Access to the text (HTML) Access to the text (HTML)
PDF Access to the PDF text

Access to the full text of this article requires a subscription.
  • If you are a subscriber, please sign in 'My Account' at the top right of the screen.

  • If you want to subscribe to this journal, see our rates

  • You can purchase this item in Pay Per ViewPay per View - FAQ : 33,00 € Taxes included to order
    Pages Iconography Videos Other
    3 0 0 0

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 


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.

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.

© 2007  Académie des sciences@@#104156@@
EM-CONSULTE.COM is registrered at the CNIL, déclaration n° 1286925.
As per the Law relating to information storage and personal integrity, you have the right to oppose (art 26 of that law), access (art 34 of that law) and rectify (art 36 of that law) your personal data. You may thus request that your data, should it be inaccurate, incomplete, unclear, outdated, not be used or stored, be corrected, clarified, updated or deleted.
Personal information regarding our website's visitors, including their identity, is confidential.
The owners of this website hereby guarantee to respect the legal confidentiality conditions, applicable in France, and not to disclose this data to third parties.
Article Outline
You can move this window by clicking on the headline