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 offer a functorial, nearly particular description of the standard Galois cohomology set $H^1(K,G)$ whilst $K$ is a bunch box.

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Ko3] R. E. Ann. 275 (1986), 365–399. [La1] R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Mathematical Surveys and Monographs, vol. , AMS, 1989, pp. 101–170. [La2] R. P. Langlands, Stable conjugacy: definitions and lemmas, Can. J. Math. 31 (1979), 700–725. [Mi1] ´ J. S. Milne, Etale Cohomology, Princeton University Press, Princeton, NJ, 1980. [Mi2] J. S. Milne, Arithmetic Duality Theorems, Acad.

1 (Kneser-Harder). Let G be a connected K-group. Then the map loc∞ : H 1 (K, G) → Π H 1 (Kv , G) ∞ is surjective. 1. See also [Kn3]. 2 (Kneser-Harder). Let T be a K-torus. Suppose that there is a place v0 of K such that T is anisotropic over Kv0 . Then X2 (K, T ) := ker[H 2 (K, T ) → Π H 2 (Kv , T )] = 0. v∈V Proof: See[Ha1], II, p. 2, Thm. 7, p. 3. 3 (Harder). Let G be a K-group. Let Σ ⊂ V be a finite set of places of K. For any v ∈ Σ let Tv ⊂ GKv be a maximal torus. Then there exists a maximal torus T ⊂ G such that TKv is conjugate to Tv under G(Kv ) for any v ∈ Σ.

Galois cohomology over local and number fields In this section we apply the results of Sections 3 and 4 to the study of the usual (non-abelian) Galois cohomology of connected reductive groups over local and (especially) number fields. 0. We will need the following fundamental results on Galois cohomology over local and global fields. 1 ([Kn1], [Kn3]). Let G be a simply connected group over a nonarchimedian local field K. Then H 1 (K, G) = 1. Another proof of this result appeared in [Br-T]. 2. Let K be a number field.