# Regular differentials on a singular curve.

Mathematics Asked by red_trumpet on February 20, 2021

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
$$begin{equation}sum_{P mapsto Q} operatorname{Res}_P(f omega) = 0 quad text{for all} fin mathcal{O}_Q’.end{equation}$$

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:

1. 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 $$f in mathcal{O}_Q’$$.
2. 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 🙂