A Note on Hodge-Tate Spectral Sequences

TL;DR

This paper demonstrates how the Hodge-Tate spectral sequence for proper smooth rigid analytic varieties can be reconstructed from infinitesimal B_{dR}^+ cohomology, providing new proofs and conceptual insights.

Contribution

It introduces a method to reconstruct the spectral sequence from cohomology data and refines the decalage functor for the proof, offering new perspectives on spectral sequence degeneration.

Findings

Reconstruction of Hodge-Tate spectral sequence from B_{dR}^+ cohomology

New proof of torsion-freeness of infinitesimal B_{dR}^+ cohomology

Equivalence between degeneration of Hodge-Tate and Hodge-de Rham spectral sequences

Abstract

We prove that the Hodge-Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal $\mathbb{B}_{\text{dR}}^+$-cohomology through the Bialynicki-Birula map. A refinement of the decalage functor $L\eta$ is introduced to accomplish the proof. Further, we give a new proof of the torsion-freeness of the infinitesimal $\mathbb{B}_{\text{dR}}^+$-cohomology independent of Conrad-Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge-Tate spectral sequences is equivalent to that of Hodge-de Rham spectral sequences.