Residue duality on the projective line ($H^i(E)\cong H^{1-i}(E^{\vee }(-2){)}^{\ast }$)

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.