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On the link, page, $ 2 $, the Berthlot-Ogus isomorphism theorem is stated as follows,

We have a canonical isomorphism, $$ \rho_{\mathrm{cris}} \ : \ H_{\mathrm{cris}}^{i} (X) \otimes_{K_ {0}} K \to H_{\mathrm{dR}}^{i} (\mathscr{X}) \otimes_{O_K} K = H_{\mathrm{dR}}^{i} (X) \tag1 $$ On the other hand, on the following pdf, page, $ 359 $, the author claims that there is an isomorphism, $$ H_ {\mathrm{DH}} (X) \otimes W \simeq H_{\mathrm{cris}} (X (s) / W) \tag2 $$.

What is the difference between the two isomorphisms $ (1) $ and $ (2) $ ?

Are the two isomorphisms the same ?

Thanks in advance for your help.

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    $\begingroup$ (2) most be $H_{dr}$ not $H_{DH}$ right? also the second isomorphism compare the De Rham and crystalline cohomology of a $\mathbb{Z}$ module integrally, while the first compare these things for a $K$ module rationally, if you tensor both side of the second isomorphism for $\mathbb{Q}$ you get a special case of the first one. in fact Berthlot-Ogus isomorphism is much more general than the thing you wrote, it is because your first source is really about rational p-adic Hodge theory.stacks.math.columbia.edu/tag/07MZ $\endgroup$
    – ali
    Commented Apr 3, 2021 at 15:25
  • $\begingroup$ Thank you @ali. :-) However, what is the difference between, $ X $ and $ \mathscr{X} $ appearing in the first isomorphism $ (1) $, on the one hand, and $ X / W $ and $ X (s) $ appearing in the second isomorphism $ (2) $, on the other side ? Is there a link between the two ? $\endgroup$
    – Angel65
    Commented Apr 3, 2021 at 16:19
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    $\begingroup$ X in the second isomorphism is $\mathscr{X}$ in the first: a scheme over the ring of integers, X in the first isomorphism is the generic fiber of $\mathscr{X}$ $\endgroup$
    – ali
    Commented Apr 3, 2021 at 16:38
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    $\begingroup$ also because the first source want to talk about rational hodge theory it start with a scheme over a field then choose a model for it over the ring of integers $\endgroup$
    – ali
    Commented Apr 3, 2021 at 16:39
  • $\begingroup$ Thank you ali. :-D $\endgroup$
    – Angel65
    Commented Apr 3, 2021 at 17:26

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