Hodge-Tate decomposition for non-smooth spaces.

We study the derived direct image of the completed pro-\'etale structure sheaf of a non-smooth rigid analytic space, over an algebraically closed $p$-adic field $K$, using the \'{e}h-differentials. We show that there exists an $E_2$-spectral sequence from the derived push-forward of the \'eh-differentials to it, in which the higher push-forward vanishes when $X$ is smooth. We use the spectral sequence to study the derived direct image, and prove that in many cases it splits in the derived category. As an application, we generalize the \'etale-de Rham comparison and the Hodge-Tate decomposition to the non-smooth setting.