Endlicher Körper/Projektive glatte Kurve/Vektorbündel/Stark semistabil Grad 0/Trivialisierbar/Fakt/en/Beweis

Beweis

We consider the family of locally free sheaves ${}F^{e*}(E)$, ${}e\in \mathbb {N}$. Because these are all semistable of degree ${}0$, and defined over the same finite field, we must have (by the existence of the moduli space for vector bundles) a repetition, i.e.

${}F^{e*}(E)\cong F^{e'*}(E)\,$

for some ${}e'>e$. By Fakt the bundle ${}F^{e*}(E)$ admits an étale trivialization ${}\varphi \colon Y\rightarrow X$. Hence the finite map ${}F^{e}\circ \varphi$ trivializes the bundle.

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