Generalization of a criterion for semistable vector bundles

Abstract

It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that both H 0 ( X , E ⊗ F ) and H 1 ( X , E ⊗ F ) vanishes. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on X such that H i ( X , E ⊗ F ) = 0 for all i. We also give an explicit bound for the rank of F.