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 ${\displaystyle \omega }$ has, at each closed point x in X, a residue which is denoted ${\displaystyle \operatorname {res} _{x}\omega }$. Since ${\displaystyle \omega }$ has poles only at finitely many points, in particular the residue vanishes for all but finitely many points. The residue formula states:

${\displaystyle \sum _{x}\operatorname {res} _{x}\omega =0.}$

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 Altman & Kleiman (1970, Ch. VIII, p. 177).

Tate (1968) proves the theorem using a notion of traces for certain endomorphisms of infinite-dimensional vector spaces. The residue of a differential form ${\displaystyle fdg}$ can be expressed in terms of traces of endomorphisms on the fraction field ${\displaystyle K_{x}}$ of the completed local rings ${\displaystyle {\hat {\mathcal {O}}}_{X,x}}$ 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 Clausen (2009).

References

• Altman, Allen; Kleiman, Steven (1970), Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, 146, Springer, MR 0274461
• Clausen, Dustin (2009), Infinite-dimensional linear algebra, determinant line bundle and Kac–Moody extension, Harvard 2009 seminar notes
• Tate, John (1968), "Residues of differentials on curves", Annales scientifiques de l'École Normale Supérieure, 4, 1 (1): 149–159