= Algebraic residue of a rational differential
{title2=$\operatorname{res}_P(\eta)=[u^{-1}]\eta/du$}
= Algebraic residues of a rational differential
{synonym}
At a smooth point with local parameter $u$, expand a <rational differential> as $\sum_j a_j u^jdu$ in the completed local field; its residue is $a_{-1}$. Formal substitution shows this coefficient is independent of the parameter, in arbitrary characteristic. For a finite separable extension of these local fields, residues commute with the differential trace, with a sum over branches. The monogenic trace formula in the <trace-dual module of a finite curve map> proves this compatibility by coefficient extraction.
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