### 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.

Ivan Panin <panin at pdmi.ras.ru >