p-adic derived de Rham cohomology
Bhargav Bhatt
TL;DR
The paper develops a non-completed derived de Rham framework modulo $p^n$ and its logarithmic extensions to unify de Rham, crystalline, and semistable $p$-adic Hodge theories. A key device is the conjugate filtration, which via derived Cartier theory yields a clean comparison between derived de Rham and crystalline cohomology for lci maps, and extends to period rings such as $A_{crys}$ and $A_{st}$. Building on Beilinson’s p-adic approach, the authors obtain a Beilinson-style construction of the crystalline ($C_{crys}$) and semistable ($C_{st}$) comparison theorems, including a Poincaré lemma on a site of pairs to handle semistable degeneration. The results illuminate the structural role of, and provide concrete descriptions for, period rings in terms of derived de Rham theory, with applications to $p$-adic Hodge theory and $G_K$-cohomology, offering conceptual simplifications relative to earlier proofs. Overall, the work bridges de Rham, crystalline, and semistable cohomologies using derived and logarithmic methods, yielding new proofs and computational tools for fundamental $p$-adic comparison theorems.
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.