A 1 -Representability of Hermitian K -Theory. 1 A 1 - - PowerPoint PPT Presentation

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A 1 -Representability of Hermitian K -Theory. 1 A 1 - - PowerPoint PPT Presentation

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A 1 -Representability of Hermitian K -Theory. 1 A 1 -Representability of Hermitian K -Theory. What do we mean by Representability? A 1 -Representability of Hermitian K -Theory. What do we mean by Representability? The A 1

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A1-Representability of Hermitian K-Theory.

1

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SLIDE 2

A1-Representability of Hermitian K-Theory.

  • What do we mean by ‘Representability’?
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SLIDE 3

A1-Representability of Hermitian K-Theory.

  • What do we mean by ‘Representability’?
  • The A1-Representability Theorem and consequences
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A1-Representability of Hermitian K-Theory.

  • What do we mean by ‘Representability’?
  • The A1-Representability Theorem and consequences
  • Outline of proof
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SLIDE 5

Representability.

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SLIDE 6

Representability. In topology, there are various ‘cohomology theories’ T n, X → T n(X)

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SLIDE 7

Representability. In topology, there are various ‘cohomology theories’ T n, X → T n(X)

  • topological K-theory of complex vector-bundles KUn

top(X),

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Representability. In topology, there are various ‘cohomology theories’ T n, X → T n(X)

  • topological K-theory of complex vector-bundles KUn

top(X),

→ e.g. KU0

top(X) = (V BC(X))+.

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SLIDE 9

Representability. In topology, there are various ‘cohomology theories’ T n, X → T n(X)

  • topological K-theory of complex vector-bundles KUn

top(X),

→ e.g. KU0

top(X) = (V BC(X))+.

  • topological K-theory of real vector-bundles KOn

top(X),

→ KO0

top(X) = (V BR(X))+.

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Representability.

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Representability. In the homotopy category HoTop

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Representability. In the homotopy category HoTop (1) KU0

top(X) = [X, Z × BU],

Z × BU = KU

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Representability. In the homotopy category HoTop (1) KU0

top(X) = [X, Z × BU],

Z × BU = KU (2) KO0

top(X) = [X, Z × BO],

Z × BO = KO

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SLIDE 14

Representability. In the homotopy category HoTop (1) KU0

top(X) = [X, Z × BU],

Z × BU = KU (2) KO0

top(X) = [X, Z × BO],

Z × BO = KO

  • KU represents the topological K-theory of complex vector bundles...
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Representability.

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Representability.

  • Representability −

→ ‘homotopy category HoTop’ (in topology).

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Representability.

  • Representability −

→ ‘homotopy category HoTop’ (in topology).

  • In algebraic geometry, what is a good notion of homotopy
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Representability.

  • Representability −

→ ‘homotopy category HoTop’ (in topology).

  • In algebraic geometry, what is a good notion of homotopy
  • Morel and Voevodsky (∼ 1999): the A1-homotopy theory
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SLIDE 19

Representability.

  • Representability −

→ ‘homotopy category HoTop’ (in topology).

  • In algebraic geometry, what is a good notion of homotopy
  • Morel and Voevodsky (∼ 1999): the A1-homotopy theory
  • [X, KU] = HomHoTop(X, KU).
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SLIDE 20

The A1-Representability Theorem and Consequences.

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The A1-Representability Theorem and Consequences. Algebraic K-groups: R = a comm ring, Ki(R), i ≥ 0

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The A1-Representability Theorem and Consequences. Algebraic K-groups: R = a comm ring, Ki(R), i ≥ 0

  • (Grothendieck- 1957)

K0(R) = P(R)+

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The A1-Representability Theorem and Consequences. Algebraic K-groups: R = a comm ring, Ki(R), i ≥ 0

  • (Grothendieck- 1957)

K0(R) = P(R)+ → group completion of the monoid P(R)

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The A1-Representability Theorem and Consequences. Algebraic K-groups: R = a comm ring, Ki(R), i ≥ 0

  • (Grothendieck- 1957)

K0(R) = P(R)+ → group completion of the monoid P(R)

  • (Quillen- 1972)

Ki(R) = πi+1(⊡)

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The A1-Representability Theorem and Consequences. Algebraic K-groups: R = a comm ring, Ki(R), i ≥ 0

  • (Grothendieck- 1957)

K0(R) = P(R)+ → group completion of the monoid P(R)

  • (Quillen- 1972)

Ki(R) = πi+1(⊡) → homotopy group of a top space.

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The A1-Representability Theorem and Consequences.

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The A1-Representability Theorem and Consequences. Hermitian K-groups: GWi(R) = higher Grothendieck-Witt groups.

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The A1-Representability Theorem and Consequences. Hermitian K-groups: GWi(R) = higher Grothendieck-Witt groups.

  • (Knebusch- 1970)

GW0(R) = W(R)+

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The A1-Representability Theorem and Consequences. Hermitian K-groups: GWi(R) = higher Grothendieck-Witt groups.

  • (Knebusch- 1970)

GW0(R) = W(R)+ → group completion of the monoid W(R)

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The A1-Representability Theorem and Consequences. Hermitian K-groups: GWi(R) = higher Grothendieck-Witt groups.

  • (Knebusch- 1970)

GW0(R) = W(R)+ → group completion of the monoid W(R) →

  • W(R) = isometry cl of nondeg sym bil forms on fg proj R-mod
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The A1-Representability Theorem and Consequences. Hermitian K-groups: GWi(R) = higher Grothendieck-Witt groups.

  • (Knebusch- 1970)

GW0(R) = W(R)+ → group completion of the monoid W(R) →

  • W(R) = isometry cl of nondeg sym bil forms on fg proj R-mod
  • (Karoubi- ∼1974)

GWi(R) = πi+1(⊟)

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The A1-Representability Theorem and Consequences. Hermitian K-groups: GWi(R) = higher Grothendieck-Witt groups.

  • (Knebusch- 1970)

GW0(R) = W(R)+ → group completion of the monoid W(R) →

  • W(R) = isometry cl of nondeg sym bil forms on fg proj R-mod
  • (Karoubi- ∼1974)

GWi(R) = πi+1(⊟) → homotopy group of a top space.

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The A1-Representability Theorem and Consequences. Examples: rk : GW0(C)

∼ =

− → Z (i+, i−) : GW0(R)

∼ =

− → Z ⊕ Z (rk, det) : GW0(Fq)

∼ =

− → Z ⊕ F×

q /F ×2 q

=        Z if q even Z ⊕ Z/2Z if q odd GW0(Z)

∼ =

− → GW0(R) = Z ⊕ Z GW0(Q)

∼ =

− → GW0(R) ⊕

p W0(Fp)

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SLIDE 34

The A1-Representability Theorem and Consequences.

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The A1-Representability Theorem and Consequences.

  • Morel-Voevodsky (1999)

For a field F Kn(X) = HomH•(F)(Sn ∧ X+, Gr)

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The A1-Representability Theorem and Consequences.

  • Morel-Voevodsky (1999)

For a field F Kn(X) = HomH•(F)(Sn ∧ X+, Gr) → Gr = Grassmannian, given by a scheme.

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The A1-Representability Theorem and Consequences.

  • Morel-Voevodsky (1999)

For a field F Kn(X) = HomH•(F)(Sn ∧ X+, Gr) → Gr = Grassmannian, given by a scheme.

  • Hornbostel (2005)

For a field F of char = 2 GWn(X) = HomH•(F)(Sn ∧ X+, a(Kh)f)

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The A1-Representability Theorem and Consequences.

  • Morel-Voevodsky (1999)

For a field F Kn(X) = HomH•(F)(Sn ∧ X+, Gr) → Gr = Grassmannian, given by a scheme.

  • Hornbostel (2005)

For a field F of char = 2 GWn(X) = HomH•(F)(Sn ∧ X+, a(Kh)f)

  • The A1-Representability Theorem (2010): If GrO is H-space,

GWn(X) = HomH•(F)(Sn ∧ X+, GrO) (char F = 2)

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SLIDE 39

The A1-Representability Theorem and Consequences.

  • Morel-Voevodsky (1999)

For a field F Kn(X) = HomH•(F)(Sn ∧ X+, Gr) → Gr = Grassmannian, given by a scheme.

  • Hornbostel (2005)

For a field F of char = 2 GWn(X) = HomH•(F)(Sn ∧ X+, a(Kh)f)

  • The A1-Representability Theorem (2010): If GrO is H-space,

GWn(X) = HomH•(F)(Sn ∧ X+, GrO) (char F = 2) → GrO = orthogonal Grassmannian, given by a scheme.

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The A1-Representability Theorem and Consequences.

  • The A1-Representability Theorem (2010): If GrO is H-space,

GWn(X) = HomH•(F)(Sn ∧ X+, GrO) It should help: = ⇒ Atiyah-Hirzebruch sp. seq. for hermitian K-theory

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The A1-Representability Theorem and Consequences.

  • The A1-Representability Theorem (2010): If GrO is H-space,

GWn(X) = HomH•(F)(Sn ∧ X+, GrO) It should help: = ⇒ Atiyah-Hirzebruch sp. seq. for hermitian K-theory = ⇒ the cohomology operations.

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The A1-Representability Theorem and Consequences.

  • Lρ∗

R, Lρ∗ C : H(R) −

→ HoTop X → X(R), X → X(C)

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The A1-Representability Theorem and Consequences.

  • Lρ∗

R, Lρ∗ C : H(R) −

→ HoTop X → X(R), X → X(C) = ⇒ Corollary 1: For Lρ∗

C : H(R) −

→ HoTop Lρ∗

CKh = GrO(C) ≃ Z × BO(R) ≃ KOtop

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The A1-Representability Theorem and Consequences.

  • Lρ∗

R, Lρ∗ C : H(R) −

→ HoTop X → X(R), X → X(C) = ⇒ Corollary 1: For Lρ∗

C : H(R) −

→ HoTop Lρ∗

CKh = GrO(C) ≃ Z × BO(R) ≃ KOtop

= ⇒ Corollary 2: Lρ∗

RKh = GrO(R) ≃ Z × Z × BO(R) × BO(R)

≃ KOtop × KOtop

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Outline of Proof.

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Outline of Proof.

  • ∆opSets = the category of simplicial sets,
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Outline of Proof.

  • ∆opSets = the category of simplicial sets

→ algebraic model for Top

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Outline of Proof.

  • ∆opSets = the category of simplicial sets

→ algebraic model for Top

  • ∆opSets = {∆ contra. functors

− − − − − − − − − → Sets}

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Outline of Proof.

  • ∆opSets = the category of simplicial sets

→ algebraic model for Top

  • ∆opSets = {∆ contra. functors

− − − − − − − − − → Sets}

  • In ∆opSets the well-known notions of topological homotopy theory

have descriptions that can be adapted in a wide variety of contexts (in algebraic geometry, in particular).

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Outline of Proof.

  • ∆opSets = the category of simplicial sets

→ algebraic model for Top

  • ∆opSets = {∆ contra. functors

− − − − − − − − − → Sets}

  • In ∆opSets the well-known notions of topological homotopy theory

have descriptions that can be adapted in a wide variety of contexts (in algebraic geometry, in particular).

  • There is a set W of morphisms in ∆opSets, called weak equivalences,

with the property there is an equivalence of categories ∆opSets[W−1] ← → HoTop

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Outline of Proof.

  • In algebraic geometry, the appropriate category is

∆opPShv(Sm/F)

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Outline of Proof.

  • In algebraic geometry, the appropriate category is

∆opPShv(Sm/F) → the category of ‘simplicial presheaves of sets’ on the category Sm/F

  • f smooth F-schemes. This is analogue of the category Top.
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Outline of Proof.

  • In algebraic geometry, the appropriate category is

∆opPShv(Sm/F) → the category of ‘simplicial presheaves of sets’ on the category Sm/F

  • f smooth F-schemes. This is analogue of the category Top.
  • ∆opPShv(Sm/F) = {Sm/F
  • contra. functors

− − − − − − − − − → ∆opSets}

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Outline of Proof.

  • In algebraic geometry, the appropriate category is

∆opPShv(Sm/F) → the category of ‘simplicial presheaves of sets’ on the category Sm/F

  • f smooth F-schemes. This is analogue of the category Top.
  • ∆opPShv(Sm/F) = {Sm/F
  • contra. functors

− − − − − − − − − → ∆opSets} → We have (full and faithful), Sm/F ⊂ ∆opPShv(Sm/F)

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Outline of Proof.

  • In algebraic geometry, the appropriate category is

∆opPShv(Sm/F) → the category of ‘simplicial presheaves of sets’ on the category Sm/F

  • f smooth F-schemes. This is analogue of the category Top.
  • ∆opPShv(Sm/F) = {Sm/F
  • contra. functors

− − − − − − − − − → ∆opSets} → We have (full and faithful), Sm/F ⊂ ∆opPShv(Sm/F)

  • In this category there is a set of morphisms WA1, the set of

A1-weak equivalences: For example, X × A1 → X is in WA1.

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Outline of Proof.

  • In algebraic geometry, the appropriate category is

∆opPShv(Sm/F) → the category of ‘simplicial presheaves of sets’ on the category Sm/F

  • f smooth F-schemes. This is analogue of the category Top.
  • ∆opPShv(Sm/F) = {Sm/F
  • contra. functors

− − − − − − − − − → ∆opSets} → We have (full and faithful), Sm/F ⊂ ∆opPShv(Sm/F)

  • In this category there is a set of morphisms WA1, the set of

A1-weak equivalences. For example, X × A1 → X is in WA1. And, the analogue of the category HoTop is ∆opPShv(Sm/F)[WA1−1]

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Outline of Proof.

  • In algebraic geometry, the appropriate category is

∆opPShv(Sm/F) → the category of ‘simplicial presheaves of sets’ on the category Sm/F

  • f smooth F-schemes. This is analogue of the category Top.
  • ∆opPShv(Sm/F) = {Sm/F
  • contra. functors

− − − − − − − − − → ∆opSets} → We have (full and faithful), Sm/F ⊂ ∆opPShv(Sm/F)

  • In this category there is a set of morphisms WA1, the set of

A1-weak equivalences. For example, X × A1 → X is in WA1. And, the analogue of the category HoTop is ∆opPShv(Sm/F)[WA1−1] = H(F) → The homotopy category of smooth F-schemes [MV (1999)].

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Outline of Proof.

  • In algebraic geometry, the appropriate category is

∆opPShv(Sm/F) → the category of ‘simplicial presheaves of sets’ on the category Sm/F

  • f smooth F-schemes. This is analogue of the category Top.
  • ∆opPShv(Sm/F) = {Sm/F
  • contra. functors

− − − − − − − − − → ∆opSets} → We have (full and faithful), Sm/F ⊂ ∆opPShv(Sm/F)

  • In this category there is a set of morphisms WA1, the set of

A1-weak equivalences. For example, X × A1 → X is in WA1. And, the analogue of the category HoTop is ∆opPShv(Sm/F)[WA1−1] = H(F) → The homotopy category of smooth F-schemes [MV (1999)].

  • ∆opPShv•(Sm/F)[WA1−1] = H•(F)
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Outline of Proof The Presheaf GrO.

  • m, n ≥ 0

GrF(n, Hm) − → smooth F-scheme

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Outline of Proof The Presheaf GrO.

  • m, n ≥ 0

GrF(n, Hm) − → smooth F-scheme → open subscheme of Grassmannian GrF(n, 2m) over F.

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Outline of Proof The Presheaf GrO.

  • m, n ≥ 0

GrF(n, Hm) − → smooth F-scheme → open subscheme of Grassmannian GrF(n, 2m) over F.

  • GrF(n, Hm)(R)

=rk n nondegen proj factors of Hm(R).

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Outline of Proof The Presheaf GrO.

  • ... GrF(n, Hm) → GrF(n, Hm+1) ...
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Outline of Proof The Presheaf GrO.

  • ... GrF(n, Hm) → GrF(n, Hm+1) ...
  • colimit

GrF(n, H∞)

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SLIDE 64

Outline of Proof The Presheaf GrO.

  • ... GrF(n, Hm) → GrF(n, Hm+1) ...
  • colimit

GrF(n, H∞).

  • Let

GrF(N, H∞) =

n≥0 GrF(n, H∞).

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SLIDE 65

Outline of Proof The Presheaf GrO.

  • ... GrF(n, Hm) → GrF(n, Hm+1) ...
  • colimit

GrF(n, H∞).

  • Let

GrF(N, H∞) =

n≥0 GrF(n, H∞).

  • addition of hyperbolic plane

... GrF(N, H∞) H⊥ − − → GrF(N, H∞) ...

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Outline of Proof The Presheaf GrO.

  • ... GrF(n, Hm) → GrF(n, Hm+1) ...
  • colimit

GrF(n, H∞).

  • Let

GrF(N, H∞) =

n≥0 GrF(n, H∞).

  • addition of hyperbolic plane

... GrF(N, H∞) H⊥ − − → GrF(N, H∞) ...

  • colimit in ∆opPShv(Sm/F)

− → GrO.

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SLIDE 67

Outline of Proof The Simplicial Presheaf Kh.

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Outline of Proof The Simplicial Presheaf Kh.

  • R = a comm ring, SR = a category.
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Outline of Proof The Simplicial Presheaf Kh.

  • R = a comm ring, SR = a category.
  • Objects: (M, φ)
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Outline of Proof The Simplicial Presheaf Kh.

  • R = a comm ring, SR = a category.
  • Objects: (M, φ)

→ M ⊂ H∞(R), a f.g. proj factor s.t. φ (= h|M) is nondegen.

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SLIDE 71

Outline of Proof The Simplicial Presheaf Kh.

  • R = a comm ring, SR = a category.
  • Objects: (M, φ)

→ M ⊂ H∞(R), a f.g. proj factor s.t. φ (= h|M) is nondegen.

  • SR = a symmetric monoidal category
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SLIDE 72

Outline of Proof The Simplicial Presheaf Kh.

  • R = a comm ring, SR = a category.
  • Objects: (M, φ)

→ M ⊂ H∞(R), a f.g. proj factor s.t. φ (= h|M) is nondegen.

  • SR = a symmetric monoidal category
  • S−1

R SR = Pℏ(R)

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SLIDE 73

Outline of Proof The Simplicial Presheaf Kh.

  • R = a comm ring, SR = a category.
  • Objects: (M, φ)

→ M ⊂ H∞(R), a f.g. proj factor s.t. φ (= h|M) is nondegen.

  • SR = a symmetric monoidal category
  • S−1

R SR = Pℏ(R),

a functor SR → Pℏ(R).

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SLIDE 74

Outline of Proof The Simplicial Presheaf Kh.

  • X ∈ Sm/F,

Kh(X) = NPℏ(Γ(X, OX))

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SLIDE 75

Outline of Proof The Simplicial Presheaf Kh.

  • X ∈ Sm/F,

Kh(X) = NPℏ(Γ(X, OX))

  • Kh

0 = the connected component of 0.

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SLIDE 76

Outline of Proof The map GrO ℏ − → Kh.

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SLIDE 77

Outline of Proof The map GrO ℏ − → Kh.

  • GrF(N, H∞)(R) = Ob SR → Kh(R)
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SLIDE 78

Outline of Proof The map GrO ℏ − → Kh.

  • GrF(N, H∞)(R) = Ob SR → Kh(R)
  • GrO ℏ

− → Kh

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SLIDE 79

Outline of Proof ℏ : GrO → Kh is in WA1?

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SLIDE 80

Outline of Proof ℏ : GrO → Kh is in WA1?

  • Commutative diagram in A1-homotopy category

BO

  • γ

Kh

  • GrO
  • ℏ

Kh

  • aNis(πA1

0 GrO) ζ

aNisGW0

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SLIDE 81

Outline of Proof ℏ : GrO → Kh is in WA1?

  • Commutative diagram in A1-homotopy category

BO

  • γ

Kh

  • GrO
  • ℏ

Kh

  • aNis(πA1

0 GrO) ζ

aNisGW0

  • Theorem:

columns A1-fibrations

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SLIDE 82

Outline of Proof ℏ : GrO → Kh is in WA1?

  • Commutative diagram in A1-homotopy category

BO

  • γ

Kh

  • GrO
  • ℏ

Kh

  • aNis(πA1

0 GrO) ζ

aNisGW0

  • Theorem:

columns A1-fibrations

  • Theorem:

γ is A1-weak equivalence

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SLIDE 83

Outline of Proof ℏ : GrO → Kh is in WA1?

  • Commutative diagram in A1-homotopy category

BO

  • γ

Kh

  • GrO
  • ℏ

Kh

  • aNis(πA1

0 GrO) ζ

aNisGW0

  • Theorem:

columns A1-fibrations

  • Theorem:

γ is A1-weak equivalence

  • Theorem:

ζ is an isomorphism.

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SLIDE 84

Outline of Proof ℏ : GrO → Kh is in WA1?

  • Commutative diagram in A1-homotopy category

BO

  • γ

Kh

  • GrO
  • ℏ

Kh

  • aNis(πA1

0 GrO) ζ

aNisGW0

  • Theorem:

columns A1-fibrations.

  • Theorem:

γ is A1-weak equivalence.

  • Theorem:

ζ is an isomorphism.

  • make A1-fibrant replacement, compare homotopy groups
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SLIDE 85

Outline of Proof.

  • A diagram of top spaces when evaluated at X ∈ Sm/F

(BO)f

  • γf

(Kh

0)f

  • (GrO)f
  • ℏf

(Kh)f

  • (aNis(πA1

0 GrO))f ζf (aNisGW0)f

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SLIDE 86

Outline of Proof.

  • A diagram of top spaces when evaluated at X ∈ Sm/F

(BO)f

  • γf

(Kh

0)f

  • (GrO)f
  • ℏf

(Kh)f

  • (aNis(πA1

0 GrO))f ζf (aNisGW0)f

→ columns fib, and γf weak eq, and ζf isom. → ℏf induces isomorphism of homotopy groups at 0.

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SLIDE 87

Outline of Proof.

  • A diagram of top spaces when evaluated at X ∈ Sm/F

(BO)f

  • γf

(Kh

0)f

  • (GrO)f
  • ℏf

(Kh)f

  • (aNis(πA1

0 GrO))f ζf (aNisGW0)f

→ columns fib, and γf weak eq, and ζf isom. → ℏf induces isomorphism of homotopy groups at 0.

  • Theorem:

If GrO is H-space, then ℏf is a weak equivalence.

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SLIDE 88

Outline of Proof ℏ : GrO → Kh is in WA1?

  • Commutative diagram in A1-homotopy category

BO

  • γ

Kh

  • GrO
  • ℏ

Kh

  • aNis(πA1

0 GrO) ζ

aNisGW0

  • Theorem:

columns A1-fibrations.

  • Theorem:

γ is A1-weak equivalence.

  • Theorem:

ζ is an isomorphism.

  • make A1-fibrant replacement, compare homotopy groups
  • Theorem:

If GrO is H-space, then ℏ is A1-weak equivalence.

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SLIDE 89