Reduction of structure for torsors over semilocal rings

TL;DR

This paper proves that for certain reductive group schemes over semilocal rings, torsors can be reduced to finite subgroups, facilitating the study of algebraic structures like loop algebras and essential dimensions.

Contribution

It establishes the existence of finite subgroups enabling reduction of G-torsors over semilocal rings and schemes, extending previous results to new contexts.

Findings

Existence of finite subgroups S for reductive G over semilocal rings

Surjectivity of H^1 maps from S to G in various contexts

Applications to loop algebras and essential dimension studies

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.