An analog of a theorem of Lange and Stuhler for principal bundles

Abstract

Let k be an algebraically closed field of characteristic p>0 and G the base change to k of a connected reduced linear algebraic group defined over Z/pZ. Let E_G be a principal G-bundle over a projective variety X defined over the field k. Assume that there is an etale Galois covering f : Y→ X with degree (f) coprime to p such that the pulled back principal G-bundle f* E_G is trivializable. Then there is a positive integer n such that the pullback (Fⁿ_X)* E_G is isomorphic to E_G, where F_X is the absolute Frobenius morphism of X. This can be considered as a weak converse of the following observation due to P. Deligne. Let H be any algebraic group defined over k and E_H a principal H-bundle over a scheme Z. If the pulled back principal H-bundle (Fⁿ_X)*E_H over Z is isomorphic to E_H for some n>0, where F_Z is the absolute Frobenius morphism of Z, then there is a finite etale Galois cover of Z such that the pullback of E_H to it is trivializable.