# Is the Dieudonne module actually a cohomology group?

One often times thinks of the Dieudonne module $M(X)$ of a $p$-divisible group (say over $k$, a perfect characteristic $p$ field) as being some sort of cohomology theory

$$M:\left\{p\text{- divisible groups}/k\right\}\to \left\{F\text{-crystals }/k\text{ with slopes in }[0,1]\right\}.$$

This is supported by the famous observation of Mazur-Messing-Oda that if $A/k$ is an abelian variety, then

$$M(A[p^\infty])=H^1_\text{crys}(A/W(k))$$

as $F$-crystals.

There are two natural questions that come from this, in my opinion. First, can one make sense of $M(X)$ as a crystalline cohomology group (in a literal sense) if $X$ 'comes from a scheme' (e.g. is the $p^\infty$-torsion of some group scheme). I've asked this here as well as asking when $M$ (an $F$-crystal) 'comes from a scheme'.

But, the other question, and the one I am asking here is the following. Is there some way to interpret the understanding of the Dieudonne module 'as cohomology' in a rigorous way? For example, is there a natural site over $X$ (perhaps with underlying category $p$-divisible groups over $X$) such that $M(X)$ is cohomology on this site? I would even be interested in knowing if there is a rigorous way of considering $M(X)/pM(X)$ as a cohomology theory (in the Mazur-Messing-Oda setup it's $H^1_\text{dR}(A/k)$).

The obvious guess is to create a crystalline/infinitesimal site on $X$, but I don't really know how to make this precise.

• One wants not a site on $X$ but a site whose "fundamental group" is in some way "dual" to $X$. The simplest picture of this seems like it should be the quotient of a point by the Cartier dual of $X$. Apr 3, 2016 at 20:46
• Doesn't Mazur-Messing interpret the Dieudonne module of a p-divisible group G as rigidified extensions? This suggests that defining a category of G-invariant D-modules should be possible. I would naively guess that the resulting cohomology would be "generated in degree 1", just as for abelian varieties, though. Apr 4, 2016 at 0:46