alternative lecture notes - Rational points and algebraic cycles

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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. 32
Definition 13.9. Let U • , V • ∈ C ∆op and let f, g : U • → V • , we say that f and g are strictly homotopic if there exists H : U • × ∆ 1 → V • (a morphism in Sh(C) ∆op ) such that H(−, 0) = f(−) and H(−, 1) = g(−). Say that f and g are homotopic if they can be connected by a finite chain in which every two adjacent morphisms are strictly homotopic. Proposition 13.10. The category of hypercovers and maps up to homotopy is cofiltered. Definition 13.11. Étale homotopy type (version 3, final): Take in Pro-Ho(Set ∆op ) the inverse system of π 0 (U • ) for hypercovers U • → X, in which the maps in the inverse system are maps up to homotopy. Let X be a smooth variety. We have a map of sites (Spec k) et → X et and Now, take L X/k : M X et M (Spec k) et : Pro(Γ k -spaces) LL X/k (∗ X ) ∈ Pro-(Γ-Set ∆op ). We call this the relative shape of X over Spec k. 14. August 15 (Harpaz) Let us recall the setup of the previous lecture. Let S be a base scheme (e.g., Spec k) Let p: X → S be a scheme over S (e.g., a k-variety). p ! : Pro(Sh(X et ) ∆op ) Define the relative étale shape of X over S to be Pro(Sh(S ∗ et ) ∆op ) Et /S (X) := Lp 1 (∗ X ) ∈ Pro(Sh(S et ) ∆op ), where ∗ X is the terminal sheaf considered as a pro-simplicial set. Special cases: (1) If S = Spec k where k = k, then Sh(S et ) = Set and Et /S (X) = Et /k (X) is a prosimplicial set that coincides with the shape of X et , so the image in Pro(Ho(SSet)) is the étale homotopy type of S. Brief digression: profinite completion. We have the Kan homotopy category Ho(SSet). Let Fin ⊆ Ho(SSet) be the full subcategory of all the simplicial sets such that all their homotopy groups are finite. It can be shown that the inclusion Pro(Fin) ⊆ Pro(Ho(SSet)) admits a left adjoint ̂•: Pro(Ho(SSet)) → Pro(Fin). For X ∈ Ho(SSet) ⊆ Pro(Ho(SSet)), we call ̂X ∈ Pro(Fin) the profinite completion of X. We have a natural map X → ̂X inducing an isomorphism of cohomology with finite coefficients. More special cases: (1) For X a variety over K = C, the image of Et /C (X) in Pro(Ho(C)) is isomorphic to the profinite completion of X(C) (Artin–Mazur comparison theorem). 33
Page 1 and 2: ARITHMETIC APPLICATIONS OF ÉTALE H
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Page 11 and 12: Definition 5.5. Let X be a space an
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Page 31: which we would like to be Quillen.
Page 35 and 36: 15. August 20 (Schlank) Let k be a
Page 37 and 38: 16. August 22 (Harpaz) Let K be a f
Page 39 and 40: Also prolong the forgetful functor
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alternative lecture notes - Rational points and algebraic cycles

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