By Yuri Tschinkel (Ed.)
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Extra resources for Algebraic Groups: Mathematisches Institut, Georg-August-Universitat Gottingen. Summer School, 27.6.-13.7.2005
As we have shown before, H ∗ (G al (K ), F ) is also an inductive limit of the cohomology of open subvarieties of X , k(X ) = K . 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 ).
The Picard group of V L /G is Hom(G, Q/Z) and its subgroup of exponent p is equal to H 1 (G, Z/p). The only elements in H 2 (G, Z/p) we can kill under stabilization belong to βH 1 (G, Z/p). Topologically this can be seen as follows. Let V be a faithful complex linear representation of the group G. The group H 2 ((V L /G), Z/p) is represented by (infinite, 40 Mathematisches Institut, Seminars, 2005 real) cycles of codimension 2 in (V L /G). On a smooth variety (V L /G) closed real codimension 2 subvarieties are dual to compact cycles of real dimension two - surfaces.
This implies (and explains) the equality. 11. The above construction can be extended to an arbitrary finite group G. , x n in V, d i mV = n with g (x i ) = λ(g , i , j )x j (g ) for any g ∈ G ( the elements of G permute coordinates x i with additional multiplication by constants) (b) The action of G is free on a subvariety T ⊂ V, x i = 0 for any i . It is clear that V L contains T which is isomorphic to the torus and T is G-invariant (see, for example, [Bog95a]). Thus T /G ⊂ V L /G is a K (π1 , 1)-space with π1 ((T /G)) being an extension of G by a finitely generated abelian group π1 (T ).