On residues in algebraic geometry

Let $f\colon X\to S$ be a dominant morphism of algebraic schemes, with $S$ integral. Let $n$ be the relative dimension of $f$ and let $x=(x_0,x_1,\dots,x_n)$ be a sequence of points of $X$ such that, for all $0\leqslant i\leqslant n$, $x_i$ is a specialization of $x_{i-1}$, has codimension $i$ and is mapped into the generic point of $S$. Under these conditions a residue mapping (of $f$ into the “chain” $x$)
$$ \operatorname{Res}_x^f\colon\Omega^*(X)\to\Omega^*(S) $$
is defined and its main properties, in particular the “residue formula”, are proved.
Bibliography: 14 titles.