Oriented Bivariant Theories, I - univie.ac.atesiprpr/esi1911.pdf · as a categorical framework for the study of singular spaces, which is the titleof theirAMS Memoir book [FM] (see

ESI The Erwin Schrodinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, Austria

Oriented Bivariant Theories, I

Shoji Yokura

Vienna, Preprint ESI 1911 (2007) April 27, 2007

Supported by the Austrian Federal Ministry of Education, Science and CultureAvailable via http://www.esi.ac.at

ORIENTED BIVARIANT THEORIES, I

SHOJI YOKURA(∗)

ABSTRACT. In 1981 W. Fulton and R. MacPherson introduced the notion ofbivarianttheory (BT), which is a sophisticated unification of covariant theories and contravarianttheories. This is for the study of singular spaces. In 2001 M.Levine and F. Morel in-troduced the notion of algebraic cobordism, which is a universal oriented Borel–Moorefunctor with products (OBMF ) of geometric type, in an attempt to understand better V.Voevodsky’s (higher) algebraic cobordism. In this paper weintroduce a notion of orientedbivariant theory (OBT), a special case of which is nothing but the oriented Borel–Moorefunctor with products. The present paper is a first one of the series to try to understandLevine–Morel’s algebraic cobordism from a bivariant-theoretical viewpoint, and its firststep is to introduceOBT as a unification ofBT andOBMF .

1. INTRODUCTION

William Fulton and Robert MacPherson have introduced the notion of bivariant theoryas acategorical framework for the study of singular spaces, which is the title of their AMSMemoir book [FM] (see also Fulton’s book [Fu]). The main objective of [FM] is bivariant-theoretic Riemann–Roch’s or bivariant analogues of various theorems of Grothendieck–Riemann–Roch type.

Vladimir Voevodsky has introduced algebraic cobordism (now calledhigher algebraiccobordism), which was used in his proof of Milnor’s conjecture [Voe]. Daniel Quillenintroduced the notion of(complex ) oriented cohomology theoryon the category of differ-ential manifolds [Qui] and this notion can be formally extended to the category of smoothschemes in algebraic geometry. Marc Levine and Fabien Morelconstructeda universal ori-ented cohomology theory, which they also callalgebraic cobordism, and have investigatedfurthermore (see [L1, L2, LM1, LM2, LM3] and also see [Mer] for a condensed review).Recently M. Levine and R. Pandharipande [LP] gave another equivalent construction ofthe algebraic cobodism via what they call “double point degeneration” and they found anice application of the algebraic cobordism in the Donaldson–Thomas theory of 3-folds.

In this paper we extend Fulton–MacPherson’s bivariant theory to what we callan ori-ented bivariant theoryfor a general category, not just for a geometric category of,say,complex algebraic varieties, schemes, etc. In most interesting cases bivariant theories suchasbivariant homology theory, bivariant Chow group theory, bivariant algebraic K-theoryand bivariant topological K-theoryare already oriented bivariant theories. We show thateven in this general category there existsa universal oriented bivariant theory, whosespecial case gives rise toa universal oriented Borel–Moore functor with products. Levine–Morel’s algebraic cobordism requires more geometrical conditions. Indeed, they call alge-braic cobordisma universal oriented Borel–Moore functor with products of geometric type[LM3]. In a second paper [Yo1] we will deal with an oriented bivariant theory of geometrictype.

(*) Partially supported by Grant-in-Aid for Scientific Research (No. 17540088 and No. 19540094), the Min-istry of Education, Culture, Sports, Science and Technology (MEXT), and JSPS Core-to-Core Program 18005,Japan.

1

2 SHOJI YOKURA(∗)

One purpose of this paper is to bring Fulton–MacPherson’s Bivariant Theory to theattention of people working on algebraic cobordism and/or subjects related to it.

2. FULTON–MACPHERSON’ S BIVARIANT THEORY

We make a quick review of Fulton–MacPherson’s bivariant theory [FM].Let V be a category which has a final objectpt and on which the fiber product or fiber

square is well-defined. Also we consider a class of maps, called “confined maps” (e.g.,proper maps, projective maps, in algebraic geometry), which are closed under compositionand base change and contain all the identity maps, and a calssof fiber squares, called “in-dependent squares” (or “confined squares”, e.g., “Tor-independent” in algebraic geometry,a fiber square with some extra conditions required on morphisms of the square), whichsatisfy the following:

(i) if the two inside squares in

X′′ h′

−−−−→ X′ g′

−−−−→ X

yf ′′

yf ′

y

f

Y ′′ −−−−→h

Y ′ −−−−→g

Y

or

X′ −−−−→h′′

X

f ′

y

y

f

Y ′ −−−−→h′

Y

g′

y

y

g

Z′ −−−−→h

Z

are independent, then the outside square is also independent,(ii) any square of the following forms are independent:

X

f

��

idX// X

f

��

X

idX

��

f// Y

idY

��

YidX

// Y Xf

// Y

wheref : X → Y is anymorphism.

A bivariant theoryB on a categoryV with values in the category of (graded) abeliangroups is an assignment to each morphism

Xf−→ Y

in the categoryV a (graded) abelian group (in the rest of the paper we ignore the grading)

B(Xf−→ Y )

which is equipped with the following three basic operations:Product operations: For morphismsf : X → Y andg : Y → Z, the product operation

• : B(Xf−→ Y ) ⊗ B(Y

g−→ Z) → B(X

gf−→ Z)

is defined.

ORIENTED BIVARIANT THEORIES, I 3

Pushforward operations: For morphismsf : X → Y andg : Y → Z with f confined,the pushforward operation

f∗ : B(Xgf−→ Z) → B(Y

g−→ Z)

is defined.Pullback operations: For an independentsquare

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y,

the pullback operation

g∗ : B(Xf−→ Y ) → B(X′ f ′

−→ Y ′)

is defined.And these three operations are required to satisfy the sevencompatibility axioms (see

[FM, Part I,§2.2] for details):(B-1) product is associative,(B-2) pushforward is functorial,(B-3) pullback is functorial,(B-4) product and pushforward commute,(B-5) product and pullback commute,(B-6) pushforward and pullback commute, and(B-7) projection formula.

Let B, B′ be two bivariant theories on a categoryV. Then aGrothendieck transformationfrom B to B′

γ : B → B′

is a collection of homomorphisms

B(X → Y ) → B′(X → Y )

for a morphismX → Y in the categoryV, which preserves the above three basic opera-tions:

(i) γ(α •B β) = γ(α) •B′ γ(β),(ii) γ(f∗α) = f∗γ(α), and(iii) γ(g∗α) = g∗γ(α).

For more details of interesting geometric and/or topological examples of bivariant theo-ries (e.g., bivariant theory of constructible functions, bivariant homology theory, bivariantK-theory, etc.,) and Grothendieck transformations among bivariant theories, see [FM]. Inthis paper we treat with bivariant theories more abstractlyfrom a general viewpoint.

A bivariant theory unifies both a covariant theory and a contravariant theory in the fol-lowing sense:

B∗(X) := B(X → pt) become a covariant functor forconfinedmorphisms and

B∗(X) := B(Xid−→ X) become a contravariant functor foranymorphisms.

And a Grothendieck transformationγ : B → B′ induces natural transformationsγ∗ :B∗ → B′

∗ andγ∗ : B∗ → B′∗.

As to the grading,Bi(X) := B−i(Xid−→ X) andBj(X) := Bj(X

id−→ X).

In the rest of the paper we assume that our bivariant theoriesare commutative(see [FM,§2.2]), i.e., if whenever both

4 SHOJI YOKURA(∗)

W

f ′

��

g′

// X

f

��

W

g′

��

f ′

// Y

g

��

Y g// Z X

f// Z

are independent squares, then forα ∈ B(Xf−→ Z) andβ ∈ B(Y

g−→ Z)

g∗(α) • β = f∗(β) • α.

(Note: if g∗(α) • β = (−1)deg(α) deg(β)f∗(β) • α, then it is called skew-commutative.)Another assumption in the rest of the paper, which is not in Fulton–MacPherson’s bi-

variant theory, is the following additivity:

B(X∐

Yf−→ Z) = B(X

f|X−−→ Z) ⊕ B(Y

f|Y−−→ Z).

When we want to emphasize this additivity, we call such a bivariant theory anadditivebivariant theory.

Definition 2.1. ([FM, 2.6.2 Definition, Part I]) LetS be a class of maps inV, which isclosed under compositions and containing all identity maps. Suppose that to eachf :

X → Y in S there is assigned an elementθ(f) ∈ B(Xf−→ Y ) satisfying that

(i) θ(g ◦ f) = θ(f) • θ(g) for all f : X → Y , g : Y → Z ∈ S and

(ii) θ(idX) = 1X for all X with 1X ∈ B∗(X) := B(XidX−−→ X) the unit element.

Thenθ(f) is called acanonical orientationof f .

Note that such a canonical orientation makes the covariant functorB∗(X) a contravari-ant functor for morphisms inS, and also makes the contravariant functorB∗ a covariantfunctor for morphisms inC ∩ S: Indeed,

(*) for a morphismf : X → Y ∈ S and the canonical orientationθ onS the followingGysin homomorphism

f ! : B∗(Y ) → B∗(X) defined by f !(α) := θ(f) • α

is contravariantly functorial. And(**) for a fiber square (which is an independent square by hypothesis)

Xf

−−−−→ Y

idX

y

yidY

X −−−−→f

Y,

wheref ∈ C ∩ S, the followingGysin homomorphism

f! : B∗(X) → B

∗(Y ) defined by f!(α) := f∗(α • θ(f))

is coavariantly functorial. The notation should carry the information ofS and the canon-ical orientationθ, but it will be usually omitted if it is not necessary to be mentioned.Note that the above conditions (i) and (ii) of Definition (2.1) are certainly necessary for theabove Gysin homomorphisms to be functorial.

Definition 2.2. (i) Let S be another class of maps inV , called “specialized maps” (e.g.,smooth maps in algebraic geometry), which is closed under composition and under basechange and containing all identity maps. LetB be a bivariant theory. IfS has canonicalorientations inB, then we say thatS is canonicalB-orientable and an element ofS is


called a canonicalB-orientable morphism. (Of courseS is also a class of confined maps,but since we consider the above extra condition ofB-orientability onS, we give a differentname toS.)

(ii) Let S be as in (i). LetB be a bivariant theory andS be canonicalB-orientable.Furthermore, if the orientationθ onS satisfies that for a fiber square withf ∈ S

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y

the following condition holds

θ(f ′) = g∗θ(f),

(which means that the orientationθ preserves the pullback operation), then we callθ anicecanonical orientationand say thatS is nice canonicalB-orientableand an element ofS iscalleda nice canonicalB-orientable morphism.

Proposition 2.3. LetB be a bivariant theory and letS be as above.(1) Define the natural exterior product

× : B(X −→ pt) × B(YπY−−→ pt) → B(X × Y → pt)

by

α × β := π∗Y α • β.

Then the covariant functorB∗ for confined morphisms and the contravariant functorB∗ formorphisms inS are both compatible with the exterior product, i.e., for confined morphismsf : X → X′, g : Y → Y ′,

(f × g)∗(α × β) = f∗α × g∗β

and for morphismsf : X → X′, g : Y → Y ′ in S,

(f × g)!(α′ × β′) = f !α′ × g!β′.

(2) Similarly, define the natural exterior product

× : B(XidX−−→ X) × B(Y

idY−−→ Y ) → B(X × YidX×X

−−−−→ X × X)

by

α × β := p1∗α • p2

∗β

wherep1 : X × Y → X andp2 : X × Y → Y be the projections.Then the contravariant functorB∗ for anymorphisms and the covariant functorB∗ for

morphisms inC ∩S are both compatible with the exterior product, i.e., for anymorphismsf : X → X′, g : Y → Y ′,

(f × g)∗(α × β) = f∗α × g∗β

and for morphismsf : X → X′, g : Y → Y ′ in C ∩ S,

(f × g)!(α′ × β′) = f!α

′ × g!β′.

Proof. The proof is tedious, using several axioms of the bivariant theory. For the sake ofcompleteness we give a proof. But, we give a proof for only (1)and a proof for (2) is leftfor the reader.

For morphismsf : X → X′ andg : Y → Y ′, consider the following big commutativediagrams

6 SHOJI YOKURA(∗)

X × Yf×IdY

−−−−→ X′ × Yp×IdY

−−−−→ pt × Y = Y

IdX×g

y

y

IdX′×g

y

g

X × Y ′ f×IdY ′

−−−−−→ X′ × Y ′ p×IdY ′

−−−−−→ pt × Y ′ = Y ′

IdX×q

y

y

IdX′×q

y

q

X = X × pt −−−−→f

X′ = X′ × pt −−−−→p

pt.

The proof of(f × g)∗(α × β) = f∗α × g∗β goes as follows:

(f × g)∗(α × β) = (f × g)∗

(

(qg)∗α • β)

(by definition)

= (f × g)∗

(

(g∗ (q∗α)) • β)

= (IdX′ × f)∗

(

(f × IdY )∗

(

g∗ (q∗α) • β)

)

= (IdX′ × f)∗

(

(f × IdY )∗ (g∗ (q∗α)) • β

)

(by (B-4))

= (IdX′ × f)∗

(

g∗(

(f × IdY ′ )∗ (q∗α))

)

• β (by (B-6))

= (IdX′ × f)∗

(

g∗ (q∗ (f∗α)) • β)

(by (B-6))

= q∗ (f∗α) • g∗β (by (B-7))

= f∗α × g∗β (by definition)

Next we show(f × g)!(α′ × β′) = f !α′ × g!β′. For this, first we observe that

(f × g)! := (θ(f) × θ(g)) • .

On one hand we have that

(f × g)!(α′ × β′) = (θ(f) × θ(g)) • (q∗α′ • β′) (by definition)

= (θ(f) × θ(g)) • q∗α′ • β′

=(

IdX′ × f)∗(IdY ′ × q)∗)

(θ(f)) • (p × IdY ′ )∗(θ(g)) • q∗α′ • β′

On the other hand we have that

f !α′ × g!β′ = (θ(f) • α′) × (θ(g) • β′) (by definition)

=(

(qg)∗(θ(f) • α′))

• (θ(g) • β′)

=

(

g∗(

q∗(θ(f) • α′))

)

• (θ(g) • β′)

= g∗(

(IdY ′ × q)∗αf • q∗α′)

• θ(g) • β′

=(

IdX′ × f)∗(IdY ′ × q)∗)

(θ(f)) • g∗(q∗α′) • θ(g) • β′

=(

IdX′ × f)∗(IdY ′ × q)∗)

(θ(f)) • (p × IdY ′ )∗(θ(g)) • q∗α′ • β′

The last equality follows from the commutativity of the bivariant theory. Thus we getthe above equality. �


Proposition 2.4. Let B be a bivaraint theory on a categoryV with a classC of confinedmorphisms. LetS be a class of niceB-orientable morphisms. Then for any independentsquare

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y

with f ∈ S andg ∈ C, the following diagram commutes:

B∗(Y′)

f ′∗

−−−−→ B∗(X′)

g∗

y

y

g′

∗

B∗(Y ) −−−−→f∗

B∗(X).

Here we remark the following fact about the covariant and contravariant functorsB∗

and B∗, which will be needed in later sections. They are what Levineand Morel callBorel–Moore functor with productsin [LM3, Mer].

Proposition 2.5. Let the situation be as above.(1-i) For a disjoint sumX

∐

Y ,

B∗(X∐

Y ) = B∗(X) ⊕ B∗(Y ),

(1-ii) for confined morphismsf : X → Y , the pushforward homomorphisms

f∗ : B∗(X) → B∗(Y )

are covariantly functorial,(1-iii) for morphisms inS, i.e., for nice canonicalB-orientable morphismsf : X → Y ,

the Gysin (pullback) homomorphisms

f ! : B∗(Y ) → B∗(X)

are contravariantly functorial,(1-iv) for an independent square

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y

with g ∈ C andf ∈ S, we have the following diagram commutes:

B∗(Y′)

f ′ !

−−−−→ B∗(X′)

g∗

y

y

g′

∗

B∗(Y ) −−−−→f !

B∗(X),

(1-v) the pushforward homomorphismsf∗ : B∗(X) → B∗(Y ) for confined morphismsand the Gysin (pullback) homomorphismsf ! : B∗(Y ) → B∗(X) for morphisms inS areboth compatible with the exterior products

× : B∗(X) ⊗ B∗(Y ) → B∗(X × Y ).

(2-i) for a disjoint sumX∐

Y ,

B∗(X

∐

Y ) = B∗(X) ⊕ B

∗(Y ),

8 SHOJI YOKURA(∗)

(2-ii) for any morphismsf : X → Y , the pullback homomorphisms

f∗ : B∗(Y ) → B

∗(Y )

are contravariantly functorial,(2-iii) for confined and specialized morphisms inC ∩ S, i.e., for confined and nice

canonicalB-orientable morphismsf : X → Y , the Gysin (pushforward) homomorphisms

f! : B∗(X) → B

∗(Y )

are covariantly functorial,(2-iv) for an independent square

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y

with g ∈ C ∩ S, we have the following diagram commutes:

B∗(Y ′)

f ′∗

−−−−→ B∗(X′)

g!

y

yg′

!

B∗(Y ) −−−−→f !

B∗(X),

(2-v) the pullback homomorphismsf∗ : B∗(Y ) → B∗(Y ) for anymorphismsf : X →

Y and the Gysin (pushforward) homomorphismsf! : B∗(X) → B∗(Y ) for confined andspecialized morphisms inC ∩ S are both compatible with the exterior products

× : B∗(X) ⊗ B

∗(Y ) → B∗(X × Y ).

3. A “UNIVERSAL” BIVARIANT THEORY

The following theorem is aboutthe existence of a universal oneof the bivariant theoriesfor a given categoryV with a classC of confined morphisms, a class of independent squaresand a classS of specialized morphisms. The construction is motivated bythat of Levine–Morel’s algebraic cobordism.

Theorem 3.1. LetV be a cateogry with a classC of confined morphisms, a class of inde-pendent squares and a classS of specialized maps. We define

MCS(X

f−→ Y )

to be the free abelian group generated by the set of isomorphism classes of confined mor-phismsh : W → X such that the composite ofh andf is a specialized map:

h ∈ C and f ◦ h : W → Y ∈ S.

(1) The associationMCS is a bivariant theory if the three operations are defined as

follows:Product operations: For morphismsf : X → Y andg : Y → Z, the product operation

• : MCS(X

f−→ Y ) ⊗ M

CS(Y

g−→ Z) → M

CS(X

gf−→ Z)

is defined by(

∑

V

mV [VhV−−→ X]

)

•

(

∑

W

nW [WkW−−→ Y ]

)

:=∑

V,W

mV nW [V ′ hV ◦kW′′

−−−−−−→ X],


where we consider the following fiber squares

V ′ h′

V−−−−→ X′ f ′

−−−−→ W

kW′′

ykW

′

ykW

y

V −−−−→hV

X −−−−→f

Y −−−−→g

Z.


f∗ : MCS(X

gf−→ Z) → M

CS(Y

g−→ Z)

is defined by

f∗

(

∑

V

nV [VhV−−→ X]

)

:=∑

V

nV [Vf◦hV

−−−→ Y ].

Pullback operations: For an independent square

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y,


g∗ : MCS(X

f−→ Y ) → M

CS(X′ f ′

−→ Y ′)

is defined by

g∗

(

∑

V

nV [VhV−−→ X]

)

:=∑

V

nV [V ′ hV′

−−→ X′],

where we consider the following fiber squares:

V ′ g′′

−−−−→ V

hV′

y

yhV

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y.

(2) LetBT be a class of bivariant theoriesB on the same categoryV with a classCof confined morphisms, a class of independent squares and a classS of specialized maps.Let S be nicecanonicalB-orientable for any bivariant theoryB ∈ BT . Then, for eachbivariant theoryB ∈ BT there exists a unique Grothendieck transformation

γB : MCS → B

such that for a specialized morphismf : X → Y ∈ S the homomorphismγB : MCS(X

f−→

Y ) → B(Xf−→ Y ) satisfies the normalization condition that

γB([XidX−−→ X]) = θB(f).

10 SHOJI YOKURA(∗)

Proof. For (1), we have to show that the three bivariant operations are well-defined, butwe show only the well-definedness of the bivariant product and the other two are clear.

Let [VhV−−→ X] ∈ MC

S(Xf−→ Y ) and[W

kW−−→ Y ] ∈ MCS(Y

g−→ Z); thushV : V → X

is confined and the compositef ◦ hV : V → Y is inS, and alsokW : W → Y is confinedand the compositeg ◦ kW : W → Z is in S. By definition we have

[VhV−−→ X] • [W

kW−−→ Y ] = [V ′ hV ◦kW′′

−−−−−−→ X].

We want to show that[V ′ hV ◦kW′′

−−−−−−→ X] ∈ MCS(X

g◦f−−→ Z), i.e.,

(g ◦ f) ◦(

hV ◦ kW′′)

∈ S.

From the fiber squares given in Product operationsabove, we have

(g ◦ f) ◦(

hV ◦ kW′′)

= (g ◦ kW ) ◦(

f ′ ◦ hV′)

.

f ′ ◦ h′V is in S, because it is the pullback off ◦ hV andf ◦ hV is in S andS is closed

under base change by hypothesis.g ◦ kW is in S by hypothesis. Thus the composite(g ◦ kW ) ◦ (f ′ ◦ h′

V ) is also inS. Thus the bivariant product is well-defined.The rest is to show that these three operations satisfy the seven axioms (B-1) — (B-7),

which is left for the reader.For (2), first we show the uniqueness. Suppose that there exists a Grothendieck trans-

formation

γ : MCS(X

f−→ Y ) → B(X

f−→ Y )

such that for anyf : X → Y ∈ S the homomorphismγ : MCS(X

f−→ Y ) → B(X

f−→ Y )

satisfies thatγ([XidX−−→ X]) = θB(f). Note that for anyf : X → Y ∈ S, [X

idX−−→ X] ∈

MCS(X

f−→ Y ) is a nice canonical orientation, i.e.,θMC

S(f) = [X

idX−−→ X].

Let hV : V → X be a confined map such thatf ◦ hV : V → Y is in S. We have that

[VhV−−→ X] = hV ∗[V

idV−−→ V ], where[VidV−−→ V ] ∈ MC

S(Vf◦hV

−−−→ Y ). Sincef ◦ hV ∈ Sby hypothesis, it follows from the normalization that we get

γ([VhV−−→ X]) = γ(hV ∗[V

idV−−→ V ])

= hV ∗γ([VidV

−−→ V ])

= hV ∗θB(f ◦ hV ).

Thus it is uniquely determined.The rest is to show that the assignment

γ : MCS(X

f−→ Y ) → B(X

f−→ Y )

defined byγB([VhV−−→ X]) = hV ∗θB(f ◦ hV ) is a Grothendeick transformation, i.e., that

it preserves the three bivariant operations:(i) it preserves the product operation: Letting the situation be as in (i), it suffices to show

that

γB

(

[VhV−−→ X] • [W

kW−−→ Y ])

= γB([VhV−−→ X]) • γB([W

kW−−→ Y ]).


Using the fiber squares given in Product operations, we have

γB

(

[VhV−−→ X] • [W

kW−−→ Y ])

= γB([V ′ hV ◦kW′′

−−−−−−→ X]) (by the definition)

=(

hV ◦ kW′′)

∗θB(g ◦ f ◦ hV ◦ kW

′′) (by the definition)

= hV ∗kW′′∗θB(g ◦ kW ◦ f ′ ◦ h′

V )

= hV ∗kW′′∗

(

θB(f ′ ◦ hV′) • θB(g ◦ kW )

)

= hV ∗kW′′∗ (kW

⋆θB((f ◦ hV )) • θB(g ◦ kW ))

= hV ∗ (θB(f ◦ hV ) • kW ∗θB(g ◦ kW )) (by (B-7) projection formula)

= hV ∗θB(f ◦ hV ) • kW ∗θB(g ◦ kW ) (by (B-4))

= γB([VhV−−→ X]) • γB([W

kW−−→ Y ]).

(ii) it preserves the pushforward operation: ConsiderXf−→ Y

g−→ Z and a confined

morphsimhV : V → X such that the compositeg ◦ f ◦ hV : V → Y is in S.

γB(f∗[VhV−−→ X]) = γB([V

f◦hV

−−−→ Y ])

= (f ◦ hV )⋆θB(g ◦ (f ◦ hV ))

= f∗hV ⋆θB((g ◦ f) ◦ hV )

= f∗γB([VhV−−→ X])

(iii) it preserves the pullback operation: Consider a confined morphsimhV : V →X such that the compositef ◦ hV : V → Y is in S and the fiber squares given inPullback operationsabove, we have

γB(g∗[VhV−−→ X]) = γB([V ′ hV

′

−−→ X′])

= hV′∗θB(f ′ ◦ hV

′)

= hV′∗g

∗θB(f ◦ hV )

= g∗hV ∗θB(f ◦ hV ) (by (B-5))

= g∗γB([VhV−−→ X]).

This completes the proof of the theorem. �

Remark 3.2. (1) By the definition ofMCS , any classS is nice canonicalMC

S -orientable.

(2) The product operation• : MCS(X

f−→ Y ) ⊗ MC

S(Yg−→ Z) → MC

S(Xgf−→ Z)

can also be interpreted as follows. The free abelian groupM(X) generated by the set ofisomorphism classes of confined morphismshV : V → X is a commutative ring by thefiber product

[V1h1−→ X] ∪ [V2

h2−→ X] := [V1 ×X V2h1×Xh2−−−−−→ X].

For a confined morphismf : X → Y we have the pushforward homomorphismf∗ :M(X) → M(Y ) and for anymorphismf : X → Y we have the pullback homomorphismf∗ : M(Y ) → M(X). Then the product operation is nothing but

[VhV−−→ X] • [W

hW−−→ Y ] = [VhV−−→ X] ∪ f∗([W

hW−−→ Y ]).

But in our case we need to chase the morphisms involved, so we just stick to our previouspresentation.


Let S be a class of specialized morphisms as above and letS be canonicalB-orientablefor a bivariant theoryB. If πX : X → pt is inS, in which case we sometimes say, abusingwords, thatX is specialized, then we have the Gysin homomorphism

πX! : B∗(pt) → B∗(X)

which, we recall, is defined to be

πX!(α) = θB(πX) • α.

In particular, if we let1pt ∈ B(pt) be the unit, then we have

πX!(1pt) = θB(πX) • 1pt = θB(πX).

This elementπX!(1pt) = θB(πX) is calledthe fundamental “class” ofX associated to the

bivariant theoryB (cf. [LM3, Mer]), denoted by[X]B.

Corollary 3.3. LetBT be a class of additive bivariant theoriesB on the same categoryV with a classC of confined morphisms, a class of independent squares and a classS ofspecialized maps. LetS be nicecanonicalB-orientable for any bivariant theoryB ∈ BT .Then, for each bivariant theoryB ∈ BT ,

(1) there exists a unique natural transformation

ΓB∗ : MCS∗ → B∗

such that ifπX : X → pt is in S the homomorphismΓB∗ : MCS∗(X) → B∗(X) satisfies

thatΓB∗[X

idX−−→ X] = πX!(1pt) = [X]B,

and(2) there exists a unique natural transformation

ΓB∗ : M

CS

∗→ B

∗

such that for anyX the homomorphismΓB∗ : MC

S∗(X) → B∗(X) satisfies that

ΓB∗[X

idX−−→ X] = 1X ∈ B∗(X).

Example 3.4. Here we recall some important examples of bivariant theories from [FM].

(1) Bivariant homology theoryH: Let V be the category of complex varieties (embed-dable into smooth manifolds),C = Prop be the class of proper morphisms and letS = Lci

be the class of local complete intersection (abbr.,ℓ.c.i.) morphisms. A Tor-independentsquare is an independent square. Then there is a unique Grothendieck transformation

γH : MPropLci → H

such that forf : X → Y ∈ Lci the homomorphismγH : MPropLci (X

f−→ Y ) → H(X

f−→ Y )

satisfies the normalization condition that

γH([XidX−−→ X]) = Uf .

In particular, we have unique natural transformation:

γH∗ : MPropLci ∗ → H∗ = HBM

∗

such that for anyℓ.c.i. varietyX, γH∗([XidX−−→ X]) = [X] ∈ HBM

∗ (X) and

γH∗ : M

PropLci

∗→ H

∗ = H∗

such that for anyvariety X, γH∗([X

idX−−→ X]) = 1X ∈ H∗(X). HereHBM∗ (X) is the

Borel–Moore homology group andH∗(X) is the usual cohomology group.

(2) Bivariant Chow group theory (or Operational bivariant Chow group theory)A: LetV be the category of schemes,C = Prop be the class of proper morphisms and letS = Lci


be the class of local complete intersection (abbr.,ℓ.c.i.) morphisms. A Tor-independentsquare is an independent square. Then there is a unique Grothendieck transformation

γA : MPropLci → A

such that forf : X → Y ∈ Lci, γA([XidX−−→ X]) = [f ].

We have unique natural transformations

γA∗ : MPropLci ∗ → A∗ = A∗ (or CH∗)

such that for anyℓ.c.i. varietyX, γA∗([XidX−−→ X]) = [X] ∈ A∗(X) and

γA∗ : M

PropLci

∗→ A

∗ = A∗ (or CH∗)

such that for anyvarietyX, γA∗([X

idX−−→ X]) = 1X ∈ A∗(X). HereA∗ = CH∗ is theChow homology group andA∗ = CH∗ is the Chow cohomology group (see [Fu]).

(3) Bivariant algebraicK-theoryKalg: Let the situation be as in (2). Then there is aunique Grothendieck transformation

γKalg : MPropLci → Kalg

such that forf : X → Y ∈ Lci, γKalg ([XidX−−→ X]) = Of .


γKalg∗: M

PropLci ∗ → Kalg∗ = K0

such that for anyℓ.c.i. varietyX, γKalg∗([X

idX−−→ X]) = [OX ] ∈ K0(X) and

γKalg

∗ : MPropLci

∗→ K

∗alg = K0

such that for anyvarietyX, γKalg∗([X

idX−−→ X]) = 1X ∈ K0(X).

(4) Bivariant topologicalK-theoryKtop: Let the situation be as in (2) withV being thecategory of complex varieties. Then there is a unique Grothendieck transformation

γKtop : MPropLci → Ktop

such that forf : X → Y ∈ Lci, γKtop([XidX−−→ X]) = Λf .

We have unique natural transformation

γKtop∗: M

PropLci ∗ → Ktop∗ = K0

such that for anyℓ.c.i. varietyX, γKtop∗([X

idX−−→ X]) = [1X ] ∈ K0(X) and

γKtop

∗ : MPropLci

∗→ Ktop

∗ = K0

such that for anyvarietyX, γKtop∗([X

idX−−→ X]) = 1X ∈ K0(X).

(5) Bivariant theory of constructible functionsF: Let V be the category of complex al-gebraic varieties,C = Prop be the class of proper morphisms and letS = Sm be the classof smooth morphisms (not local complete intersection morphisms). A fiber square is anindependent square. Then there is a unique Grothendieck transformation

γF : MPropSm → F

such that forf : X → Y ∈ Sm, γF([XidX−−→ X]) = 1f = 11X .


γF∗ : MPropLci ∗

→ F∗ = F


such that for any smooth varietyX, γF∗([XidX−−→ X]) = 11X ∈ F (X) and

γF∗ : M

PropLci

∗→ F

∗

such that for anyvarietyX, γF∗([X

idX−−→ X]) = 1X ∈ F ∗(X). HereF ∗(X) is the abeliangroup of locally constant functions onX.

Corollary 3.5. (A naıve “motivic” bivariant characteristic class) Letcℓ : K0 → H∗( )⊗R be a multiplicative characteristic class of complex vectorbundles with a suitable coeffi-cientsR. Then there exists a unique Grothendieck transformation

γcℓH : M

PropLci → H( ) ⊗ R

satisfying the normalization condition that forf : X → Y ∈ Lci

γcℓH ([X

idX−−→ X]) = cℓ(Tf ) • Uf .

Corollary 3.6. ( A naıve “motivic” characteristic class of singular varieties) Let cℓ :K0 → H∗( )⊗R be a multiplicative characteristic class of complex vectorbundles witha suitable coefficientsR. Then there exists a unique natural transformation

cℓ∗ : MPropLci ∗ → H∗( ) ⊗ R

such that for aℓ.c.i varietyV in a smooth variety

cℓ∗([VidV−−→ V ]) = cℓ(TV ) ∩ [V ].

A further discussion on “motivic” bivariant characteristic classes will be done in [Yo2]

Remark 3.7. (Riemann–Roch Theorems) We have the following commutativediagrams:

MPropLci

γKalg

{{xxxxxxxx γKtop

##GGGGGGGGM

PropLci

γKtop

{{wwwwwwwwγtd

H

""EEEEE

EEE

Kalg α// Ktop Ktop

ch// HQ

(i) α : Kalg → Ktop is a Grothendieck transformation such that for aℓ.c.i morphismf : X → Y , α(Of) = Λf .

(ii) ch : Ktop → HQ is a Grothendieck transformation such that for anℓ.c.i morphismf : X → Y , α(Λf) = td(Tf) • Uf , whereTf is the virtual relative tangent bundle ofthe morphismf and td(Tf) is the total Todd cohomology class of the bundleTf . Thecompositech ◦ α : Kalg → HQ is a bivariant version of Baum–Fulton–MacPherson’sRiemann–Rochτ : K0 → H∗Q (see [BFM1, Fu]). Thus one could say thatch ◦ α :

Kalg → HQ andch : Ktop → HQ are realizations of a “motivic” one:γtdH : M

PropLci → HQ.

4. ORIENTED BIVARIANT THEORIES

Here we introduce an orientation to bivariant theories.

Definition 4.1. Let B be a bivariant theory on a category. LetL be a class of morphismsin V, called “line bundles” (e.g., line bundles in geometry), which are closed under basechange. As in the theory of bundles, for a line bundleL → X, we simply denote it by thesource objectL, unless some confusion is possible. For a line bundleL ∈ L, the “firstChern class operator” onB associated to the “line bundle”L, denoted byc1(L), is anendomorphism

c1(L) : B(Xf−→ Y ) → B(X

f−→ Y )

which satisfies the following properties:


(O-1) identity: If two line bundlesL → X andM → X are isomorphic, i.e., thereexists an isomorphismL ∼= M overX, then we have

c1(L) = c1(M) : B(Xf−→ Y ) → B(X

f−→ Y ).

(O-2) commutativity: Let L → X andL′ → X be two line bundles overX, then wehave

c1(L) ◦ c1(L′) = c1(L

′) ◦ c1(L) : B(Xf−→ Y ) → B(X

f−→ Y ).

(O-3) compatibility with product: For morphismsf : X → Y andg : Y → Z, α ∈

B(Xf−→ Y ) andβ ∈ B(Y

g−→ Z), a line bundleL → X and a line bundleM → Y

c1(L)(α • β) = c1(L)(α) • β, c1(f∗M)(α • β) = α • c1(M)(β)

(O-4) compatibility with pushforward: With f being confined

f∗ (c1(f∗M)(α)) = c1(M)(f∗α).

(O-5) compatibility with pullback: For an independent square

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y

g∗ (c1(L)(α)) = c1(g′∗L)(g∗α).

The above first Chern class operator is called an “orientation” and a bivariant theoryequipped with such a first Chern class operator is called anoriented bivariant theory,denoted byOB. An oriented Gorthendieck transformationbetween two oriented bivarianttheories is a Grothendieck transformation which preservesor is compatible with the firstChern class operator.

Remark 4.2. In the above definition, the only requirement on the classL is that morphismsare closed under base change, and nothing else.

The following lemma shows that Levine–Morel’soriented Borel–Moore functor withproductsis a special case of an oriented bivariant theory.

Lemma 4.3. Let OB be an oriented bivariant theory. Then the orientationc1(L) on thefunctorsOB∗ andOB

∗ satisfies the following properties:(1) LetL → X andL′ → X be two line bundles overX, then we have

c1(L) ◦ c1(L′) = c1(L

′) ◦ c1(L) : OB∗(X) → OB∗(X),

c1(L) ◦ c1(L′) = c1(L

′) ◦ c1(L) : OB∗(X) → OB

∗(X),

and ifL andL′ are isomorphic, then we have thatc1(L) = c1(L′) for bothOB∗ andOB

∗.(2) For a line bundleL → X andα ∈ OB∗(X) andβ ∈ OB∗(Y ), we have

c1(L)(α) × β = c1(p1∗L)(α × β).

Also, forα ∈ OB∗(X) andβ ∈ OB

∗(Y ), we have

c1(L)(α) × β = c1(p1∗L)(α × β).

Herep1 : X × Y → X is the projection.(3) For a confined morphismf : X → Y for a line bundleL → Y , we have

f∗ ◦ c1(f∗M) = c1(M) ◦ f∗ : OB∗(X) → OB∗(Y ).

(4) For a specialized morphismf : X → Y ∈ S (here we just require thatf iscanonicalOB-orientable) and a for line bundleM → Y , we have

c1(f∗M) ◦ f ! = f ! ◦ c1(M) : OB∗(Y ) → OB∗(X).


(5) For a confined and specialized morphismf : X → Y and a line bundleM → Y ,we have

f! ◦ c1(f∗M) = c1(M) ◦ f! : OB

∗(X) → OB∗(Y ).

(6) For any morphismf : X → Y and a line bundleM → Y , we have

c1(f∗M) ◦ f∗ = f∗ ◦ c1(M) : OB

∗(Y ) → OB∗(X).

Proof. (1) follows from (O-1) and (O-2).(2) follows from the first formula of (O-3).(3) follows from (O-4).(4) follows from the second formula of (O-3).(5) follows from the first formula of (O-3) and (O-4).(6) follows from (O-5).

�

Example 4.4. All the examples, except the bivariant constructible functionsF, given inExample (3.4) are in fact oriented bivariant theories. The Chern class operator is by tak-ing the bivariant product with a bivariant element in the bivariant group of the identity

XidX−−→ X. And it follows from the axioms of the bivariant theory that this operator satis-

fies the properties (O-2) — (O-5).

(1) Bivariant homology theoryH: For a line bundleL → X, the first Chern class oper-ator

c1(L) : H(Xf−→ Y ) → H(X

f−→ Y )

is defined byc1(L)(α) := c1(L) • α

wherec1(L) ∈ H(XidX−−→ X) = H∗(X) is the first Chern cohomology class of the line

bundle.

(2) Bivariant Chow group theory (or Operational bivariant Chow group theory)A: Asin (1) above, for a line bundleL → X, the Chern class operator

c1(L) : A(Xf−→ Y ) → A(X

f−→ Y )

is defined byc1(L)(α) := c1(L) • α

wherec1(L) ∈ A(XidX−−→ X) is the first Chern cohomology class of the line bundle.

(3) Bivariant algebraicK-theoryKalg: For a line bundleL → X, the Chern class oper-ator

c1(L) : Kalg(Xf−→ Y ) → Kalg(X

f−→ Y )

is defined byc1(L)(α) := (1 − [L]) • α

where[L] ∈ Kalg(XidX−−→ X) = K0

alg(X) the Grothendieck group of algebriac vectorbundles or locally free sheaves onX.

(4) Bivariant topologicalK-theoryKtop: For a line bundleL → X, the Chern classoperator

c1(L) : Ktop(Xf−→ Y ) → Ktop(X

f−→ Y )

is defined byc1(L)(α) := (1 − [L]) • α


where[L] ∈ Ktop(XidX−−→ X) = K0

top(X) the Grothendieck group of vector bundles onX.

(5) As to the bivariant theoryF, as far as the author knows, there is not available a bi-

variant classc(E) ∈ F∗(X) = F(XidX−−→ X) associated to a vector bundleE over X,

which is just the multiplication by locally constant functions. Even if such a Chern classis available, we do not know whatc(E) would be. So, it is not clear whether there is areasonable Chern class operator in the case of the bivarianttheoryF.

In fact, following Levine–Morel’s construction [LM3], we show the existence of a uni-versal one of such oriented bivariant theories for anycategory.

Let us consider a morphismhV : V → X equipped withfinitely many line bundlesover the source varietyV of the morphismhV :

(VhV−−→ X; L1, L2, · · · , Lr)

with Li being a line bundle overV . This family is called acobordism cycle overX [LM3,

Mer]. Then(VhV−−→ X; L1, L2, · · · , Lr′) is isomorphic to(W

hW−−→ X; M1, M2, · · · , Mr)if and only if hV andhW are isomorphic, i.e., there is an isomorphismg : V ∼= W overX, there is a bijectionσ : {1, 2, · · · , r} ∼= {1, 2, · · · , r′} (so thatr = r′) and there areisomorphismsLi

∼= g∗Mσ(i) for everyi.

Theorem 4.5. (A universal oriented bivariant theory) LetV be a cateogry with a classCof confined morphisms, a class of independent squares, a classS of specialized morphismsand a classL of line bundles. We define

OMCS(X

f−→ Y )

to be the free abelian group generated by the set of isomorphism classes of cobordismcycles overX

[Vh−→ X; L1, L2, · · · , Lr]

such that the composite ofh andf

f ◦ h : W → Y ∈ S.

(1) The associationOMCS becomes an oriented bivariant theory if the four operations

are defined as follows:Orientation: For a morphismf : X → Y and a line bundleL → X, the first Chern

class operator

c1(L) : OMCS(X

f−→ Y ) → OM

CS(X

f−→ Y )

is defined by

c1(L)([VhV−−→ X; L1, L2, · · · , Lr ]) := [V

hV−−→ X; L1, L2, · · · , Lr, (hV )∗L].

Product operations: For morphismsf : X → Y andg : Y → Z, the product operation

• : OMCS(X

f−→ Y ) ⊗ OM

CS(Y

g−→ Z) → OM

CS(X

gf−→ Z)

is defined as follows: The product on generators is defined by

[VhV−−→ X; L1, · · · , Lr] • [W

kW−−→ Y ; M1, · · · , Ms]

:= [V ′ hV ◦kW′′

−−−−−−→ X; kW′′∗L1, · · · , kW

′′∗Lr, (f′ ◦ hV

′)∗M1, · · · , (f ′ ◦ hV′)∗Ms],


and it extends bilinearly. Here we consider the following fiber squares

V ′ h′

V−−−−→ X′ f ′

−−−−→ W

kW′′

ykW

′

ykW

y

V −−−−→hV

X −−−−→f

Y −−−−→g

Z.


f∗ : OMCS(X

gf−→ Z) → OM

CS(Y

g−→ Z)

is defined by

f∗

(

∑

V

nV [VhV−−→ X; L1, · · · , Lr ]

)

:=∑

V

nV [Vf◦hV

−−−→ Y ; L1, · · · , Lr].

Pullback operations: For an independent square

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y,


g∗ : OMCS(X

f−→ Y ) → OM

CS(X′ f ′

−→ Y ′)

is defined by

g∗

(

∑

V

nV [VhV−−→ X; L1, · · · , Lr]

)

:=∑

V

nV [V ′ hV′

−−→ X′; g′′∗L1, · · · , g′′

∗Lr ],

where we consider the following fiber squares:

V ′ g′′

−−−−→ V

h′

V

y

yhV

X′ g′

−−−−→ X

f ′

y

y

f

Y ′ −−−−→g

Y.

(2) LetOBT be a class of oriented bivariant theoriesOB on the same categoryV witha classC of confined morphisms, a class of independent squares, a classS of specializedmorphisms and a classL of line bundles. LetS be nicecanonicalOB-orientable forany oriented bivariant theoryOB ∈ OBT . Then, for each oriented bivariant theoryOB ∈ OBT there exists a unique oriented Grothendieck transformation

γOB : OMCS → OB

such that for anyf : X → Y ∈ S the homomorphismγOB : OMCS(X

f−→ Y ) → OB(X

f−→

Y ) satisfies the normalization condition that

γOB([XidX−−→ X; L1, · · · , Lr]) = c1(L1) ◦ · · · ◦ c1(Lr)(θOB(f)).


Proof. (1): It is easy to see that the above four operations are well-defined. Here we notethat the above Chern class operator is nothing but the bivariant product with the “motivic”

first Chern class ofL, [XidX−−→ X; L] ∈ OM

CS(X

idX−−→ X), i.e., we have

c1(L)([VhV−−→ X; L1, L2, · · · , Lr]) := [X

idX−−→ X; L] • [VhV−−→ X; L1, L2, · · · , Lr].

(2): Suppose that there is an oriented Grothendieck transformation

γ : OMCS → OB

satisfying that for anyf : X → Y ∈ S the homomorphismγ : OMCS(X

f−→ Y ) →

OB(Xf−→ Y ) satisfies thatγ([X

idX−−→ X]) = θOB(f). It suffices to show that the value ofany generator

[VhV−−→ X; L1, · · · , Lr] ∈ OM

CS(X

f−→ Y )

is uniquely determined. For that, first we notice the following:

[VhV−−→ X; L1, · · · , Lr] = hV ∗[V

idV−−→ V ; L1, · · · , Lr]

= hV ∗

(

c1(L1) ◦ · · · ◦ c1(Lr)([VidV−−→ V ])

)

.

Since[VidV−−→ V ] ∈ OM

CS(V

f◦hV

−−−→ Y ) andf ◦ hV ∈ S by hypothesis, we have

γ([VhV−−→ X; L1, · · · , Lr]) = γ

(

hV ∗

(

c1(L1) ◦ · · · ◦ c1(Lr)([VidV−−→ V ])

))

= hV ∗

(

c1(L1) ◦ · · · ◦ c1(Lr)γ([VidV−−→ V ])

)

= hV ∗ (c1(L1) ◦ · · · ◦ c1(Lr))(θOB(f ◦ hV ))

Thus the uniqueness follows.Next, we show the existence of such an oriented Grothendiecktransformation satisfying

the above normalization condition. We define the association

γOB : OMCS(X

f−→ Y ) → OB(X

f−→ Y )

defined by

γOB([VhV−−→ X; L1, · · · , Lr]) := hV ∗ (c1(L1) ◦ · · · ◦ c1(Lr))(θOB(f ◦ hV )) .

This certainly satisfies the normalization condition.The rest is to show that it is an oriented Grothendieck transformation.(i) it preserves the product operation:It suffices to show that

γOB

(

[VhV−−→ X; L1, · · · , Lr ] • [W

kW−−→ Y ; M1, · · · , Ms])

= γOB([VhV−−→ X; L1, · · · , Lr]) • γOB([W

kW−−→ Y ; M1, · · · , Ms])

Using some parts of the proof of Theorem (3.1), we have

γOB

(

[VhV−−→ X; L1, · · · , Lr] • [W

kW−−→ Y ; M1, · · · , Ms])

= γOB

(

[V ′ hV ◦kW′′

−−−−−−→ X; kW′′∗L1, · · · , kW

′′∗Lr, (f′ ◦ hV

′)∗M1, · · · , (f ′ ◦ hV′)∗Ms]

)

= hV ∗kW′′∗

(

c1(kW′′∗L1) ◦ · · · ◦ c1(kW

′′∗Lr)◦

c1((f′ ◦ hV

′)∗M1) ◦ · · · ◦ c1((f′ ◦ hV

′)∗Ms))

(θOB(f ′ ◦ hV′) • θOB(g ◦ kW ).

Here we use the property (O-4) compatibility with pushforwardand (O-3) compatibilitywith product[c1(f

∗M)(α•β) = α• c1(M)(β)], the above equality continues as follows:


= hV ∗

(

c1(L1) ◦ · · · ◦ c1(Lr) ◦ kW′′∗

(

c1((f′ ◦ hV

′)∗M1) ◦ · · · ◦ c1((f′ ◦ hV

′)∗Ms))

(θOB(f ′ ◦ hV′) • θOB(g ◦ kW )

)

.

= hV ∗ (c1(L1) ◦ · · · ◦ c1(Lr)) ◦ (kW′′)∗

(

θOB(f ′ ◦ hV′) • (c1(M1) ◦ · · · ◦ c1(Ms)) θOB(g ◦ kW )

)

.

= hV ∗

(

c1(L1) ◦ · · · ◦ c1(Lr) ◦ kW′′∗

((kW )∗θOB(f ◦ hV ) • (c1(M1) ◦ · · · ◦ c1(Ms)) θOB(g ◦ kW ))

.

= hV ∗ (c1(L1) ◦ · · · ◦ c1(Lr))

(θOB(f ◦ hV ) • kW ∗ (c1(M1) ◦ · · · ◦ c1(Ms)) θOB(g ◦ kW ))

(by (B-7) projection formula).

Furthermore, using (O-3) compatibility with product[c1(L)(α • β) = c1(L)(α) • β ]and by (B-4), it continues as follows:

= hV ∗ (c1(L1) ◦ · · · ◦ c1(Lr)) θOB(f ◦ hV )•

kW ∗ (c1(M1) ◦ · · · ◦ c1(Ms)) θOB(g ◦ kW )

= γOB([VhV−−→ X; L1, · · · , Lr]) • γOB([W

kW−−→ Y ; M1, · · · , Ms])

(ii) it preserves the pushforward operation:ConsiderXf−→ Y

g−→ Z and a confined

morphsimhV : V → X such that the composite(g ◦ f) ◦ hV : V → Y is in S. Fora generator

[VhV−−→ X; L1, · · · , Lr] ∈ OM

CS(X

g◦f−−→ Z),

we have

γOB(f∗[VhV−−→ X; L1, · · · , Lr])

= γOB([Vf◦hV

−−−→ Y ; L1, · · · , Lr])

= (f ◦ hV )∗ (c1(L1) ◦ · · · ◦ c1(Lr)) (θOB(g ◦ (f ◦ hV )))

= f∗hV ∗ (c1(L1) ◦ · · · ◦ c1(Lr)) θOB((g ◦ f) ◦ hV )

= f∗γOB([VhV−−→ X; L1, · · · , Lr]).

(iii) it preserves the pullback operation: Consider a confined morphsimhV : V →X such that the compositef ◦ hV : V → Y is in S and the fiber squares given inPullback operationsabove, we have

γOB(g∗[VhV−−→ X; L1, · · · , Lr])

= γOB([V ′ h′

V−−→ X′; g′′∗L1, · · · , g′′

∗Lr ])

= hV′∗

(

c1(g′′∗L1) ◦ · · · ◦ c1(g

′′∗Lr))

θOB(f ′ ◦ hV′)

= hV′∗

(

c1(g′′∗L1) ◦ · · · ◦ c1(g

′′∗Lr))

g∗θOB(f ◦ hV )

= hV′∗g

∗ (c1(L1) ◦ · · · ◦ c1(Lr)) θOB(f ◦ hV )

= g∗hV ∗ (c1(L1) ◦ · · · ◦ c1(Lr)) θOB(f ◦ hV ) (by (B-5))

= g∗γOB([VhV−−→ X; L1, · · · , Lr]).

�


Corollary 4.6. The abelian groupOMCS∗(X) := OM

CS(X → pt) is the free abelian group

generated by the set of isomorphism classes of cobordism cycles

[VhV−−→ X; L1, · · · , Lr]

such thathV : V → X ∈ C andV → pt is a specialized map inS andLi is a line bundle

over V . The abelian groupOMCS

∗(X) := OM

CS(X

idX−−→ X) is the free abelian groupgenerated by the set of isomorphism classes of cobordism cycles

[VhV−−→ X; L1, · · · , Lr]

such thathV : V → X ∈ C ∩ S andLi is a line bundle overV . Both functorOMCS∗ and

OMCS

∗are oriented Borel–Moore functors with products in the sense of Levine–Morel.

Corollary 4.7. (A universal oriented Borel–Moore functor with products) Let BT be aclass of oriented additive bivariant theoriesB on the same categoryV with a classC ofconfined morphisms, a class of independent squares, a classS of specialized maps and aclass of line bundles. LetS be nicecanonicalOB-orientable for any oriented bivarianttheoryOB ∈ OBT . Then, for each oriented bivariant theoryOB ∈ OBT ,

(1) there exists a unique natural transformation of oriented Borel–Moore functors withproducts

ΓOB∗ : OMCS∗ → OB∗

such that ifπX : X → pt is inS

ΓOB∗[XidX−−→ X; L1, · · · , Lr] = c1(L) ◦ · · · ◦ c1(Lr)(πX

!(1pt)),

and(2) there exists a unique natural transformation of oriented Borel–Moore functors with

products

ΓOB∗ : OM

CS

∗→ OB

∗

such that for any objectX

ΓOB∗[X

idX−−→ X; L1, · · · , Lr] = c1(L) ◦ · · · ◦ c1(Lr)(1X).

Remark 4.8. (1) Let k be an arbitrary field. In the case whenVk is the admissible sub-category of the category of separated schemes of finite type over the fieldk, C = Proj

is the class of projective morphisms,S = Sm is the class of smooth equi-dimensionalmorphisms andL is the class of line bundles, thenOM

ProjSm ∗(X) = OM

ProjSm (X → pt)

is nothing but the oriented Borel–Moore functor with productsZ∗(X) given in [LM3]. In

this sense, our associated contravariant oneOMProjSm

∗(X) = OM

ProjSm (X

idX−−→ X) is a“cohomological” counterpart of Levine–Morel’s “homological one”Z∗(X). Note that this

cohomological oneOMProjSm

∗(X) for any schemeX is the free abelian group generated

by [VhV−−→ X; L1, · · · , Lr ] such thathV : V → X is a projective and smoothmorphism.

(2) One can see that in (1)Proj can be replaced byProp. And furthermore one canconsiderOM

ProjLci andOM

PropLci , which will be treated in [Yo1]

An oriented bivariant theory can be defined for any kind of category as long as wecan specify classes of “confined morphisms”, “specialized morphisms” together with nicecanonical orientations, “line bundles”, and “independentsquares”. The above orientedbivariant theory is the very basis of other oriented bivariant theories of more geometricnatures. In [Yo1] we will deal with a more geometrical oriented bivariant theory, i.e., whatcould be calledbivariant algebraic cobordismor algebraic bivariant cobordism, which isa bivariant version of Levine–Morel’s algebraic cobordism.


Acknowledgements.The author would like to thank the staff of the Erwin SchrodingerInternational Institute of Mathematical Physics for theirhospitality during his stay in Au-gust, 2006, during which the present research was initiated.

The author gave a talk on a very earlier version of this paper at the Kinosaki AlgebraicGeometry Symposium (Oct. 24 – Oct.27, 2006): The author would like to thank the or-ganizers (H. Kaji, F. Kato, S.-I. Kimura) of the symposium for inviting him to give a talk,and also thank M. Hanamura, S.-I. Kimura, Y. Shimizu, T. Terasoma for their interests onthis subject.

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Oriented Bivariant Theories, I - univie.ac.atesiprpr/esi1911.pdf · as a categorical framework for the study of singular spaces, which is the titleof theirAMS Memoir book [FM] (see