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[parent] proof of Riemann-Roch theorem (Proof)

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.



"proof of Riemann-Roch theorem" is owned by bwebste.
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Cross-references: degree, exact sequences, cohomology, sheaf, quotient, isomorphism, injection, point, Euler characteristic, Serre duality, line bundle, divisor
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This is version 3 of proof of Riemann-Roch theorem, born on 2003-08-15, modified 2007-04-08.
Object id is 4599, canonical name is ProofOfRiemannRochTheorem.
Accessed 3279 times total.

Classification:
AMS MSC14H99 (Algebraic geometry :: Curves :: Miscellaneous)

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