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.