On Grothendieck-Serre's conjecture concerning principal G-bundles over reductive group schemes II , by Ivan Panin

In this paper we prove an interesting theorem concerning principal G-bundles, where G is a reductive group scheme. connected group scheme. Specifically, 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. Our proof is heavily based on two recent Theorems due to Panin--Stavrova--Vavilov, on a result due to Colliot-Thelene and Sansuc concerning the case of tori and on two purity theorems proven in the present preprint.

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