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:

- 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 🙂

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