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
Canonical name	ProofOfRiemannRochTheorem
Date of creation	2013-03-22 13:51:36
Last modified on	2013-03-22 13:51:36
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	6
Author	bwebste (988)
Entry type	Proof
Classification	msc 14H99