Weil cohomology theories can be considered as fibre functors from the category of motives. Given two such functors, we have an affine scheme of invertible natural transformations between them, and rational Hodge theory can be considered as providing $R$-valued points of this scheme (where $R$ is the coefficient ring). See this post, for example.

Can integral Hodge theory be viewed in a similar way? We are not over a field, so there might be Tannakian issues. Maybe it is pretty trivial, but then it would be nice if somebody provided full details.

  • $\begingroup$ When you say that rational Hodge theory provides an $R$-valued point, which two cohomology theories are you comparing? Integral $p$-adic Hodge theory is about integral coefficients, not integral base scheme, so what would be algebraic de Rham cohomology with integral coefficients? $\endgroup$ – François Brunault May 15 at 16:40
  • $\begingroup$ @FrançoisBrunault I see, I did not understand you at first. If you can answer for any comparison at all, I would be happy. $\endgroup$ – schematic_boi May 15 at 16:47
  • $\begingroup$ One can consider integral $p$-adic Hodge theory for smooth projective varieties over $\mathbb Z_p$, say. Then one can ask whether integral Hodge theory provides an isomorphism between the $p$-adic etale cohomology of the generic fiber and the algebraic de Rham cohomology, when tensored with some coefficient ring. The answer is no, I think, because integral $p$-adic Hodge theory produces a single object which specializes to those two objects. This is necessary to handle differences between the torsion in the two theories. They really are different functors. $\endgroup$ – Will Sawin May 16 at 21:46
  • $\begingroup$ @WillSawin do you think there is some other picture similar to Fontaine's but not the same verbatim? $\endgroup$ – schematic_boi 2 days ago
  • $\begingroup$ @schematic_boi I don't think there is an explanation simpler than the explanation of $A_{\infty}$-cohomology and its relation to other $p$-adic cohomology theories provided by Bhatt and Scholze. $\endgroup$ – Will Sawin 2 days ago

See the works of Bhatt--Morrow--Scholze. Possibly they provide what you want.

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