# ag.algebraic geometry – Looking for the exact and the precise statement of Ogus conjecture

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.

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