#### V. Chernousov, Ph. Gille, Z. Reichstein, * Reduction
of structure for torsors over semilocal rings, *
submitted for publication. Posted October 2007.

**Abstract:**
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.

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