Nice group structure on the elementary orbit space of unimodular rows

Abstract (1) Let R be an affine algebra of dimension d ≥ 5 over F ‾ p with p > 3 . Then the group structure on Um d ( R ) / E d ( R ) is nice. (2) Let R be a commutative noetherian ring of dimension d ≥ 2 such that E d + 1 ( R ) acts transitively on Um d + 1 ( R ) . Then the group structure on Um d + 1 ( R [ X ] ) / E d + 1 ( R [ X ] ) is nice.