# Generalization of a criterion for semistable vector bundles

Indranil Biswas, Georg Hein

Number | 27 |
---|---|

Author | Georg Hein |

Project | C10 |

Year | 2008 |

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 H0(X,E⊗F) and H1(X,E⊗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 Hi(X,E⊗F)=0 for all i. We also give an explicit bound for the rank of F.