Given a normalization of a nodal curve $\alpha : \tilde{C} \to C$ over an algebraically closed field. Assume for simplicity we only have one node $p$ with $\alpha^{-1}\left(p\right)=\left(q_1,q_2 \right)$.
We can describe the dualizing sheaf $\omega_C$ as the subsheaf of $\alpha_*\omega_{\tilde{C}}\left(q_1+q_2\right)$ consisting of sections with residues summing to zero.
I know how to get it through explicit computations of the two sheaves (using e.g. an embedding into a projective space), but is there a way to see it just from formal properties of the map $\alpha$ and without computing $\omega_C$?