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

Let ${}K$ denote a finite field (or the algebraic closure of a finite field) and let ${}X$ be a smooth projective curve over ${}K$. Let ${}{\mathcal {S}}$ be a locally free sheaf over ${}X$ and let ${}c\in H^{1}(X,{\mathcal {S}})$ denote a cohomology class. Let ${}{\mathcal {S}}_{1}\subset \ldots \subset {\mathcal {S}}_{t}$ be a strong Harder-Narasimhan filtration of ${}F^{e*}({\mathcal {S}})$. We choose ${}i$ such that ${}{\mathcal {S}}_{i}/{\mathcal {S}}_{i-1}$ has degree ${}\geq 0$ and that ${}{\mathcal {S}}_{i+1}/{\mathcal {S}}_{i}$ has degree ${}<0$. We set ${}{\mathcal {Q}}=F^{e*}({\mathcal {S}})/{\mathcal {S}}_{i}$. Then the following are equivalent.

1. The class ${}c$ can be annihilated by a finite morphism.
2. Some Frobenius power of the image of ${}F^{e*}(c)$ inside ${}H^{1}(X,{\mathcal {Q}})$ is ${}0$.

Proof, Write another proof