Doc Math J DMV

Applications of WeightTwo Motivic Cohomology

Bruno Kahn

Received July

Communicated by Peter Schneider

Abstract Using Lichtenbaums complex we reprove and extend

a little bit some known results relating the kernel of H F QZ

H F X QZ to the torsion of CH X for rational varieties X

Mathematics Sub ject Classication Primary C secondary E

Contents

Introduction

Motivic cohomology of smo oth varieties

Relative motivic cohomology I

Relative motivic cohomology II

Purity

Cohomology of pro jective bundles

The coniveau sp ectral sequence and Gerstens conjecture

Pro jective homogeneous varieties

References

Introduction

Let F b e a eld and X b e a smo oth geometrically integral variety over F In

prop ColliotThelene and Raskind pro duced an exact sequence

G

F

X K X K H H

Zar Zar

H F K F X H X K KerCH X CH X

Zar

H F H X K H F K F X H X K

Zar Zar

Here X denotes the variety X viewed over the separable closure F of F K is

the Zariski sheaf asso ciated to the presheaf U K U and G is the absolute Galois

F

group of F On the other hand in th we pro duced an isomorphism

H F K F X K F KerH F QZ H F X QZ

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Bruno Kahn

In the co ecients QZ are

lim lim if char F and lim W if char F

r

n n log

r

nchar F

where W is the weighttwo logarithmic part of the de RhamWitt complex over

r

log

the big etale site of Sp ec F see comments at the end of the introduction

F X F is When X is a complete rational variety ie the extension

purely transcendental the group H X K coincides with K F One

Zar

F K F X H X K in by may therefore replace the group H

Zar

KerH F QZ H F X QZ in this case The resulting exact se

quence has b een used in and

Moreover the left map in is injective when X is a complete rational variety

prop in characteristic prop in general Putting all this together

one therefore gets an exact sequence

G

F

X K H X K H

Zar Zar

KerH F QZ H F X QZ

KerCH X CH X H F H X K

Zar

for any complete rational variety X

In this pap er we use the Lichtenbaum complex of to recover this

exact sequence directly and extend it to the right Our main result is

Theorem Let X be a smooth variety over F

a Assume that K F H X K Let us denote by

Zar

H F QZ H X H QZ

Zar

G

F

X CH X CH

cl CH X QZ H X QZ

X

the natural maps and the divisible cycle class map Then there is an exact sequence

G

F

H X K H Ker Ker H F H X K X K

Zar Zar Zar

X H QZ is pprimary torsion where p is the b Assume moreover that H

Zar

characteristic exponent of F and H QZ is the Zariski sheaf associated to the

presheaf U H U QZ if char F this means H X H QZ

et Zar

Then the exact sequence extends to a complex

X K H F QZ Coker cl Ker H F H

X Zar

Let A resp B denote the homology of at H F H X K resp at

Zar

H F QZ Then there is another complex

Coker Zp Coker Zp H F H X K Zp

Zar

whose homology at Coker Zp resp at Coker Zp is A Zp resp

B Zp

X H QZ we can remove Zp everywhere If H

Zar

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Applications of WeightTwo Motivic Cohomology

Remark The assumptions are satised if X is a complete rational variety

but also if it is a torsor under a semisimple simply connected algebraic group

X H QZ is in If char k p in the second case the group H

Zar

general nonzero as higher logarithmic Ho dgeWitt cohomology is not homotopy

invariant hence the complicated statement of theorem However we do have

X H QZ in the rst case compare corollaries and c H

Zar

Corollary Let X be as in theorem b

Suppose cd F Then there is an exact sequence

G

F

H X K H Ker Ker H F H X K X K

Zar Zar Zar

Coker Coker H F H X K

Zar

X K after tensorisation by Zp The part of this sequence up to H F H

Zar

exists and is exact without tensoring by Zp

Suppose cd F Then there is an isomorphism

G

F

H X K H X K

Zar Zar

and an exact sequence

X K Ker H F H

Zar

H X H QZ Coker H F H X K

Zar Zar

after tensorisation by Zp The injection Ker H F H X K holds

Zar

without tensoring by Zp

X H QZ the results hold without tensoring by Zp If H

Zar

To try and get a relationship b etween theorem and the last term in we

observe that a closer examination of the sp ectral sequence used in pro of of th

yields an exact sequence

H F QZ KerH F X QZ H F X QZ

H F K F X K F H F QZ H F X QZ

How to derive theorem from sequence do es not seem obvious however

This pap er is organized as follows In section we compute the etale hy

p ercohomology of X with co ecients in this is done in theorem which

is of indep endent interest In sections and we introduce two relative com

plexes F X X over X and XF over Sp ec F Considering the

et et

Ho chschildSerre sp ectral sequence for the hypercohomology of F X X we

get back the ColliotTheleneRaskind exact sequence in a straightforward manner

see prop osition To prove theorem we similarly examine the Ho chschildSerre

sp ectral sequence for the hypercohomology of X with co ecients XF see

section In sections and we resp ectively prove a purity theorem compute

the motivic cohomology of a pro jective bundle and prove a Blo chOgus type theorem

Finally in section we lo ok at pro jective homogeneous varieties

The pro of of the isomorphism in consisted of considering the Ho chschild

Serre sp ectral sequence for the hypercohomology of F with co ecients in a relative

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Bruno Kahn

Lichtenbaum complex F X F relative to the extension F F What we do

here can b e considered as a renement of this metho d by factoring the morphism

Sp ec F X Sp ec F into

Sp ec F X X Sp ec F

Remarks on characteristic p We have to b e a little careful if char F when

dening the co ecients QZ In characteristic they are dened as lim If

n

r

char F p we set Zp W where W is the sheaf of loga

r r

log log

rithmic de RhamWitt dierentials over the big etale site of Sp ec F dened as the

subsheaf of the de RhamWitt sheaf W generated lo cally for the etale top ology

r

r

d log x I So Zp is a complex of etale by sections of the form d log x

sheaves concentrated in degree The Verlagerung maps V W W pre

n n

r

serve logarithmic dierentials hence can b e used to dene Q Z as lim Zp

p p

r

Corollaires I and I of yield exact sequences of etale sheaves

s

V

r r s s

Zp Zp Zp

hence exact sequences

r

p

r

Zp Q Z Q Z

p p p p

We now dene QZ as lim Q Z We sometimes abbreviate

p

n

nchar F

QZ by

Notation We denote by resp the complex of sheaves over the big

Zar et

Zariski resp etale site of Sp ec F asso ciated to the presheaf U U of

When necessary we denote by X resp X the restriction of

Zar et Zar

resp to the small Zariski resp etale site of a scheme X We drop indices

et

when the context makes it clear what site we are in

Motivic cohomology of smooth varieties

Let X b e a smo oth connected variety over a eld F We compute the etale hyperco

X X H homology groups H

et

et et

i

Theorem H X is

et

i for i

ii K F X for i

ind

iii H X K for i

Zar

iv H X K for i

Zar

v Coker cl for i

X

i

vi H X QZ for i

et

where cl is dened in theorem Moreover for i there is a short exact sequence

X

X H QZ X H CH X H

Zar et

As an immediate application we get

Corollary In characteristic weighttwo etale motivic cohomology is homo

topy invariant In characteristic this is stil l true up to cohomological degree

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Applications of WeightTwo Motivic Cohomology

To prove theorem we shall use the Leray sp ectral sequence

pq p pq

q

E H X R H X

Zar et

asso ciated to the changeofsites map X X For the convenience of the

et Zar

reader we prove a wellknown general lemma

j

Lemma Let X be the generic point of the irreducible normal scheme X

q

and let A be an etale sheaf over Then the cohomology groups H X j A are

et

torsion for al l q

Proof Let Sp ec K Consider the Leray sp ectral sequence for j

pq p pq

q

E H X R j A H K A

et et

Since the abutment is Galois cohomology it is torsion for p q and we have to

q

prove that R j A is torsion for all q But since X is normal it is geometrically

q

unibranch and the stalks of R j A are Galois cohomology of the strict Henselizations

of K relatively to the p oints of X hence the claim 2

q

Lemma The Zariski sheaves R are as fol lows

i for q

ii The constant sheaf K F X for q

ind

iii K for q

iv for q

q

v H QZ for q

Proof i is obvious iii is proved in th and ii resp iv is proved

in prop resp in prop but only up to torsion This partially

comes from the insistence to deal with g r K rather than with K We give pro ofs

ind

of ii iv and v

Denote by K resp H the etale sheaf asso ciated to the presheaf

ind

R K R resp R H R for etale Sp ec R X Let x X We claim

ind

that there is a chain of isomorphisms

H O H O H H O K

X x X x X x ind

et et et

H K K K K

ind ind

et

i

The rst isomorphism from the left simply comes from the fact that H

for i The last one is proven in prop see also th By

theorem if A is a lo cal ring of a smo oth variety then K A K K

ind ind

is bijective where K is the eld of fractions of A Letting j Sp ec K X b e the

inclusion of the generic p oint this shows that the map K j j K is an

ind ind

isomorphism hence the third isomorphism in Finally by prop for any

lo cal ring A whose residue eld contains more than elements there is a surjection

K A H A

ind

which is bijective if A is a eld Therefore the commutative diagram

sh sh

K O H O

ind

X x X x

o

y y

sh sh

H K K K

ind

x x

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Bruno Kahn

sh sh

is the strict Henselisationes of O and K is its eld of fractions where O

X x

x

X x

sh sh

is an isomorphism we used again for H O shows that K O

ind

X x X x

the left vertical isomorphism This proves the second isomorphism in which

proves lemma ii

We note that iv follows from iii the MerkurjevSuslin theorem for the lo cal

rings of X th the fact that R is torsion th and the triangles

r

n

p

r

Zp

n

in the derived category the second triangle in the case char F p The rst

triangle is proven exact in and only for n o dd relying on the computation of

torsion and cotorsion in K lemma However the pro of go es through just

ind

as well for n even by using the isomorphism from already mentioned The second

triangle is proven exact in lemma only for r and p this fact was

overlooked in However the pro of of lemma carries over in the same

r

for W way using ii and the Blo chGabberKato isomorphism K E p

r

E log

any eld E of characteristic p cor

Finally let us prove v By the triangle we have a long exact sequence of

Zariski sheaves

i i i i

R Q R QZ R R Q

i

so that it is enough to see that R is torsion for i For i this is iv

For i we have a long exact sequence of sheaves

i i i

R K R R K

ind

i i

so it is enough to see that R K and R K are torsion for i In view of

ind

the isomorphism see ab ove

j j K K

ind ind

i

the rst one follows from lemma We are left with proving that R K is torsion

for i As in pro of of lemma we have a Gersten resolution

a a

i Z i G K j K

m K

x x

(2) (1)

xX xX

This complex of etale sheaves is not exact but up to torsion it is Therefore up

to torsion there is a sp ectral sequence of Zariski sheaves

pq

q p pq

R C R K E

p

where C is the pth term of the ab ove resolution of K Since C is of the form

q

j F the same argument as ab ove shows that E is torsion for q The stalks of

q q

E and E are sums of Galois cohomology groups so are torsion for q This

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Applications of WeightTwo Motivic Cohomology

pq

shows that E is torsion for p q except p erhaps when q But for x X

the stalks of E and E at x are the cohomology groups of the complex

a a

Z F y K H K K

(2) (1)

y Y y Y

where Y Sp ec O Comparing with the exact sequence Gerstens conjecture

X x

a a

Z F y K K

(2) (1)

y Y y Y

and using the fact that the map K K H K K K has torsion kernel and

cokernel we get that has torsion cohomology groups which concludes the pro of

of lemma v 2

Proof of theorem As indicated ab ove we use the sp ectral sequence

i is obvious in view of lemma i and so is ii in view of the isomorphism

H X H X R

et Zar

p

and lemma ii To get further we observe that E for p since R

p

is constant and E for all p in view of lemma iv This and lemma iii

immediately imply iii and iv Still by lemma iii and Gerstens conjecture

p

E for p and E CH X this and lemma v for q gives

the exact sequence We now note that the ab ove information and lemma v

i

X is torsion for i v and vi now follow from and the imply that H

et

long exact sequence

i i

i i

H X Q H X QZ H X H X Q

et et

et et

2

Remark The same computation gives the cohomology sheaves of X

Zar

H X K K

Zar ind

H X K

Zar

i

H X for i

Zar

From this we deduce a triangle precising prop

R

Zar et

R QZ

In particular

Q R Q

Zar et

We also get the following analogue of theorem

K K if i

ind

i

i

Theorem H X

H X K if i

Zar

Zar

Zar

otherwise2

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Bruno Kahn

Relative motivic cohomology I

Let j Sp ec F X X b e the inclusion of the generic p oint and F X X b e

the homotopy bre of the morphism

X R j F X

et et

i i

Denote the hypercohomology group H X F X X by H F X X so

et

that we have a long exact sequence

i i i i

H F X X H X H F X H F X X

et et

This gives

i

Lemma The groups H F X X are for i there are exact sequences

K F X H X K H F X X H X K

Zar Zar

H F X X CH X

H X H H F X

Zar et

H F X X Coker cl H F X

X et

Proof The rst claim is clear for i for i and it follows from theorem

and the injectivity of H X H K F X For i it follows from theorem

Zar

again plus the vanishing of H F X For i we have a cross of exact

sequences

y

H F X X

y

X H X H CH X H

Zar

et

y

F X H

et

y

H F X X

y

Coker cl

X

y

H F X

et

The map H X H F X factors through H X H

Zar

et et

H F X which is injective A diagram chase concludes the pro of 2

et

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Applications of WeightTwo Motivic Cohomology

For simplicity let us denote by X K K F X the group K F X H

Zar

Using the Ho chschildSerre hypercohomology sp ectral sequence

p

q pq

H F H F X X H F X X

et

i

and the vanishing of H F X X for i we get an isomorphism

H F X X H F H F X X

and an terms exact sequence

H F H F X X H F X X

H F H F X X H F H F X X H F X X

hence using lemma

Proposition There are exact sequences

G G

F F

H X K H K F X K F X X K

Zar Zar

H F K F X H F H F X X

H F H X K H K F X F

Zar

G

F

F X X CH X CH X H F H

H F H F X X H F X X

2

The exact sequence follows immediately Moreover we also get lemma

Relative motivic cohomology II

We recall some notation

i i i

As ab ove H X j resp H j is shorthand for H X QZj resp for

et

i

H QZj

is the map H F H X H

G

F

X is the map CH X CH

We also denote by H X K the group H X K K F

Let X Sp ec F b e the structural morphism and XF b e the homotopy

bre in the derived category of the morphism

F R X

et et

i i

Denote the hypercohomology group H F XF by H XF so that

et

we have a long exact sequence

i i i i

X H XF F H H XF H

et et

This gives

i

Lemma The groups H XF are

i for i

ii K F X K F for i

ind ind

iii H X K K F for i

Zar

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Bruno Kahn

Moreover there is a complex

H X K H XF Ker

Zar

CH X H XF H F QZ Coker cl

X

This complex is exact except perhaps at H XF where its homology is

Coker In particular we have an isomorphism

H X K H X F

Zar

and a short exact sequence

X H X F H X H CH

Zar

Proof i ii and iii immediately follow from theorem and the exact sequence

The complex and the value of its homology follow from the cross of exact

sequences and

y

H X K

Zar

y

H XF

y

H F

et

y

CH X H X H X H

Zar

et

y

H XF

y

F H

et

y

Coker cl

X

and the lemma of the th 2

We now consider the hypercohomology sp ectral sequence

p q pq

H F H X F H XF

p

Note that E for p since the group K F X K F is uniquely

ind ind

divisible by prop Hence we get an isomorphism

G

F

H X K H X K

Zar Zar

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Applications of WeightTwo Motivic Cohomology

and an exact sequence

H X K H XF H F

Zar

G

F

H F X K H X K H

Zar Zar

noting that H X F H X K by lemma The isomorphism is

Zar

Suslins cor but we get it here by a formal argument in the vein of th

a The cross of complexes the ab ove exact sequence and

y

X K H

Zar

y

G

F

H F H F H X K H XF H X K H X K

Zar Zar

Zar

y

Ker

y

CH X

y

H XF

y

F H

et

y

Coker cl

X

contains via the lemma of the th all the information one can easily get in this

generality

Assume now that H X K Then the exact row in the ab ove diagram

Zar

G

F

reduces to an isomorphism H XF H hence we get a com X K

Zar

plex

G

F

Ker CH X X K H X K H

Zar Zar

H XF H F Coker cl

et X

with homology Coker at H XF and elsewhere

Moreover the sp ectral sequence and lemma give an exact sequence

X K H XF H F H

Zar

G

F

H H F H X F X K

Zar

Putting and together we get a cross of complexes the horizontal one

exact the vertical one exact except p erhaps at the crossing p oint

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Bruno Kahn

y

H X K

Zar

y

G

F

X K H

Zar

y

Ker

y

CH X

0

y

G

F

H F H H F H X K H XF H X F X K

Zar Zar

y

H F

et

y

Coker cl

X

Note that Ker Ker by We get theorem a from this cross and the latter

remark by a diagram chase analogous to the lemma of the th The same diagram

chase gives us the complex and shows that its cohomology coincides with that of

a complex

X K Coker Coker H F H

Zar

Notice the short exact sequence from

G

F

Coker Coker H X H

Zar

Using this exact sequence we easily conclude the pro of of theorem 2

Purity

In this section we establish a purity theorem for Zariski and etale weighttwo motivic

cohomology generalizing results of Recall that is dened as G and

m

as Z in b oth the Zariski and etale top ologies We also need such complexes

for i

Definition For i we dene

i

Zar

i QZi no pprimary part in characteristic p

et

The following theorem extends and precises th the metho d of pro of is

dierent

i

Theorem Let X be a smooth variety over a eld and let Z X be a closed

immersion with Z smooth of codimension c

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Applications of WeightTwo Motivic Cohomology

a There is an isomorphism in the derived category of complexes of sheaves over

Z

Zar

Z cc R i X

Zar Zar

Zar

b There is a map in the derived category of complexes of sheaves over Z

et

Z cc R i X

et et

et

whose homotopy cobre is concentrated in degree c and has pprimary torsion

cohomology where p is the characteristic exponent of F In particular if char F

this map is an isomorphism

i

Lemma Let Z X be a smooth subvariety of X of codimension c Then

p

A for al l p a For any constant sheaf A over X R i

Zar

Zar

for p c

p

K b For any n R i where K if n c

n nc

Zar

K for p c

nc

Proof a is trivial and b follows in a wellknown way from Gerstens conjecture

eg x 2

Proof of theorem a Apply R i to the triangle

K

ind Zar

K

and apply lemma noting that the Zariski sheaf K is constant

ind

For the pro of of theorem b we need some facts on etale cohomological purity

For all m there is a morphism

Zm cc R i Zm

et

For m prime to the characteristic exp onent of F this morphism is the classical

purity isomorphism of SGA eg th For char F p and m a p ower

of p it is comes from Gros thesis I I its homotopy cobre is concentrated

in degree c In the general case we dene the morphism comp onentwise on the

primetop and pprimary parts

The following rather trivial lemma is very useful

S D T the Lemma a Let f S T be a morphism of sites and R f D

functor induced from the bounded below derived category of Abelian S sheaves to that

of Abelian T sheaves Let C be a bounded below complex of Abelian groups that we

view as a complex of constant sheaves over S Then there is a natural isomorphism

of functors

L L

R f C C R f

and a natural morphism of functors

L L

f C C f

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Bruno Kahn

b Denote by i Zar or et the map corresponding to i from Z to X smal l

sites Then with C as in a there is a natural isomorphism of functors

L L

R i C C R i

Z

Proof a For A B two Ab elian groups let A B denote T or A B We note that

is left exact and its unique nonzero higher derived functor is R Hence there

is a natural isomorphism

L

R

D C D C

for all C D D Ab

Therefore the natural isomorphism of the lemma is equivalent to a natural iso

morphism of functors

R R

R f C C R f

which in turn will follow from a natural isomorphism

f A F A f F

for any Ab elian group A and any sheaf F over S Note that since is left exact

the presheaf U A G U is a sheaf for any sheaf G over any site Therefore given

U S b oth sides of evaluated on U are A F f U Finally the second

natural transformation say follows from the rst one by adjunction

b Follows from a considering the triangle of functors with j X Z X the

complementary op en immersion

i R i I d R j j i R i

X

and the fact that i is fully faithful Here we dropp ed the index for notational

simplicity 2

Note that the triangle and its analogues for i can b e reformulated as

quasiisomorphisms

L

i Zm Zmi i

et

over the big etale site of Sp ec F Note also the obvious quasiisomorphisms

i i i

Zar et

Using and lemma they give by adjunction morphisms

L

i Zm R Zmi o i

Zar

over the big Zariski site of Sp ec F

Let nally X X and Z Z b e the natural morphisms of

X et Zar Z et Zar

small sites Note the natural isomorphism of functors

R R i R i R

Z X

et Zar

over the small Zariski site of Z It can b e obtained for example with the help of

compare I I

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Applications of WeightTwo Motivic Cohomology

There is a diagram

L L

Z cc Zm R i X Zm

Zar Zar

Zar

y y

R Zm R Zm cc R i

X Z

Zar

where the vertical maps are given by the top horizontal map by theorem a

and the b ottom horizontal map is dened by applying R to and using

Z

The notation in the top right corner is unambiguous thanks to lemma

Lemma Diagram commutes up to sign

Proof As in the pro of of lemma this b oils down to the fact that the Gersten

complex for K theory is compatible with the Gersten complex for etale cohomology

via the Galois symbol m prime to char F or the dierential symbol m a p ower of

char F The rst case is wellknown see cor and pro of of lemma for

the second one 2

Proof of theorem b We rst construct the map There is a tautological

natural transformation stemming from

R i R i

Z Zar et X

hence a morphism in the derived category of etale sheaves over Z

Z cc R i X

Zar et

Z et

where we used a and On the other hand the triangle

Q

QZ

deduced from yields a map

R i QZ R i X

et

et et

Passing to the colimit in we get a morphism

QZ cc R i QZ

et

whose homotopy cobre is concentrated in degree c and has pprimary torsion

cohomology Shifting and comp osing with we get a morphism

X QZ c c R i

et

et

For c we use to dene the map of b noting that it b ecomes then

X Z cc R i

et et

et

via For c we use to dene this map

We now prove the prop erty of the map of b as claimed in the statement of

theorem It is enough to do this after tensoring and by Q and Zm for

all m in the derived sense Since R is fully faithful we may even apply this

Z

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Bruno Kahn

functor to the situation

Supp ose rst that c Using and a we see that the morphism

R Z cc Q R R i X Q

Z et Z et

et

is a quasiisomorphism On the other hand there is a commutative diagram

L L L

Z c Zmc R i X Zm R i X Zm

et Zar et

Z Zar

et

o

y y

Zm Zm cc R i

et

In this diagram the left square is obtained via and by applying adjunction

to and using lemma the right triangle is obtained via and The

left vertical map and the southwest map come from the triangle

The b ottom horizontal map is none else than its homotopy cobre is

pprimary torsion and concentrated in degree c The left vertical map and the

southwest map are quasiisomorphisms by hence the top comp osite has the

same cobre as the b ottom map This proves theorem b in the case c

Supp ose now that c We rst have

R R i X Q R i R X Q R i X Q

Z et X et Zar

et Zar Zar

by and a On the other hand tensoring by Zm and using yields

L

Zm cc R i X Zm

et

et

Using we get a comp osition

L

Zm cc R i X Zm R i Zm

et

et et

which is clearly This concludes the pro of of theorem b 2

Cohomology of projective bundles

Let E X b e a vector bundle of rank n and P X the asso ciated pro jective

bundle Our aim in this section is to compute R P and R P

Zar et

In order to state the theorem we remark that there are pairings i

L

i i

Zar Zar Zar

over the big Zariski site of Sp ec F if F has more than two elements and

L

i i

et et et

over the big etale site of Sp ec F

For i and are the pairings of prop for i they are

tautological For i and in the etale case the triangle analogous to for

Documenta Mathematica

Applications of WeightTwo Motivic Cohomology

L L

shows that for all i the morphism QZi QZ QZi

et

is a quasiisomorphism Therefore it suces to dene morphisms

L

QZi QZ QZi

for all i Z This is nothing else than Tate twists of the natural isomorphisms in

D Ab

L

Q Z Q Z Q Z

l l l l l l

for l char F Finally for i the pairing is dened similarly using the natural

map

QZ Z

et

Let L b e a line bundle over an F scheme S Let Zar or et Via its class

L H S G H S denes morphisms of complexes

m

j

L

i j j i

jS jS

where means restriction to the big site of S In particular for S P and

jS

L O we get maps

j

O

j j j

jP jP

hence by adjunction a morphism

n

a

j j R

jX jP

j

We are now ready to state the result

Theorem The morphism is a quasiisomorphism for Zar or et for

Zar assume F has more than two elements

Proof We pro ceed as in the last section rst proving the Zariski case Let A b e

a lo cal ring of X and K b e its eld of fractions The restriction of E to Sp ec A is

n

trivial hence P P Lo oking at the maps induced by on cohomology

Zar

j Sp ec A

A

sheaves and using theorem we can identify them to

K K K K T T

ind n ind

n

K A H P K

Zar A

n

A H P K

Zar A

n

Z H P K

Zar A

We have to show that all these maps are isomorphisms The rst one is an

isomorphism b ecause K is invariant under rational extensions The other ones

ind

follow from lemma

In the etale case it is enough to check that is a quasiisomorphism after

tensoring by Q and by Zl for all prime l In the case of Q we reduce to the Zariski

case as ab ove by applying R and using

Documenta Mathematica

Bruno Kahn

For Zl we rst need a lemma Note that there are pro ducts

L

Zl i Zl Zl i

For l char F they are nothing else than Tate twists of the natural pro duct in

D Ab For l char F and i they come from the pro ducts

i

i

log log

log

Lemma For any prime l and any i the diagram

L L L

i Zl i Zl

et et et

y y

L

Zl i Zl Zl i

L

commutes where the top horizontal map is Zl the bottom horizontal map is

and the vertical maps are deduced from

Proof For l char F this follows from For l char F it follows from

the denition of the logarithmic symbol since for i the etale sheaf K is

ind

uniquely l divisible 2

L

If l char F using lemma Zl b ecomes the map of th

which is a quasiisomorphism Tatetwisted twice If p char F still using lemma

L

Zp b ecomes the map

Zp R

jX jX jP

log log log

shifted which is an isomorphism by cor I 2

Corollary H X H H P H

Zar Zar

Proof We dont really need for this Consider the commutative diagram

with exact rows

X H X H CH X H

Zar

et

y y y

CH P H P H P H

Zar

et

where the rows come from Using theorem and the analogous result for Chow

groups the b ottom left horizontal map can b e rewritten

CH X CH X CH X H X H X H X

et et et

i i

The result now comes from the fact that CH X H X i is an isomor

et

phism for i 2

Documenta Mathematica

Applications of WeightTwo Motivic Cohomology

The coniveau spectral sequence and Gerstens conjecture

By the standard pro cedure we can construct a coniveau sp ectral sequence

a

pq pq

q p

H X H X E

et

x

et

(p)

xX

q p

q p

U where H X lim H

et et

x

fxg U

U x

p

Applying theorem we get for x X

q

H F X for p

q

H F x G for p and q

m

q p

H X

et

x

q

H F x Z for p and q

q p

H F x QZp for p and q

Moreover we have exact sequences

p p

H F x QZ p H X H F x F

et

x

p p

H F x QZ p H X

et

x

where F is an l primary torsion sheaf if char F l and is if char F For

p

p the map H F x F H F x QZ p has to b e so the sequence

splits into

p p

H F x QZ p H X H F x F

et

x

p p

H F x QZ p H X

et

x

p p

This shows that E for p and E is l primary torsion for p For

pq

q E for p q except for E Z X co dimension cycles Note

also that

p

E for all p

Using theorem for P P the arguments of show that Gerstens

X

conjecture holds for etale weighttwo motivic cohomology Therefore we get a Blo ch

Ogustype theorem

pq

Theorem The E term of the coniveau spectral sequence for weighttwo mo

p

p q q

tivic cohomology coincides with H X R H X H 2

Zar

Zar

Corollary For any i

i

a The functor X H X satises codimension purity for regular local

et

rings of a smooth variety in the sense of def b

i

b H X H is a birational invariant of smooth proper varieties XF

Zar

f

c For any proper morphism P X of smooth integral F varieties such that the

generic bre of f is F X rational

i i

P H X H H H

Zar Zar

Proof a follows from theorem b follows from theorem and prop

Finally c follows from b and corollary In x we give a general

pro of of these prop erties for suitable cohomology theories with supp orts 2

Documenta Mathematica

Bruno Kahn

Remark As for corollary we could prove this without having recourse to

in view of lemma More precisely we could merely use Gerstens conjecture for

K theory Quillen etale cohomology with co ecients in twisted ro ots of unity

Blo chOgus and logarithmic Ho dgeWitt cohomology GrosSuwa

Projective homogeneous varieties

Let X b e a pro jective homogeneous variety in the sense of and In

particular X is rational so the assumptions of theorem are satised including

i

X H by corollary c Moreover we have K F CH X H

j i

Zar

i i

H X K for all i j lo c cit Finally the G mo dules CH X are p ermutation

j F

Zar

mo dules hence torsionfree In particular

H F H X K

Zar

Ker CH X

torsion

Let E b e the etale F algebra asso ciated to X as in We get the following

corollary of theorem containing Theorem and th

Corollary If X is projective homogeneous there is an exact sequence

H X K E Ker CH X

torsion

Zar

and a complex

Coker Coker BrE

which is exact except perhaps at Coker where its homology is KerH F

Coker cl

X

The map in corollary is describ ed by Merkurjev there is an Azumaya

E algebra A asso ciated to X and is cuppro duct by A followed by transfer

Corollary Coker is nite

Indeed Coker is nite as a torsion quotient of the nitely generated group

G

F

2 X CH

In we show that Coker is isomorphic to Ker cl

X

Acknowledgements I wish to thank JeanLouis ColliotThelene Helene Esnault

Marc Levine Eckart Viehweg and the referee for useful comments

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Applications of WeightTwo Motivic Cohomology

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kahnmathjussieufr

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