Finite field/Smooth projective curve/Vector bundle/Finite annihilation/Harder-Narasimhan-criterion/Fact/Proof

Proof

Suppose that (1) holds. Then the torsor is not affine and hence by fact 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 ${}c_{i}\in H^{1}(X,{\mathcal {S}}_{i})$. We look at the short exact sequence

$0\longrightarrow {\mathcal {S}}_{i-1}\longrightarrow {\mathcal {S}}_{i}\longrightarrow {\mathcal {S}}_{i}/{\mathcal {S}}_{i-1}\longrightarrow 0,$

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

To fact