# proof of Riemann-Roch theorem

For a divisor $D$, let $\mathfrak{L}(D)$ be the associated line bundle. By Serre duality, $H^{0}(\mathfrak{L}(K-D))\cong H^{1}(\mathfrak{L}(D))$, so $\ell(D)-\ell(K-D)=\chi(D)$, the Euler characteristic of $\mathfrak{L}(D)$. Now, let $p$ be a point of $C$, and consider the divisors $D$ and $D+p$. There is a natural injection $\mathfrak{L}(D)\to\mathfrak{L}(D+p)$. This is an isomorphism anywhere away from $p$, so the quotient $\mathcal{E}$ is a skyscraper sheaf supported at $p$. Since skyscraper sheaves are flasque, they have trivial higher cohomology, and so $\chi(\mathcal{E})=1$. Since Euler characteristics add along exact sequences (because of the long exact sequence in cohomology) $\chi(D+p)=\chi(D)+1$. Since $\mathrm{deg}(D+p)=\mathrm{deg}(D)+1$, we see that if Riemann-Roch holds for $D$, it holds for $D+p$, and vice-versa. Now, we need only confirm that the theorem holds for a single line bundle. $\mathcal{O}_{X}$ is a line bundle of degree 0. $\ell(0)=1$ and $\ell(K)=g$. Thus, Riemann-Roch holds here, and thus for all line bundles.

Title proof of Riemann-Roch theorem ProofOfRiemannRochTheorem 2013-03-22 13:51:36 2013-03-22 13:51:36 bwebste (988) bwebste (988) 6 bwebste (988) Proof msc 14H99