Please also refer to the same question over on mathoverflow.net, where I proposed a solution which is still incomplete (and not entirely correct...).
Let $X'$ be an irreducible singular algebraic curve over an algebraically closed field $k$, and let $X \to X'$ be its normalization, and consider a (singular) point $Q \in X'$. Let $K = Q(X)$ be the function field of $X$ and $X'$.
Let $\mathcal{O}_Q' = \mathcal{O}_{X', Q}$ be the stalk of the structure sheaf of $X'$ at $Q$, and let $\mathcal{O}_Q = \bigcap_{P \mapsto Q} \mathcal{O}_P$ be its normalization. Here $\mathcal{O}_P$ is the stalk of the structure sheaf of $X$ at $P \in X$, and the intersection is over all points mapping to $Q$.
In his book Algebraic Groups and Class Fields, chapter IV §3, Serre introduces the module $\underline{\Omega}_Q'$ of regular differentials at $Q$. A differential $\omega \in D_k(K)$ is called regular, iff \begin{equation}\sum_{P \mapsto Q} \operatorname{Res}_P(f \omega) = 0 \quad \text{for all} \ f\in \mathcal{O}_Q'.\end{equation}
Similarly to $\mathcal{O}_Q$, Serre defines $$ \underline{\Omega}_Q = \bigcap_{P \mapsto Q} \Omega_P.$$ Since every differential $\omega \in \underline{\Omega}_Q$ has no poles at any point $P \mapsto Q$, clearly $\operatorname{Res}_P(f \omega) = 0$ for $f \in \mathcal{O}_Q'$, so that $\underline{\Omega}_Q \subset \underline{\Omega}_Q'$.
Now to my question: The mapping \begin{align} \mathcal{O}_Q / \mathcal{O}_Q' \times \underline{\Omega}_Q' / \underline{\Omega}_Q & \to k \\ (f, \omega) & \mapsto \sum_{P \mapsto Q} \operatorname{Res}_P(f \omega) \end{align} is clearly bilinear and well-defined. Serre claims that it is a perfect pairing, but I don't know why. I think we have to show two things:
- If $f \in \mathcal{O}_Q$, with the property that for each $\omega \in \underline{\Omega}_Q'$, one has $\sum_P \operatorname{Res}_P(f \omega) = 0$, then in fact $f \in \mathcal{O}_Q'$.
- If $\omega \in \underline{\Omega}_Q'$, such that for each $f \in \mathcal{O}_Q$, one has $\sum \operatorname{Res}_P(f \omega) = 0$, then $\omega \in \underline{\Omega}_Q$, i.e. $\omega$ is regular at every $P \mapsto Q$.
Any help would be appreciated :)