Finite field/Smooth projective curve/Vector bundle/Finite annihilation/Harder-Narasimhan-criterion/Fact/Proof
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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 is . Hence the class stems from a cohomology class . We look at the short exact sequence
where the sheaf on the right hand side has a nonnegative degree. Therefore the image of in can be annihilated by a finite morphism due to fact. Hence, after applying a finite morphism, we may assume that stems from a cohomology class . Going on inductively we see that can be annihilated by a finite morphism.