Endlicher Körper/Glatte projektive Kurve/Vektorbündel/Etale trivialisierbar und Frobenius Periodizität/Fakt/en

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 ${}E$ be a locally free sheaf over ${}X$.

Then ${}E$ is étale trivializable if and only if there exists some ${}n$ such that ${}F^{n*}E\cong E$.