Integral p-adic Hodge theory

TL;DR

This paper introduces a new integral p-adic cohomology theory for proper smooth schemes over C_p's ring of integers, unifying and extending existing theories with strong comparison theorems.

Contribution

It constructs a novel cohomology theory that specializes to all known p-adic cohomologies, utilizing Faltings's almost purity and derived functors, advancing the understanding of p-adic Hodge structures.

Findings

Unifies crystalline, de Rham, and etale cohomologies within a single framework.

Establishes strong integral comparison theorems for p-adic cohomology.

Connects the theory to de Rham-Witt complexes and q-deformations.

Abstract

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.