= Residue duality on the projective line
{title2=$H^i(E)\cong H^{1-i}(E^\vee(-2))^*$}
The coefficient of $t^{-1}dt$ gives a trace on $H^1(\mathbb P^1,\Omega^1)$. Multiplication followed by this trace pairs the Laurent bases of the <Čech cohomology of twists on the projective line> perfectly: $t^a$ pairs with $t^{-a-1}$. For any <vector bundle>, evaluation into $\Omega^1$ defines the same natural pairing. Saturate a rational rank-one subspace to a <line subbundle>, and induct on rank using the <long exact sequence in sheaf cohomology> and compatibility of evaluation with connecting maps. This proves the duality without using the later splitting theorem.
Back to article page
