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

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 -Representability Theorem and consequences

A 1 -Representability of Hermitian K -Theory. • What do we mean by ‘Representability’? • The A 1 -Representability Theorem and consequences • Outline of proof

Representability.

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

Representability. In topology, there are various ‘cohomology theories’ T n , X �→ T n ( X ) topological K -theory of complex vector-bundles KU n • top ( X ),

Representability. In topology, there are various ‘cohomology theories’ T n , X �→ T n ( X ) topological K -theory of complex vector-bundles KU n • top ( X ), KU 0 top ( X ) = ( V B C ( X )) + . → e.g.

Representability. In topology, there are various ‘cohomology theories’ T n , X �→ T n ( X ) topological K -theory of complex vector-bundles KU n • top ( X ), KU 0 top ( X ) = ( V B C ( X )) + . → e.g. topological K -theory of real vector-bundles KO n • top ( X ), KO 0 top ( X ) = ( V B R ( X )) + . →

Representability.

Representability. In the homotopy category HoTop

Representability. In the homotopy category HoTop KU 0 top ( X ) = [ X, Z × BU ] , Z × BU = KU (1)

Representability. In the homotopy category HoTop KU 0 top ( X ) = [ X, Z × BU ] , Z × BU = KU (1) KO 0 (2) top ( X ) = [ X, Z × BO ] , Z × BO = KO

Representability. In the homotopy category HoTop KU 0 top ( X ) = [ X, Z × BU ] , Z × BU = KU (1) KO 0 (2) top ( X ) = [ X, Z × BO ] , Z × BO = KO • KU represents the topological K -theory of complex vector bundles...

Representability.

Representability. • Representability − → ‘homotopy category HoTop ’ (in topology).

Representability. • Representability − → ‘homotopy category HoTop ’ (in topology). • In algebraic geometry, what is a good notion of homotopy

Representability. • Representability − → ‘homotopy category HoTop ’ (in topology). • In algebraic geometry, what is a good notion of homotopy Morel and Voevodsky ( ∼ 1999): the A 1 -homotopy theory •

Representability. • Representability − → ‘homotopy category HoTop ’ (in topology). • In algebraic geometry, what is a good notion of homotopy Morel and Voevodsky ( ∼ 1999): the A 1 -homotopy theory • • [ X, KU ] = Hom HoTop ( X, KU ) .

The A 1 -Representability Theorem and Consequences.

The A 1 -Representability Theorem and Consequences. Algebraic K -groups: R = a comm ring, K i ( R ) , i ≥ 0

The A 1 -Representability Theorem and Consequences. Algebraic K -groups: R = a comm ring, K i ( R ) , i ≥ 0 K 0 ( R ) = P ( R ) + • (Grothendieck- 1957)

The A 1 -Representability Theorem and Consequences. Algebraic K -groups: R = a comm ring, K i ( R ) , i ≥ 0 K 0 ( R ) = P ( R ) + • (Grothendieck- 1957) → group completion of the monoid P ( R )

The A 1 -Representability Theorem and Consequences. Algebraic K -groups: R = a comm ring, K i ( R ) , i ≥ 0 K 0 ( R ) = P ( R ) + • (Grothendieck- 1957) → group completion of the monoid P ( R ) • (Quillen- 1972) K i ( R ) = π i +1 ( ⊡ )

The A 1 -Representability Theorem and Consequences. Algebraic K -groups: R = a comm ring, K i ( R ) , i ≥ 0 K 0 ( R ) = P ( R ) + • (Grothendieck- 1957) → group completion of the monoid P ( R ) • (Quillen- 1972) K i ( R ) = π i +1 ( ⊡ ) → homotopy group of a top space.

The A 1 -Representability Theorem and Consequences.

The A 1 -Representability Theorem and Consequences. Hermitian K -groups: GW i ( R ) = higher Grothendieck-Witt groups.

The A 1 -Representability Theorem and Consequences. Hermitian K -groups: GW i ( R ) = higher Grothendieck-Witt groups. W ( R ) + GW 0 ( R ) = � • (Knebusch- 1970)

The A 1 -Representability Theorem and Consequences. Hermitian K -groups: GW i ( R ) = higher Grothendieck-Witt groups. W ( R ) + GW 0 ( R ) = � • (Knebusch- 1970) group completion of the monoid � → W ( R )

The A 1 -Representability Theorem and Consequences. Hermitian K -groups: GW i ( R ) = higher Grothendieck-Witt groups. W ( R ) + GW 0 ( R ) = � • (Knebusch- 1970) group completion of the monoid � → W ( R ) � → W ( R ) = isometry cl of nondeg sym bil forms on fg proj R -mod

The A 1 -Representability Theorem and Consequences. Hermitian K -groups: GW i ( R ) = higher Grothendieck-Witt groups. W ( R ) + GW 0 ( R ) = � • (Knebusch- 1970) group completion of the monoid � → W ( R ) � → W ( R ) = isometry cl of nondeg sym bil forms on fg proj R -mod • (Karoubi- ∼ 1974) GW i ( R ) = π i +1 ( ⊟ )

The A 1 -Representability Theorem and Consequences. Hermitian K -groups: GW i ( R ) = higher Grothendieck-Witt groups. W ( R ) + GW 0 ( R ) = � • (Knebusch- 1970) group completion of the monoid � → W ( R ) � → W ( R ) = isometry cl of nondeg sym bil forms on fg proj R -mod • (Karoubi- ∼ 1974) GW i ( R ) = π i +1 ( ⊟ ) → homotopy group of a top space.

The A 1 -Representability Theorem and Consequences. Examples: ∼ = rk : GW 0 ( C ) − → Z ∼ = ( i + , i − ) : GW 0 ( R ) − → Z ⊕ Z     if q even Z ∼ = → Z ⊕ F × q /F × 2 − (rk , det) : GW 0 ( F q ) = q    Z ⊕ Z / 2 Z if q odd ∼ = GW 0 ( Z ) − → GW 0 ( R ) = Z ⊕ Z ∼ → GW 0 ( R ) ⊕ � = GW 0 ( Q ) − p W 0 ( F p )

The A 1 -Representability Theorem and Consequences.

The A 1 -Representability Theorem and Consequences. • Morel-Voevodsky (1999) For a field F K n ( X ) = Hom H • ( F ) ( S n ∧ X + , Gr )

The A 1 -Representability Theorem and Consequences. • Morel-Voevodsky (1999) For a field F K n ( X ) = Hom H • ( F ) ( S n ∧ X + , Gr ) → Gr = Grassmannian, given by a scheme.

The A 1 -Representability Theorem and Consequences. • Morel-Voevodsky (1999) For a field F K n ( X ) = Hom H • ( F ) ( S n ∧ X + , Gr ) → Gr = Grassmannian, given by a scheme. • Hornbostel (2005) For a field F of char � = 2 GW n ( X ) = Hom H • ( F ) ( S n ∧ X + , a ( K h ) f )

The A 1 -Representability Theorem and Consequences. • Morel-Voevodsky (1999) For a field F K n ( X ) = Hom H • ( F ) ( S n ∧ X + , Gr ) → Gr = Grassmannian, given by a scheme. • Hornbostel (2005) For a field F of char � = 2 GW n ( X ) = Hom H • ( F ) ( S n ∧ X + , a ( K h ) f ) The A 1 -Representability Theorem (2010): If G rO is H -space, • GW n ( X ) = Hom H • ( F ) ( S n ∧ X + , G rO ) (char F � = 2)

The A 1 -Representability Theorem and Consequences. • Morel-Voevodsky (1999) For a field F K n ( X ) = Hom H • ( F ) ( S n ∧ X + , Gr ) → Gr = Grassmannian, given by a scheme. • Hornbostel (2005) For a field F of char � = 2 GW n ( X ) = Hom H • ( F ) ( S n ∧ X + , a ( K h ) f ) The A 1 -Representability Theorem (2010): If G rO is H -space, • GW n ( X ) = Hom H • ( F ) ( S n ∧ X + , G rO ) (char F � = 2) → G rO = orthogonal Grassmannian, given by a scheme.

The A 1 -Representability Theorem and Consequences. The A 1 -Representability Theorem (2010): If G rO is H -space, • GW n ( X ) = Hom H • ( F ) ( S n ∧ X + , G rO ) It should help: ⇒ = Atiyah-Hirzebruch sp. seq. for hermitian K -theory

The A 1 -Representability Theorem and Consequences. The A 1 -Representability Theorem (2010): If G rO is H -space, • GW n ( X ) = Hom H • ( F ) ( S n ∧ X + , G rO ) It should help: ⇒ = Atiyah-Hirzebruch sp. seq. for hermitian K -theory ⇒ = the cohomology operations.

The A 1 -Representability Theorem and Consequences. Lρ ∗ R , Lρ ∗ • C : H ( R ) − → HoTop X �→ X ( R ) , X �→ X ( C )

The A 1 -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 C K h = G rO ( C ) ≃ Z × BO ( R ) ≃ KO top Lρ ∗

The A 1 -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 C K h = G rO ( C ) ≃ Z × BO ( R ) ≃ KO top Lρ ∗ = ⇒ Corollary 2: R K h = G rO ( R ) ≃ Z × Z × BO ( R ) × BO ( R ) Lρ ∗ ≃ KO top × KO top

Outline of Proof.

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

Outline of Proof. ∆ op Sets = the category of simplicial sets • → algebraic model for Top

Outline of Proof. ∆ op Sets = the category of simplicial sets • → algebraic model for Top ∆ op Sets = { ∆ contra . functors • − − − − − − − − − → Sets }