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 formula_1 has, at each closed point "x" in "X", a residue which is denoted formula_2. Since formula_1 has poles only at finitely many points, in particular the residue vanishes for all but finitely many points. The residue formula states:
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 formula_5 can be expressed in terms of traces of endomorphisms on the fraction field formula_6 of the completed local rings formula_7 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 .