  Let G be a simple simply-connected group scheme over a regular local scheme
U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme
finite over U. We show that E is not necessarily trivial and give some criteria
of triviality. To this end we define affine Grassmannians for group schemes and
study their Bruhat decompositions for semi-simple group schemes. We also give
examples of principal G-bundles over A^1_U with split G such that the bundles
are not isomorphic to pull-backs from U.
