Bivariant derived algebraic cobordism: Bivariant theories June - - PowerPoint PPT Presentation

Bivariant derived algebraic cobordism

Bivariant derived algebraic cobordism: Bivariant theories

June 16,2020

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Bivariant derived algebraic cobordism

Table of contents

1

Overview

2

Bivariant theories: Definition & Examples

3

Properties of bivariant theories

4

Universal bivariant theory

5

Bivariant derived algebraic cobordism (brief introduction)

6

Summary of results

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Bivariant derived algebraic cobordism Overview

Algebraic cobordism

[V] Voevodsky’s MGL⇤,⇤(X) (Solution to Milnor conjecture) [LM] Levine-Morel construct algebraic cobordism Ω⇤(X) geometrically [Lecture 1] [LP] Levine-Pandharipande’s double point cobordism ω⇤(X) (Application to Donaldson-Thomas theory). [Lecture 1] [LS] Lowrey-Sch¨ urg’s derived algebraic bordism dΩ⇤(X) of derived schemes X. [Lecture 6 - 8] [A2] Annala’s precobordism theory Ω⇤(X) of derive schemes X (Application to Conner-Floyd over a general base) [Lecture 12] [LeeP] Lee-Pandharipande’s algebraic cobordism with bundles ω⇤,⇤(X). Remark (A summary) ω⇤,r(X) ω⇤,r(k) ⌦ω⇤(k) ω⇤(X)

[LeeP] ⇠ =

• [LP]

⇠ =

/ Ω⇤(X)

Lecture 2 ⇠ =

/ MGLBM

2⇤,⇤(X)

dΩ⇤(X)

Lecture 8 ⇠ =

O

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Bivariant derived algebraic cobordism Overview

Bivariant theories

Bivariant theories unify cohomology and homology theories and all the functorial

• properties. It is introduced by Fulton-MacPherson in [FM]. See Fulton [F] chapter 17-

18 for a version of Riemann-Roch for singular varieties.

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Bivariant derived algebraic cobordism Overview

Bivariant algebraic cobordism

[Y] Yokura’s universal bivariant theory in a general setting M⇤(X ! Y ) [Lecture 9] [A1] Annala’s bivariant algebraic cobordism Ω⇤(X ! Y ) using derived algebraic geometry (Application to Conner-Floyd theorem for singular varieties over a field

• f char. = 0) [Lecture 9-11]

[Ann-Y] Annala-Yokura’s bivariant algebraic cobordism with bundles ω⇤,⇤(X ! Y ) Remark M⇤(X ! Y )

quotient theory

'

quotient theory

w

Ω⇤(X ! Y )

the same if r = 0

ω⇤,r(X ! Y )

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Setup (The category V with confined morphisms and independent squares) Let V be a category with a final object ⇤ and all fibre products, which has a class of confined morphisms, which contains all isomorphisms and is closed under composition and pullback a class of independent squares which are fibre squares satisfying (i) whenever two smaller squares in

X0

✏ / Y 0 ✏ / Z0 ✏

Z0

✏ / Z ✏

X

/ Y / Z

• r

Y 0

✏ / Y ✏

X0

/ X

are independent, then so is the outer square, (ii) all the squares of the forms are independent:

X id ✏ f / Y id

✏

X 1 / f ✏ X f

✏

X f / Y Y id / Y

where f can be any morphism.

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Example Suppose V is the category of quasi-projective schemes over a field k. We choose all the proper morphisms to be the class of confined morphisms, all the Tor-independent fibre squares to be the class of independent squares. Recall that a fibre square X 0

/ ✏

X

✏

Y 0

/ Y

is called Tor-independent if TorOY

i

(OX , OY 0) = 0 for i > 0.

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Definition (Bivariant theory) A bivariant theory B⇤ on such a category V with values in the category of graded abelian groups is an assignment to any morphism f : X ! Y a graded abelian group B⇤(X

f

! Y ) which is equipped with the following operations: (1) (Product) For morphisms X

f

• ! Y

g

• ! Z , we have a map
• : Bi (X

f

! Y ) ⌦ Bj (Y

g

! Z) ! Bi+j (X gf ! Z). (2) (Pushforward) For morphisms X

f

• ! Y

g

• ! Z with f confined, we have an induced pushforward map

f⇤ : Bi (X gf ! Z) ! Bi (Y

g

! Z). (3) (Pullback) For any independent square

X0 f 0 ✏

/ X

f

✏

Y 0 g / Y

we have an induce pullback morphism g⇤ : Bi (X

f

! Y ) ! Bi (X 0 f 0 ! Y 0). These operations are required to satisfy the following 8 axioms: U, A1, A2, A3, A12, A13, A23, A123

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Definition (Bivariant theory (Axioms U, A1 - A3)) U (Units) B has units, that is for all X 2 ob(V), there is an element 1X 2 B(X

idX

! X). For any morphism W ! X and any α 2 B(W ! X), we have α • 1X = α. For any morphism X ! Y and any β 2 B(X ! Y ), we have 1X • β = β. For any morphism g : X 0 ! X, we have g⇤1X = 1X0 . A1 (Product is associative) Given X

f

• ! Y

g

• ! Z

h

• ! W

with α 2 B(X

f

! Y ), β 2 B(Y

g

! Z), γ 2 B(Z

h

! W ), then we have (α • β) • γ = α • (β • γ). A2 (Pushforward is functorial) Given X

f

• ! Y

g

• ! Z

h

• ! W

with f , g confined and α 2 B(X

hgf

• ! W ), then we have

(g f )⇤(α) = g⇤(f⇤(α)). A3 (Pullback is functorial) Given independent squares

X00

/ ✏

X0

✏ / X

f

✏

Y 00 h / Y 0 g / Y

and α 2 B(X

f

! Y ), we have the equality (g h)⇤(α) = h⇤(g⇤(α)).

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Definition (Bivariant theory (Axioms A12 - A123)) A12 (Product and pushforward commute) Given X

f

! Y

g

! Z

h

! W with f confined and α 2 B(X gf ! Z), β 2 B(Z

h

! W ), we have f⇤(α • β) = f⇤(α) • β. A13 (Product and pullback commute) Given independent squares

X0

/

f 0 ✏ X f

✏

Y 0

✏

h0 / Y g

✏

Z0 h / Z

with α 2 B(X

f

! Y ), β 2 B(Y

g

! Z), then we have h⇤(α • β) = h0⇤(α) • h⇤(β). A23 (Pushforward and pullback commute) Given independent squares as in A13 with f confined and α 2 B(X gf ! Z), then we have f 0

⇤(h⇤(α)) = h⇤(f⇤(α)).

A123 (Projection formula) Given an independent square with g confined and α 2 B(X

f

! Y ), β 2 B(Y 0 hg ! Z)

X0 f 0 ✏ g0 / X f

✏

Y 0 g / Y h / Z

then we have g0

⇤(g⇤(α) • β) = α • g⇤(β). 10 / 39

Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Example (Bivariant K-theory) Let V be the category of quasi-projective schemes over a field k with confined (proper) morphisms and independent (Tor independent) squares. For every morphism f : X ! Y 2 Mor(V), we recall that an f -perfect complex is a complex of quasi-coherent sheaves F• on X such that i⇤(F•) is a perfect complex on P for any factorization f : X

i

! P

p

! Y with i closed and p smooth. We define K alg(X

f

! Y ) as the free abelian group on the set of quasi-isomorphism classes [F•] of f -perfect complexes on X, modulo [F•] = [F0

• ] + [F00
• ] for each exact sequence

0 ! F0

• ! F• ! F00
• ! 0 of f -perfect complexes on X.

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Example (Bivariant K-theory) We need to give the following data on Kalg(X

f

! Y ) of f -perfect complexes (to say Kalg is a bivariant theory): (1) (Product) For morphisms f : X ! Y and g : Y ! Z, we define

• : Kalg(X

f

! Y ) ⌦ Kalg(Y

g

! Z) ! Kalg(X gf ! Z) by [F•] • [G•] := [F• ⌦OX Lf ⇤G•]. (2) (Pushforward) For morphisms f : X ! Y and g : Y ! Z with f confined (proper), we define f⇤ : Kalg(X gf ! Z) ! Kalg(Y

g

! Z) by f⇤[F•] := [Rf⇤F•] (3) (Pullback) For any independent square (Tor independent fibre square)

X0 f 0 ✏ g0 / X f

✏

Y 0 g / Y

we define g⇤ : Kalg(X

f

! Y ) ! Kalg(X 0 f 0 ! Y 0) by g⇤[F•] := [Lg0⇤F•].

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Example (Bivariant K-theory) Let V be the category of quasi-projective schemes over a field k with confined (proper) morphisms and independent (Tor independent) squares. For every morphism f : X ! Y 2 Mor(V), we recall that an f -perfect complex is a complex of quasi-coherent sheaves F• on X such that i⇤(F•) is a perfect complex on P for any factorization f : X

i

! P

p

! Y with i closed and p smooth. Recall K alg(X

f

! Y ) is the free abelian group on the set of quasi-isomorphism classes [F•] of f -perfect complexes on X, modulo [F•] = [F0

• ] + [F00
• ] for each exact sequence

0 ! F0

• ! F• ! F00
• ! 0 of f -perfect complexes on X.

Example Note that K0(X) (K-theory of vector bundles) is isomorphic to K alg(X

id

! X). G0(X) (G-theory of coherent sheaves) is isomorphic to K alg(X

p

! ⇤).

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Example (Operational bivariant Chow groups) Let V be the category of quasi-projective schemes over a field k with confined (proper) morphisms and independent (all fibre) squares. For any morphism f : X ! Y , we form the fibre diagram X 0

g0

✏

f 0 / Y 0 g

✏

X

f

/ Y

for each morphism g : Y 0 ! Y . We define

• pCHp(X

f

! Y ) as follows. An element c 2 opCHp(X

f

! Y ) is a collection of homomorphisms c(k)

g

: CHk(Y 0) ! CHkp(X 0) for all morphisms g : Y 0 ! Y and all k, compatible with proper pushforward (C1), flat pullback (C2), and intersection products (C3). This means:

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Example (Operational bivariant Chow groups)

X00 f 00 / h0 ✏ Y 00 h

✏

X00 f 00 / i00 ✏ Y 00 i0

✏ / Z00

i

✏

X0 g0 ✏ f 0 / Y 0 g

✏

X0 g0 ✏ f 0 / Y 0 g

✏ / Z0

X f / Y X f / Y

(C1) In view of the left diagram, if h: Y 00 ! Y 0 is proper, g : Y 0 ! Y arbitrary, for all α 2 CHk(Y 00), c(k)

g

(h⇤(α)) = h0

⇤c(k) gh (α) in CHkp(X 0).

(C2) In view of the left diagram, if h: Y 00 ! Y 0 is flat of relative dimension n, and g is arbitrary, for all α 2 CHk(Y 0), we have c(k+n)

gh

(h⇤α) = h0⇤c(k)

g

(α) in CHk+np(X 00). (C3) In view of the right diagram, if i is a regular embedding of codimension e, then for all α 2 CHk(Y 0), we have i!c(k)

g

(α) = c(ke)

gi0

(i!α) in CHkpe(X 00).

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Example (Operational bivariant Chow groups) We need to give the following data on opCH⇤(X

f

! Y ) (to say opCH⇤ is a bivariant theory):

X0

✏

f 0 / Y 0

✏

g0 / Z0

✏

X0

✏ / X1

f1 ✏

/ X

f

✏

X f / Y g / Z Y 0 g0 / Y1 g / Y

(1) (Product) For morphisms f : X ! Y and g : Y ! Z, we define

• : opCHp(X

f

! Y ) ⌦ opCHq(Y

g

! Z) ! opCHp+q(X gf ! Z) by c • d(α) := c(d(α)) for α 2 CHk (Z0). (2) (Pushforward) For morphisms f : X ! Y and g : Y ! Z with f confined (proper), we define f⇤ : opCHp(X gf ! Z) ! opCHp(Y

g

! Z) by f⇤(c)(α) := f 0

⇤(c(α)) for α 2 CHk (Z0).

(3) (Pullback) For any independent square (all fibre square) as in the right square above, we define g⇤ : opCHp(X

f

! Y ) ! opCHp(X1

f1

! Y1) by g⇤(c)(α) := c(α) for α 2 CHk (Y 0).

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Example (Operational bivariant Chow groups) Let V be the category of quasi-projective schemes over a field k with confined (proper) morphisms and independent (all fibre) squares. For any morphism f : X ! Y , we define

• pCHp(X

f

! Y ) as follows. An element c 2 opCHp(X

f

! Y ) is a collection of homomorphisms c(k)

g

: CHk(Y 0) ! CHkp(X ⇥Y Y 0) for all morphisms g : Y 0 ! Y and all k, compatible with proper pushforward (C1), flat pullback (C2), and intersection products (C3). Proposition (Fulton) There is a canonical isomorphism opCHp(X ! ⇤) ⇠ = CHp(X). Definition We define CHp(X) := opCHp(X

id

! X). The element 1 2 CH0(X) is defined to be the identity map on all CHk(X 0) ! CHk(X 0). Then, CH⇤(X) is a cohomology ring called

• perational Chow ring.

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Remark Set Bi(X) := Bi(X ! ⇤) and Bi(X) := Bi(X

id

! X). We have homology groups B⇤(X) which are covariant for confined morphisms. We have cohomology groups B⇤(X) which are contravariant for any morphisms. B⇤(X) is a cohomological ring via the bivariant product, and it unital by the axiom (U).

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Definition (Orientation) Let S be a class of morphisms (called specialized morphisms) in V which is assumed to contain all identity maps and to be closed under compositions (not necessarily under pullback). Suppose that to each morphism f : X ! Y in S there is assigned an element θ(f ) 2 B(X

f

! Y ) satisfying that (1) θ(gf ) = θ(f ) • θ(g) for all f : X ! Y , g : Y ! Z 2 S and (2) θ(idX ) = 1X for all X with 1X 2 B(X

id

! X) the unit element. Then, the assignment θ is called an orientation of S. Definition Assume that the class of specialized morphisms S is stable under pullbacks in independent squares. We say an orientation θ of S is stable under pullback or nice, if for any independent square

X0 f 0 ✏

/ X

f

✏

Y 0 h / Y

with f (and therefore also f 0) specialized, we have h⇤(θ(f )) = θ(f 0).

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Let V be the category of quasi-projective schemes over a field k with confined (proper) morphisms and independent (Tor independent) squares. For every morphism f : X ! Y 2 Mor(V), we recall that an f -perfect complex is a complex of quasi-coherent sheaves F• on X such that i⇤(F•) is a perfect complex on P for any factorization f : X

i

! P

p

! Y with i closed and p smooth. Recall from the above that we define K alg(X

f

! Y ) as the free abelian group on the set of quasi-isomorphism classes [F•] of f -perfect complexes on X, modulo [F•] = [F0

• ] + [F00
• ] for each exact sequence

0 ! F0

• ! F• ! F00
• ! 0 of f -perfect complexes on X.

Example (Orientations in bivariant K-theory) Let S be the class of morphisms of perfect morphisms f : X ! Y (i.e. if there is an N such that TorOY

i

(OX , G) = 0 for all i > N and all coherent sheave G on Y). Note that perfect morphisms are stable under pullback in Tor independent fibre squares, and l.c.i morphisms are perfect. Moreover, f : X ! Y is perfect if and only if the complex OX [0] of the structure sheaf in degree 0 is an f -perfect complex. We define θ(f ) := [OX [0]] 2 K alg(X

f

! Y ). Then, θ is an orientation of S. It is clear that the orientation θ of S is nice (i.e. stable under pullback).

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Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Let V be the category of quasi-projective schemes over a field k with confined (proper) morphisms and independent (all fibre) squares. For f : X ! Y , consider opCHp(X

f

! Y ). Recall an element c 2 opCHp(X

f

! Y ) is a collection of homomorphisms c(k)

g

: CHk (Y 0) ! CHkp(X 0) for all morphisms g : Y 0 ! Y and all k, compatible with proper pushforward (C1), flat pullback (C2), and intersection products (C3). Here, X 0 := X ⇥Y Y 0. Example (Orientations in Operational bivariant Chow groups)

X0

/ ✏

X f

✏

X00

✏ / X0

f 0

✏ / X

f

✏

Y 0 g / Y Y 00 g / Y 0 h / Y

Situation I. (Independent squares to be all fibre squares) Let S (specialized morphisms) to be the class of all l.c.i morphisms of pure relative dimension. For f : X ! Y l.c.i morphism of pure relative dimension d, we define θ(f ) 2 opCHd (X ! Y ) by θ(f )g := f !

g

where f !

g is the refined Gysin homomorphism of Fulton for all g : X : Y 0 ! Y (recall for the left hand side fibre

square, Fulton defines f !

g : CH⇤(Y 0) ! CH⇤+d (X 0).) The orientation θ in this case can not be nice (i.e. stable

under pullback), as l.c.i. morphisms are not stable under pullback in all fibre squares. Situation II. (Independent squares to be Tor independent squares) l.c.i morphisms are stable under pullback in Tor independent squares. If we take V0 ⇢ V the same category but with independent squares only Tor independent squares, then in view of the right hand side diagram, the orientation θ of S in V0 is nice, i.e. h⇤θ(f ) = θ(f 0), that is f 0!

g

= f !

gh from CH⇤(Y 00) to CH⇤+d (X 00). 21 / 39

Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples

Definition Any element α 2 Bp(X

f

! Y ) gives rise to (Gysin pullback) α! : Bk(Y ) ! Bkp(X): β 7! α • β (Gysin pushforward) For f is confined, α! : Bk(X) ! Bk+p(Y ): γ 7! f⇤(γ • α) If f is a specialized morphism, we may let α = θ(f ), and write f ! := θ(f )! f! := θ(f )! .

22 / 39

Bivariant derived algebraic cobordism Properties of bivariant theories

Definition A bivariant theory B⇤ is called commutative if whenever both squares W

g0 / f 0

✏

X

f

✏

W

g0

✏

f 0 / Y g

✏

Y

g / Z

X

f

/ Z

are independent squares then g⇤(α) • β = f ⇤(β) • α with α 2 B⇤(X

f

! Z) and β 2 B⇤(Y

g

! Z). Remark All bivariant theories in this talk are commutative. There is also a notion of skew-commutative, but we will not pursue this here.

23 / 39

Bivariant derived algebraic cobordism Properties of bivariant theories

Definition Let V be a category. A class of fibre squares is said to have the right cancellation property if when we have either ·

/ ✏

·

/ ✏

·

✏

·

/ ✏

·

✏

·

/ · / ·

• r

·

/ ✏

·

✏

·

/ ·

and when the large square and the right (lower) square are independent, then so is the left (upper) square.

24 / 39

Bivariant derived algebraic cobordism Properties of bivariant theories

Definition (Specialized projections) Let V be a category with confined morphisms and independent squares. Suppose that S is a class of specialized morphisms on V, and suppose that there is a subclass P of S closed under compositions and pullbacks satisfying that whenever g f = 1 with g 2 P then f 2 S. Then, we call elements of P in S specialized projections. Example Consider operational bivariant Chow groups opCH⇤ on the category V of quasi-projective schemes over a field k with confined morphisms (proper morphisms), independent squares (Tor independent fibre squares), specialized morphisms (l.c.i morphisms), specialized projections (smooth morphisms), To see the class of smooth morphisms form a class of specialized projections, we note that any section of smooth morphism is regular immersion (in particular a l.c.i morphism).

25 / 39

Bivariant derived algebraic cobordism Properties of bivariant theories

Proposition Let V be a category with confined morphisms, independent squares, specialized morphisms and specialized projections. Let B⇤ be a commutative bivariant theory on V with nice orientation θ. Suppose that all fibre squares pulling back a specialized projection are independent, and that independent squares satisfy the right cancellation

• property. Then, for all specialized projections f : Y ! Z, θ(f ) is strong, i.e.
• θ(f ) : B⇤(X ! Y ) ! B⇤(X ! Z)

are isomorphisms for all X ! Y . Example Consider operational bivariant Chow groups opCH⇤ on the category of quasi-projective schemes over a field k with confined morphisms (proper morphisms), independent squares (Tor independent fibre squares), specialized morphisms (l.c.i morphisms), specialized projections (smooth morphisms), We have seen orientation is nice in this case. The class of Tor independent squares has the right cancellation property. Therefore, all assumption is satisfied.

26 / 39

Bivariant derived algebraic cobordism Properties of bivariant theories

Definition If X ! ⇤ is a specialized morphism, the intersection product \ : B⇤(X) ⇥ B⇤(X) ! B⇤(X) is defined as α \ β := ∆!

X (p! 1α • p! 2β)

where pi : X ⇥ X ! X are two projections, and ∆ : X ! X ⇥ X is the diagonal. Proposition (Poincar´ e Duality) Let V be a category with confined morphisms, independent squares, specialized morphisms and specialized projections. Let B⇤ be a commutative bivariant theory on V with nice orientation θ. Suppose that all fibre squares pulling back a specialized projection are independent, and that independent squares satisfy the right cancellation

• property. Suppose that π : X ! ⇤ is a specialized projection. Then,
• θ(π): (B⇤(X), •) ! (B⇤(X), \)

is an isomorphism of rings.

27 / 39

Bivariant derived algebraic cobordism Universal bivariant theory

Definition A Grothendieck transformation ω : B ! B0 of bivariant theories is a collection of maps γf : B(X

f

! Y ) ! B0(X

f

! Y ) which satisfies three relations: γ(α • β) = γ(α) • γ(β), γ(f⇤α) = f⇤γ(α), γ(g⇤α) = g⇤γ(α). Definition A universal bivariant theory M⇤ is a bivariant theory such that for any bivariant theory B⇤, there is a unique Grothendieck transformation ω : M⇤ ! B⇤ compatible with all three operations (product, pushforward and pullback).

28 / 39

Bivariant derived algebraic cobordism Universal bivariant theory

Theorem (Yokura) Let V be a category with a class C of confined morphisms, independent squares, and a class S of specialized morphisms that is stable under pullbacks in independent squares. Then, there exists a unique universal bivariant theory M⇤ on V with nice orientations

• f S in the class of bivariant theories on V with nice orientations of S.

Construction (Universal bivariant theory) Let V be the category above. For any morphism f : X ! Y , we define MC

S(X f

! Y ) as the free abelian group generated by the isomorphism classes [V ! X] of confined maps V ! X such that V ! X

f

! Y is specialized. To say it is a bivariant theory, we have to give three operations (see next page...)

29 / 39

Bivariant derived algebraic cobordism Universal bivariant theory

Construction (Universal bivariant theory) We define (1) (Product) For morphisms f : X ! Y and g : Y ! Z, the product operation MC

S(X f

! Y ) ⌦ MC

S(Y g

! Z) ! MC

S(X gf

! Z) is defined by [V

v

! X] ⌦ [W

w

! Y ] := [V 0 v0 ! X] where we form the fibre square

V 0 v0

/ ✏

W w

✏

V v / X f

/ Y

g / Z

(2) (Pushforward) For morphisms f : X ! Y and g : Y ! Z with f confined, the pushforward f⇤ : MC

S(X gf

! Z) ! MC

S(Y g

! Z) is defined by f⇤[V

v

! X] := [V

fv

! Y ].

30 / 39

Bivariant derived algebraic cobordism Universal bivariant theory

Construction (Universal bivariant theory) (3) (Pullback) For an independent square

X0 f 0 ✏

/ X

f

✏

Y 0 g / Y

the pullback g⇤ : MC

S(X f

! Y ) ! MC

S(X 0 f

! Y 0) is defined to be g⇤([V

v

! X]) := [V 0 v0 ! X 0] in view of the following diagram:

V 0 v0 ✏

/ V

v

✏

X0 f 0 ✏

/ X

f

✏

Y 0 g / Y 31 / 39

Bivariant derived algebraic cobordism Universal bivariant theory

Theorem (Yokura) Let V be a category with a class C of confined morphisms, independent squares, and a class S of specialized morphisms that is stable under pullbacks in independent squares. Then, there exists a unique universal bivariant theory M⇤ on V with nice orientations

• f S in the class of bivariant theories on V with nice orientations of S.

Sketch of proof. Consider the theory MC

S constructed above (check it is a bivariant theory). There is a

canonical (nice) orientation θM on MC

S: Precisely, for every f : X ! Y of S, set

θM(f ) := [X

id

! X] 2 MC

S(X f

! Y ). One has to check θM is a nice orientation. Construct a Grothendieck transformation for any bivariant theory B with a nice

• rientation θB of S

γ : MC

S(X f

! Y ) ! B(X

f

! Y ) by setting γ([V

v

! X]) = v⇤θB(f v). Check this is a well-defined Grothendieck tranformation. Check this preserves the

• rientation and check the uniqueness.

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Bivariant derived algebraic cobordism Bivariant derived algebraic cobordism (brief introduction)

Definition (Bivariant subsets) Let B be a bivariant theory on a category C. A bivariant subset S of B consists of subsets S(X ! Y ) ⇢ B(X ! Y ) for all morphisms X ! Y in C. Definition (Bivariant ideals) A bivariant ideal I of a bivariant theory B is a bivariant subset satisfying the subsets I(X ! Y ) ⇢ B(X ! Y ) are subgroups, the subsets I(X ! Y ) ⇢ B(X ! Y ) are closed under pullbacks and pushforwards in B, the product on B restricts to morphisms

• : B(X ! Y ) ⌦ I(Y ! Z) ! I(X ! Z)

and

• : I(V ! X) ⌦ B(X ! Y ) ! I(V ! Y ).

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Bivariant derived algebraic cobordism Bivariant derived algebraic cobordism (brief introduction)

Remark Given a bivariant ideal I of a bivariant theory B, the quotient groups B/I(X ! Y ) := B(X ! Y )/I(X ! Y ) forms an inherit bivariant theory. Given a bivariant subset S of a bivariant theory B, we can take the smallest bivariant ideal of B containing S, denoted by hSiB. The bivariant ideal hSiB shall be called bivariant ideal generated by S.

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Bivariant derived algebraic cobordism Bivariant derived algebraic cobordism (brief introduction)

Construction (Bivariant derived algebraic cobordism) Step I. Let V be the homotopy category of the 1-category of quasi-projective derived

• schemes. Set

confined morphisms to be proper morphisms, independent squares to be homotopy cartesian squares, specialized morphisms to be quasi-smooth morphisms. Apply the above settings to Yukura’s universal bivariant theory M⇤, that is we defined Md(X ! Y ) to be the free abelian group of equivalent classes [V ! X] of proper morphisms V ! X so that the composition V ! Y is quasi-smooth of relative virtual dimension d, modulo disjoint union corresponds to summation.

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Bivariant derived algebraic cobordism Bivariant derived algebraic cobordism (brief introduction)

Construction (Bivariant derived algebraic cobordism) Step II. Note that the homology groups M⇤(X) := M⇤(X ! ⇤) coincides with Lowrey-Sch¨ urg’s bordism cycle group M+

⇤ (X). Lowrey-Sch¨

urg construct a surjection Θ : L ⌦Z M+

⇤ (X) ! dΩ⇤(X)

Denote this kernel by LS(X). Denote by M⇤

L the bivariant theory L ⌦Z M⇤. Now,

LS(X) is clearly a subset of M⇤

L(X ! ⇤). Construct the bivariant subset LS of M⇤ L by

letting LS(X ! Y ) = ( LS(X) if Y = ⇤ if otherwise. The bivariant algebraic cobordism group Ω⇤ is defined by Ω⇤(X ! Y ) := M⇤

L(X ! Y )/hLSiM⇤

L (X ! Y ).

Theorem (Agreement with Lowrey-Sch¨ urg) Let X be a quasi-projective derived scheme over a field of characteristic zero. There is a natural isomorphism Ω⇤(X ! ⇤) ⇠ = dΩ⇤(X) where the right hand side is the Lowrey-Sch¨ urg’s derived algebraic cobordism.

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Bivariant derived algebraic cobordism Summary of results

Theorem (Conner-Floyd for quasi-projective derived schemes) Let X be a quasi-projective derived scheme over a field of characteristic zero. Then, there exist chern classes ci(E) 2 Ωi(X ! X) which induce an isomorphism ch : K 0(X)

⇠ =

• ! Zm ⌦L Ω⇤(X ! X)

where Zm is the integers considered as an L-algebra via the map classifying the multiplicative formal group law x + y xy. Remark (Conner-Floyd for an arbitrary base [A2]) In another paper of Annala [A2], there is a notion of precobodism theory Ω⇤(X) for any quasi-projective derived scheme X over a noetherian ring A, which is basically the Lowrey-Sch¨ urg’s M⇤

+(X) modulo the ”derived double point relations”). There are

chern classes constructed on Ω⇤(X) which induce an isomorphism ch : K 0(X) ! Zm ⌦L Ω⇤(X) (see [A2]).

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Bivariant derived algebraic cobordism Summary of results

Definition Let X be a quasi-projective derived scheme over a field of characteristic zero. Define CH⇤(X ! Y ) := Za ⌦L Ω⇤(X ! Y ) where Za is the integers considered as an L-algebra via the map classifying the additivie formal group law x + y. Proposition Let X be a quasi-projective derived scheme over a field of characteristic zero. There there is an isomorphism CH⇤(X ! ⇤) ⇠ = CH⇤(π0X).

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Bivariant derived algebraic cobordism Summary of results

Theorem Let X be a quasi-projective derived scheme over a field of characteristic zero. There is a natural isomorphism of bivariant theories Ω⇤

Q ⇠

= opCH⇤

Q ⌦ LQ.

Moreover, the chern character map ch: K0(X)Q ! CH⇤(X ! X)Q is an isomorphism for any quasi-projective derived scheme X.

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