Les suites spectrales de Hodge-Tate

TL;DR

This work explores two fundamental results in p-adic Hodge theory, focusing on both absolute and relative versions of the p-adic comparison theorem and Hodge-Tate spectral sequence, with new proofs and the introduction of the relative spectral sequence.

Contribution

It provides new proofs for Faltings' p-adic comparison theorem and introduces the relative Hodge-Tate spectral sequence within the same framework.

Findings

Established absolute and relative p-adic comparison theorems

Introduced the relative Hodge-Tate spectral sequence

Extended understanding of p-adic Hodge theory structures

Abstract

This book presents two important results in p-adic Hodge theory following the approach initiated by Faltings, namely (i) his main p-adic comparison theorem, and (ii) the Hodge-Tate spectral sequence. We establish for each of these results two versions, an absolute one and a relative one. While the absolute statements can reasonably be considered as well understood, particularly after their extension to rigid varieties by Scholze, Faltings' initial approach for the relative variants has remained much less studied. Although we follow the same strategy as that used by Faltings to establish his main p-adic comparison theorem, part of our proofs is based on new results. The relative Hodge-Tate spectral sequence is new in this approach.