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Is there a crystalline analogue for the following formula, using the crystalline Frobenius $ F_v $ instead of the absolute Galois group $ \mathrm{Gal} (\overline{k} / k) $ ?

Here is the formula, which can be found in Yves André's book, Introduction au Motifs, page, $ 19 $. See here, http://tomlr.free.fr/Math%E9matiques/Andre,%20Y%20-%20Une%20Introduction%20aux%20Motifs%20%28SMF%202004%29.pdf

If $ F $ is a $ \mathbb{Q} $ - algebra, then, there exists the following isomorphism, $$ \mathcal {Z}_{\sim}^* (X)_F \simeq \mathcal{Z}_{\sim}^* (X_{\overline{k}})_F^{\mathrm{Gal} (\overline{k} / k)} $$ $ \mathcal{Z}_{\sim}^* (X)_F $ is the free abelian group of algebraic cycles modulo an adequate equivalence relation $ \sim $ over $ X $ a smooth projective variety over $ k $ a field.

Thanks in advance for your help.

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  • $\begingroup$ Chow groups are related to the motivic cohomology. In the $p$-adic world, a similar formula leads to a quasisyntomic sheaf. See the prism paper, section 14. See BMS2 section 10 a relation to the usual étale sheaf on the generic fiber. $\endgroup$
    – Z. M
    Commented Apr 12, 2021 at 19:59

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