On the Kummer pro-\'etale cohomology of $\mathbb B_{\operatorname{dR}}$

TL;DR

This paper computes Kummer pro-étale cohomology for log rigid analytic varieties over p-adic fields, introduces a new log B_{dR}^+-cohomology theory, and proves comparison theorems and spectral sequence degenerations in this setting.

Contribution

It introduces a logarithmic B_{dR}^+-cohomology theory and establishes comparison isomorphisms and spectral sequence degenerations for log rigid analytic varieties.

Findings

Computed Kummer pro-étale cohomology of B_{dR}^+ and B_{dR} for log rigid varieties.

Introduced a new log B_{dR}^+-cohomology theory as a deformation of log de Rham cohomology.

Proved the log de Rham-étale comparison and degeneration of Hodge-Tate and Hodge-log de Rham spectral sequences.

Abstract

We investigate $p$-adic cohomologies of log rigid analytic varieties over a $p$-adic field. For a log rigid analytic variety $X$ defined over a discretely valued field, we compute the Kummer pro-\'etale cohomology of $\mathbb{B}_{\mathrm{dR}}^+$ and $\mathbb{B}_{\mathrm{dR}}$. When $X$ is defined over $\mathbb{C}_p$, we introduce a logarithmic ${B}_{\mathrm{dR}}^+$-cohomology theory, serving as a deformation of log de Rham cohomology. Additionally, we establish the log de Rham-\'etale comparison in this setting and prove the degeneration of both the Hodge-Tate and Hodge-log de Rham spectral sequences when $X$ is proper and log smooth.