### Principal bundles of reductive groups over affine schemes, by Ivan Panin and Anastasia Stavrova

Let R be a semi-local regular domain containing an infinite perfect field k,
and let K be the field of fractions of R. 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 for any Noetherian affine scheme A over k, the
kernel of the map of etale cohomology sets H^1(A\times_k R,G) ->
H^1(A\times_k K,G), induced by the inclusion of R into K, is trivial. If R is
the semi-local ring of several points on a k-smooth scheme, then it suffices to
require that k is infinite and keep the same assumption concerning G. The
results extend the Serre--Grothendieck conjecture for such R and G, proved in
K-theory preprint 929.

Ivan Panin <paniniv@gmail.com>

Anastasia Stavrova <anastasia.stavrova@gmail.com>