### On Grothendieck-Serre's conjecture concerning principal G-bundles over reductive group schemes I , by Ivan Panin , Anastasia Stavrova , and Nikolai Vavilov

In this paper we prove an interesting theorem concerning principal G-bundles,
where G is a semi-simple-simply connected group scheme. Specifically, let R be
a semi-local regular domain containing an infinite perfect subfield and let K
be its field of fractions. Let G be a reductive semi-simple simply connected
R-group scheme such that each of its R-indecomposable factors is isotropic. We
prove that under the above assumptions every principal G-bundle P which has a
K-rational point is itself trivial. This confirms a conjecture posed by Serre
and Grothendieck.

Ivan Panin <panin at pdmi.ras.ru >

Anastasia Stavrova <a_stavrova at mail.ru >

Nikolai Vavilov <nikolai-vavilov at yandex.ru >