Reduction of structure for tensors over . . .

Let G be a reductive affine group scheme defined over a semilocal ring k. Assume that either G is semisimple or k is normal and noetherian. We show that G has a finite k-subgroup S such that the natural map H1(R,S) → H1(R,G) is surjective for every semilocal ring R containing k. In other words, G-torsors over Spec(R) admit reduction of structure to S. We also show that the natural map H1(X,S) → H1(X,G) is surjective in several other contexts, under suitable assumptions on the base ring k, the scheme X/k and the group scheme G/k. These results have already been used to study loop algebras and essential dimension of connected algebraic groups in prime characteristic. Additional applications are presented at the end of this paper