alternative lecture notes - Rational points and algebraic cycles

Back to algebraic geometry. Recall our D-torsor U → X, where D is the dihedral group
of order 8. Recall that U has 4 connected components on which Gal(K( √ a/ √ b)/K) acts
transitively and freely (assuming that this extension has degree 4). Then Č(U) is
U × D × D
U × D
The action of D on t and s defines a surjective homomorphism
U
D → Gal(K( √ a, √ b)/K);
let C be the kernel, a group of order 2. Taking π 0 yields
So X := ¯π 0 (Č(U)) = ED/C, so
π n (X) =
C\D × D × D
C\D × D
C\D.
{
0, if n ≠ 1
Z/2Z = C, if n = 1.
So the obstruction o 1 ∈ H 2 (Γ k , π 1 (X)) = H 2 (Γ k , Z/2Z) would be the pullback by Γ k
Gal(K( √ a, √ b)/K) of the class in H 2 ((Z/2Z) 2 , Z/2Z) related to
→
1 → Z/2Z → D → (Z/2Z) 2 → 1.
But now it is an easy calculation to see that the corresponding element in H 2 (Z/2Z ×
Z/2Z, Z/2Z) is the cup product of the two generators of H 1 (Z/2Z × Z/2Z, Z/2Z). Thus we
get that o 1 ∈ H 2 (Γ k , π 1 (X)) = H 2 (Γ k , Z/2Z) equals (a) ∪ (b).
Generalization: Given a 1 , . . . , a n ∈ K × , if there is a solution to a 0 x 2 0 + · · · + a n x 2 n = 1,
then the cup product ∪(a i ) ∈ H n+1 (K, Z/2Z) is 0.
Now let X be any k-variety. Let U • → X be a hypercover. For any set A, let Z 1 A be the
set of degree 1 formal integer combinations of elements of A. Extend this to Γ k -simplicial
sets. There is a map ¯π 0 (U • ) → Z 1¯π 0 (U).
Properties of Z 1 A for a Γ k -simplicial set A.
(1) If Z 1 A hΓ k = ∅, then A
hΓ k = ∅.
(2) π n (Z 1 A) = ˜H n (A, Z). (There is a variant: for n ≥ 1, π n ((Z/nZ) 1 A) = H 2 (A, Z/n).)
It would be nicer if Z 1 A were a group. Define ZA to be the simplicial set obtained by
taking formal integer combinations at each level. Then ZA = ⊕ i∈Z Zi A, and it is a simplicial
Γ k -module.
By the Dold–Kan correspondence, studying homotopy fixed points of ZA modulo equivalence
is the same as studying Galois hypercohomology of the corresponding complex N(ZA).
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