proof of Riemann-Roch theorem
For a divisor , let be the associated line bundle. By Serre duality, , so , the Euler characteristic of . Now, let be a point of , and consider the divisors and . There is a natural injection . This is an isomorphism anywhere away from , so the quotient is a skyscraper sheaf supported at . Since skyscraper sheaves are flasque, they have trivial higher cohomology, and so . Since Euler characteristics add along exact sequences (because of the long exact sequence in cohomology) . Since , we see that if Riemann-Roch holds for , it holds for , and vice-versa. Now, we need only confirm that the theorem holds for a single line bundle. is a line bundle of degree 0. and . Thus, Riemann-Roch holds here, and thus for all line bundles.
|Title||proof of Riemann-Roch theorem|
|Date of creation||2013-03-22 13:51:36|
|Last modified on||2013-03-22 13:51:36|
|Last modified by||bwebste (988)|