Finite field/Smooth projective curve/Vector bundle/Strongly semistable degree 0/Trivializable/Fact/Proof

Proof

We consider the family of locally free sheaves ${}F^{e*}({\mathcal {S}})$, ${}e\in \mathbb {N}$. 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.

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

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

To fact