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

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.