Journal
MANUSCRIPTA MATHEMATICA
Volume 126, Issue 4, Pages 465-480Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00229-008-0180-0
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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 and essential dimension of connected algebraic groups in prime characteristic. Additional applications are presented at the end of this paper.
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