Coordinate Transformation in Faltings' Extension

Analogue to Fontaine’s computation for ${\mathrm{\Omega }}_{{\bar{\mathbb{Z}}}_p\mathrm{/}{\mathbb{Z}}_p}$, we compute the structure of ${\mathrm{\Omega }}_{{\mathcal{O}}_{{\bar{K}}_0}\mathrm{/}{\mathcal{O}}_{K_0}}$ (here $K_0$ is the completion of ${\mathbb{Q}}_p(T)$ at place $p$) and prove that $p^{1-1\mathrm{/}{p}^n}\mathrm{d}{p}^{1\mathrm{/}{p}^n}$, $T^{1-1\mathrm{/}{p}^n}\mathrm{d}{T}^{1\mathrm{/}{p}^n}$ and $S^{1-1\mathrm{/}{p}^n}\mathrm{d}{S}^{1\mathrm{/}{p}^n}$ 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.