Biswas, Indranil ; Ducrohet, Laurent
(2007)
*An analog of a theorem of Lange and Stuhler for principal bundles*
Comptes Rendus Mathematique, 345
(9).
pp. 495-497.
ISSN 1631-073X

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S16310...

Related URL: http://dx.doi.org/10.1016/j.crma.2007.10.010

## 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 (F^{n}_{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 (F^{n}_{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.

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ID Code: | 3432 |

Deposited On: | 11 Oct 2010 09:28 |

Last Modified: | 11 Oct 2010 09:28 |

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