alternative lecture notes - Rational points and algebraic cycles

Thus
H n (|C|, A) = [|C|, K(A, n)]
= [LL(∗ MC , K(A, n)] Pro-Set ∆ op
= [∗ MC , RΓ ∗ (K(A, n))] Pro-Sh(C) ∆ op
= [∗ MC , K(Γ ∗ A, n)] Pro-Sh(C) ∆ op
= [∗ MC , K(Γ ∗ A, n)] Sh(C) ∆ op
= H n (C, A),
where Γ ∗ A ∈ Sh(C) ∆op .
Let X be a scheme. Then |X et | ∈ Pro(Set ∆op ). If A is a constant sheaf, then Het(X, n A) =
H n (|X et |, A). Also, if X is noetherian, π 1 (|X et |) ≃ π1 et (X) as pro-groups. (In particular,
π 1 (|X et |) is profinite when X is noetherian.)
Artin and Mazur (1969) define Et(X) ∈ Pro-Ho(Set ∆op ). There is an obvious functor
Pro(Ho): Pro-Set ∆op → Pro-Ho(Set ∆op ).
Theorem 13.5. There is a natural isomorphism Pro(Ho)(|X et |) ≃ Et(X).
Motivation: Let Y be a compactly generated Hausdorff space and let (U i ) i∈I be a good
cover of Y (recall that this means that for every finite nonempty J ⊆ I, the intersection
U J := ⋂ i∈J U i is a disjoint union of contractible spaces). Recall that the nerve of (U i ) is the
simplicial set N((U i )) with
N((U i )) n := π 0 (U × Y · · · × Y U) ∈ Set ∆op .
Then Y is weakly equivalent to N((U i )).
Now let X be a regular noetherian scheme. We cannot expect to find a good cover in the
sense above. For example, if C is a curve of positive genus, then there are no “contractible”
étale opens in C. (Here “contractible” should be some property that implies that C(C) in
the analytic topology is a contractible topological space.) The nerve of each étale covering
of X should be viewed as an approximation to the étale homotopy type.
Étale homotopy type (version 1, beta): take in Pro-Ho(Set ∆op ) the inverse system of N(U)
for all étale covers U → X. This presumes that the category of étale covers is cofiltered, but
this is not true (equalizers do not exist).
Definition 13.6. If U is an object of the étale site C, we obtain a functor h U : C op → Sets
defined by h U (O) = Hom(O, U) for O ∈ C, and it is a sheaf, called the represented sheaf.
Definition 13.7. A hypercover in a site C with enough points is an object U • ∈ C ∆op such
that the represented simplicial sheaf of U • is stalkwise fibrant and stalkwise constructible.
Lemma 13.8. Let U → X be an étale cover. Then the simplicial object Č(U) defined by
the fibered powers of U over Y is a hypercover.
Proof. Let x ∈ X be a point. Then the stalk of h U at x is just f −1 (x). Thus the stalk of
Č(U) at x is the simplicial set . . ., f −1 (x) × f −1 (x), f −1 (x) is Kan and contractible. □
Étale homotopy type (version 2): Take in Pro-Ho(Set ∆op ) the inverse system of π 0 (U • )
for hypercovers U • → X.
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