Interpretation of the formal groups arising from the DeRham-Witt complex

Question

In the accepted answer to this question, it is shown that for a proper algebraic variety $X$ we have that $H^{r-i}(X, W\Omega^i)[1/p]$ has slopes from the interval $[i, i+1[$, so namely is isomorphic to $H^{r-i}(X, W\Omega^i)[1/p][-i][i]$, the shift of a Cartier module with slopes in $[0, 1[$, which is thus a Dieudonne module, and thus is the Dieudonne module of some formal group, say $G_{i, r}$. For example, if $i=0$, this is $H^r(X, W\Omega^0)=H^r(X, \mathcal{W})=H^r(X, \mathbb{D(G_m)})=\mathbb{D}(H^r(X, \mathbb{G}_m))$, which is just the Artin-Mazur formal group law associated to $X$. I was wondering if there is a similar interpretations of other laws?

Answer 1

This is an old question but since it hasn't received much attention, let me just point out "the next" example beyond that given in the question:

Let $k$ be a perfect field of characteristic $p>0$, and let $X$ be a K3 surface over $\mathrm{Spec}\,k$ with finite height. Then there is an exact sequence \begin{equation*} 0\rightarrow\mathbb{D}(\Psi_{X})\rightarrow H^{2}_{\mathrm{cris}}(X)\rightarrow\mathbb{D}(\widehat{\mathrm{Br}}^{\ast}_{X})(-1)\rightarrow 0 \end{equation*} where $\Psi_{X}$ is the is the enlarged Brauer group of [NO85, $\S$3]. If moreover $X$ is ordinary (i.e. the height is $1$), then one finds that $$H^{1}(X,W\Omega^{1}_{X})\cong\mathbb{D}(\Psi_{X}^{\mathrm{et}})$$ (and $H^{0}(X,W\Omega^{2}_{X})\cong\mathbb{D}(\widehat{\mathrm{Br}}^{\ast}_{X})$ but that follows by duality from the Artin-Mazur example in the question).

I would love to know a general answer to Pax's question, or even some examples which are essentially different from the one above.

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[NO85] N. Nygaard, A. Ogus, Tate's conjecture for K3 surfaces of finite height, Ann. of Math. (2) 122 (1985), no. 3, 461–507.

Answer 2

Let me add a different answer.

Let $p$ be a prime with $p>\dim X$. Let $\mathcal{K}_{i}$ denote the higher $K$-sheaf on $X$, and let $S\mathcal{K}_{i}:=\mathrm{im}((\mathcal{O}_{X}^{\times})^{\oplus i}\rightarrow\mathcal{K}_{i})$ be the symbolic part. It is shown in [Blo77] that the slope $[i,i+1)$ part of $H^{r}_{\mathrm{cris}}(X/W(k))[1/p]$ is given by the cohomology of the sheaf of $p$-typical curves on $S\mathcal{K}_{i+1}$. In particular, if the functor \begin{equation*} F^{r-i}_{i}:A\mapsto H^{r-i}(X\times\mathrm{Spec}\,A,S\mathcal{K}_{i+1}) \end{equation*} from $k$-algebras to abelian groups is pro-representable, one finds that \begin{equation*} H^{r-i}(X,W\Omega_{X}^{i})[1/p]=\mathbb{D}(F^{r-i}_{i})\,. \end{equation*} The case $i=0$ is the Artin-Mazur case you stated in the question.

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[Blo77] S. Bloch, Algebraic K-theory and crystalline cohomology, Inst. Hautes Etudes Sci. Publ. Math. (1977), no. 47, 187-268.