In geometry of algebraic curve by Arbarello,Cornalba and Griffiths they difine trace map of dualizing sheaf of nodal curve as follows:

we choose $D=r_1 +r_2+...+r_h$ consisting of h distinct smooth points of $C$:we have following ecact sequence $...\longrightarrow H^0(C,\omega_C(D)) \longrightarrow H^0(C,\omega_C(D)/\omega_C) \longrightarrow H^1(C,\omega_C) \longrightarrow H^1(C,\omega_C(D))$ it is easy to see that last term is zero.

now define $\xi$:$H^1(C,\omega_C)\longrightarrow k$ by:$\xi(\varphi)=2\pi i\Sigma res_{r_i}\bar{\varphi}$ where $\bar{\varphi}$ is a lifting of $\varphi$.

I think at first that we have $\omega_C(D)/\omega_C$ is isomorph to $\bigoplus k$ (h times) and $\bar{\varphi}$ is h vectors $(v_1,...,v_h)$ and $2\pi i\Sigma res_{r_i}\bar{\varphi}$ means $2\pi i\Sigma v_i$. Am i right?

my main question is that why this definition is not depend on $D$?