Read e-book online Abelian Galois cohomology of reductive groups PDF

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 provide a functorial, virtually specific description of the standard Galois cohomology set $H^1(K,G)$ while $K$ is a bunch box.

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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. 5 (i), we see that locv (η) = 0 for v ∈ S as well. Thus η ∈ X2 (K, T (sc) ). 5 (ii) X2 (K, T (sc) ) = 0. We conclude that η = 0. Hence h comes from H 1 (K, T ). The theorem is proved. 8. 1) be an exact sequence of connected reductive K-groups. Suppose that the maps ab1G2 and ab1G2 are surjective.

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.

Let G be a reductive group over a number field K. 1) H (K, G)⊂ ∞ ∞ 1 is exact; 1 (ii) both the projections loc∞ : H 1 (K, G) → Π Hab (Kv , G) and ab1 : H 1 (K, G) → ∞ 1 (K, G) are surjective. 1) means that the commutative diagram H 1 (K, G)   loc ab1 −−−−→ ∞ 1 Hab (K, G)   1 Π H 1 (Kv , G) −−−−→ Π Hab (Kv , G) ∞ ∞ (in which all the maps are surjective) identifies H 1 (K , G) with the fiber product 1 1 of Hab (K , G) and Π H 1 (Kv , G) over Π Hab (Kv , G). 2. For semisimple groups this assertion was proved by Sansuc [Sa].

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Abelian Galois cohomology of reductive groups by Mikhail Borovoi


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