By Yuri Tschinkel (Ed.)
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Additional resources for Algebraic Groups: Mathematisches Institut, Georg-August-Universitat Gottingen. Summer School, 27.6.-13.7.2005
Mathematisches Institut, Seminars, 2005 46 This conjecture was proved by V. Voevodsky (see[Voe03b],[Voe03a]) for 2-adic coefficients. His proof is based on his results on the structure of motivic cohomology and integration of the main constructions of homotopy theory into the framework of algebraic geometry. However, there is another approach which relates this result to the structure of Galois groups G al (k(X )) and their Sylow subgroups (see [Bog92]). In particular, the following conjecture provides a direct approach to the proof of the Bloch-Kato conjecture and explains why it should be true.
Bogomolov: Stable cohomology 39 Thus, for any faithful complex linear representation V of G the quotient space i V /G is not rational if Hnr (G, Z/p) = 0 for some i . 7. Example. Let A be an abelian group. For any complex linear representation V i the quotient space V /A is rational and hence Hnr (A, Z/p) = 0 for any i . Therefore, ∗ for any group G and a ∈ Hnr (X , Z/p) the restriction of a to H si (A, Z/p) is zero for any abelian subgroup A ⊂ G. All the elements of H s1 (G, Z/p) are ramified since V L /G is unirational and its smooth completion is a simply-connected variety.
Since the element a on V L /G is induced from G, its restriction to W is induced from G a , but p a∗ a = 0 and hence a restricts to 0 on W = V L /G \ D which means that r ∗ a = 0. Thus K er (pG∗ ) ⊂ K er (r ∗ ). Now assume that a ∈ K er (r ∗ ) and that the restriction of a is trivial on an open subvariety W ⊂ V L /G. In order to prove that a ∈ K er (pG∗ ), it is sufficient to find an open subvariety W ⊂ W with the property that W = K (π1 (W ), 1) and π1 (W ) imbeds into the profinite completion πˆ 1 (W ).
Algebraic Groups: Mathematisches Institut, Georg-August-Universitat Gottingen. Summer School, 27.6.-13.7.2005 by Yuri Tschinkel (Ed.)