$p$-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 ?