Endlicher Körper/Projektive glatte Kurve/Vektorbündel/Stark semistabil Grad nichtnegativ/Endliche Annulation/Fakt/en/Beweis
Erscheinungsbild
Beweis
If the degree of is positive, then a Frobenius pull-back has arbitrary large degree and is still semistable. By Serre duality we get that . So in this case we can annihilate the class by an iteration of the Frobenius alone.
So suppose that the degree is . Then there exists by Fakt a finite morphism which trivializes the bundle. So we may assume that . Then the cohomology class has several components and it is enough to annihilate them separately by finite morphisms. But this is possible by the parameter theorem of K. Smith (or directly using Frobenius and Artin-Schreier extensions).