On the pro-Kummer étale cohomology of 𝔹dR

Source abstract

Abstract We investigate $p$-adic cohomology theories 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 $\mathbb{B}_{\textrm{dR}}^{+}$ and $\mathbb{B}_{\textrm{dR}}$. When $X$ is defined over $\mathbb{C}_{p}$, we introduce a logarithmic ${B}_{\textrm{dR}}^{+}$-cohomology theory, which serves 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.