Sullivan’ s question: Example: Let X=P 1 be the projective line. Then X(C) ≅ S 2 ; its p-completion, for p odd, satisfies π n ((X(C) p ) hZ/2 ) = ( π n X(C) p ) Z/2 . p-completion of X(C)
Sullivan’ s question: Example: Let X=P 1 be the projective line. Then X(C) ≅ S 2 ; its p-completion, for p odd, satisfies π n ((X(C) p ) hZ/2 ) = ( π n X(C) p ) Z/2 . p-completion of X(C) But: X(R) ≃ S 1 and π 1 ((X(C) p ) hZ/2 ) = {1} ≠ π 1 X(R) p .
Sullivan conjecture (Miller, Lannes, Carlsson): Let p be a prime number, G a finite p-group acting on a nice topological Y . e.g. finite complex or B π , π finite group
Sullivan conjecture (Miller, Lannes, Carlsson): Let p be a prime number, G a finite p-group acting on a nice topological Y . e.g. finite complex or B π , π finite group p-completion of Y Then the canonical map (Y p ) G → (Y p ) hG is an equivalence. (Thm by Miller, Lannes, Carlsson)
Sullivan conjecture (Miller, Lannes, Carlsson): Let p be a prime number, G a finite p-group acting on a nice topological Y . e.g. finite complex or B π , π finite group p-completion of Y Then the canonical map (Y p ) G → (Y p ) hG is an equivalence. (Thm by Miller, Lannes, Carlsson) In particular: if X is a variety over R, then X(R) 2 ≃ X(C) 2Z/2 → (X(C) 2 ) hZ/2 is an equivalence.
Etale homotopy in a nutshell (after Artin-Mazur, Friedlander, Sullivan,…): Let X be a scheme of finite type over a field k.
Etale homotopy in a nutshell (after Artin-Mazur, Friedlander, Sullivan,…): Let X be a scheme of finite type over a field k. Let U → X be an etale cover.
Etale homotopy in a nutshell (after Artin-Mazur, Friedlander, Sullivan,…): Let X be a scheme of finite type over a field k. Let U → X be an etale cover. We can form the “Cech nerve” associated to U → X. This is the simplicial scheme N X (U) = U •
Etale homotopy in a nutshell (after Artin-Mazur, Friedlander, Sullivan,…): Let X be a scheme of finite type over a field k. Let U → X be an etale cover. We can form the “Cech nerve” associated to U → X. This is the simplicial scheme N X (U) = U • → ← ← ← ← U × X U U × X U × X U U × X U × X U × X U ← U ← ← ... ← → ← → i.e. U n is the n+1-fold fiber product of U over X.
The idea:
The idea: Forming the Cech nerve yields a simplicial set π 0 (U • ). connected components
The idea: Forming the Cech nerve yields a simplicial set π 0 (U • ). connected components Observation: For a variety X over a field, the colimit of the singular cohomologies of all the spaces π 0 (U • )‘s computes the etale cohomology of X. i.e. simplicial sets
The idea: Forming the Cech nerve yields a simplicial set π 0 (U • ). connected components Observation: For a variety X over a field, the colimit of the singular cohomologies of all the spaces π 0 (U • )‘s computes the etale cohomology of X. i.e. simplicial sets A candidate for an etale homotopy type: the “system of all spaces π 0 (U • )’ s”.
Friedlander’ s rigidification: A “ rigid cover” is
Friedlander’ s rigidification: A “ rigid cover” is a disjoint union of pointed, étale, separated morphisms ∐ ( 𝛽 x : U x ,u x → X,x) x ∈ X(K) where each U x is connected and u x is a geometric point of U x lying over x.
Friedlander’ s rigidification: A “ rigid cover” is a disjoint union of pointed, étale, separated morphisms ∐ ( 𝛽 x : U x ,u x → X,x) x ∈ X(K) where each U x is connected and u x is a geometric point of U x lying over x. Rigid covers form a filtered category denoted by RC(X).
Friedlander’ s rigidification: The “ rigid Cech type of X” is the pro-simplicial set
Friedlander’ s rigidification: The “ rigid Cech type of X” is the pro-simplicial set X ét : RC(X) → S,
Friedlander’ s rigidification: The “ rigid Cech type of X” is the pro-simplicial set X ét : RC(X) → S, U ↦ π 0 (N X (U)). X Cech nerve set of connected components
Friedlander’ s rigidification: The “ rigid Cech type of X” is the pro-simplicial set X ét : RC(X) → S, U ↦ π 0 (N X (U)). X Cech nerve set of connected components Example:
Friedlander’ s rigidification: The “ rigid Cech type of X” is the pro-simplicial set X ét : RC(X) → S, U ↦ π 0 (N X (U)). X Cech nerve set of connected components Example: L/k finite Galois • k ét := (Spec k) ét ≈ {BGal(L/k)} k ⊂ L ⊂ K ;
Friedlander’ s rigidification: The “ rigid Cech type of X” is the pro-simplicial set X ét : RC(X) → S, U ↦ π 0 (N X (U)). X Cech nerve set of connected components Example: L/k finite Galois • k ét := (Spec k) ét ≈ {BGal(L/k)} k ⊂ L ⊂ K ; for the rigid covers of k are exactly the finite Galois extensions L of k in K.
Friedlander’ s rigidification: X ét : RC(X) → S, U ↦ π 0 (N X (U)). Let X be defined over k as before. Some interesting observations:
Friedlander’ s rigidification: X ét : RC(X) → S, U ↦ π 0 (N X (U)). Let X be defined over k as before. Some interesting observations: • X Két is a pro-space with a Gal(K/k)-action (acting on the indexing category).
Friedlander’ s rigidification: X ét : RC(X) → S, U ↦ π 0 (N X (U)). Let X be defined over k as before. Some interesting observations: • X Két is a pro-space with a Gal(K/k)-action (acting on the indexing category). • X ét is a pro-space over BGal(K/k), i.e., there is an induced map of pro-spaces X ét → BGal(K/k).
Profinite spaces with continuous Galois action:
Profinite spaces with continuous Galois action: ^ Let S be the category of “profinite spaces”, i.e. simplicial profinite sets.
Profinite spaces with continuous Galois action: ^ Let S be the category of “profinite spaces”, i.e. simplicial profinite sets. Example: the classifying space B π of a profinite group π .
Profinite spaces with continuous Galois action: ^ Let S be the category of “profinite spaces”, i.e. simplicial profinite sets. Example: the classifying space B π of a profinite group π . ^ Let S G be the category of simplicial profinite sets with a continuous action by a profinite group G.
Profinite spaces with continuous Galois action: ^ Let S be the category of “profinite spaces”, i.e. simplicial profinite sets. Example: the classifying space B π of a profinite group π . ^ Let S G be the category of simplicial profinite sets with a continuous action by a profinite group G. Example: the classifying space B π of a profinite group π with a continuous G-action.
A useful adjunction: ^ Let S BG be the category of profinite spaces over BG.
A useful adjunction: ^ Let S BG be the category of profinite spaces over BG. There are (Quillen-) adjoint functors
A useful adjunction: ^ Let S BG be the category of profinite spaces over BG. There are (Quillen-) adjoint functors profinite spaces left with G-action ^ ^ S BG S G right profinite spaces over BG
A useful adjunction: ^ Let S BG be the category of profinite spaces over BG. There are (Quillen-) adjoint functors profinite spaces left with G-action ^ ^ S BG S G right X → BG profinite spaces over BG
A useful adjunction: ^ Let S BG be the category of profinite spaces over BG. There are (Quillen-) adjoint functors profinite spaces left with G-action ^ ^ S BG S G right X × BG EG X → BG pb. EG ⟼ profinite spaces over BG BG X
A useful adjunction: ^ Let S BG be the category of profinite spaces over BG. There are (Quillen-) adjoint functors profinite spaces left with G-action ^ ^ S BG S G with G-action induced by the one on EG right X × BG EG X → BG pb. EG ⟼ profinite spaces over BG BG X
A useful adjunction: ^ Let S BG be the category of profinite spaces over BG. There are (Quillen-) adjoint functors Y profinite spaces left with G-action ^ ^ S BG S G with G-action induced by the one on EG right X × BG EG X → BG pb. EG ⟼ profinite spaces over BG BG X
A useful adjunction: ^ Let S BG be the category of profinite spaces over BG. There are (Quillen-) adjoint functors Y × G EG Y ⟻ profinite spaces left with G-action Borel construction ^ ^ S BG S G with G-action induced by the one on EG right X × BG EG X → BG pb. EG ⟼ profinite spaces over BG BG X
Profinite completion: There are profinite completion functors
Profinite completion: There are profinite completion functors ^ S S
Profinite completion: There are profinite completion functors levelwise profinite ^ S S completion of sets
Profinite completion: There are profinite completion functors levelwise profinite ^ S S completion of sets ^ pro-S pro-S complete objectwise
Profinite completion: There are profinite completion functors levelwise profinite ^ S S completion of sets ^ ^ pro-S pro-S S take fibrant replacements complete objectwise and then objectwise (homotopy) limits
Profinite completion: There are profinite completion functors levelwise profinite ^ S S completion of sets ^ ^ pro-S pro-S S take fibrant replacements complete objectwise and then objectwise (homotopy) limits For X/k and G:=Gal(K/k), we have ^ ^ ^ ^ X ét ∈ S BG and X Két ∈ S G
Continuous homotopy fixed points:
Continuous homotopy fixed points: ^ For Y ∈ S G , we can define continuous homotopy fixed points:
Continuous homotopy fixed points: ^ For Y ∈ S G , we can define continuous homotopy fixed points: Y hG := Map S (BG, R G Y × G EG). ^ BG fibrant ^ replacement in S G
Continuous homotopy fixed points: ^ For Y ∈ S G , we can define continuous homotopy fixed points: Y hG := Map S (BG, R G Y × G EG). ^ BG fibrant ^ replacement in S G For X/k and G:=Gal(K/k) as before, we get continuous Galois homotopy fixed points: ^ ^ hG X Két = Map S (BG,X ét ) ^ BG
Continuous homotopy fixed points: ^ For Y ∈ S G , we can define continuous homotopy fixed points: Y hG := Map S (BG, R G Y × G EG). ^ BG fibrant ^ replacement in S G For X/k and G:=Gal(K/k) as before, we get continuous Galois homotopy fixed points: ^ ^ hG X Két = Map S (BG,X ét ) ^ BG ^ ^ Theorem (Q.): The canonical map X Két × G EG → X ét is ^ a weak equivalence in S.
Continuous homotopy fixed points: ^ For Y ∈ S G , we can define continuous homotopy fixed points: Y hG := Map S (BG, R G Y × G EG). ^ BG fibrant ^ replacement in S G For X/k and G:=Gal(K/k) as before, we get continuous Galois homotopy fixed points: ^ ^ hG ^ X Két = Map S (BG,X ét ) ≈ (X Két ) hG ^ BG ^ ^ Theorem (Q.): The canonical map X Két × G EG → X ét is ^ a weak equivalence in S.
Back to rational points:
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