The Chow-Witt ring

The Chow-Witt ring

Jean Fasel

Received: May 18, 2006 Communicated by Ulf Rehmann

Abstract.

^{WedenearingstrutureonthetotalChow-Wittgroup}

of anyintegralsmoothsheme overa eld ofharateristidierent

from

2

^.

2000 Mathematis Subjet Classiation: 14C15, 14C17, 14C99,

14F43

Keywords and Phrases: Chow-Witt groups, Chow groups,

Grothendiek-WittgroupsandWittgroups

Contents

1 Introduction 276

1.1 Conventions . . . 278

2 Preliminaries 278

2.1 Wittgroups . . . 278

2.2 Produts. . . 279

2.3 Supports. . . 281

3 Chow-Witt groups 283

4 The exterior product 290

5 Intersection with a smooth subscheme 296

5.1 TheGysin-Wittmap . . . 296

5.2 Funtoriality . . . 300

6 The ring structure 304

7 Basic properties 307

1 Introduction

Let

A

^{beaommutativenoetherianringofKrulldimension}

n

^and

P

^aprojetive

A

^{-module ofrank}

d

^{. Onean askthefollowingquestion: does}

P

^{admit afree}

fator of rank one? Serre proved a long time ago that the answer is always

positivewhen

d > n

^{. Soinfattherst}interestingaseiswhen

P

^isprojetive

ofrankequaltothedimensionof

A

^{. Supposenowthat}

X

^{isanintegralsmooth}

shemeoveraeld

k

^ofharateristinot

2

^{. Todealwiththeabovequestion,}

Bargeand Morel introdued theChow-Wittgroups

CH g ^j (X)

^of

X

^{(alled at}

thattimegroupesdeChowdesylesorientés ,see[BM℄)andassoiatedtoeah

vetor bundle

E

^{of rank}

n

^{an Euler lass}

˜ c n (E)

ⁱⁿ

CH g ⁿ (X )

^{. It was proved}

reently that if

X =

^Spe

(A)

^{we have}

˜ c n (P ) = 0

^{if and only if}

P ≃ Q ⊕ A

(see[Mo℄ for

n ≥ 4

^{, [FS℄ for}

n = 3

^{and [BM℄ or [Fa℄ for the ase}

n = 2

^{). It}

is therefore important to provide moretools, suh as a ring struture and a

pull-bakforregular embeddings, to omputetheChow-Witt groupsand the

Eulerlasses.

Todene

CH g ^p (X )

^{onsiderthebreprodutoftheomplexinMilnorK-theory}

0 // K _p ^M (k(X )) //

M

x 1 ∈X ⁽¹⁾

K _p−1 ^M (k(x 1 )) // . . . //

M

x n ∈X ⁽ⁿ⁾

K _p−n ^M (k(x n )) // 0

andtheGersten-Wittomplexrestritedto thefundamental ideals

0 // I ^p (k(X )) //

M

x 1 ∈X ⁽¹⁾

I ^p−1 (O X,x 1 ) // . . . //

M

x n ∈X ⁽ⁿ⁾

I ^p−n (O X,x n ) // 0

overthequotientomplex

0 // I ^p /I ^p+1 (k(X )) // . . . //

M

x n ∈X ⁽ⁿ⁾

I ^p−n /I ^p+1−n (O X,x n ) // 0.

Thegroup

CH g ^p (X)

^isdenedasthe

p

^{-thohomologygroupofthisbreprod-}

ut. Roughlyspeaking,anelementof

CH g ^p (X)

^{isthelassofasumofvarieties}

ofodimension

p

^{with aquadratiform dened on eah variety. Weoviously}

haveamap

CH g ^p (X) → CH ^p (X)

^.

UsingthefuntorialityofthetwoomplexesweseethattheChow-Wittgroups

satisfy good funtorial properties (see [Fa℄). For example, we have a pull-

bak morphism

f ^∗ : CH g ^j (X) → CH g ^j (Y )

^{assoiated to eah at morphism}

f : Y → X

^{and a} push-forward morphism

g ∗ : CH g ^j (Y, L) → CH g ^j+r (X )

assoiatedtoeahpropermorphism

g : Y → X

^,where

r =

^dim

(X ) −

^dim

(Y )

and

L

^{is a suitableline bundle over}

Y

^{. Using this funtorialbehaviour, it is}

possibletoprodueagoodintersetiontheory. Thisiswhatwedointhispaper

usingthe lassialstrategy(see forexample [Fu℄ or [Ro℄). First wedene an

exteriorprodut

CH g ^j (X) × CH g ⁱ (Y ) → CH g ^i+j (X × Y )

andthenaGysin-likehomomorphism

i ^! : CH g ^d (X) → CH g ^d (Y )

^assoiatedtoa

losedembedding

i : Y → X

^{ofsmoothshemes. Theprodutisthendened}

astheomposition

CH g ^j (X) × CH g ⁱ (X) // CH g ^i+j (X × X) ^△

!

// g CH ^i+j (X)

where

△ : X → X × X

^{is the diagonal embedding. To dene the exterior}

produt, we rst note that Rost already dened an exterior produt on the

homology of the omplex in Milnor K-theory ([Ro℄). Thus it is enough to

deneanexteriorprodutonthehomologyof theGersten-Wittomplexand

showthat bothexteriorprodutsoinideoverthequotientomplex. Weuse

theusualprodutonderivedWittgroups([GN℄)and showthat this produt

passestohomologyusingtheLeibnitz ruleprovedbyBalmer(see[Ba2℄).

The denition of the Gysin-like map is done by following the ideas of Rost

([Ro℄). It usesthedeformationto thenormaloneto modiyanylosed em-

bedding to a nier losed embedding and uses also the long exat sequene

assoiatedtoatriple

(Z, X, U )

^where

Z

^{isalosedsubsetof}

X

^and

U = X \ Z

^.

The produt that we obtain has the meaning of interseting varieties with

quadratiformsdenedonthem. Itisthereforenotasurprisethatthenatural

map

CH g ^tot (X) → CH ^tot (X )

^{turnsoutto bearing}homomorphism. There is howeverasurprise: theprodutthatweobtainisapriorineitherommutative

norantiommutative. This omes from the fat that the produt oftriangu-

latedGrothendiek-Wittgroups

GW ⁱ × GW ^j → GW ^i+j

^{doesnotsatisfyany}

ommutativityproperty.

Theorganizationof this paperis as follows: Insetion 2,wereall someba-

si results on triangular Witt groups. This inludes the onstrution of the

Gersten-Witt omplex, and some results on produts and onsanguinity. In

setion3, weonstrut theChow-Wittgroups, reall someresults and prove

somebasifats. Thedenition oftheexteriorproduttakesplaein setion

4and thedenitionof theGysin-Wittmapin setion5. Inthis part,wealso

prove thefuntorialityof this map. Finallyweput all thepiees togetherin

setion6andprovesomebasiresultsinsetion7.

I would like to thank Paul Balmer, Stefan Gille and Ivo Dell'Ambrogio for

several areful readings of earlier versions of this work. I also would like to

thankthereferee forsomeuseful omments. Thisresearh was supported by

SwissNationalSieneFoundation,grantPP002-112579.

1.1 Conventions

All shemes are smooth and integral over aeld

k

^of harateristi dierent from

2

^,orareloalizationsofsuhshemes. Foranytwoshemes

X

^and

Y

^we

willalwaysdenoteby

X × Y

^thebreprodut

X ×

Spe

(k) Y

^.

2 Preliminaries 2.1 Witt groups

WereallheresomebasifatsonWitt groupsoftriangulatedategories fol-

lowingtheexpositionof[Ba2℄. Wesupposethatforanytriangulatedategory

C

^{and anyobjets}

A, B

^of

C

^thegroupHom

(A, B)

^isuniquely

2

-divisible. We alsosupposethatalltriangulatedategoriesareessentiallysmall.

Definition

^2.1

.

^Let

C

^beatriangulatedategory. A dualityon

C

^{is atriple}

(D, δ, ̟)

^where

δ = ±1

^,

D : C → C

^{is a}

δ

^-exat ontravariant funtor and

̟ : 1 ≃ D ²

^{is an} isomorphismof funtors satisfying

D(̟ A ) ◦ ̟ DA = id DA

and

T (̟ A ) = ̟ T A

^{for all}

A ∈ C

^{. A} triangulatedategory

C

^{with aduality}

(D, δ, ̟)

^iswritten

(C, D, δ, ̟)

^.

Example 2.2 . Let

X

^{bearegularshemeand}

P(X )

^{theategoryofloallyfree}

oherent

O X

^{-modules. Let}

D ^b (P (X))

^bethetriangulatedategoryofbounded omplexesof objetsof

P (X )

^{. Then theusual duality}

^∨

^on

P(X )

^{dened by}

P ^∨ =

^Hom

O X (P, O X )

^induesa

1

^{-exatdualityon}

D ^b (P(X))

^{. Wealsodenote}

thisderiveddualityby

∨

. Moreover,theanonialisomorphism

ev : P → P ^∨∨

for any loally free module

P

^{indues a anonial} isomorphism

̟ : 1 → ^∨∨

in

D ^b (P (X))

^{. More generally, if}

L

^{is anyinvertiblemodule over}

X

^{,then the}

dualityHom

O X (

^_

, L)

^on

P (X )

^{alsoinduesadualityon}

D ^b (P (X))

^.

Definition

^2.3

.

^Let

(C, D, δ, ̟)

^beatriangulatedategorywithduality. For any

i ∈ Z

, dene

(D ⁽ⁱ⁾ , δ ⁽ⁱ⁾ , ̟ ⁽ⁱ⁾ )

^by

D ⁽ⁱ⁾ = T ⁱ ◦ D

^,

δ ⁽ⁱ⁾ = (−1) ⁱ δ

^and

̟ ⁽ⁱ⁾ = δ ⁱ (−1) ^i(i+1)/2 ̟

^{. It iseasy tohekthat}

(D ⁽ⁱ⁾ , δ ⁽ⁱ⁾ , ̟ ⁽ⁱ⁾ )

^isadualityon

C

^{. It}

isalledthe

i ^th

^{-shifteddualityof}

(D, δ, ̟)

^.

Definition

^2.4

.

^Let

(C, D, δ, ̟)

^beatriangulatedategorywithduality,

A ∈ C

and

i ∈ Z

. Amorphism

ϕ : A → D ⁽ⁱ⁾ A

^is

i

^{-symmetriifthefollowingdiagram}

ommutes:

A ^ϕ //

̟ _A ⁽ⁱ⁾

D ⁽ⁱ⁾ A

(D ⁽ⁱ⁾ ) ² (A)

D ⁽ⁱ⁾ ϕ // D ⁽ⁱ⁾ A.

Theouple

(A, ϕ)

^isalledan

i

^{-symmetripair.}

Definition

^2.5

.

^{Wedenote by}

Symm ⁱ (C)

^{the monoid ofisometry lasses of}

i

^{-symmetripairs,equippedwiththeorthogonalsum.}

Definition

^2.6

.

^An

i

^{-symmetri formis an}

i

^{-symmetripair}

(A, ϕ)

^where

ϕ

isanisomorphism.

Theorem

^2.7

.

^Let

(C, D, δ, ̟)

^bea triangulated ategory with duality andlet

(A, φ)

^{be an}

i

^{-symmetripair. Choosean exattriangle ontaining}

φ A ^φ // D ⁽ⁱ⁾ A ^α // C ^β // T A.

Then there exists an

(i + 1)

^-symmetri isomorphism

ψ : C → D ⁽ⁱ⁺¹⁾ C

^suh

thatthe following diagram ommutes

A ^φ //

̟ ⁽ⁱ⁾

D ⁽ⁱ⁾ A ^α // C ^β //

ψ

T A

T ̟ ⁽ⁱ⁾

D ⁽ⁱ⁾ (D ⁽ⁱ⁾ A)

D ⁽ⁱ⁾ φ // D ⁽ⁱ⁾ A

δ ⁽ⁱ⁺¹⁾ D ⁽ⁱ⁺¹⁾ β // D ⁽ⁱ⁺¹⁾ C

D ⁽ⁱ⁺¹⁾ α // T (D ⁽ⁱ⁾ (D ⁽ⁱ⁾ A))

where the rows are exat triangles and the seondone is the dual of the rst.

Moreover, the

(i + 1)

^{-symmetri form}

(C, ψ)

^{is unique up to isometry. It is}

denotedby one

(A, φ)

^.

Proof. See[Ba1℄,Theorem1.6.

Example 2.8 . Let

A ∈ C

^{. Forany}

i

^,themorphism

0 : A → D ⁽ⁱ⁾ A

^issymmetri

andthenone

(A, 0)

^iswelldened.

Corollary

^2.9

.

^{The above onstrution gives awell dened}homomorphism ofmonoids

d ⁱ : Symm ⁽ⁱ⁾ (C) → Symm ⁽ⁱ⁺¹⁾ (C)

^suhthat

d ⁱ⁺¹ d ⁱ = 0

^.

Definition

^2.10

.

^Let

(C, D, δ, ̟)

^beatriangulatedategorywithduality. The Wittgroup

W ⁱ (C)

^isdenedasKer

(d ⁱ )/

^Im

(d ⁱ⁺¹ )

^{. RemarkthatKer}

(d ⁱ )

^isjust

themonoidofisometry lassesof

i

^{-symmetriforms.}

Definition

^2.11

.

^Let

(C, D, δ, ̟)

^beatriangulatedategorywithduality. The Grothendiek-Wittgroup

GW ⁱ (C)

^{isdenedasthequotientofKer}

(d ⁱ )

^bythe

submonoidgeneratedbytheelementsone

(A, φ) −

^one

(A, 0)

^where

A ∈ C

^and

φ

^is

(i − 1)

^-symmetri(

0

^{isalsoseenas an}

(i − 1)

^{-symmetrimorphism).}

Example 2.12 . Let

(D ^b (P (X)), ^∨ , 1, ̟)

^bethetriangulatedategorywithdu- alitydened in Example2.2. Its Witt groupsare theWitt groups

W ⁱ (X)

^of

thesheme

X

^{asdened in[Ba1℄.}

2.2 Products

Given apairing

⊗ : C × D → M

^of triangulatedategorieswith duality and assumingthat this pairingsatises somenie onditions,theauthors of [GN℄

deneapairingofWittgroups. Webrieyreallsomedenitions(see1.2and

1.11in [GN℄):

Definition

^2.13

.

^Let

C, D

^and

M

^be triangulated ategories. A produt between

C

^and

D

^withodomain

M

^{isaovariantbi-funtor}

⊗ : C × D → M

exatin both variables and satisfying the following ondition: the funtorial

isomorphisms

r A,B : A ⊗ T B ≃ T(A ⊗ B)

^and

l A,B : T A ⊗ B ≃ T (A ⊗B )

^make

thediagram

T A ⊗ T B ^{l A,T B} //

r T A,B

T (A ⊗ T B)

T(r A,B )

T (T A ⊗ B)

T(l A,B ) // T ² (A ⊗ B)

skew-ommutative.

Definition

^2.14

.

^Let

C, D

^and

M

^betriangulated ategories with dualities.

Wherethere is nopossibleonfusion, we dropthe subsriptsfor

D, δ

^and

̟

^.

A dualizing pairing between

C

^and

D

^{with odomain}

M

^{is a produt}

⊗

^with

isomorphisms

η A,B : DA ⊗ DB ≃ D(A ⊗ B)

naturalin

A

^and

B

^{whihmakethefollowingdiagramsommute}

1.

A ⊗ B ^{̟ A ⊗̟ B} //

̟ A⊗B

D ² A ⊗ D ² B

η DA,DB

D ² (A ⊗ B)

D(η A,B ) // D(DA ⊗ DB)

2.

T(DT A ⊗ DB)

δ C δ M T(η T A,B )

DA ⊗ DB

l DT A,DB

oo

η A,B

r DA,DT B

// T(DA ⊗ DT B)

δ L δ M T (η A,T B )

T D(T A ⊗ B) D(A ⊗ B)

T D(l A,B )

oo T D(r A,B ) // T D(A ⊗ T B).

Theorem

^2.15

.

^Let

C, D

^and

M

^be triangulated ategories with duality. Let

⊗ : C × D → M

^{be a dualizing pairing between}

C

^and

D

^{with odomain}

M

^.

Then

⊗

^{indues forall}

i, j ∈ Z

apairing

⋆ : W ⁱ (C) × W ^j (D) → W ^i+j (M).

Proof. See[GN℄,Theorem2.9.

Example2.16 . Let

(D ^b (P(X )), ^∨ , 1, ̟)

^bethetriangulatedategorywithdual- itydenedinExample2.2. Theusualtensorprodutinduesadualizingpair-

ingoftriangulatedategoriesandthenaprodut

W ⁱ (X )×W ^j (X ) → W ^i+j (X )

^.

Suppose that

L

^and

N

^{are invertible modules over}

X

^{. Then Hom}

O X (

^_

, L)

^,

Hom

O X (

^_

, N)

^andHom

O X (

^_

, L ⊗ N )

^{givedualities}

^♯

^,

^♮

^and

^♭

^on

D ^b (P (X ))

^.

Thetensorprodutgivesadualizingpairing

⊗ : (D ^b (P (X)), ^♯ , 1, ̟) × (D ^b (P (X)), ^♮ , 1, ̟) → (D ^b (P (X)), ^♭ , 1, ̟).

2.3 Supports

Webriey reall thenotion of triangulated ategorywith supports following

[Ba2℄.

Definition

^2.17

.

^Let

X

^{beatopologialspae. A} triangulatedategoryde- ned over

X

^{is apair}

(C,

^Supp

)

^where

C

^{is a}triangulatedategoryand Supp assigns to eah objet

A ∈ C

^{a losed subset Supp}

(A)

^of

X

^{suh that the}

followingrulesaresatised:

(S1) Supp

(A) = ∅ ⇐⇒ A ≃ 0

^.

(S2) Supp

(A ⊕ B ) =

^Supp

(A) ∪

^Supp

(B)

^.

(S3) Supp

(A) =

^Supp

(T A)

^.

(S4) For everydistinguishedtriangle

A // B // C // T A

wehaveSupp

(C) ⊂

^Supp

(A) ∪

^Supp

(B)

^.

When

I

^isasaturatedtriangulatedsubategoryof

C

^and

S

^isthemultipliative systemofmorphismswhose oneis in

I

^{,then wean onstrutasupport on}

theategory

S ⁻¹ C := C/I

^{. Thisisdonein[Ba3℄when}

C

^{hasatensorprodut.}

Howeverwewillonlyneedsomebasifats,soweprovethefollowinglemma:

Lemma

^2.18

.

^let

C

^{be a} triangulated ategory dened over a topologial spae

X

^{. Let}

I

^beasaturatedtriangulatedsubategory of

C

^andletSupp

(I) =

∪ A∈I

^Supp

(A)

^{. Supposethat Supp}

(A) ⊂

^Supp

(I)

^implies

A ∈ I

^{. Let}

S

^bethe

multipliative systemin

C

^ofmorphisms

f

^{suhthat one}

(f ) ∈ I

^andlet

I // C // C/I

bethe exat sequene of triangulatedategories obtainedby inverting

S

^{. Then}

C/I

^isatriangulatedategorydenedover

X ^′ = X \

^Supp

(I )

^{(withtheindued}

topology).

Proof. WedeneSupp

S (A) :=

^Supp

(A) ∩ X ^′

^foranyobjet

A ∈ C/I

^andshow

thatSupp

S

^{satisesthepropertiesofDenition 2.17. Itiseasytoseethatthe}

rules(S1), (S2)and(S3) aresatised. Weonlyhavetoprove(S4).

Firstobservethat if

s : A → B

^{isamorphismin}

S

^and

A ^s // B // C // T A

isan exattriangle in

C

^ontaining

s

^{, then Supp}

_S (A) =

^Supp

_S (B)

^(use(S4)

fortheategory

C

^{). ThisshowsthatSupp}

_S (A) =

^Supp

_S (A ^′ )

^if

A ≃ A ^′

ⁱⁿ

C/I

^.

Bydenitionofthetriangulationof

C/I

^{,anyexattriangle}

A ^α // B // C // T A

in

C/I

^{is isomorphito the}loalization ofanexattrianglein

C

^{. This shows}

thatSupp

S (C) ⊂

^Supp

_S (A) ∪

^Supp

_S (B)

^.

Example 2.19 . Let

D ^b (P (X))

^{be theusual} triangulated ategory. Dene the support of an objet

P ∈ D ^b (P(X ))

^{as the union of the support of all the}

ohomologygroupsof

P

^,i.e

Supp

(P) = [

i

Supp

(H ⁱ (P )).

Thenit is easyto see that

(D ^b (P(X )),

^Supp

)

^isatriangulated ategorywith support. Denoteby

D ^b (P(X )) ^(k)

^{thefullsubategoryof}

D ^b (P (X ))

^ofobjets

whosesupportisofodimension

≥ k

^{. Then}

D ^b (P(X )) ^(k)

^{isasaturatedtrian-}

gulatedategoryandthefollowingsequene

D ^b (P (X)) ^(k) → D ^b (P (X)) → D ^b (P(X ))/D ^b (P (X)) ^(k)

satisestheonditionsofLemma 2.18. So

D ^b (P (X))/D ^b (P (X )) ^(k)

^{is atrian-}

gulatedategoryover

X ^′ = {x ∈ X |

^odim

(x) ≤ k − 1}

^.

Thefollowingdenitions arealsoduetoBalmer(see[Ba2℄):

Definition

^2.20

.

^Let

(C,

^Supp

)

^beatriangulatedategoryover

X

^andassume

that

C

^{hasastrutureof}triangulatedategorywithduality

(C, D, δ, ̟)

^{. Then}

wesaythat

C

^{is a}triangulatedategorywithdualitydenedover

X

^if

(S5) Supp

(A) =

^Supp

(DA)

^{foreveryobjet}

A

^.

Definition

^2.21

.

^Let

(C,

^Supp

_C )

^,

(D,

^Supp

_D )

^and

(M,

^Supp

_M )

^{be triangu-}

latedategoriesdenedover

X

^{. Supposethat}

⊗ : C × D → M

isapairingoftriangulatedategories. Thepairing

⊗

^{isdened over}

X

^if

(S6) Supp

M (A ⊗ B) =

^Supp

_C (A) ∩

^Supp

_D (B).

Example 2.22 . Thetriangulatedategory

D ^b (P (X))

^{withthesupportdened}

inExample2.19andthepairingofExample2.16satisfytheondition(S5)and

(S6).

Definition

^2.23

.

^{Thedegeneray lousofasymmetripair}

(A, α)

^isdened

tobethesupportoftheoneof

α

^:

DegLo

(α) =

^Supp

(

^one

(α)).

Definition

^2.24

.

^Let

(C,

^Supp

)

^beatriangulatedategorywithdualitydened over

X

^{. The} onsanguinityof two symmetripairs

α

^and

β

^{is dened to be}

thefollowingsubsetof

X

^:

Cons

(α, β) = (

^Supp

(α) ∩

^DegLo

(β)) ∪ (

^DegLo

(α) ∩

^Supp

(β)).

Wearenowready tostatetheLeibnitzformula:

Theorem

^{2.25 (Leibnitz formula)}

.

^{Assume that we have a dualizing pairing}

⊗ : C × D → F

^of triangulatedategories with dualities over

X

^{. Let}

α

^and

β

betwo symmetri pairs suhthat DegLo

(α) ∩

^DegLo

(β) = ∅

^{. Then we have}

anisometry

δ F · d(α ⋆ β) = δ C · d(α) ⋆ β + δ D · α ⋆ d(β)

where

δ C , δ D , δ F

^{arethe signsinvolvedinthe dualitiesof}

C, D

^and

F

^.

Proof. See[Ba2℄,Theorem5.2.

3 Chow-Witt groups

Let

(D ^b (P (X)), ^∨ , 1, ̟)

^{be the} triangulated ategory with the usual duality ofExample 2.2and onsider itsfull subategory

D ^b (P (X )) ⁽ⁱ⁾

^ofobjetswith

supports of odimension

≥ i

^{(here we use the support dened in Example}

2.19).Thenthedualityon

D ^b (P (X))

^{induesdualitieson}

D ^b (P (X)) ⁽ⁱ⁾

^forany

i

^{([Ba1℄). Itisalsolearthat}

D ^b (P(X )) ⁽ⁱ⁺¹⁾ ⊂ D ^b (P (X)) ⁽ⁱ⁾

^forany

i

^.

Definition

^3.1

.

^{For all}

i ∈ N

, denote by

D ^b _i (X )

^the triangulated ategory

D ^b (P (X)) ⁽ⁱ⁾ /D ^b (P (X)) ⁽ⁱ⁺¹⁾

^.

Suppose that

(A, α)

^{is an}

i

^{-symmetri form in}

D ^b _i (X )

^{. Then there exists an}

i

^{-symmetripair}

(B, β)

^suhthattheloalizationof

(B, β)

^is

(A, α)

^(byloal-

izationwemeanthemap

Symm ⁱ (D ^b (P(X)) ⁽ⁱ⁾ ) → Symm ⁱ (D ^b _i (X ))

^induedby

thefuntor

D ^b (P (X)) ⁽ⁱ⁾ → D ^b _i (X )

^{). Applying2.7,wegetan}

(i + 1)

^-symmetri

form

(C, ψ)

^{. By}onstrution,

C ∈ D ^b (P (X)) ⁽ⁱ⁺¹⁾

^{. Loalizingthisformweget}

aform

(C, ψ)

ⁱⁿ

W ⁱ⁺¹ (D ^b _i+1 (X))

^{. Atrstsight,thisonstrutiondependson}

somehoiesbutin fatthisisnotthease(see[Ba1℄,Corollary4.16). Hene

wegetawelldened homomorphism

d ⁱ : W ⁱ (D ^b _i (X )) → W ⁱ⁺¹ (D ^b _i+1 (X)).

Theorem

^3.2

.

^Let

X

^{be a regular sheme of dimension}

n

^{. Then we have a}

omplex

0 // W ⁰ (D ₀ ^b (X)) ^d

0

// W ¹ (D ^b ₁ (X )) ^d

1

// . . . ^{d n} // W ⁿ (D ^b _n (X )) // 0.

Proof. See[BW ℄,Theorem3.1 andParagraph8.

Let

A

^{bearegularloal ring. Wedenoteby}

W ^{f l} (A)

^{theWittgroupof nite}

lengthmodulesover

A

^{(see[QSS℄ formore}informationsaboutWitt groupsof nitelengthmodules). Thefollowingpropositionholds:

Proposition

^3.3

.

^Wehaveisomorphisms

W ⁱ (D ^b _i (X )) ≃ M

x∈X ⁽ⁱ⁾

W ^{f l} (O X,x ).

Proof. See[BW ℄,Theorem6.1 andTheorem6.2.

Remark 3.4 . Sineweusetheisomorphismoftheaboveproposition,webriey

reallhowtoobtainasymmetriomplexfromanitelengthmodule. Formore

details,see[BW℄ or [Fa℄, Chapter3. Chooseapoint

x ∈ X ⁽ⁱ⁾

^{, anitelength}

O X,x

^-module

M

^{and a symmetri} isomorphism

φ : M →

^Ext

ⁱ _{O X,x} (M, O X,x )

^.

Let

P •

^{be aresolutionof}

M

^{byloally freeoherent}

O X,x

^{-modules. Then}

P •

anbehosenoftheform

0 // P i // . . . // P 0 // M // 0.

Dualizingthisomplexandshifting

i

^{timesgivesthefollowingdiagram}

0 // P i //

∃

. . . // P 0 //

∃

M //

φ

0

0 // P ₀ ^∨ // . . . // P _i ^∨ //

^Ext

ⁱ _{O X,x} (M, O X,x ) // 0.

Using

φ

^{we get asymmetri morphism}

ϕ : P • → (P • ) ^∨

^{. Thus we haveon-}

strutedan

i

^{-symmetripairintheategory}

D ^b (P (O X,x ))

^fromthepair

(M, φ)

^.

Sine

D _i ^b (X) ≃ a

x∈X ⁽ⁱ⁾

D ^b (P (O X,x ))

^([BW℄,Proposition7.1),weanseethepair

(P • , ϕ)

^{asasymmetripairin}

D _i ^b (X )

^.

Definition

^3.5

.

^Theomplex

0 // W ^{f l} (k(X)) //

M

x 1 ∈X ⁽¹⁾

W ^{f l} (O X,x 1 ) // . . . //

M

x n ∈X ⁽ⁿ⁾

W ^{f l} (O X,x n ) // 0

isalledtheGersten-Wittomplexof

X

^{. Wedenoteitby}

C(X, W )

^.

This omplex is obtained by using the usual duality

∨

on the triangulated

ategory

D ^b (P (X))

^{(Example 2.2). For any invertible module}

L

^over

X

^{, we}

haveadualityderivedfrom thefuntor

♯ =

^Hom

O X,x (

^_

, L)

^{andwean apply}

thesameonstrutionto getaGersten-Wittomplex.

Definition

^3.6

.

^Let

X

^{bearegularsheme and}

L

^aninvertible

O X

^-module.

Wedenote by

C(X, W, L)

^theGersten-Wittomplexobtainedfrom the dual- ity

♯

.

Theorem

^3.7

.

^Let

A

^{bearegularloal}

k

^-algebraand

X =

^Spe

(A)

^{. Then for}

any

i > 0

^wehave

H ⁱ (C(X, W )) = 0

^.

Proof. See[BGPW℄,Theorem6.1.

Let

A

^{be a regularloal ringof dimension}

n

^{. Denote by}

F

^{the residueeld}

of

A

^{. Then any hoie of a generator}

ξ ∈

^Ext

ⁿ _A (F, A)

^givesan isomorphism

α ξ : W (F) → W ^{f l} (A)

^{. Reallthat}

I(F)

^isthefundamental idealof

W (F )

^{. If}

n ≤ 0

^,put

I ⁿ (F ) = W (F)

^.

Definition

^3.8

.

^Forany

n ∈ Z

let

I _{f l} ⁿ (A)

^betheimageof

I ⁿ (F )

^by

α ξ

^.

Remark 3.9 . It iseasilyseenthat

I _{f l} ⁿ (A)

^{doesnotdependonthehoieofthe}

generator

ξ ∈

^Ext

ⁿ _A (F, A)

^.

Proposition

^3.10

.

^{The dierential}

d

^{of the} Gersten-Witt omplex satises

d(I _{f l} ^m (O X,x )) ⊂ I _{f l} ^m−1 (O X,y )

^forany

m ∈ Z

,

x ∈ X ⁽ⁱ⁾

^and

y ∈ X ⁽ⁱ⁻¹⁾

^.

Proof. See[Gi3℄, Theorem6.4or[Fa℄, Theorem9.2.4.

Definition

^3.11

.

^Let

L

^{be an invertible}

O X

^{-module. We denote by}

C(X, I ^d , L)

^theomplex

0 // I _{f l} ^d (k(X)) //

M

x 1 ∈X ⁽¹⁾

I _{f l} ^d−1 (O X,x 1 ) // . . . //

M

x n ∈X ⁽ⁿ⁾

I _{f l} ^d−n (O X,x n ) // 0.

Remark 3.12 . Inpartiular,wehave

C(X, I ⁰ , L) = C(X, W, L)

^.

Theorem

^3.13

.

^Let

A

^bean essentiallysmoothloal

k

^{-algebra. Then for any}

i > 0

^{we have}

H ⁱ (C(X, I ^d )) = 0

^.

Proof. See[Gi3℄, Corollary7.7.

Ofourse,thereis aninlusion

C(X, I ^d+1 , L) → C(X, I ^d , L)

^{andthereforewe}

getaquotientomplex.

Definition

^3.14

.

^Denoteby

C(X, I ^d )

^theomplex

C(X, I ^d , L)/C(X, I ^d+1 , L)

^.

Remark 3.15 . Foranyinvertiblemodule

L

^theomplexes

C(X, I ^d )/C(X, I ^d+1 )

and

C(X, I ^d , L)/C(X, I ^d+1 , L)

^{areanonially isomorphi(see[Fa℄, Corollary}

E.1.3),sowean dropthe

L

ⁱⁿ

C(X, I ^d )

^.

Remark 3.16 . Theomplex

C(X, I ^d )

^isoftheform

0 // I _{f l} ^d (k(X ))/I _{f l} ^d+1 (k(X )) //

M

x 1 ∈X ⁽¹⁾

I _{f l} ^d−1 (O X,x 1 )/I _{f l} ^d (O X,x 1 ) // . . . .

Remark 3.17 . As a onsequene of Theorem 3.13, we immediately see that

H ⁱ (C(X, I ^d )) = 0

^for

i > 0

^if

X =

^Spe

(A)

^where

A

^{is an}essentiallysmooth loal

k

^-algebra.

Let

F

^{beaeld and denoteby}

K _i ^M (F )

^the

i

^{-th MilnorK-theorygroupof}

F

^.

If

i < 0

^{itisonvenienttoput}

K _i ^M (F ) = 0

^.

Definition

^3.18

.

^Let

X

^{beasheme. Thenforany}

d

^{wehaveaomplex}

0 // K _d ^M (k(X)) //

M

x 1 ∈X ⁽¹⁾

K _d−1 ^M (k(x 1 )) // . . . //

M

x n ∈X ⁽ⁿ⁾

K _d−n ^M (k(x n )) // 0.

Wedenoteitby

C(X, K _d ^M )

^.

Proof. See[Ka℄,Proposition1or [Ro℄, Paragraph3.

Wealsohavetheexatnessofthisomplexwhen

X

^{isthespetrumofasmooth}

loal

k

^-algebra:

Theorem

^3.19

.

^Let

A

^{beasmoothloal}

k

^{-algebra. Thenfor all}

i > 0

^wehave

H ⁱ (C(X, K _d ^M )) = 0

^.

Proof. See[Ro℄,Theorem6.1.

If

F

^{isaeld,reallthatwehavea}homomorphismduetoMilnor

s : K _j ^M (F ) → I ^j (F )/I ^j+1 (F )

given by

s({a 1 , . . . , a j }) =< 1, −a 1 > ⊗ . . . ⊗ < 1, −a j >

^{. The following is}

true:

Lemma

^3.20

.

^The homomorphisms

s

^{indueamorphismof omplexes}

s : C(X, K _d ^M ) → C(X, I ^d ).

Proof. See[Fa℄,Proposition10.2.5.

Definition

^3.21

.

^Let

C(X, G ^d , L)

^{be the bre produt of}

C(X, K _d ^M )

^and

C(X, I ^d , L)

^over

C(X, I ^d )

^:

C(X, G ^d , L) //

C(X, I ^d , L)

π

C(X, K _d ^M ) _s // C(X, I ^d ).

Definition

^3.22

.

^Let

X

^{be a smooth sheme and}

L

^{an invertible}

O X

^-

module. The

j

^{-th Chow-Witt group}

CH g ^j (X, L)

^of

X

^{twisted by}

L

^{is the}

group

H ^j (C(X, G ^j , L))

^.

Remark 3.23 . Denoteby

GW ^j (D ^b _j (X ), L)

^the

j

^-thGrothendiek-Wittgroupof theategory

D _j ^b (X )

^{withthedualityderivedfromHom}

O X (

^_

, L)

^(seeDenition

2.11). Itisnothardto seethat

C(X, G ^j , L)

^{isisomorphito}

GW ^j (D ^b _j (X), L)

andthereforetheomplex

C(X, G ^j , L)

^is

. . . // C(X, G ^j , L) j−1 // GW ^j (D ^b _j (X ), L) ^{d j} // W ^j+1 (D ^b _j+1 (X ), L) // . . .

Hene

CH g ^j (X, L)

^{is a quotient of Ker}

(d ^j )

^{and a} subquotient of

GW ^j (D ^b _j (X ), L)

^.

Wealsohavetheexatness oftheomplex

C(X, G ^d , L)

^{in theloal ase:}

Theorem

^3.24

.

^Let

A

^{be a smooth loal}

k

^{-algebra and}

X =

^Spe

(A)

^{. Then}

H ⁱ (C(X, G ^j )) = 0

^{for all}

j

^andall

i > 0

^.

Proof. As

C(X, G ^j )

^{is the bre produt of the omplexes}

C(X, K _j ^M )

^and

C(X, I ^j )

^over

C(X, I ^j )

^{,wehaveanexatsequeneofomplexes}

0 // C(X, G ^j ) // C(X, I ^j ) ⊕ C(X, K _j ^M ) // C(X, I ^j ) // 0

induingalongexatsequenein ohomology. It followsthen from Theorem

3.13andTheorem3.19that

H ⁱ (C(X, G ^j )) = 0

^if

i > 1

^{. For}

i = 1

^,wehavean

exatsequene

H ⁰ (C(X, I ^j )) ⊕ H ⁰ (C(X, K _j ^M )) // H ⁰ (C(X, I ^j )) // H ¹ (C(X, G ^j )) // 0.

Theexatsequeneofomplexes

0 // C(X, I ^j+1 ) // C(X, I ^j ) // C(X, I ^j ) // 0

showsthat

H ⁰ (C(X, I ^j ))

^mapsonto

H ⁰ (C(X, I ^j ))

^.

Definition

^3.25

.

^Let

X

^{beasmoothshemeand}

L

^aninvertible

O X

^-module.

Wedenethesheaf

G _L ^j

^on

X

^by

G _L ^j (U) = H ⁰ (C(U, G ^j , L))

^.

Wehave:

Theorem

^3.26

.

^Let

X

^{be a smooth sheme of dimension}

n

^{. Then for any}

i

wehave

H _Zar ⁱ (X, G ^j _L ) ≃ H ⁱ (C(X, G ^j , L)).

Proof. Denesheaves

C l

^by

C l (U ) = C(U, G ^j , L) l

^forany

l ≥ 0

^{. Itislearthat}

the

C l

^{areasquesheaves. Wehaveaomplexofsheavesover}

X 0 // G _L ^j // C 0 // C 1 // . . . // C n // 0.

Theorem3.24showsthatthisomplexisaasqueresolutionof

G _L ^j

^{. Thusthe}

theoremisproved.

Suppose that

f : X → Y

^{isaatmorphism. Sine itpreserves}odimensions, itinduesamorphismofomplexes

f ^∗ : C(Y, G ^j , L) → C(X, G ^j , f ^∗ L)

forany

j ∈ N

andanylinebundle

L

^over

Y

^{([Fa℄,Corollary10.4.2). Henewe}

have:

Theorem

^3.27

.

^Let

f : X → Y

^{bea at morphism and}

L

^{a linebundleover}

Y

^{. Then, forany}

i, j

^wehave homomorphisms

f ^∗ : H ⁱ (C(Y, G ^j , L)) → H ⁱ (C(X, G ^j , f ^∗ L)).

Inpartiular, if

E

^{isavetor bundle over}

Y

^and

π : E → Y

^{isthe projetion,}

wehaveisomorphisms

π ^∗ : H ⁱ (C(Y, G ^j , L)) → H ⁱ (C(E, G ^j , π ^∗ L)).

Proof. We have amorphism of omplexes

f ^∗ : C(Y, G ^j , L) → C(X, G ^j , f ^∗ L)

whih givestheinduedhomomorphismsinohomology. For theproof ofho-

motopyinvariane,seeCorollary11.3.2in[Fa℄.

Proposition

^3.28

.

^Let

f : X → Y

^and

g : Y → Z

^{be at morphisms. Then}

(gf ) ^∗ = f ^∗ g ^∗

^.

Proof. See[Fa℄,Proposition3.4.9.

Suppose that

f : X → Y

^{is a nite morphism with dim}

(Y ) −

^dim

(X) = r

^.

Consider the morphism of loally ringed spaes

f : (X, O X ) → (Y, f ∗ O X )

induedby

f

^{. If}

X

^{issmooth,then}

L = f ^∗

^Ext

^r _{O Y} (f ∗ O X , O Y )

^{is aninvertible}

moduleover

Y

^{([Gi2 ℄,Corollary6.6)andwegetamorphismofomplexes(of}

degreer)

f ∗ : C(X, G ^j−r , L ⊗ f ^∗ N ) → C(Y, G ^j , N)

foranyinvertiblemodule

N

^over

Y

^{([Fa℄,Corollary5.3.7).}

Proposition

^3.29

.

^Let

f : X → Y

^{be a nite morphism between smooth}

shemes. Let dim

(Y ) −

^dim

(X) = r

^and

N

^{be an invertible module over}

Y

^.

Then themorphism of omplexes

f ∗

^{indues a}homomorphism

f ∗ : H ^i−r (C(X, G ^j−r , L ⊗ f ^∗ N )) → H ⁱ (C(Y, G ^j , N)).

Inpartiular,wehave([Fa℄,Remark9.3.5):

Proposition

^3.30

.

^Let

f : X → Y

^{be a losed immersion of} odimensio n

r

betweensmooth shemes. Then

f

^induesan isomorph ism

f ∗ : H ^i−r (C(X, G ^j−r , L ⊗ f ^∗ N )) → H _X ⁱ (C(Y, G ^j , N ))

forany

i, j

^{andany invertiblemodule}

N

^over

Y

^.

Importantremark 3.31 . If

f : X → Y

^{isalosedimmersion,then}

f ∗

^willalways

bethemapwithsupport:

f ∗ : H ^i−r (C(X, G ^j−r , L ⊗ f ^∗ N )) → H _X ⁱ (C(Y, G ^j , N ))

Thetransferfornitemorphismsisfuntorial([Fa℄,proposition5.3.8):

Proposition

^3.32

.

^Let

f : X → Y

^and

g : Y → Z

^{be nitemorphisms. Then}

g ∗ f ∗ = (gf ) ∗

^.

Remark 3.33 . Let

X

^{beasmoothshemeand}

D

^{beasmootheetiveCartier}

divisoron

X

^{. Let}

i : D → X

^{be theinlusion and}

L(D)

^{be the line bundle}

over

X

^assoiatedto

D

^{. Thenthere isaanonialsetion}

s ∈ L(D)

^{(see [Fu℄,}

AppendixB.4.5)andanexatsequene

0 // O X s

// L(D) // i ∗ O D // 0.

ApplyingHom

O X (

^_

, L(D))

^{andshifting,weobtainthefollowingdiagram}

0 // O X s

//

≃

L(D) //

≃

i ∗ O D //

0

0 //

^Hom

_{O X} (L(D), L(D)) _s //

^Hom

_{O X} (O X , L(D)) //

^Ext

¹ _{O X} (i ∗ O D , L(D)) // 0

whih showsthat Ext

1

O X (i ∗ O D , O X ) ⊗ L(D) ≃ i ∗ O D

^. Proposition3.30shows thatwethenhaveanisomorphism

i ∗ : H ⁱ⁻¹ (C(D, G ^j−1 , i ^∗ L(D))) → H _D ⁱ (C(X, G ^j )).

Lemma

^3.34

.

^Let

g : X → Y

^{be a at morphism and}

f : Z → Y

^{a nite}

morphism. Consider the following breprodut

V ^f

′

//

g ^′

X

g

Z _f // Y.

Then

(f ^′ ) ∗ (g ^′ ) ^∗ = g ^∗ f ∗

^.

Proof. See[Fa℄,Corollary12.2.8.

Remark 3.35 . Ofourse,in theabovebreprodutwesupposethat

V

^isalso

smoothandintegral. Suhastrongassumptionisnotneessaryingeneral,but

thisaseissuientforourpurposes.

Remark 3.36 . Itispossibletodeneamap

f ∗

^{whenthemorphism}

f

^isproper

(see[Fa℄)butwedon'tusethisfathere.

4 The exterior product

Let

X

^and

Y

^{betwoshemes. Thebreprodut}

X × Y

^{omesequippedwith}

twoprojetions

p 1 : X × Y → X

^and

p 2 : X × Y → Y

^.

Lemma

^4.1

.

^Forany

i, j ∈ N

, the pairing

⊠ : D _i ^b (X) × D ^b _j (Y ) → D _i+j ^b (X × Y )

given by

P ⊠ Q = p ^∗ ₁ P ⊗ p ^∗ ₂ Q

^{isa dualizing pairing of} triangulated ategories withduality.

Proof. Straightveriation.

Corollary

^4.2

.

^Forany

i, j ∈ N

, the pairing

⊠ : D _i ^b (X) × D ^b _j (Y ) → D _i+j ^b (X × Y )

indues apairing

⋆ : W ⁱ (D ^b _i (X )) × W ^j (D _j ^b (Y )) → W ^i+j (D ^b _i+j (X × Y )).

Proof. ClearbyTheorem2.15.

Corollary

^4.3

.

^Let

ψ ∈ W ^j (D ^b _j (Y ))

^{. Then wehave a}homomorphism

µ ψ : W ⁱ (D ^b _i (X )) → W ^i+j (D ^b _i+j (X × Y ))

given by

µ ψ (ϕ) = ϕ ⋆ ψ

^.

Reallthat we have isomorphisms

W ⁱ (D _i ^b (X )) ≃ M

x∈X ⁽ⁱ⁾

W ^{f l} (O X,x )

^(Proposi-

tion3.3).

Definition M

^4.4

.

^{For any}

s ∈ Z

, denote by

I ^s (D _i ^b (X))

^{the preimage of}

x∈X ⁽ⁱ⁾

I _{f l} ^s (O X,x )

^{undertheabove}isomorphism.

Proposition

^4.5

.

^Forany

m, p ∈ N

the produt

⋆ : W ⁱ (D _i ^b (X )) × W ^j (D _j ^b (Y )) → W ^i+j (D ^b _i+j (X × Y ))

indues aprodut

⋆ : I ^m (D _i ^b (X)) × I ⁿ (D _j ^b (Y )) → I ^m+n (D ^b _i+j (X × Y )).

Proof. Let

x ∈ X ⁽ⁱ⁾

^and

y ∈ Y ^(j)

^{. Itislearthattheprodutanbeomputed}

loally (use [GN℄, Theorem 3.2). So wean suppose that

X =

^Spe

(A)

^and

Y =

^Spe

(B)

^where

A

^and

B

^areloalin

x

^and

y

respetively. Reallthat we havethefollowingdiagram

X × Y ^{p 2} //

p 1

Y

X //

^Spe

(k).

Let

P

^bean

A

^{-projetiveresolutionof}

k(x)

^and

Q

^bea

B

^{-projetiveresolution}

of

k(y)

^{. Considerasymmetriform}

ρ : k(x) →

^Ext

ⁱ _A (k(x), A)

^andasymmet-

riform

µ : k(y) →

^Ext

^j _B (k(y), B)

^{. Then}

p ^∗ ₁ (ρ)

^{is a symmetri}isomorphism supportedbytheomplex

P ⊗ k B

^and

p ^∗ ₂ (µ)

^isasymmetriisomorphismsup- portedbytheomplex

A ⊗ k Q

^{. Theomplex}

(P ⊗ k B) ⊗ A⊗ k B (A ⊗ k Q)

^(whih

isisomorphito

P ⊗ k Q

^{)hasitshomologyonentratedin degree}

0

^{,and this}

homologyisisomorphito

k(x) ⊗ k k(y)

^{. Let}

u

^{beapointofSpe}

(k(x) ⊗ k k(y))

^.

Thentherestritionof

p ^∗ ₁ ρ⊗p ^∗ ₂ µ

^to

u

^{isanitelengthmodule}

M

^whosesupport

ison

u

^{withasymmetriform}

M →

^Ext

^i+j _(A⊗B)

u (M, (A ⊗ B) u ).

Taking its lass in the Witt group, we obtain a

k(u)

^{-vetor spae}

V

^{with a}

symmetri form

ψ : V →

^Ext

^i+j _{(A⊗B) u} (V, (A ⊗ B) u )

^{. Now hoose a unit}

a ∈ k(x) ^×

^{. Consider the image}

a u

^of

a

^{under the} homomorphism

k(x) → k(u)

^.

Thelassof

p ^∗ ₁ (aρ) ⊗ p ^∗ ₂ (µ)

^{isthesymmetriform}

a u ψ : V →

^Ext

^i+j _(A⊗B)

u (V, (A ⊗ B) u ).

Asthesamepropertyholdsforanyunit

b ∈ k(y) ^×

^{,weonludethat}

p ^∗ ₁ (< 1, −a 1 > ⊗ . . . ⊗ < 1, −a n > ρ) ⊗ p ^∗ ₂ (< 1, −b 1 > ⊗ . . . ⊗ < 1, −b m > µ)

isequalto

< 1, −(a 1 ) u > ⊗ . . . ⊗ < 1, −(b m ) u > ψ

^.

Reallthatforanysheme

X

^wehaveaGersten-Wittomplex(Denition3.5)

C(X, W ) : . . . // W ^r (D ^b _r (X )) ^d

r

X // W ^r+1 (D _r+1 ^b (X)) // . . .

andaomplex

C(X, I ^d )

^:

. . . //

M

x r ∈X ^(r)

I _{f l} ^d−r (O X,x r ) //

M

x r+1 ∈X ^(r+1)

I _{f l} ^d−r−1 (O X,x r+1 ) // . . . .

Theabovepropositiongives:

Corollary

^4.6

.

^Theprodut

⋆ : C(X, W ) × C(Y, W ) → C(X × Y, W )

indues for any

r, s ∈ N

^aprodut

⋆ : C(X, I ^r ) × C(Y, I ^s ) → C(X × Y, I ^r+s ).

Now we investigate the relations between

⋆

^{and the} dierentialsof the om- plexes.

Proposition

^4.7

.

^Let

ψ ∈ W ^j (D _j ^b (Y ))

^{be suh that}

d ^j _Y (ψ) = 0

^{. Then the}

following diagramommutes

W ⁱ (D ^b _i (X)) ^d

i

X //

(−1) ^j µ ψ

W ⁱ⁺¹ (D _i+1 ^b (X ))

µ ψ

W ^i+j (D _i+j ^b (X × Y ))

d ^i+j _X×Y

// W ^i+j+1 (D ^b _i+j+1 (X × Y )).

Proof. Let

ϕ ∈ W ⁱ (D ^b _i (X ))

^{. Let}

X ^(≥i+1)

^{bethesetofpointsof}

X

^ofodimen-

sion

≥ i+1

^,

Y ^(≥j+1)

^thepointsof

Y

^ofodimension

≥ j +1

^and

(X ×Y ) ^(≥i+j+1)

theset of points of

X × Y

^{of odimension}

≥ i + j + 1

^{. ByLemma 2.18, the}

triangulatedategories

D _i ^b (X), D ^b _j (Y )

^and

D ^b _i+j (X × Y )

^{are dened overthe}

topologialspaes

X \ X ^(≥i+1)

^,

Y \ Y ^(≥j+1)

^and

(X ×Y ) \(X ×Y ) ^(≥i+j+1)

^{. Let}

α ∈ Symm ⁱ (D ^b (P(X )) ⁽ⁱ⁾ )

^and

β ∈ Symm ^j (D ^b (P (Y )) ^(j) )

^{besymmetripairs}

representing

ϕ

^and

ψ

^{. By denition, DegLo}

(α)

^{is of odimension}

≥ i + 1

^,

DegLo

(β)

^{is ofodimension}

≥ j + 1

^and

dβ

^{is neutral. Itis easilyseenthat}

Supp

(dp ^∗ ₁ α)∩

^Supp

(dp ^∗ ₂ β) = ∅

^{inthetopologialspae}

(X ×Y )\(X ×Y ) ^(≥i+j+1)

^.

Theorem2.25impliesthat

(−1) ^i+j d(p ^∗ ₁ α ⋆ p ^∗ ₂ β ) = (−1) ⁱ dp ^∗ ₁ α ⋆ p ^∗ ₂ β + (−1) ^j p ^∗ ₁ α ⋆ dp ^∗ ₂ β.

UsingTheorem2.15,weseethat wehavein

W ^i+j (D ^b _i+j (X × Y ))

^theequality

(−1) ^j d ^i+j _X×Y (p ^∗ ₁ ϕ ⋆ p ^∗ ₂ ψ) = p ^∗ ₁ d ⁱ _X (ϕ) ⋆ p ^∗ ₂ ψ.

Thefollowingorollaryisobvious.

Corollary

^4.8

.

^Let

ψ ∈ I ^m (D _j ^b (Y ))

^{be suh that}

d ^Y _j (ψ) = 0

^{. Then the}

following diagramommutes

I ^p (D ^b _i (X )) ^d

i

X //

(−1) ^j µ ψ

I ^p−1 (D _i+1 ^b (X ))

µ ψ

I ^p+m (D ^b _i+j (X × Y ))

d ^i+j _X×Y

// I ^p+m−1 (D ^b _i+j+1 (X × Y )).

Wenow haveto dealwith theomplex in MilnorK-theory. Let

C(X, K _r ^M )

^,

C(Y, K _s ^M )

^and

C(X ×Y, K _r+s ^M )

^{betheomplexesinMilnorK-theoryassoiated}

to

X, Y

^and

X × Y

^{. In[Ro℄,Rost denesaprodut}

⊙ : C(X, K _r ^M ) ⁱ × C(Y, K _s ^M ) ^j → C(X × Y, K _r+s ^M ) ^i+j

asfollows: Let

u ∈ (X ×Y ) ^(i+j)

^,

x ∈ X ⁽ⁱ⁾

^,

y ∈ Y ^(j)

^besuhthat

x

^and

y

^arethe

projetionsof

u

^{. Let}

ρ = {a 1 , . . . , a r−i } ∈ K _r−i ^M (k(x))

^and

µ = {b 1 , . . . , b s−j } ∈ K _s−j ^M (k(y))

^{. Then}

(ρ ⊙ µ) u = l((k(x) ⊗ k k(y)) u ){(a 1 ) u , . . . , (a r−i ) u , (b 1 ) u , . . . , (b s−j ) u }

wherethe

(a l ) u

^and

(b t ) u

^{aretheimagesof the}

a l

^and

b t

^{under theinlusions}

k(x) → k(u)

^and

k(y) → k(u)

^{, and}

l((k(x) ⊗ k k(y)) u )

^{is the length of the}

module

k(x) ⊗ k k(y)

^loalizedin

u

^.

Lemma

^4.9

.

^Forany

ρ ∈ C(X, K _r ^M ) ⁱ

^and

µ ∈ C(Y, K _s ^M ) ^j

^wehave

d(ρ ⊙ µ) = d(ρ) ⊙ µ + (−1) ^j ρ ⊙ d(µ).

Proof. See[Ro℄,Paragraph14.4.

Corollary

^4.10

.

^Let

µ ∈ C(Y, K _s ^M ) ^j

^besuhthat

dµ = 0

^{. Thenthefollowing}

diagramommutes:

C(X, K _r ^M ) ^{i d}

i

X //

⊙µ

C(X, K _r ^M ) ⁱ⁺¹

⊙µ

C(X × Y, K _r+s ^M ) ^i+j

d ^i+j _X×Y

// C(X × Y, K ^M _r+s ) ^i+j+1 .