Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies

2012-02-25T03:13:51Z

Let $K$ be a $p$-adic field, that is a complete discrete valuation ring of characteristic $0$ with a perfect residue field $k$ of characteristic $p > 0$ (to simplify one could also take $K$ to be a finite extension of $\mathbb{Q}_p$). Let $\mathbb{C}_K$ be the completion of a (fixed) algebraic closure $\overline{K}$ of $K$. Then one of Faltings' theorems in $p$-adic Hodge theory says for any smooth proper $K$-scheme $X$ there is a natural isomorphism of $\mathbb{C}_K$-semilinear $\mathrm{Gal}(\overline{K}/K) = \mathrm{Gal}(\mathbb{C}_K/K)$-representations

$\mathbb{C}_K \otimes_{\mathbb{Q}_p} H^n_{et}(X_{\overline{K}}, \mathbb{Q}_p) \cong \bigoplus_{q\in\mathbb{Z}} (\mathbb{C}_K(-q) \otimes_K H^{n - q}(X, \Omega_{X/K}^q)).$

Here $X_{\overline{K}}$ is the base change of $X$ to the algebraic closure, whereas $\mathbb{C}_K(s)$ stands for the usual $s$-th order Tate twist by the cyclotomic character describing the action of the absolute Galois group on the $p$-power roots of unity.

My question is: is there an integral version of the above isomorphism? Let me be more precise and explain what I mean by this: let $\mathcal{X}$ be a smooth proper scheme over the valuation ring $\mathcal{O}_K$ of $K$ and let $\mathcal{O}_{\overline{K}}$ be the valuation ring of the algebraic closure. Is there an isomorphism similar to the one above relating, say, $H^n_{et}(\mathcal{X}\times_{\mathcal{O}_K} \mathcal{O}_{\overline{K}}, \mathbb{Z}_p)$ and the $H^{n - q}(\mathcal{X}, \Omega^q_{\mathcal{X}/\mathcal{O}_K})$ 's?

Answer by Filippo Alberto Edoardo for Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies

2012-05-30T02:00:30Z

Dear Kestutis,

your question is integral in two ways: first of all, you would like to consider schemes over a whole DVR instead of the generic fiber only. Secondly, you would like to have a comparison theorem between two $\mathcal{O}_\overline{K}$-modules and not only between $K$-vector spaces. The two are tightly connected since $H_\text{dR}(\mathcal{X})\otimes K=H_\text{dR}(X)$ , if $X$ denotes the generic fiber of $\mathcal{X}$.

As for the first question we have the so-called Crystalline Conjecture proven by Faltings and many others. It says that if $\mathcal{X}$ is proper and smooth over $\mathcal{O}_K$ (I keep your notations) with generic fiber $X$ and special fiber $\overline{\mathcal{X}}$, then $$ B_\text{cris}\otimes_{K_0}H_\text{cris}^m(\overline{\mathcal{X}})\cong B_\text{cris}\otimes_{\mathbb{Q}_p}H_\text{et}(X_\overline{K},\mathbb{Q}_p) $$ where $B_\text{cris}$ is Fontaine's ring of periods. If you extend scalars to $B_\text{dR}\supset B_\text{cris}$ you get back the comparison isomorphism that you quote because $B_\text{dR}$ admits a filtration whose associated graded ring is $\bigoplus_{q\in\mathbb{Z}}\mathbb{C}_K(q)$; and because of a basic result by Berthelot and Berthelot-Ogus saying that crystalline cohomology (which is the group $H_\text{cris}(\overline{\mathcal{X}})$ in my equation above and is a cohomology theory of the special fiber with coefficients in $W(k)$, the Witt vectors of the residue field $k$ of $\mathcal{O}_K$) is isomorphic to de Rham cohomology after you tensor both with $\text{Frac}(W(k))$. You could give a look at
1) J.-M. Fontaine, Représentations $p$-adiques semi-stables, in Périodes $p$-adiques, Astérisque 223, Section 6.1 for a discussion about the Crystalline Conjecture
2) P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University Press, Chapter 7 (the very last theorem) for the isomorphism $H_\text{cris}\cong H_\text{dR}$
Let me also remark that much of the above can be generalized assuming only that $\mathcal{X}$ be semistable instead of smooth (by Tsuji).

The second question is more subtle. As I mentioned, already the isomorphism $H_\text{cris}(\overline{\mathcal{X}})=H_\text{dR}(\mathcal{X})$ is false in general: it is true if $K/\mathbb{Q}_p$ (or $\mathcal{O}_K/\mathbb{Z}_p$, equivalently) has absolute ramification index $e\leq p-1$. In this case, at least, we dispose of an interpretation of the de Rham cohomology of $\mathcal{X}$ in terms of crystalline cohomology. We also have an "integral structure" $A_\text{cris}\subseteq B_\text{cris}$ and one can hope to have something like $$ A_\text{cris}\otimes_{W(k)}H_\text{cris}^m(\overline{\mathcal{X}})\stackrel{?}{\cong} A_\text{cris}\otimes_{\mathbb{Z}_p}Hˆm_\text{et}(\mathcal{X}_{\mathcal{O}_\overline{K}},\mathbb{Z}_p) $$ The problem is that $A_\text{cris}$ (and even $A_\text{cris}[1/p]$ !) is much smaller than $B_\text{cris}$ : roughly speaking, it cannot detect "negative Hodge-Tate weights".

[EDIT] As Keerthi Madapusi Pera and Matthew Emerton commented, in some special cases the above isomorphism holds true. They give precise references in the comments.

Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies - MathOverflow

Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies

Answer by Filippo Alberto Edoardo for Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies