Coordinate Transformation in Faltings' Extension

Analogue to Fontaine's computation for $\Omega_{\bar{\mathbb{Z}}_p/\mathbb{Z}_p}$, we compute the structure of $\Omega_{\mathcal{O}_{\bar{K}_0}/\mathcal{O}_{K_0}}$ (here $K_0$ is the completion of $\mathbb{Q}_p(T)$ at place $p$) and prove that $p^{1-1/p^n}\mathrm{d}p^{1/p^n}$, $T^{1-1/p^n}\mathrm{d}T^{1/p^n}$ and $S^{1-1/p^n}\mathrm{d}S^{1/p^n}$ are linearly dependent (Here $S := 1-T$). The main aim of this article is to find the linear equations for these three differential forms. Then we define a map which is called "differential version" of Fontaine's map to express the equations in a computable way. Finally, we prove that the coefficients in the equation can be expressed in some polynomial forms and compute some examples.