paniniv@gmail.com

Submission: 2015, Sep 7

Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. If the regular local ring R contains an infinite field this result is proved in [FP]. Thus the conjecture in the title holds for regular local rings containing a field. This preprint is a short cut of three preprints of the author: arXiv:1406.0241, arXiv:1406.1129, and arXiv:1406.0247.

2010 Mathematics Subject Classification: 14M17, 20G35, 14L30,

Keywords and Phrases: reductive groups, principal bundles, torsors

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