proof of Riemann-Roch theorem
ProofOfRiemannRochTheorem
2003-08-15 05:51:30
2007-04-08 02:08:17
Proof
Riemann-Roch theorem for curves
0
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For a divisor $D$, let $\L D$ be the associated line bundle. By Serre duality, $H^0(\L{K-D})\cong H^1(\L D)$, so $\ell(D)-\ell(K-D)=\chi(D)$, the Euler characteristic of $\L D$. Now, let $p$ be a point of $C$, and consider the divisors $D$ and $D+p$. There is a natural injection $\L D\to\L{D+p}$. 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. $\O_X$ 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.