From rational points to homotopy fixed points Chern Institute July - - PowerPoint PPT Presentation

From rational points to homotopy fixed points

Chern Institute July 25, 2016 Gereon Quick NTNU

A fundamental short exact sequence: Let k be a number field with algebraic closure K.

A fundamental short exact sequence: Let k be a number field with algebraic closure K. Let X be a smooth geometrically connected variety

• ver k with geometric point x.

A fundamental short exact sequence: Let k be a number field with algebraic closure K. π1 (X,x)

ét

étale fundamental group

Let X be a smooth geometrically connected variety

• ver k with geometric point x.

A fundamental short exact sequence: Let k be a number field with algebraic closure K. π1 (X,x)

ét

étale fundamental group

Let X be a smooth geometrically connected variety

• ver k with geometric point x.

1 Gal(K/k)

absolute Galois group ≈ π1 (K,x)

ét

A fundamental short exact sequence: Let k be a number field with algebraic closure K. π1 (X,x)

ét

étale fundamental group

π1 (XK,x)

ét

1

XK = XxkK

Let X be a smooth geometrically connected variety

• ver k with geometric point x.

1 Gal(K/k)

absolute Galois group ≈ π1 (K,x)

ét

Nice sections: π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k)

Nice sections: Let y: Spec k → X be a rational point. π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k)

Nice sections: Let y: Spec k → X be a rational point. π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k) Applying π1 (-,y) induces

ét

Nice sections: Let y: Spec k → X be a rational point. π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k) sy π1 (X,y)

ét

Applying π1 (-,y) induces

ét

Nice sections: Let y: Spec k → X be a rational point. π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k) sy π1 (X,y)

ét

an étale path p: y↝x

Applying π1 (-,y) induces

ét

Nice sections: Let y: Spec k → X be a rational point. π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k) sy π1 (X,y)

ét

p∗ an étale path p: y↝x

Applying π1 (-,y) induces

ét

Nice sections: Let y: Spec k → X be a rational point. π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k) a section after composition with p. sy π1 (X,y)

ét

p∗ an étale path p: y↝x

Applying π1 (-,y) induces

ét

Nice sections: Let y: Spec k → X be a rational point. π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k) a section after composition with p. sy π1 (X,y)

ét

p∗ an étale path p: y↝x

This section is unique up to conjugation by elements in π1 (XK,x).

ét

Applying π1 (-,y) induces

ét

The section conjecture: X(k) π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k) [sy] y ⟼ [sy] π1 (XK,x)

ét

classes of sections S(X/k) = set of

• conjugacy

The section conjecture: X(k) π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k) Grothendieck: This map is a bijection if X is a geom. connected projective smooth curve of genus ≥2. [sy] y ⟼ [sy] π1 (XK,x)

ét

classes of sections S(X/k) = set of

• conjugacy

The section conjecture: X(k) π1 (X,x)

ét

π1 (XK,x)

ét

1 1 Gal(K/k) Grothendieck: This map is a bijection if X is a geom. connected projective smooth curve of genus ≥2. [sy] y ⟼ [sy] π1 (XK,x)

ét

classes of sections S(X/k) = set of

• conjugacy

Known: The map is injective.

A motivating reformulation: X(k) S(X/k)

A motivating reformulation: X(k) S(X/k) ≈ Homout(Gal(K/k), π1 (X,x))

ét

• uter homs compatible

with projection to Gal(K/k)

A motivating reformulation: X(k) S(X/k) ≈ Homout(Gal(K/k), π1 (X,x))

ét

• uter homs compatible

with projection to Gal(K/k)

≈ [BGal(K/k), Bπ1 (X,x)]BGal(K/k)

ét

morphisms in the homotopy category

• ver BGal(K/k)

A motivating reformulation: X(k) S(X/k) ≈ Homout(Gal(K/k), π1 (X,x))

ét

• uter homs compatible

with projection to Gal(K/k)

≈ [BGal(K/k), Bπ1 (X,x)]BGal(K/k)

ét

morphisms in the homotopy category

• ver BGal(K/k)

≈ π0((XK)ét)hGal(K/k))

homotopy fixed point space of the étale topological type under the action of Gal(K/k)

A short topological detour:

Let Y be a topological space with an action of a group G. A short topological detour:

Let Y be a topological space with an action of a group G. Taking the G-fixed points is not homotopy invariant, i.e. X≃Y does not imply XG≃YG. A short topological detour:

Let Y be a topological space with an action of a group G. Taking the G-fixed points is not homotopy invariant, i.e. X≃Y does not imply XG≃YG. Example: G=Z the integers, Y=R the real line. A short topological detour:

Let Y be a topological space with an action of a group G. Taking the G-fixed points is not homotopy invariant, i.e. X≃Y does not imply XG≃YG. Example: G=Z the integers, Y=R the real line. R 1 2

• 1

n A short topological detour:

Let Y be a topological space with an action of a group G. Taking the G-fixed points is not homotopy invariant, i.e. X≃Y does not imply XG≃YG. Example: G=Z the integers, Y=R the real line. R 1 2

• 1

n n: x ↦ x+n A short topological detour:

Let Y be a topological space with an action of a group G. Taking the G-fixed points is not homotopy invariant, i.e. X≃Y does not imply XG≃YG. Example: G=Z the integers, Y=R the real line. R 1 2

• 1

n n: x ↦ x+n Y≃{pt} A short topological detour:

Let Y be a topological space with an action of a group G. Taking the G-fixed points is not homotopy invariant, i.e. X≃Y does not imply XG≃YG. Example: G=Z the integers, Y=R the real line. R 1 2

• 1

n n: x ↦ x+n Y≃{pt} and YG=∅, A short topological detour:

Let Y be a topological space with an action of a group G. Taking the G-fixed points is not homotopy invariant, i.e. X≃Y does not imply XG≃YG. Example: G=Z the integers, Y=R the real line. R 1 2

• 1

n n: x ↦ x+n Y≃{pt} and YG=∅, but {pt}G≠∅ A short topological detour:

Homotopy fixed points:

We have Y = Map(pt,Y), hence YG = MapG(pt,Y). Homotopy fixed points:

We have Y = Map(pt,Y), hence YG = MapG(pt,Y).

space of G-equiv. maps

Homotopy fixed points:

We have Y = Map(pt,Y), hence YG = MapG(pt,Y). The problem: the one point-space is not a “well-behaved” space.

space of G-equiv. maps

Homotopy fixed points:

We have Y = Map(pt,Y), hence YG = MapG(pt,Y). The problem: the one point-space is not a “well-behaved” space. The solution: Replace it by a (cofibrant) resolution EG, a contractible space with a free G-action.

space of G-equiv. maps

Homotopy fixed points:

We have Y = Map(pt,Y), hence YG = MapG(pt,Y). The problem: the one point-space is not a “well-behaved” space. The solution: Replace it by a (cofibrant) resolution EG, a contractible space with a free G-action. EG BG=EG/G

canonical G-bundle

space of G-equiv. maps

Homotopy fixed points:

We have Y = Map(pt,Y), hence YG = MapG(pt,Y). The problem: the one point-space is not a “well-behaved” space. The solution: Replace it by a (cofibrant) resolution EG, a contractible space with a free G-action. EG BG=EG/G

canonical G-bundle

space of G-equiv. maps

Define homotopy fixed points as YhG = MapG(EG,Y). Homotopy fixed points:

We have Y = Map(pt,Y), hence YG = MapG(pt,Y). The problem: the one point-space is not a “well-behaved” space. The solution: Replace it by a (cofibrant) resolution EG, a contractible space with a free G-action. EG BG=EG/G

canonical G-bundle

space of G-equiv. maps

Define homotopy fixed points as YhG = MapG(EG,Y). Example above: YhG≃{pt}. ✔ Homotopy fixed points:

Fixed and homotopy fixed points:

Fixed and homotopy fixed points: Homotopy fixed points are homotopy invariant:

Fixed and homotopy fixed points: Homotopy fixed points are homotopy invariant: X≃Y implies XhG≃YhG.

Fixed and homotopy fixed points: There is a canonical map YG = MapG(pt,Y) → MapG(EG,Y) = YhG. Homotopy fixed points are homotopy invariant: X≃Y implies XhG≃YhG.

Fixed and homotopy fixed points: But this map is in general not a homotopy equivalence. There is a canonical map YG = MapG(pt,Y) → MapG(EG,Y) = YhG. Homotopy fixed points are homotopy invariant: X≃Y implies XhG≃YhG.

Sullivan’ s question:

Sullivan’ s question: Let X be a variety over the real numbers R.

Sullivan’ s question: Let X be a variety over the real numbers R. Question: How can we recover the homotopy type

• f X(R) from the one of X(C)?

Sullivan’ s question: Let X be a variety over the real numbers R. The group Z/2 acts on X(C) by complex conjugation and X(R) = X(C)Z/2. Question: How can we recover the homotopy type

• f X(R) from the one of X(C)?

Sullivan’ s question: Let X be a variety over the real numbers R. The group Z/2 acts on X(C) by complex conjugation and X(R) = X(C)Z/2. Question: How can we recover the homotopy type

• f X(R) from the one of X(C)?

But: The homotopy type of X(R) may not be determined by the one of X(C) by taking fixed points.

Sullivan’ s question: Example: Let X=P1 be the projective line.

Sullivan’ s question: Example: Let X=P1 be the projective line. πn((X(C)p)hZ/2) = (πnX(C)p)Z/2. Then X(C)≅S2; its p-completion, for p odd, satisfies

p-completion of X(C)

Sullivan’ s question: Example: Let X=P1 be the projective line. But: X(R)≃S1 and π1((X(C)p)hZ/2) = {1} ≠ π1X(R)p. πn((X(C)p)hZ/2) = (πnX(C)p)Z/2. Then X(C)≅S2; its p-completion, for p odd, satisfies

p-completion of X(C)

Sullivan conjecture (Miller, Lannes, Carlsson): Let p be a prime number, G a finite p-group acting

• n a nice topological Y

. e.g. finite complex or Bπ, π finite group

Sullivan conjecture (Miller, Lannes, Carlsson): is an equivalence. (Thm by Miller, Lannes, Carlsson) (Yp)G → (Yp)hG Then the canonical map

p-completion of Y

Let p be a prime number, G a finite p-group acting

• n a nice topological Y

. e.g. finite complex or Bπ, π finite group

Sullivan conjecture (Miller, Lannes, Carlsson): is an equivalence. (Thm by Miller, Lannes, Carlsson) (Yp)G → (Yp)hG Then the canonical map

p-completion of Y

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. Let p be a prime number, G a finite p-group acting

• n a nice topological Y

. e.g. finite complex or Bπ, π finite group

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 NX(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 NX(U) = U• U ← ← ← ← ← ← ← ← ← ... U×XU U×XU×XU U×XU×XU×XU → → → i.e. Un is the n+1-fold fiber product of U over X. Etale homotopy in a nutshell (after Artin-Mazur, Friedlander, Sullivan,…): Let X be a scheme of finite type over a field k.

The idea:

The idea: Forming the Cech nerve yields a simplicial set π0(U•).

connected components

The idea: Forming the Cech nerve yields a simplicial set π0(U•).

connected components

Observation: For a variety X over a field, the colimit of the singular cohomologies of all the spaces π0(U•)‘s computes the etale cohomology of X.

i.e. simplicial sets

The idea: A candidate for an etale homotopy type: the “system of all spaces π0(U•)’ s”. Forming the Cech nerve yields a simplicial set π0(U•).

connected components

Observation: For a variety X over a field, the colimit of the singular cohomologies of all the spaces π0(U•)‘s computes the etale cohomology of X.

i.e. simplicial sets

Friedlander’ s rigidification:

A “rigid cover” is

Friedlander’ s rigidification:

A “rigid cover” is

a disjoint union of pointed, étale, separated morphisms where each Ux is connected and ux is a geometric point of Ux lying over x. ∐ (𝛽x: Ux,ux → X,x)

x∈X(K)

Friedlander’ s rigidification:

A “rigid cover” is

a disjoint union of pointed, étale, separated morphisms where each Ux is connected and ux is a geometric point of Ux lying over x. ∐ (𝛽x: Ux,ux → X,x)

x∈X(K)

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

Xét: RC(X) → S,

Friedlander’ s rigidification:

The “rigid Cech type of X” is the pro-simplicial set

Cech nerve set of connected components

U ↦ π0(NX(U)).

X

Xét: RC(X) → S,

Friedlander’ s rigidification:

The “rigid Cech type of X” is the pro-simplicial set

Example:

Cech nerve set of connected components

U ↦ π0(NX(U)).

X

Xét: RC(X) → S,

Friedlander’ s rigidification:

The “rigid Cech type of X” is the pro-simplicial set

Example:

Cech nerve set of connected components

U ↦ π0(NX(U)).

X

• két:= (Spec k)ét ≈ {BGal(L/k)}k⊂L⊂K;

L/k finite Galois

Xét: RC(X) → S,

Friedlander’ s rigidification:

The “rigid Cech type of X” is the pro-simplicial set

Example:

Cech nerve set of connected components

U ↦ π0(NX(U)).

X

• két:= (Spec k)ét ≈ {BGal(L/k)}k⊂L⊂K;

L/k finite Galois

Xét: RC(X) → S, for the rigid covers of k are exactly the finite Galois extensions L of k in K.

Friedlander’ s rigidification: Xét: RC(X) → S, U ↦ π0(NX(U)). Let X be defined over k as before. Some interesting observations:

Friedlander’ s rigidification: Xét: RC(X) → S, U ↦ π0(NX(U)). Let X be defined over k as before. Some interesting observations:

• XKét is a pro-space with a Gal(K/k)-action

(acting on the indexing category).

Friedlander’ s rigidification: Xét: RC(X) → S, U ↦ π0(NX(U)). Let X be defined over k as before. Some interesting observations:

• XKét is a pro-space with a Gal(K/k)-action

(acting on the indexing category).

• Xét is a pro-space over BGal(K/k), i.e., there is

an induced map of pro-spaces Xét → 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: Example: the classifying space Bπ of a profinite group π. Let S be the category of “profinite spaces”, i.e. simplicial profinite sets. ^

Profinite spaces with continuous Galois action: Example: the classifying space Bπ of a profinite group π. Let S be the category of “profinite spaces”, i.e. simplicial profinite sets. ^ Let SG be the category of simplicial profinite sets with a continuous action by a profinite group G. ^

Profinite spaces with continuous Galois action: Example: the classifying space Bπ of a profinite group π. Let S be the category of “profinite spaces”, i.e. simplicial profinite sets. ^ Let SG be the category of simplicial profinite sets with a continuous action by a profinite group G. ^ Example: the classifying space Bπ of a profinite group π with a continuous G-action.

A useful adjunction: Let SBG be the category of profinite spaces over BG. ^

A useful adjunction:

There are (Quillen-) adjoint functors

Let SBG be the category of profinite spaces over BG. ^

A useful adjunction:

There are (Quillen-) adjoint functors

Let SBG be the category of profinite spaces over BG. ^ ^ SBG SG ^

profinite spaces over BG profinite spaces with G-action

left right

A useful adjunction:

There are (Quillen-) adjoint functors

Let SBG be the category of profinite spaces over BG. ^ X→BG ^ SBG SG ^

profinite spaces over BG profinite spaces with G-action

left right

A useful adjunction:

There are (Quillen-) adjoint functors

Let SBG be the category of profinite spaces over BG. ^ X→BG ⟼ X×BGEG

EG BG X pb.

^ SBG SG ^

profinite spaces over BG profinite spaces with G-action

left right

A useful adjunction:

There are (Quillen-) adjoint functors

Let SBG be the category of profinite spaces over BG. ^ X→BG

with G-action induced by the

• ne on EG

⟼ X×BGEG

EG BG X pb.

^ SBG SG ^

profinite spaces over BG profinite spaces with G-action

left right

A useful adjunction:

There are (Quillen-) adjoint functors

Let SBG be the category of profinite spaces over BG. ^ X→BG Y

with G-action induced by the

• ne on EG

⟼ X×BGEG

EG BG X pb.

^ SBG SG ^

profinite spaces over BG profinite spaces with G-action

left right

A useful adjunction:

There are (Quillen-) adjoint functors

Let SBG be the category of profinite spaces over BG. ^ X→BG Y ⟻ Y×GEG

Borel construction with G-action induced by the

• ne on EG

⟼ X×BGEG

EG BG X pb.

^ SBG SG ^

profinite spaces over BG profinite spaces with G-action

left right

Profinite completion: There are profinite completion functors

Profinite completion: There are profinite completion functors S ^ S

Profinite completion: There are profinite completion functors S ^ S

levelwise profinite completion of sets

Profinite completion: There are profinite completion functors S ^ S

levelwise profinite completion of sets

pro-S ^ pro-S

complete

• bjectwise

Profinite completion: There are profinite completion functors S ^ S

levelwise profinite completion of sets

pro-S ^ pro-S

complete

• bjectwise

^ S

take fibrant replacements

• bjectwise and then

(homotopy) limits

Profinite completion: There are profinite completion functors S ^ S

levelwise profinite completion of sets

pro-S ^ pro-S

complete

• bjectwise

^ S

take fibrant replacements

• bjectwise and then

(homotopy) limits

For X/k and G:=Gal(K/k), we have ^ ^ ^ ^ Xét∈SBG and XKét∈SG

Continuous homotopy fixed points:

Continuous homotopy fixed points: For Y∈SG, we can define continuous homotopy fixed points: ^

Continuous homotopy fixed points: For Y∈SG, we can define continuous homotopy fixed points: ^ YhG := MapS (BG, RGY×GEG).

BG

^ ^ fibrant replacement in SG

Continuous homotopy fixed points: For Y∈SG, we can define continuous homotopy fixed points: ^ YhG := MapS (BG, RGY×GEG).

BG

^ ^ fibrant replacement in SG

For X/k and G:=Gal(K/k) as before, we get continuous Galois homotopy fixed points: ^ ^ XKét = MapS (BG,Xét)

hG

^BG

Continuous homotopy fixed points: For Y∈SG, we can define continuous homotopy fixed points: ^ YhG := MapS (BG, RGY×GEG).

BG

^ ^ fibrant replacement in SG

For X/k and G:=Gal(K/k) as before, we get continuous Galois homotopy fixed points: ^ ^ XKét = MapS (BG,Xét)

hG

^BG

Theorem (Q.): The canonical map XKét×GEG → Xét is a weak equivalence in S. ^ ^ ^

Continuous homotopy fixed points: For Y∈SG, we can define continuous homotopy fixed points: ^ YhG := MapS (BG, RGY×GEG).

BG

^ ^ fibrant replacement in SG

For X/k and G:=Gal(K/k) as before, we get continuous Galois homotopy fixed points: ^ ^ XKét = MapS (BG,Xét)

hG

^BG

≈ (XKét)hG ^ Theorem (Q.): The canonical map XKét×GEG → Xét is a weak equivalence in S. ^ ^ ^

Back to rational points:

Back to rational points: Let X be a geom. connected smooth projective variety over a number field k. Let X(k) be the set of rational points.

Back to rational points: Let X be a geom. connected smooth projective variety over a number field k. Let X(k) be the set of rational points. Functoriality of the etale homotopy type gives a natural map

Back to rational points: Let X be a geom. connected smooth projective variety over a number field k. Let X(k) be the set of rational points. Functoriality of the etale homotopy type gives a natural map (Spec k → X) ⟼ (két → Xét) ^ ^

Back to rational points: Let X be a geom. connected smooth projective variety over a number field k. Let X(k) be the set of rational points. Functoriality of the etale homotopy type gives a natural map

homotopy category

• f Skét

^

X(k) → Hom (két,Xét) Hkét ^ ^ ^ (Spec k → X) ⟼ (két → Xét) ^ ^

From rational to homotopy fixed points: X(k) Hket ^ Hom (ket,Xét) ^ ^

From rational to homotopy fixed points: X(k) Hket ^ Hom (ket,Xét) ^ ^ ≈ Hom (BG,Xét) HBG ^ with G=Gal(K/k) ^

From rational to homotopy fixed points: X(k) Hket ^ Hom (ket,Xét) ^ ^ ≈ Hom (BG,Xét) HBG ^ with G=Gal(K/k) ^ ≈ π0Map (BG,Xét) SBG ^ ^

From rational to homotopy fixed points: X(k) Hket ^ Hom (ket,Xét) ^ ^ ≈ Hom (BG,Xét) HBG ^ with G=Gal(K/k) ^ ≈ π0Map (BG,Xét) SBG ^ ^ ≈ π0(XKét) ^ hG

Obstructions to rational points: X(k) → π0(XKét) ^ hG

Obstructions to rational points: X(k) → π0(XKét) ^ hG A consequence: Homotopy fixed points define an

• bstruction for rational points (e.g. Pal, Harpaz-

Schlank, but with a different construction of the set π0(XKét)). ^ hG

Obstructions to rational points: For X and k as in the Section Conjecture, we know XKét ≈ Bπ1 (XK,x).

ét

^ X(k) → π0(XKét) ^ hG A consequence: Homotopy fixed points define an

• bstruction for rational points (e.g. Pal, Harpaz-

Schlank, but with a different construction of the set π0(XKét)). ^ hG

Obstructions to rational points: For X and k as in the Section Conjecture, we know XKét ≈ Bπ1 (XK,x).

ét

^ Hence in this case we have (XKét) ≈ (Bπ1 (XK,x)).

ét

^

hG hG

X(k) → π0(XKét) ^ hG A consequence: Homotopy fixed points define an

• bstruction for rational points (e.g. Pal, Harpaz-

Schlank, but with a different construction of the set π0(XKét)). ^ hG

Thank you!