Endlicher Körper/Projektive glatte Kurve/Vektorbündel/Endliche Annulation/Harder-Narasimhan-Kriterium/Fakt/en/Beweis

Beweis

Suppose that (1) holds. Then the torsor is not affine and hence by Fakt also (2) holds.

So suppose that (2) is true. By applying a certain power of the Frobenius we may assume that the image of the cohomology class in ${}{\mathcal {Q}}$ is ${}0$. Hence the class stems from a cohomology class ${}c_{i}\in H^{1}(X,E_{i})$. We look at the short exact sequence

$0\longrightarrow E_{i-1}\longrightarrow E_{i}\longrightarrow E_{i}/E_{i-1}\longrightarrow 0,$

where the sheaf of the right hand side has a nonnegative degree. Therefore the image of ${}c_{i}$ in ${}H^{1}(X,E_{i}/E_{i-1})$ can be annihilated by a finite morphism due to Fakt. Hence after applying a finite morphism we may assume that ${}c_{i}$ stems from a cohomology class ${}c_{i-1}\in H^{1}(X,E_{i-1})$. Going on inductively we see that ${}c$ can be annihilated by a finite morphism.

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