Sum of residues formula

In mathematics, the residue formula says that the sum of the residues of a meromorphic differential form on a smooth proper algebraic curve vanishes.

Statement

In this article, X denotes a proper smooth algebraic curve over a field k. A meromorphic (algebraic) differential form $\omega$ has, at each closed point x in X, a residue which is denoted ${\mathrm{res}}_x\omega$. Since $\omega$ has poles only at finitely many points, in particular the residue vanishes for all but finitely many points. The residue formula states:

$\sum _x{\mathrm{res}}_x\omega =0.$

Proofs

A geometric way of proving the theorem is by reducing the theorem to the case when X is the projective line, and proving it by explicit computations in this case, for example in Template:Harvtxt.

Template:Harvtxt proves the theorem using a notion of traces for certain endomorphisms of infinite-dimensional vector spaces. The residue of a differential form $fdg$ can be expressed in terms of traces of endomorphisms on the fraction field $K_x$ of the completed local rings ${\widehat{𝒪}}_{X,x}$ which leads to a conceptual proof of the formula. A more recent exposition along similar lines, using more explicitly the notion of Tate vector spaces, is given by Template:Harvtxt.

References

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