alternative lecture notes - Rational points and algebraic cycles

Also prolong the forgetful functor in the opposite direction. There is a map
and
Et /K (X) → Z Et /K (X)
X(K) → π 0 (Map der (∗, Et /K (X))) → π 0 (Map der (∗, Z Et /K (X))) = H 0 cont(N(Z Et /K (X))).
17. August 27
Theorem 17.1. Let Y be a simplicial Γ-module. Then the Dold–Kan correspondence induces
a natural isomorphism of sets
π 0 (Y hΓ ) ≃ H 0 (Γ, N(Y )).
Theorem 17.2. Let (Y α ) α∈I ∈ Pro(Γ- Mod ∆op ). Then by Dold–Kan,
π 0 (Map der (∗, (Y α ))) ≃ H 0 cont(Γ, N(Y α )).
X(K)
h
H 0 cont(K, N(Z Et /K (X)))
X(A)
loc
∏h
v H0 cont(K v , N(Z Et /K (X)))
The image of the bottom horizontal map lands in a subgroup H 0 (A, N(Z Et /K (X))). Define
Then X(K) ⊆ X(A) Zh ⊆ X(A).
X(A) Zh := {(a v ) ∈ X(A) : h(a v ) ∈ im(loc)}.
Theorem 17.3 (Harpaz, Schlank). If X is smooth and connected, then X(A) Zh = X(A) Br .
Remark 17.4. The only reason that X is assumed to be connected is that if X = Spec K ∐ Spec K,
for instance, and K has at least one complex place, then X(A) Br ≠ X(K).
Let U → X be a hypercover.
X(K)
h
H 0 cont(K, N(Z Et /K (X)))
ψ U
lim U H 0 (K, N(Z¯π 0 (U)))
X(A)
loc
h ∏
v H0 cont(K v , N(Z Et /K (X)))
loc
ψ U ∏
lim U v H0 cont(K v , N(Z¯π 0 (U)))
Proposition 17.5. For every (a v ) v ∈ X(A), h(a v ) ∈ im(loc) if and only if for all U → X,
we have ψ U (h(a v )) ∈ im loc U .
Sketch of proof. Cohomology of local fields is finite and the hyper-Tate–Shafarevich group
(kernel of the right loc U ) is finite and profinite groups do not have lim
1 .
□
←−
One can replace the products by adelic versions.
Definition 17.6. H q (A, C) is the restricted direct product of H q (K v , C) with respect to the
images of the maps H i (Ẑ, CI ) → H i (K v , C), where I is the inertia.
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