March 7, 2017

# Abelian Galois cohomology of reductive groups by Mikhail Borovoi

By Mikhail Borovoi

During this quantity, a brand new functor $H^2_{ab}(K,G)$ of abelian Galois cohomology is brought from the class of hooked up reductive teams $G$ over a box $K$ of attribute $0$ to the class of abelian teams. The abelian Galois cohomology and the abelianization map$ab^1:H^1(K,G) \rightarrow H^2_{ab}(K,G)$ are used to provide a functorial, virtually specific description of the standard Galois cohomology set $H^1(K,G)$ whilst $K$ is a host box.

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Additional info for Abelian Galois cohomology of reductive groups

Example text

Then δ(loc∞ (ζ)) = loc∞ (η). 3) η = δ(ζ). We conclude that ξ = 1. 3) is trivial. 3). 1 We prove the exactness at the term Hab (K, G) × Π H 1 (Kv , G). 3) is contained in the kernel of the double arrow. e. loc∞ (h) = ab1 (ξ∞ ). We wish to show that h × ξ∞ comes from H 1 (K, G). 7 h = ab1 (η) for some η ∈ H 1 (K, G). Then ab1 (loc∞ (η)) = 1 ab (ξ∞ ). Let ψ ∈ Z 1 (K, G) be a cocycle representing η. 9 loc∞ (η) and ξ∞ “differ” by an element of the form ψ ρ∗ (ζ∞ ) where ζ∞ ∈ Π H 1 (Kv , ψ Gsc ). ∞ To be more precise, ξ∞ = tψ (ψ ρ∗ (ζ∞ )).

Math. IHES 25 (1965), 49–80. [St2] R. Steinberg, Lectures on Chevalley Groups, Yale University, 1968. [Vo1] V. E. Voskresenskii, Birational properties of linear algebraic groups, Math. USSR Izv. 4 (1970), 1–17. [Vo2] V. E. Voskresenskii, Algebraic Tori, Nauka, Moscow, 1977 (Russian). [Wh1] J. H. C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55 (1949), 453–496.

Sansuc [Sa] showed that this abelian group structure is functorial in G. He computed X(G) in terms of the arithmetic Brauer group Bra G. 2) of [Ko2]. 1) v∈V ¯ has a natural structure of an abelian group and can be computed in terms of π1 (G). 1) as well. ¯ Let the groups T −1 (M ), Tv−1 (M ) and the corestriction map Set M = π1 (G). 7. We define the composition cor−1 ab1 v 1 µv : H 1 (Kv , G)−→Hab (Kv , G) = Tv−1 (M ) −→ T −1 (M ) = (MΓ )tors Let ⊕ H 1 (Kv , G) denote the subset of the direct product consisting of the families V (ξv )v∈V such that ξv = 1 for v outside some finite set.