I have been looking for several weeks for the exact and the precise statement of Ogus conjecture, but, I cannot find it. The only book which made me discover the statement of this conjecture is that of Yves André: Théorie des motifs. See here, http://tomlr.free.fr/Math%E9matiques/Andre,%20Y%20-%20Une%20Introduction%20aux%20Motifs%20%28SMF%202004%29.pdf, pages, $ 79 $ and $ 80 $.
To understand how this conjecture is formulated, the author of this book directs us to a paper of Ogus, the holder of this conjecture, which is entitled, Hodge Cycles and Crystalline Cohomology.
The paper can be found here, https://www.jmilne.org/math/Books/DMOS.pdf, page, $ 359 $, in the introduction.
The statement of Ogus conjecture is not very clear. I formulated it as follows, following my efforts to understand its statement.
Here is the statement that I propose,
Let $ k $ be a number field.
Let $ R $ be an étale $ \mathbb {Z} $ -algebra.
Let $ X $ be a smooth projective $ R $ -scheme.
Let $ R' \supseteq R $ be another étale $ \mathbb{Z} $ - algebra.
Let $ s $ be a closed point of $ \mathrm{Spec} \ R '$, and let $ W $ be the completion of $ R' $ in $ s $.
We have an isomorphism, $$ H_ {\mathrm{dR}}^{i} (X / R) \otimes_k W \simeq H_{\mathrm{cris} }^{i} (X (s) / W ) $$
$ H_{\mathrm{cris}}^{i} (X(s) / W) $ is a $ F_ {\displaystyle v} $ - crystal, therefore, equipped with the Frobenius $ F_{ \displaystyle v} \ : \ H_{\mathrm{cris}}^{i} (X(s) / W) \to H_{\mathrm{cris}}^{i} (X(s) / W) $ defined by, $ F_{\displaystyle v} (z) = p^r z $.
So, we can pass this Frobenius $ F_{\displaystyle v} $, to $ H_{\mathrm{dR}}^{i} (X / R) \otimes_k W $ by this isomorphism.
Let the integral class cycle map (i.e., on $ \mathbb {Z} $), be defined by, $$ \mathrm{cl}_X \: \ \mathcal{Z}_{\sim}^{i} (X) \to \displaystyle \bigcup_{v \in I} \displaystyle \Big (H_{\mathrm{dR}}^{i} (X / R) \otimes_k W \displaystyle \Big)^{\textstyle F_{v}}, $$ where, $ \Big (H_{\mathrm{dR}}^{i} (X / R) \otimes_k W \Big)^{\textstyle F_{v}} $ is the $ F_{\displaystyle v} $ - crystal of $ F_{\textstyle v } $ - invariants.
$ I $ is the collection of the closed points $ s $ of $ \mathrm{Spec} \ R'$.
So, Ogus conjecture asserts that, the rational class cycle map (i.e., over $ \mathbb{Q} $), which is $ \mathrm{cl}_X \otimes \mathbb{Q} $, as follow, $$ \mathrm{cl}_X \otimes \mathbb{Q} \ : \ \mathcal{Z}_{\sim}^{i} (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \to \displaystyle \bigcup_{v \in I} \displaystyle \Big (H_{\mathrm{dR}}^{i} (X / R) \otimes_k W \displaystyle \Big)^{\textstyle F_{v}} \otimes_{ \mathbb{Z} } \mathbb{Q} $$
is surjective?
So, is that right ?
Can you correct that statement for me to see if I got it right?
How is $ W $ defined ?
Does $ W $ vary when the closed point $ s $ of $ \mathrm{Spec} \ R' $ varies ?
See here, Berthelot-Ogus comparison isomorphism for others interesting informations.
Thanks in advance for your help.
