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$p$-adic Hodge theory gives us some comparison theorems between several cohomology theories.

It also provides a hierarchy in the category of $p$-adic representations of the absolute Galois group of a finite extension $K$ of the field of $p$-adic numbers.

Because those representations always admit a lattice stable under the action of the Galois group, it is natural to ask if one can produce an integral $p$-adic Hodge theory.

Satysfying answers for such a theory are given by the work of several people, including Fontaine-Laffaille, Breuil, Kisin, Liu among many others. They provided a theory for crystalline or semistable representations.

Now the question, hoping it is not complete non sense : is there an integral theory for de Rham representations ?

And another question : is there a "nice" integral avatar (with which we can deal with torsion representations) of the field of $p$-adic periods $B_{dR}$ introduced by Fontaine ?

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    $\begingroup$ There is a ring $\mathbf{A}_{\mathrm{dR}}$ but it doesn't seem to be used very much; I've only once encountered it (in a talk by Beilinson on the de Rham comparison isomorphism). $\endgroup$ – David Loeffler Jan 23 '15 at 14:47
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All de Rham representations are potentially semistable, so if you have a satisfactory theory for semistable representations, it should allow you to deal with the de Rham ones.

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  • $\begingroup$ This probably means that there exist a certain obstruction to this argument for integral representations. I think that this is also true for the original question: one can try to to construct such a theory (and I know of certain attempts by Breuil, Fontaine, Jannsen, and Zink), but the analogues of certain isomorphisms with rational coefficients are not necessarily isomorphisms integrally (and we can only try to bound the exponents of the "defects"). $\endgroup$ – Mikhail Bondarko Jan 27 '15 at 9:52

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