M —> Mdt is quasi-isomorphic to RΓ(F).

You can generalize this to power series rings in more variables. In fact, you can’t find exactly this statement in the preprint linked to above; it is just that the method of the proof works in this case too. Upshot: comparison with the de Rham complex works for rings with p-bases.

Computing crystalline cohomology over a power series ring is relevant in situations where one wants to do deformation theory. For example, I was recently asked by Davesh Maulik if there is an explanation of Artin’s result on specialization of Picard lattices of supersingular K3 surfaces which avoids the formal Brauer group. What Artin proves is that the Neron-Severi rank doesn’t jump in a family of supersingular K3 surfaces. It turns out that, using crystalline cohomology, given a family of K3’s X/k[[t]], you can split this question into two parts:

- When can you lift elements of H^2_{cris}(X_0/W) to elements of H^2_{cris}(X/W)?
- Can you lift an invertible sheaf on X_0 to X if its crystalline c_1 lifts to X?

Of course then you generalize (also Artin’s result is more general) and you can ask these questions for any smooth proper X/k[[t]]. It turns out that both questions have a positive answer under some conditions. I have written a short note with a discussion. Enjoy!