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(1)

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

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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

. Soinfattherstinterestingaseiswhen

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

M

p

(

k

(

X

))

/

M

x

1

∈

X

(1)

K

p

M

−1

(

k

(

x1

))

/

. . .

/

M

x

n

∈

X

(n)

K

p

M

−

n

(

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

-thohomologygroupofthisbre

prod-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

(3)

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

exteriorprodut

g

CH

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

g

CH

j

(

X

)

×

CH

g

i

(

X

)

/

CH

g

i

+

j

(

X

×

X

)

△

!

/

CH

g

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 bearinghomomorphism. There is

howeverasurprise: theprodutthatweobtainisapriorineitherommutative

norantiommutative. This omes from the fat that the produt of

triangu-latedGrothendiek-Wittgroups

GW

i

×

GW

j

→

GW

i

+

j

doesnotsatisfyany

ommutativityproperty.

Theorganizationof this paperis as follows: Insetion 2,wereall some ba-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

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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

×S

pe

(

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

)

(5)

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 denedhomomorphism

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

, ̟

)

bethetriangulatedategorywith

du-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

(6)

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

o

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

)

o

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

(7)

Proof. See[GN℄,Theorem2.9.

Example2.16 . Let

(

D

b

(

P

(

X

))

,

∨

,

1

, ̟

)

bethetriangulatedategorywith

dual-itydenedinExample2.2. Theusualtensorprodutinduesadualizing

pair-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 triangulatedategory

de-ned over

X

is apair

(

C,

Supp

)

where

C

is atriangulatedategoryand 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

(8)

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 theloalization 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

)

isasaturated

trian-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 a

trian-gulatedategoryover

X

′

=

{x

∈

X

|

odim

(

x

)

≤

k

−

1

}

.

Thefollowingdenitions arealsoduetoBalmer(see[Ba2℄):

Definition

2.20

.

Let

(

C,

Supp

)

beatriangulatedategoryover

X

andassume

that

C

hasastrutureoftriangulatedategorywithduality

(

C, D, δ, ̟

)

. Then

wesaythat

C

is atriangulatedategorywithdualitydenedover

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

(9)

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

δF

·

d

(

α ⋆ β

) =

δC

·

d

(

α

)

⋆ β

+

δD

·

α ⋆ d

(

β

)

where

δC

, δD

, δF

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

)

(by

loal-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

)

. Byonstrution,

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

(10)

Theorem

3.2

.

Let

X

be a regular sheme of dimension

n

. Then we have a

omplex

0

/

W

0

(

D

b

0

(

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℄ formoreinformationsaboutWitt 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

/

. . .

/

P0

/

M

/

0

.

Dualizingthisomplexandshifting

i

timesgivesthefollowingdiagram

0

/

P

i

/

∃

. . .

/

P0

/

∃

M

/

φ

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 have

on-strutedan

i

-symmetripairintheategory

D

b

(

P

(

O

X,x

))

fromthepair

(

M, φ

)

.

Sine

D

b

i

(

X

)

≃

a

x

∈

X

(i)

D

b

(

P

(

O

X,x

))

([BW℄,Proposition7.1),weanseethepair

(

P•, ϕ

)

asasymmetripairin

D

b

i

(

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

(11)

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

n

f l

(

A

)

betheimageof

I

n

(

F

)

by

α

ξ

.

Remark 3.9 . It iseasilyseenthat

I

n

f l

(

A

)

doesnotdependonthehoieofthe

generator

ξ

∈

Ext

n

A

(

F, A

)

.

Proposition

3.10

.

The dierential

d

of the Gersten-Witt omplex satises

d

(

I

m

f l

(

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

d

f l

(

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

(12)

Remark 3.16 . Theomplex

C

(

X, I

d

)

isoftheform

0

/

I

d

f l

(

k

(

X

))

/I

d

+1

f l

(

k

(

X

))

/

M

x

1

∈

X

(1)

I

f l

d

−1

(

O

X,x

1

)

/I

d

f l

(

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 anessentiallysmooth

loal

k

-algebra.

Let

F

beaeld and denoteby

K

M

i

(

F

)

the

i

-th MilnorK-theorygroupof

F

.

If

i <

0

itisonvenienttoput

K

M

i

(

F

) = 0

.

Definition

3.18

.

Let

X

beasheme. Thenforany

d

wehaveaomplex

0

/

K

M

d

(

k

(

X

))

/

M

x

1

∈

X

(1)

K

M

d

−1

(

k

(

x1

))

/

. . .

/

M

x

n

∈

X

(n)

K

M

d

−

n

(

k

(

x

n

))

/

₀

.

Wedenoteitby

C

(

X, K

M

d

)

.

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

M

d

)) = 0

.

Proof. See[Ro℄,Theorem6.1.

If

F

isaeld,reallthatwehaveahomomorphismduetoMilnor

s

:

K

j

M

(

F

)

→

I

j

(

F

)

/I

j

+1

(

F

)

given by

s

(

{a1, . . . , a

j

}

) =

<

1

,

−a1

>

⊗

. . .

⊗

<

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

M

d

)

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

M

d

)

s

/

C

(

X, I

d

(13)

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

b

j

(

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

M

j

)

and

C

(

X, I

j

)

over

C

(

X, I

j

)

,wehaveanexatsequeneofomplexes

0

/

C

(

X, G

j

)

/

C

(

X, I

j

)

⊕

C

(

X, K

M

j

)

/

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

j

L

on

X

by

G

j

L

(

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

i

(14)

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

/

C0

/

C1

/

. . .

/

C

n

/

0

.

Theorem3.24showsthatthisomplexisaasqueresolutionof

G

j

L

. Thusthe

theoremisproved.

Suppose that

f

:

X

→

Y

isaatmorphism. Sine itpreservesodimensions,

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 of

ho-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

)

(15)

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 ahomomorphism

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

i

X

(

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

/

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

(16)

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

p1

:

X

×

Y

→

X

and

p2

:

X

×

Y

→

Y

.

Lemma

4.1

.

Forany

i, j

∈

N

, the pairing

⊠

:

D

i

b

(

X

)

×

D

b

j

(

Y

)

→

D

i

b

+

j

(

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

b

+

j

(

X

×

Y

)

indues apairing

⋆

:

W

i

(

D

b

i

(

X

))

×

W

j

(

D

j

b

(

Y

))

→

W

i

+

j

(

D

i

b

+

j

(

X

×

Y

))

.

Proof. ClearbyTheorem2.15.

Corollary

4.3

.

Let

ψ

∈

W

j

(

D

b

j

(

Y

))

. Then wehave ahomomorphism

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

b

i

(

X

))

≃

M

x

∈

X

(i)

W

f l

(

O

X,x

)

(17)

Definition

4.4

.

For any

s

∈

Z

, denote by

I

s

(

D

b

i

(

X

))

the preimage of

M

x

∈

X

(i)

I

f l

s

(

O

X,x

)

undertheaboveisomorphism.

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

i

b

+

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

)

anda

symmet-riform

µ

:

k

(

y

)

→

Ext

j

B

(

k

(

y

)

, B

)

. Then

p

∗

1

(

ρ

)

is a symmetriisomorphism

supportedbytheomplex

P

⊗

k

B

and

p

∗

2

(

µ

)

isasymmetriisomorphism

sup-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

(

y

)

. Let

u

beapointofSpe

(

k

(

x

)

⊗

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

,

−a1

>

⊗

. . .

⊗

<

1

,

−a

n

> ρ

)

⊗

p

∗

2

(

<

1

,

−b1

>

⊗

. . .

⊗

<

1

,

−b

m

> µ

)

(18)

Reallthatforanysheme

X

wehaveaGersten-Wittomplex(Denition3.5)

C

(

X, W

) :

. . .

/

W

r

(

D

b

r

(

X

))

d

r

X

/

W

r

+1

(

D

b

r

+1

(

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

b

j

(

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

b

i

+1

(

X

))

µ

ψ

W

i

+

j

(

D

b

i

+

j

(

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

of

odimen-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

b

i

(

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

(19)

UsingTheorem2.15,weseethat wehavein

W

i

+

j

(

D

b

i

+

j

(

X

×

Y

))

theequality

(

−

1)

j

d

i

X

+

×

j

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

b

j

(

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

b

i

+1

(

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

M

r

)

,

C

(

Y, K

M

s

)

and

C

(

X

×Y, K

M

r

+

s

)

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

M

+

s

)

i

+

j

asfollows: Let

u

∈

(

X

×Y

)

(

i

+

j

)

,

x

∈

X

(

i

)

,

y

∈

Y

(

j

)

besuhthat

x

and

y

arethe

projetionsof

u

. Let

ρ

=

{a1, . . . , a

r

−

i

} ∈

K

M

r

−

i

(

k

(

x

))

and

µ

=

{b1, . . . , b

s

−

j

} ∈

K

s

M

−

j

(

k

(

y

))

. Then

(

ρ

⊙

µ

)

u

=

l

((

k

(

x

)

⊗

k

(

y

))

u

)

{

(

a1

)

u

, . . . ,

(

a

r

−

i

)

u

,

(

b1

)

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

(

y

))

u

)

is the length of the

module

k

(

x

)

⊗

k

(

y

)

loalizedin

u

.

Lemma

4.9

.

Forany

ρ

∈

C

(

X, K

M

r

)

i

and

µ

∈

C

(

Y, K

M

s

)

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

M

s

)

j

besuhthat

dµ

= 0

. Thenthefollowing

diagramommutes:

C

(

X, K

M

r

)

i

d

i

X

/

⊙

µ

C

(

X, K

M

r

)

i

+1

⊙

µ

C

(

X

×

Y, K

M

r

+

s

)

i

+

j

d

i+j

X

×

Y

/

C

(

X

×

Y, K

M

(20)

Proof. Obvious.

Nowweomparetheproduts

⋆

and

⊙

.

Proposition

4.11

.

The following diagram ommutes:

C

(

X, K

M

r

)

i

×

C

(

Y, K

s

M

)

j

⊙

/

s

(r

−

i)

×

s

(s

−

j)

C

(

X

×

Y, K

M

r

+

s

)

i

+

j

s

(r+s

−

i

−

j)

C

(

X, I

r

)

i

×

C

(

Y, I

s

)

j

⋆

/

C

(

X

×

Y, I

r

+

s

)

i

+

j

.

Proof. Let

{a1, . . . , a

r

−

i

} ∈

K

M

r

−

i

(

k

(

x

))

and

{b1, . . . , b

s

−

j

} ∈

K

M

s

−

j

(

k

(

y

))

. Let

ρ

′

beasymmetriisomorphism

ρ

′

:

k

(

x

)

→

Ext

i

O

X,x

(

k

(

x

)

,

O

X,x

)

and

µ

′

asymmetriisomorphism

µ

′

:

k

(

y

)

→

Ext

j

O

Y,y

(

k

(

y

)

,

O

Y,y

)

.

Wethen have

ρ

:=

s(

r

−

i

)

(

{a1, . . . , a

r

−

i

}

) =

<

1

,

−a1

>

⊗

. . .

⊗

<

1

,

−a

r

−

i

> ρ

′

and

µ

:=

s(

s

−

j

)

(

{b1, . . . , b

s

−

j

}

) =

<

1

,

−b1

>

⊗

. . .

⊗

<

1

,

−b

s

−

j

> µ

′

. Choose

apoint

u

in

(

X

×

Y

)

(

i

+

j

)

lying over

x

and

y

. The proof of Proposition 4.5