  We construct a new cohomology theory for proper smooth (formal) schemes over
the ring of integers of C_p. It takes values in a mixed-characteristic analogue
of Dieudonne modules, which was previously defined by Fargues as a version of
Breuil-Kisin modules. Notably, this cohomology theory specializes to all other
known p-adic cohomology theories, such as crystalline, de Rham and etale
cohomology, which allows us to prove strong integral comparison theorems.
  The construction of the cohomology theory relies on Faltings's almost purity
theorem, along with a certain functor $L\eta$ on the derived category, defined
previously by Berthelot-Ogus. On affine pieces, our cohomology theory admits a
relation to the theory of de Rham-Witt complexes of Langer-Zink, and can be
computed as a q-deformation of de Rham cohomology.
