Generalization of a criterion for semistable vector bundles

TL;DR

This paper extends a known criterion for semistability of vector bundles from algebraically closed fields to perfect fields, providing an explicit bound for the rank of the auxiliary bundle involved.

Contribution

It generalizes the semistability criterion to perfect fields and offers an explicit bound on the rank of the vector bundle F.

Findings

Semistability characterized by vanishing cohomologies with a tensor factor.

Extension of the criterion from algebraically closed to perfect fields.

Explicit bound provided for the rank of the auxiliary bundle F.

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 the cohomologies of E\otimes F vanish. 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 the cohomologies of E\otimes F vanish. We also give an explicit bound for the rank of $F$.