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 $$\operatorname{res}_x \omega$$. Since $$\omega$$ has poles only at finitely many points, in particular the residue vanishes for all but finitely many points. The residue formula states:
 * $$\sum_{x} \operatorname{res}_x \omega=0.$$

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.

proves the theorem using a notion of traces for certain endomorphisms of infinite-dimensional vector spaces. The residue of a differential form $$f dg$$ can be expressed in terms of traces of endomorphisms on the fraction field $$K_x$$ of the completed local rings $$\hat \mathcal O_{X, x}$$ 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.