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 and essential dimension of connected algebraic groups in prime characteristic. Additional applications are presented at the end of this paper.
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V. Chernousov was partially supported by the Canada Research Chairs Program and an NSERC research grant. Z. Reichstein was partially supported by NSERC Discovery and Accelerator Supplement grants.
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Chernousov, V., Gille, P. & Reichstein, Z. Reduction of structure for torsors over semilocal rings. manuscripta math. 126, 465–480 (2008). https://doi.org/10.1007/s00229-008-0180-0
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DOI: https://doi.org/10.1007/s00229-008-0180-0