By Yuri Tschinkel (Ed.)

ISBN-10: 3938616776

ISBN-13: 9783938616772

**Read Online or Download Algebraic Groups: Mathematisches Institut, Georg-August-Universitat Gottingen. Summer School, 27.6.-13.7.2005 PDF**

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**Extra resources for Algebraic Groups: Mathematisches Institut, Georg-August-Universitat Gottingen. Summer School, 27.6.-13.7.2005**

**Sample text**

The element a ∈ Hnr (X , Z/p) is called unramified if it is unramified with respect to any divisorial valuation of k(X ). This definition can be extended to the cohomology of the entire Galois group. 2. Consider a ∈ HS∗ (X , Z/p). The element a is called unramified if for every divisorial valuation νD the image of a in the decomposition group G al νD (K ) is induced from the quotient group G al νD (K )/I νD where I νD is a topologically cyclic inertia subgroup. ∗ (G al (K ), Z/p) ⊂ H ∗ (G al (K ), Z/p).

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 . . λn1 . A = .. with rk A = k. This space is contractible up to dimenλ1k . . λnk sion nk − k + 1 because M is contained in the vectorspace V of all matrices, and its complement S is : V \ M = S = {A|rk(A) k − 1} Notice, that if dim(S) < r , then V \ S is contractible up to dimension n − r − 2.

Any element a ∈ H ∗ (G al (K ), F ) is induced from a finite quotient group G a under a surjective continuous map f : G al (K ) → G, or, equivalently, from the cohomology of a sheaf F˜ on an open subvariety X a ⊂ X . We want to show that if a ∈ H ∗ (G al (K ), F ) vanishes in H s∗ (G al (K ), F ) then a also vanishes on some open subvariety X v a ∈ X a . Since a has finite order, its vanishing in H s∗ (G al (K ), F ) is equivalent to the existence of a finite group G and surjective (continuous) maps h : G al (K ) → G and g : G → G with g h = f such that h ∗ (a) = 0 ∈ H s∗ (G , F ).

### Algebraic Groups: Mathematisches Institut, Georg-August-Universitat Gottingen. Summer School, 27.6.-13.7.2005 by Yuri Tschinkel (Ed.)

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