Endlicher Körper/Projektive glatte Kurve/Vektorbündel/Stark semistabil Grad nichtnegativ/Endliche Annulation/Fakt/en/Beweis

Beweis

If the degree of ${}E$ is positive, then a Frobenius pull-back ${}F^{e*}(E)$ has arbitrary large degree and is still semistable. By Serre duality we get that ${}H^{1}(X,F^{e*}(E))=0$. So in this case we can annihilate the class by an iteration of the Frobenius alone.

So suppose that the degree is ${}0$. Then there exists by Fakt a finite morphism which trivializes the bundle. So we may assume that ${}E\cong {\mathcal {O}}_{X}^{r}$. Then the cohomology class has several components ${}c_{i}\in H^{1}(X,{\mathcal {O}}_{X})$ 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).

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