Integral p $p$ -adic Hodge theory

Abstract

We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of $\mathbf {C}_{p}$ . 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’ 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.