Sum of residues formula explained

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

\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 l O _{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