Let $K$ be a p-adic local field, $X$ a smooth projective variety over $Spec(K)$, $CH^k(X)$ the Chow group of pure codimension $k$. Then there are cycle maps

$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ and $cl^X_{et}:CH^k(X)\to H^{2k}_{et}(X,\mathbb{Q}_p)$.

There is also a comparison morphism $\alpha$ which comes from p-adic Hodge theory:

$\alpha\colon B_{dR} \otimes_K H^*_{dR}(X/K) \cong B_{dR}\otimes_{\mathbb{Q}_p} H^*_{et}(X\times_K \overline{K},\mathbb{Q}_p).$

**Question:**

(1) Are these two cycle class maps related via the comparison morphism $\alpha$? This would be analogous to what happens in the complex case: (Perhaps Fontaine's $t\in B_{dR}$ appears?)

(2) If (1) is too difficult, I would be interested in just the case where $X=Y\times Spec(K)$ where $Y$ is defined over a number field $E$ and $K$ is a $p$-adic completion of $E$.

Thanks for your help.

higher$K$-theory. My strategy would be similar, though: replace $\operatorname{CH}^*(X)$ by $K_0(X)$ (they agree after tensoring with $\mathbb Q$), and use the splitting principle to reduce to the case of $c_1(\mathscr L)$ for $\mathscr L$ a line bundle. For this case an actual argument is needed, though. $\endgroup$ – R. van Dobben de Bruyn May 7 '18 at 21:30