# Plus closure/Two-dimensional/Graded/Torsor/Strongly semistable/Section

We deal first with the situation of a strongly semistable sheaf of degree . The following two results are due to Lange and Stuhler. We say that a locally free sheaf is *étale trivializable* if there exists a finite étale morphism
such that
.
Such bundles are directly related to linear representations of the étale fundamental group.

Let denote a finite field (or the algebraic closure of a finite field) and let be a smooth projective curve over . Let be a locally free sheaf over . Then is étale trivializable if and only if there exists some such that .

Let denote a finite field (or the algebraic closure of a finite field) and let be a smooth projective curve over . Let be a strongly semistable locally free sheaf over of degree . Then there exists a finite morphism

such that is trivial.

We consider the family of locally free sheaves , . Because these are all semistable of degree , and defined over the same finite field, we must have (by the existence of the moduli space for vector bundles) a repetition, i.e.

for some . By fact, the bundle admits an étale trivialization . Hence the finite map trivializes the bundle.

Let denote a finite field (or the algebraic closure of a finite field) and let be a smooth projective curve over . Let be a strongly semistable locally free sheaf over of nonnegative degree and let denote a cohomology class. Then there exists a finite morphism

such that is trivial.

If the degree of is positive, then a Frobenius pull-back has arbitrary large degree and is still semistable. By Serre duality we get that . So in this case we can annihilate the class by an iteration of the Frobenius alone.

So suppose that the degree is . Then there exists by fact a finite morphism which trivializes the bundle. So we may assume that . Then the cohomology class has several components and it is enough to annihilate them separately by finite morphisms. But this is possible by the parameter theorem of K. Smith (or directly using Frobenius and Artin-Schreier extensions).