alternative lecture notes - Rational points and algebraic cycles

16. August 22 (Harpaz)
Let K be a field. Let X be a K-variety. Let Γ = Gal(K sep /K). We constructed Et /K (X) ∈
Pro(Γ- Set ∆op ). As we have seen, if π 0 (Map der (∗, Et /K (X))) = ∅, then X(k) = ∅.
Definition 16.1. We will say that a simplicial set X is bounded if there is an n such that
for all basepoints x and all N > n, we have π N (X, x) = 0. A pro-simplicial set is bounded if
every simplicial set in the diagram is bounded (the n may depend on the simplicial set).
Claim: There is a canonical functor X → X ♮ from Pro(Set ∆op ) to the full subcategory of
bounded pro-simplicial sets. For each X, there is a morphism X → X ♮ with the property
that for each bounded pro-simplicial set Y , there is a weak equivalence from the derived
mapping space Map der (X ♮ , Y ) to Map der (X, Y ).
Remark 16.2. Given a continuous Γ-action on X, there is a continuous Γ-action on X ♮ .
Applying the functor twice yields an X ♮♮ weakly equivalent to X ♮ . From now on, we assume
that this functor has been applied to all objects, so all pro-objects have been implicitly ♮-
ized. So we work with Et ♮ /K(X), but from now on we drop the ♮. (This possibly weakens the
obstructions.)
So now we have an obstruction theory to the nonemptiness of
X(hK) := π 0 (Map der (∗, Et /K (X)))
with obstructions that live in
Hcont n+1 (Γ, π n (Et /K (X)))
For n = 1, we get
Hcont(Γ, 2 π 1 (Et /K (X))) = Hcont(Γ, 2 π1 et (X))
Claim: The n = 1 obstruction is exactly the nonsplitting of the Grothendieck short exact
sequence.
Last week, Tomer gave the example ∑ n
i=0 x2 1 = −1 over R, for which there is a nonzero
obstruction in H n+1 (Z/2Z, π n (Et /R (X))).
There is a functor Y ↦→ P n (Y ) from Set ∆op to Set ∆op such that π k (P n (Y )) = 0 for
k > n and π k (P n (Y )) ≃ π k (Y ) for k ≤ n. Then (Y ) ♮ = (P n (Y )) n∈N ; this is called the
Postnikov tower. For each k, there is an isomorphism π k (Y ) ≃ π k ((P n (Y )) n∈N . Similarly,
π k ((Y α ) α ) ≃ π k ((Y α ) ♮ ) := π k ((P n (Y α )) α,n ).
In fact, (Y α ) ♮ is defined to be (P n (Y α )) α,n .
Let K be a number field. Let X be a k-variety. We defined
X(hK) := π 0 (Map der (∗, Et /K (X))).
There is a map X(k) → X(hK) (not necessarily injective or surjective). Because of cohomological
obstruction, this might not give an obstruction much stronger than the Grothendieck
section obstruction, however.
For each place v, consider f v : Spec K v → Spec K. Then f ∗ v Et /K (X) ∈ Pro(Γ v - Set ∆op ).
(It turns out that this is the same as Et /Kv (X Kv ); this is a general property of pullback by
field extensions.) We get
X(K v ) → π 0 (Map der (∗, f v Et /K (X)) =: X(hK v ).
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