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Comptes Rendus Mathématique
Volume 345, n° 9
pages 495-497 (novembre 2007)
Doi : 10.1016/j.crma.2007.10.010
Received : 12 July 2007 ;  accepted : 2 October 2007
An analog of a theorem of Lange and Stuhler for principal bundles
Un analogue dʼun théorème de Lange et Stuhler pour les fibrés principaux
 

Indranil Biswas , Laurent Ducrohet
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India 

Abstract

Let k be an algebraically closed field of characteristic   and G the base change to k of a connected reduced linear algebraic group defined over  . Let   be a principal G -bundle over a projective variety X defined over the field k . Assume that there is an étale Galois covering   with   coprime to p such that the pulled back principal G -bundle   is trivializable. Then there is a positive integer n such that the pullback   is isomorphic to  , where   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   a principal H -bundle over a scheme Z . If the pulled back principal H -bundle   over Z is isomorphic to   for some  , where   is the absolute Frobenius morphism of Z , then there is a finite étale Galois cover of Z such that the pullback of   to it is trivializable. To cite this article: I. Biswas, L. Ducrohet, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

The full text of this article is available in PDF format.
Résumé

Soient k un corps algébriquement clos de caractéristique positive p et G lʼextension à k dʼun groupe linéaire algébrique connexe, défini sur  . Soit   un G -fibré principal au-dessus dʼune variété projective X défini sur k . Supposons quʼil existe un revêtement étale galoisien   de degré premier à p tel que le pull-back   est trivial. Alors il existe un entier   tel que le pull-back   est isomorphe à  , où   est le Frobenius absolu de X .

Ce résultat peut être considéré comme une réciproque partielle de lʼobservation suivante due à P. Deligne. Soit H un groupe algébrique quelconque défini sur k et   un H -fibré principal au-dessus dʼun schéma Z . Si lʼimage inverse   est isomorphe à   pour un entier   convenable, alors il existe un revêtement étale galoisien de Z tel que le pull-back de   à ce revêtement est trivial, où   est le Frobenius absolu de Z . Pour citer cet article : I. Biswas, L. Ducrohet, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

The full text of this article is available in PDF format.


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