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I have a question on p-adic Hodge theory:

When e.g. $X$ is a smooth proper scheme over a finite extension $K$ of $\mathbf{Q}_{p}$ then e.g. one variant of $p$-adic Hodge theory says that there is a functorial isomorphism

$$ B_{\mathrm{HT}}\otimes_K\mathrm{gr}H^\ast_{\mathrm{dR}}(X/K) \cong B_{\mathrm{HT}}\otimes_{\mathbf{Q}_p} H^\ast_{\mathrm{\acute{e}t}}(X\times_\overline{K},\mathbf{Q}_p).$$

To my knowledge there it is not known whether such an isomorphism exists also for schemes defined over $\mathbf{C}_p$

Of course there may not be a Galois action on the left hand side, but one may still ask, whether there exists a canonical functorial isomorphism of $\mathbf{C}_{p}$-vector spaces:

$$ H^\ast_{\mathrm{dR}}(X) \cong H^\ast_{\mathrm{\acute{e}t}}(X) $$

I am thinking whether it may be possible to find such an isomorphism by approximating it modulo $p^n$ and by using $p$-adic hodge theory over bigger and bigger finite extensions of $\mathbf{Q}_p$?

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In general, you cannot expect that for a proper smooth scheme $X$ over $\mathbb{C}_p$, you have a canonical decomposition

$$ H^i_{\mathrm{\acute{e}t}}(X)\otimes \mathbb{C}_p\cong \bigoplus_j H^{i-j}(X,\Omega_X^j)\ . $$

However, there is a natural filtration on $H^i_{\mathrm{\acute{e}t}}(X)\otimes\mathbb{C}_p$ with associated gradeds $H^{i-j}(X,\Omega_X^j)$. For abelian varieties over $\mathbb{C}_p$, see Theoreme II.1.1 (and Remarque II.1.2) in Fargues' book on the isomorphism of the Lubin-Tate and Drinfeld tower.

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