= Dualizing sheaf on a smooth projective curve
{title2=$D_C\cong\Omega^1_{C/k}$}
For <vector bundles> on a <smooth projective curve>, a dualizing sheaf is a <line bundle> $D_C$ with a trace $H^1(C,D_C)\to k$ making the evaluation pairings $H^i(C,E)\times H^{1-i}(C,E^\vee\otimes D_C)\to k$ perfect for $i=0,1$. Choose a finite generically separable map to the <projective line>. The trace-dual module for this finite map, tensored with $\Omega^1_{\mathbb P^1}$, is $\Omega^1_C$ by the different–differential identity. Finite pushforward and <residue duality on the projective line> then prove the pairings, rather than assuming the curve-duality theorem to establish them.
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