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 attached 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 $H^1(K,G)$ while $K$ is a bunch box.

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Borovoi, On strong approximation for homogeneous spaces, Dokl. Akad. Nauk BSSR 33 (1989) N4, 293–296 (Russian). [Brv3] M. V. Borovoi, On weak approximation in homogeneous spaces of algebraic groups, Soviet Math. Dokl. 42 (1991), 247–251. [Brv4] M. V. Borovoi, On weak approximation in homogeneous spaces of simply connected algebraic groups, Proc. Internat. Conf. “Automorphic Functions and Their Applications, Khabarovsk, June 27 – July 4, 1988” (N. Kuznetsov, V. ), Khabarovsk, 1990, 64–81. [Brv5] M.

7. Let G be a connected reductive group over a number field K. Then 1 the map ab1 : H 1 (K, G) → Hab (K, G) is surjective. 1 Proof: Let h ∈ Hab (K, G). It suffices to construct a torus T ⊂ G such that the 1 1 image of H (K, T ) in H1 (K, T (sc) → T ) = Hab (K, G) contains h. 6 there exists a finite set S of places of K such that locv (h) = 0 for v ∈ / S. 5. We set T = ρ(T (sc) ) · Z(G)0 ; then T (sc) = T . 2) δ 1 · · · → H 1 (K, T ) → Hab (K, G)−→H 2 (K, T (sc) ) → · · · Set η = δ(h); then locv (η) = 0 for v ∈ / S.

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