Preliminaries: Let $X$ be a projective smooth curve (scheme of finite type, integral and of dimension $1$) over a perfect field $F$. Let $K=K(X)$ be the function field of $X$ and for a closed point $P$ let $K_P$ be the completion of $K$ with respect to the discrete valuation $v_P$ at $\mathcal O_{X,P}$.
By following "Tate - differential and residues on curves" one can define in a very abstract way the residue at a closed point $P$ as a map:
$$\text{res}_P:\Omega^1_{K|F}\rightarrow F$$ and moreover it is not difficult to prove the following properties :
Since $F$ is perfect, $\Omega^1_{K|F}$ has dimension $1$ over $K$. Therefore by chosing an uniformizer parameter $t\in\mathcal O_{X,P}$ for $v_P$ it is possible to write every non-zero differential form as $\omega=fdt$ where $f\in K^\times$
Let $\omega=fdt$, then $\text{res}_P(\omega)=\text{Tr}_{k(P)|F}(a_{-1,f})$ where: $a_{-1,f}$ is the coefficient of $t^{-1}$ in the expansion of $f$ as Laurent power series and $k(P)$ is the residue field at $P$.
Is it possible to define a (complete) residue map: $$\text{res}_P:\Omega^1_{K_P|F}\rightarrow F$$
such that the properties $1.$ and $2.$ hold? I'd like to have a representation $\omega=gdt$ with $g\in K_P$ and $\text{res}_P(\omega)=\text{Tr}_{k(P)|F}(a_{-1,g})$.
It seems that the key point is that $\Omega^1_{K|F}$ is one dimensional, but what about $\Omega^1_{K_P|F}$?
EDIT: As a comment suggests, it should be better to work with $\Omega:=K_P\otimes_K\Omega^1_{K|F}$, so my question can be modified in the following way: Is it possible to extend $\text{res}_P$ on $\Omega$ in order to obtain the property 2. for the elements of the form $gdt$ with $g\in K_P$? One could define directly $\text{res}_P(\omega)=\text{Tr}_{k(P)|F}(a_{-1,g})$ but now I'm not sure about the independence from the choice of $t$.
