Reduction of structure for torsors over semilocal rings

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 H^1(R, S) --> H^1(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 H^1(X, S) --> H^1(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 as well as essential dimension of connected algebraic groups in prime characteristic. Additional applications are presented at the end of this paper.