Algebraic Groups: Mathematisches Institut, by Yuri Tschinkel (Ed.)

By Yuri Tschinkel (Ed.)

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Example text

The ∞ space CP contains copies of CPn for all positive integers n and they have the same i -skeleton as CP∞ for i 2n − 1. (b) B Z/l = C∞ \ {0}/(Z/l ) because C∞ \ {0} is a contractible space with a free action of Z/l . Naturally, C∞ \ {0}) ⊃ . . Cn \ {0} for all possible n > 0. , for i 2n −2 any map f : S i → Cn \{0} S 2n−1 is contractible ( means “homotopy equivalence”), hence Cn − \{0}/(Z/l ) has the same i -skeleton as B Z/l for 0 i 2n − 2. (c) Generalizing the previous example, let M be the space of real matrices   λ11 .

Denote by G al (K ) the Galois group of an algebraic closure K¯ over K . F. Bogomolov: Stable cohomology 29 The group G al (K ) is a profinite topological group which surjects onto any Galois groups of a finite algebraic extensions of K . These surjections generate a natural topology on G al (K ). Surjections onto finite groups are continuous on G al (K ) with respect to the above topology, and we are going to consider only continuous maps and continuous cochains on the G al (K ). Since V L → V L /G corresponds to a finite Galois extension k(V L ) : k(V L /G) with G as a Galois group, we have a natural surjection pG : G al (K ) → G for K = k(V L /G).

Then there are a finite group over Fl , l = p and a ∈ Hnr G , an element b ∈ Hnr (G, Z/p) and a homomorphism h a : G al (K ) → G such that a = h a∗ (b). The proof involves a geometric argument based on the general result above. Let us discuss the example of unramified cohomology in dimension 2 in more 2 detail. Consider Hnr (X , Z/p) where X is an algebraic variety over k = Fl , k(X ) = 2 K . Any element of Hnr (X , Z/p) is induced from H 2 (G al ab (K ), Z/p) by Merkurjev2 2 (G al (K , Z/p) Suslin theorem (see [MS82],[Bog91a] [Bog92]) and Hnr (X , Z/p) = Hnr 2 and the latter coincides with Hnr (G al c (K ), Z/p) (see [Bog91a]).

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