Criterion for surjectivity of localization in Galois cohomology of a reductive group over a number field

TL;DR

This paper establishes a precise criterion for when the localization map in Galois cohomology of a reductive group over a number field is surjective, and introduces a new method for constructing abelian Galois cohomology in arbitrary characteristic.

Contribution

It provides a necessary and sufficient condition for the surjectivity of localization maps in Galois cohomology and offers a novel construction of abelian Galois cohomology applicable in all characteristics.

Findings

Derived a criterion for surjectivity of localization in Galois cohomology.

Presented a new construction method for abelian Galois cohomology.

Extended the theory to fields of arbitrary characteristic.

Abstract

Let $G$ be a connected reductive group over a number field $F$, and let $S$ be a set (finite or infinite) of places of $F$. We give a necessary and sufficient condition for the surjectivity of the localization map from $H^1(F,G)$ to the "direct sum" of the sets $H^1(F_v,G)$ where $v$ runs over $S$. In the appendices, we give a new construction of the abelian Galois cohomology of a reductive group over a field of arbitrary characteristic.