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

Indranil Biswas; Laurent Ducrohet

doi:10.1016/j.crma.2007.10.010

Algebraic Geometry

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

Presented by: Michel Raynaud

Indranil Biswas ¹; Laurent Ducrohet ¹

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

Comptes Rendus. Mathématique, Volume 345 (2007) no. 9, pp. 495-497.

Abstract
Résumé

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 $Z / p Z$ . Let $E_{G}$ be a principal G-bundle over a projective variety X defined over the field k. Assume that there is an étale Galois covering $f : Y \to X$ 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 ${(F_{X}^{n})}^{*} E_{G}$ 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 ${(F_{Z}^{n})}^{*} E_{H}$ 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 étale Galois cover of Z such that the pullback of $E_{H}$ to it is trivializable.

Soient k un corps algébriquement clos de caractéristique positive p et G l'extension à k d'un groupe linéaire algébrique connexe, défini sur $Z / p Z$ . Soit $E_{G}$ un G-fibré principal au-dessus d'une variété projective X défini sur k. Supposons qu'il existe un revêtement étale galoisien $f : Y \to X$ de degré premier à p tel que le pull-back $f^{*} E_{G}$ est trivial. Alors il existe un entier $n > 0$ tel que le pull-back ${(F_{X}^{n})}^{*} E_{G}$ est isomorphe à $E_{G}$ , où $F_{X}$ est le Frobenius absolu de X.

Ce résultat peut être considéré comme une réciproque partielle de l'observation suivante due à P. Deligne. Soit H un groupe algébrique quelconque défini sur k et $E_{H}$ un H-fibré principal au-dessus d'un schéma Z. Si l'image inverse ${(F_{Z}^{n})}^{*} E_{H}$ est isomorphe à $E_{H}$ pour un entier $n > 0$ convenable, alors il existe un revêtement étale galoisien de Z tel que le pull-back de $E_{H}$ à ce revêtement est trivial, où $F_{Z}$ est le Frobenius absolu de Z.

Received: 2007-07-12
Accepted: 2007-10-02
Published online: 2007-10-31

DOI: 10.1016/j.crma.2007.10.010

Author's affiliations:

Indranil Biswas ¹; Laurent Ducrohet ¹

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

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     title = {An analog of a theorem of {Lange} and {Stuhler} for principal bundles},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {495--497},
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     doi = {10.1016/j.crma.2007.10.010},
     language = {en},
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Indranil Biswas; Laurent Ducrohet. An analog of a theorem of Lange and Stuhler for principal bundles. Comptes Rendus. Mathématique, Volume 345 (2007) no. 9, pp. 495-497. doi : 10.1016/j.crma.2007.10.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.010/

References
Cited by

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[3] H. Lange; U. Stuhler Vektorbündel auf kurven und darstellungen der algebraischen fundamentalgruppe, Math. Z., Volume 156 (1977), pp. 73-83

[4] Y. Laszlo A non-trivial family of bundles fixed by the square of Frobenius, C. R. Acad. Sci. Paris, Volume 333 (2001), pp. 651-656

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