On the Kummer pro-étale cohomology of ${\mathbb{B}}_{\mathrm{dR}}$

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-étale 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-étale 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.

Improved results of the main theorem. Comments are welcome!