Beilinson's results (two papers, one mentioned by Keerthi and the other here) have been generalised by Bhargav Bhatt; his paper also introduces a global period ring $A_{ddR}$ for global Galois representations!! The ring $A_{ddR}$ is a filtered $\hat{Z}$-algebra equipped with a Gal$(\bar{Q}/{Q})$-action.

A (one of the many) beautiful result in this paper is the following theorem:

Let $X$ be a semistable variety over $Q$. Then the log de Rham cohomology of a semistable model for $X$ is isomorphic to the $\hat{Z}$-cohomology of $X_{\bar{Q}}$, once both sides are base changed to a localization of $A_{ddR}$ (whle preserving all natural structures on either side).

Beilinson's results (two papers, one mentioned by Keerthi and the other here) have been generalised by Bhargav Bhatt; his paper also introduces a global period ring $A_{ddR}$ for global Galois representations!! The ring $A_{ddR}$ is a filtered $\hat{Z}$-algebra equipped with a Gal$(\bar{Q}/{Q})$-action.

A (one of the many) beautiful result in this paper is the following theorem:

Let $X$ be a semistable variety over $Q$. Then the log de Rham cohomology of a semistable model for $X$ is isomorphic to the $\hat{Z}$-etale cohomology of $X_{\bar{Q}}$, once both sides are base changed to a localization of $A_{ddR}$ (whle preserving all natural structures on either side).