Residue trace on a smooth projective curve (${\mathrm{tr}}_C:H^1(C,{\Omega }_C^1)⟶k$)

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 $k$.