= Riemann-Roch via elementary modifications
{c}
{title2=$\chi(\mathcal O_C(D))=\deg_kD+\chi(\mathcal O_C)$}
For a closed point $p$ on a <smooth projective curve>, the quotient in $0\to L(-p)\to L\to L|_p\to0$ has $k$-dimension $[k(p):k]$. Additivity of <Euler characteristic> therefore changes $\chi(L)$ by that residue-field degree. Every <line bundle> is associated with a <divisor>, so iteration proves the displayed formula. The <dualizing sheaf on a smooth projective curve> identifies $h^1(L)$ with $h^0(\Omega^1_C\otimes L^{-1})$ and converts this Euler-characteristic formula into the usual divisor form of <Riemann-Roch theorem>.
Back to article page
