Annalas derived algebraic cobordism over a general base Talk I - - PowerPoint PPT Presentation

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories

Annala’s derived algebraic cobordism over a general base

Talk I July 7, 2020

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories

Bibliography

[A] Chern classes in precobordism theories, T. Annala (2019) [AY] Bivariant algebraic cobordism with bundles, T. Annala, S. Yokura (2019)

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories

Contents

1 Introduction 2 Bivariant derived algebraic cobordism with bundles

Construction Comparison theorem

3 Precobordism theories

Definition and the universal precobordism Weak projective bundle formula and Chern classes

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories

Algebraic cobordism (Voevodsky)

The bigraded theory MGL❻,❻ on Smk was defined as the theory represented by the algebraic Thom spectrum MGL ❃ ❙❍❼k➁ in the stable motivic homotopy category and was used in order to prove the Milnor conjecture.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories

Algebraic cobordism over a field k of characteristic 0 (Levine-Morel)

Ω❻ ✂ Smop

k GrRing is the universal oriented cohomology theory.

Ω❻ ✂ Sch➐

k GrAb is the universal oriented Borel-Moore homology

theory. A theorem of Levine establishes MGL2❻,❻❼X➁ ☞ Ω❻❼X➁. The graded group Ωn❼X➁ can be described in terms of cobordism cycles of the form Y

f

Ð X,L1,...,Lr✆ with Y irreducible, f projective and L1,...,Lr line bundles on Y such that n dim❼Y ➁ ✏ r.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories

Double point cobordism over a field k of characteristic 0 (Levine-Pandharipande)

For X ❃ Schk, the group ωn❼X➁ is the free abelian group on cobordism cycles of the form Y

f

Ð X✆ with Y ❃ Smk irreducible of dimension n and f projective modulo the double-point relations. A theorem of Levine-Pandharipande establishes ω❻❼X➁ ☞ Ω❻❼X➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories

Algebraic cobordism of bundles over a field k of characteristic 0 (Lee-Pandharipande)

For X ❃ Schk, the group ωn,r❼X➁ is the free abelian group on cobordism cycles of the form Y

f

Ð X,E✆ with Y ❃ Smk irreducible

• f dimension n, E a vector bundle of rank r on Y and f projective

modulo analogously defined double-point relations. In particular, one has ω❻,0❼X➁ ☞ ω❻❼X➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories

Derived algebraic cobordism over a field k of characteristic 0 (Lowrey-Sch¨ urg)

Lowrey and Sch¨ urg define algebraic cobordims groups dΩ❻❼X➁ for quasi-projective derived schemes. For a quasi-projective derived scheme X, one has an isomorphism dΩ❻❼X➁ ☞ Ω❻❼tX➁, where tX denotes the truncation of X.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories

Bivariant derived algebraic cobordism over a field k of characteristic 0 (Annala-Yokura)

Derived algebraic cobordism as a bivariant theory over the homotopy category of quasi-projective derived k-schemes with groups Ω❻❼X

f

Ð Y ➁ for any morphism f between quasi-projective derived schemes over k. Associated cohomology theory: Ω❻❼X➁ ✂ Ω❻❼X

id

Ð X➁. Associated homology theory: Ω❻❼X➁ ✂ Ω✏❻❼X pt➁. The associated homology theory coincides with the derived algebraic cobordism as defined by Lowrey-Sch¨ urg. Next step: Bivariant derived algebraic cobordism of bundles over a field of characteristic 0!

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Construction Comparison theorem

Setting

k: field of characteristic 0 dSchk: homotopy category of quasi-projective derived k-schemes with

• proper morphisms as confined morphisms,
• all homotopy Cartesian squares as independent squares,
• quasi-smooth morphisms as specialized morphisms,
• smooth morphisms as specialized projections.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Construction Comparison theorem

Step I: Construction of ▼❻,❻

L

Let ▼i,r

L ❼X f

Ð Y ➁ be the quotient of the free L-module on the set

• f cycles of the form V

h

Ð X,E✆, where

• h ✂ V X is proper,
• the composite f ❳ h is quasi-smooth of virtual relative

dimension ✏i,

• E is a vector bundle on V of rank r,

by the relation that taking the disjoint union on the sources corresponds to summation. Pushforward: For f ✂ X X ➐ proper and g ✂ X ➐ Y , one defines the pushforward map f❻ ✂ ▼i,r

L ❼X g❳f

Ð Ð Y ➁ ▼i,r

L ❼X ➐ g

Ð Y ➁ by f❻❼V

h

Ð X,E✆➁ V

f ❳h

Ð Ð X ➐,E✆.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Construction Comparison theorem

Step I: Construction of ▼❻,❻

L

Pullback: For f ✂ X X ➐, g ✂ Y ➐ Y and X ➐ Y ➐ ✕R

Y X, one

defines the pullback map g❻ ✂ ▼i,r

L ❼X f

Ð Y ➁ ▼i,r

L ❼X ➐ f ➐

Ð Y ➐➁ by g❻❼V

h

Ð X,E✆➁ V ➐ h➐ Ð X ➐,E ➐✆, where V ➐ Y ➐ ✕R

Y V and

E ➐ ❼g➐➐➁❻❼E➁ is the pullback of E along the projection g➐➐ ✂ V ➐ V . V ➐

h➐

• g➐➐
• X ➐

g➐

• f ➐

Y ➐

g

• V

h

X

f

Y

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Construction Comparison theorem

Step I: Construction of ▼❻,❻

L

Bivariant products: We define two bivariant products ❨❵ ✂ ▼i,r

L ❼X f

Ð Y ➁ ✕ ▼j,s

L ❼Y g

Ð Z➁ ▼i✔j,r✔s

L

❼X

g❳f

Ð Ð Z➁ ❨❛ ✂ ▼i,r

L ❼X f

Ð Y ➁ ✕ ▼j,s

L ❼Y g

Ð Z➁ ▼i✔j,rs

L

❼X

g❳f

Ð Ð Z➁ as follows: For cycles V

h

Ð X,E✆ and W

k

Ð Y ,F✆, we form the following homotopy Cartesian diagram V ➐

h➐

• k➐➐
• X ➐

k➐

• f ➐

W

k

• V

h

X

f

Y

g

Z

We let E ➐ ❼k➐➐➁❻❼E➁ and F ➐ ❼f ➐ ❳ h➐➁❻❼F➁ and define V

h

Ð X,E✆ ❨❵ W

k

Ð Y ,F✆ V ➐ h❳k➐➐ Ð Ð X,E ➐ ❵ F ➐✆, V

h

Ð X,E✆ ❨❛ W

k

Ð Y ,F✆ V ➐ h❳k➐➐ Ð Ð X,E ➐ ❛ F ➐✆.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Construction Comparison theorem

Step I: Construction of ▼❻,❻

L

Proposition [AY, Proposition 5.3] ▼❻,❻

L

is a commutative bivariant theory with respect to both products ❨❵ and ❨❛. We may also choose natural orientations along quasi-smooth morphisms: If f ✂ X Y is quasi-smooth of relative virtual dimension ✏i, then we may define the two orientations θ❵❼f ➁ ✂ X

idX

Ð Ð X,0✆ ❃ ▼i,0

L ❼X f

Ð Y ➁ θ❛❼f ➁ ✂ X

idX

Ð Ð X,❖X✆ ❃ ▼i,1

L ❼X f

Ð Y ➁ for the two bivariant theories ❼▼❻,❻

L ,❨❵➁ and ❼▼❻,❻ L ,❨❛➁

respectively.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Construction Comparison theorem

Step II: Relations

Recall that Annala defined his bivariant derived algebraic cobordism Ω❻ as the quotient bivariant theory Ω❻ ✂ ❼▼❻

L,❨❵➁⑦❵❘LS❡❼▼❻

L,❨❵➁.

But ❼▼❻

L,❨❵➁ can be identified with ❼▼❻,0 L ,❨❵➁ ❜ ❼▼❻,❻ L ,❨❵➁.

The bivariant ideal ❵❘LS❡❼▼❻

L,❨❵➁ is generated by a bivariant subset

❘LS. Definition [AY, Definition 5.7] Bivariant algebraic cobordism with vector bundles is defined as the quotient bivariant theory ❼Ω❻,❻,❨❵➁ ✂ ❼▼❻,❻

L ,❨❵➁⑦❵❘LS❡❼▼❻,❻

L

,❨❵➁,

where we regard ❘LS as a bivariant subset of ❼▼❻,❻

L ,❨❵➁ via the

canonical identification ❼▼❻

L,❨❵➁ ❼▼❻,0 L ,❨❵➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Construction Comparison theorem

Basic facts about Ω❻,❻

The bivariant theory ❼Ω❻,❻,❨❵➁ is obviously commutative and inherits natural orientations θ❵❼f ➁ ✂ X

idX

Ð Ð X,0✆ ❃ Ωi,0

✔ ❼X f

Ð Y ➁ along quasi-smooth morphisms f ✂ X Y of relative virtual dimension ✏i. Associated cohomology theory: Ω❻,❻❼X➁ ✂ Ω❻,❻❼X

id

Ð X➁. Associated homology theory: Ω❻,❻❼X➁ ✂ Ω✏❻,❻❼X pt➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Construction Comparison theorem

Basic facts about Ω❻,❻

There are two canonical Grothendieck transformations ❩ ✂ ❼Ω❻,❨❵➁ ❼Ω❻,0,❨❵➁,V

h

Ð X✆ ✭ V

h

Ð X,0✆, ❋ ✂ ❼Ω❻,❻,❨❵➁ ❼Ω❻,❨❵➁,V

h

Ð X,E✆ ✭ V

h

Ð X✆, which give a canonical isomorphism Ω❻ ☞ Ω❻,0. The theory Ω❻,❻ also becomes bivariant with respect to the bivariant product ❨❛ on ▼❻,❻

L . Then the subtheory ❼Ω❻,1,❨❛➁ of

❼Ω❻,❻,❨❛➁ automatically becomes a bivariant theory equipped with an inclusion ❼Ω❻,❨❵➁ ✵ ❼Ω❻,1,❨❛➁,V

h

Ð X✆ ✭ V

h

Ð X,❖V ✆. The subtheory ❼Ω❻,1,❨❛➁ is called bivariant algebraic cobordism with line bundles.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Construction Comparison theorem

Comparison theorem

Theorem [AY, Theorem 5.28] For any quasi-projective derived scheme X, there is a natural isomorphism Ω❻,❻❼X➁ ☞ ω❻,❻❼tX➁, where tX denotes the truncation of X. One considers a natural cross product map ω❻,❻❼pt➁ ❛L Ω❻❼X➁ Ω❻,❻❼X➁ defined on cycles by V pt,E✆ ❛L W X✆ ✭ V ✕ W X,pr❻

V ❼E➁✆

and then identifies ω❻,❻❼pt➁ ❛L Ω❻❼X➁ ☞ ω❻,❻❼pt➁ ❛L ω❻❼tX➁ with ω❻,❻❼tX➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Setting

Question: What is the ”correct” definition of bivariant algebraic cobordism over a general base scheme? A: Noetherian ring of finite Krull dimension dSchA: homotopy category of quasi-projective derived A-schemes with

• proper morphisms as confined morphisms,
• all homotopy Cartesian squares as independent squares,
• quasi-smooth morphisms as specialized morphisms,
• smooth morphisms as specialized projections.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Construction of ▼❻,❻

✔

Let ▼i,r

✔ ❼X f

Ð Y ➁ be the quotient of the free abelian group on the set of cycles of the form V

h

Ð X,E✆, where

• h ✂ V X is proper,
• the composite f ❳ h is quasi-smooth of virtual relative

dimension ✏i,

• E is a vector bundle on V of rank r,

by the relation that taking the disjoint union on the sources corresponds to summation. Pushforward: For f ✂ X X ➐ proper and g ✂ X ➐ Y , one defines the pushforward map f❻ ✂ ▼i,r

✔ ❼X g❳f

Ð Ð Y ➁ ▼i,r

✔ ❼X ➐ g

Ð Y ➁ by f❻❼V

h

Ð X,E✆➁ V

f ❳h

Ð Ð X ➐,E✆.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Construction of ▼❻,❻

✔

Pullback: For f ✂ X X ➐, g ✂ Y ➐ Y and X ➐ Y ➐ ✕R

Y X, one

defines the pullback map g❻ ✂ ▼i,r

✔ ❼X f

Ð Y ➁ ▼i,r

✔ ❼X ➐ f ➐

Ð Y ➐➁ by g❻❼V

h

Ð X,E✆➁ V ➐ h➐ Ð X ➐,E ➐✆, where V ➐ Y ➐ ✕R

Y V and

E ➐ ❼g➐➐➁❻❼E➁ is the pullback of E along the projection g➐➐ ✂ V ➐ V . V ➐

h➐

• g➐➐
• X ➐

g➐

• f ➐

Y ➐

g

• V

h

X

f

Y

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Construction of ▼❻,❻

✔

Bivariant products: We define two bivariant products ❨❵ ✂ ▼i,r

✔ ❼X f

Ð Y ➁ ✕ ▼j,s

✔ ❼Y g

Ð Z➁ ▼i✔j,r✔s

✔

❼X

g❳f

Ð Ð Z➁ ❨❛ ✂ ▼i,r

✔ ❼X f

Ð Y ➁ ✕ ▼j,s

✔ ❼Y g

Ð Z➁ ▼i✔j,rs

✔

❼X

g❳f

Ð Ð Z➁ as follows: For cycles V

h

Ð X,E✆ and W

k

Ð Y ,F✆, we form the following homotopy Cartesian diagram V ➐

h➐

• k➐➐
• X ➐

k➐

• f ➐

W

k

• V

h

X

f

Y

g

Z

We let E ➐ ❼k➐➐➁❻❼E➁ and F ➐ ❼f ➐ ❳ h➐➁❻❼F➁ and define V

h

Ð X,E✆ ❨❵ W

k

Ð Y ,F✆ V ➐ h❳k➐➐ Ð Ð X,E ➐ ❵ F ➐✆, V

h

Ð X,E✆ ❨❛ W

k

Ð Y ,F✆ V ➐ h❳k➐➐ Ð Ð X,E ➐ ❛ F ➐✆.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Construction of ▼❻,❻

✔

Proposition ▼❻,❻

✔

is a commutative bivariant theory with respect to both products ❨❵ and ❨❛. If f ✂ X Y is quasi-smooth of relative virtual dimension ✏i, then we may define the two orientations along f θ❵❼f ➁ ✂ X

idX

Ð Ð X,0✆ ❃ ▼i,0

✔ ❼X f

Ð Y ➁ θ❛❼f ➁ ✂ X

idX

Ð Ð X,❖X✆ ❃ ▼i,1

✔ ❼X f

Ð Y ➁ for the two bivariant theories ❼▼❻,❻

✔ ,❨❵➁ and ❼▼❻,❻ ✔ ,❨❛➁

respectively (and, in particular, for the subtheories ❼▼❻,0

✔ ,❨❵➁ and

❼▼❻,1

✔ ,❨❛➁).

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Euler classes

Associated cohomology theory: ▼❻,❻

✔ ❼X➁ ✂ ▼❻,❻ ✔ ❼X id

Ð X➁. Associated homology theory: ▼✔

❻,❻❼X➁ ✂ ▼✏❻,❻ ✔

❼X pt➁. If f ✂ X Y is proper and quasi-smooth of relative virtual dimension ✏i, then the orientation θ❵❼f ➁ along f induces a Gysin pushforward homomorphism f! f❻❼✏ ❨❵ θ❵❼f ➁➁ ✂ ▼k,r

✔ ❼X➁ ▼k✔i,r ✔

❼Y ➁. Definition Let E be a vector bundle of rank r on a quasi-projective derived A-scheme X with zero section s ✂ X E. Then we define its Euler class by e❼E➁ s❻❼s!❼1X➁➁ V ❼s➁ X,0✆ ❃ ▼r,0

✔ ❼X➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Definition of naive cobordism theories

We now consider the bivariant theory ❼▼❻

✔,❨❵➁ ✂ ❼▼❻,0 ✔ ,❨❵➁.

Definition [AY, Definition 6.1(1)] Let ❼B❻,❨❵➁ be a quotient bivariant theory of ▼❻

✔. We say that

❼B❻,❨❵➁ is a naive cobordism theory if, given a morphism f ✂ X Y and a projective morphism W P1 ✕ X such that W P1 ✕ X

idP1✕f

Ð Ð Ð P1 ✕ Y is quasi-smooth of relative virtual dimension d, then W0 X✆ W➟ X✆ ❃ B✏d❼X

f

Ð Y ➁, where W0 and W➟ are the homotopy fibres of W P1 ✕ X over ➌0➑ ✕ X and ➌➟➑ ✕ X respectively.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Definition of precobordism theories

Definition [AY, Definition 6.1(2)] Let ❼B❻,❨❵➁ be a naive cobordism theory. We say that ❼B❻,❨❵➁ is a precobordism theory if, given line bundles L1 and L2 on a quasi-projective derived scheme X, we have e❼L1 ❛ L2➁ e❼L1➁ ✔ e❼L2➁ ✏ e❼L1➁ ❨ e❼L2➁ ❨ P1 X✆ ✏ e❼L1➁ ❨ e❼L2➁ ❨ e❼L1 ❛ L2➁ ❨ ❼P2 X✆ ✏ P3 X✆➁ in B1❼X

idX

Ð Ð X➁, where ❨ ❨❵ and P1 ✂ PX❼L1 ❵ ❖➁, P2 ✂ PX❼L1 ❵ ❼L1 ❛ L2➁ ❵ ❖➁, P3 ✂ PPX ❼L1❵❼L1❛L2➁➁❼❖❼✏1➁ ❵ ❖➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Equivalent definition of precobordism theories

Proposition [AY, Proposition 6.3] A naive cobordism theory ❼B❻,❨❵➁ is a precobordism theory if and

• nly if, given a quasi-smooth morphism W P1 ✕ X with

homotopy fibres W0 and W➟ of W P1 ✕ X over ➌0➑ ✕ X and ➌➟➑ ✕ X respectively such that W➟ is equivalent to the sum of divisors A W and B W , the double point cobordism formula W0 X✆ A X✆ ✔ B X✆ ✏ P X✆ ❃ B1❼X➁, holds, where P ✂ PZ❼❖❼A➁ ❵ ❖➁ holds with Z the derived intersection of A and B in W .

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Associated precobordism with line bundles

Note that we have an inclusion ❼▼❻

✔,❨❵➁ ✵ ❼▼❻,1 ✔ ,❨❛➁ given by

V

h

Ð X,0✆ ✭ V

h

Ð X,❖X✆. Definition [AY, Definition 6.6] For a precobordism theory ❼B❻,❨❵➁ ❼▼❻

✔,❨❵➁⑦I, its associated

precobordism with line bundles is defined as the bivariant theory ❼B❻,1,❨❛➁ ✂ ❼▼❻,1

✔ ,❨❛➁⑦❵I❡❼▼❻,1

✔ ,❨❛➁.

We obtain an induced inclusion ❼B❻,❨❵➁ ✵ ❼B❻,1,❨❛➁ and an induced forgetful Grothendieck transformation ❋ ✂ ❼B❻,1,❨❛➁ ❼B❻,❨❵➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Associated precobordism with line bundles

Theorem [AY, Theorem 6.12] Let B❻ be a precobordism theory. Then the group B❻,1❼pt➁ has an B❻❼pt➁-linear basis given by ❼Pi pt,❖❼1➁✆➁i❈0. Theorem [AY, Theorem 6.13] Let B❻ be a precobordism theory. Then the natural cross product map B❻,1❼pt➁ ❛B❻❼pt➁ B❻❼X Y ➁ B❻,1❼X Y ➁, ❼V pt,L✆,W X✆➁ ✭ V ✕ W X,π❻

V ❼L➁✆,

is an isomorphism, where πV ✂ V ✕ W V is the canonical projection.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Examples

Example 1 [AY, Example 6.4] If A k is a field of characteristic 0, then Annala’s bivariant derived algebraic cobordism Ω❻ is a precobordism theory. The associated precobordism with line bundles is just Annala’s bivariant algebraic cobordism Ω❻,1 with line bundles. Example 2 [AY, Construction 6.5] One easily obtains the universal precobordism theory Ω❻ as a quotient bivariant theory from ▼❻

✔ by enforcing the relations from

the definition of precobordism theories. Any precobordism theory B❻ is a quotient of Ω❻.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Weak projective bundle formula

Theorem [AY, Theorem 6.22] Let B❻ be a bivariant precobordism theory and X Y a morphism between derived schemes over a Noetherian ring A of finite Krull

• dimension. Then the assignments α ✭ e❼❖❼1➁➁i ❨ θ❼π➁ ❨ α induce

an isomorphism ❃n

i0 B❻✏i✔n❼X Y ➁ ☞ B❻❼Pn ✕ X Y ➁

• f B❻❼pt➁-modules, where π is the projection Pn ✕ X X.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Weak projective bundle formula

Corollary [AY, Corollary 6.24] There is a natural isomorphism B❻❼Pn ✕ X➁ ☞ B❻❼X➁t✆⑦❵tn✔1❡ of rings, where t e❼❖❼1➁➁. Proof. Since B❻ is a commutative bivariant theory, the homotopy Cartesian square Pn ✕ X

π

• id

Pn ✕ X

π

• X

id

X

implies that θ❼π➁ ❨❵ α π❻❼α➁ ❨❵ θ❼π➁ for all α ❃ B❻❼X➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Weak projective bundle formula

Proof (continued). Since the orientations along quasi-smooth morphisms are all nice (i.e. stable under pullbacks), the homomorphisms ✏ ❨❵ θ❼π➁ ✂ B❻✔n❼Pn ✕ X➁ B❻❼Pn ✕ X X➁ are isomorphisms. So the previous theorem implies that the homomorphisms ti ❨❵ π❻ ✂ B❻✏i❼X➁ B❻❼Pn ✕ X➁ induce an isomorphism ❃n

i0 B❻✏i❼X➁ B❻❼Pn ✕ X➁.

Since ti Pn✏i ✕ X Pn ✕ X,0✆, it follows that tn✔1 0.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Formal group law

Now let n,m ❈ 1 and consider the line bundle ❖n,m❼1,1➁ ✂ π❻

1❼❖Pn❼1➁➁ ❛ π❻ 2❼❖Pm❼1➁➁

• n Pn ✕ Pm. Using the previous result, one obtains an isomorphism

B❻❼Pn ✕ Pm➁ ☞ B❻❼pt➁x,y✆⑦❵xn✔1,ym✔1❡. In particular, there are unique coefficients an,m

ij

❃ B❻❼pt➁ such that e❼❖n,m❼1,1➁➁ Pi,j an,m

ij

xiyj ❃ B❻❼pt➁x,y✆⑦❵xn✔1,ym✔1❡. These coefficients do not depend on n,m as long as i ❇ n and j ❇ m.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Formal group law

Therefore one obtains the following result: Corollary [AY, Corollary 6.25] Let B❻ be a precobordism theory. Then there is a formal group law given by FB❻❼x,y➁ ✂ Pi,j aijxiyj ❃ B❻❼pt➁x,y✆✆ such that, for any two (globally generated) line bundles L1,L2 on a quasi-projective derived scheme X, one has e❼L1 ❛ L2➁ FB❻❼e❼L1➁,e❼L2➁➁ Pi,j aije❼L1➁ie❼L2➁j ❃ B❻❼X➁. The formal group law above induces a homomorphism of rings L B❻❼pt➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Construction of Chern classes

For all vector bundles E of rank r on a quasi-projective derived A-scheme X one constructs

• 1. a projective quasi-smooth morphism πX,E ✂ ˜

XE X of relative virtual dimension 0 which is natural in the sense that, for any morphism f ✂ Y X, there is a homotopy Cartesian square ˜ Yf ❻❼E➁

πY ,f ❻❼E➁

• f ➐

˜

XE

πX,E

• Y

f

X

• 2. π❻

X,E❼E➁ has a natural filtration 0 E0 ❜ E1 ❜ ... ❜ Er π❻ X,E❼E➁

with line bundles L1,...,Lr as the associated graded pieces;

• 3. a class ηX,E ❃ Ω0❼ ˜

XE➁ pushing forward to 1X ❃ Ω0❼X➁ which is natural in the sense that f ➐❻ηX,E ηY ,f ❻❼E➁ for f and f ➐ as above.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Construction of Chern classes

Definition Let E be a vector bundle of rank r on a quasi-projective derived A-scheme X. Its ith Chern class is defined as ci❼E➁ ✂ πX,E!❼si❼e❼L1➁,...,e❼Lr➁➁ ❨❵ ηX,E➁ ❃ Ωi❼X➁ for 1 ❇ i ❇ r and its total Chern class is defined as c❼E➁ 1X ✔ c1❼E➁ ✔ ... ✔ cr❼E➁ ❃ Ω❻❼X➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Euler classes in the universal precobordism theory

Lemma [A, Lemma 4.1] Let X be a quasi-projective derived A-scheme and 0 E ➐ E E ➐➐ 0 be a short exact sequence of vector bundles over X. Then e❼E➁ e❼E ➐➁ ❨❵ e❼E ➐➐➁ ❃ Ω❻❼X➁

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Euler classes in the universal precobordism theory

Lemma [A, Lemma 4.2] Let E be a vector bundle of rank r on a quasi-projective derived A-scheme X. Then its Euler class e❼E➁ ❃ Ω❻❼X➁ is nilpotent. Sketch of proof. If r 1, we can write L L1 ❛ L✲

2 with L1,L2 globally generated.

By the defining relation of a precobordism theory, we can hence assume that L is globally generated. In this case, we can find global sections s1,...,sn which generate L. Since the derived intersection of the vanishing loci V ❼s1➁,...,V ❼sn➁ is empty and e❼L➁ V ❼s1➁ X,0✆ ... V ❼sn➁ X,0✆, the claim follows. If r ❆ 1, one constructs a surjection E n L on a line bundle L for some n ❆❆ 0, so e❼E➁n e❼E ❵n➁ is multiple of e❼L➁.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Construction of Chern classes

One constructs πX,E ✂ ˜ XE X, L1,...,Lr and ηX,E ❃ Ω0❼ ˜ XE➁ by induction on r: If r 1, simply define ˜ XE X, πX,E idX, choose the trivial filtration 0 ❜ E and ηX,E 1X ❃ Ω0❼X➁. If r ❆ 1, we let Z be the vanishing locus of the zero section of E. Furthermore, we let ˜ X BlZ❼X➁ and denote by π ✂ ˜ X X the structure morphism of ˜ X. Then there is a canonical exact sequence 0 ❖❼❊➁ π❻❼E➁ Q 0

• f vector bundles on ˜

X, where ❊ is the exceptional divisor.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Construction of Chern classes

One proves that the element η➐

X,E ❃ Ω❻❼ ˜

X➁ defined as P➟

i0 e❼E➁i ❨P ˜ X❼❖ ❵ π❻❼E➁➁✆i ❨❼1 ˜ X ✏e❼❖❼❊➁➁❨P ˜ X❼❖❼❊➁❵❖➁✆➁

with ❨ ❨❵ satisfies π!❼η➐

X,E➁ 1X and defines:

˜ XE ✂ ❐ BlZ❼X➁Q πX,E ✂ ❐ BlZ❼X➁Q

πBlZ ❼X➁,Q

Ð Ð Ð Ð Ð BlZ❼X➁

π

Ð X ηX,E ✂ ηBlZ ❼X➁,Q ❨❵ π❻

BlZ ❼X➁,Q❼η➐ X,E➁

We obtain the desired filtration of π❻

X,E❼E➁ by combining the

pullback of ❖❼❊➁ to ˜ XE and the pullback of the filtration of π❻

BlZ ❼X➁,Q❼Q➁ along the surjection E Q.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Construction of Chern classes

One then checks that all the desired properties of ˜ XE are satisfied, e.g. πX,E!❼ηX,E➁ π!❼πBlZ ❼X➁,Q!❼ηBlZ ❼X➁,Q ❨❵ π❻

BlZ ❼X➁,Q❼η➐ X,E➁➁➁

π!❼πBlZ ❼X➁,Q!❼ηBlZ ❼X➁,Q➁ ❨❵ η➐

X,E➁➁

π!❼1BlZ ❼X➁ ❨❵ η➐

X,E➁

π!❼η➐

X,E➁

1X.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Projective bundle formula

Theorem [A, Theorem 5.16(1)] Let B❻ be a bivariant precobordism theory and X Y a morphism between derived schemes over a Noetherian ring A of finite Krull

• dimension. Let E be a vector bundle of rank r over X. Then there

is a natural isomorphism B❻❼P❼E➁➁ ☞ B❻❼X➁t✆⑦❵f ❡

• f rings, where t e❼❖❼1➁➁ and f Pr

i0 ❼✏1➁icr✏i❼E ✲➁ti.

Talk I Annala’s derived algebraic cobordism over a general base

Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes

Bivariant projective bundle formula

Theorem [A, Theorem 5.16(2)] Let B❻ be a bivariant precobordism theory and X Y a morphism between derived schemes over a Noetherian ring A of finite Krull

• dimension. Let E be a vector bundle of rank r over X. Then the

assignment α ❛ β ✭ α ❨ θ❼π➁ ❨ β induces an isomorphism B❻❼P❼E➁➁ ❛B❻❼X➁ B❻❼X Y ➁ ☞ B❻❼P❼E➁ Y ➁

• f B❻❼P❼E➁➁-modules, where π is the structure morphism

P❼E➁ X.

Talk I Annala’s derived algebraic cobordism over a general base