Cambridge Journal of Mathematics

This article is a direct continuation of [B]. It contains a simple proof of comparison theorems in p-adic Hodge theory (the Fontaine-Jannsen conjecture). Different proofs were found earlier by Faltings, Niziol, and Tsuji, the case of open varieties treated by Yamashita. An alternative approach, based on an identification of the log crystalline cohomology for lci maps with the noncompleted (for the Hodge filtration) derived de Rham complex, was developed by Bhatt [Bh2].