mod p etale cohomology of the special fiber and the generic fiber

Question

Let $O$ be a valuation ring with fraction field $K$ of characteristic zero and residue field $O/m=k$ of characteristic $p>0$, and $X$ be a proper smooth scheme over $O$. Then can we control the mod $p$ etale cohomology of the special fiber by that of the generic fiber? Namely, do we have $\dim_{\Bbb F_p}H^i_{et}(X_k,\Bbb F_p) \leq\dim_{\Bbb F_p}H^i_{et}(X_K,\Bbb F_p)$ in general?

I am interested in the case $O=O_{\Bbb C_p}$, and it turns out mod p etale cohomology vanish for $i>\text{dim}(X_k)$ by Artin-schreier sequence while the $2\text{dim}(X_K)$'s etale cohomology can be nonzero for the generic fiber, see this answer. So they are not of the same dimension in general and it's natural to ask for a bound.

Also, smooth base change theorem is good for $\Bbb Z/\ell$ ($\ell \not=p$) but don't hold if the coefficient sheaf has $p$-torion. From BMS we know $\text{dim}_{\Bbb F_p}H^i_{et}(X_K,\Bbb F_p) \leq \text{dim}_{k}H^i_{dR}(X_k)$, so maybe a lower bound is also possible in our setting.

Edit: The case $i=0,1$ is true. What about the case $i=2$? Moreover, $\text{dim} _{\Bbb　F_p}H^i_{et}(X,\Bbb F_p)$ is finite and less than $\text{dim}_k H^i(X_k, O_k)$, see Lemma 0A3L for a related semi-linear algebra result (so maybe we need to understand the Frobenius action).

Edit: Note that $LHS \leq \text{dim}_{k}H^i_{k}(X_k,O_{X_k})\geq \text{dim}_{C}H^i(X_C,O_{X_C}) \leq \text{dim}_{C}H^i_{sing}(X_K^{an},K) \leq RHS. $ The only $\geq$ is due to upper semi-continuities, and we use an abstract isomorphism between $\Bbb C$ and $K$ to apply singular cohomologies, Hodge-de Rham decomposition and the universal coefficient theorem. So if $\text{dim}_{k}H^i_{k}(X_k,O_{X_k})=\text{dim}_{C}H^i(X_C,O_{X_C})$ then what we want holds, however this is not true in general. For example, one considers lifting of a singular Enriques surface in char $2$, but even in this case the inequality still holds (because the second Betti number is $10$, which is much larger than $h^{2,0}=1$).

Answer 1

(Not an answer, just a suggestion as to where to look.)

When $O$ is a Henselian DVR, the groups $\mathrm{H}^i(X_K, \mathbb{Z}/p)$ were studied by Bloch and Kato in "p-adic étale cohomology", Pub. math. IHÉS 63 (1986). They use Leray + proper base change to get a spectral sequence

$E_2^{p,q} = \mathrm{H}^p(X_k, i^* \mathrm{R}^q j_* \mathbb{Z}/p) \Rightarrow \mathrm{H}^{p+q}(X_K, \mathbb{Z}/p)$

where, as usual, $i$ and $j$ are the inclusions of the special and generic fibres, respectively.

They then study the sheaves $i^* \mathrm{R}^q j_* \mathbb{Z}/p(q)$ by putting a filtration on them and relating the graded pieces of the filtration to various spaces of differentials on the special fibre.

I suppose that to answer your question you'd need to understand not only the groups in this spectral sequence but also the maps...

Answer 2

There was a nonsense answer here earlier. It is now removed.