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Additional info for Algebraic Groups: Mathematisches Institut, Georg-August-Universitat Gottingen. Summer School, 27.6.-13.7.2005
Proof. Assume first that pG∗ a = 0 for some a ∈ H ∗ (G, F ). Then there is a finite quotient G a of G al (K ) with a chain of surjections G al (K ) → G a , p a : G a → G and p a∗ (a) = 0. Thus, there is a finite chain of Galois extensions k(V L /G) ⊂ k(V L ) ⊂ k(X ) with Galois groups G,G a respectively, and p a : G a → G is the corresponding homomorphism between the Galois groups. There is a Zariski open subvariety W = V L /G \ D ⊂ V L /G such that the extension k(X ) : k(V L /G) is nonramified over W .
1. Let G be a compact profinite group with a countable fundamental system of open subgroups given as kernels of surjective homomorphsims f i : G → G i , where each G i is a finite group. Define H s∗ (G, F ) for a finite module F as an inductive limit of H s∗ (G i , F ) over finite quotient groups G i . 2. Define Hnr (G, Z/p) as an inductive limit of groups Hnr (G i , Z/p). F. Bogomolov: Stable cohomology 41 Example. Let G = Zp . Then H si (Zp , F ) = 0, i > 1, H s1 (Zp , F ) = H 1 (Zp , F ) for any finite module F .
10. The spaces V L /G as universal spaces We have mentioned above a result which shows that the spaces V L /G play the role of universal spaces for the stable cohomology of algebraic varieties. In fact, a version of this result also holds for unramified cohomology. Let f : X → V L /G be a map of algebraic varieties. Then the induced map ∗ f : HS∗ (G, F ) → HS∗ (X , f ∗ F ) maps unramified elements of G into unramified ele∗ ments of X . The question is whether all the unramified elements Hnr (X , Z/p) can be induced from the unramified cohomology elements of finite groups.