= Residue trace on a smooth projective curve
{title2=$\operatorname{tr}_C:H^1(C,\Omega^1_C)\longrightarrow k$}
Use the <rational principal-parts resolution on an algebraic curve> to represent a class by finitely many rational differential principal parts. Sum their <algebraic residues of a rational differential>. The sum of residues of a global rational differential is zero: trace to the <projective line>, where partial fractions prove the assertion. Hence the sum descends to cohomology. Pairing an <invertible sheaf> with its differential-valued dual gives the perfect <Serre duality> pairing. Since the global regular functions on an integral projective curve over an algebraically closed field are $k$, evaluation at the constant function $1$ identifies this trace canonically with an isomorphism to $k$.
Back to article page
