Cohomological proof of Riemann-Roch for curves
= Cohomological proof of Riemann-Roch for curves
{title2=$\chi(\mathcal O(D))=\deg D+1-g$}
On a <smooth projective curve>, the exact sequence $0\to\mathcal O(D-P)\to\mathcal O(D)\to k(P)\to0$ increases the Euler characteristic by one. Iteration for positive and negative coefficients gives $\chi(\mathcal O(D))=\deg D+1-g$. <Serre duality> identifies $h^1(\mathcal O(D))$ with $h^0(\mathcal O(K-D))$, proving $\ell(D)-\ell(K-D)=\deg D+1-g$.