p-adic derived de Rham cohomology

TL;DR

This paper develops a p-adic derived de Rham cohomology framework, establishing isomorphisms with crystalline cohomology and providing new proofs for key conjectures in p-adic Hodge theory.

Contribution

It introduces a natural isomorphism between derived de Rham and crystalline cohomology for lci maps, extending to logarithmic variants, and offers new proofs of Fontaine's conjectures.

Findings

Derived de Rham cohomology describes period rings in p-adic Hodge theory.

Constructs natural isomorphisms between derived de Rham and crystalline cohomology.

Provides a new proof of Fontaine's crystalline and semistable conjectures.

Abstract

This paper studies the derived de Rham cohomology of F_p and p-adic schemes, and is inspired by Beilinson's recent work. Generalising work of Illusie, we construct a natural isomorphism between derived de Rham cohomology and crystalline cohomology for lci maps of such schemes, as well logarithmic variants. These comparisons give derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory. Placing these ideas in the skeleton of Beilinson's construction leads to a new proof of Fontaine's crystalline conjecture and Fontaine-Jannsen's semistable conjecture.