La suite spectrale de Hodge-Tate

TL;DR

This paper revisits Faltings' p-adic Hodge theory using the Hodge-Tate spectral sequence, clarifying its construction, abutment, and initial terms to deepen understanding of p-adic cohomology comparisons.

Contribution

It provides a detailed analysis of the Hodge-Tate spectral sequence, connecting it to Faltings' comparison theorems and elucidating its relation to differential forms and the Cartier isomorphism.

Findings

Clarifies the construction of the Hodge-Tate spectral sequence.

Links the spectral sequence to Faltings' comparison theorems.

Relates the initial term to the sheaf of differential forms.

Abstract

The Hodge-Tate spectral sequence for a proper smooth variety over a p-adic field provides a framework for us to revisit Faltings' approach to p-adic Hodge theory and to fill in many details. The spectral sequence is obtained from the Cartan-Leray spectral sequence for the canonical projection from the Faltings topos to the \'etale topos of an integral model of the variety. Its abutment is computed by Faltings' main comparison theorem from which derive all comparison theorems between p-adic \'etale cohomology and other p-adic cohomologies, and its initial term is related to the sheaf of differential forms by a construction reminiscent of the Cartier isomorphism.