An algebraic approach to the residues in algebraic geometry (Q1115491)

Let K/k be a finitely generated field of transcendence degree n and v a discrete valuation of rank n of \(K/k\). For any \(\omega\in \Omega^ n(K/k)\), we define the residue of \(\omega\) at v, \(Res_ v(\omega)\), as the sum of all the residues of \(\omega\) at the extensions of v in the inequivalent composites of K and \(\bar k\) over k. The paper shows that \(Res_ v\) is a well-defined k-map from \(\Omega^ n(K/k)\) to k and proves a theorem about the residues under a finite extension of the algebraic function field. Then, we define the residue of a differential form at a chain of points in a variety over an arbitrary field as the sum of the residues of the differential form at all the discrete valuations which have the chain of points as center.