panin at pdmi.ras.ru
Submission: 2009, May 9
Let R be a regular semi-local ring containing an infinite perfect subfield and let K be its field of fractions. Let G be a reductive R-group scheme satifying a mild "isotropy condition". Then each principal G-bundle P which becomes trivial over K is trivial itself. If R is of geometric type, then it suffices to assume that R is of geometric type over an infinite field. Two main Theorems of Panin's, Stavrova's and Vavilov's preprint state the same results for semi-simple simply connected R-group schemes. Our proof is heavily based on those two Theorems, on a result of Colliot-Thelene and Sansuc concerning the case of tori and on two purity theorems proven in the present preprint.
2000 Mathematics Subject Classification: 11E72; 14L17
Keywords and Phrases: reductive algebraic group, principal G-bundle, Grothendieck-Serre's conjecture
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