motive sb.exe

Transcription

motive sb.exe

Lecture Notes in
Mathematics
Edited by A. Dold B. Eckmann and R Takens
t
Subseries: Mathematisches Institut der Universitat und
Max-Planck-Institut fur Mathematik, Bonn - vol. 14
1400
Uwe Jannsen
Mixed Motives and
Algebraic K-Theory
Springer-Verlag
Berlin Heidelberg NewYork London ParisTokyo Hong Kong
Author
Uwe Jannsen
Max-Planck-Institut fur Mathematik
Gottfried-Claren-Str. 26
5300 Bonn 3, Federal Republic of Germany
Untv.-Bibtiothak
R&genshuf§
Mathematics Subject Classification (1980): Primary: 14A20,14C30,14G13,18F25
Secondary: 12A67, 14C35, 14F15
ISBN 3-540-52260-3 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-52260-3 Springer-Verlag NewYork Berlin Heidelberg
This w o r k is subject to copyright. All rights are reserved, w h e t h e r the w h o l e or part of the material
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© Springer-Verlag Berlin Heidelberg 1990
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Preface
This is an almost unchanged version of my 1988 Habilitationsschrift at Regensburg. My original plan was to completely rewrite it for
publication; in particular I wanted to make it more readable for the
non—expert. Finally I chose to rather publish it like it is than turn it
into a long range project. So I have only made some minor corrections
and added three appendices. Thefirstone reproduces a letter from S.
Bloch to me and the second one consists of an example by C Schoen. I
thank both for the permission to publish this material, and the latter
for the effort of rewriting the example, which alsofiguredin a letter to
me. The third appendix contains some remarks and complements
written in 1989.
Uwe Jannsen
Bonn, November 1989
This
text
category
In
part
of
II
algebraic
mixed
we
in
(very)
Chern
Grothendieck
of
algebraic
to
£-adic
cycle
of
trivial
structure
to
of
cycles
motives
of
Hodge
in
not
and
j can
algebraic
§10
that
tensions
a
mixed
£-adic
of
structures
cohomology
varieties
or
gives
- and
[D5]
rise
to
cohomology
prove
these
Tate
,
For
example
1
H (X)
i , the
and
the
i s that
realization
non-compact
of
[D9]
mixed
zero
was
introduced
as
rise
to
shows t h a t
structures,
of
the
an
algebraic
from
the
sending
of
the
a
faithful.
which
shows
- this
and
often
arise
i n [.D5
non-trivial
d i f f e r e n t weights
treatments
the
equivalence
Deligne
re-
formulation
functor
varieties,
give
theories
smooth
a morphism
is fully
these,
a
c l a s s of
parallel
the
i n both
of
modulo h o m o l o g i c a l
to
[SR]
i n p a r t i c u l a r those
interpreted
in his
,
way,
be
of
homologous
structure
the
some
il-adic
and
between
i n d i f f e r e n t cohomology
weight
geometry. Concerning
pure
cycles.
in
present
into
,[Klj
of
2
singular
cycles
a
relations
fields,
cohomology
H -^(X)(J),
and
[Ma]
weights.
the
i t s cohomological
cover
Hodge
structures
I I I we
global
coherent
A l l t h i s o n l y concerns c y c l e s
does
mixed
part
define
absolute
K-theory
phenomena
i s pure
into
or
I we
cases.
explain
codimension
part
possible,
and
In
from
fields
Hodge t h e o r y
variety
conjectures
motive
to
as
K-theory,
varieties in a
the
projective
finite
concept
algebraic
and
general
characters
specific
The
by
lated
as
In
s e t t i n g of
arbitrary varieties.
on
Background
parts.
i n the
algebraic
v a r i e t i e s over
some
three
investigate,
of
conjectures
of
motives
cycles,
cohomology
for
consists
is
]
ex-
called
Hodge t h e o r y
and
cohomology
arbitrary
too.
Indeed,
of
both
facts
are
directly
Chow g r o u p
the
r e l a t e d , and
and
a
s e t t i n g of
work o f
a category
a
K-theory,
absolute
f
[DMOS]
problematic,
Part
-
has
Hodge c y c l e s
cations
and
I
In
i n the
of
factor
and
natural
.
1
for
a
AH-motives
the
with
[Sch
1]
of
mixed
forms,
these
construction
by
[Bei
2].
only
motives
algebraic
the
for
cycles
appli-
motives
is
corresponobvious
way.
observation
(AH-motive)
c o r o l l a r y we
,
standard
of
i n an
crucial
constructed
constructed
the
higher
but
s u b r e a l i z a t i o n of
modular
representations
simple,
suffices for
Hodge c y c l e s
to
1]
Finally,
that
the
reali-
is a
direct
show t h a t
there
having
£-adic
as
Deligne
[D1]
are
(Re-
motives a l g e b r a i c a l l y ) .
of
direct factors
in
the
cohomology.
In
§2
we
realizations
category
MR^,
allowed.
some o f
of
- a
[Bei
q
.
to
so-called
extend
but
associated
a p p l i c a t i o n i s the
k
simple
related
(algebraic
nor
a
Another
In
apply
s u b m o t i v e . As
Scholl
are
methods
later
absolute
cently,
£-adic
the
introduced
a motive
realizations
neither
K
replacing
definition
whole
, [D10]
definition"
which o f t e n
Grothendieck s
start
hence
ones
4]
are
the
"working
latter
[Bei
i s quite
with
the
arbitrary varieties in
motives
motives
a
algebraic
idempotents)
we
given
of
related to
together
(the
An
language
zation
d e s c r i p t i o n of
motives
are
of
definition),
since
§1
a
mixed
cycles
definition
Deligne
Grothendieck s
dences
like
mixed
that
s a t i s f a c t o r y theory
conjectures.
in
of
suggests
Grothendieck's
gives
expects
s a t i s f a c t o r y treatment
Beilinson
algebraic
one
their
§3
we
make a
of
precise
definition
AH-motives over
of
These
mixed
prove
field
k
a category
live,
r e a l i z a t i o n s , i n which
obviously
formal
a
of
are
Tannakian
by
also
i n which
defining
mixed
categories,
and
a
bigger
structures
we
study
properties.
that
for a
c h a r a c t e r i s t i c zero
smooth
i t s &-adic,
variety
U
deRham a n d
over
Betti
a
the
field
cohomolo-
Vll
qies
all
define
taken
showing
an
o b j e c t H(U)
from
papers
t h a t one
has
with
tients
AH-motives.
In
§4
the
Tannakian
that
subcategory
Deligne's
subcategory
objects
and
can
i n MM^,
be
. This
ginal
one,
hull,
inverting
the
If
G
gives
and
MG
with
§5
Deligne
mainly
case
M^
cohomology),
, and
with
of
the
quo-
the
prove
Tannakian
projective
of
than
pure
the
ori-
pseudo-abelian
the
commutation
"Galois groups"
MM^
( f o r some
embedding
above
is
pure
. We
o f M^
associated
the
the
subcategory
changing
then
which
i s d e f i n e d as the
smooth,
full
and
k
H(U)
t a k i n g the
o b j e c t and
pro-unipotent
quotient of
i s , except
contained
MG
U,
identifying
of
fibre
M^-»MM^
is reflected
in
an
5.13
1
Hp(X,ZZ ( 1 ) )
The
and
or
conjectures
i n those
G
with
the
maximal
.
f o r theorems
cohomology
motivational.
are
the
definition
of
categories
that
in
of p r o - a l g e b r a i c groups
connected,
II
by
the
simpler
-* G
and
are
-> G -> 1 ,
pro-reductive
Part
with
the
a h o m o m o r p h i s m MG
with
a
theory
m o t i v e s ov*?r
realizations
are
Betti
i n each
identified
processes
defines
1 -> U -> MG
be
identified
given
sequence
can
the
functor
exact
mixed
generated
Lefschetz
n e u t r a l Tannakian
by
M^
a p p l i e d here
main p o i n t c o n s i s t i n g
isomorphisms,
of
MR^
by
a v o i d i n g the
constraints.
the
of
category
generated
varieties,
MM^
the
techniques
filtration
comparison
category
. The
Deligne,
a weight
compatible
are
the
of
i n MR^
formulated
5.15
etale
cohomology
s t a t e d here
later
(comparing
O(X)*
1
H^ (X,(1))),
t
f o r the
smooth
for arbitrary
varie-
ties .
In
§6
a very
and
Ogus
[BO],of
the
aspects
of
important
tool
appears,
a twisted Poincare
a cohomology
theory
the
duality
and
an
n o t i o n , due
theory,
to
Bloch
axiomatizing
a s s o c i a t e d homology
theory.
Vlll
In
this
n
setting
the Poincare
-* H
1
(0.1)
H (X J)
f
between
with
cohomology
values
of
weights
or
mixed
2 d
i n a tensor
modeled
£-adic
Tate
to arbitrary
( 0 . 1 ) . We
if
the classical
true
-
also
X
f o r mixed
The b a s i c
structures
& - a d i c , deRham a n d B e t t i i n part
values
i n MR^
the right
being
reobtained
i s true
i s , and t h a t
.
i s that
formulations
conjecture
I - that
o f Hodge a n d
observation
the c l a s s i c a l
Hodge c o n j e c t u r e
Hodge
the conjectures
Hodge
a version
i n t r o d u c i n g t h e concept
with
how t o e x t e n d
this
. We d e f i n e
the results
theory
varieties.
f o rthe Tate
i f and o n l y
t h e same
i s basically
characters
and Riemann-
conjectures.
§ 8 we r e c a l l
In
f o r smooth
extending
that
d = dim X ,
After discussing
duality
show
i s an isomorphism
the situation
i s t h e homology,
by
,
category,
sheaves.
§7 we p r o p o s e
setting
(X,d-j)
after
i s a Poincare
In
i
and homology
c o h o m o l o g y we p r o v e
there
^
duality"
some p r o p e r t i e s o f C h e r n
Roch t r a n s f o r m a t i o n s
assuring
(0.2)
H^(X,$(b) )®$£
(0.3)
H?J(X,®(b) )->
that
t h e maps
t
-> H ^ ( X x k , ( J ( b ) )
k
r H ( x « n ,©(b))
H
k
,
£
,
a
char
k * I ,
k = C ,
M
(where
and
i s the motivic
denotes
ties
on
H
the group
to arbitrary
In
§9
duality
varieties,
thus
relations
and
a l g e b r a i c c y c l e s homologous
why
a naive
extension
particular,
[Bei
the
3
proving
between
t o zero.
of the conjectures
disproves
a Hodge-theoretic
non-injectivity
of the Abel-Jacobi
map
CH (X)
o
-> H
"
2 j
(X,<r)/H ~
1
(X ZZ ( j ) ) + F
f
5.13 a n d
for
j
show
t othe
a,b € TZ i s f a l s e . I n
conjecture
examples
.
curves.
of realizations
o f Hodge a n d T a t e
from
1
theorems
A s a c o n s e q u e n c e we
the counterexample
2 j
conjectures
the conjectures
extensions
t
a l l functoriali-
t h e o r i e s . We s t a t e
deduce
j
satisfy
o f (0.2) a n d (0.3) f o r a r b i t r a r y
this
2] .We
b y B e i l i n s o n v i a K J (X)
o f (0.2) a n d (0.3) a n d e x t e n d
we d i s c u s s
surjectivity
defined
o f Hodge c y c l e s ) ,
o f morphisms o f P o i n c a r e
the surjectivity
5.15
homology
by B e i l i n s o n
o f Mumford on
Then
C
by
we
H
J
(
extend
X
)
In
H
o
using
everything
"* c o n t
results
§10
we
for
any r e a s o n a b l e
H
2
J
"
1
In
t o o many
f
f
i
£<3)))
recall
the fact
results
to higher-dimensional
are non-injective quite
duality
parameters.
of Bloch
theory
for finite
fields,
some i d e a s
of Beilinson
that
h i s philosophy
- o n t h e s t r u c t u r e o f Chow g r o u p s
over
Chow g r o u p .
Part
is
arbitrary
an e x t e n s i o n
some
We
a weak
imply
over
form
since
for
H
a
Tate
they
i t would
these
standard
fields,
would
,
'
with
[ B e i 4]
sheaves
earlier
should
f o l l o w from
would
ones by
be
regarded
t o t h e whole
the i n j e c t i v i t y
f o l l o w from
homology
of positive
M
X
a
over
similarities
i n [D.E.]
finite
prove
fields
i t i n some
several "classical"
at least
The
3F
p
conjecture
and f l a t
.
Note
f o r a smooth, p r o j e c t i v e
X
would
varieties
X
by w r i t i n g
explain this
k
con-
i f we a s s u m e
of arbitrary
characteristic,
= I i m H ,(X , Q ( b ) ) . We
a ot
a
of a global function f i e l d
even
over
i n § 1 2 we
of singularities.
varieties
(X,Q(b))
the case
motivic
conjectures
forvarieties
of motivic
fields
non-proper
the
that
of resolution
arbitrary
M
the s i -
o f smooth p r o j e c t i v e
I think
(0.2) i s a n i s o m o r p h i s m ,
a description
a
i s that
on mixed m o t i v e s
- extending
on smooth, p r o j e c t i v e v a r i e t i e s ,
X = Iim X
+a
the base
map.
a n d show t h a t
jectures
fields.
Our b a s i c c o n j e c t u r e
t h a t here
cases
conjectures
of Grothendieck's
remark
cycle
III
principally,
global function
o f mixed
Bloch
of
varieties
theme o f o u r c o n j e c t u r e s
and B e i l i n s o n ,
some q u i t e e x p l i c i t
as
- provided
The main
imply
varieties
,
fields.
§11 we
stress
^'
maps
Poincare
i s different
number
X
Bloch's
of several conjectures
and
(
maps
[ B i 1] .
Abel-Jacobi
contains
tuation
We
k'
extend
show t h a t
and
G
of Bloch
and
field
(
t o t h e £-adic A b e l - J a c o b i
transition
i n more
t h a t we
, and
and t h e d i f f e r e n c e s t o t h e a p p r o a c h
maps,
c
detail
need
observe
o f A r t i n and
We
but
don't
i n §13
(0.4)
have
we
(where
range"
gical
discuss
êt
a
> dim
only
pieces
our
Final
conjectures
learnt
lectures
by
cated
my
to
It
latter
The
for
a
in
recent
and
four
paper
work on
I would
Galois
cohomolo-
mixed
p o s s i b l e , i . e . , Tate
In
§14
with
we
by
define
linear
this
structures
objects,
a class
spaces,
property,
ideas
like
so
of
like
and
prove
most
in
inspiring
varieties.
(Harvard
of
the
on
to
N.
Schappacher
from
paper
thank
Hodge c y c l e s
198 3/84), a n d
my
own
investigations
whether
the
such motives
(see
by
Ogus, and
Bloch
K-homology
chapters
by
I I and
Beilinson*s
"stable
them
and
and
extension
for this
realiza-
§ 1 ) . A.
Scholl
communi-
classes
inspiration
(cf.
and
the
end
of
1985
I t should
be
noted
further discussions.
to
a
exist
few
our
It will
also
1-motives
[D5]
in
.
[D10]
form
category
be
i n f l u e n c e d by
I would
to
in this
mathematicians.
to
Deligne.
I I I are
B e i l i n s o n , but
Deligne's
as
for absolute
construction similar
much p a r t s
certain
structures are
stratified
f o r m s come
communicated
that
pure
remain.
a question
some i d e a s
first
were
of
simple
motives
i s a pleasure
two
to
i n the
.
varieties)
attention to
me
of
acknowledgements
f o r modular
brought
[J3]
Anderson
s t a r t e d by
tions
is related
f o r these
about
G.
bijectivity
*
e t a l e homology)
modified
as
flag
the
fields,
£
phenomena
or
on
f o r number
t
. This
are
r e m a r k s and
I
a
H^ (X,© (b) ) ,
(containing those
Grassmannians
and
-
counterpart
mixed
varieties
were
+ b
conjecture
conjecture
certain
X
extreme
whose p u r e
that
is a
general
a
investigations in
The
§6).
similarly
H^(X,Q(b)
r
of
a
like
and
thank J . Neukirch
to
his
MR^
clear
work
since
also
to
and
the
ideas
s t r e s s the
appears
reader
of
how
Bloch
influence
of
r e i n t e r p r e t a t i o n of
heartily
for his
constant
Xl
enthusiasm
their
at
was
interest
encouragement,
and
support.
Bonn, w h e r e t h i s
written.
Gazo
for
and
program
Special
f r o m t h e MPI,
a phantastic
Also
was
t h a n k s go
and
typing
and
a l l friends
I thank
started
t o K.
in particular
under
the
i n Regensburg
Max-Planck-Institut
and where
Deutler,
t o M.
b i g pressure
for
the
final
M.
Grau
and
Pertl
from
Regensburg
of time.
H.
part
Wolf-
Contents
Preface
III
Introduction
PART I
V
-
MIXED MOTIVES FOR ABSOLUTE HODGE CYCLES
§1
Some remarks on absolute Hodge cycles
1
§2
The category of mixed realizations
9
§3
The mixed realization of a smooth variety
25
§4
The category of mixed motives
43
PART II
-
§5
ALGEBRAIC CYCLES, K-THEORY, AND EXTENSION CLASSES
The conjectures of Hodge and Tate for
57
smooth varieties
§6
Twisted Poincare duality theories
§7
79
The conjectures of Hodge and Tate for
107
singular varieties
§8
Homology and K-theory of singular varieties
121
§9
Extension classes, algebraic cycles, and the
139
case i = 2j — 1
§ 10
On the non-injectivity of the Abel-Jacobi map
§ 11
Chow groups over arbitrary
PART III
-
169
fields
178
K - T H E O R Y AND t-ADIC COHOMOLOGY
§ 12
Finitefieldsand global function
fields
§ 13
Number
§ 14
Linear varieties
214
Appendix A:
A letter from Bloch to the author
222
Appendix B:
An example by C. Schoen
224
Appendix C:
Complements and problems
227
fields
189
205
Bibliography
235
Notations
241
Index
244
1
j
!
PART I
MIXED
§1.
MOTIVES
Some r e m a r k s
Let
k
beddable
G^
i n
these
H (M)
1
H (M)
Then
-
( f o r each
Hodge
1
Co
of
i s em-
k
and l e t
with motives
[D6],
k
f o r ab-
see also
notations
over
asi n
has r e a l i z a t i o n s
a descending
1)
which
filtration
a Q -Vector
1
F^
space, on
continuously,
o: k «-* (E)
structure
on
a Q-vector
space
H ( M ) 8 3R , i . e . , a Q 0
structure,
o f t h e same f i n i t e
oarison
M
number
embedding
a Hodge
k
by D e l i g n e i n
with
acts
( f o r each
with
all
prime
c
closure
we d e a l
a motive
a k - v e c t o r space
Gj
zero,
we u s e s i m i l a r
-
-
0
as d e f i n e d
§6, i n p a r t i c u l a r
which
cycles
of characteristic
. In the following
references.
(M)
D R
Hodge
(T . F i x a n a l g e b r a i c
Hodge c y c l e s
[DMOS]II
HODGE C Y C L E S
on a b s o l u t e
be a f i e l d
= G a l (Ji/k)
solute
H
FOR ABSOLUTE
dimension. Furthermore,
there
a r e com-
isomorphisms
:
V
r 0
M
>
•c
C
H
~
DR
( M )
®k,c «
and
for
each
If
an
H
D
extension
X
integer,
R
( M )
=
H
D
R
a
: k
i s a smooth
projective
the motive
M = h (X)
( X )
=
H
H
d r
(
X
x
=
H (M)
= H (X)
= H (Xx
G
g
e t
(x/k)
= H (X)
1
variety
n
H (M)
1
(E o f
k ^
N
k
/
i sgiven
( d e Rham
Q
l
)
C,Q)
0
over
(
1
^
a
d
i
c
k
and
n ^ O
by t h e r e a l i z a t i o n s
cohomology)
cohomology)
(singular
cohomology)
.
The
comparison
ones
X
x
v
between
n$
isomorphisms
are obtained
the cohomology
over
theories
(E . N a m e l y
I
n
H (Xx ^Ce ) ^ ^ ( X x
k
and
I
i
H
t
k
f
-
1
a
from
of the v a r i e t y
i s given
C,
C
l
the canonical
)
aX =
by
it ^ (Xx E
H
t
k
f
C
l
)
i s i n d u c e d bv
M
H CaX (J) - S a s . n
n
7
If
i
we
summand
H
l e t h(X) =
of
h(X)(m)
is
rather
1.1.
Let
k-subspace
U
0)-subspace
such
U
M
M
( )
R
5 H
morphisms. Then
q
(M)
there
such that
indices
Proof
U
(this
sub-d)-Hodge
say.
DR,
1
Then
Y
giving
rise
o
r
=H
summands,
a
and
1
—
(M)
a
5
1
a : k «-* <C
a
structure,
the comparison
M = M^
where
a
© M^
runs
iso-
i n mothrough
a
a .
U
are compatible with the weight
i n the statement that
assume
a moronism
M
t o non-degenerate
* H (M)
pairings
- H (l(-r))
a
pure
the
U
are
g
of weight
r ,
of motives
(M = d u a l
a
given
a Q -Subspace
i s a sub-Q-Hodge
M(-r)
a
. Suppose
and f o r each
(M.) c H
exists
H (M)
1
each
s t r u c t u r e s ) , we may
: M
k
c o r r e s p o n d under
i s implicit
there
direct
h ( x ) t f o r some
the possible
i s a decomposition
As t h e subspaces
gradings
f
, which
a
the
*
over
i s a G^-submodule,
G
of
i s a
.
describes
be a m o t i v e
D
M
important f o r the following.
that these subspaces
tives
m € ZZ
lemma, w h i c h
£E
R
(X) , any m o t i v e
a n d some
H
D
, which
1
dim X
©
h
n=o
easy b u t very
Lemma
H (M)
X
following
m
, the m-fold Tate-twist
smooth p r o j e c t i v e
The
R ( a V ( C )
of
for
=
M)
a G
Q
1
{DR,1,a}
( T )
Q ( T )
a
= DR
a
= 1
a
= a
which
are
for
a
compatible
=
under
1
and
the
with
structure for
comparison
isomorphisms.
of
real
Hodge
H (M)SlR
$
a
fact,
of
the
to
r
motive
for a
over
k
giving
smooth
. Then
theorem
one
a
by
may
r
: h (X)
->
the
pairings
combining
with
the
Poincare
r
H (X)
a
t w i s t s by
morphism
Let
U
, U
DR
^DR
then
V
h
2
d
~
r
and
H
2 d
a
. Or:
2 d
h
-
that
of
M
n
divisor
and
Hodge c y c l e
above
the
in
C
hard
2 d
i n Hodge
d
ô
the
H
2 d
a
are
(X)
Poincare
r
and
1
V
V
be
Cf
the
, respectively,
*
B
y
t
n
under
of
e
c
o
the
m
P
then
H
~
Lefschetz
(XxX)
see
obtained
by
(l(-d))
a
pairings
, whose
give
an
iso-
composition
orthogonal
with
a
t
i
b
i
l
i
t
with
complements
respect
o
V
comparison
the
V
follows
is a
g
Q-Hodge
H
to
the
of
pairings
f
t
n
e
l
^
t
sub-Q-Hodge
like
e
s e
(x,Cx)
spaces
the
U
: the
G învariance
structure,
Q-Hodge
> O
[D4]
also
as
shows
^
g
that
p.
44,
t h a t any
structure is a direct
for a l l
O
*
G, ~
k
of
k
and
is a
U
polariza-
fl V
O
argument
the
a
the
This
n
a
isomorphisms. A l s o
(M)
from
structures.
Deligne's
a polarized
r
r
theory,
71
(2TTi) iJ;
d
pairings
r
h (X) (-r)
substructures of
invariance
of
is
dimension
a
(compare
powers
r
^^
a
of
with
~ ( X ) (d-r)
~ (X)-
(X) ( d - r )
U
correspond
tion
induce
g
O
a
i);.^ , a n d
^
correspond
.
I
are
®
d-r
V__
, V
D R l
'
X
'""-operator"
The
Y
the
twisting
motives
ample
absolute
6.2.
is
e t c . , and
3R(-r)
variety
very
[DMOS] I I
$
G^-action
homomorphism
motivic v e r s i o n of
and
a
- by
other
projective
a
=
Moreover,
assume
adding
using
a
-
a
c o n s t r u c t s an
$
the
and
structures like
structures.
H (M)®3R
f i x i d e a s we
Tate
h (X)
various
Hodge
polarizations
In
the
x
€ H
(M)
8
sub-structure
factor):
3R
=O
O
one
, where
has
C
is
the Weil
Hodge
operator:
C
s t r u c t u r e , see
=
i € S (3R ) = C
[D4]
(2.1.14).
acting
As
C
on
every
IR-
respects the
sub1
Hodge
structure
claimed.
By
and
0 V__
DR
U DR
rk
induce
U
the
= O
for
as
a
:
conclude
and
an
the
U-
and
element
= h(X)
H (M)
= h(X)
we
by
that
M
then
^
® V
=
1
O
then
a
the v a r i o u s
this
6.7
and
p
is
of
(g)
r e s p e c t s t h e Hodge
the
the
or
p
a
6.1
filtration
i s a homomorphism
of
~ O
=
are
f o r any
with
[DMOS]II
O
M^
(M)
H ( X
family
i s a projector
X,Y
.
T h e r e f o r e the
Im
p
= (h(X),p)
1
If
and
and
i n the
M
gives the
=
2
Im(1-p) ;
n o t a t i o n of
smooth v a r i e t i e s
morphism
f : Y -» X
wanted
[DMOS].
over
and
for
k
g:
with
X -+ Y
X
the
of
r e p r e s e n t e d by
r e p r e s e n t e d by
factors
of
The
r
h (X)
r
r
H (X)
a motive
g*
and
as
H V
= U
a
as
(see
p
which
f*:
Proof
=O
get
isomorphisms,
p
with
taking
have
Corollary
projective,
is
also
are compatible
V-spaces.
, note
decomposition
is
IR )
g
H (M)
a
P r ^ t i o n
a
p € End(M)
Hodge s t r u c t u r e s ) ,
kernel
IR D ( U ®
we
r
comparison
i t i s compatible
1.2.
®
g
decompositions
c
M
U
isomorphisms
, which
f
f o r the
M
. The
{DR I,a}
structures
gives
we
endomorphisms
a €
case
IR
comparison
p
for
8
g
-> H ( Y )
Kerf* c
r
h
» H (X)
Im
g* c
{DR,l,a}
(X)
r
: H (Y)
a motive
a €
and
a €
r
h (X)
the
image
of
{DR,l,a}
, and
these
are
direct
.
cohomology
groups
f * and
g*L a r e m o r o n i s m s
o
o
•
[D4] . So
Ker f*
and
Im g *
r
H (Y)
have mixed
of mixed
are
Q-Hodge
(pure)
Q-Hodge
structures,
structures
sub-Q-Hodge
structures
of
the pure,
Im
g*
i n the other
Im
g*
under
rial
polarized
lemma
In
U
Y
= Ker f*
we
) •
to
K
e
r
f*
Ker f *
as t h e s e
, and o f c o u r s e
or
X
and
are functo-
i n the 1-adic
S o we
and
reali-
can apply the
Im g * ) .
by a c o n j e c t u r e o f G r o t h e n d i e c k - S e r r e
on t h e semi-
get a result
of the action
Corollary
of
G^
r
r
r
r
which
on t h e 1 - a d i c
In the situation
above,
cohomology.
the kernel of
: H (x) - H (Y)
t h e image o f
g*
are
direct
Of
maps
factors
course,
like
: H (Y) - H (X)
r
of
H (x)
similar
of
t h e cohomology w i t h
U
of
. This
1.4.
M(f)
Proof
into
to other
natural
i s needed
form
r
->
H (X)
compact
support
o f an open
subvariety
t h e cohomology o f a smooth p r o j e c t i v e
Corollary
modular
.
G y s i n maps o r t h e c a n o n i c a l map
r
X
as G^-modules
considerations apply
H (U)
c
N
isomorphisms,
(
be t r u e more
f*
X
Q
should
simplicity
and
for
H
correspond
( ^ - i n v a r i a n t subspaces.
particular
generally
1.3.
exist
one g e t s
(with
realizations
the comparison
and a l s o
zations
Q-Hodge s t r u c t u r e s
i n the proof
The r e a l i z a t i o n s
f
by Deligne
o f the next
attached
([D6] §7)
variety
corollary.
t o an
belong
elliptic
t o a motive
.
Let
f
and c h a r a c t e r
b e a new
e
for
form
o f weight
k+2
(k = 0 ) ,
conductor
r
There
I
(
N
)
=
{
(
]
c d
€
S
L
2
(
E
)
i s a smooth p r o j e c t i v e
1
(
c d
}
s
curve
(
X
0
1
}
(N)
1
m
o
d
N
}
*
over
<D
and an
open s u b v a r i e t y
j:
such
that
the C-valued
Y (N)
X (N)
1
1
p o i n t s c a n be i d e n t i f i e d
=
^ (N)
where
1
Jfy i s t h e P o i n c a r e u p p e r
Let
N = 3 ; then
there
g:
E
halfplane.
i s the universal
->
Y
(N)
1
Deligne describes the r e a l i z a t i o n s
the
"universal
1
( X (N) , J
1
one has t o form
this
of
M(f)
as p a r t s o f
cohomology),
k
3lt
1
Sym (R
the 1-adic,de
gjtc
Q) )
Rham a n d s i n g u l a r
namely as k e r n e l o f
T
- a
n
prime
on
to
N
, where
t h e cohomology
a^
are not i n
sense:
Let
T
Q(a.,a_,...),
I z
Q
the
and
T^
By
t h e n we
1
n
2
T
T
i
z
k
by t h e
T -•> E
T^
by
T
n
1
( Y (N) , S y m ( R g 0 ) ) )
+
1
»H
n
. I f
k
1
1
N
k
1
denotes
K a
E =
( X (N) , j * S y m ( R g*Q) )
1
H^
and
b y oc .
1
H (Y (N),Sym (R g Q))
which
acting
diagram
;
in
n
morphism, d e f i n e t h e r e a l i z a t i o n s o f
annihilated
1
generated
have a morphism
t h e commutative
H
fora l l
n
I
a q
, q = e
. I f the
\ A
n
n-1
, one h a s t o t a k e t h e k e r n e l i n t h e f o l l o w i n g
be t h e Q - a l g e b r a
as the p a r t
versions
a r e t h e Hecke c o r r e s p o n d e n c e s
f (z) =
OL i s t h e k e r n e l o f t h i s
M(f)
curve
cohomology"
H
of
elliptic
,
and
(i.e.,
c o m p a c t i f i c a t i o n by
adding the cusps
,
T (N) \
1
with
cohomology w i t h
+
compact
support
,
and the
maps
of
are the canonical
M(f)
o n e s , one c a n a l s o
t o be t h e k e r n e l
of the
T
define
- a
n
the r e a l i z a t i o n s
i n the parabolic
n
cohomology
k
HpCY
Sym
a
k
1
(N) , S y m ( R
1
(R g*<R)
direct
factor
k-fold
formula),
of
fibre
where
Finally
Q) ) = Im(H^(Y
R
E
k
factor
(g^) Q
=
E
x
q
k
E
( Y (N) , . . . ) ) .
1
which
i n turn i s
Y
(
1
N
)
version
o f the Kunneth
= Y (N) .
q
1
H
k
with
+
*
E
(relative
(N) , R ( g ) * < D ) =*
H (Y (N),R (g ) Q)
1
sequence
and moreover,
1
g
q
1
k
(R g*$)*
1
by d e f i n i t i o n
P
1
of
(N) •• •"Y ( N )
1
of
the spectral
degenerates
Y
(N) , . . .) S H
1
,f o r
+
product
H (Y
P
g 3 l 5
i s a direct
:
the
1
P + Q
(E ,Q)
k
as remarked by Lieberman,
the subspace o f
H
P + Q
(E ,Q)
k
q
m
identifies
• id_
i n d u c e s t h e m u l t i p l i c a t i o n by
m
k
p . 1 6 8 . T h e same i s t r u e f o r t h e c o h o m o l o g y
, on which
, compare
[D1]
E
Altogether
of
the r e a l i z a t i o n s of
M(f)
with
compact
are direct
support.
factors
the cohomology
H
which
K
+
1
(E ,$)
are defined
dences:
= Im(H
K
the
T
n
k + 1
(E ,Q)
as t h e k e r n e l
are also
-> H
k
k + 1
(E ,Q))
k
of several
defined
algebraic
as c o r r e s p o n d e n c e s
corresponof
E
k+1
and
so o f
which
E
[D1 ]
(3.16),
the subquotient
of
E
H^
^ J '®)
C
corresponds to
HpCY
via
, see
K
1
1
k
(N) , ( R g * Q ) ® )
the s p e c t r a l
sequence
5 H^Y
k
(N) , R ( g ) *Q)
1
k
c a n be i d e n t i f i e d
with
the subspace
k+1
of
Hp
(E ,Q)
where
k
t h e morphism
induces
t h e m u l t i p l i c a t i o n by
ing
Sym
of
to
k
1
(R g * ® )
the symmetric group
If
one l i k e s
1
i n
(R g*Q)
S
K
on
In i d x . . . x m i d
1
Iti . . . Iti
1
®k
£
k
K
£
(nu €
, and t h e p a r t
by t h e a c t i o n
E ^ .
- and i n p a r t i c u l a r i f one does
S )
correspond-
not like
to
elaborate
this
as the d e f i n i t i o n :
o f t h e above
the realizations
a
k+1
„k+1
H
( E ) = Im (H
(E )
p,a
k
a,c k
T
in
as
t h e d e Rham v e r s i o n s
1
TT
k+1
the kernel
sufficiently
T
of the
many
- one c a n t a k e
M(f)
are obtained
, f o r a € {DR I O} ,
k
a
~
n
of
(E ))
1
steps
f
' (m^ i d x . . . xrnîdg) * - m
n
£
€ ZS , a n d
a*-1
. . .it^
1
fora l l a € S
r
for
c Aut(E ).
k
k
k+1
They
are substructures,
sub-Q-Hodge s t r u c t u r e s
i . e . , G^-submodules
k+1
of
H^
the
with
(P
OT
compact
k
support
smooth
E
k
[D1]
projective
E
containing
still
variety.
of
k
E
k
E
have
Now
that
as an open
+
1
compatible
cohomology
with
with
exists
subvariety
k
by a
a smooth
projective
(either
1
o r by D e i i g n e s
i n positive
E
direct
comvariety
by H i r o n a k a ' s
construction
characteristic),
a n d by
diagram
by t h e commutative
k
e
to replace
a
appears as a subquotient
H
r
d e Rham
there
H
k
a
, i . e . , a smooth
k
of singularities
the commutative
Tj
^
f o r the coho-
i n [HL] .
5.5, w h i c h a l s o w o r k s
k+1
H^
(E )
a n c
^ K^
exist
under
and t h e c o m p a r i s o n i s o m o r p h i s m t o s i n g u l a r
g e t a m o t i v e we
resolution
C
of algebraic
c a n be found
pactification
also
E
support
. A definition
cohomology
To
compact
( Jc^'
( E ) e t c . , and correspond
t h e c o m p a r i s o n i s o m o r p h i s m s , as t h e s e
mology
e
of
(I )
k
H
+
1
< V
of
H^
( E ) . We
k
diagram from Poincare
k + 1
(E )
k
.
H
2 ( k + 1
remark
duality
>(E )
k
f
H
we h a v e
Im \\> -
( K e r p)
k + 1
(E )
k
t
2
H
c
(orthogonal
(
k
+
1
)
(E
V
k
)
complement) . T h i s
shows
that
H
we
could
express
the subquotient
( E ) = p ( i m ty) * Im i|i/Im ij; n K e r p =
k + 1
(Ker p ) / ( K e r
by
lemma
give
subquotients
1.1
(applied
compatible
again
for a l l
b y lemma
twice),
define
More e x p l i c i t e l y :
M(f)
in
lemma
H
1.1,
k + 1
(E )
( K e r p)^
p)"*" D K e r p
(Ker
For
with
bigger
a motive,
§2.
The c a t e g o r y
The
a
mixed
the
by
the motive
contains
2.1.
define
M(f)
p (H(M(f)))/
t o be
from
under
by
1
M(f) .
a motive
a
finite
again
p. 206.
section
suggest
to define
do t h i s
also
which
covers
t h e cohomology o f
i n general
by f o r m a l i z i n g
the weight
a l l
gives
rise
to
the properties
of
graduation
o f motives
filtration.
k
be a f i e l d ,
closure
Definition
absolute
a motive
i n [D1 ] p . 1 5 8 . T h i s
and t h a t
r e a l i z a t i o n s and r e p l a c i n g
algebraic
M(f).
the r e a l i z a t i o n s of motives with
non-proper v a r i e t i e s ,
again
M(f)
can c a l l
and so
the fixed part
[DMOS]
structures
a weight
a motive
o f mixed r e a l i z a t i o n s
s t r u c t u r e s . We
Let
an
that
extra
(smooth)
compare
w h i c h we
like
now
gives
i s a motive,
considerations o f the previous
category
their
n Ker p
SL^fffi/N'ZZ )
n Ker
1
be t h e r e a l i z a t i o n s o f
1
v i a taking
gives
w h i c h we
p~ (H(M(f)))
one g e t s
N'
subgroup o f
l e t H(M(f))
i s a motive,
N = 1,2
define
:
i n a l l i t s r e a l i z a t i o n s and so
a motive,
, then
k
{DR,1,a}
and the r e a l i z a t i o n s o f
substructures
1.1
a €
p)
1
k
These
i n terms o f p
entirely
of
k
which
, and
The c a t e g o r y
Hodge c y c l e s )
over
G
MR
k
k
i s embeddable
k
= Gal(k/k)
o f mixed
consists
in
C
,
k
.
realizations
of families
(for
p.
f
DR' 1' O
oo '
l , a ' l prime
t0
a: k
number
C
where
a)
H_
UK
creasing
i s a finite-dimensional
(F ) ^
( t h e Hodge f i l t r a t i o n )
and an
n t LL
filtration
(W )
(the weight
filtration).
m mtffi
H^
c)
Q^-vector
space w i t h
a
con-
G - a c t i o n a n d an i n c r e a s i n g f i l t r a t i o n
(W )
k
^
m m€5Z
weight f i l t r a t i o n ) , which i s G ^ - e q u i v a r i a n t .
H
1
C
i s a mixed
0
creasing
and
a decreasing
on
H
$ C
a
W
Gr H
ma
with
H^'
a
= Hy
a
I
the
filtrations
the
weight
,a
: H
a
tration
n
f
of
p
and
-»
~
Q
. that
1
i s
F^GrH
ma
r t
H__ ®.
£
DR
k,a
i
W
Gr H
ma
0 C =
on both
/ such
for
filtrations
of
H
0
p € G^
H-,
0
V
l
H '
a
m
q
©
H*
• ^ a
P-P
1,0,-'
H
©
_. ^
p+q=m
P
identifying
^
J
(respectively,
sides.
filtration
that
* C =
of weight
i s an isomorphism
^
i n d u c e d b y t h e Hodge
the weight
H
f
i n d u c e s a (J)-Hodge s t r u c t u r e
filtrations)
transforming
i s ani n -
C r r }
= W H /W
-H
m a m-1 a
4
i . e . , there
m
(W )
(the weight f i l t r a t i o n )
on
H^
m mcZZ
o
filtration
(F )^
( t h e Hodge
filtration)
, which
d)
00
Q-Hodge s t r u c t u r e ,
filtration
3
on
r77
tinuous
(the
n
filtration
increasing
b)
k - v e c t o r space w i t h a de-
P
into
the weight
fil-
commutes.
T
H
E
1
a
OOâ
n
d
1
I
A morphism
a
Q
r
e
f: H
c
a
l
l
^ d
H'
the comparison
o f mixed
f
f
isomorphisms.
realizations
f
^ DR' I' O^ 1
prime
is a
family
number
a: k ^ C
where
1. )
f
D
: H
R
filtrations
2. )
f^
the
4.)
and
F
O
of degree
zero
f o r the
.
i s a Q - I i n e a r G^-morphism w h i c h
1
-•H
1
i s a morphism
o f mixed
<p-Hodge
respects
,f,
structures,
*
O
compatible with the
f
and
filtrations.
: H
O
i s k-linear
r
: H ^ -* H |
f
i.e.,
H^
R
W
weight
3.)
D
and
f
filtrations
W
c o r r e s p o n d under
and
F
.
the comparison
isomor-
phisms.
2.2.
1. )
1
2. ) '
Remark
f
D
f
1
Assuming
3.)
4.)
,we
only
have
to
require
i s k-linear.
R
i s a Q - I i n e a r G^-morphism.
1
This
follows
2.3.
Proposition
Proof
This
mixed
Hodge
from
the p r o p e r t i e s
MR^
i s clear
i s an
from
structures
p a r t i c u l a r the morphism
[D4](1.1.5)
cokernels
and
with the
filtrations
and
the remark
f__
DK
D
above
abelian
, f,
and
±
filtrations
comparison
isomorphisms.
abelian category.
form an
are the obvious
of the comparison
W
and
category
f_
O
and
(componentwise)
isomorphisms.
the f a c t
[D4]. In
are s t r i c t l y
F
that
, and
compatible
kernels
and
ones w i t h the i n d u c e d
2.4.
of
Remark
de
and
Of
combine
structures
do
m
If
6 MR
W
m
realizations
category
of
of
vector
( W
I
realization,
H
W
H
W
H
m DR' m l' m a
i
I m
r
'
stands
7
w
I
o o
and
a
H
1
Q-Hodge
)
. Note
spaces with
we
define
; I
c»,o|
'
w
1
I ^ I
• m
of
mixed
H
how-
filtrations
the
subobject
by
=
where
categories
categories.
i s a mixed
r
H
1-adic
define
Hodge r e a l i z a t i o n s
categories
abelian
H
the
the
separately
and
d r
these with
i n general
form
2.5.
H
(containing
that
not
WH
could
Rham r e a l i z a t i o n s
then
ever
c o u r s e we
I
for
H
®^(r
ma
<R
the
-> w
~
>l,o,J
' m
w
restriction
H__
m
®,
DR
a
k,a
-I
1
W
i s the r e s t r i c t i o n
m
HI m G CQ - W H
m a
Q l ~
m l
m
1
1
IO
F
2.6.
if
ii)
of
Definition
W H = H
m
The
MR^
We
i)
and
W
category
of
A mixed
successive
that
realizations
are
i n general
extension
H
i s pure
of
the
i s the
direct
any
sums o f
object
pure
H
2.7.
and
full
pure
in
weight
realizations
subcategory
realizations.
MR^
c
m
of
.H=O.
m-1
, whose o b j e c t s
see
realization
is
w
Gr H
m
a
:=
m-1
There
taking
is a
the
natural
natural
tensor
structures
law
on
on
the
MR^
by
components,
defining
namely
m,
the
induced
filtrations
W
l
F
V
(H
DR
D R
and
similarly
for
by
p(x
=
the
natural
®
x')
spaces
tensor
category,
G^
on
the
comparison
Q
Q
, Q
1
=
1
zero
Q)
the
(the
" i n t e r n a l Horn")
1
n
F Hom <H ,H' >
b)
k
( H
D R
H
DR' DR
)
1
H
o
m
k
=
MR
H
DR
p
€
acting
. Furthermore,
gets
k
1.2,
; Id
,
o0f0
1
k
, F k
=
Q-Hodge
H
the
with
by
taking
for
structure
identity
of
a
object
Id ^ )
1
0
5
, trivial
structure
induced
1
€ MR
=
<
h
D
R
c
by
Q -*
of
action
type
C ^ k
of
(0,0)
,
and
=
pf(p
f
define
k
H
'
d
R
P
{
f
'
f
^r
Hom
1
D R
W
w
)
)
i
F
S
W
^
Horn(H,H )
t
€
MR
R
P
h
+
n
H'
H
f
with
1
for
R
r+m DR
(H ,H£)
m
-1
c)
H|
1.1,
(fIf(F H
=
1
weight
3
by
-
R
H (Horn(H,H ))
(pf)(h)
®
1
that
Q
H,
=
D R
H o n
<
associativity constraints
1
For
** (Horn(H,H ))
m
F
•
and
isomorphisms
Definition
W
0 R
.
1
k
H
H
unique
2.6.
a)
P
© W H
s DR
on
Q ,
(F°k
1
*
r e a l i z a t i o n s , and
[DMOS] I I
(k,
f
J-
p+q=n
other
px'
cf.
=
weight
Q
®
1.1.12)
T
WH
r DR
r 4-s=m
=
i t is clear,
1
of
the
DR>
commutativity
vector
pure
H
®
px
[D4]
®H' ) =
DR
t t
H
^
(cf.
°
r
a
1
a l l p}
1
r
}
,
'
G -action
k
^
h)
for
p
€
G
and
k
h
€ H
1
, and
similar
filtration,
1
H (Horn(H,H ))
g
= Hom (E
m
structure,
i . e . , with
above,
[D4]
cf.
,
a f
H )
a
with
Hodge and
the
weight
induced
mixed
filtration
Q-Hodge
like
d)
the obvious comparison
of
H
Then
and
we
H
have
(2.9)
Horn (T,
2.10.
For
1
a natural
H € MR
n
d
jection
the
Q
Horn(H,H')
we
(2.11)
In
H
I
F
O
I
X
Q
Y
J
for
x
f
I
o
r
a
x
, DR
1
1
€
a
F
c
^
f
f
1
(2.9)
€ MR
H
° DR
we
obtain
[SR]
0
Hodge c y c l e s
and
W
H
o DR
that
by
x^
€ T(H)
properties
€ H^
d)
n W H^
}
'
. From
s e e s by
and
and
Q
of H
o r k ^ C
Q - v e c t o r s p a c e , a s one
x
e)
proof
€
g
the d e f i n i t i o n
1
r(Horn(H,H ))
of
implies
.
functorial
isomorphisms
i s represented
MR
and
k
i s a neutral
[DMOS] I I 2 . 1 9 ) ,
tensor
category
(fibre
functors)
with
exact
by
1
H,H )
, i . e . , the c o n t r a v a r i a n t
k
® H,H')
Theorem
ones
see
Tv~»Hom(T
easily
the
.
- C
Hom(T,Hom(H,H')) ^ Horn(T ®
T H H
(see
x
to a
. Note
, H')
the s e t of absolute
=
Horn(H,H') =
(2.12)
2.13.
a
q
particular,
for
Horn (T ® H
isomorphisms
(E) n W H
a
isomorphism
finite-dimensional
comparison
F°(H ®
(functorial)
define
k
1
t o one
by
.
restricting
is a
induced
Horn (H ,H ' ) )
a
This
isomorphisms
functor
Horn(H,H').
With
Tannakian category
namely
faithful
a rigid
Q-Iinear
this
over
abelian
tensor
Q
Q-Iinear
functors
by
ly
"
MD
.
_>
each
The
category
of
finite-
dimensional
*
H
Q-vector
spaces
0
a : k «-» C .
proof
i s routine
axioms o f a T a n n a k i a n
vector
—
• ttf^
a
H
for
T/p«
spaces
additional
a n d we
and r a t h e r
a r e modeled
s t r a i g h t f o r w a r d , as the
after
are dealing with
structure),
the properties of
vector
where a l l f u n c t o r i a l
spaces
maps
(with
some
a r e the ob-
v
vious
is
ones.
We
only
note
that
the dual
H
of
H € MRj
g i v e n by
H
= Horn (H,1)
v
where
1,
-
(I
00
, a
)
v
and
, respectively.
taken
tely
H
componentwise,
determined
by
by
f »* ( f
and
only
2.15.
of
f
D R
There
MR^
1
i f
H
0
contains
R^
R^
and d u a l s ,
exactness
of subobjects
v
as
00
, o
and
and c o k e r n e l s a r e
f G Horn(H,H')
isomorphism
i s comple-
H o m ( I H ) ^ T(H)
f
/ i np a r t i c u l a r
the object
i s a
(neutral)
have
subcategory
of
functor
t o prove
subquotients.
the formation
the identity
f o l l o w s by a p p l y i n g
(twice) . But q u o t i e n t s
Tannakian
the formation
and i t c o n t a i n s
T(H) * O i f
1 .
i s c l o s e d under
only
I
O
3
of the inclusion
s t a t e m e n t , w h i c h we
H h~»H
O
Obviously
as k e r n e l s
of
-» H' .
O
i s a canonical
1
, which
products
case
: H
(D Zf (I) rf (D)
Proposition
Proof
The
i s exact,
-
Remark
are the transposes
-
and f a i t h f u l ,
J
2.14.
v
(I. - )
1, a
follows
of tensor
object
from
the second
f o r quotients,
the exact
o f pure o b j e c t s
1 .
as t h e
functor
are obviously
pure,
and t h e g e n e r a l
quotient
weights
2.16.
H © H' -» H"
we
must
Im H n Im H
There
a)
the base
b)
= H__ ® , k '
DK
K
= Res_
H
Gj^, I 1
k
k
1
1
H,H
=0
are natural
k
with
1
as f o r a
G
with
different
.
base
H
of fields
1
^k
l
extension
G
as above
G MR^,
the induced
and r e -
,
1
up t o c o n j u g a c y
G^)
C
and
H €
MR^
by
G,
K
given
closures extending
i n
and
filtrations,
K
of the algebraic
defined
x
= H
v i a t h e map
1
1
k'
extension
H'
DK
H
I
1
induction,
functors:
F o r an e x t e n s i o n
define
f o l l o w s by
of realizations
have
Definition
striction
i)
case
, with
by an
k ^
k
inclusion
1
(well
t h e same w e i g h t
f i l -
tration .
c)
1
H
1
= H
for
0
mixed
d)
1
I
Q-Hodge
for
00, o
1
— r = I
1
1, a
ii)
H
a)
For a
= R ,
k
H
D
1
o = a ' I,
IK
/ k
H'
1,a
c
1
and
H'
of
J
® ,,a'
R
C
a
as above,
O
T
€ MR
V a
7
: k "
extension
H'
<T
k'/k
t h e same
v i a the canonical
1
'
rr
and
a = O L- •
IK
define the
restriction
by :
k
(restriction
j r
H
= DR
k
for
1
finite
= H I
R
, with
°°, o
isomorphism
I
: k
structure,
. = I
1
e)
a'
0
of scalars
to
k)
with
t h e same
filtrations,
b)
H,
= Ind
l
H
G ,
to
G
k
1
, i . e . , the representation induced
from
l
k
K
(we may
assume
k = k'
and
G ,
k
5
G^)
, with the
G
weight
k
IndG ,
01 k ^
filtration
W
H
=
O
{T:
R
©
H'
- - T
x€Ja
k
1
(T I
for
x|
k
= 0}
C
H' .
1
, with
1
m
k
c)
(direct
I
for
,0
00
01 k ^
C
J
0
=
sum o f t h e m i x e d
structures),
d)
G, ,
i s g i v e n by
Q-Hodge
via
the c a n o n i c a l isomorphism
k ' ®,
'
K
e)
I
-
1
L
F
:
©
T€J
0
for
a l l
for
o
G,
- I n d * H' =
~
k'
H'
T
Q
1
G
a
p' G G ^ }
: k = k
<-+ C
{ f : G,
1
(with
1
7
C -^C
k
c|
= a
k
k' #, ,
'
<E,
K
a
T
- H' | f ( p ' p )
= p ' f (p)
(pf)(p )
f(p P))
1
the G^-action
with
©
T€J
=
Q
and
a|
1
= a
Q
i s given
by
A
A
=
5
< T>x€Jo
p
V
)
1
= Iâp-I
(
a
c ' p ^ ^
k
which
lies
i n
Ind
n
H'
G , 1
by p r o p e r t y
e)
of the
comparison
k
isomorphism.
2.17.
Definition
1(1)
is
pure
=
i)
The T a t e
(MD CD (I),
f
o f weight
-2
k(1) = k
b)
Q ( I ) = Z Z (1) ®nn; Q
1
1
P
i s t h e group
natural
Qd)
e)
I d
o
^ d ) , i d
o
1
of
filtration
/ where
n
l -th
F°k(1)
ZZ
roots
i s t h e (D-Hodge s t r u c t u r e
( f o r any
i d ^^(1)
,o
i d , -(1)
0:
i s induced
for
a
^ -
1
= 0,F~ k(1)
of unity
i n
a
k ^
by
: k
2??i Q
k
<C
Q
<n
H(n)
k ,
i s induced
(a "
1
by
(exp
realization
H
H
1(1 ) ®
®
, o f Hodge
(E) ,
2iri
and
n
n
) )
n
n € 2Z we l e t
n = 0
=
n
Hom(l(1)®~ ,H)
Tate
twist
of
H .
n < 0
=k(1),
and
n
1
F o r any mixed
n-fold
^
l
(1) = ^ i m y
n
1
1
1
the
^ ( 1 ) )
with the
k
2fri ffi - ffi (1) , 2-rrit H
ii)
l
G -action,
(-1,-1)
d)
Hodge
1
n
Q(I),
with!
a)
c)
with
1
realization
type
2.18.
W^
m
Remarks
(H(n))
=
i)
Twisting
( W H ) ( n )
m+Zn
The
Tate
objects
above d e f i n i t i o n
sentations
and
of
Tate
G
the
[d33.
We
of
Rham r e a l i z a t i o n s :
de
also obtain
weight
i s compatible
twists
and
k
the
by
-2
so
that
.
9
ii)
shifts
a
i n the
of
the
category
category
notion
with
of
usual
notion
1-adic
repre-
o f Hodge s t r u c t u r e s ,
Tate
twists
i n the
of
see
category
H„(n)
equals
H_
as
k-vector
DK
DK
s p a c e w h e r e a s t h e Hodge f i l t r a t i o n i s
F^H .-. (n) = F ^ H . - .
DK
DK
W i t h t h i s we c a n w r i t e
H(n) = (H
(n),H (n),H (n);I
( n ) , I , -(n)
DK
1
O
oo,0
i,U
D
+ n
r
r
1
The
the
next
sequel
is
P(x
p
P
to
Aut
D R
,x
l f
acts
(k«)
k
x )
on
H
D R
X ,
k
, then
factorizes
DR
®k
lies
in
T(E)
*
€
n
DK
x
k
c
t
a
^
k'
D R
^
t
via
1
r
k
on
X
= k
,
as
image °f
a
where
to
; note
k
that
=
algebraic
on
k
reader
by
k
x
i s an
= G
the
lifting
-» G
for
1
quotient,
e
€ MR
1
Aut (F )
G a l (ic/k)
n
H
k
H
so
reading.
Aut Ck )
on
1
e),
in
the
r (H
x^k)
map
basis
r (H
of
extension
closure
X
c
x
F(H
k'/k
c
J Ô *
k
.
^)
Obviously
.
Pf /
1
f
Hom(Hx k',Kx k'),
k
a n <
k
e
P
needed
r
k
=
k
z
r
f o r some f i n i t e
x
and
e
above a p p l i e s
(pf
o
4.7
first
for
and
If
a
of
not
K,
k)
k
=
f
)
proof
^')
trivially
finite
v
x
k'
x^k ).
r(Hom(Hx k',K k'))
pf
p
a c t i o n of
e
®,
o
1
r(H
J
x
Z
t e c h n i c a l and
on
projection
acts
the
H__
r(H
The
the
k
I
1
2.20
r (H
via
c
through
k
X
ZP
R
very
i n the
and
on
J ^'
G ,
x^)
1
D
and
=
,
of
H
0
k
GaMkVk')
X
(P
D R
r
2.19
acts
=
a
are
partially
skip
€ Aut (JT )
(x
sections
except
advised
2.19.
two
)
for
f
, and
for
1
p € Aut (k J
1
one
has
=
H,K
and
k
(Pf )(X)
1
Horn(Hx^k ,Kx^k )
r(Horn(H,K)x^k')
with
k
similar for
a p
1
i n p a r t i c u l a r to
Pf (P
l
- 1
X)
€ MR
f =
for
Hom(H,K) -+ Horn
=
. Here
k
^ d r ' ^ ' V
x
€
(Hx
H
€
1
k,kx
k)".
2.2Q.
a)
Proposition
The base
adjoint
b)
\vk
extension
( H
'
d)
(
1
€ MR ,
k
v
k
»
R
k t / k
Hom(H,R , H')
iii)
Hom(R
k
e)
R
(R
( H
= R
k v k
k
k
,
)
k
H'
in
H
^
a)
i
m
H
)V
isomorphism
IHx Ic'
H € MR
k
V
There
i s a natural
If
k'/k
i s a Galois
V/k
( H X
k
under
T h e maps
k , )
k
Hom(H',Hx k') .
k
k
the f i r s t
- will
t
i
P
l
i
c
be
called
the familiar
r
i f we w r i t e
-* S p e c
l
MR ,
k
J
c
v
^
=
a
<P*
n
d
k .
we h a v e
u
H' €
Kom(Hx k',H')
k
1
and
k
9 H')
k
by t h e l e f t and
f o r H € MR
a
i s canonically
i s canonically
the trace
k
t
i
o
n
b
k
Y
and
k
^ '
:
H' €
MR ,:
k
r
i s the identity.
k
f)
Proof
s
H ' ) x k' B H '
particular,
and
is
k 7 k
- especially
f o r <p: S p e c
x
.
*k '
( H
f o r m u l a s ; n o t e t h a t we o b t a i n
adjointness
k'/k
H' i
In
k
F o r t h e c o m p o s i t i o n o f t h e maps g i v e n
right
H i
v
isomorphisms,
= R
H',H)
isomorphisms
= cp*
k
H*
and r i g h t
isomorphisms
- »*k ' =
H ® tP*H' = (p* (tp*H ® H')
x k'
k V k
k .
= T(H')
= R
/ k
projection
form
of
i s left
H'*—>R
functorial
)
H'
ii)
kl / k
HHiHx^k'
functor
) V
are functorial
H
the
1
H
extension
.
k
i)
H
i s a canonical
There
These
be a f i n i t e
functorial
= V/k '
) V
H R
H
f
functor
are canonical
There
for
k
to the r e s t r i c t i o n
There
c)
Let
a direct
a direct
action
factor
of
extension,
9
H i
Aut (k')
k
of
Aut (k')
k
t h e map
R . (Hx k')
k
/ k
factor
k
= Gal(k'/k) .
R
of
R
c
H
x
^ J »/k ' ^
c
on
R
k f
] '/k^
k
K
H x
' •
^ (Hx k').
k
k
k
k '
(2.20.1)
such
H ^ R
that
right
7
k
1
DR
where
space
/
k
(h
of
H ' ^ = *
k
t
^ '
o
DR
®
V
H^
h ® a
k
^k '
r
h'
denotes
Q
o
n
w
n
i
c
I
G
I
1
k
IndJ
H
k'
1
f: p
ph
G
±
-»
H»
1
®
k
k
'
1
Q
t o t h e sub-
coincide
which
-
gives
k'
this
a
.
G
k
Ind*
k'
I
<==^
J
1
V
k' ^ k '
P
c
^F=*
q
1
1
1
morphism
w
f°ll° s:
(h' e 1 )
k -structures
k' ®
s
\ h' 3 a'
h
of
a
^
DR
the projection of
to the m u l t i p l i c a t i o n
H
H
>
both
(j,q) the
DR
i
n
K'
R
DR
7 = >
a'-h' «
p r o j e c t i o n by t h e idempotent
section
DR
/y
and
i n e c
D
1)
K
e
DR
<—»
1
r
P
H
I—» h 9 1
a
a
^•/k
•) X
(R ,H
adjointness
DR
^
h
k
k ' )
k
the l e f t
x
of
H
G
h.tr ,
( H x
(i,p) give
adjointness
N
the
k
G
1
f(1) ^
1
f: G
k
H'
1
1
—•H^
'ph' , P € G
1
I pf(p~ )
p€G /G
k
<
1
f: G
where
G
the functor
€ G /G ,
k
=
( t :k
1
d
a
H
rr
p • G ,
>f: f ( p )
a s i n 2.16
0
for
1
a : k'
I T I . = a}
Ik
©
H
i
we
C
,
a
s
,
k
P^G
HV,
of representatives i n
s
a
l
I j
a
c
n
•
>
*
h-,
*
a
H
1
T€J
.
(h;)
a
TtJ
o f the
^
T ,
T
and
define
x€J
> (h)
ii)e),
( t h e sum i s i n d e p e n d e n t
k
rr
q„
h
i s defined
means c h o o s i n g a n y s y s t e m
k
choice) . F i n a l l y ,
J
i
k
Ind^
f o r the cosets
k
h'
1
k l
G
P
- H
v
T
a
a
a
h^
= 0
f o r T *a•
k
It
i s lengthy
with
but easy
the comparison
r
For
x
J '/]^
c
(
©
H
k
k '^
t o check, t h a t
isomorphisms, which
'
B
Y
2
1
-
) ® <E
6
1
( H
*
0
T€J
t h e s e maps a r e c o m p a t i b l e
O
i
a
0
DR
s
^
)
V
Q
are given
i
v
e
n
b
as
^
h#a®c
C
k , a
follows.
o
©
I
T€J
©
T
and
I
€
(HOCt)
J
a
-
1
©
0,
H__
DR
k, a
<C
(h®x(a)-C)
T
F
T€J
for
a
: k
<C
with
a
1
= a L
by
G
©
k
Ind* H
G , 1
(H ® Q , )
1
T € J
k
A
a » ( a
T
)
f
T
f
For
H"
1
=
(R ,^ H )* k'
k
k
a
a
: G
1
-> H.
1
= I Sp-I V p - 1 >
( p )
(
1
f
, I^
k
k
i s d e t e r m i n e d by
the
commutative
diagram
(
© H')
(
©
®
C
( H
DR®k
k , )
®k',a'
C
h
' '
H^) 0 C
h' ®
oo
(h»
where on t h e r i g h t
9
T€J
onto
1
I
( ^
the factor
( H i . ® (D )
T
h
side
-»
'
®
c
a'(a')c
V
/T
(H^ * (E)
morphically
a
•
k
r
®k 'ô
H'
OR
®k' a'^
s
8, , _,<T, w h i l e
K ,u
I n d *
H'
G ,
1
K
i
m
S a'(a')c)
a
PP
e
I V I,O
d
isoi s
T
,
a
=
(a;)
a
f
H
T
f
with
t h e same n o t a t i o n s
Finally
2.2)
that
i t i s clear
t h e above
filtrations,
the
a)
Galois
Once
i
H
-
= I Sp-^ -P- ) '
1
)
a
1
f
0
the d e f i n i t i o n s
respect
action
i , j , p and
q
realizations,
morphisms
may
example,
be
(especially
a l lrelevant
e t c . . We
R
give
k
the d e f i n i n g
now
using
structures
prove
that
x
k'/k
like
the claims
DR
d
H
®
'
the
of
R
the r e a l i z a t i o n s .
i
,
k /k
( H
>
k
(
(
R
,
seen
DR
H-
H
)
k /k '
x
H
k
*
Rham
— ^
H
\
*
>
k
k
For
H
— - ^
i n t h e de
® k'
k
k
)
k '
a'
x
-
k k l
V
(h <& 1) 8
-f
R
(H* k-))x
/ k
i s easily
>
1
adjunction
P
k
-~^>
a
of
of
of
compositions
(R .
V
h
properties
morphisms
k
the i d e n t i t i e s ,
H
to define
i n anyone
H K '
_ J U L ^
.
k
are proved
checked
the f a c t
i
HX
n
p
k
above.
from
maps
(
a
G
proposition.
mixed
a
as
!
R
k
k ' / k
H
'
realization
DR
V
(h ®
H'
« k'
1)-a* =
h
R
DK
h'
That
>
the f o l l o w i n g
H
V A
H
can
i
R
.
k
®
q
tej
*
checked
H
,
T
0
8
1
>
wk
( (
T
i.^
f9
x€J
(
a
give
V/k '
H
>
h' • 1
the
identities
k
H
) x
k , )
(^,/j^tHXj^k'JJx^'
i n t h e Hodge
J
i
compositions
'
x k'
be
h'
e
p€J
H
1
*
)
p
a
>
®
H'
h'
\ k '
H
7
)
realizations:
=
H
X Tc'
k
,
(h;
)
I
T
H
>
((h;-6
•—-—>
©
TGJ
T p
) )
p
,
T
H
>
--^
I
(h;6
T p ) p
=
( h
.
)
H
a
°
a
(h-6
.)
TO
b)
The f i r s t
iii) ,as
I^
general
= I
1
x
f c
^
j c
'
•
i s the s p e c i a l
T
n
e
second
which
i s o b v i o u s and a l s o
(see
[DMOSlII
€ M
fc
functor,
1.9)
x
f c
of the fact
from
in
f c
d)
t h e more
that
k
k
')
a formal
HH-)H x k
1
consequence
is a
tensor
isomorphism
DR
by
V D R
1
V
( H
^
(x 0 y) 8 a ' - b ' <
and
the i d e n t i t y
c)
By
2.14,
given
DR
V
'
>
1 (x 0 a') 0
i n the other
this
k
-
W
(y 0 b')
realizations.
i s a special
case o f the adjunction;
t h e map
by
(x
i
x
DR'
I t i s easily
x
l)
K
I ' o l,o*
checked
^
-
' W R '
h ® h ' i-» ( h ® h') ®
(X
DR'
ill'
( X
)
T x€J
)
a
l,a
.
that
(2.20.3)
n
— * t p * ( (K x k')
i s given
sle
case
1
fc
by t h e o b v i o u s
(2.20.3)
H ©CP H'
. = h
TO
follows
x k' -> Horn (H X J c ,K
H,K
d)
X h6
T
Horn (H, K)
for
is
>
isomorphism
(2.20.2)
which
isomorphism
/
T
+
+
f o r example
V
2
1 K
(h ®
( H
DR
1) ®
V
>V
(h' ®
( H
k
DR®k '>
1) v* (h ®
-
( H
DR®^
1) ® h '
, ) < 8
H
k' '
t
is
obviously
an isomorphism.
Writing
Horn(T,Horn(H,RH'))
R =
R
i /
k
Horn(T,R Horn(Hxk',H'))
|j\
||\ad j u n c t i o n
1
1
Horn (T ft H,RH )
f[\
Hom((T 0 H)xk',H')
a n y T € MR
=•
(2.20.3)
, which
fc
gives
Hom((Txk') ft (Hxk') ,H ' )
i i ) . Similarly,
1
Horn(T,Horn(RH , H) )
||\ a d j u n c t i o n
1
Horn(T ft RH',H)
111
D
for
any
T € MR
e)
The
first
i s immediately
3
3tt
: J I/]^**
R
C
H
gR
Ind
where
k
H
G
T
J c
"
-
1
i d
R
H
D R
Ind
Galois.
G
k
G
H
x
V
H
on
i
S
'
1
i s given
j4c
. Explicitely,
e
s
c
r
i
b
e
d
ky
t
n
e
M
A
P
S
x f t a » * x f t T ( a ' )
,
a lifting
^
Cp Cp*
x)* , inducing
+
k
cate-
i n $ .
T) ( S p e c
k '^
i s a Q-Iinear
fc
f c
the d e f i n i t i o n s
q.e.d.
M
T ) * C O * = cp^tp*
"* k ' / k ^
a l s o denotes
from
that
T € Aut (k')
(Spec
T ) * (Spec
k '^
V
directly
k'/k
x
note
c l e a r from
i s invertible
the adjunction
P Jc P* -* CP ( S p e c
T
part
the action o f
t
Hom((Txk') ft H', Hxk')
, and therefore i i i ) .
[k' : k]
Formally,
by
t
fc
ftH*),H)
adjunction
t h e maps. F o r t h e s e c o n d
f)
1
Horn(Txk ,Horn(H ,Hxk'))
Hom(R((Txk')
gory and
we h a v e
Horn(T,R Horn(H',Hxk'))
||\
of
1
Hom(Txk' ,HomCHxk ,H ) )
||\ad j u n c t i o n
for
we have
k
f -
i n G
that
fc
T
f :
T
f(p) =
. With
io q =
this,
(i~ p)
1
i f
i t follows
z
T for
TEGal(k'/k)
§3.
The
mixed
realization
Let
k,
k
o
V
be
the
let
fc
and
G
of
be
fc
category
a
as
of
smooth
in
the
smooth
variety
previous
paragraphs
quasi-projective varieties
n
over
n
k
€
. We
to
n
3.1
Let
.
n
(U)
=
of
the
de
we
only
3.2.
By
Hironaka's
there
exists
i
=
a)
x
Each
is etale
nates"
for
X
1 f
some
union
b)
of
€
f.
=
3
H
n
( U
Z
Rham c o m p l e x ) .
show
that
result
c
j : U -^X
on
^
r
u
For
construction
to
a
of
/
, such
structure)
, with
i s the
normal
following
X
over
=" v
the
a f f i n e space
T(V O )
x
^min(d,N)
v
is a
=
union
a f f i n e open
r
Y
eauivalent
an
A
, and
coordinate
local
equation
Hodge
i s defined
over
=
dim
Y.
X
€
part
at
3.3.
The
along
O
a
by
Y
regular
sheaf
i s not
system
ft^<Y>
i s defined
1
P.
, via
in
x
€
this
X
fc
"coordi-
by
of
A
that
X ...x
1
the
.
and
J
=
I
{1,...,N}) I f .
i
of
re-
such
pull-back
1
Ci
an
hold
i s defined
hyperplanes
for
[Hir] ,
the
that
V
the
of
of
as
a
unit
at
parameters
x}
differentials
the
subsheaf
of
, then
at
x
with
j * ^
X
and
dlog
J
3 4
A
7
/ w h e r e we
write
.
smooth d i v i s o r s
neighborhood
V D Y
k
and
(with
Recall
, d
over
k
to
fc
and
singularities
XMJ
of
F
structures
conditions
, i.e., is
first
filtrations
mixed
crossings.
has
€
d
that
for
fc
(Zariski-hyper-
the
everything
- * MR
'
)
k
X
the
the
If
(U/k)
smooth p r o j e c t i v e v a r i e t y
...,x
O
R
f c
realization
1 ,a"* 1, 0 ,0
of
1,...,N
of
n
•V
i t s n-th
fc
Q
Q O
°
H
resolution
subscheme
means o n e
V
have
immersion
, i
H
Deligne's
a
€ V
n
H ^
follow
duced
U
f
[D4],
open
each
functors
o
(H^(U) ,H (U) H (U) ; I
cohomology
we
to
n
=
Y
construct
TL , a s s o c i a t i n g
H (U)
W
want
and
Q
X
( f . ) _ _
i i€J
is
x
.
logarithmic
generated
1
and
Q
1
poles
over
for
the
1
^ /k
sheaves
dlog
a n c
x
f = ^jr
1
Û/k
^
fora unit
r
f
^ l
a
t
l
. Then
v
e
d i f f e r e n t i a l s , and
<
^
Y
>
i
s
l
o
c
a
l
l
Y
free,
namely
<Y>
for
an open
with
affine
dx
V
i n 3.2a),
basis
— -
kr
|V
x
v
X^
O
ftl
like
dx
,
1
X
,...
i
s
a
f
r
e
e
0 -module
v
<Y>
• By d e f i n i n g
V+ I
rf^
=
A
Cl
1
A ft <y> , a n d t a k i n g
oQf<Y>
thos
:e
for
the d i f f e r e n t i a l s
j^ft^
one o b t a i n s
d
t o be t h e r e s t r i c t i o n s
the logarithmic
de Rham
base
k
complex
X
3
0
x
Its
formation
base
change
1
ft <v>
x "
2
ft <v>
x "
i s compatible
i n
3.4.
Lemma
———————
3
with
change
in
and e t a l e
X .
T h e map
*
ft*<Y>
x
j * f tu
*
induces
isomorphisms
i n the
Zariski-hypercohomology
B
n
n
(X,Q^<Y>)
Proof
and
As
IH (X,
Y
i s a divisor
U
^ u
}
^
(
with
normal
=
H
( U )
*
crossing,
j
i s affine,
therefore
P
R J*û
This
implies
The
= 0
the second
first
P > 0
and a l l q .
isomorphism.
isomorphism
follows
by base
fact
methods,
[D3] I I 3.14. One c a n a l s o
proof
see
along
polydisk
criterion
these
over
for
corresponding
3.5.
DR
lines,
by s i m i l a r ones
(E , w h i c h
change
from the
c a n be proved
by r e p l a c i n g
give
by a n a l y t i c
a purely
the considerations
f o r the a f f i n e
filtration
W
on
^C
for a
space A , using the
fc
3.2 a) .
The w e i g h t
algebraic
< Y >
i
s
d
e
f
i
n
e
c
^
b
Y
m < O ,
m
ftP" A^v<Y>
X
X
m X
O ^ m 
m ^ o
fi
The
we
differentials
d
g e t an i n c r e a s i n g
respect these
filtration
subspaces,
of
<
^
Y
>
b
W Q"<Y>
(Note t h a t
W .
0 ,
W
= Q*
o
X
ITl X
-1
when
d = d i m X) . F o r a n i n t e g e r
m
0
1
i„
 ,
a
X
indices
c o n s i d e r t h e map
m x
which
i s locally
g i v e n by
dx.
i
a A
x.
a h
where
for
a
dx.
x.
i
i s a holomorphic
Y.. T h e i n d u c e d
i
(p-m)-form and
x^
i s a local
equation
map
W
6r flP<Y>
m X
does n o t depend
for
other
on t h e c h o i c e o f t h e l o c a l
equations
x|
equations
x^
, as
one has
x.
dx!
i
x!
i
dx.
i
i
X .
1
which
i s holomorphic.
Also
i t factorizes
through
p-m
(b^
1
ITl
where
b.
V
1
Y. f). . . DY.
I
In
1
m
immersion,
A
as
dx.
i
where
m-fold
Y
(m)
(I ) ftm * (m)
rn
+
intersections
i s the closed
x.
• a
a r e mapped t o z e r o
.
v
-»
Y
\y
1î«<. . .
X
.
V m
i s the d i s j o i n t
Y^,...,,
which
union
of the
i s also the
normalization
of
m
Y
=
Y.
.
I ' * 'Sn
: Y
i
3.6.
is
, and where
m
(m)
X
Lemma
(3.6.1)
c x
1
1î..<. . . 
that
(K'tn])
1
= K
1
+
n
f o r a complex
K') .
Proof
By c r i t e r i o n
3.2 a)
only
be checked f o r t h e c a s e
and
Y
i
= 1,...,v
for
U
= U
1
X = X
x U )
x X
as i n
2
and
2
1
the exact
0
3 4 t
isomorphism
l
x
need
x^,..-x^
x X
U X
2
1
o f both
x Y
case
(i.e.,
2
reduces
sides
to the
m = 1 ^ and
'
Q
Q
"* X
<Y>
x "*
^
t o the exact
i
1
U
0
Y
sequence
works
characteristic
zero
t h e lemma a l s o
sponding statement
about
0
of
k[x]
-modules
-^ - k - 0
also
P^
m
1
an
proof
u s i n g base
decomposition
one e a s i l y
Q <Y>/Q
y
0 -> k [ x ] d x - k [ x ]
by
coordinates
i n theonly interesting
O
this
sequence
corresponding
This
change
Y^ = ( x ^ = 0} f o r
Y = Y
[D2] I I 3.6
we o b v i o u s l y h a v e
(I )
via
1
base
with
fc
o f the hyperplanes
d = 1 , where
= 1
X = A
. By u s i n g t h e c a n o n i c a l
products
case
p
the union
and e t a l e
i ncharacteristic
analytic
extension to
(D
follows
sheaves
p
from
proved
, while
for
the correi n [D3l I I 3.6,
a n d t h e GAGA p r i n c i p l e ,
as
the
are linear.
K
3.7.
Remark
I f
(x )
n n t cci
i s the canonical
increasing
filtration
(coitiDare
T
[D4] (1.4.6.))
o f a TL - q r a d e d
n-1
K'
->
complex
K'
Ker d
n
ni
n-1
. - K
K*
we h a v e
T
n
K
n
+
K
1
n+2
T ft"<Y> e W ft*<Y>
n X
—
n x
n
-
ft*<Y>
X
n
n-1
X
1
ft ~ <Y>
Ker
d
i
n
X
However,
the
i n contrast
i d e n t i t y map
complexes
i s not
T
general:
which
Gr
n
(ft^<Y>,x)
trated
3.8.
ft*<Y>
X
(Q*<Y>,W)
n
= H (Q'<Y>)
x
not true
t h e complex
i n dimension
n
( s e e [D4]
n
complexes i n
i s concentrated
analytic version
n
£
7
Z
i n dimension n
( i
n
. . [-n]
of the latter
, quasi-isomorphic
(a )
(3.1.8)),
of the algebraic
W
~
f o r G r Q'<Y> =
n A
The " s t u p i d " f i l t r a t i o n
O Q'<Y>
X
case
a quasi-isomorphism o f f i l t e r e d
i s i n qeneral
Cwhereas
to the analytic
to
of
^
^

i
s
n
( ) t ~ ^ )•
n
^
i
v
e
n
0
nl
fi < >
x
n
3.9.
of
1
->ft ~ <Y>
X
Y
Recall
that
a complex
K"
P,q
=
E
P+q
n
f o ra decreasing
o f sheaves
sequence f o r t h e hypercohomology
E
-+ ft .<Y>
X
(x,Gr^K-)
-
on
X
bireqular
there
P
+
q
(X,K')
p
(F ),
p€
i s an a s s o c i a t e d s p e c t r a l
([D 4] (1.4.5.))
H
filtration
,
where
the
short
exact
The
filtration
for
which
+
We
apply
(
p
Gr E
P
F
n
B
filtration
n
f P
P
- F KVF
o
) £2Z
p
= E '
n
p
to obtain
H
t
h
e
n
(U)
n
B
n
Defintion
induced
(3.10.1)
Ê '
P
r
associated
F
By
P
B
n
{X Q'
f
q
=
<Y>)
as
(3.10.3)
also
K
F
E
P
n
=
B
B
K
1
P
+
q
q
can apply
{X K')
r
.
f
and
t h e Hodge
.
F
on
H
n
(U)
P
n
=>
B
P
+
q
(X Q'<Y>)
f
A
11
(a )
P
n
-> B
A
(X,fl'<Y>)) .
X
P
=
ft <Y>[-p]
with
i * 0)
the complex
the s p e c t r a l
P
=*
B
P
to the i n c r e a s i n g
filtration
but i n a d d i t i o n
i.e.,
f
(X,a ft*<Y>)
= B (X,ft <Y>)
3.9
^
complexes
for
q
i s the
sequence
(X,Gr ft'<Y>)
i s identified
= 0
'
n
(X K'))
filtration
the s p e c t r a l
= Im(B
the d e c r e a s i n g
back,
n
by
P
to the f i l t r a t i o n
a sheaf
written
to
by
P
and
E
filtration
Gr ft^<Y>
K° = K
term
.
O A
(3.10.2)
We
Hodge
the isomorphism of
(where
-> O
r
(X,ft*<Y>)
I
A
morphisms f o r
A
The
filtration
Gr K-
(X, F K ") -+ B
the weight
=
p
-
liniit
JJK
3.10.
K-
, i s given
= Im(B
f
on
n
P + 2
d
(X K*)
this
are the connecting
sequences
G rr ^ V
O -
q
d^'
there
(W
n
+
q
K"
sequence
(X,ft^<Y>)
filtration
= W
) ^
and
-n ntzz
i s a shift involved.
3.11. D e f i n i t i o n
The w e i g h t f i l t r a t i o n
—
•
t a i n e d from the f i l t r a t i o n
W'
induced
by
on
that
can
be
.
W
, by
passing
translating
rrt
W
such
H
n
(U)
UK
the s p e c t r a l
i s obsequence
(3.11.1)
of
P
E
w
'
q
=
3H
the f i l t r a t i o n
1
W [nj
,
W
n
+
3.12.
k
H
n
By
of
: G A ; < Y >
<
^
Y
>
b
y
a
=*
n
n
~
P
H
f
o
l
d
+
q
(X , Q^<Y>)
shift:
W =
that
i
m
P+
W
the
isomorphism
-
( i )
ft*
m
and t h e r e f o r e
we h a v e
2
=
K P
+ c
F
which
induces an i s o -
an isomorphism
^ ( Y ^ , ^ ,
i n [ D 4 ] ( 3 . 2 .4 . 1 ) )
p'
[- ]
m * y(in)
fi-<Y»
filtrations
. On b o t h
differ
i s an i s o m o r p h i s m
.)
sides
by a s h i f t ;
of filtered
there
i n fact
complexes,
m
one t a k e s
.
k
i sfinite
i sa misprint
natural
n
( X , W ^ < Y > ) - 3H ( X , ^ < Y > ) )
hypercohomolgy,
H <* ( X , G r
are
n
(3.6.1)
=
m X
i n Zariski
(3.12.1)
if
( X , G r ^ f t V Y>)
the isomorphism
the fact
(there
Q
(X,ft^<Y>) = I m ( 3 H
m
morphism
W
+
i.e.,
p'
and
P
"
induced by
a
on t h e l e f t
side
(m)
and
(af-m]
1
= a
1 m
)
a(Y
from
)[-m] o b t a i n e d
the stupid
by
filtration
shifting
o f ft*, , . T h u s
we c a n r e w r i t e t h e s p e c t r a l s e q u e n c e
( 3 . 1 1 . 1 ) as
(3.12.2)
EP'<* = K P S ( Y - P ) S r ^
Mp) * H
(X,fi-<Y>)
2
+
t
P
W
which
p
Y^
^
2.18:
F
P
+
m
A
write
i s compatible with
, i f
of
(p)
o
t h e Hodge
indicates
A(m) = A
filtrations
the shift
as a b e l i a n
group,
the spectral
w
E
P
'
q
sequence
= H
P
(Y
The f i l t r a t i o n s
the particular
choice
( q )
q
of
X
and t h e
of the f i l t r a t i o n
with
P
E '
W 1
q
= .E ^
W 2
p
asi n
P
filtration
r 7
,
F (A(m) =
+
q
'"
p
to
as
,ft*
) (-q)
and
W
(
F
+
j
. I t i s c o n v e n i e n t t o renumber
(3.12.3)
3.13.
f
of
X
on
B
P
+
q
X
H ^ (U)
T
(X,ft <Y>)
T 3
.
a r e independent
and a r e f u n c t o r i a l
i n
U ,
as one
sees
like
zation
of
3.14.
F o r t h e 1 - a d i c r e a l i z a t i o n we l e t
U
in
[D4]
cohomology),
functoriality
creasing
i.e.,
P
n
q
fined
q
n
. Then
as
k
id
^
P
t
k
x k,R J
k
n
G r
by
n+k
3.18
E n
E
below
3.15.
For
»
'
and
a;
k ^
C
3tt
(J) )
O = W
aU
= U
x
W
for
P
E '
q
by
w
P + q
= E
.
b
e
t
h
K
R
n
W
Gr '
q
H (U)
c
—
n
X
and
f o r H (U)
+
q
=
i s de0
of
P
E
E
... c: W
E
—
zx\
n
result
=
1
n
of
_
i n
sequence
with
c W E
—
n
e
X JJ Q )
(U
n
W
independent
n
*
k
a
C
defined
over
n
=
E
n
functorial
([D4](3.2.11)).
c*fl)
sheaf cohomology) w i t h
by
D e l i g n e [D4]
<E . I f
ax
= X
f o r the
x
C
K.
e t c . , t h e n by
i n 3.14.
P
from
q
F
t h e Hodge f i l t r a t i o n
and
d e f i n i t i o n the weight
->
H
P + Q
(0U,Q)
H (au,(l)) ® C
F
on
H
n
(aU
smooth
simi-
filtration
sequence
,
n
on
the
, O
the Leray s p e c t r a l
= H ( 0 X , R j*<T))
t h e Hodge f i l t r a t i o n
given
s
t
0
, Y ^
i s o b t a i n e d as
(3.15.1)
and
Y
n
n
,E
1
cohomology or a n a l y t i c
JC ,
larly
i
e
H
filtration
= H (u
Q-Hodge s t r u c t u r e
variety
-
1
... c W E
—
n
t
L
we l e t
n
mixed
reali-
induced v i a
fc
the Leray s p e c t r a l
the analogous
H (U)
(singular
T
p € G
#
1
c
—
, i.e.,
=
of
y
n
with
t h e de Rham
)
Q l
x
_> P
q
(X
the weight
W'tn]
x E,
i n d u c e d by
c WE
—
o
1
q
x
U
= H
0 = W E
-1
r
E ~ '
from
E '
= H^(U
w i t h the a c t i o n
filtration
(3.14.1)
gives
.
H^(U)
(etale
(3.2.11). T h i s
i s just
a n
)
the
one
( a n a l y t i c de Rham
DK
cohomology o f t h e complex
via
the
isomorphism
analytic
space
<jU
an
associated
to
a
U)
n
(3.15.2)
H (au,<J)) ® <C
induced
from
filtration
H
n
(au
a n
,ft'^
the cruasi-isomornhism
i s defined
filtration
GAGA
-
Q
F
a n
)
=H
n
R
(aU
a n
C-^ft' ^ . This
*—
.••.^•n
aU
a
)
Hodge
i n t h e same way a s t h e a l g e b r a i c
defined
above
and c o m p a t i b l e w i t h
Hodge
i tunder t h e
isomorphism
H
(3.15.3)
with
DR
( a l j a n )
t h e a l g e b r a i c d e Rham
3.16.
Combining
H
~
DR
( a U / ( E )
cohomology.
these isomorphisms
w i t h the base
change
iso-
morphism
(3.16.1!
we o b t a i n
the comparison
(3-16.2)
I„
As m e n t i o n e d ,
spects
H
p
= I^ (U)
f a
0 0
a
p
: H
respects
,a
the weight
spectral
IH
I
isomorphism
(compare
( U )
[D4] ( 3 . 1 . 7 ) ,
q
a n
H
W
a n
,gr ft*
q
<aY >)
QX
-
B
P + q
B
The
p
(aX
a n
and i t r e -
isomorphism o f
(3.1.8))
(aU,C)
,ft*
GX
<aY
a n
»
->
~
a n
GAGA
(ax, g r ^ ST <aY>)
x
first
p + q
a n
GAGA
.
filtration,
by the c a n o n i c a l
( a x , R j * (T)
(ax
. UîV)^^
v
t h e Hodge
filtration
sequences
a
isomorphism
an
follows
(ft- _ < Q Y > , W )
GX
B
from
(ft*
ax
P
+
q
D
GAGA
(aX,ft^ <0Y>)
X
2
the quasi-isomorphisms
a n
P
<aY >,x) ^
(J QaU
jlf
, T)
q
H + (aU
R
a i
of
filtered
duces
complexes, as t h e q u a s i - i s o m o r p h i s m
-
+
Gr
[q] = H
3.17.
The l a t t e r
<oY >[q] -
a
q
R J*S
( i ) ,,ft'
^
^
a n
$
Q-structures,
considerations
spectral
E
w
q
->
+
i n [D4] (3.2.7)
identified
E
2'
q
with
P
'
induces
3.18.
fined
I
=
P
H (0"Y
( Q )
q
• (2iri)
- (3.2.10)
,Q) (-q)
(3.18.1)
c C
. This
and the
show t h a t we
q
= 3H
P
(
(°Y
q
sequence
,n*
)
H
P + q
have
t h e (2iri)
1
^
5
=
I
i
/
a
(
q
U
)
between
( q )
c a n be
(3.12.3)
)(-q)
m
-
P
+
q
P
)
: H
(
u
a
®
c
l
~
x
and those
n
1
H
i
(
u
)
f
o
r
of the canonical
and e t a l e
H (OU Q )
f
)
^
H
n
t
cohomology
(OU Q )
7
1
q
(aX,ft; <QY>)=H ^ (aU)
of
C Y ^
.
complex
the isomorphisms
(aU,$)
whose c o m p l e x i f i c a t i o n
the s p e c t r a l
t o be t h e c o m p o s i t i o n
morphism
an i s o m o r p h i s m
q
the comparison isomorphisms f o r c u
(regarding
with
this
(i )*<^(-q)
Q(-q) = Q
m i x e d (D-Hodge s t r u c t u r e s ,
w
that
: R j <J)
where
sheaves
sequence
(3.17.3)
via
) <c
M
q
shows i n [D4] ( 3 . 1 . 9 )
(3.17.2)
of
the quasi-isomorphisms
an i s o m o r p h i s m o f a n a l y t i c
Deligne
a
aU
combined w i t h
a n
^ '
(3.17.1)
of
ina n
)
aU
gives
ft
QU
R^j C
g
C
5
:
k
<C
i s de-
comparison i s o -
(3.18.2)
(3.18.3)
p
For
H
G
w
^ k
e
n
a
v
e t
e
(
a
a*
tr
"k^'V
U
=
J
0
=
a p
k
J]^
a n c
r T
H
n
e t
(
U
"^,(B )-H (O)
1
^ therefore
1
a
commutative
diagram
1
:
1,5
H
t
o
u
H^ (OD Q )
£-*
Hj (OU Q )
4^-*
t
^
f
t
I
1
of
-
respects
spectral
E
q
2'
P
2 '
by
for
is
f
1
the weight f i l t r a t i o n s
H
( U x
et
k
k '«l
Hg (Ux k
t
as there
k
f f f i l
)
)
sequences
=
q
1
p
H
(OX R
t
q
f
P
j
f
1
q
= H (0X R j (D)
f
p
H ^(OU Q )
^ )
® Q
+
«*
1
t h e c o m p a r i s o n theorem between
constructible
constructible
n
H (U)
=
P + q
(oU,Q)
complex
GQ
,
1
and e t a l e
cohomology
s h e a v e s , s e e [SGA 4] XV 4 . N o t e
by
3.19. D e f i n i t i o n
quasi-projective
H
1
[SGA 4] XV 5.1
The mixed
variety
n
U
n
n
(H (U),H (U),H (U);
R
q
R J
j f c
.
n
realization
over
that
k
H (U) o f a
smooth
i s
I , ,^ , > ,
0 0
a
1
5
p
c
r
Oik -+
i
m
e
C
a rlc^-* <E
n
where
H ^(U)
3.14,
H (U)
Clearly,
n
this
i s defined
by
3.15
by
, I
o o Q
3.1, 3.10 a n d 3.11,
b y 3.16
and
I ^ -
n
H (U)
by
b y 3.18 .
a s s i g n m e n t i s f u n c t o r i a l , s o we g e t t h e d e s i r e d
Q
1
functors
(n € 2Z )
n
H
: ^
-
U
the following
k
n
U
In
MR
h->>H (U)
we w r i t e
U, j
e t c . f o r the base
extensions
X ^k , j x i d ^ e t c .
1
3.20. P r o p o s i t i o n
There
are canonical
isomorphisms
of 1-adic
sheaves
q
(3.20.1)
such
: R IU(I)
that
weight
v i a these
filtration
(3.20.2)
(note
w
that
where
be
on
the
i
t t
J)
/ C ] L
) (-
^ are acyclic
D,q
7
H P ( Y
s
t
^ Q
f
1
)
giving the
with
H ^ U , ^ )
q)
f o rthe etale
(-q)
given
as f o l l o w s : L e t
(5.)„
: H P ( T ^ Q ) (-q)
1
t
the Gysin
f
Then
P
d
'
q
: y
H
|J
2
2
(Y
cohomology),
>,
D
Y.
X
q
immersions
*
...x.
1
(-q-1 )
( -q+1)
)
(Y
by t h e c l o s e d
,
... x
1
q
=
I
(-1) (6.)
j= 1
1
Proof
induced
: Y .
X
-
-. H ^
1
morphism
3
the
1
t
6.
6*
H (U)
(3.14.1)
d
are
the s n e c t r a l sequence
f ' ^ = RP (Y
E
(i UQ^-q)
^
1
...x
g
.
+
3
Vfe c l o s e l y f o l l o w t h e arguments o f Rapoport and Zink i n [RZ] § 2 . L e t
(
r
+
1
)
- Y
(
r
)
, 1 ^ j ^ r+1
, b e t h e map
i n d u c e d by
inclusions
Y
n...n Y
i
1
X
Y
r+1
1
I
n
. _
n
$
1
n
J
n
Y
Sr+1
for
1 -
<
... Y
we g e t f u n c t o r i a l
i s t h e c a n o n i c a l map.
By a d j u n c t i o n
morphisms
r
r
1
(6 .)*(£> .) F - F
for
any e t a l e
sheaf
8
for
J
any s h e a f
...
is
exact,
Proof
on
r
+
1
I
i s injective
1
r
+
1
) I 2
(a )
r
d = I(-1)^3.
l e t I"
(O
1
on
(
a
r
)
!
F
on
Y
, then
l
J | t
(a )4-...-(a ) (a ) I
r
1
+
1
- I - O
.
be an i n j e c t i v e
X
. There
0 - I
(3.20.4)
I I'
1
J 1 T
By a p p l y i n g t h e a b o v e
get a r e s o l u t i o n
^
(3.20.5)
total
- I* -
lemma
(note
j^j*!-
complex
i
to
o f the constant
sequence
j*j*I*
~> 0 .
~t
i " I " and by
(3.20.4)
we
= i a )
m
m
...
-(I )^(I ) I--I--.
I
1
1
- 0 .
sC''
(i
resolution
i s an e x a c t
...-(Ij„(Ij V -
The
(V*
Y .
)*(a
where
"
! p
morphisms
A s i n [ R Z ] Lemma 2.5.
Now
sheaf
a
If
-(a
, and t h e r e f o r e
<W*< r+1>
:
F
3 . 2 0 . 3 . Lemma
on
a s s o c i a t e d to the double
J J i J *
I
1
q
• p.q =
q
= 0 ,
complex
with
quasi-isomorphic
Now
to
( i
q
by p u r i t y ,
1
the following
0
0
c '
1
'
(3.20.5)
, i s therefore
i s quasi-isomorphicto
1
therefore
double
c
from
.
= IR ( i ) ' Q
Q (-q)[-2q]
to
induced
complex
0
c
2
C '
1
2
'
C'*
i s quasi-isomorphic
C**
0
c
3
3
/ I m 3»
0
'
C '"
c
1
4
'
4
C '"
4
0
C
1
C
2
C '~ /Im3'
As
C
P
/
q
=0
6
f o r p + 2q < 0
s C * ' :=
9
q--r
r
T.
the canonical
of
s C''
J
3 . 2 0 . 6 . Lemma
and
Proof
q
0
£
C
t h e second
r
, T)
^
5
C
5
have
P
,
q
£ T s
(s C
C "
r
p+q^r
B
( s C"*
morphism o f f i l t e r e d
P
C '
, we
5
(increasing)
' , 6)
i sa
filtration
quasi-iso-
complexes.
A s i n [ R Z ] Lemma 2.7.
This
induces quasi-isomorphisms
q
R J ^
3
1
q
-
( G r 5*5*1 •) [ q ]
-
(Gr s ~ ) [ a ] =
=
(I )
q
q
slt
IR I ^ Q [2q] 1
((I
>*igl'[q]) [ql
(I )
Q
I S
Q C^)
1
et
and
therefore
Leray
p
E '
the
q
t h e wanted
spectral
= E
2
p
second
+
q
'
p
sequence
isomorphisms
(3.14.1)
- be i d e n t i f i e d
filtration,
where
d>
. Furthermore, the
c a n - up t o r e n u m b e r i n g
with the spectral
sequence fo]
dP'
are
=
=
q
p
q
q
( Y
H
(
"
p
(I_ )
1
P
)
,Q )
q
the
, we
morphisms
morphism
IH
(X,C-'
P
-
this
P + 1
C*'
)
P + 1
. As
H
q
(X C
w
7
p
)
(X, ( i _ ) ^ C ( P ) [ 2 p ] )
p
t
P
that
d '
.
( V
1
1
q
i s induced
) ^ V
(i )*(i )"I*
=
r
g i v e s the
by
) * ! -
1
(I )
r
3 l t
alternating
•
IR ( i )
'Q
r
sum
of
1
the
q
as
r
6_.
for
q
see
quasi-isomorphisms
r
q
9=M-1)V
3>
q
"* (- ) *Q-_ (-*") [ - 2 r ]
Gysin
K
C"
=
1
(-p)
1
Q )
,
( i ) * ( i ) - r
Via
-
the morphism
(X,(J)+TR
+
P
: H (X,C" )
i n d u c e d by
3H
2
q
c l a i m e d , as
i
= 6
^ i . , and t h e G y s i n
_______
_____ Pi
3
Q '
(r+1)
(r)
: Y
-*• Y
i s g i v e n by t h e q u a s i - i s o m o r v
phism
(6j)^Q (-1)[-2]
-
1
followed
by
1
(6j) ]R
(6*) Q
+
1
the a d j u n c t i o n
1
(6*)+TR ( 6 * ) Q
I n m o r e down t o e a r t h
- Q
1
terms:
.
1
i f
J"
i s an
injective
resolution
(r)
of
Q
1
on
Y
then
the
G y s i n morphism
i s g i v e n by
the
isomorphism
Q_(-1)
the
2
-
1
R (6^) Q
quasi-isomorphism
2
1
R (Sj) Q
and
the
2
=
1
!
tf ((6j) J*)
canonical
cohomology
P
H (Y
( r )
this
1
J '
corresponds
, ,_(-!))
P
= H (Y
P
second
(6j) J'[2],
map
= H |
Y'
(the
I
->
(6j)^ ( 6
In
,
1
isomorphism
( r )
2
+ 1 )
from
»
to
2
,
( Y
the
!
R (6^) (23 )
1
l
r
' ^ )
C
spectral
n
$
H
P + 2
(Y
sequence
( r )
,Q )
1
for
,
!
(6j) ),
the
at least
map
strict
3.21.
6 ^
r * O
. For
i s not a closed
to the
Lemma
quences
for
Y
c
±
a)
Y
The
(3.12.3)
Gysin
morphisms
b)
The
isomorphism
immersion,
differentials
d
(3.17.3)
as
p
and
,
q
Y
one
( 0 )
=
has
X
to re-
i n the s p e c t r a l
are given
i n proposition
(3.17.2)
as
R (aj)*0) -
se-
alternating
sums
3.20.
of a n a l y t i c
q
:
, i.e.,
(1)
and
of
r = O
sheaves
(Oi ) Q(-q)
#
q
et
corresponds
structible
to
<J>
q
(smooth)
v i a the comparison
sheaves
f o r the e t a l e
l y t i c t o p o l o g y [SGA 4] XVI
4.1.
c)
There are c a n o n i c a l isomorphisms
5
_'
q
=
Het
-
H
P
t
( Y ( q )
(aY
'G>l
( q )
) (
-
isomorphism
and
f o r con-
t h e complex
of spectral
ana-
sequences
q )
p
q
H ^ (OU Q )
,(D )(-q)
f
1
1
I
^P'
1
P
=
given
H (cY
by
spectral
which
sequences
a)
reference.
via
,Q(-q))
the comparison
cohomology,
Proof
( q )
plies
Q
isomorphism
correspond
The
+c
->
1
to the
HP 2(aU,Q) *
claim
of
isomorphisms
though
t
the isomorphism
(3.6.1)
f o r the s i n g u l a r
cohomology,
and
etale
of the
Leray
<{> .
q
I could
f o r t h e de Rham c o h o m o l o g y
1
n
(J)^ a n d
not
c a n be
and
f o r t h e G y s i n m o r p h i s m g i v e n i n [Ber] V I
the c l a i m
Q
between complex
v i a the isomorphisms
seems t o b e w e l l - k n o w n ,
formula
•
as
the
find
good
checked
explicit
3.1.3. T h i s
Gysin
a
im-
morphisms
correspond
under
Another
is
given
proof
a x
on
approach
n
. Then
we
a canonical
(3.21.1)
which
E
P
'
n
q
: R (°j)*Q
spectral
q
P
= H (cY
( q )
a s i n lemma
the
c o r r e s p o n d i n g complex
process
cal
the functor
s p e a k i n g , we
to limits
on each
base
change
structible
et
<!>_
i nthe
sheaves
(aU,Q)
spectral
sequence
arealternating
and Q - Sheaves
1
1 1
the etale
q
(Gj
e t
e*
we
the comparison
et
) ^Q
a n
and there
(3.20.2)
v i athe canoni-
we
'^g
see that
1 1
a
n
t o prove
'
=
^
^(-q)
change
( O i J ) , < B _ <-q)
1
between
the spectral
(base
isomorphisms.
remains
c
isomorphism, i . e . ,
® Q
t h e whole
n
- 2 — ^
a n d (3.21.1)
sheaf
E -sheaves and
base
v
1
f o r con-
t
),«
* l
Q
isomorphisms
e* (Oi^ )
1
with
etale
c a n compare
change
a
n
construction
i s exact,
a j , and
( s e e [SGA 4] X V I 4 . 1 ;
ZZ/ I
) ,
tensoring
t o each
sheaf
for
sums o f G y s i n
after
associating
n
R (aj
It
P + q
to consider
and as
base
change)
'
q
analytic
have
a r e compatible under
sequences
o f b)
(°iJ*Q(-q)
q
morphisms. As t h e s e g i v e
sheaves
commutative,
p
e*
step with
e*R
is
z
3.20. M o r e o v e r ,
applying
pass
construction
out f o ranalytic
,Q) (-q) = > H
d
and
strictly
t h e whole
proof
sequence
morphisms
natural
g e t isomorphisms
i s isomorphic to the Leray
where
then
that
3.20 c a n b e c a r r i e d
*4>*
q
and
isomorphisms.
and p r o b a b l y t h e most
by t h e o b s e r v a t i o n
o f lemma
a
the comparison
• There
i s an isomorphism
depending
on
the
fixed
ordering of
the
Y_.
, and
a canonical
isomorphism
q
a n
R (CTj ) <i)
given
1
by
an
<j>
the
are
to
show
ox
by
cup-product
compatible
'4>
an
A R
-
+
=
a n
open
(Oj
(compare
with
these
. The
<$>an
1
D
)*Q
[D4]
3.1).
d
is local,
, with
D
=
v
and
suppose
that
aY
=
cY.
3
j=1
that
at
OU
0
X
=
(D )
D
x D "
q
of
finition
V
R (aj
an
a
, with
) Q
D
X
=
i s isomorphic
+
1
(cb )
n
V
with
maps
the
class
an
Both
<j>
and
so
we
only
have
and
we
can
isomorphisms,
question
polycylinder
a n
{z€C|
]z| <
-1
= p r . (0)
D
l
aY.
DMO}
.
H (aU,Q)
of
oY .
dx.
—2.
_
x.
d
i n the
\}
,
such
Then
q
to
replace
the
fibre
, and
fibre
by
de-
of
3
(Oi
a n
)
j t c
Q
at
0
to
the
class
of
H
n
(au,Q)(1)
, where
3
X^
are
the
canonical generator
the
canonical coordinates i n
D
.In
the
same
setting,
of
j
is
the
image u n d e r
H
1
the
connecting
(oxôY
,Q) (1) - ^ H
3
of
the
morphism
2
canonical generator
(OX Q)(I)
y
~
f
J
ou
of
i
H
which
sends
positive
( O X ^ o Y j ,Q-2-rri)
the
choice of
of
OYj
(w.r.t.
i = V^l)
the
image o f
to
as
dx .
1
J
=
the
a^
1
x
Yj
2iTi
H (oXôYj,Q(1))
and
1
the generating path
orientation
to
= Hom(fT ( o X ^ o Y j ) , Q - 2 i T i )
2 TTi
around
orientation
an
. Now
under
-
OYj
' <f>
the
1
ox
sends
restriction
H (aU Q(1))
r
of
which
,
has
given
the
map
by
class
we
see that
we
nave
From
quence
of
=
t h e above
coincides with
^
i t i s clear
0
the weight
t h a t we
have
a spectral se-
)(-q)
filtration
P
~H
+
q
(U)
on t h e r e a l i z a t i o n
a
r
e
v
9 i
e
n
attached
^Y
sums o f G y s i n m o r p h i s m s .
d
Ker
to asubquotient
degenerates
for
by
l
/Im d
2
2
§4.
f
a
H
s
a t the E^-terms.
[D4]
n
k
G r
n
( Y ^ ) (-k)
the s p e c t r a l
The l a t t e r
and i s proved
+k
n
H
i
s
i
s
o
~
, namely t o
sequence
has o n l y
(3.22.1)
t o be
f o r t h e Hodge
proved
realization
(3.2.13).
The c a t e g o r y
With
of
*
one r e a l i z a t i o n ,
Deligne
In p a r t i c u l a r ,
to
alternating
W
morphic
dx.
—^- , i . e . ,
of
.an
4^
^P' - = H ^ Y ^
, where
the class
realizations
(3. 2 2 . 1)
U
image
,,an
' cf)^
shown
3.22.
giving
this
of mixed
motives
the notations o f the previous
section
we
define the
functor
H=
by
H(U)
4.1.
=
9
n=0
n
^
H (U)
Definition
-
MR
for
U
Hodge c y c l e s ) o v e r
of
generated
MR^
4.2. P r o p o s i t i o n
if
and o n l y
smooth q u a s i - p r o j e c t i v e o v e r
The c a t e g o r y
absolute
k
MM^
k
i s the Tannakian
by t h e image o f
A mixed
o f mixed motives
(for
subcategory
H .
realization
i f i t i s a subquotient
k .
of
H € MR^
i s a mixed
H(U) ® H ( V )
V
=
motive
Horn(H(V),H(U))
and
V
over
Proof
a)
f o r some s m o o t h q u a s i - p r o j e c t i v e v a r i e t i e s
k .
We
quotients.
first
Let
U V
f
i)
H (U) € M M
of
an idempotent
n
n
ii)
W H (U)
m
in
H
1.1: B y 3.22
n
of
over
a n d some
the language
Lemma
subquotient
n
So
Gr H (U)
m
mixed motive
iii)
(Y)(n")
H(U)
f
i . e . , as k e r n e l
.
by i n d u c t i o n on
WH (U)EMM
implies
m
j*c
n
W
v
n
, as
k
Gr H (U) CMM
m
n
Gr H (U)
MM
fact,
k
iv)
If
Y
of
H (Y)(r)
f o r some s m o o t h
i s smooth p r o j e c t i v e
, n , r E ZZ
n
factor
projective
over
variety
Y
lemma 1.1
k , then any
, i s a direct
n
of
H(r) E M M
, then
r
i s canonically
By t h e l a s t
so
W
v)
If
H c H =
—
o
max
by
H '(Y)(n")
factor.
, which
i sa
by t h e f o l l o w i n g remark.
and
over
m
t o a sub-
E Z Z , a n d we c a n r e f o r m u l a t e
i s a direct
1(-1)
m
n
,H (U) =
m— I
i s isomorphic
m
: for
f o ra l l
r
f o r a l l r E ZZ
k
2
In
m
of realizations:
H E MM
If
n',n"
of
m € ZZ
n
m
a n d lemraa
4.3.
f o ra l l
k
contains a l l these sub-
k
factor
End(H(U))
=H (U) ,and
quotient
in
n € ZZ .
and
k
-> G r H ( U ) ) E M M
n
m
k
MM
o
€ V
n
WH (U)
in
K e r ( W H (U)
3.22
show t h a t
as a d i r e c t
k
€ MM
n
m » 0
U
(H(U)
formation
argument
V
0 H(V) )
of tensor
n
also
E MM
H(U) ft H ( V )
isomorphic
V
fft
V
H
=
of
m € E
H'
H E MM
— k
we show
with
n!
WH
mo
Gr H
m
nI
»
and d u a l s .
n
V
m
E MM
R
m E Z2 .
f o ra l l
f)
) , and
(W_ H (V))
1
W
Gr H * 0 . L e t
m
(nEzz | Gr^H * 0 } and c o n s i d e r t h e cartesian
W
t h e number
1
(~?
products
W (H (V) )
k
to
.
Gr*H
mo
by
induction
m =
square
Then
the quotient
GrÛ^/Gr^B
i s in
MM
, as i t i s a
k
direct
W
factor
Gr^H
G
of
r
H
m
€
MM^
Q
b y lemma
i s a subquotient
o
projective
of
H
1.1
n
o r r a t h e r 4.3. N a m e l y
(Y)(n")
f o r some
smooth
Y
and
n ' , n " G ZZ
b y t h e same a r g u m e n t s a s i n i i ) ,
2dimY - n
as
H (Y ) & H
*
<Y )(dim Y )
by P o i n c a r e d u a l i t y ,
n.
n«
n.+n*
H '(Y ) ® H ^(Y ) c H
^(Y x Y )
by t h e K i i n n e t h f o r m u l a ,
2
2
2
1
1
1
H (Y1K-I)
n
n
c H
+
2
1
( Y xJP )
Therefore
H
varieties
1
€ MM
W
W H
mo
-> G r ^ H / G r H
m o m
T
the
commutative
W
n
and
1
2
Y
H
(Y J 9 H
and
i
n
n
(Y ) = H
n
2
1
, a s i t i s t h e k e r n e l o f t h e map
k
. By i n d u c t i o n
W
.H
m-1
J
exact
is in
MM,
— k
, and
H
.H
m-1 0
A
w
Gr H
>0
vv
> Gr H
m
H = K e r (H'
H /W
H
0
m-1
b)
Now
of
a
we
have
t o show t h a t t h e f u l l
a l l subquotients
Tannakian
i)
Let
B'/A'
the
MR
of
H
1
canonical
of
€ MR
i s (isomorphic
i s in
0
MM.
— k
consisting
0
for
U
and
V € V
H € MR^
,
A c B c H
r
k
A' 5 B
,
1
c H'
to) a subquotient
, and
. Then
of
H ® H
1
, via
isomorphism
B O B '
v
H(U
(H(U ) ® H ( V ) J
1
1
x U )
2
®(H(U )
1
S H(V
x V
2
2
)
V
B
~
A ® B' + A' ® B
ii)
/
7
V
® H(V ) )
3
= H(U )®
1
is
.
k
be a s u b q u o t i e n t
a subquotient
B/A ® B'/A'
of
.H)
m-1
->
subcategory
v
H(U) ® H ( V )
of
subcategory
B/A
from
diagram
H
see that
3
€ ZZ .
i
m-1
we
(Y JLY ) f o r
•
A
B
V
7
"
A'
i s isomorphic
H(U ) * H ( V )
2
1
V
®
to
H(V )
2
V
iii)
If
(B/A)
V
B/A
V
= A /B
i s a subquotient
V
= H(V) 0 H ( U )
Now
it
V
subcategory
of
formation
i t contains
identified
with
of
the i d e n t i t y
H(Spec
MR^
k)
absolute
a fully
The f u n c t o r
faithful
subcategory
tensor
M^
of
realizations,
D
in
V
formed
MR^
and
object
H: M^
k
remarks
show
that
t which can
a
Tannakian
which
t o any m o t i v e
i t s realization,
f u n c t o r and i d e n t i f i e s
MM
k
M^
with the
, whose o b j e c t s a r e d i r e c t
i . e . , with
the c a t e g o r i a l
sums
intersection
.
o
V
and
smooth q u a s i - p r o j e c t i v e v a r i e t i e s ,
k
diagram
a r e t h e c a t e g o r i e s o f smooth
projective
respectively,
we
have
of functors
o
H
Bk
where
v
and d u a l s .
associates
If
commutative
H(V) )
by t h e above s u b -
1 € MRj
-» R^
b)
k
v
(H(U) 0
. q.e.d.
Hodge c y c l e s ) o v e r
MM^
, and
, so i t i s indeed
(for
pure
v
of tensor products
a)
of
H
MR^
4.4. T h e o r e m
full
of
, then
i s o b v i o u s l y a b e l i a n , and t h e above
subcategory
is
H € MR^
.
Finally
be
i s a subquotient
the f u l l
quotients
of
means
a fully
faithful
functor giving
an
embedding
a
of
a subcategory
and
where
c)
I f we
^
which
means
an e q u i v a l e n c e
identify
with
we w i l l
always
do from
is
also
the Tannakian
by
t h e image o f
H
now
Y
on, then
' Y
k
d)
H € MR^
i s a motive
of
H(X)(r)
f o r some s m o o t h
and
some
H(X)(r)
Proof
r € ZZ
.
and
d ) : H:
the
left
was
already mentioned
are
to
just
direct
some
the
MR^
MR
H
also
, which
M^
which
of
motives
i s generated
.
k
i fi tis a
projective
-* MR^
subquotient
variety
a direct
X
over
summand
i s the unique
i n the diagram
k
of
of
b)
that i t i s fully
f u n c t o r making
commutative,
faithful
choosen
i n such
a way
s e e [DMOS]
category
( p a s s i n g from " f a l s e
I I p . 203) t h a t
and i t
(compare
g ) ) . The c o n s t r a i n t s o f t h e t e n s o r
"true motives",
H
M^
motives"
becomes a
functor.
For
is
triangle
I I 6.7
tensor
of
i f and o n l y
I t i s then
under
.
a)
[DMOS]
M^
the category
-
k
of subquotients,
of categories.
i t s image
subcategory
|
formation
a motive
factor
r € ZZ
of
over
H(X)(r)
, compare
a sum o f p u r e
form
M
k
, the r e a l i z a t i o n
d)
i s a
f o r some s m o o t h p r o j e c t i v e
[DMOS]
realizations,
stated i n
H(M)
I I 6.7 b)
. I n particular,
i.e., lies
. Furthermore
we
2
H(X)
0 H CIP
H(X)
0
X/k
in
R^
and
H(M)
, and i t i s o f
have
1
)®
l r l
H(X) ( r )
so
H(X)(r)
any
subquotient
shows
d.) .
and t h e r e f o r e a l s o
of
H(X)(r)
2
(H (1P
H(M)
1
V
) )®
lies
r
in
r > 0 ,
MM^
" i s " a m o t i v e b y lemma
. Conversely,
1.1/4.3,
which
Finally,
l e t H
subquotient
V
k
. By
~
=
W
©
G r (B/A) , s o we o n l y
me TZ
a l l m € TZ . B u t G r (B/A)
W
V
® H(V) )
=
f
V
are
t o show
W
©
U
w
G r (B/A) € H ( M )
m
-k
i s a subquotient of
w
Gr (H(U)
m
as a
assumption,
have
m
for
, given
f o r A c B 5 H(U) ® H ( V ) , w h e r e
B/A
smooth q u a s i - p r o j e c t i v e o v e r
B/A
MM^ n
be an o b j e c t o f
1
V
G r H(U) 0 G r ^ H ( V )
s o b v u s i n g 4.3
p
q
p+q=m
\
a s b e f o r e , we o n l y h a v e t o show t h a t
G r (H(U) ® H ( V ) ) € H ( M )
m
—k
W
f o r a l l m € TZ , i . e . , t h a t G r H ( U ) € M,
f o r a l l smooth q u a s i v
1
P
projective
products
U/k
M^.
as
and d u a l s ,
K
~
W
and
v
Gr (H(V) )
4
X 3 u
W
we o b t a i n t h a t
G r H(U)
q
Y =
i
=
1
i sa direct
^
sum o f t h e
W
n
b y 3.22 t h a t
and
therefore a motive
the
summand o f
M^
M£
inclusion
e
lemma
N
and
R^ ^*MR^
to
4.5.
contains
f o l l o w s from
2.15, t h e c o r r e s p o n d i n g
1.1/4.3,
H
subcategory
M£
f o r X € V^
subcategory.
of
2 n
p
~ (Y
( p
~
n )
)(-p+n
b y lemma 4.3.
by
rest
i s a subquotient
be t h e Tannakian
H(X) ( r )
. The other
Tannakian
Gr H (U)
H: V ^ -» M R ^ . T h e n
image o f
(U) ,
P
and
and c ) : L e t
_
Gr H
P
b)
of tensor
v
(Gr (H(V))
. Choosing
"
Y^
as i n s e c t i o n 3 ,
N
and
formation
W
=
generated
every
f o r M M ^ <^-» M R ^
direct
r € ZZ , a n d s o i t c o n t a i n s
the fact
that
i s closed w.r.t.
statement
by
f o r M ^ °—> M R ^
i t f o l l o w s from
M^
i sa
subquotients
i s equivalent
4.2 a n d t h e
i s clear.
Remark
In the proof
we h a v e
the subobjects
W^N
N
In particular,
r
seen,
that
a r e mixed motives
f o r any mixed
motive
and t h e s u b q u o t i e n t s
W
Gr
m
a r e motives.
extension
of motives,
any mixed motive
and t h e s p e c t r a l
= H P t Y ^ ) (-q)
is
a spectral
sequence
o f mixed
sequence
=» P
motives.
H
+ (
3(U)
i s a successive
(3.22.1)
4.6
By a b a s i c
Tannakian
theorem
on Tannakian c a t e g o r i e s ,
categories
and
MM^
are equivalent
gories of representations of certain pro-algebraic
These
a
arise
as automorphism
neutralization:
the
automorphism
H
: MM
0
by t h e r e s t r i c t i o n
G(o)
t h e automorphism
H
Rep G
denotes
representations
valences
the inclusion
M
as
MR
M
k
and
MM
M
Rep G
k
-» MR 1
M-
k
for
k
MM
l e t MG (a)
be
functor
Vec^
Rep
^
k
-+ V e c ^
MM
k
of (finite
group
dimensional) algebraic
G/Q
, we
k'/k
equi-
G(o)
motives
morphisms
IT: G (a)
(loc.cit.).
Finally,
. Thus
: M G ° (o)
have
c o r r e s p o n d s t o a morphism
k
of Artin
k
-> MM-
k
, and
.
-> G
inclusion
U
extension
G°(o)
k
i
functor
functors
j
over
i n
MTT : MG(a) ~* G^
H(U)X ^k
g e t homomorphisms
, where
and
extension
, as c a n o n i c a l l y
MG(a)
k
the base
induces base
we
The
ofpro-
( s e e [DMOS] I I 6.17)
a smooth q u a s i - p r o j e c t i v e v a r i e t y
Mi
( s e e 2.13)
R e p MG(a) ,
( s e e 2.16 i ) )
and
extension
and
M
gives
k
o: k <-* C
i): MG(a) -* G ( a ) , s e e [ S R ] .
the category
k
giving
of the r e s t r i c t i o n
k
k
k
M
groups
Q .
functors
: MR
q
over
categories
MM
of
H
groups
c o
of a pro-algebraic
o f tensor
algebraic
Vec
the category
M
and
of
group
: M
q
-+
k
to the cate-
of the f i b r e
of the f i b r e
given
If
groups
F o r an embedding
group
the neutral
1
-
x
H(U
and a
: G°(o)
^')
field
-+ G(o)
= G(o) = A u t ® ( H - | _ )
M
and
I —k
o
MG
the
—
®
(o) = MG(a) = A u t ( H J J
fibre
embedding
functor
o
: k
H-
on
CE
with
m m
.)
M
k
a
r
e
t
and
o I,
Unto-BiMothek
n
e
automorphism
MM-
= o .
groups o f
r e s p e c t i v e l y , f o r some
4.7.
b)
Theorem
Via
a)
\j; i s a n e p i m o r p h i s m ,
\\) , G(o)
quotient
of
i s identified
MG(O)
; the kernel
i . e . ,faithfully
with
t h e maximal
U(o)
of
\\)
flat.
pro-reductive
i s connected and
pro-unipotent.
c)
With
with
t h e above
oj
=
k
algebraic
o)
notations
there
(involving
a choice
i s a commutative
exact
of
o
: k
<C
diagram o f p r o -
groups
U(o)
U(Q)
i
MG (O)
->
0
d)
G (O)
of
G(o)
and
reductive
e)
and
HOL
MG(a)
quotient
F o r any
T
€
G
restrictions
f)
F o r any p r i m e
TT o S p
g)
1
1
a r e c o n n e c t e d and t h e i d e n t i t y
G°(a)
on
to
1
MMr
K
i s the maximal p r o -
-1
, and
TT
, regarding
®
( T ) = Horn
(H-,HCT
, there
: G^ -+ G ( O ) ( Q )
1
1
= i d
are canonical
and
Msp :
and
Sp
section
G
1
1
r
continuous
for
I : G(o)
Gr
= TT , a n d
I © Sp
1
= Msp
1
w
m
homomor-
1
= ^ o Msp
m £ ZZ
Pictorially:
)
MG(O)(Q )With
1
.
MG(o)
of
to the s e m i - s i m p l i f i c a t i o n functor
MTT O I
OT
H-
.
MM,
One h a s
components
.
T
i s a canonical
corresponding
G(g)
' (MTT)" (T) = Horn®(H-,H- )
= i d , MTT O M s p
There
(O)
MG°(a)
as f u n c t o r s
Sp
•I
o .
, respectively.
of
MTT
MG(O)
1
k
the
phisms
G
MG°(a)
and
Mi
0
->
N
f o r any
1 .
\p ,
MTT
-
MG(Q)
^
Msp
-
G
k
1
(4.7.1)
G(O)
SP
is
commutative
Proof
ful
II
a)
i s faithfully
and s a t u r a t e d w.r.t.
equi-directed horizontal
flat,
as
M
subquotients
<-> MM
k
G(c)
II
6.22
Let
i s p r o - r e d u c t i v e , as
i s fully
k
b y 4.4 b)
be t h e maximal
faith-
, s e e [DMOS]
to subquotients.
w
N = ©
m€ZZ
Gr N
,i.e.,
For
N € R
m
= G(o)
.
We
quotient of
pro-unipotent
radical
MG (O)
c)
i s semi-simple
k
N € Rep G(a)
MG(a) .
t h e r e f o r e have
N € MM
H R =
K
R e p G ( a ) =Rep G ( a )
MG°(a)
G°(o)
, and by
o f the connected
- f ) • The statements
the proofs
c)
Mi
K
and
thus
i s t h e maximal
c) , U(o)
i sthe
p r o - a l g e b r a i c group
for
for
MG(o)
G(O) a r e p r o v e d
i n [DMOS] I I 6.23,
are similar:
i s a c l o s e d immersion
as any o b j e c t
N
of
MMj- i s
see t h e
a s u b q u o t i e n t o f an o b j e c t
N
x k
with
N
€
c r i t e r i o n i n [DMOS] I I 2.21 b ) ) . I n f a c t , i t s u f f i c e s
v
case
models
with
.
and
the
groups).
and c l o s e d
we
and t h e r e f o r e
t h e same a r g u m e n t s ,
pro-reductive
0
v
conclude
With
s e e [DMOS]
f o r non-connected
K
= Rep G(a) .
G(a)
i s semi-simple,
k
pro-reductive quotient of
R e p G ( a ) c R e p MG(a) = MM
respect
k
M
( h e r e we u s e " r e d u c t i v e " a l s o
G(o)
Then
we
maps.
2.21 a) .
b)
M
\p
i n a l l ways w i t h
1
N = H(U) 0 H ( V )
U'
can take
and
N
q
V
1
over
for
k
U V
a finite
= R ,^ (H(U')
k
to consider
°
f
€ V^
; but
extensions
v
0 H(V) )
U
k'
and
of
k
= H(Res y U')
0
k
k
V
have
, and
H(Res^,/
U'
-
v
k
Spec
MTT
')
k
1
t where
-
Spec
w.r.t.
o
1 -* MG
U (o)
(o)
the
in
M G ° (a)
any
non-trivial
ducts,
see
= MM-
must
are
of
X
be
the
of
have
a
2.22.
=
k
MG(Q)
^
k
faithful
MM
X
which
to
stable
such
an
object
zero,
as
the
G°(Q)
e)
F
i n N,
, and
under
tensor pro-
N
G Re£
weights
we
MG°(o)
occurring
therefore
have
reduced
i s the
in
bounded.
i n [DMOS] I I
MG°(Q)
for
category of
and
i s proved
t o any
g € MG(o)
element
T = MTT (g)
for
M,N
of
. We
€ MM
=
(R)
Horn
to
the
6.22.
full
identity
and
k
= Horn
(H ®R,H ®R),
a
a
( H - , H ~ ) (R)
=
t
write
H
q
(M,R) = H
(M) ® R
q
f € Horn(M,N)
there
A
M
— ^
H (M R)
F
g
f
M
clear
the
H (M R)
a
=
f
H-^. (M
F
H-(N R)
map
is
show t h a t
f-
by
. Finally,
diagram
H (M R)
5
have
MG°(a)
i s not
occurring
of
.
k
of
g
fact,
and
of
case
TT O \\J
MTT =
M G ( a ) we
But
canonical
. Then
commutative
In
surjective.
.
for
k
special
, as
, which
to associate
* k
(4.7.2)
restriction
is
the exactness
is a
,
Rep
0
T
M=M
V
is fully
k
0
of
- O
G (Q)
Horn® ( H - ® R , H _ ® R )
is
M^ <=-• M
of weight
€ M
a (D-algebra, a
and
Grothendieck
S H(U')®H(V')
i s t o t a l l y disconnected,
k
component
R
1
M ° <^ MM
-* 1
k
same a s t h o s e
of
We
as
= k e r MTT
, n
pure
connectedness
e)
n
[DMOS] I I
N
G
N ^ k
representation
particular
As
G
connectedness
subquotients
i s the
k
s u b q u o t i e n t s , and
factorization
the
^ U'
MTT
For
In
k t
, as
-> MG (o)
lies
n
k
Mi
d)
N
Res
is faithfully flat
saturated
from
V
= H (N R)
applying
x Horn(M,N) -* N
a
f
the
N
_;
l
^
functoriality
one
sees
that
H (N R)
a
of
=
f
the
g
M
H
5 7
R)
f
QT
(N R)
f
to the
t h e r e i s a commutative
evaluation,
diagram
H (M R)
a
-i
f
H (M R)
a
f
(4.7.3)
H (N R)
a
for
any
On
?
€ H
o
the other
be r e g a r d e d
Rep
G
the
choice
and
we
of
f
v i athe action
of
G^
p
the extension
Horn(M,N)
: k
o
J
(C
.
This
. Then
Horn(M,N)
H
and
i s induced
Horn
D R
(Hom(M,N) ) =
H CHom(M N)
by the p r o j e c t i o n
1
F
(H (M) H (N))
1
CB
f
J-
1
1
f
=
a
If
the projection
MTT (g) = T
to
0 k)
K
F
F
(Tf)
G^Q
(M)
i s a
morphism
on
1
, H
r r
(N))
and
and on
Horn(M,N)
by d e f i n i t o n
. So
(4.7.2)
g
acts
follows
H (M)
Q
like
from
H
5
Q
x
on
(4.7.3),
as
( M )
T
5
H (N)
H-(N)
M
r r
on
<E ,
j
0
Hj(M)
g
F
k
M° =
: k
there
k
of
Horn(H-(M),H-(N)) = H o r n ( H ( M ) , H ( N ) ) .
, then
Hom(M N) c Hom(M N)
is
F
=
Hom(M N)
t o Hom ( H
respectively,
p
G
F
J-
H (Horn(M,N))
is
(Hom(M N)
A
H
5
T
(M)
commutative.
The
diagram
(4.7.2)
i n
Hom(H-(M,R)
shows t h a t ,
i f we
, H- (MR)) by t h e upper
OT
T
/JurK T
Hom(M N)
depends
such that
-
u
1
F
f o r any
a
for
g = g
,
Hom(M N)
( s e e 2.19)
[DMOS] I I 6.17 a n d 6.18)
motives
where
f
m o t i v e , i . e . , an o b j e c t
o f an e x t e n s i o n
choose
f
rr
as an A r t i n
(compare
mixed
a
(Horn(M N ) ) = Horn(H^ (M) ,H ( N ) )
a
0
o
hand,
can
k
H (N R)
f
define
line
t h e image o f
of
(4.7.2),
we
g e t elements which
compatible
Horn
with
( H - ® R, H ~
factor
o f an
are functorial
tensor
T
® R)
object
products.
in
These
, a s any o b j e c t
ft
for
M
in
M
and
define
in
MM^
is
we
have
bijective
looking
f)
Msp
it
defined
a map
a s o n e may
at
(4.7.2)
i s shorter
MM^
We
maps
a
only
MM
unique
t o note
to
K
Sp
just
=
1
(T)
in
1
Horn
.
(H-,H~ )
, which
T
the construction,
[DMOS] I I 6.23
to define
o Sp
that
(d),
but
i t by
.
1
M>~~>s.s.M
b y 4.5, a n d t h a t
i s a tensor
f o r any
H € R^
functor,
there i s
isomorphism
~
H=
W
Gr H
e
m
m € TZ
inducing
the i d e n t i t y
on the graded
for
realizations
of different
pure
direct
$
(MTT)
like
f o r us h e r e ,
have
is a
again.
Msp
g)
and
, s e e 2.20 e)
see by r e v e r s i n g
c a n be d e f i n e d
1
,
elements i n
-1
So
R
pieces,
1
as
weights.
Horn(H,H ) =
With
respect
0
to
this
isomorphism,
M
is
the i d e n t i t y ,
MM
R
s
K
s
A -
M
a n d s . s . commutes
K
with
H
. Therefore
s.s.
a
induces
t h e homomorphism
As
s . s . commutes
we
have
4.8.
[SR]
with
I
a n d we
the inclusions
MTT © I = IT , a n d t h e r e s t
For the d e s c r i p t i o n of
I V §2 on f i l t e r e d
functors
W H „ =
ma
H
HW
am
q
on
MM
k
Tannakian
M°
o Z =
MM
K
id .
and
M^
M
K
,
i s clear.
U(a)
we
can use Saavedra's
categories.
are f i l t e r e d
Namely t h e
by t h e w e i g h t
results
fibre
filtration
, and i f
$
1
Aut
is
have
the subfunctor
of
Aut
(H )
Q
(H )
a
formed
by those
tensor
automorphisms
of
H
which
o
w
Gr H
ma
induce
= W H ZVJ
,H
ma
m-1 a
Proposition
Gr H
o
1
, t h e n we
rr
4.9.
t h e i d e n t i t y on
U(a) = A u t ®
k
C
Proof
, respectively
From
the proof
canonically
1
(H
•
) = Aut®
a: k
Gr H
we
U(a) = Ker(MG(a)
have
= Aut®
1
g
(H ,
a
4.10.
i n 4.7
. With
to see that
G(a)
is
group o f the f i b r e
this
identification
-• G ( a ) ) = K e r (Aut® (H .
) a j MM^
M X i r
apply
to
inconvenient
varieties,
A u t ® (GrH
W
and
k
smooth
proper
can
define
the
k
, MG°(a)
and
to r e s t r i c t
G°(a)
by t h e
to projective or
a n d we w i l l
show t h a t
o
W,
•K
be t h e c a t e g o r i e s
o f smooth
v a r i e t i e s over
this
isi n
exactly
quasi-projectivity
3,
the v a r i e t i e s
not
necessarily
.We
by
W
t h e same way
construction
now
k
separated
, respectively
. Then
we
functors
H:
motives
l
a
not necessary.
and
[D4]
aI£ = a .
c) .
I t i s often
Let
in
MM^
for
— k
quasi-projective
fact
on
)
MMr;
) .
same c o n s i d e r a t i o n s
diagram
with
|
MM,
1
The
= H .s.s.
(H- •
a
t o the automorphism
functor
q
(C
1
k
o f 4.7 g) i t i s e a s y
isomorphic
W
Gr H
m o
©
crrf
have
JMM
a:
=
was
k
3, b e c a u s e n o w h e r e t h e
u s e d . Of c o u r s e ,
projective
claim
MR
as i n s e c t i o n
X,Y,Y..,Y^
o f mixed
this.
-
k
are only
then;
this
Hodge s t r u c t u r e s
that
we
with
the notations
smooth
of
and p r o p e r and
corresponds
to
Deligne's
f o r smooth v a r i e t i e s
d o n o t g e t new m i x e d
motives or
w
w
MM
k
4.11.
Proposition
H
: W_
-> R
k
factorizes
k
through
and
o
H:
MR^
Proof
is
Let
U
an open
result
is
By Chow's
a proper
Then
be a smooth
o
on r e s o l u t i o n
Hironaka
MM
we may
.
k
variety
over
lemma
there
X
X
we
q
f : X -> X
q
and b i r a t i o n a l
1
that
X
variety
X
U = f
1
o
Hironaka s
, and a g a i n
Let
[N], U
, and by
c a n assume
t o be smooth.
i s proper
. By N a g a t a
i s a projective
morphism
assume
k
variety
of singularities
birational
f : U -+ U
induced
through
subvariety of a proper
smooth.
and
factorizes
Q
by
( U
Q
) .
and t h e r e f o r e t h e
map
f*
: H(U )
q
H(U)
o f mixed r e a l i z a t i o n s i s i n j e c t i v e - i n f a c t , i n a l l t h r e e cohomology
theories
duality
support.
i.e.,
we
there
conclude
i n v e r s e by t h e t r a n s p o s e
of the corresponding
So
with
have
i s a left
U
f*
identifies
map
= X
o
and
b y 4.4. a) a s
2
U = X
Poincare
f o r t h e cohomology w i t h
H(U )
q
with
a m i x e d m o t i v e b y 4.2. F o r
o
under
U
a subobject
q
and can use
H ( X )€ R
.
o
—k
1
of H(U),
smooth
and
4.4
,we
d)
compact
proper
can also
PART I I
ALGEBRAIC
§5.
CYCLES,
K-THEORY,
The c o n j e c t u r e s
The
the
common
o f Hodge
object
description
a smooth p r o j e c t i v e
k
be a f i e l d
projective
Chow g r o u p s
linear
5.1.
whose
r
r
=Cl Z
H
under
G^
group
over
5.2.
For
whose
r
x
:
2 r
r
CH (X)
r
2 r
—>
H
X
k
and Tate i s
i n t h e cohomology
conjectures, l e t
k , G^. = G a l (k/k) , X
a
r
, and
CH (X)
r
X
of codimension
k
there
r
map
on
the
modulo
2 r
k
(X)(r)
,
part
i
(H
that
i s a cycle
H
I * char
£
2 r
2 r
( X ) (r) )
t h e image o f c l ^
i s finitely
—>
for
(X,ffi (r)) = H
=: T
conjectures
: CH (X)
(X((E) ,(B)
2 r
i n the f i x e d
Q^ , i f
=cl '
these
k
cycle
f
k = (E
cycles
k , X = X >< k
cycles
(X Q^r) )°
. Tate
o f Hodge
(see, e.g., [Kl]§2) .
image c o n s i s t s
H
over
i s a canonical
image l i e s
C l
variety
To r e c a l l
closure
CLASSES
f o r smooth v a r i e t i e s
of algebraic
algebraic
equivalence
and T a t e
variety.
of algebraic
There
C l
with
EXTENSION
of the conjectures
o f the group
of
smooth
AND
generated
generates
as a f i e l d
map
(X«E),Q)
,
of ( r , r ) - c l a s s e s , i . e . , i s contained
r
r
n H ' (X,CE)
= H ( X ( C C ) ,(D)
2r
([T
r
0 F H
2 r
(X,(E) .
in
this
1]).
The
group
over
5.3.
In
map
by
Cl
into
Q
our
r
j
=
r
C l
the
r
i t is better
2iTi
and
r
: CH (X)
r-fold
the
image o f
el
generates
this
Tate
to
regard
renormalize
i t as
2 r
—>
H ( X ( ( E ) ,Q-
twist
of
the
a
the
last
cycle
map
r
(2iri) )
Q-Hodge
= H
2 r
( X ) (r)
structure
2r
(X)
(Note
If
x
^
that
[Gr]).
setting
powers o f
2r
H.
(cf.
states
:=
the
one
(X((E),Q)
first
cl^
= C
, whose
F ° ( H ( r ) ) 0 (C = F H
Chern
Chern
i s the
classes,
®
<E
of
for a
t h i s amounts
(O, O) - c l a s s e s .
Hodge
to
structure
using
the
H
)
more
class
1
: Pic(X)
1
image c o n s i s t s
r
formula
works w i t h
natural
which
H
2
= H (X,0*) — >
connecting
morphism
H (X,Z-2Tri)
for
the
exact
,
sequence
of
analytic
sheaves
O —>
Then
no
choice
Z-27Ti — >
of
r
and
C l
complex
Finally,
cycle
=
/^T
is
P
>
x
O
—>
involved,
O
and
.
moreover
are
g
and
compatible
etale
f o r any
under
the
comparison
cycle
0
C
^>R
image l i e s
H
r
field
'
X
)
k
of
c h a r a c t e r i s t i c zero
~>
H ^ ( X Z k ) (r)
=
( F V j i ( X ) ) (r)
=
in
F ° ( H ^ ( X ) (r»
isomorphisms
cohomology.
map
whose
the
r
maps c l ^
between
i
e x
O
3
,
H^(XXr)
there
is
a
and
which
e
f o r any
o : k —>
(E
i s compatible
c l
cj*
r
r
=- c i '
0
X
r
r
: CH (X)
CH (GX)
£'
with
t h e map
a X
—
> H
2 r
(aX (CE) , (D) ( r ) = H
0
2 r
(X) (i
0
2r
under
t h e comparison
5.4.
Thus
isomorphism
I
o oQ
( X ) (r) .
f o r c h a r k = 0 we o b t a i n a c y c l e map
c l
= c l ^
: CH (X) — > T ( H ( X )(r)
r
r
X
r
2 r
H
A H
2r
into
the group
o f a b s o l u t e Hodge c y c l e s
in
H
(X)(r)
(denoted
2r
T(H
(X) (r)) i n t h e f i r s t p a r t ) . A s a c o m b i n a t i o n a n d w e a k e r f o r m
o f 5.1 a n d 5.2 o n e may c o n j e c t u r e t h a t t h e i m a g e o f c l
generates
2r
CD-vector
the
r
r
H ' (X)
0
r ^ ( H
space
n H
(X) ( r ) )
. In f a c t ,
(X)(r)
H
0
this
T
A H
(H
T
a l g
2 r
(H
i s implied
2 r
(X)(r))
by e i t h e r
r
:= I m c l '
we
5.5. Lemma
a) L e t
be a f i n i t e l y
X
true
k^ c k
i s defined over
and every
for X
finite
and
k^
G
: K —>
(E , t h e n
X
0 Q
have
. I f Tate's
extension of
generated
field
such
conjecture i s true f o r
k^ , t h e n
c o n j e c t u r e 5.4 i s
k .
b) I f t h e H o d g e c o n j e c t u r e i s t r u e f o r
a
k
t h e Hodge o r t h e T a t e
More p r e c i s e l y ,
X
(X)(r)°
(X)(r))
conjecture.
that
2 r
**
(5.4.1)
we s e e , t h a t
by the i n c l u s i o n s
aX
f o r some
c o n j e c t u r e 5.4 i s t r u e f o r X/k .
embedding
Proof
a)
It i s clear
defined
over
is
f o r every
true
2.19
kQ
finitely
over
absolute
element
k^
in
H
r r
5.4.1,
the
same
(X/k)
under
, see
It suffices
5.4.1,
k'
which
5.6.
We
r
a)
for
k
note
that
2 r
= k
: k —>
: H
2 r
over
(X)
should
k
kg
k
is finitely
the
assumption
are algebraic,
by t a k i n g
fixed
closed.
i s algebraic
over
hence
modules
k
, a n d we
By
assumption
over
some
g e t an
field
algebraic
s e e [DV] e x p . 0 .
r
finitely
"inclusion"
case
2 r
2 r
imply
some
other
-^->
an
0 H
2 r
H
2 r
case
h a d Hodge
s h o u l d be
surjective
i s D e l i g n e 's " e s p o i r e "
that
see [D6].
( X ) (r)
k
s h o u l d be
surjective
for
k
field.
( X ) (r) f o r
c
o
: k ^—>
CE
with
isomorphism
(X) ( r ) ] ® Q
£
^ H
2 r
and s u f f i c i e n t l y
i s known, b u t a) w o u l d
we
( X ) (r)
CE ; t h i s
H
generated
" . In the f i r s t
second
H H
the prime
®
induce
r
£
that
extension
a r e t h e same. By
over
i s a b s o l u t e Hodge
[H ' (X)
k
r
c
generated
-(r)
= a
r
( r ) ) 8 ® g ->
AH
finitely
H Z (X)
CL
T (H (X)
for
. By
generated
t h e above c o n j e c t u r e s would
a
and
b)
k
k
algebraically
generated
( X ) ( r ) ) ^=—>
2r
I
k
by s p e c i a l i z a t i o n ,
Hodge c y c l e
c)
and
k
i n [DMOS]I 2 . 9 t h a t
a b s o l u t e Hodge c y c l e
every
2 r
k^
®£
Q
this
ones:
(H
A H
k
0
is
, since
(X /k )
finitely
proved
, and f o r
to consider
is finitely
over
k
D R
k
2.19.
every
cycle
weaker
for
= H
some
a l l a b s o l u t e Hodge c y c l e s
statement
and
over
over
extension of
to consider the case
. I t i s then
Hodge c y c l e s
and
11
a b s o l u t e Hodge c y c l e
generated
, i.e., i t suffices
generated
a|
every
i t i s t h e r e f o r e defined over
of
b)
some
that
we
imp]y
(X) (r)°
b i g . I n g e n e r a l , no
" c
"
c o u l d conclude Tate
Tate.
k
a n d b) w o u l d
imply
=» H o d g e , i n t h e
d)
dim
(H
f J 3
r , r
(X)
2 r
n H
( X ) (r)
s h o u l d be
independent
of
a
.
2r
e)
dim_
f)
These
and
[H
(X)(r)]
0
dimensions
sufficiently
5.7.
Remark
[DMOS]I
2.11;
and
s t r o n g e r form
The
results
In
in
the
fact,
hold
t h e work
stated
be
of
k
&
.
finitely
the
for abelian
generated
in
varieties
varieties
Shimura-Taniyama
[Se
2]
§
to the
3,
and
have
associated
6.27
,
see
complex
S e r r e , even
also
Pohlmann
of motives
f o r example
X
with
and
compare
category
[DMOS] 6.26
conjectures a)-c)
of
a)
containing
compare
setting
for
for abelian
of
extended
varieties,
hypersurfaces,
equal
proved
a)-c)
by
can
independent
big.
the
abelian
s h o u l d be
D e l i g n e has
multiplication
by
s h o u l d be
in
[P].
generated
K3-surfaces
and
Fermat
.
convenient
Tannakian
interpretations
categories.
For
example,
2r
let
H
2 r
(X)
in
u
MT(H^
be
a l l products H
e Z
§
the
, and
3.
On
thus
by
the
fibre
Then by
a)
b)
r
(X)
0
S
®(H
H
c)
v
(X) )
G(H
of
2 r
0 t
(X))
< G(H
have
similar
of
(X) )
0
®
Q (1)
the
(X)
q
(X),a)
M^
2 r
2 r
of
2r
H
(X),0)
imply
image
the
Hodge
fixing
U
structure
a l l Hodge
for a l l
category
and
, see
be
Q(1)
the
by
of
cycles
s,t CJN
Tannakian
generated
f o r a l l t e n s o r s we
2 r
group
(H
), i . e . , the
q
f o r a l l t e n s o r s would
and
fjj
group"
let
subcategory
(5.4.1)
2 r
GL
structures
o t h e r hand,
functor
of
"Galois
Hodge
MT(H
and
2
Mumford-Tate
subgroup
the
the
Tannakian
(with
the
. I t i s the
generated
I
(X))
Q
,
[DMOS]
" G a l o i s group" of
2r
H
(X)
and
1(1)
2r
—>
G(a)
GL
f f l
(H
q
(X))
have
;
equality.
interpretations.
However, one
does
not
2r
know
i n g e n e r a l , whether
conjectured
by
G^
Grothendieck
acts
and
semisimply
S e r r e , so
one
on
also
H^
j
(X)
has
as
to consider
subquotients
of the tensor
any
case,
i s related
and
MT(H
c)
products
for H
2 r
to the conjecture
(X)
([DMOS]
that
Im(G,
I . 3.2). In
—>
\
±
5.8.
(X))(Q„ )
a
I
Similar
new,
U
smooth
a
smooth
at least
R
before,
(
D R
r^(H
T (H
H
2 r
2 r
H
D R
(
U
)
r
f
f
r
having
(
R
)
)
=
(U) (r)J= H
U
^
0
2 r
(
H
variety
r
(
D R
U
( U ) (r)
q
(H
2 r
)
(
R
)
k
—>
T
A H
)
N
F
° <
(H
q
2 r
H
2 r
( f o r example
H
2 r
U
)
and
to
give
In f a c t , l e t
l e t
X
are cycle
be
a
maps
r
c l ^ with
(
R
)
=
W
H
2 r
images i n
D R
(
U
)
" ^ " D R ^
'
(U) ( r ) ) ,
(CU,Q)
by
(
D R
seem
( U ) (r))
and
r
(1H ( X ) ) )
3.
zero.
k
there
2 r
(H
r
r
CI^ ,cl^
W
N
over
Then
(U) ( r ) ) n F ° ( H
(27Ti) W
[ S e 2]§
of c h a r a c t e r i s t i c
: CH (U)
R
=
respectively
k
components
W
( U ) (r) ) - W
(5.8.1)
for
compactification.
= c l
see
f o r non-proper v a r i e t i e s
quasi-projective
projective
Cl
as
a r e commensurable,
conjectures
nothing
be
1
GL-
using
2 r
( U ) (r) 8(E)
r
n F H
2 r
(aU,(E)
,
fundamental classes
i n the
2r
relative
cohomology
r
then
). But
r (W H
?
(see
W H
Q
Q
3.22,
2r
(U)(r)
2 r
r U
c l '
or
[D4]
surjective.
H
?
3.2.17,
(X)(r)
On
f o r a prime c y c l e
= r (Im
i s a direct
2 r
(U)
z
factorizes
(U)(r))
H
are
H
2 r
—>
the other
r
9
of
codimension
through
(X)(r)
—>
f o r the l a s t
factor
Z
of
H
2 r
H
2r
H
2 r
(U)(r))
equality),
(X)(r)
and by
,-so t h a t
lemma
1.1,
t h e maps
(U)(r)
hand,
the r e s t r i c t i o n
r
CH (X)
—>
r
CH (U)
is
also
surjective,
and
the
Cl^'
CH
(X)
V
and
of
= F H
r
2 r
T
2 r
(H
1
the
(U CE)
U
H W
f
(U) (r)) = H
algebraic
implied
by
H
9
2 r
T
>
T
H Z
9
H
9
for
(U, (D)
and
k
the
those
for
X
2 r
( U ) (r)
the
= CE
conjectures of
k
T
u
s h o u l d be
same t i m e ,
Hodge
2 r
(H
(U) ( r ) ) =
n
ia
f o r f i n i t e l y generated
G
k,(D^(r))
(U
At
- ( X ) ( T )
following.
formulation of
cycles.
J
V
i s that
2 r
>
U
-
only possible
for
'
7
T h i s shows
Tate
r
G l
r
The
X
=
CH (U)
commutes.
diagram
these
k
generated
by
conjectures are
immediately
.
2r
The
same h o l d s
and
also
f o r the
has
resolution
f o r c o n j e c t u r e 5.4
Tate
of
Y^ (H
involving
conjecture i n characteristic
singularities
and
semi-simple
(U)(r))
U
p
>
0
action
of
in § 7).
So
, i f
G^.
,
one
on
2r
H^
(X)
(this
will
least
m o r a l l y , we
5.9.
However,
to
study
the
those
and
proper
1
H (X)(j)
(5.9.1)
for
any
d i s c u s s e d more g e n e r a l l y
obtain nothing
f o r smooth
again,
at
new.
non-proper
varieties
U/k
i t makes
sense
space
r
and
be
1
A f /
(H (U) (j) )
d e f i n e d i n 5.8.1
over
T(H)
mixed
1
k,
i s pure
TfH (X)(j))
of
=
for arbitrary
weight
fj
realization
H
2
i s zero unless
i-2j
T(W H) <^>
i , j€
and
. For
i
= 2j
X
,
smooth
because
i n general
T(Gr H)
0
. But
otherwise
the
space
above
can
be
non-zero
for
i
^ 2j
, and
i s indeed
connected
with
interesting
questions.
For
example,
x
*
y
are
two
U
=
X^{x,y}
X
is a
smooth p r o j e c t i v e c u r v e
k - r a t i o n a l p o i n t s , we
1
0 —>
i f
H (X)
1
—>
H (U)
—>
get
an
exact
H ° ( { x , y } ) (-1)
which
6(1)
factorizes
we
get
exact
sequence
an
1
0 —>
H (X)
through
1
—>
H (U)
the
2
— >
H (X)
—>
c y c l e map
1(-1)
—>
k
sequence
1
in
over
for
—>0
,
X
cl '
0
and
.
Therefore
,
(5.9.2)
weight
where
1(-1)
whether
Galois
Hodge
and
by
this
a
sequence
s t r u c t u r e of
H
non-trivial
if
shall
(x)-(y)
This
1
K-theory.
is a
, and
First
described
or
i n the
(U)
by
the
extension
and
of
Hodge r e a l i z a t i o n .
weights,
a
1
there
point
look
recall,
of
realization
= Hom(1,H (U)(1))
there
Chern
think
question
s e c t i o n of
the
Since
5.9.2
of
is
mixed
kernel
given
in
torsion
to
not:
i n the
2
* It is a non-trivial
1-adic
show i n § 9 t h a t
suggests
T(H (U)(j))
be
splits
element
1
we
1
^x~ y
have d i f f e r e n t
Hom(1(-1),H (U))
and
weight
"basis"
representations
cokernel
a
has
1
for
are
that
=
1
T(H (U)(1))
i s such
i n the
a
,
s e c t i o n i f and
Jacobian
of
X
" a l g e b r a i c elements"
indeed
the
characters
some, g i v e n
rational
by
.
in
higher
c y c l e maps
only
c l
algebraic
r
can
also
Ch.:
on
the
K (U)
>
Q
Grothendieck group
of
2 j
T (H
?
(U) (j) )
locally
free
(^-modules,
v i a the
isomorphism
K (U)
®
N
Q
s
of
Y KQ(U)
[Sou
(5.9.3)
on
[Sche]
1]
above.
there
ch.
Quillen's
®
N
Y
Q
=
U
and
: K
In the c i t e d
CH (U)
[SGA
Gillet
are higher
higher
1
©
0
Q
,
i>0
i s t h e y-filtration
Schechtman
Soule
Gr K (U)
i>0
1
where
1
©
U
(U)
K-grouos
exp.
0.
[Gi] g e n e r a l i z i n g
Chern
characters
>
1
Now
by
earlier
results
work
of
T (H (U)(j))
9
such
references
6]
that
ch- . . coincides with
O i 3
ch. . a r e d e f i n e d f o r t h e
r3
the
ch.
3
1
singular,
into
are
the e t a l e
the group
of
and
absolute
c o m p a t i b l e under
characters
so by
i , j
reducing
principle
one
:
K
Hodge
by
cycles
2j-1
(
U
has
)
>
to
V H
Chern
1
class
already
used
For
propose
j
ch
a
to
. First
can
be
for
each
2
in
-J=Ch
1
= c l
t o check
that
( U ) ( J ) )
classes
: Pic(U)
—>
the
these
Chern
,
and
using
r (H
the
splitting
f o r the
first
2
(U)(D)
?
, which
we
5.8.
study the
by
realization
of
image
investigate
non-zero,
morphism
classes
show t h e c o m p a t i b i l i t y
generalization
we
has
to get a
isomorphisms. But
1
Chern
one
means o f C h e r n
to universal
only
Rham c o h o m o l o g y ;
the comparison
are defined
c
t h e de
of
t h e maps
f o r which
studying
H
t h e Hodge a n d
-
i
5.9.3
and
the weights
i n the
sense
the Tate conjecture
of
f o r general
j
of
§
the
target
i
and
groups
the r e a l i z a t i o n s .
3,
o r an
we
£-adic
Namely,
one,
or
a Hodge
structure,
filtration,
W
Gr^H
5.10.
Then
Proof
5.11.
o r a d e Rham r e a l i z a t i o n - we h a v e a
and say t h a t
Lemma
Let
U
be a smooth v a r i e t y
occuring
in
1
f o rthe weights
w
i-2j
< w < 2i-2j
, i f 0 < i < d
i-2j
< w < 2d-2j
, i f d ^ i ^ 2d .
occurs
in
H
,
of dimension
H (U)(J)
we
d
i f
over
have
S e e [D4] 3.2.15 b) a n d [D9] 3.3.8.
Corollary
I n view
i-2j
or
w € TL
* 0 .
One h a s
0 < j S d
Proof
the weight
weight
1
T(H (U)(j))
and
* 0
a t most f o r
j ^ i < 2j .
o f 5.9.1 we m u s t
have
< 0 < 2i-2j
and
0 ^ i ^ d
i - 2 j < 0 < 2d-2j
and
d ^ i < 2d .
(5.11.1)
N
/
1
d
2d
k
5.12.
For the study
consider
o f t h e maps
the a c t i o n
K-groups
K^(U)
5.9.3
i t i s convenient
o f t h e Adams o p e r a t o r s
( s e e [ S o u 3]
for this
ip , k
and
^ 1
to
,on
the
t h e f o l l o w i n g ) . I f we
set
K
(U)
m
( j )
{x
=
€ K
(U) ® _ ( D | Y k ( x )
a
m
(U) ® (D =
S K
(U) ^
m
j*0
isomorphic
to the graded term
then
(
K
and
= kj-x
K
m
t
( U )
for a l l
^
k
CH}
,
i s canonically
m
Gr^K^(U)
® Q
f o r the
y-filtration.
k
i
S i n c e c h . . (ty (x) ) = k c h .
. (x) , t h e map c h . . v a n i s h e s o n
/J
i / l
i / l
K .
.(U) ^
for v * j
and i t s u f f i c e s t o c o n s i d e r the r e s t r i c t i o n
zj— i
J
1
9
c
Following
of
U
h
:
i , j
K
2 j - 1
Beilinson
)
(
J
)
>
V
define
H
1
( U ) ( J ) )
•
the "motivic
cohomology"
by
:=
f
(denoted
"absolute
study
K
2 j
(
^ (U)
j
)
i
cohomology" H^(U QHj))
i n [ B e i 1 ] ) , so
f
that
t h e morphisms
C h
from
U
[ B e i 2] we
HjJ(U OMj))
we
(
u
1
: HjJj(U,Q(j))
j
the motivic
> T (H (U)(j))
?
cohomology t o the v a r i o u s
other
cohomology
theories.
We
5.13.
the
can d e s c r i b e
Theorem
Chern
C
Let
class
1
U
be
induces
X
1
their
: 0(U) /(E
X
S1
- ->
image
a
i n the following
smooth
an
connected
case.
v a r i e t y over
(E , t h e n
isomorphism
r ( H g ( U , X ) (1) )
H
= 27Ti-W H
2
1
(U Z)
f
n
1
1
F H (U CE)
f
in
particular,
ch
is
: K (U)
( 1 )
>
1
r ( H g ( U ) (1))
H
surjective.
Proof
We
shall
get
two
proofs of t h i s
fact.
The
one
1
uses
the B e i l i n s o n - D e l i g n e
cohomology
H (U Zfj))
[Bei
1],
one
g i v e n i n S .11,
the
[EV]);
theorem
It
the
of Abel-Jacobi,
i s shown i n
0(U)
so
the claim
O —>
second
X
—
follows
and
that
>
Hp(U,Z(D)
from
t h e r e i s an
j
X
i s based
with
(D)
—>
x
x
> o(u) /a:
x
O
> o
and
J
a
v
> Hp(U KD)
f
I
'1,1
r (H^(u,KD))
H
together
with the
K
1
(U)
fact
M
)
— >
that
det
0(U)
X
i n d u c e s an
0 Q
,
isomorphism
on
(5.9.2).
diagrams
H
here
(see
A
> o(u)
—
U
isomorphism
r (Hg(U,I)
—>
f
of
,
the commutative
Hp(U Zd))
A
be
shows t h e r e l a t i o n
[EV]
1/1(1) — >
o —> a:
will
r
given
compare
5.14.
the
[Sou
3].
Remark
The
proof
which
can
for
smooth p r o p e r
a
this
use
be
i s very
s i m i l a r to
f o r d i v i s o r s by
r e i n t e r p r e t e d as
a
using
holds
Beilinson-Deligne
the
the
true
proof
of
exponential
quasi-isomorphism
v a r i e t y . However,
quasi-isomorphism
the
above
Z(1)p
f o r non-proper
no
longer,
so
we
cohomology
instead
of
the
U
sequence,
—
® [~1]
as
above
really
m
have
to
exponential
sequence.
The
Prop.
Theorem
over a
#
generalizes
a
result
of
Friedlander
Let
finitely
char k
U
be
generated
. Then
a
smooth, g e o m e t r i c a l l y
field
k
the
connecting
y
—>
. and
let
morphism
£
induce
n
(E
m
—>
-> (Em
O
isomorphisms
< —
<—
n
£
n
and
in
(0(U) /k )
X
X
particular,
A
=
the
(0(U) /k )
X
first
: K
is surjective.
1
(U)
X
Chern
0 I
1
0 Z
—
1
class
—>
H
(U,»
£
connected
be
f o r the
n
O —>
b)
[Fr]
3.6.
5.15.
1
following
(1) )
a
variety
prime,
Kummer
sequences
Proof
a) f o l l o w s b y p a s s i n g
to theinverse limit
over
the
exact
sequences
0 —> 0(\J) /i
X
since
Weil
b)
— >
Pic(U)
1
(U, p
t
for finitely
we u s e c o n t i n u o u s
H
H
i sf i n i t e l y
theorem
- cont«vV
0
n
1 ) )
-
H
n
generated
generated
cohomology
Lt
( U
) —>
'V
1 ) )
n
Pic(U)
— >
0 ,
by t h e g e n e r a l i z e d M o r d e l l k , cf.
and the
five
[La]
I I 7.6.
term
exact
H
— > H V V D > ^ "*
For
sequence
cont W
(
1 ) )
TT*
V
H
induced
TT
by the
i s induced
k-rational
general,
we
have
Hochschild-Serre
by the
morphism
TT : U
p o i n t , TT h a s a s e c t i o n
let
K/k
H (G,Z (1)
G k
£
)
sequence
Spec
k
and hence
exact
(
U
'V
[J1]
. I f
U
1
)
)
3.5. Here
hasa
TT* i s i n j e c t i v e . I n
be a G a l o i s e x t e n s i o n w i t h
a commutative
2
spectral
ccnt
G a l o i s group
G
. Then
diagram
= O
A
H
c o n t
(
U
x
K
' V
1
)
)
—
G
>
( H
1
( U
f
V D ) V
- >
H
l
<
G
'
H
c o n t
(
G
K ' V
1
)
)
Q
H
(
c o n t
U
' V
1
)
)
-
j
z
^
H
1
( U ,
V
1 ) )
k
A
H
1
(G,Z (1)
Since
Kummer
£
*) = O .
1
H
.(G
cont
R
v
theory,
f
I
(D) =
x, ^
H
0
n
1
(G ,y
x
) = I^m K V ( K ) *
n
a n d K /(K"®ZZ . ) i s u n i q u e l y
p
divisible,
=: K
x
by
)
1
H (G,H
1
. ( G
COnt
surjective.
We
0 —>
H
V
T Z
F
1
(1 ) ) ) = H (G,K* )®ZZ
0
Ja
36
g e t a commutative
1
cont
(G, , Z
^
( 1 ) ) —>
exact
HL
. (U X
cont
l
f
k
> k
which
be
restrictions are
— >
(D)
0
H
1
(U Z
I
f
k
(D)
0
—>
0
6
0
can
so both
diagram
(5.15.1)
in
=0,
x-
A
seen
-> O ( U ) '
= O(U)
from
/k
-> A
is finitely
g e n e r a t e d and t o r s i o n
This
the diagram
0 —>
k
—>
O(U)
—>
x€X
0
free.
—> k
( 1 )
— > k(u)
x€X
Û
( 1 )
(D
Pr
®
Z =
(D
x€U
for
a normal
field
of
of
U
compactification
(and o f
codimension
differential
d(f)
1 in
X ), U
U
and
of the Quillen
= Iv (f)
where
x
(
v
x
1
x€U
X
)
of
and
X
U
X
(D
. Here
( 1 )
k(U)
are the sets
, r e s p e c t i v e l y , and
spectral
i s the function
sequence
i s the valuation
at
d
of
i sthe
([Q1] 5 . 4 ) ,
x
points
1
€ X* *
i.e.,
or
1
U* *
A
Hence
pletion
A = A ® Z^
, a n d we
i n 5.14.1. The r e s t
1
'
1
g e t b) b y p a s s i n g
follows
^ H
1
( U
r
I
from
i
H )
t o the
t h e commutative
x>comdiagram
.
note
however,
isomorphism
5.16.
a)
§ 9,
related
b)
F o r smooth
remains
k
finite
f o r 5.13
we
f o r the proof
to extension
U
true
is a
Like
theory
in
X
A
> (O(U) )
1
unless
Remark
cohomology
O ( U ) * ® TL
that
i n n o t be
w
field.
used
above
a
suitable
and
shall
"absolute"
get another
proof
classes.
, not necessarily geometrically
without
an
change,
and
instead
o f b)
connected,
we
5.15
a)
have
G
X
(0(U) /
9
k V
X
=
(0(U) /
9
(
x
where
k^
assume
of
k
a
Ind ^
k
that
p
q
H (U
U
q
x
k
g^f2
Corollary
embeddable
in
d
, we
U
in
tc (x)
. For this
k
k
be
H
q
the separabel
x
(u
o n t
p,q
(G ,H (U
closure
J ^,(J))
* k,S (j)))
k
> 0
we
c
q
H^
1
H (U,E,(D)
( 0 )
. Since
have
for a l l
U
be
. Then
: K (U)
1
of
— >
k
, s o we
£
may
replace
k
(
1
a
smooth
the
map
)
>
1
T
v a r i e t y over
A H
(H
1
a field
k
which
5.13,
the
(U) (1) )
surjective.
Proof
map
First
assume
ch^., : K
a
fixed
x
€ K
1
of
® X,
considerations.
Let
Ch^
K
field
(j)) )
£
i n the above
5.17.
closure
i s irreducible; l e t
x~k,z^(j))
s H (G^,H (u
is
X6U
i n the function
c
is
u °'
i s the separabel
may
by
e
k*)
1
(U x
1
(U
(C)
embedding
(U x
R)
R
(C)
—>
that
1
)
Jc —>
c
lies
K
(
(U x
k
—>
i s algebraically closed.
r
( E . On
1
A
H
( H (U X
R
( E ) ( I ) )
the other
i n t h e image
of the
hand,
By
i s surjective for
every
element
restriction
( C ) f o r some f i n i t e l y
generated
k-algebra
k
R/ k c R c I , s e e [ Q 1 ] § I
f
Choosing
a closed
2.2.
point
connected
component
as the "generic
image o f
x
( H (U x^CC) ( 1 ) )
—>
K (U)
1
since
= a* : r
Kiinneth
k
5.18.
i n tlio
1
A H
(H (U
we s e e t h a t
image o f
X (E)(D)
,
k
X (E)(D)
k
,
i n t h e £-adic r e a l i z a t i o n
v i a the
extension
U
the trace
with
K/k .
and discuss
If
we may a p p l y
the following
i s a smooth v a r i e t y
over
a number
field
f o r every i , j
h
i , j
:
K
2 j - i
(
U
)
® ®
>
r
A H
(
H
l
(
U
M
j
)
}
surjective.
I
also
true,
5.19.
i,j
f o r example,
to state
Conjecture
C
U
T
a^ :R c (C
r^lHÛ
R
t o some f i n i t e
, then
be
x R ) (D ) —>
(H (U
i snot algebraically closed,
I want
is
lies
—>
(U) (D)
1
point"
i n t h e same
formula.
respect
k
(H
A W
1
A H
c a n be checked,
If
r
k ^ — > (C
we h a v e
a*
as
1
in
c
: R —>>
think
replacing
Conjecture
that
T
If
the following
a
^
k
by
t h e map
version"
of i t should
.
i sa finite
i s a smooth v a r i e t y o v e r
"Tate
k , then
field
or a global
f o r every
£ *
f i e l d and
char(k)
and
c
is
view
state
for
it
a
K
=
2
(
- i
j
of
5.15
conjecture
finitely
becomes
The
u
«A - >
9
)
k
be
= <E -
[Bei
2],
but
arguments
as
the
H
it
( u
x
5.19
discussion
(like
the
field
k
in general,
said
obvious
for
and
generated
false
same c a n
The
be
i , j
K
k
k '«i<3))
surjective.
In
to
h
for
i f
in
Tate
we
of
5.18
in a
conjecture
we
that
this
I
too
more
tempting
generally
show
in
§ 9
that
many
parameters.
5.18.
i s contained
above.
shall
contains
" Hodge v e r s i o n "
shall
i t i s very
conjecture)
, but
k
5.8
see
think
that
-
stated
is false
the
T^
replacing
by
Beilinson
i n general
following
Y
by
U
by
U
in
the
s p e c i a l case
same
should
true.
5.20.
Conjecture
defined
over
a
Let
number
U
be
field
a
smooth
v a r i e t y over
k
. Then
for
a l l
CC
that
i , j EZ
the
can
be
Chern
character
c
is
h
:
i , j
2 j - i
(
u
)
® "*
®
(27TI)
J
1
W H (U Q) n F
2I
j
F
1
H (U,
(E)
=
T (HB(U)(J))
H
surjective.
The
to
K
the
next
cases
statement
k
= (D
shows
or
k
that
= F
we
(t)
may
or
restrict
k
= F
P
5.21.
Lemma
conjecture
only
Let
5.18
K/k
(resp.
i f i t i s true
Proof
closure
Galois
By
applying
of
group
be
K/k
G
finite
5.19,
resp.
K
this
to
, we
. For
may
a
for
a
attention
prime
p
.
P
a
for
our
separabel
5.20)
extension.
i s true
for
k
Then
i f
and
.
N/K
and
consider
variety
the
U
N/k
where
case
that
over
k
let
N
K/k
U
i s the
normal
i s Galois
x K
be
the
with
base
extension,
dieck
restriction
5.18
K
K
follows
2 j - i
(
2j-i
(
V
R
R
K
/
uses
K
->
K/k
1
H
V —>
from
)
following
=
and f o r a v a r i e t y
V
)
->
from
Spec
over
K —>
t h e commutative
r
r
V
A H
AH
(H
(v)
( H
(
( j )
l
W
K
Spec
1
f c f
K._.(U
)
j
n
K
2j-i
and
X
H
( U x K)(j)
x k
k
/ f f l j l
For
over
(j))
k
=
(
l e t V
t h e number
1
H
T
K
field
K
such
6
that
ff-morphism
c
y
: V — > V = U
R
0
the
inclusion
X
K
ff
' 0
o f t h e component
r
A H
^K
o
x K)(j):
k
i
. For
y
K
(H (u)(j))
5.19
r
0
one
,
G G
)
.
CC , l e t V
X
A
=
q
R
K
/
v
k
ff
X f f ,
K
F
O
be a
q
variety
f o r some
K , OQ
•
0
T
n
e
n
K
t
n
®
canonical
<E , w h i c h i s
R
r
(
v
0
u
l
1
=
M
V . x
6:K<^->ff
'
6
V
>
(H
U
V = V
U
A H
k
over
: K ^—> (C , a n d l e t
Q
r
H (R --V)(J)
K/K
U
embedding
->
1
?
be a v a r i e t y
Q
r e p l a c e d by
G
"^,QI (J))
and
A
( U x K x k (jj^(j))
K
the claim f o r
(D
H (U)(J)
K
5.20
( U )
k
1
=
K
G
X
H (U
x K)
2
diagrams with
G
^ , ( j ) )
= H ^ t R ^ V
V
. Then
2 . 1 9 , 2.20 a n d t h e r e l a t i o n s
( V ) ( j )
H
k
be t h e G r o t h e n -
diagrams
the corresponding
since
l e t R
K/K
6
induces a
commutative
0
diagram
K
with
V
2
J
.
±
( V )
= i d . Hence,
. The c o n c l u s i o n
The
that
from
conjectures
properties
->
^ ( H ^ ( U )( j ) )
->
r (H (v)(j))
1
H
i f5.20 i s true
k
to
above
have
K
for U
, i ti s true f o r
i s trivial.
some r e m a r k a b l e c o n s e q u e n c e s , i n
o f t h e K - t h e o r y would
imply s i m i l a r
ones
f o r the
,
realizations.
[Bei
The
following
1]; i t i s based
5.22.
Theorem
a
HjJj(F,QKj) ) = 0
b)
H (F Qd))
1
for
= K
f
M
M
l
l
n
o
r
3]3)
i >
(F)
scheme
that
X
f o r any
the
1
presheaf
=
by
U
Suslin:
field,
then
K-theory).
x
these
terms,
1
Suslin S
Corollary
HjJj(U,Q(j))
j
spectral
U
f o r the
i s defined
Ker(G(X)
—>
>
be
f
sequence
of
G(U))
topology
on
by
.
i
theorem
= HjJj(U QKj))
a result
Zariski
coniveau
a
implies:
smooth v a r i e t y
has s u p p o r t i n c o d i m e n s i o n
N ~ H^(U,Q(j))
By
Let
G
open
COdim (XÛ)
Proof
is a
0 Q} ( M i l n o r
U c X
1
F
of
j ,
filtration
N G(X)
5.23.
If
result
Beilinson
L
Recall
In
i s c o p i e d from
fundamental
([Su 2][Sou
a)
a
on
argument
Soule
over
a field
k
,then
i - j , i.e.,
.
([Sou
i n K-theory
3]
([Q1]
theoreme
5.4)
4)
induces
the
a
Quillen
spectral
sequence
(5.23.1)
P
q
E ' (U)(j)
6
1
where
K(X)
i s the
for
p
U ^
K
( p )
i s the
, For
of
-p-q
 K
-p-q
set of points
residue f i e l d
j - p > -p-q
(K(x))
x
. By
5.23.1
( j )
,
-p-q
of codimension
= 2j-i
(U)
we
5.22
see
a)
we
p
of
p
have
that
E
contributing
to
P
,
U
and
q
E ' (U)(j)
q
= 0
for
1
H (U C(J))Iives
7
M
in
codimension
5.24.
^ i - j.
C o n j e c t u r e s 5.18
to
5.20
predict
the
same b e h a v i o u r
for
T
1
a
h
( H ( U ) ( J ) )
which
is a
(
H
for
and
of
(5.24.1)
H
(
e t
£
highly
property
Tate
r
7
U
Q
'
(
£
j
)
)
}
A
non-trivial
T^
and
Hodge,
T
N
y
*
J
H
(
B
U
In
consider
v
2i
> H (X)(j)
*
J
y
(
H
Q
'
(
j
> H
)
)
fact,
equivalent to
respectively:
2i
(X)(j)
R
question.
is
u
D
the
21
J
}
'
respectively,
for
i =
2j
this
the
conjectures
exact
sequence
of
(U)(j)
v
c
for
M
: Y —>
for
the
X
c l o s e d of
considered
canonical
codimension
cohomology
theory.
2
isomorphism
H ^(X) (j) =
y
object:
Hodge
with
structure
5.24.1
Q
induces
(5.24.2)
Hence,
with
trivial
f o r the
exact
Betti
sequence
D
T(HY (X)(J))
i f
Tv*
support
For
an
G^-action
= 0
on
Y
general
>
j
By
@
y
1
(
0
U
purity,
= X\Y
there
(where
—>
X
,
is a
1 i s the
trivial
)
e
i n the
£-adic
case,
the
trivial
cohomology) . T h i s
shows
(T =
respectively)
T^
(X)
T (H
2
, then
and
HH ^(X)(j))
or
T
,
u
(j) )
T ( H °
>
i s generated
that
by
(U)
(j ) )
cycles
.
i
and
j
the
situation
i s more
complicated,
since
i
T(H U)(J))
is
not
necessarily
picture
that
Y
1
H (X J)
7
diagram
where
we
i s smooth,
1
s H "
2
(
l
>
exact.
write
of
Tv*
±
1
(5.24.3)
1
HH (X)(J))
Nevertheless
1
H (X J)
f
"^ (Y,j-(i-j))
get
1
for
codimension
we
H (X)(j)
i - j , then
2
1
1
>
= H ^" (Y,2j-i)
the
HH (U)(J))
following
. Assume
we
have
and
a
rough
for a
an
moment
isomorphism
commutative
F y
2i-i
-
i
cb
ch.
i/j/X
2j-i 2j-i Y
f
K
with
f
2
j
(2j-i)
^(Y)
the usual Gysin
Gysin
morphism
involves
">
morphism
y,
f o r the motivic
t h e Riemann-Roch
surjectivity
of
of
chu .
.
to
Milnor
. .
9
y,
ch.
.
. By
K-theory,
2 j
^ ( X ) (j)
i n the cohomology
cohomology
t h e o r e m , c f . 7.1
reduces
v
K
5.22
0
i n any
case
(whose
below).
By
to the s u r j e c t i v i t y
b), K .
we
.(Y)
(
2
^
_
l
)
and
a
construction
5.23
of
Ty
i s strongly
can c o n s t r u c t
certain
the
and
f
related
some e l e m e n t s i n
(1)
this
K-group
= 0(-)
X
by
using
® (D . T h e
c
generic
Km m Y
v
:
K
m
f
is
related
for
any
symbols and
f
surjectivity
Y
J m*
—
to the theorem of
field
F
and
/ T n
an
this
in
isomorphism f o r
should
be
true
m
S 2
for a l l
the f o l l o w i n g drawing
TH (Y In)
f
m
H
c h a r (F)
f
TT
m
E T
'
m
>
n
K^(-)
of
Merkurjev-Suslin
integer
_.Milnor
v ,
K
(F)/n — >
is
elements i n
[MS
\ n
1] s a y i n g
, the Galois
that
symbol
,,-,„./ / * v
(F,Z/n(m))
, and
to the conjecture
m
.We
I O
can
of Kato
incorporate
a l l
that
this
1
H (X J)
7
(2i-j,2
1
where
the
Of
above.
triangle
course,
First
of
surjectivity
of
singular,
to
be
and
non-exactness
p
the
next
study
of
TH (X J)
7
i s not
really
vanishing of
the
a r g u e by
Tv*
Gysin
Hom(1,-)
, i . e . , the
p
> 0
t r u e as
we
does not
morphisms.
varieties
for
(possibly).
subvariety will
singular
T
* O
as
remarked
imply
i n general
Hence
well,
the
and
be
i t turns
also
out
the
derivatives
. This w i l l
be
discussed
in
chapters.
Twisted
Poincare
suitable
"twisted
[BO]
just
T=
1
with
. Secondly,
cannot
7
A
not
lf
p
= Ext CI -)
Ogus
Fp^
area
picture
a l l , the
of
R T
§6.
this
we
u s e f u l to
i s the
setting
Poincare
1.3
duality
f o r our
duality
. We
need
in abelian
theories
theory"
purposes
as
i s the
n o t i o n of
i n t r o d u c e d by
a version with
values
in a
Bloch
tensor
Definition
Let
over
a
c o n t a i n i n g a l l q u a s i - p r o j e c t i v e ones,
k
and
category,
groups.
6.1.
field
a
I/
be
a category
of
schemes o f
finite
and
type
let
T
be
an a b e l i a n
identity
1)
category
i n t h e sense
Poincare
duality
by a c o l l e c t i o n
theory
of objects
every
in
I/
(homology)
a
(X,b)
X
\J
of
c
Y
^
> X
the
f o l l o w i n g ones;
c)
Z e Y c X
-
H (X J)
functorial
d)
immersion
in
c
Y -—>
Y )
X
that
respect
to cartesian
precise description of this
concentrate
with
here
rather
squares
there
i s a. l o n g
1
exact
on t h e n e c e s s a r y
-
with
respect
to the contravariance
for
1
H (X J)
F
i f the diagram
Z e X
H^ (XN.Z,J)
Z
c l o s e d and
1
H (U J)
F
below
i s an
on t h e l e f t
-
morphisms
I/ ,
sequence
H (X J)
F
p r o p e r t y and
T ) ,
-
F
(excision)
morphism
e)
we
f o rworking
1
...
closed
support
(X,b) i s c o n t r a v a r i a n t w i t h r e s p e c t t o e t a l e
a
c o v a r i a n t w i t h r e s p e c t t o proper morphisms i n
for
is
1
V ( s e e [BO] f o r a m o r e
and
T
V
1
in
modifications
in
> X
I
H
such
i s c o n t r a v a r i a n t with
F
Y
b)
and every
€ TL
i,j,a,b
values
T
H
and every
H (X J)
with
(cohomology w i t h
object
1
f
H (X J)
F
for
on
of
1
a)
o f [DMOS] I I 1.15, w i t h
o b j e c t 1_.
A twisted
given
tensor
H
L
+ 1
U c X
(X j)
f
-> ... ,
in a),
open w i t h
Z c U
the
isomorphism,
i sc a r t e s i a n , with
proper
f ,g
and
etale
a,3 / t h e n
Y'-
f)
i f
Y
-> H ( X ,n)
-> Y
H (Y M)
-> H
c
i
> X
i s a closed
open
immersion,
> H
with
i
compatible
h)
Y
c
for
H ^(Y Iti-Ii)
on t h e r i g h t
i
,m)
v
c
: X -Y —>
X
H
,(Y,b)
a—i
a
i s the
sequence
—>
... ,
Y
c
> X
closed
,
f
f o ra cartesian
H (X'
f
morphisms,
diagram
on t h e l e f t
with
i s commutative
D
-> H
1
H , ( X ,n)
i
^ (Y
1
,m-n)
v
c
Y =
i)
formula)
(Y n ) ,
morphisms,
(cap-product)
i
1
i
i s a long exact
the contravariance f o r etale
> X'
v
there
a
(X Y,b) — >
to proper
D
with
and
a
respect
of the diagram
c
> H
0 H ( X , n ) -Q->
(projection
proper
then
(X,b)
i s a morphism
H (X,m)
f
immersion
a
functorial
1
i
i
(Y,b)
there
commutes
H (Xjn)
a
g)
on t h e r i g h t
-> X
corresponding
-> H
the diagram
> X
H (X In)
i
(fundamental
irreducible
ri
which
x
class)
f
®
f o r each
of dimension
d
D
H (X,n)
variety
i s functorial
f
2 d
f
with
j
X
in
I/ , w h i c h i s
, there i s a canonical
€ H o m ( 1 _ H ( X d ) ) =: r ( H
T
-> H ^ ( Y , m - n )
respect
2 d
(X d))
f
to etale
morphism
,
morphisms,
,
j)
(Poincare
dimension
duality)
d
i f
X
c
and
Y —>
6 ob(l/)
X
is a
n
closed
the
of
morphism
H (Y
i
)
f n
by
1
H^
" (X,d-n) s 1 ®Hy
is
an
k)
i n the
(X d-n)
—
f
> H
2 d
( X d ) ® H^
(Xjd-n)
f
\
H (y,
±
isomorphism,
•-H^
- 1
situation
> H
of
j ) , for
Z c Y
i
closed
d
the
j
diagram
i
O N Z f d - J ) - H ^ ( X j d - J ) - H ^ " ( X d - j ) -> H ^ ( X S Z j d - J )
f
Vz
is
immersion,
smooth
n
.2d-i,„ ,
,
X
(X,d-n) — - — >
given
is irreducible,
i + 1
n
^x
(Y--Z J)
> H (Z J)
f
commutative
i
(this
/|Vz
> H (Y J)
f
i s not
n
i
> H (INZ J)
f
postulated
n
i
in
[BO],
—>.
f
but
will
be
needed
below).
2)
A
morphism
morphism
the
of
6.2.
definition,
Since
abstract,
we
a)
cases
a
category
the
b)
the
tensor
If
T
Poincare
duality
i s compatible
theories
with
the
is a
pair
axioms
of
a)-k)
in
sense.
Remark
In
twisted
f u n c t o r s which
obvious
By
of
like
of
we
the
to
i s an
1
let
H (X J)
f
definition
remind
are
"vector
law
we
the
of
1
= H (X J)
X
f
tensor
reader
of
i n t e r e s t e d i n , the
spaces
i s given
by
with
the
categories
the
i s quite
following.
category
T
is usually
some a d d i t i o n a l s t r u c t u r e "
tensor
abelian category
.
with
product
tensor
of
vector
product,
and
spaces.
i . e . , where
for
each
C I—*—>
is
two o b j e c t s
Bil(A B C)
7
the functor
= {bilinear
f
representable
A,B
by an o b j e c t
morphisms
f
(A B)I
f
~> A 0 B
f
with
f
f
the obvious
associativity
constraints i sa tensor
[SR]
s o i t o n l y needs an i d e n t i t y
I 2.1.1,
C }
A 0 B :
Horn(A 0 B C ) = B i l ( A B C )
then
f : A © B —>
commutativity
law with
object
and
constraints
t
1_
o
obtain
AC
a
tensor
category.
6.3.
Definition
tensor
W
m
a)
is
o f exact
T
be a f i e l d .
has a weight
subfunctors
of
An F - l i n e a r ,
filtration,
i d : T -> T
rigid
i fthere
f o r m € TL
abelian
i sa
such
sequence
that
W
cz W
, and f o r every o b j e c t
A
i n T the f i l t r a t i o n
W A
u
J
m
m+1
m
f i n i t e , exhausting, and separated, i . e . , WA = 0
f o r m << 0
and
b)
category
Let F
W A = A
m
f o r m >> 0 ,
'
f o r objects
W
(note
that
m
A,B
of
(A® B ) =
T
one h a s
V W A 0 W B
.
P
Q
p+q=m *
M
t h e sum i s f i n i t e
by a ) ) .
W
G r A = W A/W
.A , s a y t h a t
m
m
m-1
Letting
t h eweight
m € Z
occurs
W
in
A
the
only
6.4.
, i f
G
r
A
m
weight
Lemma
*
0
/
a
n
^ that
occuring
A
i spure
o f weight
m
, i f m
i n A .
The f o l l o w i n g p r o p e r t i e s f o l l o w from
t h e axioms.
i s
i)
Horn(A,B)
e.g.,
=O
i f A
and
i i ) j_ i s p u r e
W
iii)
Gr (A
W__ (A )
a r e pure
o f weight
s
m
Proof
B
0 B) s
V
iv)
i fthe weights
W
for
C = A
(A/WÂ^
and
B
and
0 Gr B
(where
B
V
W
> W C
I
m
> Gr C
m
i tsuffices
o f weights
m * n
ii)
B y 6.3 a) a n d d e c o m p o s i n g
W
f = Gr A
m
iii)
iv)
we may s u p p o s e
This
This
= A
follows
follows
1 ®
from
from
, hence
that
B
([DM0S]II 1.6)).
> 0
the case
that
f : A —>
B
A
W
Gr
m
that
f = 0 .
1_ a n d
T
l i s pure
1 - 1
of
, say.Since the functors
f = Gr Ker
m
the isomorphism
are distinct,
weights,
to consider
Ker
and
B
sequences
e x a c t , t o o , we g e t f o r a m o r p h i s m
II.1.17)
and
i s the dual
are
W
A
,
on t h e e x a c t
C = B
a r e pure
of different
W
©
Gr A
p+q=n
^
> W
,C
m-
in
0,
i ) By i n d u c t i o n
0
occuring
i f necessary
o f weight
we c o n c l u d e
m
( c f . [DMOS]
, s a y . B y 6.3 b)
m = 0 .
6.3 b) a n d t h e e x a c t n e s s o f t h e t e n s o r
the exactness of
A h~~~> A
v
product.
and t h e f a c t
that
v
A
i s pure
case
of weight
we h a v e
W
-A
-m-1
V
= 0 , and
m
let
field
category
V
morphism
V
Hom(X ,X )
= Hom(X
l e t R
be an a b e l i a n ,
T 0 F
sets
Hom(W
= 0 , since
m
of fractions
i s pure
V
the following
T
, i f A
= 0 , since
:= A /W_ A
and hence
For
and
X
-m
v
of
, which
R
V
X
o f weight
v
-A ,A )
-m-1
V
V
c
A
0 X,1) = 0
, we o b t a i n
m
= Hom(W
= A
V
„ A ® A,1)
-m-1
i s pure
o f weight
. q.e.d.
ring
(with
category. I f
a new a b e l i a n ,
h a s t h e same o b j e c t s
: i n this
V
be a commutative
R-Iinear tensor
a r e defined by
V
as
F
F-Iinear
unit)
i s the
tensor
T , a n d where t h e
H O X I
where
that
6.5.
T
and
a)
A
i s the object
p
T
WW
has a weight
Definition
=
i f
of
The w e i g h t s
w
m
T (A B)
T ® F
filtration
T ® F
i f the following
o
R
T ® F
occuring
in
H
(X,b)_
r
a
2b-a
< w
^ 2b-2(a-d)
for
a > d .
w
^ d
occuring
in
< i-2j
for
i < d = dim X ,
2(i-d)-2j
^ w
S i-2j
for
i < d .
smooth
in
filtration,
= dim X ,
< w
X/k
values
satisfy
-2j
For
with
hold.
for
the weights
ob(T) . Say
has.
and has a weight
< 2b
6.6. C o r o l l a r y
A G
i s rigid
conditions
proper
to
d u a l i t y theory
< w
X/k
f
Poincare
2b-a
For
F
associated
i f
a
b)
®
f
A twisted
has weights,
H
the weights
w
1
H (X^) r
t
occuring
satisfy
in
H
1
(X,
satisfy
i-2j
< w
< 2i~2j
for
i < d = dim X ,
i-2j
< w
< 2d-2j
for
i < d .
In
particular,
of
weight
This
6.1
j ) . We
for
i-2j
X
smooth a n d p r o p e r
over
1
k, H ( X J ) r
f
i s pure
.
i s c l e a r from
6.5
now
examples.
give
some
and the P o i n c a r e
duality
isomorphisms
6.7.
Example
finite
G^.
type
a
different
be
from
generated
over
operation;
TL^
note
category
k
with
i t s absolute
char(k)
, and
the
t h a t we
We
get
a
= Hom
c
by
K
1
H (X J)
the
, let
g
let
c
£
be
(G^. ,2
action
tensor
object
k
is
a
of
of
law
of
G^
is
i s the
with
tensor
trivial
k
by
duality
cp I
>
on
cp(1)
with
.
values
in
closed
,
letting
= H
F
Rep
continuous
identity
=M*
A
twisted Poincare
Rep (G ,Z^)
and
schemes
have
(1 ,M)
c
closure
category
category:
G
T(M)
a l l separated
G a l o i s group,
. The
tensor
of
separabel
Z^-modules w i t h
abelian, Z^-Iinear
product
the
field
be
g
finitely
I/
over
= Gal(k /k)
prime
an
Let
-(X,Z^(J))
1
t
c
for
Z ->
X
(6.7.1)
H (X,b)
= H
a
t
a
U,Z
(b))
J L
!
= H7^(X,Rf Z^(-b))
(£-adic
etale
cohomology
V I I I ) , where
the
X
= X
x^k
algebraic closure
equivalent
and
the
to
the
finite
and
——>
k
generation
Spec
k
of
of
X
homology,
of
category
for
k
Spec
cf.
[BO]
denotes
. The
constructible
the
2.1
the
category
k
and
base
Rep
c
exp.
extension
(G^.,Z^)
%^-sheaves
above Z^-modules
[DV]
on
to
is
Spec
follows
k
,
from
i
Deligne's
(complexes
Rep
C
result
of)
(G, ,Z ) ®
K
Jc
that
i n the
constructible
QB
can
be
above
sheaves,
identified
Q^-vector
G
(equivalent to
category
k
) and
with
[SGA
the
4^]
and
Rf"
the
is rigid:
the
spaces
of
with
category
continuous
constructible
internal
Horn
Rep
by
(G,
K
action
Q^-sheaves
i s given
respect
[finitude]
C
finite-dimensional
Spec
see
Rf*
x.
of
k
situation
on
2.9.
,Q )
x,
0
of
Hom(V W)
= Horn
7
with
the
6.8.
Example
V
£
G,-action
ob ( R e p
c
Now
(V W)
7
( g f ) (u)
let
( G ^ , G}^) )
k
= gf ( a
be
has
^v)
finitely
a weight
generated.
filtration,
Say
that
i f there
exists
an
1
integral
domain
fractions
F
k
over
U
A
of
such
= Spec
finite
that
A
V
type
over
extends
, which
has
an
to
Z[-]
a
with
field
constructible
increasing
of
(D^-sheaf
exhausting
and
separating f i l t r a t i o n
WF
by
G r F = W F/W
.F
i s pointwise
m
m
m-1
c o n s t r u c t i b l e subsheaves such t h a t
pure of weight
m
s e e [D9]
1.2.2.
Since
tordue")
on
U
amounts
m
W
F
generic
i)
i s smooth
point
there
or
IF
n
is a
("constant
= Spec
k
connected,
(char k
= p
>
of
smooth
0)
7
c
such
, this
scheme
a
U'
t h a t the
neighbourhood
to
over
2
saying
z
^ J
factorizes
ii)
there
i s an
...cW
—
m-1
point
x
i
V
G U
i
c ...
—
the
on
f
absolute
1
r e p r e s e n t a t i o n of
2
, Nx
every
in
Gal(K(x)/K(x))
Nx
I
=
TT ^ ( K (x) ,K (x) ) — >
conjugacy.
a
=
archimedean
a
for
of
W
V = W
m m
Tr^(U',n),
exhausting
G
1
n
and
V/W
of
.V
m-1
= Spec
such
that
a geometric
are
k
separating
-submodules
k
eigenvalues
Gr
for
global
>>
=
G,
K
0)
on
,
filtration
f o r each
Frobenius
algebraic
that
Fr
numbers
closed
at
x
a
x
with
value
I ct I = N x
> a
G^
increasing
c W V
—
m
TT (U Ti)
1
in
through
the
(char k
P
V
of
, and
For
field
finite
a
k
valuation
i s the
i t s image
field
ring
of
primes
S
the
the
via
= TT^(U Ti)
we
have
integers
including
Frobenius
Frobenius
,
G a l (« (x) / K ( X ) )
i s w e l l - d e f i n e d up
7
k
geometric
arithmetic
•^(U',^
in
1
with
|| . H e r e
inverse of
Tr (U, «(x) )
finite
set of
,
#K(X)
O
U'
jc
we
= Spec
have
a l l primes
k
U
, and
1
to
for a
= Spec
above
%
O^yS
. Then
1
Tr^(U Jn)
i s the G a l o i s group
extension
of
A
k
a) A w e i g h t
WRep (G^,Q^)
filtration,
Let
be t h e f u l l
by
the representations having
c
o n WRep (G^,Q^)
weight
of
Rep
c
filtrations,
(G^. ,QJ^)
and
, closed
weight
filtration
i fi texists,
a weight
of
for
V .
i s unique.
formed
c
compatible
Then
every
( [ D 4 ] 1.1.5)
i s an a b e l i a n
c
S .
Rep (G^,Q^)
filtration.
WRep (G^,Q^)
with
S-ramified
i s unramified outside
subcategory
i s strictly
c
the
V
as i n i i ) i s c a l l e d
b)
morphism
o f t h e maximal
, a n d i ) means t h a t
filtration
6 . 8 . 1 . Lemma
G^
with
subcategory
respect t o taking subobjects or
quotients.
c)
WRep (G^,Q^)
c
i s a rigid,
abelian,
Q^-linear tensor category
with
weights.
Proof
no
I t f o l l o w s immediately
non-trivial
hence
that
filtrations
and
every
c
k
isomorphism
obtain
functors
remark
i s an a b e l i a n
after
Horn(V,W)
\
filtration
~> W V
m
i t ) . The c l a i m s
From
the given
weights,
t h i s one
weight
map).
Subobjects
and t h e
hence
Rep (G^,Q^)
. Since
c
on
V ® W
every
o b j e c t s by t h e above,
This
i n turn
( c f . [D4] 1.1.11
i n c) a r e now c l e a r :
the
implies
c W.
W
—
l+m
f o ra l l
we
that
and t h e
filtration
i s g i v e n by
6.3 b) .
with
of
morphism.
W Hom(V,W) = { f : V - W l f ( W . V )
m
i
and
of different
( c f . [D4] 1 . 1 . 8 ) ,
are exact
there i s
by t h e i n d u c e d
of filtered
f o r every
that
sets of weights.
subcategory
i s an isomorphism
V
modules
(look a t the i d e n t i t y
respectively
the strictness
pure
i s compatible
and t h e u n i c i t y
WRep (G ,Q^)
on
morphism
filtration,
the definitions
f o rdistinct
q u o t i e n t s o b t a i n a weight
quotient
the
G^-morphism between
t h e same s t a t e m e n t
deduces
from
i } ,
i s g i v e n by t h e formula i n
6 . 8 . 2 . Lemma
Rep
Denote by
(G Z )
w
weigth
formed
0
filtration.
WRep ( G Z ^ )
c
Then
the f u l l
c
by those
and form
w
WRep (G^,Z^)
objects
M
subcategory o f
f o r which
t h e f u n c t o r s 6.7.1 h a v e
a twisted Poincare
M ®_ QJ
has a
0
image i n
duality
theory
with
weights.
Proof
For a closed
there
Fp
i s a smooth
(if
closed
of
char
UQ
By
U -schemes
v
by base
k-schemes
( i f char k = 0 ) o r over
generic point
Deligne's
change
arbitrary
n = Spec
1
1
F = H (Rh*Rv Z
constructible
associated
This
every
the
base
Spec
change
sheaf
complexes,
S — > U
change
UQ . I n p a r t i c u l a r ,
bounds
such
k
, and a
(j))
a
G = H
Q^-sheaves
, and
i
mixed
sheaf
change
H
X
H
(
X
X
K
case
follow
p r o p e r t y we
= C
char(k)
(
X
from
)
'
on
V
b
)
)
U
U
. Moreover, t h e
result
[D9] 6 . 1 . 1 1 .
has a weight
loc.cit.
have
with
to the
= p > 0 , since
also
and
subscheme
extends
£
by Deligne's
for
open
1
to the constructible
( X , Z (b) )
I
i n this
on t h e w e i g h t s
and a r e compatible
(Rf*Rf Z^(-b))
a r e mixed
shows t h e c l a i m
A
i s
( c f . [SGA 4^]
Rf*/ Rf,/ R f , Rv
extends
H
C
k .
f o ra suitable
H|(X,Z^(J))
Z —> X
that
theorem
1.5 a n d 2.9) t h e o p e r a t i o n s
base
The
type
to
g e n e r i c base
respect constructible
3.4.1
of algebraic
over
of finite
q
from
[finitude]
sheaf
—> X
dimension
obtained
of
scheme
Z
k = p > 0 ), with
separated
Rv*
immersion
3.3.8,
by
loc.cit.
filtration.
since by
(cohomology
x
of
U
t
Hj U Q
7
(Q^-dual)
char k = 0
concepts
o f mixed
6.11 t o g e t h e r
g
= TT^ ( S p e c
of
H
we c a n n o t
i
V ^
below).
including
Instead
r^
(
for
G
and
s
,Rf Q ^)
s
j
2
1
.(G Q (m))
cont
S
a
o f
extensions
(6.8.5)
0
0
we s h a l l
are different
g e t the weight
and t h i s
f o r absolute
will
be p r o v e d
Hodge c y c l e s
as b e i n g reduced by t h e
a l l primes
<
2
n
+
1
above
one e a s i l y
)
)
field,
(note
topological
>=
r
i
+
r
2
'
lemma
S
be a
finite
at infinity i s
computes
1
a r e t h e numbers o f r e a l
( c f . [J3]
let
I , and l e t
(ramification
1
s E x t ^ (Q„,Q„(m))
Gg
x, x
and complex
places
2 ) , so by t h e isomorphism
there
exist
non-trivial
of continuous Gg-representations
0 —>
Q
way, s i n c e t h e
filtrations
b e a number
n € TL
' ^
r
i n this
of singularities,
a s above
k , respectively
Since
1
s Rf *RHom(Q
cohomology).
Oy^S,n)
t
argue
and weight
i) Let k
a l l o w e d ) . Then
where
B ) ) V
j
with the result
of etale
o f primes
d
point
t
we may a s s u m e a l l s h e a v e s
6.8.4. R e m a r k s
G
(
^ ,c *'V
RHom(Rf ,Q^,Q J
by r e s o l u t i o n
invariance
set
H
=
sheaves
(see t h e remark
filtration
that
and every
.
j l
For
in
(b))
j l
by t h e d u a l i t y
1
here
support) f o r H = R f X
, and since
(6.8.3)
Rf^Rf Q
a
w i t h compact
and
Q (2n+1) — >
0
Q.(2n+1)
E —>
a r e pure
Q
0
—>
of weights
0 .
0 and
-2(2n+1) ,
respectively,
E
corresponds
n<0
, however,
give
a splitting
ii)
Let
of
k
E
for
, then
a l l
there
E
s
M
n
are exact
passing
Spec
O ^S
. For
v
since
WQE
would
s
( G
->
n
^ Cl
n
by D i r i c h l e t ' s
C l
cont
and C l
is finite,
g
1 ) )
S'V
a
over
E
S
8
n
%
s
->
1 ) )
k'V
group
0 ,
theorem
we
E
finitely
l '
. On t h e o t h e r
s
is a
g
conclude
A
A
( G
cont
be t h e S - c l a s s
g
sequences
H (G ,^ )
t o the l i m i t
H
on
a weight f i l t r a t i o n
1
_>
g r o u p and
H
sheaf
o f 6.8.5.
n ^ 1 . Since
generated
by
cannot have
be t h e group o f S - u n i t s
G
0 _>
E
t o a mixed
x
= Iim k /k
hand
we
have
n
X
.
n
Since
k
z> k
*
® X
n
&
= I i m E_ ® X „
S
I
,we
conclude
that
>
S
(6.8.6)
Iim
( G , Q ( I ) ) c:
S
£
*
Q
cont
cont
(G,,Q.(1))
K x,
(non-trivial)
extension
0
.
S
This
shows t h a t
there
is a
of continuous
G^-representations
(6.8.7)
which
does
0 —>
E' —>
Q (1) — >
£
n o t come
j
from a G - e x t e n s i o n
g
corresponds
t o a Q ^-Sheaf
Spec(O^S)
f o r any
The
i s t h e same a s S e r r e ' s
example
Q^ — >
j
S
on
, this
0 ,
f o r any
S
Spec k
which does
answers
the question
example
. Hence
n o t come
E
1
from
i n [D9] 6.1.1
i n [ S e 1] I I I 2.2.
6.9.
Example
A-Hodge
A = Z
structures
weights;
(pure
Let
the
of
1
I (H)
i s an
identity
weight
(E
the
the
Q
zero),
and
Betti
I/
cohomology
1
H (Xo)
=
category
tensor
trivial
A-MH
category
Hodge
0
= W H
n F H , =:
Q
of
mixed
with
strcuture
A
T (K)
fl
>
f(1)
u
.
a l l separated
and
the
have
i
of
, then
i s the
we
M K
category
3R
A-Iinear abelian
= Hom _ (A,H)
A
or
object
f
For
7
schemes
of
finite
type
over
homology
(X«E) ,A) ( j )
(6.9.1)
H
with
a
a
(X,b)
the
mixed
twisted
has
the
a
Hodge
8.3.8),
(X(C) ,A ) (-b)
A-Hodge
Poincare
a mixed
([D5]
= H
structure
f o r the
as
with
to
i t by
weights.
relative
Borel-Moore
H
Deligne
Note
forms
1
that
H (X A)
f
1
cohomology
(or B e t t i )
a
(X«E) Q)
= H ( X ( ( E ) Q)
C
f
H (X X^Z A)
f
homology
we
f
may
use
(Q-dual
of
the
mixed
compact
support)
and
compactification
[J2]
§ 2 for a direct
Hodge t h e o r y
Hodge
[D5]
structures
8.1.25 a n d
proved
in
Q-Hodge
the
X c X
[D4]
with
structure
with
f
The
2.1.13, and
exact
H (X A)
closed
approach.
sign
the
8.2.4, t h e
on
bounds
by
the
a
f
(m)
Z
the
= X\X
products
weights
f o r homology
of
; see
the
the
Tate
as
proper
follow
also
twist
mixed
follows
for a
with
for a
f
denotes
compatibility
and
cohomology
= H (X Z A)
complement
sequences
bounds
given
a
relation
8 . 3 . 9 . The
[D5]
V
f
cL
are
associated
structure
homology)
isomorphism
(6.9.2)
in
(Borel-Moore
via
in
scheme
6.7.9
from
are
in
and
corresponding
f o r cohomology
m e n t i o n e d i n [D7] 8 a n d f o l l o w f r o m
question
i s t h e cohomology
proper
c o m p o n e n t s , and t h a t
variety
6.10.
\J
bounds
of pure dimension
Example
Let
k
be t h e c a t e g o r y
finite
type)
H
Z
( X
'^>
=
H
k
compact
the fact
of a simplicial
K^(U Z)
= 0
f
support,
which
t h a t t h e cohomology
scheme w i t h
smooth
f o r a smooth
affine
d > i .
be a f i e l d
of varieties
over
with
of characteristic
(i.e.,
reduced
zero,
separated
and l e t
schemes o f
. Then
( X
DR,Z
k
/ >
(6.10.1)
H
(x,b) =
a
(de
K
*r (x/k)
a
Rham c o h o m o l o g y w i t h
twisted
finite
Poincare
duality
dimensional
vector
support
theory
a n d d e Rham h o m o l o g y )
with
spaces
values
over
k
gives
i n the category
. If
X
i s smooth
a
Vec^
of
o n e may
set
H
and
N
DR
i f
Z ^
X
(e.g.,
x
'
=
H
Z^
the
i f
general
above
components
6.11.
^x/k^
i s embeddable
X
(Zariski
i n a smooth
i s quasi-projective),
H f ( X A )
In
X ,
a
= H^ (MZk)
( s e e 6.11
Example
M
o f pure
dimension
o n e may s e t
of singularities
simplicial
below, b u t a l s o
The f o l l o w i n g
scheme
.
one has t o use r e s o l u t i o n
groups f o r s u i t a b l e
hypercohomology)
schemes
the approach
with
in
and c a l c u l a t e
smooth
[Harts]).
g e n e r a l i z e s t h e c o n s t r u c t i o n sf o r
absolute
Hodge c y c l e s
finitely
generated
category
cycles
JMR^
over
following
W.
by
objects
of
2.1
b)
H
like
SL Z
-^—>
TL7 I0
should
defining
in
of
Theorem
is a
twisted
that
the
the
Betti
replace
generated
a
the
the
Poincare
—>
result
of
A H , Z
de
(
X
' *
)
'
H
*
H
(
X
Rham c o m p o n e n t s
components
H
H
for
q
: k —>
c
smooth q u a s i - p r o j e c t i v e
Proof
coincides
after tensoring
We
define
with
with
functors
the
6.9.1
a
that
for a
and
, the
and
3.19
isomorphisms
Finally,
Gr^
m
H^
,
with
3.19.
over
weights
€ TL
® 3R
category
varieties
there
k
and
. Then
values
' * >
6.7.1
H
of
structures
generalizes
with
in 6.10.1,
H^ (U,0)
of
sense
i . e . , as
tensor
functors
such
the
comparison
structures
category
Hodge
1_^(-m) f o r a l l
W
^-linear
the
filtration
Q-Hodge
the
a
define
i n the
mixed
Hodge
following
be
be
for
the
TL , - m o d u l e s .
I
W
G r ^ H^
real
JMR^
we
except
that
defined,
the
k
Q^-representations
integrally
®
then
filtration
we
H^
Let
V
on
K
filtered
2,
that
W
Gr^
\J
§
and p o s t u l a t e
finitely
Let
in
, requiring
w
are
and
MR^
a weight
morphisms
H
and
e)
filtration,
IMR
such
I
2.1
part.
r e a l i z a t i o n s for absolute
the
Furthermore,
i s clear that
6.11.1.
there
0
mixed
(G Z^)
p o l a r i z a t i o n s of
It
weight
c
structures
of
H
exist
Rep
first
c h a r a c t e r i s t i c zero,
replace
W
i n the
category
is in fact
m i x e d Hodge
or
the
We
of
above.
isomorphisms
of
integral
changes:
e x a m p l e b)
by
field
of
k
c a r r i e d out
mixed
Q
i
H^
for
ax
Ji-adic
components
(E
are
= X
><
k
given
a
(E
variety
realization
N
H (U)
by
H
the
, respectively,
U
over
defined
.
U
_(X)
and
H
AH
(X)
and
0
then
let
k
in
H
A H
li
(X,b)
= H ^ ( X ) (-b) .
a
The
£-adic
3L
and the B e t t i
H
et,Z
(
X
realizations
a
'V
n
H
d
f
(
X
i
H
„ (X)
An , L
of
' V
and
7vtj
A H
H
(X)
a
<
(6.11 .2)
H
respectively,
are
as
clear.
1
z
(OX Z)
then
M
and
f
H^ (aX,Z)
the claimed
The comparison
,
compatibilities
isomorphisms between
i n § 3, b y t h e c a n o n i c a l c o m p a r i s o n
H
(
X
aZ * '
Z )
Z
•
* —
>
H
with
6.7 a n d 6.9
them a r e d e f i n e d
isomorphisms
J t , a z
(
a
X
' V
'
(6.11.3)
M
H^ (aX,Z)
which
i n fact
® TL -^->
1
are integrally
E
K ^(OX TL )
t
d e f i n e d and a l s o
cohomology
and homology
as i n d i c a t e d
VI
For defining
t h e d e Rham
2.8.4).
isomorphisms,
comparison
and t h e weight
i t suffices
simplicial
varieties
cohomology
f o rarbitrary
variety)
of
this.
f . : X. — >
variety
U.
Cone(f)
for relative
compatible
comparison
with the
resolutions"
as i n
[D5].
t o d e f i n e homology f o r
varieties,
maps a n d
s i n c e homology o f
of the associated constant
support
a n d [DV]
the other
f a c e and d e g e n e r a t i o n
simplicial
and cohomology w i t h
simplicial
filtrations
proper
(as t h e homology
a morphism
realizations,
a n d i s more g e n e r a l
with
exist
( c f . [SGA4] X V I 4.1
i s o m o r p h i s m s we u s e " s i m p l i c i a l
First
varieties
,
9
(as r e l a t i v e
simplicial
cohomology f o r
and hence as cohomology
of the
, s e e [D5] 6.3) a r e s p e c i a l
cases
For
and
a simplicial
the B e t t i
simplicial
by
realizations
versions
the s i m p l i c i a l
the
is
H „(Z.)
o f 6.11.3
and
H
define
AH
a
(for their
F o r a morphism
this
(Z.)
the
by
5,-adic
the
existence
—>
TT : U.
i s functorial
functorial
there
exists
Z.
compare
o f two
i n a contravariant
i n a covariant
a proper morphism
isomorphism i n the etale
hypercoverings,
ment
Y.
a s a b o v e we
way
f o r homology
such
way f o r
i f
TT
proper.
Now
an
of
versions
varieties
cohomology, and
Z.
i
o f 6.11.2, a n d t h e c o m p a r i s o n i s o m o r p h i s m s
arguments below).
simplicial
variety
c f . [D5] 6.2
i n a smooth
with
cohomology
and
and p r o p e r
8.3)
simplicial
normal c r o s s i n g s . B a s i c a l l y ,
simplicial
suffices
version
o f § 3. N a m l e y ,
i
H^
to define
by
—>
TT : U.
and homology
such
that
U.
variety
X.
everthing
Z.
inducing
(e.g.,
proper
i s the compleof a
reduces
t h e arguments
by
divisor
to a
i n [D5] i t
AH
and
H
f u n c t o r i a l l y f o r U. , s i n c e b y
a
J
r
J
c o m p a r i s o n i s o m o r p h i s m s a morphism w i l l i n d u c e an i s o m o r p h i s m
i f i t induces
an
tt
AH
the
of
a l l groups,
isomorphism i n etale
i
homology.
for
In f a c t ,
pairs
those
(6.11
(X.,Y.)
a s above
to define
such
that
H^
FI
and
H^
functorially
realizations
i n [ B e i 1] §
are
1.
.4)
a
R ( Z
'>
:
=
H
DR
( Z
/ C
'
) V
(
k
"
d
U
t h e de Rham c o h o m o l o g y w i t h
relative
(6.11.5)
a
l
)
compact
support
cohomology
H DR,c (Z.)
:=
H
A
DR,c
(U.)
:=
H
A
and
AH
the B e t t i
of
U. , c f . B e i l i n s o n ' s c o n s t r u c t i o n s
T h u s we d e f i n e
H
where
i t suffices
cohomology
(X.,Y.)
i s defined
as
by
the
these
remark
groups
above;
by
isomorphisms
computing
H
one
a similar
i n the
1
gets quite
approach
smooth
as
complexes
i n [ J 2 ] 2 . 9 . By
case, generalized
a
(aU . , Q j ^ / k ^
n
explicit
n
d
1
H (OU Z)
to the
, we
w
computing
the
comparison
complexes
obtain
the
comparison
isomorphisms
i
(6.11.6)
I ^
If
we
the weight
in
[D5] 8.1.12
define
comparison
in
4ar
§
for
X
F
F"
W.
on
filtration,
by
•>>
z
<E
> H^
see
that
formula
1
(U. , Jl * _
r
filtrations
:
1
1
( Q U
e
m
term
, we
M
obtaining
Q
£
the
% I
i n the
is a
with
>
H
§
(
s e n s e o f b)
" '
2
)
£
ffl
of d e f i n i t i o n
6.1
(n m))
+
exact
sequences
b).
in loc.cit.
above
[ B e i 2 ] , Namely, by
in fact
and
'
the
as
groups
weight
since
t h e Hodge
cohomology
results
ascending
above,
for
§ 4 one
the
subquotient of
, c f . [D5] 8.1.19
long
G
T
the capproduct i t i s convenient t o proceed l i k e
in
by
isomorphisms
and
theory
the
formulae t o put a
get a G^-equivariant
\
X )
H ^ - ^ f T ^ ' ,
tensoring
formulae
isomorphisms
the mentioned
filtration
graded
and
<C .
a
^ a r
use
H
the
respected
[D5] 7.1.6.5
compatible with the comparison
I-
S^
by
they are
the comparison
V o
also
®
t h e Hodge
we
t
i s a weight
For
and
H^ (U.,Z^)
(6.11.7)
after
< Y
. I f we
(i+n)-th
Z)
8.1.15,
, and
^ X ./k
filtration
It
and
l f
isomorphisms - by
3 for
<
: H (OU
0
the
g e t s much more
the comparison
Beilinson
same
than
arguments
just
isomorphisms
the
between
them. By D e l i g n e ' s
to
theetale
each
and thea l g e b r a i c
object
(X.,Y.)
(6.11.8)
Tf
l
constructions
K *
d e Rham c o h o m o l o g y ,
a s above
= K^ (X.,Y.)
i n [D5] 8 . 1 , now a l s o
:
H
TT
<'K^ ,w)
a k ^->C
a
<
:
6 = TT 6
y - TT Y
TT
Tf(K-,W)
a mixed
following
TL^-modules
f
K ^
without
iii)
but
iv)
A,
with
finitely
ii)
a
continuous
, f o r each
of
(MAH-complex),
of the
below,
G^-action
filtered
such
that
A
and each
i k <^—> I , i s a s i n i ) ,
0
'K*
) , f o r each
o : k
VL , 0
r e p l a c e d b y TL ( r e s p . (E ) ,
i s a bounded
homology,
filtration
F ,
f o r each
f
but
below complex
an ascending
-> (E , i s a s i n i i ) ,
of k-vector
spaces,
filtration
W
o : k <=—> (E ,
ol
t h e homology groups
ffi^-modules.
(resp.
Z^
complex o f
G^. ,
dimensional
v)
Hodge c o m p l e x
i s a bounded
generated
action
K'
0
with
(K£,W,F) ,
u
kind:
F o reach
are
absolute
r
(K-,W)
c
a :k ^ - X E
i)
one g e t s f o r
a diagram
B = T T f 'A,a
l,o
called
applied
1,0
A
.
K
C
>
(
E
/
A
^
=
A
with
finite
and a descending
is
a
family
of f i l t e r e d
a
1
i,-a
with
vi)
a
( K
- P ^ a
0
f o r each
^'
-
Jt O
:
a
vii)
filtered
is
a
(
;
K
( K
o'
a'
ix)
o
W
O
a filtered
H
c
for
p e G
>
(
,
v
9
%
z
i —
k
w)
' ;, '
0
CC ,
* — >
\
)
f o r each
i
W )
(
K
' i,o'
W
)
quasi-isomorphism,
6
is
w )
: k —>
a
filtered
viii)
K
' A,o'
quasi-isomorphism,
f o r each
^0
(
~*
,
1
is
W )
(homotopic)
io,l)
e
quasi-isomorphisms
: k —>
c
a
:
T
K
k'
W
)
(C ,
\,o
1
~>
(
K
' i,o'
W
)
quasi-isomorphism,
W
(
G r
K
m
^)
i
s
pure
of weight
m+i
, and f o r each
: k — > CC
c
defines
a mixed
Hodge c o m p l e x
i n the sense
of
[D5] 8 . 1 . 5 .
The
topology
actually
better
G^-modules
instead
then
The c o m p a r i s o n
must be i n t e r p r e t e d
L
n
TLfl
has proved
a
structure.
Hodge
cohomology
in
^MRfc
the
graded
. We d e f i n e
pieces
® H
9
compatible
zations
notion
that
the
From
complex
inii)
in vi)
letting
o f pro-£-systems
are quasi-isomorphic to
i t i s obvious
that
the
by r e q u i r i n g
® Q
i n each
9
the comparison
Hodge
f o rthe weight
i n t h e sense
isomorphisms
TL , a n d T a t e
H^
homotopy,
and quasi-isomorphisms
we h a v e
and d e f i n e
Following
Hodge c o m p l e x e s
category
that
bilinear
polari-
i s an o b v i o u s
of tensor
f o rp o l a r i z e d
object
which a r e
2] § 3 ) , o n e h a s a n o t i o n
a triangulated
an
i - t h
f i l t r a t i o n of
. There
twists.
f o rp o l a r i z e d mixed
by him i n [Bei
that
realization
structures
complexes
obtains
W[i] i s
MAH-complexes
o f morphisms between MAH-complexes,
constructions
and
polarized
W i
> Q (-m)
MAH-complex
F
defines
are polarizable
under
this
by
Hodge c o m p l e x
m
9
induced
o f a mixed
absolute
H = Gr H
of the real
trivial
thus
way, b y
i - t h cohomology
the f i l t r a t i o n s
o f a mixed
objects
H
groups
quasi-isomorphism
(in t h e pro-36-direction)
with
forms
filtered
6.9) o f d i s c r e t e
of abelian
i n an a p p r o p r i a t e
be a c a n o n i c a l
Hodge c o m p l e x
these
([J1]
one; i t i s
.
Deligne
mixed
pro-36-systems
- and pro-I-systems
o f Z^-modules.
whose components
®
t o i n i ) i s t h e £-adic
to consider
instead
(K",W) ®
%
O
ch 36
Kj
referred
products,
Beilinson's
(called
p-Hodge
of cones,
MAH-complexes, and
o f MAH-complexes
up t o
quasi-isomorphism.
For
and
a simplicial
proper
normal
crossings
MAH-complex
(6.11.9)
simplicial
object
(X.,Y.)
v a r i e t y and
- and
K * (X.,Y.)
U. = X.^Y.
as above
Y. c X.
as the diagram
- X.
a divisor
one o b t a i n s
a
a
smooth
with
the polarized
JJ(Rr
l
( u . ,Z ) ,w)
et
TT(Rr
0
jl
where
W
and
F
are
the
maps come f r o m
the
comparison
extending
their
the
to
comparison
derived
in
the
wrong
complexes
By
upper
working
one
copy
the
H
and
u
AH
H^
= U.
giving
H
1
[Bei
whose
Z.
of
a
(Z.)
f i
——>
1
H
with
£
canonical
1.6.5). H
in
canonical
cohomology
compact
as
above
of
support,
i s computed
complexes
(Z.)
varieties,
(cf.
Y..
[J2]
W
' '
F )
and
a
n
^
ones
of
notice
that
defined
morphisms
2]
p.
55,
the
(X.,Y.)
in
[Bei
gives
of
by
ones.
complexes
. Starting
2]
the
§
4
to
from
get
previously
by
one
f i
i s computed
by
and
coskeleton
~
AY..
of
may
i s the
proceed
—>
AH
K.
(Z.)
=
R
HomIK^
j
(Z.),I)
H
,
properties
as
K'
version
in
[Bei
(AY..
AH
a
(Z.)
Y
1]
))[-1]
An
normalization
diagonal.
or
for
sophisticated
Cone(K:„(X.)
the
varieties,
required
Hxl
i s the
2.9)
the
K^(X.,Y.)
( f o r a more
K^ (Z.)
simplicial
by
(6.11.10)
)
i
will
obtains
with
An, C
where
>
quasi-isomorphic
in
arguments
,
quasi-isomorphisms
Beilinson
then
Y
8.1,
naturally
honest
one
<
computation
as
sites
[D5]
latter
only
involve
* '^x
TL —>
U.,
standard
are
x
of
them
by
(
j^pj
example,
X.^Y.
r
a
c a r e f u l reader
they
define
as
the
(contravariant)
MAH-complexes
versions
For
big
r
Z
— — >
6.11.9
Beilinson's
i
defined
in
complexes
functorial
may
polarized
on
R
procedure
oU.
The
since
proceed
(
r e s p e c t i v e l y , the
5.2.7).
category,
the
the
schemes by
d i r e c t i o n . To
K^(X,,Y.)
this,
([D5]
may
by
morphisms
isomorphisms
one
replacing
the
isomorphisms,
cohomology
the
( a u . , % ) ,w)
defined
simplicial
in
a n
Y.
—>
.
is
computed
where
K*
^(Z.)
i s a complex
computing
previous
take
the
internal
Horn
long
carefully
by one w i t h
map
f o r open
sequences
suffices
assures
a r e morphisms
K ^ ( X )
resolutions
that
JMR^
A
H
the mentioned t r i a n g l e s
K
A H
(
X
)
K
® i
(
C
a
(
Z
c) and f ) ,
and e x a c t
(
Z
(
X
"
and
now
including
come
from
triangles of
i n the exact
. For defining
cohomology
a capproduct i t
pairing
)
->
K ^
c
( X )
.
and t h e f o l l o w i n g
A H , C
Horn
a s i n [ B e i 1] 1.8, c f . a l s o
t h e maps
in
«£ K
i n 6.1
immersions i n homology
b y 6.11.10 t o d e f i n e
(6.11.11)
internal
complex).
b y t h e same a r g u m e n t s
1.19. T h i s
derived
i n j e c t i v e components
sequences r e q u i r e d
chosen s i m p l i c i a l
MAH-complexes
By
TL
i s the obvious
f o r each s i n g l e
exact
restriction
[J2]
R Horn
t h e complex
The
^(Z.) - e.g., the
An , C
one, and
(replace
1
H
Arl, C
)
K
AH,C
(
X
AH,c
<
X
lemma, a p p l i e d
to
>
(6.11.12)
K
we
see that
functorial
By
AH
(
X
^
Z
)
K
• i AH,C
K
^ >
"
Z
a pairing
)
'
\p a s
indicated
i n t h e s e n s e o f 6.11.12 f o r a n a r b i t r a r y v a r i e t y
X
simplicial
case
that
[Bei
2] 4.2; e v e r y t h i n g
6.11 .13. Lemma
i s smooth,
Given
G^-modules,
canonical
)
i t suffices to define
( c a r e f u l l y chosen)
groups,
Z
pairing
and then
carries
making
o n e may
over
compatible
. . . ) , ip
tp
resolutions
and
1
proceed
to the
as B e i l i n s o n i n
t o MAH-complexes.
pairings
y
one r e d u c e s
X .
o f complexes
, a s indicated,
the following
(of abelian
there
i s a
diagram commutative
'A[-1] 0
1
->
B *[1]
C o n e ( a ) [-1] ® C o n e (3)
A
0
B
1
C•
-
->
C
->
c*
Y
A'
f
Here
p
•A "
\jJ
1
0
Proof
on
'B
q + 1
Let
cp((a',a)
1
is
in
1
H
A K
= \p (a 0 b)
p
=
\p' : o n
('A "
1
+
P
© A )
P
1
q
©
(-1 " y
0
(B
ip'(a*
'B
q + 1
0 b')
)
i t i s s t r a i g h t f o r w a r d t o check
(X)(-b)
have
form
a twisted Poincare
.
that
duality
immersions
i n v o l v e d , we
can
be o b t a i n e d
and
then
restriction
i n the sequel,
give
of Bloch
holds
the fundamental
i n the A-adic
to hold
of
remark
checked
i n the etale
apply.
d e f i n e d by t h e
i s an isomorphism
realization.
a s i n [BO] 2.1, b y r e d u c i n g
t a k i n g the preimage
c a n be
in loc.cit.
morphism,
class,
i s only
L e t us j u s t
(this
a n d Ogus
duality
map
map f o r
and s i n c e i t s c o n s t r u c t i o n
i t here.
a n d i s known
the Poincare
this
o f [BO] 1.4.2
a l l results
with
Since
don't
any o f t h e r e a l i z a t i o n s
Similarly,
not defined the r e s t r i c t i o n
i n homology.
the c o m p a t i b i l i t y
capproduct
q
by
•
(b.b ))
definition
t h a t we
f o r open
theory),hence
is
0
induced
a
except
somewhat
that
-> • c "
(-1) "V
0 Cone(3)
and
morphisms
needed
P
these
„(X)(j)
Aii i L
etale
1
P
i t i s
Cone(a)[-1]
theory,
B
i s the canonical pairing
With
H
0
since i t
The f u n d a m e n t a l
t o t h e smooth
1 E r (H^ ( X O ) )
f
= TL
class
case
under
O ^ ™
H
Y< '
X
d
i
H., , .
2dim
All
other
case
properties
o f the A-adic
6.12.
Example
theoreme
over
x
0
>
0
® H (X O)
> H (X O)
7
f
„ ( X , d i m X) .
X
f
required
i n 6.1
realizations,
easily
follow
from the
isomorphisms.
I t i s i n d i c a t e d i n [ B e i 1] a n d p r o v e d
9, t h a t
a regular
universally
m
on t h e c a t e g o r y
noetherian
catenaire)
I/
X
<Z< '<»>
o f schemes q u a s i - p r o j e c t i v e
irreducible
the motivic
=
K
2 j - 1
(
X
base
S
cohomology
)
l
i n [ S o u 3]
J
)
( i t should
a l s o be
and homology
'
(6.12.1)
H
form
H^j
Z
a twisted
(X,(D(j))
(In
for
with
without
coherent
K
with
m
r
(
2 b
(x) -
duality
Z c X
i s perhaps
weights).
sheaves,
fora l l
k
b )
theory, with
to the previous
( X ) ® ffl i w h e r e
k
= K;_
Poincare
the contrast
theory,
in
J(x,ffl(b))
Here
K (X)
m
defined
examples,
K
1
and
only
1
the r e s t r i c t i o n
this
denotes
(r)
f o r smooth
i s an
X
"absolute"
Quillen's
i s defined
that
K-theory
as t h e subspace
fc
the Adamsoperators
(cf. the d e f i n i t i o n
'
of
a c t by
K
(X)
multiplication
( r )
i n 5.12).
m
k
For
the
reader
into
t o the c i t e d
a scheme
W
smooth
X
operators
on
K
1
Q„\
Wj b
of
c
Actually,
k
(called
papers,
over
<j>
i n [ S o u 3] ) we
i t involves
S
and then
t h e embedding
changing
1
K (X)
m
by m u l t i p l i c a t i o n
t o make e v e r y t h i n g
Soule
i n h i s paper
independent
defines
motivic
h o m o l o g y b y means o f t h e d e s c e n d i n g y f i l t r a t i o n
a s c e n d i n g y f i l t r c i t i - c m on
K' ® Q , s e t t i n g
-
refer
of
X
t h e Adams
~
(W) <
m
classes
1
of the
with
of
cannibalistic
W .
cohomology and
on
K
m
and an
H
M,Z
(
X
'®
(
j
)
)
=
G
r
K
y
2 j - i
(
X
)
0
® '
(6.12.2)
H^(X,Q(b))
Both
approaches
K
m
= Gr^(K _
a
are equivalent,
(
X
)
(
r
)
~
S
1
- >
G
r
2 b
(X)
0 (D) .
by t h e c a n o n i c a l
K
(
y m
X
)
0
isomorphisms
« '
(6.12.3)
(
K (X)
r
)
— - — > Gr^ (K (X)
m
r
induced
by the i n c l u s i o n s
K
E
(
m
has
X
^-
K
( x )
m
r
t obe c a r e f u l
discussion
In
lower
1
es
a)
H
the
Chow " h o m o l o g y "
(X,Q(b))
by
Fulton
b)
H
2
b
u
2
this
*
8
a
n
d 7
-
2
* However, o n e
identification,
c f . the
- one h a s t h e f o l l o w i n g
© K
xOC,
(P)
to
r
f
+
points
field
of
sequence
([Q1]§
7,
5.4)
K ( x ) ) => K ' ( X ) ,
q
p
of
X
+
q
with dimension
x € X , induces
p and
isomorphisms
2
M
+
1
= E
f a
(b+2)
b
( X ) 0 Q = CH (X)
b
group
ofcycles
0 Q , where
o f dimension
CH (X)
b
i s
b , as defined
[Fu],
(X,Q(b) ) s E
©
X € X
e
i s the residue
2 b
o
niveau spectral
p
n
S
(co-)homology.
(X) =
t
s
[
- f o r t h e K-groups
q
( p ) -^
*
3.7.
Quillen's
E
P'
f
with using
o fm o t i v i c
(6.12.5)
K(X)
c
® '
degrees
6 . 1 2 . 4 . Lemma
X
0
i n [J2]
calculation
where
Z
(r)
r
K (X)
c y K (X) 0 Q a n d
m
—
m
u
)
0 Q) ,
m
K
2
(K(X))
^
+
bn
_ (X)
b
>
0 Q , with
©
x
€
X
(b 1)
+
K
( X )
X
E
+
b1
b
( X ) = homology o f
^ A Y _ >
©
x
€
X
(b)
Z ,
where
c) H
"tame" d e n o t e s t h e tame s y m b o l
^ (X,Q(d-2))
2 d
s
2
E
d
2-d
( X )
=
K
e
r
(
6
a
particular,
field
k
for
X
(
x
= HP
f
)
o
r
d
=
a r
d
i
x E X
smooth
spectral
w
'
i
t
K(X) )
X
h
.
(d-1)
of dimension
d
over
sequence
=, K _ _ ( X )
q
sheaf
X
©
S
^ ->
(X,K_ )
m
map,
p
,
q
associated
to
U I—> K
m
(U)
,
isomorphisms
,2j
H
(X,Q(j) ) s H ( X , K
e)
H^ "
3
3
3
) ® (B = C H ( X )
j u
1
(X,(D( j ) )
3
1
s H ~ (X,K.)
2
3
xex' " '
xex
s H°(X,K )
K
( 3
-
® Q ,
Q ® homology
e
2
Hjjj(X,ffl(2))
3
® Q
K ( K f x ) ) £22«>
0
f)
)
Q
irreducible,
d)
3
k
0
" d i v " the d i v i s o r
(d)
i s the Z a r i s k i
m
induces
G
X
Ef^(X)
K
(
)
t h e Brown-Gersten
(6.12.6)
where
k
X
2
x
In
(
-d
2
and
(
X
X
)
®
K-(MX))
©
xex
® (D = flj ® k e r n e l
2
îX>
1 )
of
z,
( 3 )
of
K(X) ,
X
(1
xex >
where
k(X)
Proof
The
i s the f u n c t i o n
first
three
f r o m 6.12.5 b y
of
that
(6.12.7)
0
= Gr K (F)
K (F)
= Gr K (F)
= F
K (F)
= Gr^K (F)
= K
1
2
X
0
1
1
2
.
from the s p e c t r a l
S o u l e i n [ S o u 3]
K (F)
0
of
statements follow
constructed
the f a c t
field
theoreme
8 i v ) , i n view
= Z ,
X
,
l l n
2
r
° (F)
(Milnor
sequence
K-theory)
for
a field
since
5.6).
F
(cf. also
1
d
.
P it~P
= div
+
and
= Coker (
fa
©
x € X
view
4 and remarque), and
i n 6.12.5
+0
K (x)
X
resolution
P
©
x € X
(b+1)
of the construction
Gersten
o f 6.12.6
Z)
= E
b
_
q
= E
* ,
(X) , X
d-p,a-q
irreducible
= dim X
of
X
the
class
X
, i s defined
in
CH^(X)
.
= X^
. ,
(d-p)
a
reformulation.
the fundamental
as the element
v i a 6.12.4
of
(X)
f r o m 6.12.5 b y means o f t h e
( P )
s t a t e m e n t s d ) , e) a n d f ) a r e j u s t
For
b
(b)
M
d
( c f . [Q1]§ 7
( s e e [Q1] § 7, 5 . 8 ) , i . e . , b y t h e f o r m u l a e
E ' (X)
r
the
theoreme
2
d
= tame
P ^/-p
F o r t h e saiae r e a s o n o n e h a s
CH (X)
In
loc.cit.
in
class
n
x
M
E H
2 d
(X QHd))
7
corresponding to the class
a ) . In Soule's d e s c r i p t i o n
Gr^(K^(X)
® Q)
, however,
this i s
i n general the
(-d)
class
§7.
in
K'(X)
corresponding to this
The c o n j e c t u r e s
For
a twisted
tensor category
(7.1.1)
5"
different.
varieties
Poincar^ d u a l i t y theory with values
t h e fundamental c l a s s e s
» TH
2 i
from t h e group o f c y c l e s
induce a c y c l e
map
f
17
z
(Z,i)
under
-»
TH
of dimension
i
2 i
in a
(X I)
•—» image o f
rH
sections"
be
o f Hodge a n d T a t e f o r s i n g u l a r
c l . : Z.(X)
[Z]
will
o f t h e homology group
2 i
H-.(X,i).
(X,i)
i n t o t h e "group o f
Here
is
identified
closed
with
the free
subvarieties
x •—• Z = { x } . F o r
7.1.1
reads
2 i
f
on t h e i r r e d u c i b l e
i ,
irreducible
v i a
of dimension
d
( c f . 6.1 j ) )
2d-2i
(X,d
= H'
(X I)
c l ^ : Z^(X)
where
j = d - i
plained
image
- i)
2 : i
claim
maps 7 . 1 . 1 .
consists
cycle
T 7
(X,d - i)
o f Hodge a n d T a t e
f o r an a r b i t r a r y v a r i e t y
Note t h a t
reasonable
of the cycles.
As ex-
concern the
map.
that
generalization
(X,j)
i s now t h e c o d i m e n s i o n
of the last
2d-2i
>TH
i n §5, t h e c o n j e c t u r e s
I
H
o f dimension
smooth,
duality
group
as
(7.2.2)
a
X
by Poincare
H
Z C X
abelian
into
the correct
similar conjectures
f o r singular
map
X
i n stating
X
varieties there
cohomology,
since
f o r the
i s n o t even
t h e morphism
0
•H
2 i
(X,i)
i s n o t an isomorphism i n
general.
7.2.
X
Conjecture
(Hodge
conjecture
i s a variety
(i.e.,
a separated,
over
C, t h e n
Cl
i
fora l l
i £ 0
f o rsingular
reduced
varieties) I f
algebraic
scheme)
t h e map
r^(H .(X,Q)(i)))
0 Q : Z (X) 0 Q
2
i
=
1
(2TrV^l)" W
H
- 2
X
Q
i 2i( ' )
0
F
1
H
2 i
CX,C)
is
surjective
7.3,
k
(the homology b e i n g
Conjecture
(Tate
i s a finitely
then
f o r each
conjecture for singular
generated
a
prime
the Borel-Moore one).
field
* char
and
(k)
X
varieties)
i s a variety
and e v e r y
If
over
i I 0
the
k,
map
G
C l
is
surjective
notations
We
want
this
we
schemes
pal
divisor
which
inducing
cl.
cf„
a
[BO] p .
k,
weights
through
If
\ ,
map
reduce
varieties.
7.1.1
the
to the
For
f o r a twisted
on a c a t e g o r y
duality
i : D —>
theory
X,
X
H
of
i n addition
then
i s a smooth
i *
7
^
i n t h e examples o f §6, t h e n
rational
equivalence
map
197,
z
a s d e f i n e d i n 6.5.
i n a smooth v a r i e t y
1
(b) ) *
basically
the cycle
with
triviality)
: CH (X)
ft
we
1.5):
i s satisfied
factorizes
a n d 7.3
I f f o r our Poincare
(principal
(X,Z
f o r smooth p r o j e c t i v e
theory
a field
( c f . [BO]
a
o f 6.7).
conjectures
over
et .
k
H^V(X,Q (i))
>
a
those
duality
a
(X,Q (b)) =
t o show t h a t 7.2
Remark
assume
: Z.(X) 0 Q
more g e n e r a l l y s t u d y
Poincare
7.4.
a
(where
being
classical
Q.)
* Q
i
• TH
step
1.
2 i
(X,i),
-
princi-
O
r
the cycle
map
as d e f i n e d by F u l t o n ,
7.5.
Lemma
Let
U -^-» X
be an open
immersion. Then t h e
restriction
W
is
H
X
b
2b-a a( ' >F
W
^
H
U
b
2b-a a< ' >F
surjective.
Proof.
then
Let
Z C X
be a c l o s e d subscheme
b y 6.1 f ) a n d t h e e x a c t n e s s
of
W
with
U = X - Z ,
we h a v e a n e x a c t
sequence
W
W
since
H
X
b
W
2b-a a< ' >F
H
2b-a a
( Z
'
b )
F
=
0
b
y
6
H
'
5
I n t h e f o l l o w i n g we a s s u m e
Poincare
m)
If
duality
f : X
>Y
same d i m e n s i o n ,
Also
F = H
is
we a s s u m e
0
satisfied
theory
7.6.
11
(Spec
theory
that
a
i s a proper
=
J
b
)
—
°'
'
the following property
(compare
then
U
2b-a a< ' >F
[BO] 7.1.2)
map b e t w e e n v a r i e t i e s
Tj
(cteg f )
y
ofthe
.
i s an F - I i n e a r c a t e g o r y
k,0), which i s o f c h a r a c t e r i s t i c
f o r t h e Hodge t h e o r y
f o r our
( F = Q)
for the f i e l d
zero.
A l l this
and t h e a-adic
(F = Q ^ ) .
Lemma
Assume t h a t
a r e reduced.
If
k
TT : X'
i s p e r f e c t a n d t h a t a l l schemes i n
•X
i s proper
then
W
H
X
b
2b-a a( '' > —
W
X
b
2b-aV ' >
and s u r j e c t i v e ,
is
surjective for a l l
Proof.
in
a,b € Z.
T h e r e e x i s t s a smooth s u b v a r i e t y
e v e r y i r r e d u c i b l e component,
TT : U' = ^ (U)
1
—•U
such t h a t t h e r e s t r i c t i o n o f
is faithfully
c o n n e c t e d components
flat.
s e p a r a t e l y we may
c o n n e c t e d . Then t h e r e
U C X, open and dense
By t r e a t i n g t h e
assume t h a t
U
is
i s a commutative diagram
g
with
h
faithfully
flat,
q u a s i - f i n i t e separated
( c f . [Mi] I
2.25). By Z a r i s k i ' s Main Theorem and p o s s i b l y making
U')
s m a l l e r we may
by assumption
Tj
(note
that
image o f
h
is flat
and f i n i t e .
(and
Then,
m),
=
u
assume t h a t
U
(deg
g
h)"
1
h^,,
=
1
(deg
h) " T r ^
i s n e c e s s a r i l y proper),
(g^v )
xjn
i . e . , n^
i s i n the
T T ^ . By t h e commutative diagram
.
(deg h)
X
H (U',J)
g*T7 „n
y
lT
H
> 2d-i
( U ,
d
' ~
j )
Tr
, d = dim
it
U,
A
H (U,j)
we see t h a t
H
W
IT
U
d
2d-i< ' -
3 )
has a r i g h t i n v e r s e , so t h e c l a i m i s t r u e f o r
For
case
X
we
dim X = O
Otherwise
= Tr^ (Z)
U
be
exact
H
z
b
,2Hb - a* a( ' ' >
Z
TT^
left
the
right
in
the
7.7.
b
2b-aV ' >
The
the considerations
as above,
-
-
H
Z = X\U
6.1
x
, h
2 b-- A
a <a ' -
W
H
5
The
above.
and
f ) and
X
b
)
b
h2b -aa <a ' < >
7.5
by
W
—
W
—
a
i s surjective
TT^
we
obtain
a
H
U
H
2
b
2b-a a< '' >
U
b
b2-b - a* (a '' ' >
a
assumption of the
i s S O by t h e f i r s t
step,
induction,
hence the
surjectivity
middle.
Remark
proper
by
on t h e d i m e n s i o n .
diagram
a
W
induction
= X'\U'. T h e n b y
commutative
w
by
i s covered
let
1
Z'
proceed
and
Let
X'
X' —
smooth
X
be p r o p e r
and p r o p e r .
and
Then
surjective
by
the
with
X
above
**
Im
In
H
a
(H (X',b)
particular,
(-,b)
H^(-,b),
one
_ H (X,b) .
a
a
and
6.9,
with
f t
->
H£ (X',Q )>
t
where
compact
-W^ H
f t
1
1
f
The
second property
i t to etale
= W^H
i s [D5]
cohomolgy
the assumption of r e s o l u t i o n
1
1
( H (X (C) ,Q) — • H ( X f ( C ) Q )
generalizes
without
2 b
suport
obtain
t
respectively.
o f 6.8
of the cohomology
Ker(H* (X,Q )
Ker
= W
Q
i n the s i t u a t i o n s
i s the dual
we
>H (X,b))
a
and
of
t
(X Q
1
8.2.5,
f
f t
)
f
(X ( C ) , Q ) ,
and t h e
arbitrary
first
fields,
singularities.
7.8,
Now
c o n s i d e r a diagram o f
X
7
varieties
X"
TT j
(7.8.1)
X
TT
with
p r o p e r and s u r j e c t i v e ,
a
smooth and p r o p e r . Then by 7.5
(7.8.2)
H
2
I
( X
is surjective,
T H
2
I
C l
M )
-
L
£
L
->
n
o Ta
0
H
2
1
the
( X M )
composition
->
W
T W
0
H
2
1
( X M )
•
H
2
I
( X
(
1
)
0
2
=
T H . ( X , i )
2
C l .
1T
x
Z (X ) ® F
• Z (X)
i
is still
the l e f t c y c l e map
Q
TW H .(X,i)
C l .
i
Tirit
W
7.6
X"
g e t a commutative diagram
1
*
Z ( X ) ® F -2—>
If
•-»
and we
( X M )
(7.8.3)
1
and
an open immersion and
® F
i
surjective,
t h e n the s u r j e c t i v i t y
i m p l i e s t h a t o f t h e r i g h t one.
not e x a c t , i t i s i n g e n e r a l d i f f i c u l t
Since
of
r
t o d e c i d e when t h i s
is
will
be the case, but i t i s o b i o u s l y t r u e , i f
a)
H
2
W H (X,i)
Q
1
2 i
is
( v i a TT^ o a*)
a direct
f a c t o r of
( X M ) .
In t h i s case we
proper c o v e r o f
say t h a t 7.8.1
(or
X",
f o r s h o r t ) i s a good
X ( f o r t h e c o n s i d e r e d homology t h e o r y
In p a r t i c u l a r t h i s h o l d s t r u e , i f
b)
H (X",i)
2 i
i s a semi-simple
o b j e c t of
3.
H) .
ie
7.9.
Theorem
The
varieties,
i f i t i s true
Proof.
Chow's lemma a n d
By
given
X/C
jective
d
there
X".
= dim
X",
simple.
7.8.3
Similarly,
we
p r o j e c t i v e ones.
singularities,
7.8.1
H
2 d
with
"
2 1
the
smooth and
(X"(C),Q(d
for
for
for a
X"
pro-
- i ) ) ,
semi-
implies
the
X.
have:
T h e o r e m a)
characteristic
for
a diagram
arbitrary
i s a p u r e p o l a r i z e d Hodge s t r u c t u r e , h e n c e
By
varieties
for
r e s o l u t i o n of
H ^ ( X " ( C ) , Q (i) ) =
7.10.
f o r smooth and
exists
Then
i s true
For
zero
i s true
a
finitely
the
Tate
i f the
generated
conjecture
original
Tate
field
7.3
for
k
of
arbitrary
conjecture
5.1
i s true
smooth p r o j e c i t v e v a r i e t i e s .
b)
For
the
a
finitely
Tate
proper
generated
conjecture
cover
X"
7.3
field
i s true
such
that the
k
of
characteristic
for
X,
i f
X
has
Tate
conjecture
a
p
>
0
good
i s true
for
have t o
show
,f
X .
Proof
the
k =
2
i
0.
by
By
As
a b o v e , we
(XSQ U))
=
a
lemma
2 d
H
we
"
2 1
(XSQ
can
the
map
so
still
in characteristic
1.1,
get
a diagram
conjecture
G^-module,
knowing t h i s ,
cycles
above arguments,
singularities,
a general
semi-simple
the
o f good p r o j e c t i v e c o v e r s
r e s o l u t i o n of
tive.
H
is clear
existence
char
and
b)
7.8.2
a
with
zero.
has
7.8
7.8.1
b)
a)
Namely,
a right
lemma
projec-
Grothendieck,
X",
should
as
Chow's
s m o o t h and
and
by
case
by
d - dim
t h a t 7.8
get
i n the
X"
of Serre
(d-i)) ,
s o we
apply.
using
we
inverse
should
seen
f o r the
a
Not
absolute
have
be
Hodge
in
corres-
ponding
realizations
ticular
f o r the
So
f a r we
conjecture,
7.11.
the
the
cycle
A)
cl-*8 Q
B)
A-* (X)
j
A (X)
*
Q
:=
j +
is
=
in
par-
f i r s t part of
following three
X
field
be
k
smooth,
and
Tate's
parts.
projective
let
> H (X,Q (J))
2j
a
0. ?
for
char
k.
then
surjective,
cl-*
H
dim
generated
map,
Im
of the
[Tl]) Let
j
a
rk A (X)
=
(Tate
: Z (X)
ft
j
C)
s
3
discussed the
consists
finitely
so
realizations.
have o n l y
which
cl
be
Q.-adic
Conjecture
over
f o r a b s o l u t e Hodge c y c l e s and
2
j
is finitely
generated
and
G
(X,Q (j)) \
a
order
of pole
of
L(H
2 j
( X , Q ) , s)
at
a
k.
a
7.12.
Here
dim
arithmetical
L-function
=
H
x
e
(or Kronecker)
L(V,s)
t( /Q&)
scheme
t r . deg(k)
(+1
i f char
dimension
of
a s s o c i a t e d to the
k,
S
k = p
extends
0)
and
ft-adic
is
the
"the"
representation
1
S
over
d
e
f
l
n
e
d
a
s
follows: there
Z[ir] ( i f c h a r
k =
0)
exists
or
F
to
> 0)
a
smooth
(if
P
M
char
k =
—
2i
V
k =
a
with
generic point
a pure Q -Sheaf
a
3
r\ =
Spec k
of weight
2j
sucht
on
S
and
let
L(V,s)
= TT
ye|SI
1
ZT
1
,
1
det(I-Fr (Ny)
y
s
|? )
y
s €
that
C,
we
where
y €
|s|
i s the
set of c l o s e d
Ny
i s the
cardinality
S,
>c(y)
of
y,
Frobenius,
fibre
with
and
€
y
?
the
extends
may
action
fact,
i s the
y
fibre
t o a smooth,
S
as
projective
2
= R -^f^Q
change theorem
of
?
for
residue
field
(geometric)
at
y
S
(meaning
the
over
y
together
that
X —• S p e c
G a l (tc ( y ) / K (y) ) ) .
of
?
is a
and,
f
finite
S p e c K (y) —>
point
t h e r e i s an
take
of the
G a l (K (y)/*c (y) )
at a geometric
In
we
Fr
S
points of
: By
ft
( c f . [Mi] VI
above such
morphism
the proper
3
4.2)
f
: 9t — •
and
smooth
i s smooth
and
S,
and
k
then
base
one
has
a
G a l (*c (y) /K (y) ) - i s o m o r p h i s m
?
This
i s pure
conjectures
L(V,s)
[D9]
( 9 C
x
S
'V
of weight
[ D 8 ] . The
2j
•
by
Deligne's proof
property of the weights
for
Re(s)
> dim
d e p e n d s on
S + j = dim
doesn't,
the choice of
because
argument
Part
varieties,
7.13.
B)
of the
Weil
assures
that
k +
a
j , cf.
X
be
for
but
i n the
C)
can
also
following
choices the
Re(s)
be
generalized
S
-
1 +
j
(cf.
to
arbitrary
form.
over
the
finitely
generated
let
G
i
> dim
at
quotient of
(Tate c o n j e c t u r e s f o r s i n g u l a r
a variety
Cl
the pole order
loc. cit.).
and
Conjecture
S,
f o r two
L-functions i s convergent
Tate's
Let
6t
1.4.6.
+ diin k
a
the
H
converges
This
j
=
y
: Z.(X)
>
et k
H^V(X,Q (i))
a
varieties).
field
k
and
be t h e c y c l e map, f o r Q, * c h a r k. Then
A)
c l ^ftQ
i s surjective,
B)
A ( X ) = Im c l ^
a
i s f i n i t e l y g e n e r a t e d and
i
A.(X) ft Q
H .(X,Q (i))
a
2
,
a
2 1
C) r k A ( X ) = o r d e r o f p o l e o f
L(H (X,Q ),s)
i
a
at
s = i + dim k.
a
7.14.
21
L(V,s)
f o r V = H (X,Q )
w i t h a mixed sheaf
S
9
o f weights
R e ( s ) > i + dim k
a
choice of
: 3t — • U
S. The e x i s t e n c e o f such a
extending
=
U
L(V,s)
V
over an
convergent
9
of the
f o l l o w s as i n t h e
X —> Spec k t o a morphism
and u s i n g D e l i g n e ' s g e n e r i c base change theorem
q
and h i s r e s u l t t h a t f o r a f i n i t e
weights
as above, b u t now
and t h e p o l e o r d e r independent
p r o o f o f 6.8.2, by e x t e n d i n g
S
i 2i
a s b e f o r e . A g a i n t h i s s u f f i c e s t o make
for
f
i s formed
ft
field
H*
i s mixed o f
i a. I n f a c t , w i t h t h e n o t a t i o n s t h e r e we may t a k e
and
For
9
=
R
2
1
^ Z
Q
J
U
.
smooth and p r o j e c t i v e
e q u i v a l e n t : we may assume t h a t
X, 7.11 and 7.13 a r e
X
i s connected,
o f dimension
d, t h e n we have
j
A (X)
and
2 j
=A ^ (X),
d
H (X,Q (J)) = H
j
a
2 d - 2 j
(X,Q (d-j)),
Q
furthermore
2 i
H (X,Q ) = H
2 1
a
(X Q ) £ H
f
a
2 d
"
2 i
(X,Q (d - 2i))
a
by h a r d L e f s c h e t z , hence
2 l
L(H (X,Q ),s) - L ( H
a
case we show:
2 ( d _ l )
(X,Q ),s
a
+ d - 2 i ) . For the singular
7.15.
Theorem
t
Assume t h a t
let
has
a good
0).
If
for
H^
i)
Tate
B)
i s true for
X"
and
ii)
Tate
A)
i s true for
X"
x X"
then
Tate
B)
i s true for
X
and
Proof
(e.g.,
X
Consider the
char k =
proper
dimension
and
cover
i , and
d"
=
dimension
dim
X",
i .
diagram
G
£
:
G
H
2 l
(X",B
f t
X"
(i))'
K
*'
*
• ». H
2 i
(X,Q
(i))
f t
1
k
"X
G
By
A
i
(
x
">
assumption,
End.
(H
0
there
. (X" ,Q
End ^(H
G
2 i
H
a
e
H
H^
2 ( d
""
Ker
_ l )
cycle
on
X"
By
the
a
k
morphism
- i)))
(Poincare d u a l i t y
(Kunneth
a
)
(X»,Q (d'
a
H
to a Tate
"
x
in
(X",Q )(d")]
(X xX ,Q (d"))
corresponds
2 ( d
(
TT a * .
Ker
G
l )
i
e
e =
= End ^(H
a
2 l
idempotent
with
(X»,Q (i)))
=[H (X",Q ) 0
C
i s an
(i) ))
n
A
'
)
formula),
x X",
so
by
i i ) i s the
algebraic
correspon-
d"
image o f
dence
a cycle
E
on
X"
construction
cular,
E
E €
z
operates
compatible
operates
11
(X"
x X ) . The
u
on
with
CH (X )
e
u
on
(cf. [Kl] §4),
i
A (X ),
i
v i a the
and
we
v
cycle
map.
In
have
a
*
Ker
E
=
Ker
e
fl A
1
( X
H
)
=
Ker(A (X
1
L F
)
* A
1
( X ) ) .
Since
by
parti-
A
1
( X
N
—
)
X
» A^(X )
is
obviously
surjective,
and
*
—
X
A (X )
1
the
is
X
> A (X )
proof
an
of
7.6,
we
isomorphism,
E) ft Q
(Im
7.16.
=
F T
Remark
assures
i s surjective
1
that
get
Im
E
=
e =
H
In
short
2 L
(X,Q
F T
terms,
similar
TTâ*
Im
i t i n d u c e s an
Im
by
=
r e a s o n i n g as
A (X).
If
1
in
Q
ft
F T
isomorphism
(i))
, which
shows t h e
the Tate conjecture
the decomposition of H
2 L
(X")
into
claim.
for
W H
Q
2 1
X"
x
X
(X)
F L
and
*
K e r Trâ
i s algebraic.
et
7.17.
If
i)
Theorem
T a t e C)
Let
X
i s true
have a good
for
X"
and
proper cover
dimension
f o r no
L-function
composition factor
of
W
has
a
of
W
zero at
s =
H
C
for
H
it
.
i , and
2i
ii)
X"
—
(X",Q
i + dim
F T
k
)
the
(e.g., i f
a
char k = p
iii)
> 0),
and
Tate
B)
i s true
for
X
and
then Tate
C)
i s true
for
X
and
Proof
First
jectured
weight
tive
and
we
remark t h a t
for the
2i
characteristic
p.
by
.
y€|s|
for
i .
i i ) i s generally
of motives
conj.
i ,
which
( a ) ) , and
D e l i g n e ' s fundamental
are pure
conof
holds for posiresult
in
[D9]
Grothendieck's formula
.
TT
319,
f o r dimension
dimension
condition
L-functions
( c f . [D6]
X",
2 dims
1
ZZ-
det(l-Fr Ny
y
s
= TT
|? )
a constructible
y
.
,
J=O
Q^-sheaf
1
1
N
det(l - Fr p ^ l H . f S , ? ) ) * - '
$
on
S
(see
[SGA
4
j+l
[rapport]
3.1). Here
S = S x
F
to
S
mixed
of
and
f
Fr
most
for
eigenvalue
most
a
for
Y =
t h e n by
[D9] 3.3.4
det(l
Re(s) =
m € Z
o f weight
m
?
of
Fr
F
. If
H (S,J)
- Fr P
with
m £ w + j
IH CS,?))
has a t
c
one h a s
ap
is
i s mixed
c
- s
s
?
( f o r an
= 1
at
Re(s) = | ) .
Instead
iii')
£ w,
£ w + j , so t h a t
zeroes
i s the pull-back of
i s the absolute Frobenius over
of weights
weights
, ?
P
P
o f i i i ) we
A (Y)
® Q
i
> H
ft
shall
only
assume
(Y,Q (i))
2 i
*
a
i s injective for
X,X".
For
pole
a subquotient
order of
L(W,s)
W
of
at
2 1
H
(X",Q )(i)
l e t
q
C(W)
be t h e
s = d i m k. T h e n a s s u m p t i o n i i )
a
implies
c(W)
± dinu W
,
^k
where
W^
G
K
i s t h e module
1
a
= K e r Trâ* C H . ( X S Q
3
n
K = Ker(A CX )
sum
decomposition
2
a
by d u a l i z i n g ,
a
i
a
By t h e a s s u m p t i o n
and
and
by assumption
of
we
have a
G^-modules
= W H
Q
2 i
(X,Q (i))
a
©
K ,
a
decomposition
Hj (XSQ U))
equalities
(i) )
i
H .(X",Q (i))
and,
a
•> A ( X ) ) , t h e n
i
direct
of coinvariants. l e t
k
= H
2
i
( X
i i ) and i i i ) '
inequalities:
f
Q
we
a
( I ) ) ^
1
e
K^.
get the following
system
of
c ( H
2
I
( X
,
\ «
(
A
I
)
)
)
c(H
=
2 I
(X,Q (i)/W_ )
A
V/
dim
H ^ ( X
+
x
1
Q
1
( I ) )
= dim
G
( H
2
1
( X
F
Q
A
( I ) ) ^
1
)
(X*\Q (i)
I
Q
If
lt
=
1
on t h e r i g h t .
c(H (X,Q (i) ) = C ( H
a
product
F T
(i))
k
formula
G
K
+ dim K
fi
V/
+
1
and dimension
rk K
i , the l e f t
i n e q u a l i t i e s , hence we a l s o have e q u a l i -
2 1
( X Q ( I ) ) ^ ) , since the L-function of
f
a
has no z e r o o r p o l e a t
—i
(X,Q
1
Finally,
2 l
W
2
rk A (X)
T a t e C) i s t r u e f o r X"
i n e q u a l i t i e s are both
ties
k
V/
A (X )
G
H
G
= dim W H
k
a
V/
rk
a
II
G
2
+dim(K )
E
k
II
H
a
V/
k
dim
C(K )
V/
i n this
1
s = dim k by t h e c o n v e r g e n t
a
Euler
range.
7.18 Remark I f i s perhaps no s u r p r i s e t h a t homology i s t h e
right setting
f o r c y c l e s on s i n g u l a r v a r i e t i e s ,
F u l t o n Chow groups form
since the
a homology t h e o r y . One way wonder
whether o t h e r t h e o r i e s c o u l d work f o r cohomology, e.g., t h e
groups
H-J^ fX,K.)- However, i n a note t o t h e a u t h o r
zar
3
(2/11/87), B l o c h has g i v e n an argument t h a t t h e r e i s no
c o n t r a v a r i a n t Chow t h e o r y c o i n c i d i n g w i t h
CH*(X)
f o r smooth
X
and g e n e r a t i n g a l l Hodge o r T a t e c y c l e s i n t h e cohomology
of
singular varieties
§8.
8.1.
Homology
tic
and K - t h e o r y
We now p r o c e e d
category
K
(see appendix A) .
of singular
varieties
t o h i g h e r K-groups. We f i x a f i e l d
o f v a r i e t i e s over
k, a f i e l d
z e r o , an F - I i n e a r t e n s o r c a t e g o r y
F
k, a
of characteris-
y, and a P o i n c a r e dua-
Iity
theory
In
(H*(
the
the
: CH (X)
d,
then
the
map
general,
we
can
have
K ^ (X)^
classes
by
f o r our
a kind
ded
suring
[Bei
down a
the
1]
There
theorem
2
(
d
[SGA
l
"
)
Grothendieck
is a
of
pure
XIV
4,
c h a r a c t e r s on
K
cohomology
group
theory,
c h a r a c t e r " on
set of
C l
K^,
K'-groups.
axioms
of
and
q
. In
for a
i
,
coherent
reasonable
following (cf. also
theory
again
and
can
this
Namely,
Poincare
Chern
be
expressed
be
[Gi]
duality
Schechtman's paper
ch
Z
relative
Z
: K (X)
m
compatible
with
(X',Z') — »
(X,Z)
the
as
Chern
»
c h a r a c t e r on
©
j>0
rH
2 j
a).
m
" (X,j),
Z
contravariance
i n 6.1
higher
f o r morphisms
on
of
can
Gillet
sheaves
extenhas
theory
[Sche],
2.3):
is a
3".
(X,Q ( d - i ) )
6]
= H^.(X,Q(i))
Q
I f there
Chern
1
0
K (X)
to Quillen's higher
written
a)
n
of
map
i s smooth
K (X)^ "* ' = H
f
=
i s the
c f . 6.12.4 a ) .
cycle
in
isomorphism
d
=
values
isomorphism
Q
i
X
d e f i n e d v i a Chern
an
CH (X) 0
X,
i s an
1
be
If
i
CH ^ (X)GQ
and
(X,i)
Grothendieck-Riemann-Roch
cycle
where^
2 i
weights
studied the
CH (X).
there
d
=
i
with
we
> rH
i
Ch (X)SiQ
the
section
F u l t o n Chow g r o u p
dimension
by
,*))
it
previous
cl.
on
,*) ,H I
K-groups
asand
b) I f
sion
X
i s embeddable i n a smooth v a r i e t y
N
(e.g., i f X
i s quasi-projective),
f i n e d by t h e commutative
o f pure dimen-
t h e map
T
de-
diagram
X
K-(X)
(8.1.1)
M
b
™ -2b< ' >
m
/ J
[ Td(M) n
X
K
m
( M )
** 9 rH "" (M, j )
8
~
2j
m
jez
i s independent o f
(resp.
e ©
K
TH
(M,u) = ©
X
K'(X)
m
= K (M)
m
[Ql],
i tcoincides
ch
and
M
i s t h e one proved by Q u i l l e n
w i t h t h e cappropduct w i t h [0^] € K^(M).
o f Baum, F u l t o n
K-theory
and Mac Pherson, g e n e r a l i z e d
f
T )
functors
: (K*( ),KJ( ))
> (rH*( ,*),rH„( ,*))
i s compatible with a l l the f u n c t o r i a l i t i e s of twisted
duality theories,
K-theory,
This
except t h a t t h e fundamental
class
i s c l o s e l y r e l a t e d t o t h e Riemann-Roch theorem
f o r smooth
A compatibility,
X
which
one has
T([0^])
T] .
X
w i t h de-
= Td(X).
i s not e x p l i c i t l y s t a t e d
but which w i l l be o f some importance
Poincare
class of
[ 0 ^ ] , i s n o t mapped t o t h e fundamental
nominators;
to
(see [Gi] 4 . 1 ) .
f a c t , t h e morphisms o f
( C h
2u
s a t i s f y t h e c o m p a t i b i l i t i e s as i n t h e Riemann-
higher algebraic
In
2N
TH " (M,N-u)
M [SGA 6 ] , and t h e isomorphism
f o r smooth
T
Roch theorem
e t a l e mor-
and homology. Here
ro
i s t h e Todd c l a s s o f
c)
and c o m p a t i b l e w i t h t h e c o v a r i a n c e
c o n t r a v a r i a n c e ) f o r p r o p e r morphisms ( r e s p .
phisms) o f
Td(M)
M
i n [Gi],
f o r us, i s t h e commuting
of
t h e Chern
and
character
f o r cohomology
immersions
K
the relative
(6.1 c ) ) . Namely,
one has a commutative
2j-l< >
X
(8.1.2)
Z
|ch
2J-i< >
K
-
X
(1)
|ch
1
for Z C Y C X
2j-i< - >
X
(2)
Y
K
Z
(3)
Z
Jch ~
1
f
closed
2j-i-l( >
X
|
1
^-rH
Z
of (2) i sthe functoriality
commuting
f o r K-theory
diagram
K
-
Y
sequence
T H (X, j ) — • T H ^ ( X - Z , j )
TH (X J)
The
with
+
1
(X,j)
8 . 1 a) a b o v e , a n d
2
the
o f (1) a n d ( 3 ) f o l l o w s
commuting
is
induced
long
b y a map b e t w e e n
exact
homotopy
Similarly,
plement,
•••
(
-* ;
(8.1.3)
T
y
)
—
rH
Y
b
r
•••" ,n-2b< ' >-
The
k
J ( I ) ' T
Y
commuting
and
a commutative
>
rise
;
(
x
)
t o t h e t o p and bottom
—
*
V2b<
X
'
b
>-
(
u
)
m
J (2)'
X
T
r H
t h e open
U
—
>
—
J
K
m-i
(3)'
U
b
m-2b< ' >
^
T
r H
(
Y
i
K
J
Z
Y
K > ° >
from 8.1b),
2
1u
LI
I
J
9
• T
T HH
J€Z
Y
;-I< >
k
,(M°,j)
W
c h
Y
^ f f i rH?^
n + 1
(M,j)
u
D
I"
u
Td(M )
r H
b€Z
U
b
m-2b< ' >
•
b€Z
r H
b
m-l-2b< ' >-
/1
/I
com-
)
o f ( 3 ) ' i s the commutativity o f
U
whose
diagram
K
K (U)
ch
ch
fibrations,
U = X - Y C X
o f ( 1 ) 'and ( 2 ) ' f o l l o w s
commutativity
that
c f . [Gi] 2 . 3 4i i ) .
for Y C X
we h a v e
k
t h e two homotopy
sequences give
i n 8.1.2,
sequences
from t h e f a c t
Td(M)
m-l-2b
( Y
'
b )
'
and t h e
which f o l l o w s from t h e c o m p a t i b i l i t y o f t h e capproduct
with
(this
Td(M)
i s i m p l i e d by
r e s t r i c t s to
variety
8.2.
M
vii))
T d ( M ° ) . Here we
and
f
[Gi] 1.2
Remark
let
M
0
and t h e f a c t t h a t
have embedded
X
i n a smooth
= M H U = M X Y .
Our g e n e r a l s e t t i n g f o r c e s us t o a p p l y
o b t a i n i n g a b e l i a n groups from t h e o b j e c t s o f
u s u a l l y not e x a c t so t h a t
(TH* (, *)
r
T is
9", and
,THie ( ,*))
for
w i l l not form
t w i s t e d P o i n c a r e d u a l i t y t h e o r y . However, i n t h e c a s e s we
interested
: 2T
> Vec
i n t o the category of
(vH
F
( f i n i t e - d i m e n s i o n a l ) v e c t o r spaces
(compare remark 6.2a))
*
( ,*),vH (
and,
A
since
t i o n , we
(the
are
i n t h e r e i s an e x a c t , f a i t h f u l t e n s o r - f u n c t o r
v
F
6
,*))
T(A)
may
with
over
v ( l ) = F. Then
.
.
.
.
i s a t w i s t e d Poincare d u a l i t y
= Hom (1,A) C Hom (F,vA) = vA
y
F
r e g a r d a l l elements i n
T(A)
theory,
i s an
injec-
as elements i n
vA
" u n d e r l y i n g F - v e c t o r space") which happen t o l i e i n t h e
subspace
TA.
In p a r t i c u l a r , we may
regard
(ch,T)
as
mor-
*
phisms i n t o
(vH
( ,*),vH^( , * ) ) , and t h e y a r e morphisms o f
t w i s t e d P o i n c a r e d u a l i t y t h e o r i e s , except t h a t t h e
l i t y between t h e fundamental c l a s s e s
[# ]
x
and
r/
compatibii s mis-
x
sing.
8.3.
Examples, where t h e s e "Riemann-Roch t r a n s f o r m a t i o n s "
exist,
are
a) the d - a d i c t h e o r y
(6.7 and
b) the Hodge t h e o r y
(6.9,
c) The
(6.10, [Gi] 1.4
deRham t h e o r y
6.8,
[Gi] 1.4
iii)),
[Gi] 1.4 i v ) ) ,
i ) , no
weights),
d) r e a l i z a t i o n s f o r a b s o l u t e Hodge c y c l e s (6.11, by
combining
a
a)
- c)).
The
fact
T-subspaces
phisms
t h e images
2i
under
and a r e c o m p a t i b l e
c a n be c h e c k e d
C.
€ H
i, n
and
that
under
theories
k ) , Deligne
the
geometric theories.
cohomology,...)
I n t h e above
morphisms
examples,
of twisted
Poincare
l i e i nthe
isomor-
classes
from
the construction
c l a s s . One may
of
X
over
k
and
T
give
also
use
(instead
and t h e r e s t r i c t i o n
ch
maps
rise
into
to actual
dualities
1
r
Chern
( e t a l e cohomology
r
Chern
.
where i t i s c l e a r
of
8.4.
and
the comparison
f o r the universal
.
(B.GL / k , i ) ,
iv
'
t h e known c a s e o f t h e f i r s t
absolute
ch
1
: H ^(X QCj))
» VH (X J),
f
F
(8.4.1)
r'
tfJ(X,Q(b))
:
> vH (X,b),
a
Z
since,
by t h e p r o p e r t i e s
o f Chern
Z
to
characters,
ch
restricted
( : i )
H^ „ ( X Q ( j ) ) = K
.(X)
has t h e only
non-trivial
Jk , L
Zj —1
component i n V H ( X J ) and s i m i l a r l y T ( H ( X , Q ( b ) ) ) C vH
(X b),
tt
a
a
c f . [ J 2 ] 3.6b). M o r e o v e r , s i n c e f o r smooth
X
we h a v e
f
1
i
F
Td(X)
and
= T ([0])
since
there
f
= n +
terms
in
©
v>dim
i s a commutative
vH
(X u)
f
diagram
©
CH-(X)H
x
i*o
(8.4.2)
f
x
Cl
K£(X)
VH
2 i
(X I) ,
f
i>0
we
have
r'(TJ )
71
-
€ CH
X
Here
W
TJ
X
d i m
t h e upper
X
T
X
f o r the motivic
(X)» « =
KJ(X) -
i n 8.4.2
(
d
i
B
1
X
fundamental
)
S
H^
d i f f l
i s Riemann-Roch
x
class
(X,«(dim X ) ) .
transformation
for
Chow t h e o r y
induce
generalization
t o study
(8.4.3)
8.5.
an
is
the
r^ <8>
algebraic
A,b
\
8
surjective,
over
a number
r
:
a,b
then
Hf(X,«(b))
If
field,
If
in
i s known
to
6.12.4a).
and
§ 7 we
pro-
® Q F —» T H ( X , b ) .
a
for
• Q
Il *
X
a,b
—
a
or
char
finite
€ Z
the
t
field
X
is
map
Hj (X Q Cb))
f
and
G
)
C
a
k.
i s a variety
over
C
that
i s defined
then
for a l l
Conjecture
i i ) ) , which
o f t h e maps
i s a global
—» r j ( H ( X ( C ) , Q ( b ) ) )
H^(X,Q(b))
surjective
8.7.
k
provided
Conjecture
1.4
o f t h e c o n j e c t u r e s i n §5
: H^(X,Q(b))
If
:
and
isomorphisms
images
k-scheme,
8.6.
is
F
b
Conjecture
r
[ G i ] 4.1
inverses f o r the
In
pose
([Fu] and
a
a,b
X
e
Z.
i s a variety
over
a number
field
k,
then
r
is
a,b
surjective
8.8
lent
Remark
As
:
x
b
<( '«>( >>
for a l l
a,b
x
—
€
Z.
e x p l a i n e d above,
to the s u r j e c t i v i t y
of
b
( < >>
these
conjectures are
equiva-
G
6t k
~* *V (X,
X
8
«
T
a,b
8
«
:
K
b-2a< >
X
8
«
b
—
(2ir>/=T)" W_ H (X(C) ,Q) fl F
-
r
2b
r f 3 f ( H
f
a
(X,b)).
The next r e s u l t shows, t h a t a t l e a s t
forprojective
X
t h e s e c o n j e c t u r e s a r e i m p l i e d by t h e c o n j e c t u r e s 5.18, 5.19
and
5.2 0 ( f o r smooth schemes). S i n c e we a r e i n t e r e s t e d
i n the
g e n e r a l p r i n c i p l e , we j u s t assume t h a t we have morphisms o f
t w i s t e d P o i n c a r e d u a l i t y t h e o r i e s as i n 8.4.1, and c o n s i d e r
the s u r j e c t i v i t y
of
o f pure dimension
r ' . as i n 8.4.3, which f o r smooth
X
a, D
d by P o i n c a r e d u a l i t y agrees w i t h t h e map
2d
a
2d
a
(r = ch) ® F : H ~ (X,<Q(d-b) )®F — * TH "" (X,d-b)
s t u d i e d i n §5. C o n s i d e r t h e f o l l o w i n g p r o p e r t i e s f o r o u r
Poincare d u a l i t y
n)
theory.
( p r o j e c t i v e bundle
isomorphism) The c a p p r o d u c t
i n d u c e s an
isomorphism
© P nf
n
©
H
u=0
for
2 n
a
" "
2 u
u
( S p e c k,n-b-u) —
• H (PJ\b)
a
~
a,b e TL
1
where
f
u
€ TH
(ip!\n-u)
^ n~ct\) x
0
i s the cycle
class
m
v
of
H
the
projection.
f o r a hyperplane
H
of
I P ^ and
p : P £ —• Spec k
is
o)
i
1
( g e o m e t r i c chomology) One has
for
0
* 0, and
Usually
H (Spec k,j) = 0
T H ( S p e c k,0) = F.
n) i s one o f t h e c o n d i t i o n s
needed t o o b t a i n
o f Chern c l a s s e s as above, and b o t h
n)
and
o)
a theory
hold
f o r the
examples i n 8.3.
8.9. Lemma
Assume t h a t
X —-—>
be a c l o s e d
U = IP^NX, and l e t
surjective for
Proof
Step 1
r
2b b ®
the
F
(D>£,b) = 0
x
1
S
a
n
fact that
0,..,,n
Step 2
^
s
o
m
r' ,8
a, D
F
o
for
odd
for
a * 2b,
r
Phi
s
m
f
w i t h open
r
a
complement
+
o
r
a
n
d
a
1
1
F
is
n
a=
*
H ~
2c,
b
for
b =
as b a s i s .
a £ 2d, d =
dim X, t h e morphisms
n
» H (PJJ,b)
a
are s u r j e c t i v e . I n f a c t , by
n)
e v e r y element o f
H
a
i s of t h e form
X.
Here we used
fa
i
H (X,b)
®
a,
b
F o r even
1 b
i s surjective for
H^ (Ip£,<Q (b) ) = CH (D>£) ® Q = Q
with the c l a s s of
a
are s a t i s f i e d . Let
n ) , o) and 6.6 we g e t :
a
a
o)
be i n t e g e r s . I f
U, t h e n
From
and
subvariety
a,b
H (Pwb) = 0
a
x
TH
n)
n
(P£,b)
x
c
a n f ~ , and by t h e p r o j e c t i o n formula i t
s u f f i c e s t o prove t h a t
f
has a commutative diagram
n
~
c
i s i n t h e image o f
i^.
But one
n
TH (X C)
2 c
TH (P k,c)
l
2 c
I
r'j
Hj (X Q(c))
c
r'
» H^ (IP^Q(c))
f
c
/
so we o n l y have t o prove t h a t t h e c l a s s o f
image o f t h e map
has
=
T
I^ J
x
d e <
3
Let
n
c
i s i n the
below, which i s w e l l known ( f o r c = d
x
*
[ H
N
~
C
]
* 0, and f o r
use t h e p r o j e c t i o n formula
Step 3
H
a
be even,
0 £ c £ d
one
one
may
again).
0 £ a £ 2d, t h e n by s t e p 1 we
have
an e x a c t sequence
0 - H
a + 1
( U , b ) - H (X,b) —> H (D>£,b)
a
a
and hence a commutative e x a c t diagram
0
> TH
a + 1
(U,b)
TH (P^,b)
> TH (X,b)
a
Q
F®r'
F®r '
T
F®r'
IP
1
F * J f J ( U , Q ( b ) ) -+ F 8 H^(X,Q(b))
F • H^ (tf>£,Q (b) ) ,
+1
where e i t h e r
TH
( I P b ) = 0, o r e l s e
a x
an isomorphism and
Step 4
If
.(U,b)
a+j.
of
r'.
a
H
We
~
w
and
F ® r'
jpfi
is
i ^ i s s u r j e c t i v e . This implies the claim.
i s odd,
0 < a < 2d, we g e t an isomorphism
• H (Z,b), so t h e c l a i m i s c l e a r by
a
q.e.d
can extend t h e theorems 5.13,
arbitrary varieties
dimension
a = 2b
d, and l e t
X. L e t
X
5.15
functoriality
and 5.17 t o
be o f (not n e c e s s a r i l y pure)
E
2
d -d-nW
d
x
=
K e r
9
<
x
i c
A
W
(d)
i
v
e
>
z
>
(d-1)
x
2
be t h e E -term o f t h e Q u i l l e n s p e c t r a l sequence 6.12.5 f o r
If
E
X
i s smooth o f pure dimension
2
d -d+l^
^
G
=
x
' ^
u
t
i
n
e
9
n
e
r
a
l
d, we have
there i s only a
homomorphism
° W
X
E
—
x
d,-d i< >
+
which might be n e i t h e r i n j e c t i v e nor s u r j e c t i v e .
8.10 Theorem
Let
X
be a v a r i e t y o f dimension
then t h e r e i s a c a n o n i c a l
*
where
:
E
x
d,-d i< >
^
+
H
25
d
over
C
isomorphisim
H ^
1
( X
K d - I ) ) ,
f
i s t h e D e l i g n e homology [ B e i 1 ] , [ J 2 ] . I n
p a r t i c u l a r , there i s a surjection
r
: KJ(X)
r
a f
(H
2 d - 1
(X Z(d-l)))
#
= (2irv^l) ^
i n d u c i n g t h e map
r
2
d-i
d
i
n
c o n j e c t u r e 8.6 i s t r u e f o r
8
+ 1
*
H
1
-
2 d
1
f
.!
o
r
(a,b) =
n
B
e
t
t
^
F~
d + 1
H
2 d - 1
(X,C)
homology, and
(0,0),(2d-2,d-1),
( 2 d - l , d - l ) and (2d,d), hence i n g e n e r a l f o r c u r v e s .
Proof
If
isomorphism
X
i s smooth o f pure dimensin
d,
<
*
>
i s the
a
=
C
:
l , l
HJ(X,2(1)) = H ^ _ ( X Z ( d - l ) )
{X)X
°
d
used i n 5.13. F o r g e n e r a l
subvariety
ment
U C X
o f pure d i m e n s i o n
Y = X X U
tain
(reduced) X
f
t h e r e i s a smooth open
f
d
such t h a t t h e comple-
f
has d i m e n s i o n s t r i c t l y
a commutative
1
l e s s than
diagram
0 -> H ^ _ ( X , Z ( d - U ) - H^
2d-l
'
d-1)) - H ^ ( U Z C l ) )
(8.10.1)
J
/ I a
d
1
v
v
1
0 - *l
(X)
d,d l^ '
H
+
d. We ob-
f
-
H^ (Y Kd-U)
/ I Cl
2
•2 _
(U)
~d -d l<
1
*H
E
+
f
f
+
x
€
x6Y
inducing the dotted
second c l a i m
isomorphism we needed,
see [ J 2 ] 3.1. The
f o l l o w s from t h e diagrams
d+
0 ^ H ( X , C ) / H ( X , Z (d-1) )+F ^H^^j.j^CX,Z (d-1) J - T
2d
2d
(8.10.2)
0
J /
»
x
/ J V
*
9
C
A
(d)
,2
(X)
E
The upper sequence
2
1
2
•E
_
d + 1
(X)/
H^
d - 1
x
•
C -O
(d)
(X Z(d-l))
f
X
d,-dfl< >
i n 8.10.2 i s t h e c a n o n i c a l e x a c t
D e l i g n e homology
sequence
(see, e.g., [ J 2] 1.18 b ) ) , and t h e
f a c t o r i z a t i o n and b i j e c t i v i t y o f t h e l e f t
be seen by r e s t r i c t i n g
do n o t change,
H ^ C X , Z (d-1) )-
/ [
I
E
easily
j f
X
(8.10.3)
for
( d _ 1 )
to
vertical
map can
U, whereby s o u r c e and t a r g e t
and u s i n g 5.13. The v e r t i c a l
induced by t h e Q u i l l e n s p e c t r a l sequence,
map i n 8.10.3 i s
since
1
E _(X) = 0
P/4
for
p > d = dim X.
i f we
define
T
with
T
I t i s s u r j e c t i v e by lemma 8.11
by c o m m u t a t i v i t y
s
2 d - i d-1
"*" P
r
o
v
e
d
l
n
t
J
2
o f 8.10.3, t h e
J
compatibility
r
c f
2 d - i d-1
^ *
and t h e r e m a i n i n g c l a i m s a r e o b v i o u s : t h e
(a,b) =
(0,0), (2d-2,d-l)
and
(2d,d)
remark
cases
amount t o t h e Hodge
c o n j e c t u r e f o r t h e known c a s e s o f d i m e n s i o n
d
d-1
c f . theorem
( L e f s c h e t z ' theorem),
Finally,
(a,b) =
8.11.
X
f o r c u r v e s we
(0,0), (1,0)
Lemma
o f dimension
E
d
d
have
0
(trivial),
(trivial),
7.9.
(X,b) ^ 0 a t most f o r
a
(2,1) by lemma 8.12 below.
and
(compare
2
and
and
3.4.
T h i s shows t h e s u r j e c t i v i t y o f
8.8),
below,
TH
[Sou 3] theoreme 4 i v ) ) F o r a v a r i e t y
over a f i e l d
k
one
has
Co
-d+l^
=
E
d
i
-d+1^
n
t
n
e
^
u
i
l
6.12.5, so t h a t t h e edge morphism
l
e
n
spectral
KJ(X)-*
sequence
_
x
d + 1
i
( )
s
surjective.
P r o o f We
know t h a t a l l d i f f e r e n t i a l s
r
d
from t h e
line
P^q
p + q = 1
t o the l i n e
tensoring with
Q
p + q = 0
( S o u l e ' s r e s u l t used
6.12.4 a ) ) . On t h e o t h e r hand we
T
-
<-d i<*>>
+
-
x
€
;
x t A
where
k
x
i s t h e group
r I 2
after
f o r the proof of
have
-vex)
=
x
e
x
x
(d)
>x
(d)
<
of r o o t s of u n i t y i n a f i e l d
i s the a l g e b r a i c c l o s u r e of
normalization of
vanish for
k
in
X, t h e n t h e commutative
K ( X ) . Let
diagram
L
X
and
be
the
©
X
K (Ic )
1
> KJ(X)
x
> KJ(X)
(d)
1
x
s
•
<l<
*(d)
P
e
c
k
I
> 5,-d l<*>
E
x>
E
> d,-d+lW
+
o b t a i n e d from t h e c o v a r i a n c e o f Q u i l l e n ' s s p e c t r a l
for
sequence
f i n i t e morphisms and i t s c o n t r a v a r i a n c e f o r f l a t mor-
phisms, shows t h a t
8
jx
x€X,,
x
(d)
lies
i n t h e image o f t h e r i g h t
x
v e r t i c a l map, hence t h e d i f f e r e n t i a l s a l s o v a n i s h on t h e t o r s i o n subgroup o f
8.12.
Lemma
weights
Proof
d + 1
(X)
f o r r > 2.
F o r a t w i s t e d Poincare d u a l i t y t h e o r y w i t h
one has
b 2 a - d,
_
FH (X,b)
a
* 0
a t most f o r
a I 2b I 0 and
d = dim X, hence i n t h e t r i a n g l e i n d i c a t e d below:
(compare 5.11) I n view o f 6.5 we must have
2b - a
0
£
2b
2b - a < 0
£
2b - 2 ( a - d)
£
Note, t h a t f o r smooth
8.12.1 i s t r a n s f o r m e d
by u s i n g P o i n c a r e
and
X
a £ d, o r
and
a
I
o f pure dimension
d.
d
t h e diagram
i n t o 5.11.1 v i a (a,b) •—» (2d - a,d - b)
duality.
8.13.
Theorem L e t X
be a v a r i e t y o f dimension
generated
a) There
a r e c a n o n i c a l homomorphisms f o r a l l
*
1
E
field
k, and l e t
a *
finitely
( x ) / a n
d,-d+i
— ' H^
char k
n
- 1
d
(X,Z/a (d
over a
be a prime,
n 2 1
- 1) ) ,
c o m p a t i b l e w i t h t h e t r a n s i t i o n maps f o r d i f f e r e n t n, such
that
t h e i n d u c e d map
*
i s an
:
E
x
1
d,-d i< >~ = ^
m E
+
x
d,-d i( >/
a n
+
- H^ (X Z Cd-I))
1
f
a
isomorphism.
b) The r e s t r i c t i o n
H
is
2d-i
(
x
d
'V ~
1
)
)
H
— * 2d-i
( x
z
( d
' a
~
1 ) ) G ) C
surjective.
c) The i n d u c e d map
K
x
i< >
—
E
x
d,-d+i( >
coincides with
In
particular,
(2d-l,d-l)
— * SS-I < ' V - ^
H
x
d
i
6 f c
T_,, o , •
2a-l,a-1
c o n j e c t u r e 8.5 i s t r u e f o r
and
(a,b) = (0,0),
(2d,d), hence f o r c u r v e s i n g e n e r a l .
Proof T h i s f o l l o w s as i n 8.10:
Choosing
a r e d e f i n e d by t h e commutative e x a c t
U
diagram
as t h e r e , t h e

+
E
—
n
u
j <f
(8.13.1)
0
n
(u, z / a ( d - i ) ) ^ H * _ ( Y , z / a ( d - i ) )
1
j Cl
5.15
U
d,-d l< >
'v€ Y
+
x
t
where we have used t h a t
2
n
H^ (Y Z/Jl (b)) = O
a
Z
<
*(d-1)
for
f
a > 2d - 2 > 2 dim Y ( c f . 8.13.3 below). P a s s i n g t o t h e
inverse l i m i t
over
n
i n t h e t o p row and t h e pro-Q.-com-
p l e t i o n s i n t h e bottom row, we o b t a i n a commutative diagram
0^5_ (X,Z (d-l)
1
I
O-Hj^
G ) c
a
1
r e s
- H^
- : L
1
(X,Z (d-1)
- H*^
a
E
-
d
f
r e s
(U Z
f
/
->
G k
a
X
(8.13.2)| 
r e s
Y
(Y Z
2
G k
a
ft
(d-1))
j Cl
•
\
(d-1)
w i t h e x a c t tows: t h e t o p row i s e x a c t by t h e l e f t - e x a c t n e s s o f
t a k i n g f i x e d modules, t h e middle one by t h e l e f t - e x a c t n e s s o f
passing to the inverse l i m i t ,
and t h e e x a c t n e s s o f t h e bottom
row f o l l o w s from 8.13.1 and t h e f a c t t h a t
©
XE
Z
f i n i t e l y g e n e r a t e d t o r s i o n - f r e e . The b i j e c t i v i t y
f o l l o w s from t h e f a c t t h a t
e t
is
V-U
of
res
y
n
H (Y,Z/a (b)) = 0 for
a
a > 2 dim Y, v i a t h e H o c h s c h i l d - S e r r e s p e c t r a l sequence
(8.13.3)
n
= H (G ,H^(Y,Z/a (b)))
q
P
fiP'
* H^_ (Y Z/a (b) )
n
k
q
f
The isomorphism
(8.13.4)
©
v€Y
Y
Z/a
(a)
n
—
n
> H^(Y,Z/a (a) ) ,
~
a 1 dim Y,
i s well-known;
v i a t h e r e l a t i v e homology sequence
and P o i n c a r e d u a l i t y
that canoincally
i t can e a s i l y be reduced t o t h e statement
Z/a
Puttirg this together
reSy
( c f . 6.1 f ) )
n
——> H^ (Y,Z/Q. )
n
for Y
t
smooth and
connected.
we o b t a i n t h e statements a) and b) , s i n c e
i s surjective,
as we have seen i n t h e p r o o f o f 5.15
( t o g e t h e r w i t h 5.16 b ) ) .
lity
The
first
of
T
c l a i m i n c) f o l l o w s as i n 8.10: by c o m p a t i b i -
and t h e o t h e r maps i n v o l v e d w i t h r e s t r i c t i o n t o
and w i t h P o i n c a r e d u a l i t y ,
x
K (U)
is
5
-
1 5 a
1
o(u)
the f i r s t
Chern c l a s s
U
i t s u f f i c e s t o remark t h a t
1
)—>
n
1
n
H (u,z/a (i) — » H (u,z/a (i))
C
1
«. . F o r t h e f o l l o w i n g ,
k
diagram
8.10.2 i s r e p l a c e d by t h e diagram
0
1
"* c o n t V 2 d
H
(
(8.13.5)
0
H
G
( X
Z
' a
( d
"
1 ) )
j /
( X
2
' a
( d
"
1 , )
"* 2 d - l
H
j *
E
~ " v,V
(d)
x
H
"* 2d-l
X
' d,-d.l( )^
""
E
-
X
( d,-d l( )/®
(d)
+
H
" cont«V
k
'V
l j
1
i
x t x
( G
V **
(d)
9
> ^
(d)
i n d u c e d by Kummer t h e o r y ( c f . 5.16 b ) ; n o t a t i o n s as i n t h e
p r o o f o f 8.11). The e x a c t n e s s o f t h e lower sequence
f o l l o w s from t h e f a c t h a t
2
E,
,
k
1 } )
v e r t i c a l map comes form t h e isomorphisms
1
H
( d
x e x
Hjont^k'^d^'V*- )))
• cont
(d)
Z
' a
/ I
x
i n which t h e l e f t
=
( X
x
k
(X)/ ®
X t X
(d)
i n 8.13.5
isa
finitely
"> °>
g e n e r a t e d , t o r s i o n - f r e e group,
0
—
E
d
-d+l
( X ) /
9
k
x e x
x
(d)
E
—
d
which v i a t h e e x a c t sequence
( U
-d+l >/
9
k
x t u
f o l l o w s from t h e c o r r e s p o n d i n g s t a t e m e n t
proof of
9
x -»
(d)
for
x t Y
(d-l)
U, p r o v e d
i n the
5.15.
Hence the c o m p o s i t i o n
K (X)
• Z
i
- ~
a
is surjective,
#
d + 1
(X) • Z
(0,0)
for
or
(a,b) =
— ~
(2d,d)
Q
H
ft
2 d
_ (X,Z (d-l))
we
again are t r i v i a l
t i o n s o f s i n g u l a r i t i e s , we
i n s t e a d . The case
1
G k
a
o b t a i n the
( 2 d - l , d - l ) . The
o f t h e T a t e c o n j e c t u r e . S i n c e we
directly
a
and by t e n s o r i n g w i t h
o f c o n j e c t u r e 8.5
(a,b) =
Ej _
claim
cases
cases, t h i s
time
do not want t o assume r e s o l u -
cannot a p p l y 7.10
(2d,d)
but
proceed
f o l l o w s from t h e
isomor-
phism
•
X € X
\
(0,0)
we
know the c l a i m f o r
p r o c e e d by
d > 1, choose
UCX
i n d u c t i o n on
d = 0 (above) and
have r e s o l u t i o n o f s i n g u l a r i t i e s and
exact
G k
a
(d)
used above. For
We
> H^(X,Z (d))
7.8
d = 1
b)
as above, and a f f i n e .
for
d = dim
X.
(by 7.10;
we
i = 0). For
Then we
have an
sequence
H
1
C U
f
I
A
( O ) )
-
H (Y,
Q
Z (O)) a
H (X,
Q
Z ( O ) ) -> H (U,Z (0) )
a
I/
H
2 d
ft
q
I/
1
" (U,Z (d)) =0
a
0 = H
2 d
(U,Z (d))
a
1
since
H (U Z
f
reduces
§9.
(J)) = O
Extension
classes,
interesting
non-trivial
weight
9.0.
affine
and
i > d i m U.
and hence t o s m a l l e r
algebraic cycles,
f e a t u r e o f mixed
extensions
F o r example,
Poincare
let
closed
p)
Y
U
This
dimension.
and t h e c a s e
i = 2j - 1
structures i s the existence
- i n particular
those
belonging
to the
filtration.
twisted
and
for
the question to
The
of
f t
z
l e tas b e f o r e
duality
be a c y c l e
subvariety
Z
theory
( H * ( ,*) , H*(
with
of dimension
H
(Z,b) = 0
be a
weights,
let X
i
, supported
on
X
o f t h e same d i m e n s i o n .
( c f . [BO] 7.1.1)
,*))
for
I f we
be a
T-valued
variety,
on a
assume
a > 2 dim Z
a
(by 6.5
in
a) t h i s
then
we
with
2 i
*H
U = X\Z
(Z,i)
valent
case
0
i
+
1
^H
. The c y c l e
f
on
X
2 i
(Z D)
, then
0 —>
and
d e f i n e s an e l e m e n t
thus
the f i e l d
f o r t h e examples
(X,i)
i
+
1
(U,i)
> H
c l ( z ) on
, and i f
f
(9.0.2)
2 i + 1
2
class
z
F
i n § 6),
—>
rise
t o an
E —>
i
Ci
(z)
1 —>
in
Z
2 j
.(Z i)
f
6(c£(z))
extension
0 ,
H ( X
f
I )
i s an e l e m e n t i n
i s homologically
v i a pull-back gives
H
tensoring with
sequence
(X,i)
= Hom^d H
to zero
z
2
true after
6.5, a n d i t i s t r u e
g e t an e x a c t
(9.0.1)
TH
i s always
equi-
= 0 . In
this
,
Ext}(1fH
Let
(X)Q
are
(9.0.3)
while
oV
: Z (X)
i
the c y c l e
(9.0.4)
9.1.
ci
map
H
2 j
where
"
1
2 j
1
V
H
i s now
map
on
a space
the cycles
which
prescription
2 j
"
,
rH
2 i
(X,i)
of pure dimension
corresponds to the
1
(U,j) — ^
H
2 j
(X,j)
d
, then
diagram
— ^
H
2 j
(X,j)
0.
l
(X,j)
Abel-Jacobi
In view
by
2
class
b)
formed
homomorphism
t h e above
U
j = d - i
c£'
(X,i)
t h e n t h e above
(X,i)
i s smooth
Il
H
2 i + 1
>
0
X
(X,j) — >
fundamental
(9.1.2)
If
duality
(9.1.1)
0 —>
i
a)
Poincare
0 —>
1
R rH
: Z (X)ZZ (X)
Remarks
Z^(X)
to zero,
induces a
i
2 i + 1
map
—>
Q
= : R TH
of
equivalent
an A b e l - J a c o b i
1
(X,i))
the subgroup
homologically
defines
by
be
2 i + 1
of
z
i s an
element
c a n be w r i t t e n
3
: Z (X)
of the
(site
the codimension of the c y c l e
1
0
^R TH
formula
...)
H ( T , A ) : = E x t i l ( 1 , A)
T
of
the
, and
the
as
2
j
1
(X,j)
V
H (T,F)
= E x t ^ ( Z,F)
, i t i s sometimes
f o r an
2
TH -Mx,j)
and
object
A
of
f o r a sheaf
suggestive
T
, while
to
F
write
V
RY
reminds
of
taking
If
by
T
the
has
v-th
enough
right
derivative
i n f e c t i v e s , one
injective resolutions;
can
otherwise
of
T(
in
fact
we
have
)
= Horned,
compute
to
take
)
=
H°(T,
Ext^d -)
f
the
Yoneda
Ext
groups.
c)
The
above
can
already
For
9.2.
r e l a t i o n between
be
found
Hodge
Lemma
in
theory
For
[D6]
we
cycles
extensions
(or
torsons)
4.3.
recover
a mixed
and
Hodge
the
classical
structure
H
Abel-Jacobi
there
is a
map.
canonical
isomorphism
(9.2.1)
and
for
Ext J
J
a
J W
( Z,H)
smooth
j
of
coincides
means t h e
Q
9.2.1
or
i s to
by
We
it
also
can
First
preimage
be
note
be
that
assume t h a t
(9.2.3)
side
Ext J
J
H
=O
f
hand
W
by
to
of
( Z,H)
1
"
1
of
W H
Q
in
the
—>•
H
this
from
a
®
Q
,
the
homomorphism
Abel-Jacobi
H
sense
(C)
from
H / T o r (H)
, and
that
the
H
results
of
papers
, and
Then,
i f we
(
E
W
' Q
)
— *
J(
on
for
H
in
® (C
9.2.2) .
Carlson
H'
write
a morphism
H
1
[Ca],
[Bei 2].
E x t ^ ( 2Z,H')
lattice.
MH
"
quotient
[ B e i 1] ,
structure
E x t
2 j
(Here
acts
; similarly
Hodge
obtain
+F H
map.
torsion
9.2.1, we
j
Z(2TT/=T) )
Beilinson's
©
G
Z
CC
in
®
®
J
(X,
classical
= Tor(H)
—>
2 j
over
the
deduce
is a
Q
X
(X,(C)/H
Im(H
for
H
Q
either
obtained
= Ext^k( Z H )
right
H
show how
1
may
taken
replacing
Proof
with
0
(C/W H + F W H
Z
variety
H ""
0
®
Q
2j
Z (X)
W H
W H
proper
(9.2.2)
9.1.2
s
W
Q
of
H)
, hence
J(H)
functors
=
J(H)
for
we
the
by
sending
an
extension
0 —>
to
the
serving
W H
the
0
is
split
the
be
> H'
> E'
W E'
the
of
9.0.3
classical
9.3.
Remarks
exact
E x t
0
>
and
an
MH
(
[EV]
first
left
map
,
map
in
to
(C ^ »
9.2.3
n
=
Q
9.2.3
0
class
(E ,
iso-
any
extension
,
i s an
isomorphism
l o c . c i t . 2d
7.11,
where
E l Zein a n d
Zucker,
and
9.2.3, a n d
[J2]
map
mixed
is
§
; a
the
Abel-Jacobi
which
1,
complete
amounts
where
the
to
8
(A,B)
proof
structures
A
and
B
Ext Jj (A B)
j
w
E x t ^ (A,B)
7
Q
> Horn (A,B)
of
Ext^_
and
a
>
E
M W
(A
®
maps
> TL
©/B
an
®
(B)
.
extension
>
0
can
map
is
the
com-
one
>• 0
has
,
de-
with
shown.
Hodge
i s obvious,
0
push-out
®
by
agreement
isomorphism
iw
pre-
of
i s an
Z,H/W H) :
W H'
the
inverse
in
w
0
®
n
Ext^ (
of
7
morphism
the
W E
is a
z
=0=
compare
^Hom(A B))
j
to
of
then
[Ca].
claim
For
and
sequence
Ext J
The
in
Abel-Jacobi
a)
>
second
combining
v i a a method
the
an
as
second
by
> TL
TL — 0
r
The
w
. The
Q
obtained
position
4).
Hom ^ ( Z , H / W Q H )
same a r g u m e n t s
fined
lemma
0
inverse
and
since
by
For
right
filtration,
j
—>
Q
is a
T
( c f . [Ca]
Q
morphism
S^
Hodge
W E
Q
—>• W E
Q
, where
TL — >
E
0 —* W H
extension
r_os„
of
H
O
> Hpm(AzB) ® A
via
the evaluation
b)
One h a s
map
= O
f
also
is
c l e a r from t h e e x p l i c i t
smooth;
for
i ^ 2 . This
from t h e r i g h t - e x a c t n e s s
now t u r n
to the
t h e n we o b t a i n
> O
B .
f
it
We
follows
> A
Hom(A B) ® A — >
ExtjJj (A B)
w
> E ® A
description
£-adic
of
i s proved i n [Bei
Ext J (A B)
j
w
2];
, which
f
above.
realizations.
First
let X
be
a homomorphism
Z (X) -Êxt I
j
c
0
( Z
H
£ f
2 j
"
1
~
1
(X, Z
j l
X
(
(J)))
(9.4.1)
=
9.4.
Lemma
T h e map a b o v e
z
j
(
x
(see
K e r ( H
(cf.
P'<3 = H P
E
( G
o n f c
cont
(
G
k'
H^
n t
spectral
k
H
( X
'
V
J ) )
i
the discussion
This
will
A
' V
(
Z
'
*
j
)
)
)
map
Z (J))
i
J ) )
. H ^ f X , Z (J))
4
morphism
*
H
Iont
( G
i n [ J 1 ] 6.15).
follow
•
sequence
, ( X , Tl (J)))
^
J
by t h e c y c l e
(X,
[ J 1 ] 3.5 b ) ) , i . e . , b y t h e e d g e
cont
2
i s induced
_>
)
and t h e H o c h s c h i l d - S e r r e
(9.4.1)
H
from t h e next
general
fact.
k'
H 2 J
"
1 (
^'
V
J ) ) )
9.5. Lemma
Let
A
be an a b e l i a n c a t e g o r y
with
enough
injectives
and l e t
0 —>
be
f
an e x a c t
: A — >
and
sequence
8
o f bounded
be a l e f t
d e n o t e by
terms
A" •—>• B" — ^ - C" — >
F
1
exact
P
q
(9.5. 1 )
= R fH (K')
for
below
homology
of
K*
Fix
an
n G %
exact
at the
P + q
K*
q-th
i n
another
filtration
spectral
=>IR
complex
complexes
functor into
the decreasing
by t h e h y p e r c o h o m o l o g y
a bounded
below
0
A .
Let
abelian
induced
category
on t h e l i m i t
sequence
fK-
i n
A
(here
q
H (K*)
i s
the
place) .
and l e t X = K e r d
and
Y = Im 3
i n the long
seguence
Let
Y' = p
1
n
(fY)
1
= a * ( F 3R fB')
n
i n t h e commutative
diagram
F
1
n
F
>nR fB*
n
>
fH (B*)
/T
A
(9.5.3)
n
IR fB'
— • > IR f A
IR f A
\
^ fH
(A )
Ul
\
Y'
with
a*
exact
, and
from
rows.
let
As
:F
the spectral
Then
indicated,
1
n
]R fB*
sequence
let
1
—>
> fY
T
n
R fH ~
1
be
the arrow
(B*)
be
induced
t h e edge
morphii
9.5.1.
the diagram
fY
(9.5.4)
^ W f X
1
Y'
R f (TT)
^
commutes, where
5
F
1
n
3R fB'
i s the connecting
— ^ -
T >f
VH
u
R
morphism
n
-
1
/u'(B")
f o r the exact
quence
(9.5.5 )
Proof
We
by
0 —>
X —>
may
assume
components.
Consider
n
H "
1
that
(C') — >
A"
,
Y —>
B*
the commutative
and
0 .
C"
have
diagram with
injective
exact
rows
se
.n+1
(9.5.6)
d
A
d
n-1
This
induces
n-1
a commutative
C
n-1
diagram with
exact
a"'
n
1
Im d
1
0 - H " (B-)
rows
B
Ul
Ul
(9.5.7)
n
0 + Ker d V i m d
B
B
n 2
B
+ (d* S
+
We
o b t a i n a commutative
film
1
Im a/Im d "
n _ 1
2
B
Ker d / I m d " "
C
C
^ Im a fl Ini d "
1
^ R
1
n
f H
_
1
( B - )
JD
j
(9.5.8)
Il
1
f ( l m a n Im d £ ~ )
1
n
1
R f H " (B*)
JD
1
R f (TT)
V
fY
V
1
-^- R fx
1
B
^ 0
2
diagram f o r the connecting
d!T >
n-1
morphisms
By
the assumption
0 +
1
F
1
g e t a commutative e x a c t
n
->
1
= f (a)"
d
< B~
one
+
1 )
for
we
have seen
1
f(dg)/Im
f(dg~ )
+ f(Ker
, g i v i n g a canonical
n
1
= f (Im a
9.5.4
d£/Im
isomorphism
fl Im d £ ~ ) / f ( a ) (Im f ( d ~
1
n
A
JD
i s obtained
from
1
dg~ )
9.5.8
via factorization,
once
that
is
commutative.
B"
, then
1
n
}R fB-
Let
n
1
n
n - 1
B
#
1
#
(B )
increasing
filtration
by t h e i s o m o r p h i s m
= Im ( l R f T
.B'
n-1
nR "" f(B'/T
n
R fH "
n
]R fB'
n
1
-Û
be t h e c a n o n i c a l
i s given
nR fx
Il
A
F
by
r
1
the diagram
(note t h a t
e
(f Im d J T ) / l m f ( d ~ )
So
1
K
A'
B
F
-> f H ( B ' )
M
f
diagram
n
IR fB*
^
a similar
induced
we
n
0 + f Im d g ' V l m
Y
B*
3R fB*
(9.5.9)
and
on
-B*)
n-1
—JÎR f(H "
n
n
= 0)
1
-
n
*!R fB')
n
s 3R f T
B*
n-i
1
and t h e morphism
( B ) [-n+1])
=
1
n
1
R fH ~ (B')
of
T
^.
,B
n-1
^n-3
: ... B
^n-2
—>- B
v
—>
v
,n-1
Ker d
fcs
_
O
D
v
v
v
v
n
H "
1
(B* ) [ - n + 1 3 : ... O
Diagram
(9.5.10)
i s now
diagram
i n the derived
—>
O
obtained
by
n
H "
1
(B * )
n
applying
3R f
O
to the
category
n-1
T
B*
n-1
Im
obtained
Im
T
d
dg
1
9.6.
-n]
diagram
Im
.n-2
a
(B ) [ - n +
where
[
(B') [ - n + 1 ]
[ - n]
B
B
n
1
from the commutative
n-1 *
H
H
1
—^>
To
stands
obtain
_ n-1
B
/Ker
-,n-1
d
R
f o r a quasiisomorphism.
9.4,
we
apply
9.5
n-1
B
n-1
Im d:
B
.n-1
1]
d
to the exact
n-1
Im d:
B
q.e.d.
triangle
commutative
(9.6.1)
Rg*v*Rv
associated
!
TL (J)
Rg* Z ( j ) — ^
— ^
i
Rg*RP*P* Z
£
j l
(J)
to the diagram
v
y
Z
X
U
=
x\z
Speck
If
Z
( j ) <—>
etale
I*
i s a resolution
s h e a v e s on
[J1]
6.9),
X
9.6.1
(i.e.,
by
injective
injective
i s represented
by
objects
pro-£-systems
in
S(X
the short exact
)
g t
of
, see
sequence
of
complexes
(9.6.2)
!
0 —>
g*v*v I*
—>
g*I*
—>
g ^ y * I '
—>
0
TL
1
whose c o m p o n e n t s
fying
are i n j e c t i v e
noetherian
R e p ( G ^ , TL )
c
, the long
... E ' (X, TL (j))
N
£-adic
1
—^H
N
V/
1
objects
sheaves on
exact
JL
H
J
g t
9.5.2
)
e t
.
Identi-
with
objects i n
then
is
H"(X, 2Z ^( j ) ) — > K (X , % (j) ) — 5 - ...
%
S
the continuous
. (X, Z „ ( j ) )
cont
x,
S(Speck
V
R
while
Speck
sequence
CU, B (J)) — >
in
and
cohomology
groups over
H*
. (U, Z ( J ) )
cont
x,
r t
, and
k
,
their
H
c
o
n
t
z
(X
relative
r
,
cohomology
sequence,
9.6.1
i s obtained
by
R H
applying
G
t C >
cont* k'~*
0
f = I^m H ( G , - )
t o the sequence
n
a s s o c i a t e d long exact cohomology sequence.
the
(i.e.,
Now
0 — >
an e l e m e n t
n
R —>
G
via
the pull-back
H "
(U, Z
1
of the
h
e
t
r
i
a
n
9.6.2) a n d
k
t
l
S
e
taking
extension
S — >
(j) ) — >
0
k
z f. S
l
= Hom_
( Z S)
G
l
morphism
corresponds
nf
to the
image
k
of
z
under the c o n n e c t i n g
k
S
and
by
9.5
this
sequence
(which
sequence
for
9.7.
( 9
7
- *
image
by
Remarks
definition
a)
2Z/£ (j)
H
cont
(
G
k'
R
n
(
and u s u a l
Z
'
r
£
(
j
,
)
,
0
k
v i a the Hochschild-Serre
i n the claimed
from t h i s
X
= E x t I ( 2Z ,R)
G
£
i s the hypercohomology
Similar results
r
c a n be d e d u c e d
1 )
i s obtained
Rg^ Z^ ( j ) )
coefficients
above
> HL.
(G,,R)
cont
K
hold
spectral
way.
of course
for finite
e t a l e cohomology,
s i n c e one
=
spectral
and
the
result
has
1
n
r
H (G ,H (X, 2Z/* (j)))
k
r
(because
1
1
I^m
r
1
n
r
H ° ( G , H ( X , Z/£ (j))
k
n
r
H " ( G , H ( X , Z/£ (j)))
is infinite
k
this
is
formula
cohomology
of
H
1
passing
9.5,
limit).
H
= Ijm H
r
, c f . [J1 ] 2 . 1 ) . I f
(e.g.,
G
cont* k'~^
sequence
1
limit
f o r number
'
1
>
f o r the
r
(X, 2 Z / £ ( j ) )
F o r t h e same r e a s o n ,
to the inverse
so I have
for
spectral
(X, Z ( j ) )
£
the inverse
with
of
becomes f a l s e
no H o c h s c h i l d - S e r r e
= 0
one
9.4.1.
'
a
n
d
t
"naive"
n
e
r
e
£-adic
(by t h e n o n - e x a c t n e s s
i n the various
p r e f e r r e d t o work w i t h
1
fields),
has
t o be
exact
careful
sequences
b)
For possibly
for
homology,
(9.7.2)
and
the
E?^ = P
H
finite
Q
n
t
a
similar
o
n
t
<G ,H
k
2 i+ 1
(X,
spectral
r
Z/£ (i)
defined
as h y p e r c o h o m o l o g y
- H°°^(X, Z,(i))
, etale
dualizing
complex
is
as above,
proved
Rg*
is
TL/1
(- i)
defined
by
H
(X,Rg*
TLfl
the
sequence
H
.
R
TLfl (I))
(X,
r
theory
i n 9.6
; the spectral
homology
1
duality
replacing
map
sequence"
q
coefficients
holds
£
<G ,H_ (X, I , ( i ) > >
k
i n 9.4
Z (i))>
—a
is
s t a t e m e n t as
of the Abel-Jacobi
Hj
0
X
"Hochschild-Serre
(9.7.3)
For
i n terms
.(X)
Z
singular
( - i) )
of the
( c f . 6.7).
sheaf
9.7.3
twisted
Everything
Z^ (j)
by
(for finite
coefficients)
by
d e f i n i t i o n the hypercohomology s p e c t r a l sequence f o r
r
O
TL/l ( - i )
and ( t h e d e r i v a t i v e s o f ) t h e f u n c t o r
H (Gj '"")
t
i
i
one u s e s t h e r e l a t i o n s
Rg*v*Rv*Rg" = R(g )*Rg^*
and
1
R
R
g* g*
and
c
1
Rg*Ry*y*Rg
For
!
= R(g )*Rg2
2
TL ^ - c o e f f i c i e n t s o n e
h
in
D
•
(S(X
complexes
e t
^ P
)
)
Rg*
uses
a certain
complex
b
, whose components
TL/1
( - i)
above,
in
and
D
Rg"
Z^(-i)
))
are the
r
(S (X
fifc
, TL/1
i t s continuous
etale
hyper-
a cycle
map
i
cohomology.
Cl
= Z (X)
1
we
refer
c)
The
G Z
H I IX,
Hodge
of
has
fits
shown
i s associated
n t
(X,Rf
and
!
J
4
I - D I
paper [ J 5 ] .
into
that
a
Z^ ( - i )
- H-2
1
to a future
also
Rg*
Z U))
2
theory
[ B e i 2]
there
nt
0
—
the reader
Beilinson
j
For the existence
t h e scheme o f
t o each v a r i e t y
complex
9.5.
X
Namely,
over
(C
and
i
b
RT(X, Z ( j ) )
of
mixed
is
just
Hodge
s t r u c t u r e s such
the singular
structures
(9.7.4)
and such
that
cohomology
that the
H
r
1
f
o
N
the exact
1
r
i =
coming
2
N
- H£(X,Z(J))
spectral
r ^ H ( X , TL(J))
sequence
( c f . 9.3 b ) ) c o i n c i d e w i t h
from
the usual d e f i n i t i o n
- O
and t h e v a n i s h i n g
the exact se-
of Deligne
cohomology
(*) a n d [ E V ] 2.10 a)) , v i a t h e i s o m o r p h i s m s
9.2.1 a n d
relation
(9.7.6)
T H = Hom
H
a mixed
There
Hodge
M H
( TL(O) ,H) s W H
Q
structure
RT
2
are versions with
b
D (MH)
= r^H^(X,
(X, TL(J) )
> RT (X
n F ° H 0 CC
H .
support,
H o d g e v e r s i o n o f 9.1.1 i s i n d u c e d
in
Z ( j ) ) f o r n ^ 2j .
sequences
t h e hypercohomology
1.6
of
cohomology
H
( [ B e i 1]
for
case
1
quences
the
the Deligne
O + R r H ~ ( X , TL(J))
T ^
Hodge
D
i n this
from
i t s mixed
n
D (MH)
with
(9.7.5)
with
H
( TL (o) ,RT (X , TL ( j ) ) )
w
Moreover
n - t h homology
(X, TL) ( j )
R r R T (X, S ( j ) ) : = RHom .
coincides
of
n
i t s
n - t h homology
"
coming
€ D (MH)
f
and by t h e c y c l e
Z ( j ) ) . The l a t t e r
by an e x a c t
TL(J))
class
f o r homology
e t c . , andt h e
triangle
> RT ( U Z ( J )
f
of
z
in
defines a cycle
H^
z
map
(X, ZZ ( j ) )
Cl
whose c o m p o s i t i o n
mology
discussed
commutative
O
d
with
Hp (X, Z(J))
e
before.
i s t h e c y c l e map
By
9.5
compare
2 j
1
We
1
hold
cont
( X
map
f
into
a
j
f o r homology
2 j
-> T H
w
If
X
(X, Z(J))
i s no
( l o c . c i t . and
(compare
1
Cl
Z(J))
[ B e i 2] 5.2.
with
[ B e i 4]
G^-action
longer
O
,
smooth,
similar
[J2]).
3.0):
H
1
(X(CC) , Z ( j ) ) w i t h
theories
J)
H
' V >
P
( X
'
a
^ ) )
connection
short
Hochschild-Serre
spectral
fits
theories
(X, 2 L ^ ( j ) )
absolute
H
1.9,
•+ HpiX
get the analogy
geometric
H
( X , Z(J))
[Bei1]
statements
s i n g u l a r coho-
Z (X)/Z (X)
Cl
1
w
j
+
j
Cl
1
the Abel-Jacobi
Z (X)
Z (X),
O + R T H
into
diagram
j
+
j
Z (X)
:
v
sequence
9.4.1
exact
sequence
9.7.5
Hodge s t r u c t u r e
with
1
the difference that
R T.
Vj
k
=
. (G, , -)
conu.
1
vanish
to
for
this
i ^ 2 , e.g.,i f k
point
later
Now we w a n t
and
on
with
setting
weights
i s a global field.
of the previous
o f an
(F
F-Iinear
a field
does n o t
We s h a l l
return
i n 11.4 c ) o r 1 2 . 1 9 ) .
t o show t h e c o n n e c t i o n
the conjectures
general
(e.g.,
i n general
between
chapters.
twisted
extension
We g o b a c k
Poincare
of characteristic
classes
t o the
duality
zero)
theory
and t r a n s f o r -
mations
r
=
r'
as
c
h
: H
i , j
= T
f
a
M,Z
( X
'
Q ( j ) )
r
-+
z
: H^(X,Q(b))
(
X
'
j
)
TH (X,b)
a
i n 8.4, a n d we s u p p o s e t h a t
map
H
these
a r e compatible
with
the cycle
a s i n 8.4.2.
9.8.
Lemma
Let
X
be smooth
and proper
let
Z c X
be o f c o d i m e n s i o n
^ j , then
o f pure
there
dimension
d
and
i s a commutative
diagram
TH
2 j
1
(U,j)
rH
2 j
(x,j)
ci
(9.8.1)
1
R rH
(3)
CH^(X)
(1)
^
0
0
2 j
1
" (X,j)
ci '
® F
CH
j
(X)
Q
® F
2D-1
ch
(2)
ch
2j,j
2 j
H "
where
1
(U (DCj) ) ® F -> H
f
c i ' i s induced
2j,j
2 J
z
(X,Q(j) )
Q
® F
by t h e A b e l - J a c o b i
H
2 j
(X,(D(j) )
map 9.0.3,
Q
® F ,
A^
r
and
TH ( X , j ) )
f o rthe various
t h e t o p e x a c t sequence
sequence
(Note
- 1)
ch
associated
2j
0 — >
H ""
that
TH
2 j
1
i s part
to the short
(X,j) — >
1
" (X,j)
H
2 j
= 0
"
o f the long
exact
1
groups
2 j
H
f
H
i n t h e diagram,
exact
Ext^(1,-)-
sequence
(U J)
since
A
2 j
(X,j)
1
" (X j)
— ^
Q
0 .
i s pure
f
o f weight
.
Proof
T h e c o m m u t a t i v i t y o f (1) f o l l o w s
with the relative
= ch
by
J
= K e r (A — >
.
0
by a s s u m p t i o n .
definition
= CH
.(Z)
n
Hence
© F =
Similarly,
/ M
the compatibility of
ci ° ^ 2 j j
i s commutative,
map 9.0.3 o n
(explicit
n
c
since
CH^(X)
n
and
6 o ci
© F
description
of the con-
(
6
i n the long
(3) i s c o m m u t a t i v e
sequence
(2)
( Q
» ) 0 F
x€X 3»nZ
°
morphism
factorizes
sequences,
i s the Abel-Jacobi
0
necting
cohomology
from
once
through r a t i o n a l
we h a v e
seen
that
the Abel-Jacobi
e q u i v a l e n c e . B u t by t h e b o t t o m
2j
a n d (2) t h e i m a g e o f
(
®/-\
Z) © F
X€X J'DZ
e x a c t E x t - s e q u e n c e , compare 3.$ a b o v e )
H *"
i s t h e subgroup
1
(U ,(D ( j ) ) © F
map
exact
in
g e n e r a t e d by c y c l e s
linearly
v
equivalent
Since
Z
cycles
in
9.9.
to zero,
i s arbitrary,
Z^(X)
Lemma
the
examples 8 . 3 ) .
q)
For irreducible
"
1
— *
shows t h e f a c t o r i z a t i o n
The m i d d l e v e r t i c a l
assumption
Z
this
H
for a l l
map i n 9.8.1 i s a n i s o m o r p h i s m
p) a n d t h e f o l l o w i n g
Z
6 ° ci .
.
n
the
n
a n d b y (2) i t i s mapped t o z e r o u n d e r
e
2e^ ' ^
Z
i
s
a
one (which
o f dimension
n
i
s
o
m
o
r
P
n
i
s
m
'
e
a
n
d
i s satisfied for
t h e fundamental
r
(1) = F .
class
under
Proof
We
have
TH
2 j
(X J)
= TH
f
..(Z,d- j) =
2(d- )
©
0
D
-^-
(
xEX
reduces
j
©
TL) % F = C H ( X )
( j )
® F
(by u s i n g
x
e
z
6.1
F
(
Q
)
f ) a n d p)
one
OZ
t o the case
that
Z
i s disjoint
union
of i t s irreducible
components).
9.10.
let
Corollary
Z cz X
be o f p u r e
c h
is
Assume
2
surjective
j
_
n
. :H
on
injective
: CH
j
and q ) . L e t
codimension
"
1
j
X
be s m o o t h
, and l e t
2j
(U ,Q(J) ) ® F — >
i f and o n l y
C^
is
2 j
p)
(X)
1
® F
on t h e subgroup
> R TH
generated
2 j
proper,
= X-Z
. Then
1
TH "* ( U J )
f
i f the Abel-Jacobi
0
U
and
"
1
map
(X j)
f
by t h e c y c l e s
with
support
Z .
Proof
This
jectivity
follows immediately
of the middle
from
vertical
diagram
map
9.8.1
(without
and t h e b i -
restriction
X
is
connected).
9.11. C o r o l l a r y
F o r smooth v a r i e t i e s
U
over
(E
the Chern
character
C
h
2j-1 j
: H
f
is
i n general
jecture
Proof
J
M "
1
( U
(
'® J>>
not s u r j e c t i v e
of Beilinson,
The complex
— ^
for
T (H
h
j ^ 2
2 j
"
1
(U Q(J)))
f
(This
disproves
a
con-
[ B e i 2] C o n j e c t u r e 6 ) .
A b e l - J a c o b i map
i s known
t o be
injective for
j
=
but
1
(this
Mumford
gives
h a s shown
codimension
and
be
proper
j ^ 2
(9.11.1)
For
AQ(X) e
k
= CC
i t i s i n general
CHQ(X)
1
2 j
= H
j-th
intermediate
of
the Abel-Jacobi
(9.11.2)
p
(X)
(in
T(X)
O
=>
: For a
k
let
smooth
Alb(X)
1
T(X)
j
for
®
of c y c l e s of degree
(X, Z ( J ) )+ F H
Jacobian.
and
,
zero.
J ^ ( X ) , where
(X,CC)/H ~
map,
>
([Mu])
a field
Alb(X)(k))
i s the subgroup
J (X)
the
over
A l b ( X ) (CC)
"
not i n j e c t i v e f o r
to torsion
d
—>
Q
is
up
Q-rationally),
and l e t
= Ker(A (X)
2 j
for 5.13, at least
of dimension
one has
J
that
variety
T(X)
proof
, n o t even
variety
the Albanese
where
a new
(B *
d
Hence
= 2
2 j
"
1
(X,(C)
T(X)
Mumford
i s the kernel
shows
O
9
fact,
is
the geometric
by
Roitman
i s huge
[Ro 1 ]
(9.11.3)
H°(X,^)
*
should
1
Mumford s
fined
fields
and
i t i s expected
number
fields.
conjectured
fields
O
k
X
p
that
>
5 . 2 0 be
counterexample
of higher
the following
. This
( X ) = dim.H
g
(C
was
2
T(X) ®
true?
transcendence
The
Q
and B e i l i n s o n have
)
f
x
and
zero:
*
O
.
answer i s t h e
("generic
i s completely
([Bei4] 5 . 6 ) :
d
c y c l e s which
degree
(X O
generalized
for arbitrary
involves
the s i t u a t i o n
Namely, B l o c h
p
of characteristic
f o r some
conjecture
following.
over
where
to the following result
uncountable
why
case),
genus o f t h e s u r f a c e
arbitrary
Now
i n this
a r e decycles")
different
independently
over
9.12.
Conjecture
complex
For
Abel-Jacobi
a
injective
up
to
proper
variety
X
over
Q
the
map
CH
is
smooth,
j
(X)
>
0
torsion
J
j
(XX CC)
Q
1
(here
CH-
(X)
i s the
Chow g r o u p
over
Q) .
Since
and
Z
i n the
are
jecture
d e f i n e d over
5.20
Actually,
for
Bloch
algebraically
be
no
Let
(X)
k
>
=
Q
2j - 1
fields
the
i s defined
zero,
5.21
consider
but
from
i t i s the
Q
i f
see
the
our
9.18
con-
below.
subgroups
approach
same t o
X
of
cycles
there
should
consider
.
£-adic
theory
Q
converse
instead of
Hodge
9.12
over
case.
and
Bloch
has
remarked
L e f s c h e t z ' theorem
on
that
1-
equivalence
2
H (X CC)
F
NS(X)
only
c o n s i d e r the
situation
U
, for a
Beilinson
A l s o , by
9.10
, conjecture
e q u i v a l e n t to
now
0
of
i s the
* H
1 , 1
«
NS(X)
Neron-Severi
following generalization of
®
2
CC *
group
H (X,CC)
of
Mumford's
X
, and
theorem
to
he
has
proved
arbitrary
fields
.
9.13
Theorem
surface
I
us
give
where
the
and
number
Mumford's
cycles
p
i
difference.
arbitrary
in
situation
*
over
char
T(Xx K)
v
( [ B l 1]
k
®
a
field
be
Q) *
a
0
1.24)
k
prime.
.
Let
, with
If
X
be
function
NS(X)
®
(D
£
a connected,
field
* H
2
f c
K
smooth,
, and
(X ,Q) (1) )
£
let
,
then
proper
This
the
i s not stated
p r o o f . We
£-adic
now i n v e s t i g a t e
Lemma
T(X)
lies
Let
X
i n l o c . c i t . , but follows
the relations
be smooth
i n the kernel
A (X)
= CH (X)
0
* char(k)
0
injective,
Proof
and p r o p e r
of the
between
T(X)
- £ * ^
*4
0 TL
i
observe
—+
H
n
t
that
(G ,H
from
and t h e
N
T
d
1
- (X,
2
(G ,H
D
K
"
= K e r cl
, then
£
(X,
,
the natural
map
Z (d)))
£
0 ZS
£
D
= CH (X)
q
Z (d)))
then
2
1
£
A (X)
g i v e n by t h e &-adic
2
generated,
O
d
A b e l - J a c o b i map
k
1
T ( X )® 2
of dimension
&-adic
i s finitely
so t h a t
First
eguivalence
0
. I f k
A l b ( X ) (k)
is
form
A b e l - J a c o b i map.
9.14.
I
i n this
N
f o rthe homological
cohomology, by t h e commutative
diagram
H
2 d
(X,Z (d))
£
D
CH (X)
where
t r
i s the canonical
groups
A (X x
theorem
o f Roitman
q
commutative
k
k)
and
trace
map. I t i s w e l l - k n o w n
Alb(X)(k)
( [ R o 2]
3.1
are d i v i s i b l e ,
) the f i r s t
that the
and by a
vertical
map
i n the
diagram
->
A
q
(X)
( X )
-> O
A l b ( X ) (k)
> O
-> A
Q
(9.14.1)
O
>
n
A l b ( X ) (k)
> A l b ( X ) (k) - — >
is
an
to
isomorphism.
Pic^(X)
, one
Alb(X)(Jc)
by
Poincare
Since
has
Alb(X)
dual
abelian
variety
isomorphisms
1
s Hom(H (X,y
duality,
i s the
and
),y
one
)
easily
s H
2 d
shows
1
"
( X ,Z/£
(see
n
(d) )
[J5])
that
the
map
Q
A (X)
> Alb(X)(K)
Q
= Alb(X)(Ic)
k
>
Iim
H
1
(G ,H
k
' ~
induced
shows
by
the
the
Kummer
statement
Alb(X) ( k )
A
:=
for
Iim
K
If
k
is finitely
generated
and
is
up
(9.12)
the
Alb(X) ( k ) / S
generated,
by
® Z
^-divisible
to
on
shows
global
that
torsion,
the
other
(implying
following
9.15.
n
<=
an
>
H
c
(X,Z/£
n
(d)))
I!
c o i n c i d e s with
gives
then
the
c l
f
. This
injection
o
n
(G
t
k
,H
2 d
"
1
(X ,Z
Alb(X)(k)
is a
(d) ) )
%
finitely
generalized Mordell-Weil
-^->
£
by
> Alb(X)(k)
while,
and
1
Alb(X)(k)
1
Roitman s
A
. We
theorem,
remark
in
theorem,
that
T(X)
particular,
k
This
even
9,14.1
"
n
Alb(X)(k)
uniquely
G
T(X)
a b e l i a n group
hence
AQ(X)
sequences
2 d
the
k
£-adic
for fields
hand,
T(X)
0
conjecture
Conjecture
field
is surjective
Let
, then
the
Q
Abel-Jacobi
k
of
for a
for a
number
for
be
smooth
I
* char
tensoring with
map
higher
conjecture
= O
X
after
of
Bloch
and
k
and
X/k
k
and
proper
over
the
map
£
be n o n - i n j e c t i v e ,
transcendence
variety
field
can
Z
degree,
Beilinson
) would
j
a
= d
imply
:
finite
or
ci'
9 QJ
: CH^(X)
4
» ^
0
— >
i n d u c e d by the A b e l - J a c o b i map
H^
2
n f c
9.4.1
fc
i s injective.
B e f o r e we d i s c u s s some examples, we
s h i p between
closely.
j
c o n j e c t u r e 9.15
I f i n 9.1
i n v e s t i g a t e the r e l a t i o n -
and the s u r j e c t i v i t y o f
we drop the assumption t h a t
, we g e t the f o l l o w i n g r e f i n e d
We
z e r o ) , which
). Let
.
1
more
i s of c o d i m e n s i o n
X
condition
be a smooth, p r o p e r v a r i e t y of pure dimension
1
1
N H (X J)
f
r
d
r
. Let
> H (X J)) ,
f
N CH (X) = Im(CH _ (Z)
U = X^Z
d ,let
1
= IiMH (X J)
r
2
( F a f i e l d of c h a r a c -
r e c e i v e s Chern c l a s s e s and s a t i s f i e s
be a c l o s e d subscheme and l e t
2
9
p l a c e o u r s e l v e s a g a i n i n the s i t u a t i o n o f a g e n e r a l weighted
teristic
Z c X
Z
Ch .
version.
F-Iinear twisted Poincare d u a l i t y theory
p
1
(G , H ^ " (X ,Q, (j) ) )
r
> C H ^ ( X ) = CH (X))
n
,
r
cl
where now
z
CH
(Z)
= Ker(CH
n
of c o d i m e n s i o n
commutative
O —>
H
2 j
1
on
-> H "
1
(X,j)
Z
2 j
—>
with support
-> T H
2 j
1
2 j
-> H "
Z
n
c Z
we have a
1
H (X J)
( X ^ Z , j ) ->
H (X J)
2 j
C
2 j
—>
2 j
H (X,j)
2 j
-> H ( X J )
f
f
1
-> R T ( H
0
2 j
"
H
(X) / C H _ . ( Z J )
Q
d
1
,
Z
(X j ) / N )
f
cl'
cl
j
(
(X J)
f
2j-1,j
(3 ) _ >
f
0
-> T H
f
( U )
1
2 j
—>
e x a c t diagram
(U J)
ch
1
H " (U,j)
we o b t a i n a commutative
K
X
TH~2nT(Z,m)) . S i n c e f o r a c y c l e
diagram
(X,j)/N
2 j
j
(Z) -=—>
0
j
ffl
Z
> (CH (X) /N )
Q
ffl
9.16.
Corollary
(9.16.1)
is
c l ®
still
c l ' ®
injective
on
F
If
on
ch
9
.
is surjective
(CH (X) ZCH ^ (Z) )
0
d
j
® F
0
and
>
TH
2 j
(X J)
f
n
then
F
j
:
the
Z
(CH (X) /N )
®
Q
subgroup
F
1
>
generated
by
R T(H
the
2 j
1
Z
~ (X,j)/N )
cycles
supported
Z .
b)
Conversely,
on
Z
and
For
i f 9.16.2
9.16.1
the
following
v
is injective
is surjective,
1
N H (X J)
we
observe
=
f
ZcX
is
the
coniveau
v
j
{z
£ CH-^(X) 13
supported
is
can
the
global
a)
map
Let
Z
state
a
Z e X
of
k
be
zero
by
version
X
and
let
a
closed
be
I
supported
„
is
0
that
by
codimension
described
Let
subgroup
ch .
. ® F
surjective.
definition
2
1
N H (X J)
f
v
2
U
codimension
of
maps t o
refined
Conjecture
field
X
and
filtration
now
9.17.
on
Z c
then
the
while
=
ZeX
on
U
codimension
of
filtration,
N CH (X)
=
. ® F
j
:
injective,
(9.16.2)
is
a)
in
v
Ogus
conjecture
smooth
* char
such
that
z
is
T H ^ ^ ^ ^ (Z,d-j)}
B l o c h and
of
j
N CH (X)
v
and
k
subscheme,
be
proper
a
then
in
[BO]
(7.2) .
We
a
finite
or
j
Z O
9.15.
over
prime.
for a l l
the
j
(9.17.1)
induced
b)
Z
(CH (X) /N )
Q
v
(9.17.2)
j ^ O
Lemma
Conjecture
c)
I f Tate's
X, t h e n
of
X
d)
If
i fchar
cycles
9.15
for
Proof
1
~
that
finite
j
:=
space.
for
subquotient
every
mutative
proper
$£-^H
over
f o r t h e £-adic
1
diagram
H
j
1
2
(X,Q (j))/N )
£
2j
1
"
R
(X ,Q ( j )) )
£
2 j
Since
~
1
c
and open
factor
o
n
t
subvarieties
j
and
(e.g.,
of H
conjecture
X x Z
r
2 j
(G ,H
2
k
j
~
1
X .
i f char
"
1
(X,$ )
£
A i s true
then
conjecture
and
a l lZ e X
£
j
subscheme o f
9.17 a) f o r X
(X, $ ( j ) )
2 i-1
f o rcycles of
J, -> Z
on
9.17 b ) .
9.15.
closed
cover
cohomology
H
V
conjecture
9.17 a ; f o r
9.17 a) i m p l i e s
to the limit
2 j
(G , Gr^H
t
i = 2j - 1
conjecture
y
H
2
k
maps
conjecture
d = dim X
Q
$ -vector
£
n
k = 0 ) , and i f T a t e ' s
(CH ( X ) / N ) ®
since
o
(X,<& ) i s a d i r e c t
a) C o n j e c t u r e
by p a s s i n g
c
to conjecture
implies
(G ,H
( s e e 7.13) i s t r u e
5.19 f o r
o f dimension
X
(9.18.3)
note
2 j
t
9.17 a) i m p l i e s
B
Z c X has a good
2
n
injective
(possibly singular)
conjecture
o
O < v i j .
conjecture
d - j on
c
i s injective.
> H
£
9.17 b) i m p l i e s
i s equivalent
(e.g.,
* d}
and a l l
k = 0) , i f N H
for
j
a) C o n j e c t u r e
b)
dimension
map
map i n d u c e s
Gr CH (X)
a l l
9.18.
> H
by t h e A b e l - J a c o b i
The A b e l - J a c o b i
for
®
Z .
of
y
(X, <£ ( j ) ) / N )
£
o f codimension
this
limit
y -
i s actually
G
i s pure o f weight
o f i t ; hence
-1, V
k
=0
t h e t o p r o w i n t h e com-
° - H c o n t ^ '
G
2
^
J
"
1
) - c o n t ^ ' H
2
J
1
-
/
V
+
1
) ^ H ;
N
o
n
( G
t
k
, H ^ - V N - ,
(9.18.4)
v
® CD
j
O^Gr CH (X)
j
•(CH (X) /N
£
0
V + 1
Q
)
j
® <&
® (fe^-O
V
>(CH (X) /N )
£
Q
v
is
exact,
1.5),
and
b)
Using
v
by
c)
Tate's
d
and
t h e map
injective
9.18.4, o n e
conjecture
Z, we
Z
which
B
for
for
by
(2j -
o f 9.17.1.
a r e done
Z
and
the b i j e c t i v i t y
£
injectivity
for
(F = $ ) . T h u s
5.19
is injective
v +
[ B i 3]
1.
o f 9.18.3
for a l l
v .
assures
realization
on
shows
conjecture
just
i s w e l l - d e f i n e d (compare
i f 9.18.3
i n d u c t i o n on
- j
a
by
9.16
1,j) and
Let
9.16.
a e
cycles
of
o f 9.16.1
we
only
a l l
CH^(X)
Otherwise
f o r the
have
let
a
£-adic
t o show
U c X
. I f
q
dimension
implies the
a
is
supported
be
supported
1
Then
i s of codimension
j
in
X, a n d l e t Z
= Z U
Z i
Z'
i
o b v i o u s l y we h a v e
N CH (X) = N
C H ( X ) , a n d we may
9.16
to
q
j
d)
If
2
N H
=
I
m
2 j
that
Z'
and
Z-Ix
U'
has
= X
Z'
the good
Z
j
on
.
Q
apply
.
proper
cover
Z 1
Z, we
have
1
' (X,$ (j))
£
(
H
2(d-j)+1
(Z'V ~"
d
the
assumptions
the
target
this
\ if
of
+
+
j ) )
— ! L J
i s a direct
( f o r char
polarizations
for absolute
are
sections
s
The
composition
formula
- be
H
^ 2(d-j)+1
and
t
ts
can
interpreted
d
of both
k = 0 the l a t t e r
Hodge
as
factor
^'V -J)) '
cycles
as
a n d
b y
the source
and
follows v i a
i n § 7 ) . Thus
there
indicated,
- v i a Poincare
a s an e l e m e n t i n
duality
and
Kunneth
H
By
Tate's
of
dimension
Z
i t just
it
also
posing
^
d
exact
(
k '
G
so
j
N
Z
9.19.
we
W'
^ C H
we d o n ' t
with
vertical
tions,
class
of a cycle
. As a c o r r e s p o n d e n c e
ts
i n t h e £-adic
g e t t h e maps
w
and
from
X
W
to
cohomology, b u t
o f t h e c o r r e s p o n d i n g Chow
j
(
0
H
2
J
" ^
1
H
groups.
w'
and
c o n t
Com-
i n t h e com-
from
'
G
k
H
2
Z
" /
J
>(CH ( X ) / N C H
1
j
Q
w'
i s compatible
w'
Z
with
of
)
- -
0
w .
i t s com-
s i m i l a r r e s u l t holds
N
(X) )® ( D ^ ^ O
i s a section,
s u f f i c e s t o deduce
map
(
j
®
know w h e t h e r
w
course,a
k '
G
( X )
i s a section
patibility
Of
X
) ^ C n t
(X) QQ
w
Although
right
Z
be t h e c y c l e
diagram
1
that
.
A
t h e map
I*TT*
.
z
k
on
induces
with
O-N CH
x X,<6 (d))
c o n j e c t u r e i t would
c o n t
H
(Z
g i v e s u s a map
mutative
0
2 d
of the
c l ' ® fl> .
£
f o r t h e Hodge
realiza-
a n d we g e t
Corollary
mension
d
over
Let
X
be a smooth a n d p r o p e r
a number f i e l d .
The f o l l o w i n g
variety
of d i -
statements are
equivalent.
i)
Conjecture
5.19
(resp.
and
a l l open
ii)
T h e A b e l - J a c o b i map
tive
iii)
subvarieties
f o r zero cycles
T(X)
« 4 = 0 .
on
5.20) i s t r u e
U
of
( i , j )=
(2d - 1,d)
X .
c l ' 6 (t)
£
X .
for
(resp.
c l ' ® (D)
i s injec-
Moreover,
open
subvarieties
jective
Proof
true
j
conjecture
of
X
. We
D
CH (X)
cycles,
have
d
t o show
1
d
Q
1. T h e n we
d ) , s i n c e we
i)
conclude
i s true
Z
o f t h e A b e l - J a c o b i map
conjecture
for
O
9.20
Examples
in
chapter
nerated
field
to the
Hodge
a) f o r
curve,
cases
N
E
T(X)
^
A
S
T
C
I
O
and
(the proof
of a f i n i t e
field
be-
i s the
A
c) and t h e f a c t
plus
and the
For the equivalence
Q(X) * ^
o f 9.18
analogue of
1
^
M
fol-
that the
(trivial
o f ) 7.9).
k
k
and l e t
Kummer
variety
0
A = Pic (X)
over
i s discussed
a finitely
be t h e P i c a r d
sequences
I
0 —>
for
are
case
a
be a smooth a n d p r o p e r
. The e x a c t
(& * c h a r
B)
reduces
f o r codimension
The c a s e
i n -
12.
X
X
a)
is
i i ) . Since
t o be a smooth
on
and L e f s c h e t z ' theorem
b) L e t
of
i s true
® (I)
conjecture
that i n both
t h e Hodge a n a l o g u e
c l
and a l l
for divisors.
kernel
1
X .
by t h e a b s o l u t e
i i ) and i i i ) o b s e r v e
Hodge
i f
(3,2)
i s e q u i v a l e n t t o 9.17
one e a s i l y
tween
from
2 on
that this
can choose
(i,j)=
and t h e T a t e
hence
= N " CH (X) ,
0
Z =
lows
for
i f and o n l y
The Hodge c o n j e c t u r e
= d
9.18
U
i s true
f o r c y c l e s o f codimension
f o r zero
dim
5.20
A —» A
k) i n d u c e
A(k)
— ^
a l l
n € ]N
A—»0
the exact
1
A(k)-^H (k,
and hence
A)
cohomology
>
H
the exact
1
sequences
(k,A)
>0
sequence
ge-
variety
a
0—>lim
n
where
T B
A(k)/£ -^H
=
(G
c o n t
k /
T A)--^ T H
£
1
(k,A)^ 0
£
,
Ijm
B
i s the T a t e module o f a group
B . By t h e
n
£
o f M o r d e l l - W e i l A(k)
i s a f i n i t e l y generated abelian
£
n
n
theorem
group,
hence
A(k)
TZ^
0
+ Iim A(k)/£
n
canonical
isomorphism
isomorphism
of
exact
0
A(k)
T A
=
and
v i a these
Abel-Jacobi
Cl'
see
©
ZZ
i
0—>H
identifications,
(X ZZ,
: CH
1
(X)
0
©
the
ZZ
„
l<
z—+
A(k)
> 0
6 agrees
(U,ZZ (1))
£
G k
0
=
i
Pic (X) ©
latter
(compare
with
,
the
&-adic
2Z —> H
£
is injective,
c
o
n
and
t
(G^ ,H
the
1
(X,ffi
(1)))
£
commutative
9.8.1)
—>H|(X,ZZ
£
k
(D)
H
> cont
0
( G
k'
H l
{
X
f
2
Z
l
j c l '€>
O-^(O(U) A )
x
x
1
© ffi ->( ©
zz ) —>CH (X)
£
i
xex
Z c X
b) , a t
c)
a
(1)) v i a t h e
f
'1 ,1
for
have
map
diagram
1
1
H
we
sequences
1 A(k)
[ J 5 ] . Hence
exact
. Furthermore
Pic(X) =
T
If
»
n
Let
closed
least
X
projective)
be
a rational
over
torsion-free,
case
the
that
®
0
Z,
U
variety
global
finitely
Q
s
£
,
*nz
U = X \
and
i n the
( 1
field
gives
has
a
another proof of
smooth
compactification.
of dimension
d
k
(X)
. Then
generated abelian
Pic
group
5.15
and
(smooth,
is a
hence
H
( 1 ) ) = 0 . Byduality
(X,(D/E
tures
9 . 1 5 and 9 . 1 7 a r e t r u e
A (X)
= CH (X)
D
q
Thelene
[C] i n f a c t
possibly
for
i s a torsion
0
proved
for p-torsion
H
£
j = d
group.
that
f o r the case
X
)
conjec-
i f and o n l y i f
For
Q (
a
= 0 , so
(X,ffi (d))
i
2
d =
s
that
Colliot-
finite,
char
except
k = p > O .
2
d)
in
X
over
Birch
[ B i 3 ] Bloch
a number
where
and
a t t h e same
2
modulo
prove
H
which
those
(tensored
H
(see 9 . 2 0 . 1
0
of
there
0
3
below).
cycle
E in
homologous
are algebraically equivalent
Bloch
(X,<D ( 2 ) ) )
£
t o zero
to zero
1 . 1 and below.
i n f a c t proves
under
0
s = 3
at
£
c i t . Lemma
of
He e x h i b i t s c a -
3
of cycles
threefold
conjecture
G r H ( X , fl) ) h a s a z e r o
Q), see l o c .
. (G, , G r H
cont
k
IS)
for a certain
i s a non-zero
i s non-zero
1
(X)
the generalized
i s t h e group
which
with
that
time
CH
to test
the L-function
Gr CH (X)^,
in
field
and Swinnerton-Dyer
ses
0
considers
that
the Abel-Jacobi
To
i t s image
map i s
non-zero.
e)
Bloch
the
following
over
and
cycles
the
field
Swinnerton-Dyer
Bloch
of
conjecture
a number
9 . 2 0 . 1 .
the
[ B i 4] a n d B e i l i n s o n
Conjecture
proves
that
f o r a smooth
k, g e n e r a l i z i n g
Birch
case
v
j
rk Gr CH (X)
j
1
j
Gr ~ CH (X)
i s strongly
D.
projective
proposed
variety
the conjecture
With
= ord _jL(Gr^H
j
1
j
= N ~ CH (X)
to zero
X
of Birch
( X , ( J ) ) ,s) .
£
i s the group o f
and t h a t
for this
to the o r i g i n a l
for abelian
t h e same
2 j
g
related
and Swinnerton-Dyer
j =
independently
:
algebraically equivalent
conjecture
[ B e i 4]
arguments
varieties
we
part
conjecture
(which i s
can prove
9.20.2.
Lemma
Assume
that
the B i r c h and Swinnerton-Dyer
con-
jecture
(9.20.3)
for
r k A(k)
abelian
Then
conjecture
jecture
the
varieties
9.17
£-adic
A
9.20.1
b) f o r
=
Proof
In [ B i 4 ] 1.4 Bloch
gether
from
with
L(T (A)
module
a surjection
j
1
1
CH (X)
is
commutative,
6
i s induced
(see
b) a b o v e ;
out,
under
for
v = j - 1
vertical
§10.
®
constructs
T W
®
£
Gr^"
1
CH
sions .
D
i s true.
t o con-
c l ' comes
o f 9.20.3
i s equivalent
extend
"
1
2
H
J
~
1
( X , ( j ) ) to. I t follows
. (G T W
cont
k' l
® (DJ
V
1
1
2
1
H!
. (G, , G r r H ^ cont k' N
'<X,VJ>)>
( G
from
G r
and hence
see [ J 5 ] ) .
f o r W,
H
the Abel-Jacobi
for B
f o rthe commutativity
map, s o t h e c l a i m
j
W
the diagram
b y Kummer s e q u e n c e
assumption
on t h e subgroup o f
—»W(k)^
(X) ^
>
where
us f i r s t
k
i s equivalent
= G r
£
On t h e n o n - i n j e c t i v i t y
Let
field
an a b e l i a n v a r i e t y
H
Q
£
t o zero.
t
Gr
Q ,s)
F F I
t h e number
c
®
O
£
equivalent
the construction that
W(k)
over
1
map up t o t o r s i o n
algebraically
Tate
=
v = j - 1, i . e . , t o t h e i n j e c t i v i t y o f
Abel-Jacobi
rational
g
for v = j - 1
cycles
with
ord
map a n d
i s injective
As B l o c h
conjecture
to the b i j e c t i v i t y
points
9.20.1
of the l e f t
follows.
of the Abel-Jacobi
!
Bloch s
theorem
(9.13)
map
t o higher
dimen-
10.1,
Theorem
dimension
field
d
2
V
N (£
Proof
We
be
d
closely
First
we
follow
d
fact,
1
CH (U X U)
does
for
some N
supported
that
this
U'
would
+
[ r
XxD,
£
be
d
of
non-empty,
= 0
smooth
, since
surjective,
by
Y
Z
we
f
= TT^ (Y ),
1
on
and
that
A
Y
3
d
= ^
and
X
to zero i n
, hence
2
D xX
D,D'
by
on
}
V
' 3
H
d
=
->
see
V
Y
" 3
Y
2' 2
t o see
to
£
D'
l o c . c i t . . Now l e t
of
. L e t D"
, let
let
(X ,Q
X >
. I t i s easy
be
the
1
as b e f o r e
1
X
T2
a cycle
maps H ( X , Q )
2
d
and
and
d
T
Tr :
Z
3
that
)
a
c D"
= D"\V^
([Mi] 7.2). I f
£
Y
1
on
i n the support
affine,
(
U c
£
(X ,<£ ) / N
2
of
under
restricts
open
1
< d
H
c f . also
k
T ^ : Z -* D'xX
1
1,
made, t h e c l a s s
to zero
(d) ) ) c N H ( X , Q ) ,
jl
can conclude
i s zero
1
2
dim V
lecture
.
induced
(X Q
under
co-
.
i n [B11]
supported
irreducible
Z
by
1
Pr
image
* 0
function
(U x K) ®Q
f o r some d i v i s o r s
I m ( H ( D , $ ( d ) ) -> H
Q
of
]
2
£ ]N , a c y c l e
on
D'xx
X
imply
the correspondence
d
c
d
f o r some n o n - e m p t y
[T1]
N-[A]=
open
®
variety
filtration
the assumption
CH
the
f o r the
not r e s t r i c t
k
® Q
k
under
proper
l e t K be
(X, Q^)
k
d
non-empty
d
and
T(Xx K)
- CH (Xx K)
k
k,
smooth,
the arguments
show t h a t ,
d
In
* N H
then
k
any
1
(X,^)
diagonal A c Xx X
for
a connected,
the f i e l d
* char(k)),
CH (Xx X)
Z c
X
over
. If H
niveau
the
>
of X
[BS].
Let
d
V
2
.
Then
pr
D'xX
-+X
open,
d
H (V ,Q^)
1
i s not
the correspondence
. If
n
be
IT
i s surjective, l e t
2
=
given
U
' V
{ V
3
)
'
Since
dim
jection
in
the
Z = d, we
formula,
have dim
the
commutative
Y
< d
3
correspondence
, hence V
g i v e n by
* 0
3
Z
. By
i s the
the
pro-
bottom
way
diagram
(V ,Q )
0
">
0
H (V ,Q (d))
d
2
£
H (V ,Q )
d
d
H (Z,<£
d
T
£
d
H (X,Q (d))
d
d
H (X,$ )
d
as
the
to
the
that
T
d
with
£
would
prove
the
such
theorem choose
that
is
surjective,
to
the
the
-> A
w
and
n - f (c)
such
c
1
= N H
£
d
(X ,Q )
acts
, contrary
£
the
to
element
K.
n € CH (Xx^K)
o
K -» X
to
z e r o i n CH
(U*x
image
may
everything
curve
and
a
morphism
the
O-cycle
is a c
z e r o i n A £ b ( X ) (k)®Q
T(XxJ)I
K)<
for
U'
is
corresponding
G Pic(C)
, and
for
non-zero,
= X^f(c)
such
any
since
, but
f*(c)
n does
not,
K
of
Remarks
on e
be
. Then t h e r e
n-f*(c) £
O
the
a smooth p r o p e r
(Xx K ) 8 Q -» AilL(X) (K) ®Q
O
let
i s mapped
+
restricts
and
d
H (X,Q )
A
composition
g e n e r i c p o i n t Spec
that
then
too. Since
£
conclude
zero.
assumption.
C -* X
10.2.
is
£
d
to N H (X,Q ),
£
we
3
1
maps H ( X , Q )
1
P i c ( C ) (K)(
as
d
H ( X , Q ) -> H ( \ ? , Q )
d
identity,
To
f:
£
i t s composition
find
£
H (X,Q )
£
We
£
H (Z,Q (d)
0
H (D",$„
hence
3
fin?
a)
If
k
i s uncountable
*
replace "T(Xx^K)®Q
t o a model X
using the
inclusion
q
of
X
0"
over
T(X X^
by
a
K )®Q£->
Q
Q
and
algebraically
"T(X)®Q
finitely
T(X)®fl)
* 0",
by
generated
induced
by
closed,
applying
field
an
k
Q
em-
0
bedding
K ^—
b)
a
For
q
k
field
of the
function
field
k of c h a r a c t e r i s t i c
of
X
q
z e r o one
into
has
k
(cf. [Bi1 ] ) .
(10.2.1)
Iar
argument
consider
we
1
H° ( X , ^ )
as
the
f
i n a)
case
There
as
k
i s embeddable
that
a
Grothendieck
1
N
H (X Q)
case
defines a
of
N
V
effective
1
C N
lized
the
c)
In
[D.E.JVI
conjecture
finitely
of
weight
F
call
to
10.3
an
w on
a
object
such
follows
from
l
= N
over
v
claim
k
has
3.3.2) a n d
over
follows
from
for
f
V
entire
f
inclusion
states
the
of
which
W(v)
(see
the
[D6]
effective
i f and
as
as
imply
, pure
Q^-sheaf
3.3.8
that
For
or e f f e c t i v e ,
For
thm.
I
i f i t
X/k
subrepresentation W
+char
Y
only
parts of
a pure
smooth,
1 ) , and
we
for
H (Y,$^(e))=
a
f o r the
H (Y $^)
c
f
H
a
(X Q (Cl))
F
j l
=
H
2 d
a
(-e)
are
G a l (IF ^ / I F ^)
Hence
~ (X,Q^))
e
v
occuring p-adic
d e f i n e d i n I o c . c i t . 3.3.7).
Im(H (Y,Q (d) ) -
dimension
—
a
^ - r e p r e s e n t a t i o n of
i f r , s _> 0
of
(d-e)
is
Hodge
for Y c X
k
extends
i f effective
for a variety
F
effective
i s entire.
of
Tate
extended
ZZ [1]
[J3], proof
converse
be
G^-representation
£
the
genera-
a generalized
a model U o f k as i n 6.8.
union
f o r the
shows t h e
T h i s can
over
F ^(W)
_> 0
f
=
(note t h a t
a
k = <X, w h e r e
W
p q
constructive
i f
i n Rep (G^ Q^)
c
with
fields.
type
the weight-graded
(r,s)
to
,
formulated
finite
effective
types
simi-
i n [Gr].
structures
—
over
then
- which would
finite
is effective
f
)
v
scheme S o f
V
1
N
a
([D9]
H (X G)^)
(X,Q
a
setting
- and
. Call
a Q^-sheaf
1
by
then
fields
. _
V H
V
those
10.2.1
Grothendieck
generated
is entire
jective
It
are
(p,q). Grothendieck
for varieties
to
the
by
i t suffices
investigations
i f effective
structures
Hodge c o n j e c t u r e t h a t
10.2.1.
the
sub-Hodge
- which o b v i o u s l y proves
of
if
Hodge
Hodge t y p e s
v
of
f i l t r a t i o n *N
w h i c h W(v)
occuring
Then
f
special
cycles
i n <I, a n d
H (X(CJ) Q).
fact,
0
=
f
where
of
1
by
union
1
f
specialization
r e p l a c e H ^X,Q^)
Hodge t h e o r y ,
1
* H ( X Q ) . In
0
and
3
may
1
* 0 =* N H ( X Q )
the
prolet
is
f o r d = dim X , which
effective,
The
g e n e r a l i z e d Tate
10.3
f
By R o i t m a n s
conjecture
n o n - i n j e c t i v e complex
on
varieties
states the e q u a l i t y N
(9.11.3)
theorem
of
shows t h e i n c l u s i o n
of arbitrary
dimension
v
l
l
c N
= N
v
v
maps
.
.
1 0 . 1 one g e t s
and theorem
o r £-adic A b e l - J a c o b i
v
N
examples
f o r O-cycles
d. S e v e r a l questions
naturally
arise.
The
fields
examples
provided
of higher
transcendence
degree
X , a n d o n e may a s k w h e t h e r
for
[ S c h o e ] has produced a
cycles
As
for
a finite
f a r a s I know t h e r e
but I think
injective
Hodge c y c l e s .
injective
higher
10.4.
transcendence
2
intersection
q
Z ^
Let
i t turns
X
E
2
However,
o f ®(t)
Schoen
and t h e
( see appendix B ) .
f o r cycles of higher
below.
the Abel-Jacobi
map
j > 2
c o u l d be
of absolute
the Abel-Jacobi
i n codimension
degree),
of definition
the variety
examples
i n 10.4
out that
(over
map
i s non-
fields of
i n view o f the f o l l o w i n g c o n s t r u c t i o n .
be a smooth p r o j e c t i v e v a r i e t y .
are c y c l e s which
Z
where b o t h
over
be p o s s i b l e t o c o n s t r u c t t h e s e , t o o ,
described
quite principally
Z GCH^(X)
the f i e l d
t h e o r i e s , e.g., f o r the theory
However,
Principle
cycles defined
i s essential.
extension
o n e may w o n d e r w h e t h e r
f o r other
used
than
do n o t e x i s t
i t should
example by t h e method
Finally,
this
counterexample,
a r e defined over
dimension,
and
so f a r always
1
CH *-* ( X )
q
a r e homologous
lies
to zero,
If
then
1
Z ^ C H ( X )
their
i n the kernel of the Abel-Jacobi
map.
We
cases,
can turn
the f i r s t
applying
this
principle
into
a theorem
i n t h e f o l l o w i n g two
one e x p l a i n i n g t h e i d e a b e h i n d
f o ra l l theories of
interest.
10.4 a n d
probably
O
10.5.
Assume
homology
T
of
(not
that
the
cohomology
some c o m p l e x
(10.5.1)
through
f
some
RT RT(X J)
r
with
"absolute
=RHom,
D
f
H*(X,j)
b
RT(X J)
n e c e s s a r i l y equipped
factorizes
theory
i s obtained
i n D (T)
f o r our
weights),
and
cohomology"
(1,RT(X J))
tensor
that
- the
the
is fulfilled
9.6)
and
the
the
f o r the
Hodge
cycle
homology
Abel-Jacobi
map
(T
map
of
;
f
(T)
£-adic
theory
the
category
721
this
as
theory
(T =
=
MH
, c f . 9.7
i s induced
by
the
S((Spec
c))
I
k)^ )
cf.
t
. Then by
hypercohomology
9.5
spectral
se-
quence
q
(10.5.2)
E?'
where by
If there
quence,
e.g.,
(10.5.3)
one
the
the
F
V
r
for
F H^(X I)
the
these.
deduce
a
=
i - t h homology
with
this
of
spectral
se-
pairing
RT(X,i+j)
2]
r
e
1
cl(z^)
to
€ F H
using
and
the
[J1]
absolute
the
a
on
,
2 1
1
T
r
H
and
method,
cl(z )
2
cupproduct
2
j
implies
' ~
1
such
complexes
also valid
of
for
H*(X,*)
1
€ F H
2 j
+ j
6.11;
^ (X,i+j) ,
definition
cupproduct
10.4
proceed
for
with
2
2
a
f
f
c l (z ^ • Z ) € F H ^
by
to
(X j)
2
principle
could
associated
i s compatible
( X , i + j )
hence
c y c l e s one
Hodge
terms
equivalent
theory
and
(X,i+j)
limit
(X,i)
£-adic
§6),
+ n
Hy
the
10.5.4
in R
Hodge
absolute
different
v
r + S
If the
then
zero
Hodge and
F
F
homologically
2
product,
4.2
-
f
cycle classes.
For
i t by
F H^(X J)
a
f
i t i s mapped
by
+
filtration
Z-J Z
definition
[Bei
way,
from
compatible
s
®
f
. For
(see
,
Ext (1,RT(X,j))
cupproduct
L
( X , j)
1
=
f
® RT(X,j)
intersection
hence
r
P + q
has
absolute
the
H (X J)
i f i t comes
I f now
by
=> H
r
is a
descending
10.5.2.
then
r
RT(X,i)
(10.5.4)
for
q
R T H (X J)
definition
10.5.1.
then
p
=
of
exists
is a
in a
theorem
similar
i n s t e a d we
f o r more g e n e r a l
shall
coho-
mology t h e o r i e s .
10.6.
To
prove
10.4
without
using
an
absolute
cohomology
theory
I
need
the following compatibility
duality
r)
twisted
Poincare
theory.
For
maps
for the considered
Y c X
c l o s e d a n d U = X^Y
the f o l l o w i n g diagram
of natural
commutes
H (X,b)
a
1
* H (X J)
H _
f
a
(Y,b-j)
±
V6
H (U,b)
a
s)
For
X
® H
1 1 -
smooth
the
Z^ a n d Z
>H _
f
map i s c o m p a t i b l e
cycles
-(U J) —
a
and p r o j e c t i v e
with
dimension
d
the cycle
i n t h e f o l l o w i n g sense: F o r
i and j on X i n t e r s e c t i n g
properly
diagram
r H
(
d
2(d-i) ^r -
i )
® ™^(X,j)
tcl
0
(Here
H^(X,j)
-> H ^
For
6.11:
Let
n z
example,
this
checked
the upper
2
(Z
1 f
j)
both
z
v = 1,2. A s s u m i n g
true, e.g., i f
a
s
V
2
map
i s induced
hold
true
we g e t a c o m m u t a t i v e
1
2
0
Z (Z OZ )
1
2
by t h e r e s t r i c t i o n
for the absolute
i n any o f t h e r e a l i z a t i o n s ,
Hodge
theory
and i s e a s i l y
theory.
n
a n c
-^ I O * ^ /
^ ^
e
t
r ) a n d s ) , and t h a t
T
(Z OZ , d-i-j)
2
toterse0ti0t
Z (Z )
things
c a n be c h e c k e d
i' 2
TH ^ , . j
and t h e c a p p r o d u c t ) .
f o rt h e £-adic
z
»
t c l
0
9
1
commutes
—
icl
Z (Z )
is
f
o f pure
t h e capproduct
o f dimension
2
(U b-j)
± + 1
i sa rigid
diagram with
z
v
he t h e s u p p o r t
® i s an e x a c t
tensor
exact
category
rows
o f z^,
functor
(this
[DMOS] I I 1 . 1 6 ) ,
0
H
2(d-i)
- 2 (d-i)
0
H
(
V
( Z
®
)
1
h 2 J
'
(
^
1
( X )
H
j
^ 2 ( d - i ) (^ W -
1
Z
1
^
H
2
2 (d-i) ^ l ) ® ^ "
H
2(d-i)
(
Z
X
1
Z
)
i d
(XXZ )
1
e
1
H
Z
l
NZ
( 1 >
2
J
"
1
( Z
1
6
S H
d
1
j
(X)
e
S <d-i) 1
5
2
^(Z ) H
2
( Z
) € H
( Z )
?
X Z )
in
°-
> H
2(d-i-j) 1
( Z
+
1
H
)
(Z -Z)
>
1
2(d-i-j)+1
H
2
w
^
j
j
Il
+
0 - H
2
(
d
^
j
)
+
( X )
1
w h e r e we h a v e
get
H
omitted
a commutative
z
j
^ ( d - i ) ( I) ^ "
1 ( X )
(10.6.1)
-
t h e t w i s t s and s e t Z = Z
diagram with
0
(X-Z)
2(d-i-j)+1
exact
z
-*2(d-i) ( I ) ^
2 j
'
0
1( X
( Z )
">
H
2(d-i-j)
n Z
1
2
(
Z
)
. Hence
'
we
rows
H
^2)- 2(d-i)
(
Z
l ) o ^
( X )
o-°
O
V
^(d-i-j)+!
where
of
(
( X )
)
means
the l e f t
pull-back
But
in
Hom
7
"homologous t o z e r o
2(d-i-j)
( Z
)o
o f the bottom
(1,H
t h e same i s t r u e
2 t d - i
_j^
the pull-back
E x t 1j-( 1 , H
exact
2 i + 2 j
(Z,d-i-j) )
Q
sequence
via
D
, s i n c e by
(X,i+j))
under
Thus, t h e
i s trivial,
f o r the pull-back
s)
by c l ( z ^ • Z ^ ) by d e f i n i t i o n
1
0
^
o n X" a n d t h e v a n i s h i n g
maps f o l l o w s v i a t h e p r o j e c t i o n f o r m u l a .
consequently,
€
H
^(d-i-jHl*^
via
and,
cl(z^•Z^)
i t factorizes
i s t h e image o f
the Abel-Jacobi
map,
through n .
z
which
^
z
2
proves
10.4.
10.7.
To deduce
10.4 i t r e m a i n s
is
non-zero
iomologous
the non-injectivity
to find
in C H
1
t o zero,
^
j
elements
( X ) .
i fthis
z
First
^
of the Abel-Jacobi
z
a
2
note
i s true
s
there
that
map
z
such t h a t
cycles
from
^
z
2
0
i n Pic (X) are
f o r a l l smooth, p r o j e c t i v e
C=X
curves
via
(use
algebraic
metric"
that
0
elements
in Pic (X)
correspondences).
theories;
i n any
case
This
will
i t i s true
come f r o m P i c
be
the
under
case
0
of
curves
for a l l
each of
the
"geo-
following
2
conditions:
a)
H
(U,1)
2
rational,
for
the
and
an
for
(A^,1) = O
of
=
absolut
5
Pic (X)
abelian
0
a f f i n e curves,
b)
H*(-,*)
is
Q-
1
0
Assuming
for
H
theory
=
1
H
( S p e c k,0)
Hodge
1
CH (X)
surface
X
over
course,
i t is
true
cycles.
, we
q
. Of
may
use
Bloch's
result
an
algebraically closed
-
T(X)
that
field
k
the
map
0
(10.7.1)
0
Pic (X)
x Pic (X)
induced
by
one
has
T(Xx^K)®Q
* 0
theorem
(9.11.3),
since
in
i n t e r s e c t i o n i s s u r j e c t i v e , see
for
suitable
H (X O )
=
p o s i t i v e c h a r a c t e r i s t i c one
has
(9.13) f o r X
=
xE
2
r
* E^
2
yi
field
[ B i 4]
extensions
A H^(X O )
2
r
* 0
x
T(Xx^K)®Q =
elliptic
. For
curves
char(k)
0
Roitman S
by
1
. As
an
example
0 by
Bloch's
3F
, not
over
=
theorem
both
of
2 them
s u p e r s i n g u l a r , and K t h e f u n c t i o n f i e l d
G
H (X,Q (1))
= N H (X Q^d)) , I * p .
2
*
k
1
Remarks
a)
p
> 0
Fano
In
are
[Bi
Mumford's
since
H
(X,Q
(1))
f
10.8.
(X)
X,
2
£
g
b)
of
1]
or
Other
surfaces,
lecture
his
examples where
1 Bloch
theorem
(see
see
[Bi
10.7.1
1]
conjectures
9.11.2 a n d
is surjective
and
1.7.
that
the
9.13)
converse
i s true,
or,
of
equi-
2
valently,
that
only
and
is
torsion-free
c)
It i s quite
d)
Using
of
abelian
of
a
i f A (X)
if
reasonable
H
Roitman's
likely
Abel-Jacobi
Zarchin's
X
i s generated
i s representable
Q
by
surface
theorem
that
T(X)
map,
but
proof
of
the
lies
following:
If
k
p
O
i s an
is a
I h a v e no
finitely
]
note
cycles
that
T(X)
).
i n the
kernel
proof
conjecture
varieties in characteristic p
the
algebraic
(by A l b (X);
[Ro2
Tate
by
> O
for
of
every
this.
on
endomorphisms
, i t i s easy
generated
field
to
show
2 >
and
X
abelian
surface
over
k
, then
H
(X Q
f
(1))
=
-
2
H
G
k
(X -A)^(I))
i f and o n l y
F
singular
i f
X
i s isogenous
e l l i p t i c curves d e f i n e d over T .Hence
only
to a product
of super-
i n this
one e x -
case
P
pects
representability
§11.
Chow g r o u p s
over
In
this
section
pected
over
fields
smooth,
Gr?
r
CH (X)
Gr^
CH (X)
Gr
and
we
= CH
2
2
= A
2
X
2
= F CH (X)
would
recalled
be t r u e ,
C
C, B l o c h
F
v
2
i H (X,2Z (2))
~
V
H
3
(X,(C) /H
degree.
such
For a
1)
that
,
( X , 2Z ( 2 ) )
3
ex-
( [ B l 1] L e c t u r e
on C H ( X )
4
(X)
( X ) A ( X )
+ F
2
,
=T(X) ,
2
2
b e t w e e n G r _ C H (X)
r
conjectures a relation
conjecture
over
C
O
A b e l - J a c o b i map.
o f Chow g r o u p s
transcendence
filtration
( X ) /K
2
CH (X)
r
Q
the structure
arbitrary
surface
2
and an i n j e c t i v e
fields
discuss
a descending
2
(X)
arbitrary
with
projective
considers
of A
i n 10.8 b) a b o v e .
i f the action
and H
He s h o w s
of algebraic
that
2
, c f .also h i s
the latter
correspondences
one
on
2
Gr^CH
(X)
The
factorized
following
through
homological equivalence.
conjecture of Beilinson
i s a generalization of
this.
11.1.
Conjecture
projective
F
V
on
varieties
X
over
Let
k
k
there
be a f i e l d .
F o r smooth,
i s a descending
filtration
3
CH (X)
V
1
. ..CF ^ CH
such
( [ B e i 4] 5.10)
1 1
V
(X)CF CH
1 1
0
(X)C. . . C F C H
L J
3
(X) = C H (X) ,
that
v
3
= O
1
3
= CH (X)
a)
F CH (X)
b)
F CH (X)
for
v >>
3
O ,
3
=
{z€CH (X)
I z
numerically equivalent
num
to
zero}
r
c)
1
s
3
F CH (X) F CH (X)
l
d)
f:
,
F
X
v
i s functorial,
+ Y ,
c F
r
+
S
CH
1
+
3
(X)
under
the intersection
i . e . , r e s p e c t e d by f * and f
+
product,
f o r morphisms
e)
(note
t h a t by
1
GrpCH- (X)
1
last
motive
2 j
H
V
statement
I
Part
H-
1
V
factorizes
a c t on
through
numerical
motive
(X) .
factor
since the existence of the
o f t h e motive
standard
homological
H*(X,Q^(*)),
action
i s conditional,
( X ) , as a d i r e c t
equals
2
equivalence
1
A(r,s)
this
correspondences
o n l y depends on t h e G r o t h e n d i e c k
depends on G r o t h e n d i e c k s
valence
d) a l g e b r a i c
b)
Gr^CH- (X)
numerical
The
and
and t h a t by
equivalence)
w.r.t.
c)
H(X), see [ K l ] ,
conjectures that numerical
equivalence
( f o r &-adic
equi-
cohomology
+ c h a r k, s a y ) a n d t h a t t h e K i i n n e t h c o m p o n e n t s
of the cycle
class
of the diagonal
A cXxX
are algebraic.
e) t h e n means t h a t
id
, ( r ,s) = ( 2 d - 2 j + v , 2 j - v )
,
A(r,s)
G
where
The
r
d
F
C
H
:
(
Q
O
c o n d i t i o n s a)-e)
11.2.
Lemma
b)-d),
Proof
H (X)
of
,
X
otherwise
(X
connected,
a r e n o t independent;
i m p l i e s G r ^ CH^(X)
t
(Assuming
m
= O
the standard
,
without
for
v >j :
c o n j e c t u r e s ) Under
J
i s z e r o o n G r ^ CH~ (X)^
fors < j .
(compare
[ B i 1] 1.9) By t h e h a r d
Lefschetz
= H
2 d
s
( X ) ( d - s ) , the correspondence
= r «
T
3lc
+
o r
+ /
with
T € CH
h a s image i n CH
d
S
+
j
2 d _ S
(XxX)
restriction)
f o r e x a m p l e , p r o p e r t y e)
A(r,s)
A(r,s)*
then
)
i s t h e dimension
automatically
s
X
A(r,s)
+
P
and
assumptions
isomorphism
factorizes
s
€ CH (XxX); b u t
(X) :
j
CH (XxX)
CH
2 d
"
S + j
(XxX)
\(pr )
2
j
CH (X)
and
this
group vanishes
as
^
fors < j .
+
d
CH "
S + j
(X)
11.3.
of
There
are i n fact
t h e above
there
and
filtration
should
exist
complexes
with
versions
parts
the
derived
"with
forms
that
a)
€ Ob(D (MM))
:= H
a twisted Poincare
smooth
axioms
( i n s h o r t , analogues
of a twisted
1
Poincare
inducing a
( R T ( X J ) ) EOb(MM)
z
f
(Rr (Xfb))
1
—a
duality
X
H
equivalence
o f Q-motives
© H
i>0
W W
cupproduct
(X O)BQ
,
with
weights,
and such
holds:
€ Ob(MMfcQ) i n d u c e s a
f
c a t e g o r i e s between
(w.r.t.
o f semi-simple
objects
numerical
equivalence)
b)
the cycle
c)
J ) = Horn
(1,RT(X,j)[2j]),
D (MM)
a n d t h e ( h y p e r e x t ) s p e c t r a l s e q u e n c e ( c f . 10.5.2)
J
degenerates
Then
5
H
2
J
( X
after
sequence
provided
the cupproduct
cycles
(compare
3
CH (X)
we
num
have
(11.3.1)
,
hence
F
v
on CH
c ) , which
10.5).
,
b
tensoring with
spectral
Q .
3
(X)
assures
would
the properties
i s compatible
By
a)
be d e f i n e d b y t h e
with
we h a v e
1
3
F CH (X)=Ker(CH (X)+
the formula
v
3
=
R V F
H
MM MM"
V ( X
'
11.1 c ) a n d d)
the intersection of
3
11.1 b ) , a n d 11.1 e) i s c l e a r ;
Gr CH (X)SQ
and
an isomorphism
F
i n MM
1
subcategory
induces
weight-
Grothendieck s
the
CH (X)
duality
Ob(MM)
the following
k
map
i n
m
of tensor
over
€
theory
and p r o j e c t i v e
The a s s o c i a t i o n
category
counter-
so t h a t
m
preserving
MM
and h o m o l o g i c a l
® R T ( X , j ) -* R T ( X , i + j )
^rf (Xih)
a
X
weights
i n [ G i ] a n d [ B e i 1] 2 . 3 , i n p a r t i c u l a r
M
for
with
that
together
z
o f t h e axioms
conjectures
a s i n 10.5, w h i c h
RT (X,j)
certain
^ H J J I ( X , j ) := H
X
category
L
t h e cohomology)
F
tensor
support"
category
Beilinson
b
RT(X,j)
RT(X,i)
(X Z)
. Namely,
a suitable
c f . t h e axioms
pairings
v
F
Rr' (X,b) s a t i s f y
theory,
on
more p r e c i s e c o n j e c t u r e s on t h e o r i g i n
3 }
®
4
'
i n fact,
TH^j ( X ,
by
c)
w h i c h makes more
=h^
(X,0)
(part
1(1)
i n MM
of
, cf.
the
G r ^ C H ( X ) ®Q
r
axioms
[DMOS] I I
depends
i s the
existence
such
t h a t RT (X J)
§5,
t7
V
A
expects
MM
motives
a
as
above
over
k,
could well
and
HjJj(X,0>(j) ) =
for
the
its
name. T o
be
motivic
cohomology
deduce
1
(X ZZ
J
M
MM
®
J )
F
Q
d e f i n e d by
a l l this
(J))
= M(Spec
from
=
E x t J
RT (X O)
r7
Remarks
of
X
f
E
)
) and
( f f i
M
as
of
V
(X)
=
a
b)
category
above
Beilinson
A
similar
thus
justifying
< 0 ) , Z Z
M
over
=
Rf
j k
in
E ^ ( j ) f o r f:X-^Spec k
that
( J ) )
Hj (X,j)
=
.
M
interpretation
mixed motives
ZZ
K-theory,
RT(X,j)
M
M
a)
a category
[D10]
2 j
(j)
f
Beilinson's formulation
Ext£ <1,Rf,ZZ (J))
=
11.4.
(
k,2Z
, i , j €
B e i l i n s o n ' s n o t a t i o n i n l o c . c i t . , so
H
, let
H ^ ( X
with
k was
of
also
K (X)
^
m
given
in
by
terms
Deligne
.
The
f o l l o w i n g axiom
a')
X rv^> e
j>o
the
standard
H e r e one
(like
has
Over
, see
i.e.,
also
j
would
work
(X,Q(j))
the
t h a t the
field
usual
Weil
Over
there
which
[ B e i 2]
are
a
k
global
"no
8.5.1,
and
as
11.3
cohomology
satisfied
definition
cohomology
a):
for
which
( c f . [Kl] ) .
i n an
now
obvious
takes
way
values
in
a
category.
one
expects
field
motivic
is implicit
equally well
i s a Weil
conjectures are
abelian tensor
finite
§12.
that
statement
so
rigid
a
H^
to modify
i n 6.1),
Q-Iinear,
cf.
=
f
in generalization
4]
0
object
regarded
[Bei
=
a Tate
H
formula
(11.3.2)
c)
motive
.
category
mixed
b)
of
the
RTJ_(X,b) = RTJ_(X,0) (-b) ; i n p a r t i c u l a r H ^ ~ ( X , j )
H ^ " ( X , 0 ) (j))
of
on
v
and
of
p r e c i s e how
k
F
one
1
HjJj (X, © ( j ) ) =
expects
2-extensions"
up
2
F HjJj (X
to
in
[D10]
. Here
(j ) ) =
torsion,
in Beilinson's conjectures
explicit
HjJj (X >Q ( j ) )
in
the
[Bei
a
1],
spectral
q
0,
sequence
0
11.3 s h o u l d
RF
1
1
H
become a s h o r t
-MM^M < ' < H J > )
like
f o r the Deligne
mological
HJ(X,«(J) ) - r
-
X
exact
cohomology,
M M
sequence
H^(x,$(j))
-> o
c f . 9.7 c ) . S i n c e
d i m e n s i o n two, one h a s t o have
G^.
a vanishing
,
has coho-
or a truncation
2
for
H
(Gj /-)
here,
d)
As B e i l i n s o n r e m a r k s ,
v
CH (Y)
a motive
= O
1
2
N*
to motivic
cohomology
and
results
11.5.
Z H
should
1
2 j
. Here
V
L*
"
j
(X)/N "
i fH
2 d
N"
,
i s the level
11.1
d
~ (X,®
shows
) = N ~
V + 1
would
H
2 d
imply
V
~ (X,$ )
0
1
= dim X
filtration
V + 1
. Thus
V
, i n generalization of Bloch s
does
not exist
one) o f a c a t e g o r y
above,
MM
so one d e f i n e s
be g i v e n
by t e n s o r
T = WRep (G^,Q^)
c
complexes
(absolute)
that
for
conjectures
z
Poincare
, RT j(X,b)
duality
HjJj(X,ffi(j))
5
motivic
k
cohomology
realizations.
weights
generated
1
(not even a
motives with the
by v a r i o u s
categories with
forfinitely
RT^ ( X , j )
a definition
o f mixed
and s t u d i e s approximations
twisted
V
" (X))(j))
f o rd = 2 .
conjectural
K-theory
2 j
by coniveau
Up t o now t h e r e
properties
j
c a n be r e f i n e d t o
i n [ B e i 4]) , and one e a s i l y
i s representable
2,3,...,d
of
V
L - H ^ (X)
v =
and
close
11.3.1
= R T((L " H
v
(X)
formula
f o r dim Y < j
(called
j
(11.4.2)
for
j
Gr CH (X)®Q
3
since
of
cohomology
c f . §12 and § 1 3 .
(11.4.1)
A
t o g e t £-adic
c
T
These
(e.g.,
, T = MH
as above
by
f o r k = C,...)
together
with
morphi
theories
H^(X,j)
(similar
with
supports),
(11.5.1)
a
into
be
(X,©(b))
r
+
H^(X,b)
a.
the associated absolute
induced
b y 11.3.2 a n d f u n c t o r s
transforming
For
(co-)homology
the Rr^-complexes
smooth,
projective
X
of tensor
into
(Ideally,
categories
Rl^-complexes).
the spectral
r and r
sequence
f
would
<J>:MM •+ T
P
( 1 1 . 5 . 2 )
should
( 1 1 . 5 . 3 )
Gr^
I
(
R
)
1
F(r)* = r
1 1 . 5 . 2 .
(
M
X
F*
two s t e p s
one
must
+
Q
( X , j )
-
R
^
H
^
are given
Q
'
(
j
)
)
(X, j )
I
on H ( X J )
g i v e n by
F
by t h e C h e r n
characters
T H (X J)
R
*
o
t h e maps
I
-
F
R
1
T
F
H
T "
1
(
X
'
J
also consider
maps"
'
)
or a global
1 1 . 4c ) . For fields
remark
P
for the filtration
§ 9 .Fora finite
cf.
H
i n § 5 and § 8 , and " A b e l - J a c o b i
discussed
H
H (X,$(j))
The f i r s t
HjJj(X,fi(j))
=>
T
a s i n 1 1 . 3 c ) , a n d we a r e l e d t o s t u d y
degenerate
where
R r H£(X,j)
Q
E!?' =
field
o f higher
this
transcendence
11.5.3
t h e maps
should
for v >
_ 2
suffice, c f .
degree,
however,
2
, since
F
i n general
2
does
not vanish
(e.g.,
Hodge
theory^ and hence
catch
motivic
F
CH (X)
= T(X))
Q
Deligne
cohomology,
. I nparticular,
cohomology,
0
0
since
RT=
u
f o r k = (C
i s not sufficient t o
for v >
_ 2
( s e e 9 . 3 b) ) .
2
The
k
non-vanishing
of F
f o r smooth,
has t h e f o l l o w i n g consequence
composition
reso r
f o r smooth
i n the (conjectural)
r
(11.5.4)
M M
H
i M
(
U
' ^
(
j
)
R
)
non-surjective
cular
(look
jective,
for
geometric
r
the
i
In
for certain
T = MM)
and absolute
could
T H (U J)
F
H
( U
j
' >
H^(U,j)
U c X
the left
(and c e r t a i n
restriction
11.5.2
smooth U/k. I t i s i n t h i s
image o f
r
diagram
T T
fres
r
and the s p e c t r a l sequence
general
and
at
big fields
By 9 . 1 0 t h e
varieties.
res f
HjJj(U,ffi(j) )
is
p r o j e c t i v e X over
theories
be i s o m o r p h i s m s ;
separately.
i n any case
Tty a s t h e s u b s p a c e
map w i l l
will
case
i,j).
n o t be s u r -
not degenerate
better
Note
to treat the
that
we s h o u l d
of algebraic
In parti-
both
rather
Dj>
regard
elements i n
.
the rest
realization
of this
functors
s e c t i o n we d i s c u s s
over
arbitrary
fields.
£-adic
Note
a n d Hodge t h e o r e t i c
that
the existence
of
injective
the
"regulator
maps" HjJj (X,® ( j ) ) 5
standard c o n j e c t u r e s f o r the
would
imply
Beilinson's
F(r)'
defined
11.6.
Beilinson
faithful
and
on
MM
imply
associated geometric
c o n j e c t u r e 11.1,
conjectures that
- this
D (M(S,Q))
would
together
by
taking
with
theory
the
H^ ( X , j )
1
filtration
above.
b
on
H^(X,j)
the
corresponds
for a
scheme
£-adic
to the
S
of
realization
functor i s
i n j e c t i v i t y of
f i n i t e type
Vty -
over
E -
this
, see
§12
for
that
H Jj(X O)(J) ) + H^. ( X , Q ( J ) )
j
is
f
e
injective
for varieties
over
a
finite field
more p r e c i s e
conjectures. For
is
( e . g . , f o r k = k) , a n d
too
small
Write
X = Iim
J.
finte
type
over
(11.6.1)
H
i
e
is
well
K
and
(X)
this
ture
to
= Iim
a?A
carries
§7,
the
K
(X
m
a
11.7.
Remark
including
proceed
as
of
over
maps.
schemes
of
Then
a
H^
t
(X , Q
(* ) £
a
a
ky
A
[EGA
IV]
map
£
(j ) ) ,
t o t h e Adams e i g e n s p a c e s .
r®Q^)
not
a treatment
11.6.2
(in fact,
f o r smooth X
commute w i t h
Beilinson's
. Note
one
should
that
inverse limits,
conjec-
ZZ ^
in
even
-
contrast
coefficients.
extending
follows.
of
c
construction.
i
finite
For
following
) ,
case
cohomology
2.2
Q^-cohomology does
the
etale
(X ) .
a a£A
transition
obtains a regulator
1]
the
2
e
amounts t o t h e
expect
or
[Qui
use
k
Iim H
(X ,(JK(J))
ot?A
^
independent of the c h o i c e of
r : HjJj(X,4fe(j) ) -
by
we
fields
i n v e r s e system
( X , Q , ( j ) ) :=
L
one
(11.6.2)
an
TL [ j ] , w i t h a f f i n e
d e f i n e d and
8.13.5, a n d
since
, with
a
arbitrary
k
H^
t
to a
satisfactory
o f more g e n e r a l s h e a v e s t h e n
t h e o r y as
Q^(J),
one
in
11.5,
may
Let
R
be a r i n g ,
q
the i n d u c t i v e system
FT/R
type o v e r R
€ R
, R
on X/R
one
R
I
be an R - a l g e b r a , and
)
L
with
1
3 R^
t h e r e i s an R
I
X = X. x
R
I R«l
1
^
f o r some
r
and
. Let
. Let
q
type over
a scheme
X^
X _ , = X.x_ R
R
R
^R'^R'^R
€ R , R
2
o
3 R
2
n
t
h
e
.By
1
i
n
v
e
of
L
for
I R^
r
s
e
R,
F
system
u s i n g [ E G A IV] 8 . 8 . 2
d e f i n e s morphisms o f t h e s e as morphisms o f I n d - o b j e c t s and
t
=
V
Iim
H*
—
o n l y depends on
X
o n t
<X
,
R l
F
R I
t
and
F
where the l e f t
the f o l l o w i n g
a) C a l l
for this).
From
sheaf F =
(F
are c o n s t r u c t i b l e and
isomorphisms f o r
1
R
F
R
F
i
3 R
-> p . i
i d e n t i f i e d with a f u l l
£-adic sheaves
on
X
p , : X -* X
3
. If
k
K
_
t 3 l
on X/R
constructible
^R2
R
such
2
sheaves
on
generated
X € ob(FT/k)
s u b c a t e g o r y of the c a t e g o r y of
i s the
F
that
p: Spec R " -+ Spec R'
to i t s l i m i t
are
field,
can
be
constructible
Iim p*,F
g,
R
, ,
R
projection.
i s a morphism i n FT/R
£-adic sheaf on X/R
=
P
q
H (Y/R;R ,R f*F)
O
=* H
P + Q
Q
i s a Dedekind
If
ring,
R = k
and
F
, then there i s a s p e c t r a l
(X/R;R ,F)
q
with c e r t a i n c a n o n i c a l R - p o t e n t i a l
sheaves.
e a s i l y obtains
K
f : X -> Y
Q
)
is a finitely
, by s e n d i n g
,
D
E^'
have
does not
(R
t h e r e i s an R^
for
then t h e c o n s t r u c t i b l e p o t e n t i a l
If
n l
K K
the t r a n s i t i o n maps P *
where
11.6.1
[ E G A IV] § 8 one
O
R
and o b v i o u s
= 2Z [ j ]
Q
properties.
an R - p o t e n t i a l
allF ,
For R
,
hand s i d e i s d e f i n e d as i n
have t o be a f i e l d
that
)
, i n a f u n c t o r i a l way.
= H^. (X/R;E
£
, such
ob-
2
s i m p l y t a l k o f p o t e n t i a l £-adic sheaves
H^ (X,Q (j))
b)
be
Q
Hj (X/R; F)
if
type over R
R
, and d e f i n e an R - p o t e n t i a l £-adic sheaf
as an £-adic sheaf
r
€ R
1
t a i n s an a b e l i a n c a t e g o r y , f i b r e d over F T / R
we
let
Q
be the c a t e g o r y o f s e p a r a t e d schemes o f f i n i t e
finite
( X
R
of i t s s u b r i n g s o f f i n i t e
then f o r X € o b ( F T / R )
R'
let
q
R f*
i s an
,
£-adic sheaves
generated
Q
sequence
respects constructible
is a finitely
R -potential
on Y/R.
If
R
Q
R -potential
field,
Q
q
R f Q^(j)
+
can
be
identified
neric
c)
base
the usual
change
theorem
(-/R;R ,CD ^ (*) )
H|
FT/R
with
.
(Define
t
H^
homology
(X/R; R , ^
[SGA
=
f
D
, : X
transition
[SGA
4
ge-
] [finitude]) .
Poincare
duality
on
R
H~* ( X ^ R f
(-b) ) ,
^
1
i s the s t r u c t u r a l
maps a r e i n d u c e d
obtaining
complexes
by
the
Spec
k one
(11.7.1)
f r o m b)
the Poincare
in a suitable
11.5.2 w i l l
may
"base
be
E^'
q
= H (k/k R
?
above,
for
Y=
L e t us
above,
Deligne
mology
for varieties
parameters.
briefly
= R
in
k
, R
morphism
and
the
change" morphisms
of
, and
a filtration
motive
H
1
v
= Q
X ,/R'
F' on
Q
11.6
R )
would
for X
we
may
q
(X/k ; R
sheaves
sequence
,Q
a good
(j ) )
V
As we
remarked
absolute
k
coho-
with
many
with
G r ^ H p (X ,<£ ( j ) )
T
finite
type
such
only
depends
on
the
that
(X)(J))
H*
theory
f o r variations
projective
R (f^,) .Q
£
but i t remains the q u e s t i o n f o r
from a s u i t a b l e
5 4
q
over the Q-algebras of
such that
q
£-adic
,
Q
Hodge c o h o m o l o g y )
such that
+
t r y t o work
x C,Q(j))
running
and
P
, but not f o r f i e l d s
geometric r e a l i z a t i o n
smooth
RT-
spectral
t h e Hodge r e a l i z a t i o n s .
=Ext^d,H^"
follow
. The
by
.
( X ) , or even a c a t e g o r y
(or a b s o l u t e
as above
H^
[J4]
£
a s i n 11.7,
R
of
of p o t e n t i a l
i s e x p e c t e d t o be
over
, R'
the a s s o c i a t e d
Namely,
Spec k
Iim H ( X
R
Gr^H*(X,Q(j))
This
category
(X $ (j))) - H
f
1
=
o f c) a s i n 11.5,
sequence
discuss
In view of
Q
q
o / H
cohomology
Hp(X,Q(j))
X/k
derived
the s p e c t r a l
q
duality
use the t e c h n i q u e s
11.8.
for
4
(Use t h e
] 3.1.14.2 ) .
For
on
-> S p e c
1
v i a a)
+
of a twisted
Iim
R
where
R f Q^(j)
by
(b))
o
of
i s part
j
Q
q
sheaf
.
of a Deligne
o f Hodge
over k there
i s a variation
cohomology
structures.
i s a n R' =
o f Hodge
R^
structures
d
for
1
R
=> R^
(C
(over
a
spectral
E
which
for
P'
1
Q)
q
of
H
P
"
,: X
R
S ,
R
R |
-+ S p e c
R
provide
t h e wanted
by p a s s i n g
= S ,
. The t h e o r y
R
C
spectral
sequence
to the l i m i t
s i t i n a short
exact
q
(S T
over R
should
give
q
R
that
+
there
c
i s a canonical
(t
s (t
groups
H (S
R (f i)*(b(j))
K
K
full
generality).
and f i l t r a t i o n
1
. The
initial
sequence
,R (f ,) Q(j)^(S^,R (f^), Q(j))
R
assuming
1
extension to
Hp(X«,,a(j,) ,
q
1
i s the base
R
S
should
H
, : X ,
R
= HP( J,,R (fJ,),Q(j))
Hp(X,Q(j))
T
f
f
(C
sequence
would
terms
R
, where
(C
mixed
-» T
F F
H
P
(sj, ,»P(f ,) *Q(j))
R
Hodge
structure
,
on t h e
r
T 5 L F
11.9,
L e t us p o i n t
spectral
(this
D
sequence
has not y e t been
out the analogy with
11.7.1
i s obtained
t h e £-adic
as the l i m i t
established i n
c a s e . The
over the
spectral
sequences
E
2'
Instead
q
q
q
= 3? <s ,,R (f ,)^ (j)) -S£ (x
t
R
R
o f t h e above
£
short
exact
sequence
R 1
,e <j>)
.
£
there
i s a spectral se-
quence
E
2 '
but
S
S
=
base
e
c
H
(
«' et ^'
R q ( 5
too ( c f .
of f
R
D
,
to
Q
(over
etale
(11.9.1)
R
1
K(TT,1)
H| (S
t
I
t
Q ) .
apparent consider
has
k
i s c o n n e c t e d and an A r t i n
R
'
^^
x
make t h e a n a l o g y e v e n m o r e
S ,
( j
o f H^ .
should i n f a c t give short exact
et
remark 1 1 . 4 c ) ) . Here
f
: X_, + S _
i s the
K K
K
f i n i t e l y generated, that
that
)
R' **£
modification
extension
To
an
p
a further
sequences,
is
S
e V
the case that
as f i e l d o f f r a c t i o n s and
n e i g h b o r h o o d o f Spec
k
, hence
. Then
R l /
R^(f ,)^)
R
= H^
o n t
(TT (S
1
r 1
),H^Xx K O.,))
k
f
,
while
(11.9.2)
where
TT
H
5
Q
(sj, ,R (F^ )
1
denotes
3 t c
Q)
S
P
q
= H (7T ° (S^,),H (Xx cr,Q))
the algebraic
1
and Tr^
k
o p
,
the topological
funda-
k
mental
point
group
Spec
profinite
the
(with
k
respect
Spec
1
R )
. Note
completion of
actions
on
H
g
to the
TT^
(Xx^,Q^)
O P
base
that
(S , )
and
H
TT ( S
and
R
q
points
o l
that
(X x^C! ,$)
)
i n d u c e d by
the
generic
i s isomorphic to
this
the
i s compatible with
v i a the
comparison
isomorphism.
Thus
and
the
produces
structures
dence
Of
over
potential
" o v e r Q"
between
course the
and
treatment of
the
$
.
latter
as
( v i a the
G^-representations
11.9.2,
solution
11.9.2 a p p e a r
parameters
of
and
two,
this
11.9.1
the mystery
i . e . , i n the
mystery
realizations
of
lies
be
i s quite
and
mixed
parallel
Hodge
i n the correspon-
crucial
should
the
S_,)
K
case k
that
same m i x e d
=
11.9.1
motive
Q
.
PART I I I
K-THEORY
§12.
AND
Finite
Let
fields
k
a prime
4-ADIC
COHOMOLOGY
and g l o b a l
be a f i n i t e
different
v a r i e t y over
function
field
fields
from p , and l e t
X
p , let £
be
be a smooth, p r o j e c t i v e
k. Then one c o n j e c t u r e s (see[Bei 4] 1.0)
12.1. C o n j e c t u r e
The c y c l e map i n d u c e s an isomorphism
Q
J
CH (X)
for a l l
® Q
^ H
£
2 J
k
(X,Q (j))
£
j > 0 .
Obviously t h i s
conjecture
B)
i s e q u i v a l e n t t o the conjunction of the Tate
and t h e c o n j e c t u r e t h a t
, t h e subgroup
3
CH (X)
q
g e n e r a t e d by t h e c y c l e s homologous t o z e r o , i s a t o r s i o n
In f a c t , one s h o u l d expect t h a t
i s known t o be t r u e f o r j = 1
Pic°
t h e group
/ k
points
i s finite)
q
(since f o r the a b e l i a n
Pic° (k) = P i c ( X )
0
/ k
and f o r
i s a f i n i t e group. T h i s
3
CH (X)
group.
1
= CH (X)
j = d , i f X
q
variety
of k - r a t i o n a l
has pure dimension d
(see [CSS] theoreme 5 ) .
By G r o t h e n d i e c k ' s
K
Q
(X)
conjecture
(
j
)
isomorphism
3
= CH ( X ) 0 $ ,
12.1 can a l s o be r e f o r m u l a t e d i n terms o f
o r t h e m o t i v i c cohomology
K-groups
3
Hjjj ( X
(j ) ) = K ( X ) ^
Q
K-theory,
. For the other
one has (see [ B e i 1] 2.4.2.3)
12.2. C o n j e c t u r e
(Parshin)
A g a i n , these groups
K
m
(
x
)
i
s
a
t o r s i o n group f o r m * 0.
a r e i n f a c t expected t o be f i n i t e ,
since
by
a general
This
of
finiteness
Quilien
One
12.3
conjecture
o f Bass
they
i s known t o b e t r u e
[Q2] a n d H a r d e r
c a n combine
Conjecture
be
finitely
f o r dim X <
1 ,by
generated.
results
[Har 1 ] .
conjectures
For a l l
should
i, j
12.1 a n d 12.2
G S
into
one has
G
Hj|j(X,<fe(j) ) 0 Q
Here
t h e map
character
12.1
G
1
the Weil
K(X)OQ
or given
by t h e C h e r n
To s e e t h e e q u i v a l e n c e
with
k
= O
for
i * 2 j
c o n j e c t u r e s , and t h a t
=
m
8
K
j>0
(X)
( j )
=
j
©
j>0
m
Hf! ~ (X,Q(j) ) ,
[ B e i 1] 2.2.
As
i n previous
varieties
Z over
Friedlander
chapters
and B e i l i n s o n
Conjecture
Let
U
a)
f
H e r e t h e map
suggest
to extend
( s e e [ B e i 2]
Let
Z
£
this
S H^ (Z,© (b))
k
£
variety
over
this
to
arbitrary
goes back t o
8.3.4b)).
be a v a r i e t y
t
be a smooth
i
HÛ Q(J))
we
k; f o r s m o o t h v a r i e t i e s
H^(Z,Q(b))®Q
b)
t h e z e r o map
that
£
12.4
f
i s either
a n d 12.2 n o t e
see
S
on t h e a l g e b r a i c K-group.
i H (X,Q (j))
by
£
• v
H (X Q^Cj))
1
over
k
, then
.
k
,
then
G
~
i
k
- H (U Q^j))
®
e t
i n
b)
f
.
i s g i v e n by t h e C h e r n
c h a r a c t e r and i n
a)
by t h e "Riemann-Roch
t r a n s f o r m a t i o n " as c o n s t r u c t e d by
cf.
8.3. O f c o u r s e ,
f o l l o w s from
cf.
8.1
clusion
known
b)
b ) , a n d b) t r i v i a l l y
i n the converse
a) b y P o i n c a r e
implies conjecture
direction
we
recall
Gillet
duality,
12.3. F o r a
con-
the following well-
f
12.5.
of
Semisimplicity Conjecture
1
G,
on H ( X C )
f
i s semi-simple
p
More g e n e r a l l y ,
12.6.
(Grothendieck/Serre)
one
The
f o r X/k s m o o t h a n d p r o p e r .
expects
Semisimplicity Conjecture
forarbitrary
varieties
( c f . [D7])
W
a)
For a variety
is
semi-simple
b)
Z
over
k, t h e a c t i o n
f o r a l l a,m
In particular,
Theorem
f o r a , b € ZZ
jectures
a)
+ CH*(X) ®$ = O
ties
b)
U
of dimension
Z
= Z\Z'
12.4
(that
also
c)
I f 12.4 a n d 12.6 b) h o l d
b)
X)
, Z
1
c Z
In the following
we
L
a
the
a closed
varie-
b
s u b v a r i e t y , and
£ TL , c o n j e c t u r e s
f o r two o f t h e v a r i e t i e s
f o r a l l smooth v a r i e t i e s
o f dimension
s e t T = G,
= Im a
/ Y
a
, H
Z,Z'
a n d U, t l
of
dimension
<d .
(Y) = H
a
= Im 3
•§
H ( U ) H-H
a
-,a
ra
(Y,$ (b))
0
a
f o r s h o r t . In the exact
. . .+H
(U)H-H ( Z ) 2 H (Z)
a+1
a
a
o
+ Parshin
A s s u m e 12.4 a n d 12.6 b) f o r Z' a n d U, t h e o t h e r c a s e s a r e
f
a
con-
a r e e q u i v a l e n t t o con-
I f , f o r a fixed
hold f o r a l l varieties
M
M
H' (Y) = H (Y,Q(b) )®<&,
a
a
x.
X
+ Tate
i f one o n l y c o n s i d e r s
K
let
holds, then
f o r t h e t h i r d one.
k, t h e y
similar.
k
t h e open complement.
are
Proof
of singularities
i s , Grothendieck/Serre
over
a) a n d 12.6 b) a r e t r u e
<d o v e r
1 of the Frobenius
<d .
be a v a r i e t y
true
a
i s semi-simple.
12.4 a n d 1 2 . 6 . T h e same h o l d s
Let
m
f o r smooth, p r o j e c t i v e
o
jectures
o f G^ o n G r H ( Z , © ^ )
the eigenvalue
I f resolution
12.3 a n d 12.5
-
E ZZ .
e n d o m o r p h i s m o n H^(Z,$^(b))
12.7.
action
, and
JC
sequence
, ( Z » ) -v ...
a— I
a
as i n d i c a t e d .
t o p row i n t h e c o m m u t a t i v e
diagram
By 12.6 b) f o r U a n d Z
1
0
y
* I
H
+
H^(Z)
a
i s exact. By
H
a
D
)
r
12.4
H
*
HNU)
a
-
(Z) + H
a - 1
+
-
a i
f o r U and
U — p
U
the map
(
a
(Z)
a
Z'
(
F
,
r
X
"
a-1
0
*
(Z')
and e x a c t n e s s
o f the bottom
row
p
•> Y
i s s u r j e c t i v e . T h i s shows the
a
exact-
ness of
(12.7.1)
0 +
X
r
+
H
(Z)
F
+ Y
F
-> 0
.
a
a
a
Hence the top row i n the commutative
->
H .
(U)
4
r
a. i I
H^, -(U)
a> I
-
H
+
T
(Z )
1 ,
+
a
H
H^(Z') +
a
i s exact, and we
3
T
(Z)
diagram
-* H^ (U)
a
T
+
a
H^(Z)
a
-
12.6 b)
, (Z )
for H
(Z)
F
-
a— i
(U) a
deduce the isomorphism H
5-lemma. C o n j e c t u r e
r
H
H^
M
a
.(Z')
a— i
~
(Z) -> H
-1
a
(Z)
f o l l o w s from
with
12.7.1
the
with
the
a
lemma below.
c)
now
f o l l o w s by i n d u c t i o n on the d i m e n s i o n ,
zero being t r i v i a l .
Given a v a r i e t y
the case o f
Z , the i n d u c t i o n s t e p
i n a p p l y i n g b) t o a smooth, open, dense s u b v a r i e t y U e Z
ZMJ
. Note t h a t we may
a)
I f one
has
take
U
dimension
consists
and
Z
f
resolution o f s i n g u l a r i t i e s ,
0
one may
use
simplicial
to
0
H ^(Y,Q^ (v) )
for suitable
v a r i e t y Y o f dimension
even assume t h a t
simplicity
for
Conjectures
Y
Z
y, v G 2Z
< dim
and
i s p r o j e c t i v e . Thus one
12.6 b)
may
can deduce the
and
12.6 b)
and
semiSerre.
U,
be
f o r smooth, p r o -
a g a i n by i n d u c t i o n on t h e dimension.
First
we
by a p p l y i n g b) t o a smooth
( r e s o l u t i o n of s i n g u l a r i t i e s )
get the r e s u l t
proper
f o r a r b i t r a r y v a r i e t i e s can
t r e a t the c a s e of a smooth v a r i e t y
compactification X
smooth and
from the c o n j e c t u r e o f G r o t h e n d i e c k
12.4 and
jective varieties,
a certain
Z . By u s i n g Chow's lemma one
deduced from t h e c o n j u n c t i o n of 12.4
Then we
=
a f f i n e , hence q u a s i - p r o j e c t i v e .
v a r i e t i e s as i n [ J 2 ] §2 t o get a s p e c t r a l sequence c o n v e r g i n g
W H (Z Q ) and i d e n t i f y i n g Gr H (Z,Q ) w i t h a s u b q u o t i e n t of
X/
m a
Jo
5lc
1
and
f o r a r b i t r a r y v a r i e t i e s by c ) .
Z' = XMJ
.
12.8.
Lemma
Let k
be a f i n i t e
(^-representation V of
eigenvalue
0
T =
1-semi-simple,
1 i s semi-simple
+A
+
B
f i e l d . Call a (finite-dimensional)
i f the Frobenius
on V . L e t
C+ O
+
be an e x a c t sequence o f ^ - r e p r e s e n t a t i o n s o f T
a)
B
i s 1-semi-simple i f and o n l y i f A and C a r e 1-semi-simple
and t h e sequence
O H- A
(12.8.1)
r
H- c
R
B
H- o
r
i s exact. In p a r t i c u l a r , i f
...H-B^ H-B H-B
1
m+1
m
m-1
i s a l o n g e x a c t sequence o f Q^-T-representations,
w i t h B^
1-semi-
simple f o r a l l m € TL , t h e n
r
. . . H-B
H-Bmr H-Brm-1 H- . . .
m+1
i s exact.
„
b)
If
B
„
has a weight
filtration
(i.e.,
i s a mixed (j^-sheaf on
W
Spec k) and Gr B
Proof
a)
i s semi-simple,
then
exact
1-semi-simple.
Fr € T
i s the F r o b e n i u s
f
we have
sequence
O H- v
for
is
I t i s c l e a r t h a t 1 - s e m i - s i m p l i c i t y c a r r i e s over t o quo-
t i e n t s and s u b r e p r e s e n t a t i o n s . I f
an
B
F
H- v
F
5*"
V H- V
1
P
H-
O
any r e p r e s e n t a t i o n V, so t h e snake lemma shows t h a t
O H-
A
P
H-
B
P
H- Cp H- o
i s exact i f and o n l y i f 12.8.1 i s . The c l a i m now e a s i l y f o l l o w s
from t h e commutative e x a c t
A
r
fa
A
r
*
B
-
B
diagram
r
r
+
c
r
r
and t h e f a c t t h a t
an isomorphism.
V
i s 1-semi-simple i f and o n l y i f
V
H- Vp
is
b)
One has a T-isomorphism
1 o n l y appears i n G r B
(note
Q
12.9
Remark
W
B = ©
that
G r B , and t h e e i g e n v a l u e
E x t J (B,B
1
1
1
) = Horn
(B B )
f
r
) .
L e t us s i n g l e out t h e two p r i n c i p l e s u n d e r l y i n g
theorem 12.7:
a)
The s e m i - s i m p l i c i t y c o n j e c t u r e
implies
that
G
i k
(X Z) h->H|(X,Q^(J))
f
x
r v ^ H (X Q
a
forms a t w i s t e d
b)
f
jl
(b))
Poincare
k
duality
theory.
A morphism between t w i s t e d P o i n c a r e
duality theories
i s an
isomorphism i n homology i f i t i s an isomorphism f o r smooth v a r i e t i e s .
I f one has r e s o l u t i o n o f s i n g u l a r i t i e s ,
reduced t o smooth and p r o p e r
In
b)
the l a s t question
can be
varieties.
i tactually sufficies
t o have a weak form o f r e s o l u t i o n
of s i n g u l a r i t i e s :
every smooth v a r i e t y
dense s u b v a r i e t y
V
U
has t o c o n t a i n an open,
which has a smooth c o m p a c t i f i c a t i o n . Homology
i s b e t t e r behaved t h a n cohomology, s i n c e by t h e r e l a t i v e
sequence 6.1 f ) one can c u t any v a r i e t y i n t o p i e c e s
exact
t o study i t s
homology. F o r cohomology one e n c o u n t e r s t h e problem t h a t
H^(X,j)
i n general
i s singular
(besides
depends on the embedding o f
t h e problem t h a t
HjJj
z
(X,Q(j))has
Z
in
X
i f X
not y e t been
defined
for singular X).
For
t h e a p p l i c a t i o n o f the 5-lemma i t i s v e r y
important t o
have an isomorphism f o r smooth and p r o p e r v a r i e t i e s ;
has
an i n j e c t i o n o r a s u r j e c t i o n i t i s not a t a l l c l e a r how
extends t o a r b i t r a r y v a r i e t i e s . One e n c o u n t e r s t h i s
if
i f one j u s t
1
one t r i e s t o extend B e i l i n s o n s c o n j e c t u r e s
varieties,
this
problem,
to a r b i t r a r y
1
s i n c e i t i s not c l e a r how t o extend B e i l i n s o n s groups
HjJj(X/E ,Q(j))
to arbitrary varieties.
From theorem 12.7 we can deduce t h e f o l l o w i n g
12.10. Theorem C o n j e c t u r e s
for
rational
Proof
surfaces
12.4
and
12.6
are t r u e f o r dim
(not n e c e s s a r i l y p r o p e r or
For a smooth and
and
3
the
f i n i t e n e s s o f CH (X)
conjecture
, j = 0,1,
and
smooth).
p r o p e r v a r i e t y X o f dimension <1
j e c t u r e i s t r u e as remarked above, T a t e ' s
true,
Z<1
Parshin's
is
con-
trivially
i s a l s o known. F u r t h e r -
o
more, the
H
2
conjecture
(trivial)
The
of G r o t h e n d i e c k and
f o r H (X,Q ) = T P i c ( X ) ^
£
5
case o f a r b i t r a r y c u r v e s
More e x p l i c i t e l y ,
the
i s t r u e f o r H°
1
and
now
r e s o l u t i o n of s i n g u l a r i t i e s
be
Serre
follows with
f o r dim
M
Z
let
b
f f i
be
Z<1
the
12.7,
T
a
t
e
T 2
^ ^
•
s i n c e one
has
.
normalization
r e g u l a r l o c u s of Z, T = ZMJ,
Y
and
v
and
T = Z -U.
o f Z,
let U=Z
Then the
recr
^
exaxt
sequences
...-H
(T) - H (Z) - H (U) - H
.(T)
a
a
a
a-i
...-H (T) - H (?) - H (U) - H
.(T)
a
ci
a
a— 1
(12.10.1)
a
where we
have s e t H
Gr^ H (Z) =Gr^
1
Gr
Now
W
1
H (Z)
O l
1
a
(Y) = H ( Y Q )
a
Xy
o
f
H (U)
1
c GrJV(U) c H
0 0
—
w
Gr H (Z) = 0
m 1
—
H (Z)
(T)
f o r m * 0,-1
i
£
H (Y Q )
a
f
Q
£
c
ji
a v a r i e t y Y , where C(Y)
... ,
.
C (Y)
'
K
C <*>
0
for
-
and
Q (d)
(12.10.2)
.
,
1
O
...
f o r s h o r t , show
0
=
1
-
,
a=d:= dim
Y ,
,
a - 0
,
,
a>2d
or a < 0 ,
( r e s p . C ( Y ) ) i s the
c
set of
( r e s p . compact connected) components o f dimension d o f Y
the
G a l o i s a c t i o n on t h e s e ! ) . T h i s shows the
for
12.4
and
12.6b) one
A rational
3P
2
all
by
surface
may
the
first
t w i c e . For
the
12.7b) t o
obtain
Z;
12.10.1.
to a v a r i e t y P =
3
i s some dense open s u b v a r i e t y
are known f o r P (see
s t e p , we
(regarding
semi-simplicity for
Z i s b i r a t i o n a l l y equivalent
2
1L. . . IL IP ^ , i . e . , t h e r e
conjectures
j u s t apply
irreducible
12.4
14.1), and
and
12.6
b)
v
Zz>UcP . S i n c e
f o r Y=Z -U and
1
f o r Z by a p p l y i n g
s e m i - s i m p l i c i t y l o o k a t the exact
sequences
v
Y =P -U
12.7
b)
... - H (Y) -> H (Z) -> H (U) ... -* Ha.a (P) 1* HSia (U) -•H da-1,(Y )
. . .
f
We
may
since
choose U a f f i n e and
H
Then H
2,3,;
2
2 ^ k ' ^ £ ^ ^
a
1 S
(U) = 0 f o r a<1
3
W
GrmH (Y)
a
, and
= 0
note t h a t H(P)
t
^
H
a
. We
^
l e
(U) + H
Gr
H (Z)
a
W
W
W
Grm H 2(Z)
9
f
.(Y )
a I
f
W
is injective
for
for a =
m € TL ,
f o r m = 0,-1
f
9
W
a=2,
obtain
W
0
for
c l a s s of a hyperplane s e c t i o n .
Grm H-(U)^
Grm H20 (Y )
3
G rm HZ ( U ) W G r H
(Y )
ml
W
Grm H 3(Z) <^
j * i s zero
f o r a = 0 , 1 , m € ZZ ,
W
m
...
such t h a t the map
e n e r a t e
9
-+
1
,
W
Gr H (Y) -~ Gr H (Z) ,
2
2
2
Gr H (Z) = 0
2
using
2
for m * 0,-1,-2
W
the bounds f o r the weights g i v e n by the
g e t h e r w i t h 1 2 . 1 0 . 2 we
of Y and
Y
f
over
12.4
or
k
b))
have reduced the q u e s t i o n
For a v a r i e t y
To-
t o the known cases
, the
( r e s p . a smooth v a r i e t y ) of dimension
r i g h t hand s i d e i n c o n j e c t u r e
v a n i s h e s f o r a < 2b
i < j
formulae i n 6.5a).
.
12.11 . Remark
d
,
or
j > d ) , see
or
8.12
i s known f o r
(resp. i > 2j
or
j < 0
and
S o u l e , see
[Sou
has
t o be
replaced
For any
field
a > 2d
(resp.
3]
by
or
a)
b < 0
a < 2b
i > j+d),
or
b > d
(resp.
( r e s p . i > 2j
( r e s p . 5.11). For the
s i d e the v a n i s h i n g
or
a > b+d
12.4
left
or
B e i l i n s o n and
i < 0),
see
(resp. I o c
3]
2.9
and
"j<-n"
c i t . proposition
Soule c o n j e c t u r e
[Sou
b > a
by r e s u l t s of S u s l i n
Theoreme 8 - where the c o n d i t i o n
" j > n"
hand
a vanishing
for
[ B e i 1] 2.2.2.
5).
12.12. Theorem
For a v a r i e t y
d o v e r the f i n i t e
field
i s t r u e f o r (a,b) =
=
k , c o n j e c t u r e 12.4
(2d,d) , (2d-1,d-1)
(0,0), (1,1), (2d,d))
Proof
We
( r e s p . a smooth v a r i e t y ) o f
f a c t , the c y c l e c l a s s i n d u c e s an
d
case
we
0
0
f f i
Z^
=CH (Z)S ZZ
d
(2d-1,d-1)
H
+
(
( G
H
k' 2d
k
•
xez
>
f
f
( 2
i
'
f f l
(0,0)
b))
(resp. ( i , j )
case
12.4
a ) , the o t h e r
(2d,d)
i s clear, i n
isomorphism
=H
j l
for
2 d
(Z E Id)) ~ H
l
i
(Z , ZZ
2 d
(d) ) ^ .
%
f o l l o w s from 8.13.5: w i t h the n o t a t i o n s t h e r e
diagram
*
( d
-
i
1 ) ) )
* H
E
-
2 d
(Z, ZZ ^ d - D )
d , - d + i i
-
-
( E
H
2 d
oU+i
.
] (
( z ) /
?,^(d-1))^
l
®
Q
xez
( d )
s i n c e the groups
an
2 2
have a commutative
* cont
( r e s p . 12.4
.
case f o l l o w s by P o i n c a r e d u a l i t y . The
The
or
have o n l y t o show the statement
Z (Z)S
a)
dimension
are f i n i t e .
For the same r e a s o n t h i s
m
C
induces
Q
2d-1
( Z
'®
( d
-
1 ) )
V£
3
E
d,-d+1
( Z )
9
ZZ^
Z
H
2d-1
( g
'°£
( d
-
1 ) )
k
c o i n c i d i n g w i t h the t r a n s f o r m a t i o n T
The case
of
B l o c h and
(a,b) =
Ogus
(0,0)
f o l l o w s w i t h the s p e c t r a l
'
( d )
isomorphism
H
0
sequence
E
( 0 )
p,q
=
where
by
X
(
' ^
0
)
)
" V q
(
' ' ^
(
0
)
)
'
(P)
definition
H (x,2Z (b))
a
the
(
V q
X
1
limit
m
= ^
£
running
H (U,2Z (b))
a
£
over
a l l open
for
q
subvarieties
U c: ( x )
. Now
one
has
E
(b) = 0
2
<
0
2
as
follows
(cf.
easily
from
[ G i ] § 8 ) , where
ciated
the
i t holds
Ur^>H (U,ffi
q
to
smooth
case
by
since
(ffi^(j))
j)) , vanishes
f o r the E
excision
for
q
, the
> d by
sheaf
-terms
asso-
([Mi] VI
7.1).
2
E
Therefore
hand,
E
Bloch
o,o
(
0
q
q
R
0
=
P
Supp(a)
1
Z
{a €
Supp(Z)
c
to H ( Z , ZZ^(O))
i s isomorphic
the
other
P
o
(Z)|
and
there
exists
c l (a) = O
torsion.
Obviously
a
But
then
this
has
t o be
i t follows,
Y
in
of
P
+
R
( Z )
such
(see
that
[BO]
( 7 . 2 . 2 ) h a s t o be r e p l a c e d
N„Z (Z)
i n C H (Z)
is
I o
o
checked
e.g.,
£ Z
(Y, TZ ^(p) ) }
in
7.2, w h e r e i n f a c t t h e i n d e x
p+r
by
p + r - 1 ) . I c l a i m t h a t t h e image
curve.
. On
Q
a n d Ogus show t h a t
= Z (Z)ZN Z (Z)
)
N Z (Z)
with
(0)
from
for
12.10.
d
=
We
1
, i.e.,
obtain
an
for
iso-
morphism
CH (Z)
Q
which
0
implies
®
5
£
H (Z,<£ )
Q
,
£
the claim,
since
G
CH (Z)
(CH (Z)
Q
12.13
Now
function
Q
let
field
k
0
Q)
k
*
£
.
be
a global
function
i n one
variable
over
a
field,
finite
that
field
i s , an
IF
algebraic
. Without
q
restriction
we
may
assume
that
IF
i s algebraically closed
in
q
k
. Then
there
C
over
curve
If
variety
Z
exists
JF^
a
with
i s a variety
W c c
and
smooth,
a
function
over
flat
projective,
k
model
field
, then
Z
geometrically
k
.
there
of
connected
Z
exists
over
an
open
sub-
W
(that i s
f:
Z^-W
models
[EGA
flat,
of f i n i t e
become i s o m o r p h i c
IV]
over
Z^x^Spec
a suitable
• F
. Call
I
Let
a
* p = char
k
(finite-dimensional,
q
V
of
arithmetic,
W
as a b o v e
In t h i s
(that
such
1
W
(see
and l e t k^ =
continuous) Q -representation
x,
0
a smooth
G^ - A u t ( V )
Q^-sheaf over
factorizes
through
some
TT^ (W,Spec k) ) .
case l e t
V
where
subvariety
be a prime
i f i t comes f r o m
i s ,i f
H (k,V) = I i m
H
open
k = Z ) . Two
8.8.2.5).
12.14. D e f i n i t i o n
k
type, with
v
U
o o
, V)
H^
H
= Iim
the inductive
( T i (U, S p e c k) ,V) ,
1
o n t
cont
limit
(TT
(
1
^
,
S
p
e
C
i s over
k o o )
'
V
)
'
a l l open U c W
a n d U = Ux
f
IF
12.15. Remark
sheaf
In the notation
V
v
H (K
o o f
=
V)
(Spec
V
H
= H
12.16. T h e o r e m
v
k/k;3F
Vk
(Spec
Let
Z
c o
,V)
Jff
potential
I f c o n j e c t u r e 12.4
cofinal
system
i s true
o f open
;
over
k
, and f o r U c W
open,
with
1
T7
U
a
a
,V) .
be smooth
n o t a t i o n s o f 12.13, l e t Z
a)
V i s just
o n S p e c k, a n d
H (k,V)
the
o f 11.7,
= Z x U = f ~ (U) .
WW
T7
T7
f o r ( i , j ) and Z , f o r U r u n n i n g
jj
subvarieties
o f W,
then
through
t h e r e i s an e x a c t
sequence
r
0
-
with
b)
H
1
(K
1
H ""
a j f
1
F
( Z G ( J ) ) ) ->HjJj(Z, Q ( J ) ) © $ £
f
Let
U c W
over
be open
U
-
u f
surjective
x
U IF
G
i
H (Z,Q (j))
£
k
,
%
'
t
such
that
Z
1
rj
i s smooth and R fQ
+
is
. If
Hj(z c(j)) ®
2
e
T = G a l ( k /k) = G a l (3F /JF ) .
«>'
q
q
smooth
is
.
q
q
and H|
fc
h
e
n
H
it
( z
u'V
( Z ,Q, (j ))
y
j
j > ) r
i s 1-semi-simple
(12.8),
for Z
u
=
r
is
:
i , j
h
i
M
(
Z
(
' Q
3
)
)
® ®l ~*
i
^
(
G
'
Z
t
Q
£
(
j
)
k
)
surjective.
c)
I f f o r some
U
as i n
(Hi(Z ,«(j))/Hj(Z
x
is
=
H
injective,
N R
q
r
if
j
«(j)) )^
-
0
closed point x € U
H* <Z ,« <j))
t
Z x <(x)
U U
the fibre
, then
t h e same
i s true f o r
TT
TT
x
£
of
f
over
x
and Z
x
.
(H^(Z, Q(J)
S
a n d some
Z
=
x
J
Z X , w
x IF
q
x /
b)
)/HjJj (Z,<D(j) ) )
- H^. ( Z G ( J ) )
® ^
Q
f
e
,
G
q
similarly
one
Z
i
j
Z^
, x € U
closed,
Q
Hj(Z,4(j))
:
isomorphisms
true
K
)
5
9 ^
0
S
£
1
H (k
f o r a l l i , j € TL
then
H ^ . (Z , $
G,
(j ) )
K
£
,H^ (Z,G (j)))
1
T
O
and
r
£
, and a l l T a t e
conjectures are
for Z .
Proof
(see
a),b ) :
We
may
assume
[ M i ] V I 7.2) a n d h e n c e
gives
an e x a c t
1
1
O-HH
We
f
H ^ . (Z ,(I^ ( j ) )
12.4 a n d 12.6 b) a r e t r u e f o r
(H^j(Z,G(j))/H^j(Z,Q(j) ) ) ® G
r ^ . :
are
i f conjectures
and a l l f i b r e s
?
: HjJj (Z , Q ( j ) )
±
f o r Z^ .
d)
u
= Ker ( r ^
s e t HjJj (Z , Q ( j ) )
w h e r e we
(U R "^
1
i
i s affine;
Leray
spectral
then
cd^(U)
sequence
1
1
) - n H ( Z , ^ ( j ) ) - H° ( U R ^ Q
1
o b t a i n an e x a c t
OH-H
the
U
for
<1
f
sequence
I^ (J)
f
that
u
f
e
(j) ) ^ O
.
sequence
1
R
i
(U,R " f^ (j)) H-H (Z ,Q (j) )
£
u
(12.16.1)
R
£
i
H- H ° ( U , R f ^ ( j ) )
F
+
|ch
H^(Z S)(J))
u f
with
map
By
the Chern
c h a r a c t e r as i n d i c a t e d ,
i s surjective
Deligne's
i fH
generic
1
( Z , Q^ ( j ) )
base
y
change
where
the right
vertical
i s T-semi-.simple .
theorem
([SGA 4
•^][finitude])
V
the
sheaves R f * Q^
a r e smooth i f
U
i s s m a l l enough; then they
can be i d e n t i f i e d w i t h t h e TT (u,Spec k) = : TT^ ( U ) - r e p r e s e n t a t i o n s
1
V
(R f Q^)
+
one
V
£ = H ( Z , Q ) and t h e diagram 12.16.1 w i t h t h e f o l l o w i n g
S p e c
£
( c f . [Mi] V 2.17)
1
O-H^
O
N
(TT
T
i-1 T
(U) ,H ' ( Z Q ^ ( J ) ) )
1
i
1
H
f
1
1
U
(12.16.2)
-
T
T
, Q (J))'
J
i
*1
-H (Z Q ^j))
(
U
)
1
f
i
j
Ich
Hj(Z ,«(j))
0
By p a s s i n g t o t h e l i m i t
over the U
we g e t a commutative
exact
diagram
Q
1
O-H ( K
1
cof
1
r
H " " (Z Q (J)
t
(12.16.3)
1
)) -lim
l
H (Z ,Q (j))
T J
K1
0+H
1
(Z Q(J) ) m
f
using
o
r
/
£
k
£
I J
9
f
f
1
i
A
1
- ( H ( Z , Q ( j ) )/H (Z,©(j) ) J ® Q
HjJj (Z,CB
-
z
®Q
1
- H (Z,Q (j))
£
- 0 ,
that
K
(12.16.4)
cf.
Ch
r
£
2
j - i
(
Z
)
(
J
)
s s l
J
m
K
2j-i
( Z
U
) ( J )
[Q1] §7, 2.2. The assumption o f a) i m p l i e s t h a t ch ®Q^ i s an
isomorphism,
For
so t h e statements o f a) and b) a r e c l e a r .
c) we use t h e commutative
H^Z
x f
Q(J))
diagram
-fi-U
SPJ
(Z
x f
Q^j))
fsp
M
Hjjj(Z Q(j))
i
/
j
f
sp
t
v
(12.16.5)
where
1
H
y
Hj (Z,Q (j)) ,
t
£
i s t h e s p e c i a l i z a t i o n map i n e t a l e cohomology - which can
be o b t a i n e d a s t h e d u a l o f t h e g e n e r i z a t i o n map f o r cohomology w i t h
compact s u p p o r t , c f . [DV] exp. 0 , 4.4 - and where
sp^ i s the s p e c i a
z a t i o n map i n m o t i v i c cohomology. The l a t t e r one can be d e f i n e d as
follows
for
( c f . G i l l e t ' s c o n s t r u c t i o n and remark on K
V c U
open,
1
x
s u f f i c i e n t l y s m a l l , l e t t € 0(V)
meter a t x € U , and a l s o denote by
t
i n [ G i ] 8.6
ff.):
be a l o c a l p a r a -
t h e c o r r e s p o n d i n g element i n
1
H (V Qd))
M
class
0(V)
=
f
X
$ Q
o f x under
...
6 i n the
- H J (U Qd))
i
v
Z = U -V
...
one
j
Hj(Z ,Q(j))
6
comes
U f
v
HjJj
+ 1
f
f
defines a
t
-
exact
- H J (V Qd))
f
and
f
sp :
where
f
( c f . 6.12.4 e ) ) . T h e n
from
*
$ Hj^ (U Qd))
z
-
f
J (Z ,4(j+1))
fundamental
H^(U Qd))
f
-Hj(Z
v
exact
Q(j))
x f
f
sequence
+1
HJ (^^(J + DJ^HJ^
-
u
the
sequence
+ 1
H
(Z ,<J(j + U )
maps t o
map
( t )
the
t
( Z
u
^ ( J t D )
-
...
IIZ
This
depends on
the
H JjU Q(J))
=
j
does
not.
with
for
f
t
the
The
c h o i c e of t , but
i
Iim
HJ(Z
induced
map
,4(j))
sp
on
H (Z ,Q(j))
M
v
same c o n s t r u c t i o n
r e p l a c e d by
the
e
its first
commutativity
of
6
H ^ t y V J ) )
can
be
Chern
12.16.5
carried
class
t ^
i t remains
out
in etale
cohomology,
€ H^ (V Q^(I))
t
t
to
show
f
and
f
that
H^,^),
* ' - " * ' " * * : "
H| (Z Q^(J))
t
commutes.
T h i s f o l l o w s from
cupproducts,
Now
c)
with
similar
i s clear
from
compatibility
arguments
the
Hi<Z ,<fi(j)>/Hj<Z <B(j))
x
the
f
x f
as
i n [DV]
commutative
c_
o
e t
Hj(Z Q(j))/Hj(Z Q(j))
f
d)
that
the
f
sp
i s an
i s now
x
phism
i f the
clear
from
12.16.3 c h
right
12.17. Remarks
£ ( j
a)
and
a)-c)
®Q„
Q
J '£
by
and
the
VI
3.6.
f
))
(J))
similar
is
b ) : by
for a l l
is bijective,
map
'
smoothness o f
12.7
12.6b) h o l d
vertical
A
(5
T
Q
isomorphism
c o n j e c t u r e s 12.4
diagram
exp.
with
i+sp
H
which
specialization
diagram
H (Z ,g>
J-
in
of
then
I^f^Q^
induction
, V c
r*. . ® Q „
U
.
i t follows
, and
i s an
i f in
isomor-
injective.
theorem h o l d s
for
arbitrary
varieties
and
homology. The s p e c i a l i z a t i o n
the
generization
motivic
one h a s t o u s e t h e p u l l - b a c k
14.4 b e l o w ) .
H U,®(b))
i
= Iim
a
and
follows
homology
map f o r c o h o m o l o g y w i t h c o m p a c t
homology
map f ( c f .
map f o r e t a l e
from
Formula
H
a
+
12.16.4
i s the dual o f
s u p p o r t , and f o r t h e
morphism
f*
f o r the f l a t
becomes
(Z QCb-M ) )
ir
2
[Q1] § 7 ( 2 . 4 ) ,
and e v e r y t h i n g
c a n be p r o v e d by c o n s i d e r i n g
cohomology with s u p p o r t s ,
b)
If
Z
i s smooth and p r o p e r ,
then
G
i
k
(Z,<S.,(j))
H
=O
f o r i * 2j ,
'
1
cont k'
H
• ^1
i-1 V
H (^,H
(Z Q^(J)))
l
1
1
( G
H
i-1 < '<Vi>>
1
Z
i
'
<
2
j
_
1
. _ ,.
or 2.
Zj
1
7
H
1
(k,H
l _ 1
(Z,Q (j)))
,
£
0
In
fact,
O-H
1
one h a s a H o c h s c h i l d - S e r r e s p e c t r a l
(Z,Q^(j))
H
the
vanishing of
for
H
i-1
- o ,
i s pure
o f weight
r-2j
G
c
(T,H " (Z,Q (j))
* 2 j . T h e map
1
o
n
1
t
i
1
1
1
f
injective,
Finally,
k o
£
H (k,H - U Q^(J)))
H
S
which
global
1
1
(Z Q (J) ) )
1
oo
The
f
H
(H "
1
above
f
r
j l
= 0
function
and extends
field k).
the f i r s t
G
(Z,Q^(j)) ~ )
c l a i m and
k
r
(G ,H " (Z,Q (j)))
1
k
£
f o r i * 2 j - 1 , s e e [ J 3 ] lemma 4 .
1
o
n
t
of weights
investigations
sharpens
1
, implying
1
o n t
f o r U a s i n 12.16 b) , H
Ck ,H ""
1
1
°) =
and an isomorphism
( U , R " f * Q „ (j) ) i s m i x e d
l _
H
sequence
—
and
1
i > 2j .
( ^ " ^ , ^ ( j ) , ^
o n t
r
is
i = 2j-1 ,
(Tr (U)
1
, H "
1
>i-2j
1
(Z , Q ^ ( j ) ) ) =
([D9] 3.3.5),
f o ri > 2j .
suggest the following
the conjectures
conjecture,
8.5 a n d 9.15
(for a
12.18. C o n j e c t u r e
€ TZ
a,b
let
( c f . 9.4).
a)
Let
H^(Z,Q(b))
a
Z
be
H
a
( Z
isomorphisms.
b)
® V
H
M
for
(X
'
<C(
^
1
j
Cl ^CH (X)
Remark
)) 8 > < C
saying
Formally,
(k^ "
, and
k ,
(X Q (J) ) )
f
j l
the A b e l - J a c o b i
2j
c o n t
1
(G , H "
r
1
p r o p e r over
1
£ -
H^ CZ,^ (b)))
1
map
(X,C (j)))
k
A
part
b)
can be
e x p r e s s e d f o r a l l i,j€22
0 Q
Q
£
-
H
1
1
(k,W^ H "
1
1
(X Q (J)
f
l
) )
t
should
be an isomorphism f o r a l l i , j G Z . I f H
semi-simple, one
H
o o f
that
HjJj(X,fl(j) )
is
(K
isomorphism f o r j _> 0 .
12.19
by
1
£
smooth and
» C, - H
o
a
H
X
k
- H (Z,Q (b))
£
«
0
i s an isomorphism f o r i < 2j
i s an
for
k
b
<®< >>
are
h
, and
Then
a,b®V
^ i,j
k
= K e r ( r ' . : H ^ (Z ,Q (b) ) - H (Z ,Q (b) ) )
a ,o
a
a
u
(H^(Z,(J)(b) )/H"(Z,(|>(b)) )® «
?
a v a r i e t y over
1
where
1
(k,W
1
.H "
1
1
(X,Q^(j))
has
(X QAj)))
=
r
S , S
S (k,Q )
a
36
0
i s the
Ext
1
1
(©,,H "
category
1
(X QAj)))
,
r
of a r i t h m e t i c a l Q - S h e a v e s
0
X-
W
F
w i t h weight f i l t r a t i o n on
simple.
be
The
motivic
t h a t the
12.20. Theorem
b)
and
=
Conjecture
Proof
By
(a,b)
12.12
a)
Conjecture
(2d-1,d-1)
12.18
or
(up t o
12.18
are
semi-
(compare §-]-j ) would
functor i s f a i t h f u l
2-extensions over k
a)
f o r which the Gr^F
i n t e r p r e t a t i o n of t h i s
£.-adic r e a l i z a t i o n
are no m o t i v i c
X
Spec k,
and
that
there
torsion).
i s t r u e f o r smooth v a r i e t i e s
(2d,d)
, d = dim
X .
i s t r u e f o r X = Spec k .
T h i s f o l l o w s from 12.16.3 f o r
the middle v e r t i c a l map
i s an
(i,j) =
(1,1), ( 0 , 0 ) :
isomorphism, and
for (i,j)
=
(1,1)
the
b)
This again
t h e r e we
is
an
r i g h t v e r t i c a l map
may
follows
take
Z^
= U
isomorphism. I f
all
groups v a n i s h .
For
i = 1
from
, and
case
isomorphism by
12.16.3. With the
by
i * 0,1
The
12.10
, or i f
(i,j) =
i = 0
(0,0)
have H ( S p e c k , Q ^ ( J ) ) = 0
M
)
and
the
claim
5.16.
notations
the m i d d l e v e r t i c a l
1
we
= Hl(Spec k,$(j))
and
has
j * 0
, then
been t r e a t e d
, hence H^(Spec
map
above.
k,$(j))
follows.
O
§13.
Number
fields
In a n a l o g y w i t h the
f i e l d s we
13.1.
be
i s an
conjecture
Conjecture
conjecture
the
Let
12.18b) f o r g l o b a l
following.
k
be
a finite
extension
a smooth p r o j e c t i v e v a r i e t y o v e r k
Then the
r
i
f
j
:
Hj<X,«(j))» 8
fi
the
- H
£
i s an
(0,0)
f u r t h e r d i s c u s s i o n and
[J3]
§2, where t h i s
with
12.14
we
(G
c o n f c
Chern c h a r a c t e r
for i = j , ( i , j ) *
For
, and
of Q
let
f
I
let
be
X
a prime.
map
i n d u c e d by
jective
function
1
k 7
1
H "
(X,Q) (j)))
£
isomorphism f o r i < j
in-
.
motivation
conjecture
and
we
i s stated
r e f e r the
for i < j
reader
to
. I f i n analogy
define
V
1
V
i
H ( k , H ( X , Q ( j ) ) = H (k/k;2Z , H ( X Q ( j ) ) )
£
/
(13.1.1)
where
U
=Iim
r i n g o f i n t e g e r s and
all
those above
I
1
f
[J3]
i
lemma 4.
S
a finite
o r where
H (k,E (X Q (J)))
see
Q N T
(TT
runs over a l l open subschemes o f
the
1
H^
1
S
f
H^
X
o n t
has
(G
In t h a t paper we
£
i
1
(U) , H ( X , Q ( j ) ) )
£
Spec O^S
reduction,
1
k f
, with
s e t o f primes o f k
bad
H (X Q (J)))
f
i
also discuss
,
containing
then we
have
f o r i * 2j-1
the
following
,
13.2.
Conjecture
2
For
X/k
as above
1
H ( k , H ( X Q ^ j ) )) = O
for
7
In
view
P
E '
q
of the spectral
P
q
q
H (X,Q (j))
K
= O
£
conjecture
1
7
, ( i , j)
H
(X Q(J))
7
induced
for
dimension
2
q * 2j ,
a
<d
13.1 a n d 13.2 t o
X
a s above
/
then
(X,Q (b)))
&
<d,
then
£
to arbitrary
13.2 i s t r u e
= d+b .
i s an
isomorphism
= 0
for
varieties.
f o r smooth,
f o r any v a r i e t y
X
a > d+b+1
f o r smooth, p r o j e c t i v e
X
of dimension
+ H ( X / k , $ ( b ) ) =:
a
( c f . 11.7.c)) i s a n i s o m o r p h i s m
a
( c f . 11.6.2)
fori = j .
this
f o r any v a r i e t y
H (X,Q(b))®^
£
character
If
I f 13.3 i s t r u e
t h e map
-> H ( X G ( j ) )
l
0
( X , < ^ ( j ) ))
1
Q Jb
a)
l _ 1
one has
7
For
q
as above
(k,H
combine
b)
a
(J))
j l
(O O) .
t o extend
Theorem
H (k,H
1
X
and i n j e c t i v e
We w a n t
of
*
by t h e Chern
i < j
13.4.
may
Conjecture
1
for
^ H
j
H e n c e we
13.3.
(X,G
that
For
H (X G ^j))
i < j
P + q
13.2 i s e q u i v a l e n t t o
13.2'. C o n j e c t u r e
for
=> H
£
and t h e f a c t
G
i+1 < j .
sequence
= H (k,H (X,Q 'j)))
(cf.11.7.1)
one has
£
projective
of dimension
d
varieties
over
.
varieties
d over
of
dimension
k t h e map
H (X,G (b))
a
f o r a > d+b
k
£
and i n j e c t i v e f o r
For
13.5
be
Lemma
If
we u s e t h e f o l l o w i n g
Let X
a closed
a)
of
theproof
be a v a r i e t y
subvariety
such
a
thev a r i e t i e s
f
X
U = XMT
= 0
£
and
of dimension d over
that
H (k,Gr^H (Z Q (b)))
lemma.
i s dense
f o r a > d+b+1
U , then
this
and Z = Y
vanishing
also
other one.
b)
I f t h e map i n 13.4 b) h a s t h e p r o p e r t y
stated
and
one o f t h e v a r i e t i e s
property
for
t h e other one.
Proof
a)
Gr
We h a v e
H
{
( b )
this
belongs
• • -+ m a * > ® l
and
since
a long
Gr
exact
(
^ m a *'®*
absolute
Hodge c y c l e s
of
short
split
H
and
exact
( b )
U , this
Y c X
i nX .
the
X
k, l e t
a n d one
holds f o r
there
for Y
also
holds
sequence
Gr
H
( b )
>^ ma
t o a sequence
( c f . 6.11.10/
sequences.
Gr
H
>+ m a-1
V
b
)
> '
of p o l a r i z a b l e motives f o r
i t c a n be s p l i t
This
shows t h a t
into a series
t h e sequence
remains
2 ~
exact
after applying
since
a > d+b+1
b)
We h a v e
the functor
implies
H
diagram
of exact
. . . - H ( Y / k , Q ( b ) ) ^ H ( X / k , £ ( b ) )-*H
£
a
£
+
f
a
(UA Q
f
i
f
a
so
theclaim
I.
3.1) a n d t h e f a c t
dim
Y + b
Proof
13.4
easily
e
(b) ) - H ^
a) b e i n g
2
H (k Gr^H
f
f
follows
that
with
(resp.
that
f
t h e weak
a > d+b
induction
(X Q,(b)))
1
= 0
(resp.
£
d
f
-> H ^
1
four-lemma
a = d+b)
a > dim Y + b
on
(Y/k,$ (b) )+ . . .
£
t
f
f
a n d a-1 > d+b
o f 13.4: W i t h
the claim,
sequences
. . .->H (Y Q(b) ) ® Q ^ H ( X Q ( b ) ) ® Q ^ H ^ ( U Q ( b ) ) ® Q
a
implies
a-1 > d i m Y +b+1 .
a commutative
a
(k -) . This
(Y , Q (b) ) O Q ^
(see
[ML]
implies
a-1 >
a n d a-1 > d i m Y+b) .
theinduction
claim f o r
for
every
and
for d > 0
varieties
and
m € 71 p r o v i d e d a > d+b+1
we f i r s t
U . Choosing
. The case d = 0
is trivial,
show t h e c l a i m s f o r smooth q u a s i - p r o j e c t i v e
a smooth, p r o j e c t i v e c o m p a c t i f i c a t i o n X z> U
l e t t i n g Y = XÛ, t h e i n d u c t i o n s t e p i s g i v e n by 13.5, s i n c e the
claims coincide with
13.2 and 13.3, r e s p e c t i v e l y ,
d u a l i t y and p u r i t y . F o r an a r b i t r a r y v a r i e t y
X
f o r X , by P o i n c a r e
we choose a dense,
open smooth q u a s i - p r o j e c t i v e s u b v a r i e t y U f o r which we getthe
T
by t h e f i r s t
s t e p . The i n d u c t i o n from Y = X\ J
to
X
result
now a g a i n
f o l l o w s w i t h 13.5.
13.6.
Remark
statements
By t h e H o c h s c h i l d - S e r r e s p e c t r a l sequence f o r homology,
a) and b) o f theorem 13.4 t o g e t h e r would imply t h a t
the map
H^(X,Q(b)) ® Q
Q
1
£
- H (*,H
a + 1
i s an isomorphism f o r a > d+b
*(2d,d)
. Conjecture
function f i e l d
13.7.
(X,Q (b)))
£
and i n j e c t i v e
f o r a = d+b, (a,b)
12.18 a) would imply e x a c t l y the same i n t h e
case.
A t t h e end o f t h i s
section,
l e t us d i s c u s s the r e l a t i o n
a g e n e r a l c o n j e c t u r e o f B e i l i n s o n on K-theory
([Bei
H ^
j
and £-adic cohomology
4] 5.10 D ) v i ) , b u t note t h e m i s p r i n t i n t h e d e f i n i t i o n o f
(S, ZZ/ Z (i) )
N
ine
be r e p l a c e d
i n the l a s t
by x ^R'n^7Z / Z
N
<
For a r e g u l a r scheme
(i) )
X
r e f e r e n c e , where
which i s i n v e r t i b l e on
Zariski
AA (J)
n
n
RTr Z Z / £ ( i )
+
should
.
l e t a: X>
-> x
et
& ar
map from t h e e t a l e t o t h e Z a r i s k i s i t e o f
of
with
be t h e c a n o n i c a l
X, and l e t
X . B e i l i n s o n notes
Z
be a prime
t h a t the complexes
sheaves
:= T
Ra E A
jlt
n
(J)
s a t i s f y G i l l e t ' s axioms f o r a t w i s t e d P o i n c a r e d u a l i t y
1
provided Grothendieck s
purity conjecture
theory
( s . [SGA 5] I 3.1.4) i s
t r u e . Assuming t h i s one
H j
(X, S /I (J))
n
i
n
e
factorizing
canonical
x
:= H ^
map
induced
R a ffi A ( J )
N
i n t o H^ (X,TL /I (j))
that
H*
N
i n e
( X , T L /I (*))
n
expressed i n three
Gillet's
ones) such t h a t v i a the
i.e.,
H
H
1
ar
mology. The
"motivic
coho
statement
of complexes o f
s a t i s f y i n g c e r t a i n axioms
can
Chern
Zariski
(including
characters
( X , Z Z ( j ) ) * Q = Hjjj(X,Q(j) )
M
{X TL Aj))
t
meaning of t h i s
existence
ffi^(j)
(13.7.1)
X
the
sheaves
1
i s the
( r e l a t e d ) ways.
B e i l i n s o n conjectures
on
the
.
n
mology w i t h ZS / £ - c o e f f i c i e n t s " . The
1)
via
t
by
B e i l i n s o n conjectures
be
into
34t
n
- Ra*2Z A ( J )
+
from K (X)
(X,AA (J) )
ar
the Chern c l a s s e s
n
<:j
g e t s Chern c l a s s e s
can
i
be
r e g a r d e d as an
above c o n j e c t u r e
then claims
i n t e g r a l motivic
that there
are
coho-
canonical
q u a s i - i somorphi sms
AA (J)
n
S (j))
~Cone(2Z (j) £
M
In p a r t i c u l a r t h i s would g i v e
(13.7.2)
2)
For
higher
...^H
ar
M
X
a r
(X,2Z
sequences
(j)) ^
M
over a f i e l d k B l o c h
Chow groups CK*(X,*)
(13.7.3)
long exact
(X,2Z ( j ) ) ^ H j
a variety
dence f o r the
i s the
1
.
M
conjecture
, and
that
i n [B15]
(X, SA (J))^...
n
i
n
e
and
Landsburg
Bloch
defined
g i v e s much e v i -
f o r a smooth v a r i e t y
j
HjJj(X,ZZ ( j ) ) : = C H ( X , 2 j - i )
c o r r e c t v e r s i o n of i n t e g r a l m o t i v i c
cohomology. B l o c h
in
p a r t i c u l a r proves that
(13.7.4)
and
HjJj (X,ZZ
that there
(j))
$
i s a c y c l e map
r e a s o n a b l e cohomology t h e o r y
in
[Bi 6]
one
n
t h a t t h i s map
(j ) )
from the
[ B i 6]
groups 13.7.3 i n t o
. By
the
for regular
i s an
be
as
X
n
can
any
same arguments
(X;2Z A ; 2 j - i ) - H ^
j
Beilinson's conjecture
conjecture
HjJj (X,Q
gets a c y c l e map
Hjjj(X,2Z A ( J ) ) := CH
and
=
U
i n e
(X, TL/ l
(j ) )
s t a t e d more c o n c r e t e l y
as
isomorphism f o r a l l n , i , j € TL
the
and
a l l smooth v a r i e t i e s
3)
One c a n show t h a t
n
(
K . . (X TZ /l )
23-1
' '
0
and
up
j
one a c t u a l l y
)
1
+ H
f
standard
gets
Chern
(X,ffi/J6 (j))
' '
n
fine
t o "small
i n [MS 2 ]
X, c f . t h e s t a t e m e n t s
V
claims
J
that
factorials"
classes
,
'
M
this
(those
.
map i s a n i s o m o r p h i s m
necessary
to define the
Chern c h a r a c t e r ) .
can
Grothendieck's
purity
conjecture
u s e Thomason's
result
that
true
X
f o rreasonable
of finite
H
type
(
f i n e
X
schemes
i s unproved
the purity
yet,
conjecture
[Th] . I t i m p l i e s t h a t
part
Chern
' V 3 ' >
:
U i *
=
4
(x,2Z A ( J ) ) ) ®
n
i n e
2j-i
(
X
)
H
K-groups
or similar
Bass'
motivic
three
imply
9
and t h a t
ffl
)
)
there are
•
0
on t h e f i n i t e
t
would
- H
i
i
generation
on t h e f i n i t e
generation
n
e
f o ra l l
imply
that
of the
of
inte-
( i n any o f
the induced
map
(X,^(j))
i , j £ ZZ a n d X r e g u l a r , o f
TZ . F o r a s m o o t h
S f
i
n
e
variety
n
Ra ZZ / £ ( j )
jlt
rise
- Hj
( X ,
X = Iim X
a
a
Now t h e e x a c t
gives
theory
X
over
a field
finite
k this
would
isomorphisms
where
T
j
' V
conjectures
HJ(X,«(J)>
for
duality
conjecture
formulations)
an isomorphism
over
X
cohomology,
Hjj(X,(B(j))
type
(
* f i n e
with
is
£
of a twisted Poincare
Together
the
f o r schemes
characters
K
gral
i s "almost"
o v e r ZZ
n
is
b u t one
Q (J))
X
( X Q (J))
f
,
l
i s d e f i n e d a s i n 11.6
JL
with
i n e
of finite
a
type
over
triangle
+ Ra 5 Z / £ ( j )
to a long
n
+
exact
-> T
>
sequence
N
Ra 2Z /I
+
(j
, as I i m H
a
TL .
1
±
n
e
(X , Q (j))
a
£
... - H ^
n
(X,T Ra 2Z / S ( J ) ) + H ^
ar
>j
J|c
Il i f i < j + 1
1
(X Ra
a r
f
E /I (J))
n
jlt
Ra„ZS /I
-
H
fine
H
Zar
( X
( X
'
f f i /
T
' >j
*
R c t
n
H (X,ffi /£ (j))
H
1
( X , S / £ ( j ) ) - H ( X f f i /I (J)
1
Beilinson S
char k
If
X
- H
and
injective
1
and
injective
N
1
i n e
(X , ZZ /I
f o r i = j+1
n
(j ) ) and
H
N
1
fc
(X, 2Z /I
(j ) )
spectral
i s j u s t the s p e c t r a l sequence f o r the
f i l t r a t i o n N*([BO]6.4 and
f o o t n o t e ) , the t r u n c a t i o n x ^
means t o f o r c e the
1
and
.
f o r i > j . S i n c e the hypercohomology
+
1
k,
13.3.
e
sequence f o r Ra ffi/£ (j)
i n a c e r t a i n way
sharpening
( X , Q (J) )
r e l a t i o n between H
(13.9.1)
for i = j + 1 . Concluding,
then
for i < j
is highly n o n - t r i v i a l
coniveau
)
i s a smooth v a r i e t y over a f i e l d
i s a prime,
i s an isomorphism
The
n
/
f o r i <_ j
HjJj(X,Q(j))
13.9.
i<j
c o n j e c t u r e suggests the f o l l o w i n g one,
Conjecture
if
1
t
i s an isomorphism
13.8.
S
map
n
i n e
*
>>
O
t
Hence the
n ( j
H if
Il
1
(j ) ) -
Il
O
- H
n
(X , x
ar
j
<
condition
1
H (X,j) = N " H (X,j)
upon the c o n s i d e r e d t w i s t e d P o i n c a r e d u a l i t y t h e o r y . T h i s s h o u l d
be compared w i t h the remarks i n 5.24:
condition
13.9.1 i s a n e c e s s a r y
c o n d i t i o n t o g e t a cohomology t h e o r y c l o s e t o m o t i v i c cohomology.
The v a r i o u s c o n j e c t u r e s I made f o r f i n i t e o r g l o b a l f i e l d s t r y
to c a l c u l a t e
H*
, i . e . , the c o n i v e a u s p e c t r a l
sequence, i n terms
of the £-adic r e a l i z a t i o n s - compare the Tate
g e n e r a l i z a t i o n due
and
no
i = 2j
and i t s
In f a c t , the case of c y c l e s
i s the extreme case, where B e i l i n s o n ' s c o n j e c t u r e
gives
information at a l l .
To
to
to Grothendieck.
conjecture
see
U I-* H^
H
P
l e t H (2Z A ( J ) )
(U , TZ/ I
t
(X, H
n
1
this,
(j) )
N
be the
, then we
q
(ZZ A ( J ) ) )
P
(ZZ A ( p ) )) = CH
n
= O
have
,
Zariksi
sheaf
associated
(assuming p u r i t y )
p > q ,
O CL I.
H
P
^ (X, H
Zi a r
by(the
H
proof
n
of)
[BO]
7.7,
{X TZ A ( J ) )
n
1
i n e
jecture
the mystery l i e s
H J
13.10
(X,SA (J))
Remarks
spectral
a)
quence one
- H?J(X, TZ/ I (J))
n
I f one
conjecture
for
H^
n
FA
( F , ZZ /I (j))
1
(F, TZ/ £
for
i > j)
"arithmetic"nature, while
-
H (X,ZZ/I (j))
1
t i o n - i s of
N
for
of sheaves T ( j )
b r a i c K-theory
n
.In
this
CH
nature,
(formulation
N
(F; TZ/ I
2))
j)
T
of H
1
i n e
( X , TZ
t o the c o n i v e a u
=
with
filtra-
complexes
c e r t a i n r e l a t i o n s to a l g e -
constructed candidates
the
N
/I (j))
5.24.
the e x i s t e n c e of c e r t a i n
( l o c . c i t . ) and
=O
sense, the c o n j e c t u r e i s
c f . the p i c t u r e i n
f o r the etale t o p o l o g y
[ L i 1] and
1
the d e t e r m i n a t i o n
conjectured
By the t r i a n g l e axiom
s p e c t r a l se-
(j ) ) , i < j ,
j < i < 2j - r e l a t e d
"geometric"
Lichtenbaum has
a Bloch-Ogus
( t h i s i s sometimes r e f e r r e d t o
(Note t h a t t r i v i a l l y
of
b)
has
( c f . [MS2])
(F;ZZ/ £ ,2j-i) ~ H
N
i n e
assumes p u r i t y , then one
e a s i l y deduces t h a t B e i l i n s o n ' s c o n j e c t u r e
a l l fields
here
.
f o r e t a l e cohomology). From t h i s
i s e q u i v a l e n t t o having
j
t r u e f o r i = 2 j , and
map
sequence f o r etalecohomology
as G e r s t e n ' s
CH
1
. C o n s e q u e n t l y , B e i l i n s o n s con-
is "trivially"
i n the
n
n e
n
B e i l i n s o n i n [ B e i 4]
( f o r m u l a t i o n 2))
2
f o r i > 2j ,
j
n e
n
hence
=CH (X)A
n
H ^ ( X , f f i /£ (j) )
as remarked by
(X)/I
and
=O
t
P
for j < 2
i m p l i e d sequence
[ L i 2].
...-H
1
t
(x,r(j))
the groups
can
be
H
1
f c
H
quite different
and
( J )
«
(x,r(j))
are
a l c
(i.e.,
and
i n c o m p a t i b i l i t y with
Lichtenbaum c o n j e c t u r e s
quasi-isomorphisms
T(J)
his
imply B e i l i n s o n ' s c o n j e c t u r e
The
...
TL - c o e f f i c i e n t s " )
cohomology. But
showsthat this together with
13.11. Theorem
(x,E/i6 (j))-.
n
t
r a t h e r r e l a t e d t o e t a l e cohomology
from m o t i v i c
T^Ra
1
- H
" e t a l e cohomology w i t h
M
t
(X,T(j))
thought of as
S
1
" H i l b e r t 90"
(formulation
conjectures
axiom would i n f a c t
1)).
s t a t e d i n t h i s chapter
are
true
f o r X = Spec k .
Proof
One
has
H
(k,$ (n)) = O
for i > 3 ,
0
Q (J)
, i =
ji
1
O
,
4
H ( S p e c k,CB (J)) =
jl
, i * O ,
and
i t f o l l o w s from r e s u l t s of B o r e l and
K-
Zn
Soule that
n > 1 ,
(k) 6 Q = O
ch.
K
2 n - 1
(
k
)
(
n
)
K
V ^
2 n - 1
(
k
2
H (k,$ (n)) = O
£
cf.
[J3]
, example 3.
)
^
Q
£ ^
e
1
(
k
' ^
(
n
)
)
'
n
13.2
i s true
r
H JtSpec k , 4 ( j ) )
j
H° (Spec k , ^ )
1
* Q
1
H^(Spec k , $ ( j ) ) = O
showing c o n j e c t u r e
G,
2'°
13.8.
'
f o r Spec
and
£
1
for n > 1 ,
Hence c o n j e c t u r e
H^J (Spec k,Q(0))<8>(D
^
H (k,Q (j))
£
, otherwise,
,
, j > 1 ,
k,
are
§14.
Linear
varieties
In t h i s
s e c t i o n we
study c e r t a i n v a r i e t i e s , whose cohomology
groups are s u c c e s s i v e e x t e n s i o n s
of Tate o b j e c t s , and prove most
of the c o n j e c t u r e s s t a t e d i n t h i s paper f o r them. We
general
14.1.
s t a r t with
observations.
Lemma
For every c o n j e c t u r e
s t a t e d i n t h i s paper the f o l l o w i n g
p r i n c i p l e h o l d s : I f the c o n j e c t u r e i s t r u e f o r a smooth v a r i e t y
i t a l s o holds
f o r every
a f f i n e or p r o j e c t i v e f i b r e bundle over
14.2.
T h i s i s due
theory
s a t i s f i e s the f o l l o w i n g axioms f o r a smooth v a r i e t y
[Gi] 1.2
t)
two
t o the f a c t t h a t every
( i x ) , ( x ) , and
8.9
(homotopy i n v a r i a n c e )
p: V(E)
-> X
considered
Poincare
X,
X.
duality
X (cf.
above):
If
E
i s a v e c t o r bundle on X
and
i s the a s s o c i a t e d a f f i n e f i b r e bundle, then the mor-
phisms
i
p*:
i
H (X J) -
H (VfE),j)
f
are isomorphism f o r a l l i , j € TL .
n')
( p r o j e c t i v e bundle isomorphism)
of rank n on X and
bundle,
•
p: P(E)
X
If
E
i s a v e c t o r bundle
i s the a s s o c i a t e d p r o j e c t i v e f i b r e
then
H
a 2v-2n
+
( X
'
b + V
"
n )
H
ft
(P(E) ,b)
V=O
2
i s an isomorphism f o r a l l a,b
E TL , where
£ € TH
Chern c l a s s of the c a n o n i c a l l i n e bundle 0(1)
14.3.
i.e.,
Here the p u l l - b a c k
p*
(X,1)
on P = P(E)
i s defined v i a Poincare
as the arrow making the diagram
i s the
.
duality,
H _ (X,b-n)
a
nH
TT
f
n
/v
commutative, and
£
V
—*
p*
.
^
T T
H
n
' d=dim X ,
2d+2n-a, ,
(P,d+n-a)
x
i s the v - f o l d cupproduct
of
£ , where the
i s d e f i n e d by P o i n c a r e d u a l i t y , too, by commutativity
1
H CX^) 9 H
f
(X J )
f
t
Qlf IX J )
i
P r o o f of 14.1.
l l
t
H _ (X,d-j)
2 d
a
^ P
2d+2n-a
, , ,.
(X,d+n-b)
cupproduct
H (P b)
^
x
H
>
2n
r
—-
>
H
—
>
H
1 + i
2
'
d
of
(X, j+j • )
^
i
, (X,d-j-j') .
The axioms t) and n') iirmediately give the claim for the semi-
simplicity conjectures i n §12 and for Tate's conjecture C), since L(V(r) ,s)=L(V,r+s) .
A l l other conjectures concerned morphisms between twisted Poincare duality theories,
their bijectivity,
i n j e c t i v i t y or s u r j e c t i v i t y , and o b v i o u s l y
morphisms r e s p e c t the above isomorphisms. The
axioms t '
are well-known f o r the £-adic t h e o r y and
follow for absolute
Hodge t h e o r y , and
then
i n p a r t i c u l a r the Hodge and
n')
f o l l o w from Q u i l l e n ' s c o r r e s p o n d i n g
Remark
The
f*: H
a
(Y,b)
of f i b r e dimension
ii)
a cupproduct
<n
t')
If
E
-> H
i)
eigenspaces.
by
a p u l l - b a c k morphism i n
,
(X,b+n) for f l a t morphisms f : X -> Y
a~r ZTX
0
,
i n cohomology, both compatible
d u a l i t y as i n 14.3.
these e x i s t , and
K-theory
above can be extended t o s i n g u l a r v a r i e t i e s
i n t r o d u c i n g the f o l l o w i n g c o n c e p t s :
homology
(co -)homology t )
r e s u l t s f o r the
([Qui 1] § 7 , 4 ) , s i n c e the maps r e s p e c t the Adams
14.4.
n')
the deRham t h e o r y ,
by the comparison isomorphisms. For the m o t i v i c
and
and
these
with
Poincare
F o r the c o n s i d e r e d P o i n c a r e d u a l i t y t h e o r i e s
i n s t e a d of
t)
on
has:
i s a v e c t o r bundle of rank n on a v a r i e t y X and
p:
V(E)
-+ X
p*:
i s the a s s o c i a t e d f i b r e bundle,
then
H (X,b) - H . - ( V ( E ) , b + n )
i s an isomorphism f o r a l l a,b €ffi.
Axiom n') h o l d s l i t e r a l l y
back and cupproduct
f o r a r b i t r a r y v a r i e t i e s X, w i t h t h e p u l l -
mentioned above
t h e o r y , which c a r r i e s over
t o m o t i v i c homology by t h e methods o f
3 ] , and [DV] V I I I 5 f o r £-adic
[Sou
The
rieties,
14.5.
with t h i s ,
for XX^K,
X
Proof
i.e.,
X
over
a field
k, t h e c o n j e c t u r e
G a l o i s e x t e n s i o n K / k , then i t holds
for a finite
itself.
A l l conjectures involved functors with
rational Galois
descent,
we have c a n o n i c a l l y
(14.5.1)
H (X^)O
a
G = Gal(K/k)
2 2
Q=
(H (Xx ^b)O
a
k
f f i
Q)
. Here we r e g a r d X x ^ K
t h a t we have an a c t i o n o f G on Xx^K o v e r
the homology. The p r o p e r t y
for
i n terms o f homology.
F o r e v e r y c o n j e c t u r e s t a t e d i n t h i s paper t h e f o l l o w i n g
p r i n c i p l e holds. I f , f o r a variety
for
1 4 . 1 f o r a r b i t r a r y va-
hence one o b t a i n s
s i n c e a l l c o n j e c t u r e s were f o r m u l a t e d
Lemma
holds
homology)-.
c o n s i d e r e d Chern c h a r a c t e r s and Riemann-Roch t r a n s f o r m a t i o n s
a r e compatible
for
(See [Qui 1 ] I o c . c i t . f o r K -
14.5.1
then
G
,
as a v a r i e t y over k so
k and an induced
f o l l o w s from t h e f a c t
the functors
P*
H (Xx ,K,b) J H (X,b)
a
k
^ a
induced
by t h e e t a l e p r o j e c t i o n p: X x ^ K -> X
we have
p p* = [K:k] • i d,
+
(14.5.2)
P
*p
+
=
I
a€G
a
We make t h e f o l l o w i n g i n d u c t i v e d e f i n i t i o n .
one on
that
14.6.
Definition
O-Iinear,
for
.
{U,X,Y}
immersion and
is
Call
of
be
Z
flat
U c X
(n-1)-linear,
linear,
S
i f i t i s empty or
some N > 0
tripel
Let
n-linear,
S-scheme
affine
f o r n •> 1
Z
and
o t h e r member i n
f o r some n>0
(and
S-space
i s a closed
open complement, Y
i s the
Z
, i f there i s a
S-schemes such t h a t Y c X
i f i t i s n-linear
implies
i s o m o r p h i c t o the
i s the
and
a scheme. C a l l a f l a t
one
{U,X}
of
S-
{U,X}
. Call Z
note t h a t
n-linear
(n+1)-linear).
Examples are
S-schemes t h a t are
s t r a t i f i e d by
affine
S-spaces,
N
e.g.,
Z = IP
. Over a f i e l d
s
N
N
complements i n JP^
(or A^)
k,
examples of
of
a u n i o n of
linear k-varieties
linear
are
subspaces, or
sue-'
N
c e s s i v e blow-ups of
P ^
in linear
v a r i e t i e s , and
varieties
14.7.
a)
Theorem
nerated f i e l d
k and
If
s t r a t i f i e d by
X
I
subspaces, Grassmannians, f l a g
such
i s a linear variety
*char(k),
et
a G^-module, f o r a l l m,a
,
m
of
Q
A H
Gr H
m a
v
a,m
linear
c)
r
a,b
: H
,
of
1 (-v) 's
, m = 2v
r e a l i z a t i o n f o r a b s o l u t e Hodge c y c l e s
the
( c f . 6.11)
for a l l
c o r r e s p o n d i n g statement h o l d s f o r
(X ($),(&) , i f
Hodge c o n j e c t u r e and
, v 6 TL ,
the
X
i s a linear variety
Tate c o n j e c t u r e s are
over
true
varieties.
If
X
a
( X
is a linear variety
'^
e 2z ,
f77
€ TL . In p a r t i c u l a r ,
The
odd
v
, then
(X) O Q
ffi
Hodge s t r u c t u r e s H
b)
2v
m
€ ZZ . I f c h a r ( k ) = 0
sum
f o r the
odd
(-v)'s,
0
m
W
ge-
sum
as
over a f i n i t e l y
then
0
W
varieties.
( b )
]
® \
~
over a f i n i t e
H (X,<^(b))
a
k
field
k,
then
for
the
i s an isomorphism
a)
and
12.4
I
for
+char (k)
€ 7L . Hence i n view
and a l l a,b
b ) , a l l stated conjectures are true f o r X .
and
12.6
b) a r e more g e n e r a l l y t r u e f o r l i n e a r
Conjectures
C-varieties.
C a curve over k .
d)
If
X
i s a l i n e a r v a r i e t y over a g l o b a l
function f i e l d
k ,
then
r
?
a,b
:
a,b
:
( H ( X ^ C b ) )/H^(X,Q(b) ) )®<^
a
0
Hj(X,«(b))
0
» Q
£
H
Z
Z
jl
1
*V
(k^H^
( n o t a t i o n s as i n 12.18) a r e isomorphisms
(X Q Cb))
f
(X
K
(b)))
r 4 j l
I
for
* c h a r ( k ) and a l l
a,b
€ ZZ . Hence a l l s t a t e d c o n j e c t u r e s a r e t r u e f o r
e)
If
X
: H
a,b
a
( X
'
Q ( b ) )
Q
® i
i s an isomorphism
+ b-1
~*
2
£
t
f
a)
r
namely,
H^(X,Q (b))
f o r a > dim X + b
, and H ( k H ^ ( X Q (b))) = 0
a
)L
Proof
X .
i s a l i n e a r v a r i e t y over a number f i e l d k, t h e n a l l con
j e c t u r e s s t a t e d i n §13 are t r u e f o r X,
r
and
jt
0
and
injective
f o r a = dim X
f o r a > dim X + b + 1
I t f o l l o w s from t h e r e l a t i v e homology sequence
(6.1 f ) )
W et and the i n d u c t i v e d e f i n i t i o n o f l i n e a r v a r i e t i e s t h a t Gr H
(X Q )
m a
*l
W AH
or
Gr H
(X)^ f o r c h a r ( k ) = 0 , have a f i l t r a t i o n such t h a t
ma
Q
r
f
the
graded terms have the wanted p r o p e r t y . For c h a r ( k ) = 0
use the p o l a r i z a t i o n s f o r a b s o l u t e Hodge c y c l e s , l i k e
to
A H
see t h a t G r ^ H ( X ) ^
i s i n fact a direct
sum
rt
we
may
i n lemma
1.1
of t h e s e graded
terms, and hence g e t the r e s u l t . For c h a r ( k ) = p > 0
we have t o
W et proceed d i f f e r e n t l y . A g a i n i t s u f f i c e s t o show t h a t Gr H
(X, Q )
ma
X/
0
i s semi-simple.
[FW]
For t h i s we
t r a n s p o r t F a l t i n g ' s arguments i n
VI §3 t o our s e t t i n g . There e x i s t s a smooth, g e o m e t r i c a l l y
irreducible variety
U
over a f i n i t e
field
IF
, with generic
Si
point
i)
n = Spec k, such t h a t
X/k
extends t o a l i n e a r U-scheme
from t h e i n d u c t i v e d e f i n i t i o n and
ii)
a
i
R~ f Rf Q
*
v.
A
0
i s smooth over
[EGA
f: X
U
(this
follows
IV] § 8 ) ,
U, commutes w i t h a r b i t r a r y
base
change,
and has a weight f i l t r a t i o n
(compare
t h e p r o o f of 6.8.2).
One g e t s an e x a c t sequence o f fundamental groups
1 + TT (Ux
1
Jt ^
JF ) + TT (U) + G a l (3F
1
q
1
( o m i t t i n g t h e base p o i n t s
q
/W
) + 1
q
i n t h e n o t a t i o n ) , and by F a l t i n g s ' a r g u -
ments i n I o c . c i t .
i t s u f f i c e s t o show t h a t TT ( U X
JF ) and the
I
lb
Q
d e c o m p o s i t i o n group
of a closed point
x o f U a c t semis i m p l y on
Gr^H
ma
e t
(X,Qx,)
0
. For
Tr (Ux
JF )
I
lb
q
fundamental r e s u l t o f D e l i g n e
the
D
base change p r o p e r t y i n
= G a l (IF
where
[D9] 3.4.1
X^/ (x)
of
the f i n i t e
t h e case o f a f i n i t e
over
x € U
K(X),
field,
of
and
x . By
I
CH (X)
i
is surjective
OQ-*
which i s t r e a t e d
in
0
2I
f
0
2i
tel.
Y e X
definition
-
that
e x a c t diagram
1
CH (U)
i
- O
t C l .
1
CH (X)
c l o s e d and U = X\Y
O ,
i
(compare
7.5), and t h e i n d u c t i v e
N
space Jh^, which
15.6, we r e d u c e t o t h e case o f an a f f i n e
i s known by 14.1. The T a t e c o n j e c t u r e A) i s s i m i l a r ,
c
generation of
A. (X)
i)
f
t e l .
1
i
=
c) .
- W 0 H 2 i ( X f Q d ) ) - W Q H 2 i (U,Q(i))
CH (Y)
x
W H (X Qd))
f o r a l l i > O . By t h e commutative
W H (Y Qd))
X
so we have reduced t h e q u e s t i o n
a) t h e Hodge c o n j e c t u r e i n t h i s case means
Cl OQ:
for
X/U
residue f i e l d
i s a l i n e a r v a r i e t y over
By
hand,
i i ) above g i v e s an isomorphism o f
i s the f i b r e
K
x
b)
i i i ) . On t h e o t h e r
/K (x) ) - r e p r e s e n t a t i o n s
X x , JF
, K,(X)
x K(X) q
to
t h i s i s known by a
i
and i n j e c t i v i t y
of
1
f o l l o w s from t h e case o f a f i n i t e
field
A2(X)OQ
.
~
1
0
l
and t h e f i n i t e
i®^
et H . (XFQ ( D )
9
1
v i a t h e commutative
A,
dia-
gram
A (X)
>
C
i
A (X )
i
where
X
x
i s as i n a)
c
H|^
r
jt
H§t(X , i))
>
and
(X C (I))
x
sp
V
i s t h e s p e c i a l i z a t i o n map
(compare
the p r o o f
of 12.16). For a f i n i t e
N
k
i n d u c t i o n s t a r t i n g from
field
k
one
, t h a t the groups K (X)
m
(from Q u i l l e n ' s s p e c t r a l sequence 6.12.5) are
for
a linear variety
of a P o i n c a r e
shows i n f a c t
1
X/k
and
finitely
by
2
E
(X)
p, q
generated
(use t h a t t h e s e are the homology groups
d u a l i t y theory,
c f . [Gi] f o r the E
2
). In p a r t i c u l a r ,
P tI
c
this
The
i s true f o r E
injectivity
2
_
(X) = CH (X), and
±
jl
of cl^fcQ^
o r d
C),
i n view of a) and
s=dim (k)
L (
<V
s )
1
"
the Tate c o n j e c t u r e
(X)
f o r X = Spec k
i s well-known f o r f i n i t e
we
L(Q^,s)
replace
.
the proved
by
the
or g l o b a l f i e l d s ,
zeta function
Cj (s)
c
of
proof
F
of 7.17.
If
i s the prime f i e l d of
L($£,s) = f ( s ) • C p ( s - t r . d e g ( k ) )
with
f(s)
k
the
holomorphic and
non-
s=
c)
the
£-adic cohomology of a l i n e a r v a r i e t y
dim^k .
extension
b)
, and
that
at
12.6
k
, i t implies
vanishing
and
where
r
from G r o t h e n d i e c k s formula r e c a l l e d i n
successive
con-
(Note t h a t L ( V ( i ) , s ) =
follows i n general
Since
.
'
C)
L(V,i+s)). This
can
±
have to show t h a t
a
i.e.,
for A
i s a s p e c i a l case o f c)
For T a t e ' s c o n j e c t u r e
j e c t u r e B), we
a fortiori
of r e p r e s e n t a t i o n s
are e q u i v a l e n t
X/k
is a
Q^(n), c o n j e c t u r e s
i n t h i s case. I f
C
12.6
a)
i s a v a r i e t y over
N
k
w i t h dim
by
12.10
and
C <A
14.1
C - v a r i e t i e s by
d)
(14.3
12.7
b)
l i n e a r scheme
above, U now
being
12.4
and
12.
6b)
hold
for
A
c
f o r s i n g u l a r C ) , hence f o r a r b i t r a r y l i n e a r
and
f o l l o w s from t h i s by
t o the
an
, then c o n j e c t u r e s
induction.
applying
X/u
theorem 12.16
and
satisfying properties
a curve over a f i n i t e
remark 12.17
i)
a)
and i i )
f i e l d . Note t h a t we
have
isomorphism
H
a
( X
x'
Q ( b ) )
® ®l ~
for
every c l o s e d x G U
e)
By
we
get
13.11, 14.1
that
and
H
a
by
t
(
X
x'^
c)
(
b
)
)T
.
i n d u c t i o n w i t h the
5-lemma
(cf remark
12.9)
r
a,b
:
f
H^ (X CKb))
r
® Q^
H
z
f o r a l l a,b 6 ZZ , where
to
H ^
j
, and H
ine
f i n e
X
(X,Q (b))
£
H
f i n e
i s t h e homology t h e o r y a s s o c i a t e d
j _ o b t a i n e d from i t as i n 11.7. F o r a q u a s i s
p r o j e c t i v e scheme
bedding
f i n e
X
over Spec 2Z ,
i n t o a smooth scheme
M
H
f i n e
may be d e f i n e d by em-
o f pure f i b r e dimension N
over Spec 72> and s e t t i n g
Hf
for
n
e
(X, b))
V
a: M,
-» M
6u
H^
l n e
zar
=H^ (M,T
x
. The Q - p u r i t y
0
36
(X Q (b)))
r
a
b
M t
(N-b))
[Th] then
,
shows:
e t
r
r
0
£
for a >
_ d+b
r e l a t i v e dimension of
H (k,H
. R
(X Q^(b)) - H (X Q (b))
i s an isomorphism
2
i N
X
and i n j e c t i v e f o r a = d+b-1 , d t h e
o v e rffi. The c l a i m e d v a n i s h i n g o f
f o l l o w s from 13.11, 14.1 and i n d u c t i o n w i t h
36
13.5 a ) , o r from 13.11, 6.5 a ) , and a) above.
M
~et
One may deduce t h e statement on t h e map from H
to H
a
a
a l s o without r e f e r i n g to £ f ^
by u s i n g 13.11, 14.1 and i n d u c t i o n
a
as i n t h e p r o o f o f 13.5 b ) , r e p l a c i n g d by d-1 i n t h e argument.
o
i n e
f
ra
Appendix A:
A letter from Bloch to the author
2/11/87
Dear Jannsen,
The homological formulation of the Hodge conjecture for singular varieties which you
gave in your talk is the only one possible. In fact, some years ago Mumford suggested to me
that one should look for a counterexample to the corresponding cohomological conjecture.
Here is one.
Let SQ C P be a smooth hypersurface of degree d > 4 defined over Q . Let x e So(C)
3
be Q-generic and let S/C be the blowup of So at x. Let P = BLps(x),
so S C P. Let
z
W = P lis P > Since H (S) = (0), we have an exact sequence of Hodge structures
4
0 -+ H (W,Q(2))
2
4
-> H\P,Q(2))®
tf (S,Q(2))
-> 0.
2
Note H (P, Q(I)) has generators e = class of exceptional divisor and h = pullback of
hyperplane from P . We have
3
2
CH (P)
A
q
2
H\P,
2
Ql(2)) S Q • h © Q • e
2
(e • h = 0).
2
4
Also H\S, Q(2)) S Q with Zi • 5 = <f, e • 5 = - 1 . It follows that # (V/,Q(2)) ^ Q
2
2
2
4
(as a Hodge structure). In particular ((d - l)/i , d(h + e )) G # (P, Q(2))
from a Hodge class on W.
comes
But on the level of Chow groups, this class restricts to
2
d(x) — h • S € CHo(S) = CHo(So )'
Because x is Q-generic and P (So) > 0, this
c
9
2
class is of infinite order. Thus Ker(Ctf (P)|
Remarks
02
e 3
2
2
CH (S))
SQ
e 2
.
1. This discussion shows that no contravariant Chow group can provide the
extra element needed.
2. I guess the Hodge conjecture is true for divisors on complex projective varieties because
one has the exponential. This presupposes, however, that ^ j j
2
o d g e
2
H ( X , C) <—• H (X, Ox)
always. Is this o.k. ?
3. Note the counterexample is for curves on a 3-fold, where the classical Hodge conjecture
is true!
4.
With a bit more work, one can get a hypersurface example.
Namely take So =
P H T C P where T is a smooth 3-dim. hypersurface defined over Q. Let
3
4
0
0
Q = BL <({x}\
P
T = BL ({x}),
To
P = BL ,({x}),
P
so we have a cartesian square of strict transforms
i
I
S =
BL ({x})
So
Take W — P U T . To show this works, the key point is to show the image of
2
2
CH (To )^CH (So )
c
c
is "small". One can do this by using cycle classes in
2
CH (T )
I®?/Q)
2
H(
:
2
•
0c
CH (S )
0c
1
1
i
(0) =
I
2
2
ff (T ,OTo )®cft
0c
c
c/5
Mumford's differential techniques for showing
2
2
•
prove that the image of d(x) — h in H (So Os )
i
0c
0
Soc
c/li
is large can be reinterpreted to
2
CH (SQ )
2
2
H (S ,0 )®n ).
C
<8> ft* r is non-zero for x Q-generic.
Best,
Spencer Bloch
/ ?
Appendix B : An example by C. Schoen
An example is given of the following phenomenon: A smooth projective surface V over a
field k satisfying:
i)
rank (Ker: C H ( V )
ii)
V cannot be obtained by base changing a variety V ' / k ' for a field k' C k with trans,
0
— Alb (k)) = a> .
deg Q
y
deg. (k/k') > O .
2
3
13
Begin with the elliptic curve E/(f with equation
zy = x + ^
^ origin
e = (0 : 1 : 0) . The group of cube roots of unity,
, acts on E via multiplication on the
3
3
3
x—coordinate. Thus (/^) acts on E . The largest subgroup H < (/^) which acts trivially on
z
a n c
0 3 3
0
1 ®3
3
H (E ,fl ) 2 H (E,fl )
contains the diagonal ^ ^ A < ^ as an index 3 subgroup. In fact, if
M < H is the largest subgroup which operates trivially on thefirstfactor, then H = M x A .
3
Projection on thefirstfactor pr^ : E
> E induces a commutative diagram of morphisms
E
P
R
I
>
3
j
3
P
E
>
E /A
R
3
Pr
I
>E//i
E /H
~ P
3
1
E//*
3
a IP
1
,
1
in which f is the canonical quotient map. Let k = ^(IP ) £ $(t) . Write Fj (respectively Tj )
t
for the generic fiber of
Tj
Fj /(H/A)
t
is a singular
c
^k^sing ' ^
e t
pr^ (respectively
t
K3
p T^) . Then
Fj
t
t
is an abelian surface, and
surface whose singularities are resolved by blowing up
ie
^k denote ^ resulting non-singular K3 surface. Thus Alby = 0 .
To check that (ii) is satisfied it suffices to show that Vj is not the base change of a variety
t
V
.
If such V ' were to exist, Gal(I/k) would act trivially on H (V ,Q ) ~ H ( V Q ) .
2
7
2
p
j
We claim however that Gal(k"/k) does not even act trivially on the quotient H (F ,Q
2
p
p
.
E
Since the base change of Fj by the field extension ^(E)/k yields the constant abelian surface
t
ExE
over If(E) , the Galois action factors through Gal($(E)/k) . The natural action of this
group on Fj X^jf(E) is induced by an isomorphism Gal(^(E)/k) £ A C Aut(E^ ) . Since A acts
t
non-trivially
on
H (F ,Q ) / .
H
2
p
A
[A H (e x E x E , Q ) ]
2
2
p
M
,
Gal($(E)/k)
acts non-trivially on
For any variety
W / ^ write
B (W) for the group of
1
modulo algebraic
1—cycles
Q
equivalence. Let 7 : U
07
p:=
> E / H be a resolution of singularities such that the genericfiberof
is isomorphic to
Vj . It is possible to deduce (i) if one knows that
c
rank BÛ^) = 00 . In fact,
CH (V ) = IimCH ( U - p - ^ D ) )
k
as D ranges over reduced effective divisors on P . Hence, by the commutative diagram with
exact row
1
Iim CH (U -P^ (D))
D
Ji
1
1
1
Iim B (P- JD))D
1
A
B
iV
I i m B (U -p-^D))
D
$
(
1
1
one need only to show that the image of c hasfiniterank. This is true because an open subset of
U is dominated by a constant family of surfaces. More precisely, there are birational morphisms
of non-singular projective
varieties f : Z
>E
and a : Y
(a)
the exceptional locus of a maps to afinitesubset of E ,
(P)
there is a commutative diagram of morphisms
(id x £) 0
Let
IP C IP
be
a
> E x Z where
a
non-empty
Zariski
open
subset
over which
and
q := f 0 Pr 0 (id * () 0 a = p 0 f are smooth. Denote the base change to IP of an object over
1
I
o
0
0 0
IP by adding
to the notation. Thus Y ~ E x Z , and restriction B (U)
> B (U) is
surjective withfinitelygenerated kernel. The vertical maps in the following commutative diagram
1
are surjective
1
O
1
Iim B Cq- ID))® O
4
1
B (E x Z) 8 Q
1
o
DCIP
o
C
1
lira B Cp- CD))® «
1
o
B (U)
®Q
1
DCP
o
(note that (?)° is proper and surjective). Because E xZ/E is a constant family, c' has finite
dimensional image: by the definition of algebraic equivalence, corresponding cycles in different
o o
o
fibers of E x Z/E have the same image in B ^ E x Z) , and the Neron-Severi group of Z is
finitely generated. Thus c hasfiniterank image as desired.
Finally, the fact that rank BÛ^) = OD may be deduced from a similar statement for the
elliptic modular 3-fold W(3)
[1, p. 778]. The correspondence between
constructed in [2, § 1]
(where
Q e CBL(W(3) x U ) such that
W(3)
is called
Q*:H°(U ,n )
3
W(3)
and E°
W ) gives rise to a correspondence
» H°(W(3) ,n )
3
is an isomorphism. Since these vector spaces are not zero (in fact they are one dimensional) [1,
Thm. 4.7] implies that rank B J U
) = OD .
[1] C. Schoen: Complex multiplication cycles on elliptic modular threefolds, Duke Math. J. 53
(1986), 771-794.
[2] C Schoen: Zero cycles modulo rational equivalence for some varieties over fields of
transcendence degree one, Proc Symp. Pure Math. 46 (Algebraic Geometry, Bowdoin 1985),
Part 2, A.M.S., p. 463-473.
Appendix C: Complements and problems
Cl. I was asked to add some clarifying words on the weightfiltrationon
i-adic cohomology.
Let U be a smooth variety over afinitelygenerated field k of characteristic zero. It was not
mentioned but is true that thefiltrationon
(TT,^) constructed in 3.14 is a weight filtration
in the sense of 6.8. Namely, with the notation of § 3,
H?
n—m
m
n
(Y^ "" \<)^(n — m)
y(n—m) .
g s m o o t n
n
G r ^ H (TT,^)
t
is a subquotient of
as a G^—module by 3.20, and this is pure of weight
m , since
p per.
r0
1
That H (XC)^) is pure of weight i , for X smooth and proper over k , follows as in 7.12:
j
Choose a smooth and proper extension f : <%
> S over a model S of k (an integral scheme of
finite type over TL [1/t] , with function field k) . Smooth and proper base change gives a
Gal(/c(y)/«(y))—isomorphism
for every y e S . If x e S is closed, then Deligne's proof of the Weil conjectures for the smooth
and proper variety
K(X)
over the finite field /c(x) ([D 8] (1.6), as amplified by [D 9]
(3.3.9)) shows that the eigenvalues of F r on the above cohomology group are pure of weight i .
x
By 6.8.2 there are weight filtrations with the correct weights (i.e., those of 6.5) on
1
H
(X,Qp) and H (X,Q ) for arbitrary varieties X and closed subvarieties Z C X over any
et,Z
*
a
IL
finitely generated field k of characteristic ^ I . These weightfiltrationsare unique and functorial by 6.8.1. In particular, they coincide with the one in 3.14 in the case above. This also follows
from the construction in 6.11.
p
C2. It is a problem what realizations one has to take to get the good category of mixed motives.
Probably it is not enough to consider smooth varieties as in 4.1. Perhaps one also needs singular
varieties (see C3 below) or other "geometric" realizations like those constructed by Deligne in
[Dll].
The construction of 6.11 attaches "cohomological" and "homological" realizations to arbitrary varieties, morphisms of varieties and, most generally (as noted in 6.11), to simplicial varieties. The technique of § 4 always gives Tannakian categories and associated "Galois" groups with
the properties stated in 4.4 and 4.7, provided one takes a Tannakian subcategory of MR^ generated by a class of mixed realizations with the following two properties:
n
a)
It contains all H (X) for X smooth and projective,
b)
The pure quotients are in Mu (and hence polarizable).
This is still the case for the realizations attached to simplicial varieties.
Recall from 11.3 that for any field k one would like to define a ^-linear tensor category
with weights JCJC= JCJC^ such that
i)
the pure objects are semi-simple, and the subcategory of semi-simple objects is equivalent
to Grothendick's category JC= JC^ of (pure) motives with respect to numerical equivalence over
k,
ii)
there is a twisted Poincare duality theory with weights (H2(X,j),H (X,b)) with values in
a
JCJC , and
iii)
there is a spectral sequence (1 = identity object)
q
E§'9 = Ext ^ ( L H ( X j ) ) 4 Hy-I(X QU))
1
converging to the motivic cohomology defined via K—theory and degenerating for a smooth and
projective variety.
We can add here that iv) for a field k of arithmetical dimension d the cohomological
dimension of JCJC should be d , i.e., one should have Ext
O for p > d . For afinitefield
this would mean that JCJC coincides with Grothendieck's category of motives. For a global field
this would imply the short exact sequence
i
1
i
O - 4 Ext ^ ( ! , H - C X J ) ) - * H ^(X,<}(j))
Horn ^ ( I H ( X j ) ) - » O
l
mentioned in 11.4 c). In contrast to this, for a smooth projective surface X over C one expects a
non—trivial group
Ext^
2
(i H (x,2)) =
f
T
W
for p (X) ^ O . To deduce this formula from the spectral sequence iii) note that
O
2
E x t ^ ( l , H ° ( X , 2 ) ) = Ext ^(1,1(2)) C H ^(Spec C,Q(2)) = CH (Spec
and
ExtJ
1
r<<(r
1
1
2
(l,H (X 2)) C Ext ^ ( ! , H ^ ) ) C CH (C)^ = O ,
I
1
using a curve C with H (X) <=—» H (C) .
= O
For a field k of characteristic zero, the category
constructed by absolute Hodge cycles
coincides with JC^ if (and only if) every absolute Hodge cycle is algebraic (and this would follow
from either the Hodge or the Tate conjecture, cf. 5.4). Therefore the category
MMj of 4.1 is exc
pected to satisfy i) , but I don't know if it satisfies ii). Certainly it satisfies ii) (and still i)) after
suitable enlargement, e.g., by adding all mixed realizations in the image of the twisted Poincare
duality theory of 6.11.1, but it seems hard to guess which enlargement should satisfy iii) and iv).
Of course it would be desirable to have a purely algebraic description of JCJC^ , even a conjectural one, and perhaps this is not out of reach: the idea would be to use Beilinson'sfiltrationon
the Chow groups (cf. § 11) to construct JCJC^ as the heart of a certain triangulated category. In
any case one conjectures that the realization functor H to
Mfij is fully faithful and identifies
c
JCJC^ with a Tannakian subcategory of MR^ , which makes the construction of § 4 reasonable.
For a characterization of the image of H , i.e., of the motivic realizations, it will be useful to consider further realization functors (e.g., crystalline ones as in Grothendieck's original approach to
motives) and further comparison isomorphisms (e.g., those conjectured in [Fo] and partly proved
in [FM] and [Fa]). While it seems a complete mystery how to characterize the pure motivic
realizations it may be easier to predict those extensions of given pure motives which are motivic,
cf. [Bei 1] and [BK].
C3. The spectral sequence iii) of the previous section arises the question for the relation between
motivic cohomology and motivic extensions, i.e., extensions in the category
MMj of 4.1 (or simic
lar ones). First consider zero—extensions, i.e., motivic morphisms. The Chern characters give maps
C h : H ^ ( X , < K J ) ) —-» r H | ( X j ) = H o
ij
H
m M f i k
(l,H| (X,j)) ,
H
and by definition the target group is Hom^j^ (1,H^g(XJ)) if H ^ g ( X J ) is in
MMj (e.g., if
c
X is smooth). Hence ch. . can be regarded as the first edge morphism in iii), and the degeneration of iii) for smooth and projective X is related to the surjectivity of ch. . for i = 2j (the
only case where TH^g(XJ) ^ 0) , which was discussed in 5.4.
Now let
= ker cL j . The constructions of 6.11 give maps
H^(X,Q(J))Q
c
Mj
:
V M))
E
X
0
—
1
Brtia OHil (Xj)) •
k
I have not shown that these have image in
art
1
x
1
MM CÂH ( J)) G ^ M B k ^ i i ^ J ) ) '
k
at least not in general. Only for smooth X and i = 2j the results in § 9 show that one gets motivic extensions: for the considered Poincare duality theory the object E in 9.1.1 is in MMj as a
c
2 i —1
subobject of
(UJ) (cf. 4.2). It is possible to extend this somewhat to the case i > j .
For example, let X be smooth of dimension d , and let z be an element of H (X J^ ) . It
1
j
2
is represented by a finite family (f.) , with f• ^ O in the function field of an irreducible divisor
Y C X and E div(f) = O on X (cf. 6.12.4 e)). If Y = U Y. , then (f) defines an element in
i
E
i
h e n c e
d-1 d + 2 M >
'
a
n
e l e m e n t
i n
H
2d-3(
Y > 2 ?
£(
d
H
"
= ^
X
2
2
( > £.( ))
et, Y
cf
(-
8 1 3
)-
I f H
i s
3
2
Y
mapped to zero in H^(X,2?^(2)) , one obtains a motivic extension of TL^ by H^(X,ff^(2))/N
t
via pull—back from the exact sequence
O
H2 (X,2 (2))/N
t
Y
3
Hj (X=T I P)) — H
(X,B (2)),
et ,Y
t
Y
t
1
t
t
2
where N
is the image of H
(X,ZL(2)) as in 9.16. This construction amounts to the one
et,Y
*
communicated to me by A. Scholl in a letter from September 1985 (cf. the introduction). One can
O
show that the element in H^(X,Q^(2)) associated to z by the above prescription is ch^ (z)
t
via
the
2
1
identfication
H ( X ^ ) ® Q = H^(X,Q(2))
j
of
2
6.12.4 e)
and that, for
2
z e H^(X,t)(2)) , ch^ (z) e Ext^ (d)^,H (X,Q^(2))) induces the above extension via push0
2
2
out to H (X,<} (2))/N
Y
(e.g., by using 9.5).
t
More generally, for
X
z ' e H W( ><KJ))
for
s
o
m
i>j
e
every
ze H^(X,(|(j))
Y C X of codimension > i - j
TH^ (XJ) , then the image of ch^ j(z) in E x t ^
fl
comes from an element
(cf. 5.23). If ch. .(z) = O in
Y
R
(!,H^g^X j)ZN ) is the class of the pull-
back extension
0
1
y
— * HIH (XJ)ZN —
H
I
1
CX-Yj) — 4 H |
U|
0
1
'Hlg (Xj)ZN
Y
i
(where N = Im(H^
X
Z
>
( J)
Y
( X , j) —->
H^ (XJ)
h
j z'
E
i
1
H
V
H
>
1
>
0
,
1
> H ^ (Xj))
as in 9.16) and hence motivic. The class ob-
viously vanishes if z is in
Y
N H ^(X,«Kj)) = Im(H ^ ( X ( K J ) )
Y
I
0 0
— - H ^(X O(J))) ,
1
where
H^ (X Q(J))
Y
1
= Ker(H^X O(J)) — , T H ^ ^ X J ) ) .
0 0
J
We thus obtain maps (for v > 0)
^(X-O(J))0/
H
r
a.HAi1(X,j)/NI/) ,
> E x t
M M k
where N* is the coniveau filtration and S f = U f t
v
varieties of codimension v in X , cf. 9.16 (where
, with Y running through all closed subN
V
and W were also denoted N
V
and
17
N , with ariskof confusion). If k is a numberfieldand X is smooth and proper, then one
expects injectivity of
Hi (XJ(J)) / N
0
r
and
the above
1
Hj^ (XJ) = N
Y
maps
by similar
1
Y
—
Y
Ext^LHji-!(X J)ZN )
1
arguments
as in 9.16. By using
a splitting
v
e H ^ g ( X J ) Z N one then has to study if
c
:
Mj
H
V
x
E x t
' ^ ) ) —
MR
( 1
k
H
x
' ii!Y( ^))
has image in the motivic extensions, but here the above method will not apply in general: For
example, if (i,j) = (4,3) and Y is a smooth irreducible division, then by Poincare duality isomorphism we study
H
i(
Y
2
-<K ))—
1
E x t ^ l . H ^Y )),
l 2
i.e., a situation with i < j .
Concerning the case i = j = 1 one has H^(X,Q(1))Q <-£- H ^(Spec k,^(l)) for a smooth
geometrically connected variety
Spec k , and one may use 1—motives to construct geox
1
metric extensions associated to elements in H ^(Spec k,Q(l)) = k ® <) : One can show that for
xek
x
the 1-motive ([D5] 10.1)
I
M:
><G
m
givesriseto an extension of realizations
0
»1(1)
>T(M)
»0,
»1
Q
whose class in Ext^j^ (1,1(1)) is ch^ ^(x) . T ( M ) ^ is also obtained as relative cohomology of
smooth varieties or as cohomology of a singular, non—proper variety, by the isomorphism
1
T(M)Q ™ H (G (X) I) and the commutative exact diagram
j
m
diag^
1 ( 1 )
1
2
>
1
H (G Od(I X) I)
mm
1
, H (G I)
i
» 0
ml
(a,b)
I
a-b
0
1
> 1(1)
,H (G (X) I)
,
1
m
1
• O
1
1
where ^ M is G = Spec(k [M" ]) with 1 and x glued together. The isomorphism follows
m
m
with the methods of [ D 5] 10.3.1 don't know if T(M)Q is an object of MMjc as defined in 4.1.
Concerning i < j , I do not even know if the extensions of 1 by l(j) attached to elements in
1
7
H ^(Speck IKj))
by ch . appear in some kind of cohomological realization for j > 1 . In
j
[D 11] Deligne has shown that certain canonical extensions appear in the realizations attached to
?r (IP^\{0,l,aD}) , but it is unknown, if they come from some elements in K—theory . It may be
1
possible to use Bloch's description of ch. . via cycle maps on higher Chow groups [Bi 6] and a
construction analogous to the one in § 9 to produce geometric extensions from elements in
H^(X,<Kj)) for arbitrary i and j .
Of course, it is as important to study the relation between motivic cohomology and extensions in the single realizations. For example, for a smooth, projective variety X over a number
field k one expects that the map
c h
i,j®^
:
H
i r ( - ^ J ) ) o * ' 4 — " ^ G ^ i ' ^ ^ W ) ) = ( k- a ( '^<L(J))
Hl
x
G
H
1
X
is injective and that its image can be described by explicit local conditions (see [BK] (5.3), and
[J3] § 6 as well as a forthcoming paper; this extends the conjectures in § 13 to all i,j e I) . This
in turn would imply that the map
E x t
iM^'
H i
X
( 'J)) —
E x t
H
X
G mot^- L( ^(J))
k l
obtained by "passing to the t—adic realizations" is an isomorphism, where the "motivic G^-extensions" forming the target group are characterized by local properties related to Fontaine's
theory of p—adic representations.
C4.
It may be useful to recall how one can calculate the Yoneda-Ext-groups in a neutral
Tannakian category in terms of group cohomology of the associated "Galois" group: Let G be a
linear algebraic group over afieldof characteristic zero, let U be its unipotent radical, and let
Rep G be the category offinitedimensional algebraic representations of G (these are the rational modules of [Ho 1] ). Then for objects V , W in Rerj G one has
Ext
V
W
Rej) G ( ' ) = ^ , H o j n ^ W ) ) ,
1
where H (G,—) is the cohomology theory defined in [Ho 1] , and isomorphisms
i
i
C
U
H (G V)^H (U V) Z ,
1
1
1
i
H (U V)^H (U V)
1
j
1
where u = Lie U is the Lie algebra of U and Hû,—)
denotes Lie algebra cohomology
([Ho 2]).
All this extends to pro—algebraic groups and continuous representations by passing to the
limit. Putting this together we obtain the isomorphisms
EXI
(1,M)
MM
H"(Iae
MG{ff)
V{a)M )
ff
,
where M^. is a MG(a)—module by definition.
Let me also mention that one may use injectives to calculate the Yoneda—Ext-groups, even
MMj does not have enough injectives. But for any neutral Tannakian category Rep G the
if
c
ind-category has enough injectives and these may be used to calculate H^G,—). In fact, these are
the rationally injective modules of [Ho 1] .
C5. Let
MMj be defined as in 4.1 or as a suitable enlargement. While it seems out of reach to
c
describe this category, completely, i.e., to determine the whole group
MG(a) for one
a : k « — • C , it may be possible to describe certain subcategories, i.e., to calculate certain
quotients of MG(a) . By Deligne's result that every Hodge cycle on an abelian variety A is
absolute Hodge ([DMOS] I 2.11), one can in principle determine the Tannakian subcategory of
1
Mj generated by H (A) in terms of the Mumford-Tate group of A . In any case, one has a nice
c
description of the category P C M
c
of motives of potential CM—type over f) (generated by
Artin motives and abelian varieties of potential CM-type) , in terms of Hecke characters and the
Taniyama group ( [DMOS] IV).
In the mixed case, Bryhnski has proved an analogue of Dehgne's theorem for a 1—motive M
([Br] 2.2.5). However, it is not clear to me whether his theorem is sufficient for the determination of the quotient of MG(cr) corresponding to the Tannakian category generated by T(M)^
and 1(1) . Bryhnski only considers Hodge and absolute Hodge cycles in tensor products
T(M)JP • T ( M ) J ^ • K r )
whereas for non—reductive groups one a priori has to consider cycles in subquotients of such
spaces, too ([DMOS] I 3.2 (a)).
j
j
Another object of interest is the category JCJ= JCJ^ of mixed Tate motives over k : these
!
are mixed motives whose pure quotients are sums of pure Tate motives l(r) , r e TL . There are
now several proposals for an algebraic description of such a category ( [BMS] , [BGSV] ). It may
be interesting to compare them with other candidates
MT^ constructed in MR^ by the
techniques of § 4. I don't know how close the category Lj generated by realizations of linear
c
varieties (see § 14) comes to such a category MT^ .
Suppose JCJC and JCJ exist. What are the properties of the maps
Ext ^jOi(J)) — * Ext
I
(Lm)
fcjc
for j > 1 ? Will the expected maps
H ^(Spec k,Q(j))
factorize through Ext
> Ext^(UQ))
for i > 1 ? In particular, can one associate successive extensions of
Tate realizations (e.g., t—adic ones) to elements in K-groups of k ?
Bibliography
[BFM]
P. Baum, W. F u l t o n , R. MacPherson: Riemann-Roch
f o r s i n g u l a r v a r i e t i e s , P u b l . Math. I.H.E.S. 45
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[Bei 1]
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4]
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des I n t e r s e c t i o n s e t Theoreme de Riemann-Roch, S p r i n g e r
L e c t u r e Notes 225 (1971).
6]
Notations
1 and TL^ are theringsof rational and t-adic integers, respectively.
^ , R, C,
F
R
are thefieldsof rational, real, complex, and £,-adic numbers, respectively.
is thefieldwith q elements.
x
is the multiplicative group of a ring R .
A
, for a group G and a G—module A , is thefixedmodule: A = {a e A | ga = a for all
geG}.
If X is a variety over a field k , then X(k) is the set of k—rational points, X *^ k' (or
X x ^ k ' ) is the base extension via afieldextension a : k «—• k' , and Pic(X) is the Picard
0
0
group of X . If X is smooth, Pk (X) is the Picard variety, and Pic (X) = Pic°(X)(k) .
1
1
H (XjA) = H (X(C) A) , for an abelian group A and a variety X over C , is the singular (Betti)
j
cohomology of the topological space X(C) with coefficients in A . If F is a sheaf for the analytic
topology on
1
X
1
an
H (X F) = H (X F)
j
j
(i.e., a sheaf on the associated complex analytic space
is the analytic sheaf cohomology. Since canonically
a n
X ) , then
1
an
H (X A) =
5
1
H (X(C) A) for the constant sheaf A associated to A , we mostly write A again for A without
j
risk of confusion. Hg(X,2)(j) = H (X ZZO))
1
J
Hodge structure given by
1
H (X ZZ)
J
is the j—fold Tate twist of the integral (mixed)
.
H^(X F) denotes etale cohomology of an etale sheaf F on a scheme X (see, e.g., [Mi]). If L
j
is a prime invertible on X , we let as usual
HL(X,ZZA(J))
ei
'L
n
=
1 im FLÂX fj^l) , where fi _ is the
ex
Qn
«n
y
sheaf of £ - t h roots of unity, and put Hl (X,fy(j)) = H^(X,^(j)) ® ^
t
. For these groups
we often omit the index "et" . Similar notions apply for more general Zf^— or
— "sheaves",
for which we refer to loc. cit. or [SGA 5] .
*JU TJ) , for a scheme U and a geometric point rj
i
> U , is the algebraic fundamental group
(cf. [Mi] I § 5).
Cone (a) is the cone of a morphism a : A *
> B" of complexes, A * [r] is the r—fold shift
of A* (cf. [Mi] p. 167, 174).
As usual, D^(A) is the derived category of bounded complexes in an abelian category A . We
also use other standard notations connected with derived categories (like Rf* , Rf' , ®^ ,
RHom , ...) , which can be found, e.g., in [SGA 4] XVII and XVIII.
tr. deg(k) is the degree of transcendence of afieldk over its prime field.
Other notations are introduced at the following places:
H
DR( )> S R ( )
x
U
h
1,25
Hj(X) , Hj(U)
1,32
Hj(X) , Hj(U)
1,32
1,10,33
25
^k
0
X'°X
<
Y
25,26
>
(two meanings)
(TU
27,76
32
an
I
L,a
aX , aV
1,10,34
H (U)
35
2,32
ME
9
UX
MM , M
43,46
10
G(<T) , MG(ff) , RgE G
49
H
10,43,46
U(<r) , sp^ , Msp^
50
W
10,83,87
CH (X)
12
c
H
d
k
,H ,H
r
t
7
m
G r
m •%
H « H ' , Hom(H,H')
12,13
1 (identity object)
13,80
r
14,81,125
n
k
H
Vec
F
Hx k' , R /
k
k
/ k
57
*t 'l ' D R ' < #
R
T
R
57,62
58,59,62
V
65
) - i,j
X
ch
( ) ' !*( 'W>)
67
i
68
K
U
15
H
15,125
H
cont
16,75
k
r
m
v
36,57
1
a )
H
U
i
70
A = Iim A / i
n
70,71
n
G
k
Indp
k'
l(n) , H(n)
lc
17
i
NH
n
76,162
G
17
76
RPr
r, sr
BtyXj) , H (X,b)
A
*
U ,a ,
I
H (X J)
n , ?7
1
A
X
F
K
79
80
N H
2
I
,
, WRep (G Q )
157
p (X)
G
2
, N CH
J
I
, N CHJ
161,162
81
167
82
172
85
174
178
J
CH (X) ^
T
I I T Y I
JU ^ TT
86
t
157
0
I
Hf(X,I (b))
G
A ( X ) , Alb(X)
T(X)
86
Rep (G -)
C
79
I
TTMM
180
H V f o r J= JCJtj
180
, H (X,-)
184,206
88
H
WRep (G B )
89
H
K-JlJf
92
H^X/RIRQ.F)
94
H"(k,V) , H^k V)
199
94
r. .
IJ
200,205
s
c
c
H
A H
kl
H
'
kj
t
t
a
J=
(X-)
E T
a
185
ffll
204
98
r
101
V^ (J) , H
, K (X)W
104
Z^(J) CH (XJ)
H^X,«(b))
X / x , div , tame
104
K
A H
A H
K
(X.,Y.)
(Z.)
K^(X)
,
K
A
H
C
(Z.)
A
105
105,107
P
Ch (X)
B
106
m
107,108
Z (X) , ZJ(X)
I
A (X) , A (X)
115,117
dim k , L(V,s)
a
115
J
I
Z
K (X),ch
, ch ,
Td(X)
122
Z
123
T
126,154,182
r,r'
r
127,128
r
a,b ' a,b
k
133
X
Z (X) ,
I
ct
ZJ(X)
0
140
0
, c£,j
CHJ(X)
0
140,156
, CH (Z)
I
0
155
a,b
N
I
J
208,209
1
F J N E
209
Index
Abel-Jacobi map 140-143, 151, 153-178,
183, 204
absolute Hodge cycle 3, 14, 59-65, 72,
73, 127
absolute (co)homology theory 153, 174,
182, 183
Albanese variety 157-160, 177
Chowgroup 57-59, 105-107, 109, 118,
121, 122, 154-182, 189, 204, 212,
220
comparison isomorphism 1—5, 11, 14, 33,
34, 40, 41, 58, 59, 65, 97, 126
continuous etale cohomology 70,
149-151, 153
Adams operators 67, 104
cycle: see algebraic cycle
algebraic cycle 57, 107, 108, 115-121,
cycle map 57-59, 107-109, 113, 115, 117,
139-141, 155, 165, 168, 170,
173-180
arithmetical dimension 115-117
arithmetic C)^—representation,—sheaf
199, 204
Artin motives 49, 53
base change 41, 89, 116, 117, 186,
200, 218
base extension 16, 75
Beilinson
complexes 209
122, 126, 140, 143, 151-154, 162,
174, 175, 189, 219
Dehgne (co)homology 68, 69, 131, 132,
152, 153, 182, 183, 186
de Rham (co)homology 1, 8, 25, 30, 58,
93, 96, 125
de Rham complex 25-31, 96, 97, 101
effective (Hodge structures or £-adic
representations) 172
extension class 139-143, 150, 176,
180-183, 204
/Bloch conjecture 158, 168
fibre functor 14, 49, 125
conjecture on Chow groups 178-182
filtration
Betti (co)homology 92
Bloch's conjecture on zero cycles 177
Bloch's theorem on zero cycles 158
generalizations 170, 182
Bloch—Ogus theory: see Poincare duality
theory
by coniveau 76, 162-164, 168-173,
182, 211
on Chow groups 178-182
fundamental class 81, 107, 110, 123, 126
Galois descent 74, 75, 216
geometric cohomology 129, 153, 177, 183
Birch and Swinnerton-Dyer 168, 169
good proper cover 113, 114, 118, 119, 163
canonicalfiltration28
Grothendieck
Chern class, character 65, 67-75,
motive 180
122-126, 154, 190, 200, 205, 206,
- Riemann- Roch 122
209, 210, 216
/ Serre conjecture 5, 61, 191
higher Chow groups 209, 212
mixed realization (for absolute Hodge
homologous to zero/homological
cycles)
equivalence 139, 140, 168, 176,
abstract 9-24, 43, 94
179
of a smooth variety 35, 55
Hodge conjecture
classical 58, 77
for arbitrary varieties 63, 108,
114
generalized (Grothendieck) 172
Hodge
of an arbitrary variety 94—104, 115,
125, 175, 177, 217
motive
as defined by Grothendieck 180
attached to a modular form 5—9
for absolute Hodge cycles 1-4, 46-49,
cycle 60, 61, 121
filtration 10, 18, 30, 32, 33, 97
structure 1, 58, 172
identity object 13, 80, 83
intermediate Jacobian 141, 157
internal Horn 13
intersection of cycles 174—178
K-cohomology 106, 121
K-theory, K'-theory 65,67-79,
56
motivic (co)homology 67, 104-107, 181,
182, 189, 209-213, 215, 216
and t-adic (co)homology 127, 184,
189-221
and other (co)homology theories
126-130, 154-156, 182-184
Mumford's counterexample 157
Mumford-Tate group 61, 62
104-107, 121-128, 131-138, 181,
numerical equivalence 178—180
189, 190, 210, 220
one—semi-simple 193, 199
t-adic
Parshin's conjecture 189
Chern character 69, 74, 190, 200,
205, 210
(co)homology 1, 4, 32, 36-40, 69,
Poincare duality 3, 8, 45, 82, 108, 118,
128, 134, 140, 190, 215
Poincare duality theory (twisted)
86, 89-90, 97, 101, 115-117, 126,
79-107, 121-126, 173-176, 182, 186,
149-151, 172, 184-186, 191,
194, 208, 210, 214-216
215-219
£-adic (co)homology and motivic
(co)homology 127, 184, 189-221
with weights 85, 89, 92, 94, 109-113,
129, 130, 139, 154-156, 161, 180
potential
adic sheaf 185, 199
level filtration 182
projection formula 19, 81, 111, 130, 171
L-function 115-121, 168, 169, 220
pull back morphism (in homology) 215
Lichtenbaum complexes 212
pure (of weight m)
linear varieties 217-221
Hodge structure 10
mixed
t—adic representation/sheaf 87, 116
absolute Hodge complex 98—102
object 83-85
Hodge structure 10-13, 32, 34, 64,
realization 12,15,46,63,64
92, 93, 100, 141-143, 152, 186-188
t-adic sheaf 89-91, 117, 120
motive 43-S6, 181-184, 188
purity 38, 77, 208, 210
realizations
smooth t-adic sheaf 87, 116, 199, 201,
attached to a modular form 5—9
218
Betti (Hodge), de Rham, £-adic and
standard conjectures 179, 181
others 1, 12, 73, 95, 96, 103,
Tannakian category
115, 182-184, 186
Tate conjecture
14, 15, 4 3 - 5 6 , 61
for absolute Hodge cycles (cf. mixed
classical 57, 77, 115
realization) 12, 15, 16, 46, 55
for arbitrary varieties 63, 109, 114,
regulator map 184
116-121
representable A ( X ) 177, 182
generalized (Grothendieck) 172
Q
resolution of singularities 25, 56,
Tate
93, 95, 112, 114, 191, 192, 194,
realization, object 17, 181
195
twist 2, 17, 18, 58, 92
restriction 16, 75
tensor category 14, 80, 82, 83, 86
Riemann-Roch
with weights
theorem 122, 123
transformations
83^85, 88, 92, 94,
180-183
(see also Poincare duality theory)
123-128, 154, 190,
Toddclass
204, 206
j
RoitmanS theorems on zero cycles 157,
123-126
twisted Poincare duality theory: see
Poincare duality theory
159, 160
semi—simplicity (see also Grothendieck/
variety 93
Serre conjecture) 50, 51, 63,
weight: see pure of weight m
113, 180, 191-195, 199, 204, 218
weight filtration
simplicial variety
on a mixed realization 10, 12, 94
93, 9 5 - 9 7 , 100-102,
on an I—adic representation 87-91
192
specialization map
on a tensor category 83, 84
201—203, 219
spectral sequence
on (co)homology
associated to afiltration29
3 0 - 3 2 , 89, 92, 97
weights occurring
Bloch-Ogus 197
in an object 66, 83
Brown--Gersten 106
in (co)homology 66, 85, 89, 92, 116,
Hochschild-Serre 70, 143, 150, 151,
117
weight spectral sequence
153, 203, 206
hyper cohomology/ext 144, 151, 152,
Weil cohomology 181
180-183, 211
Leray
32, 41, 185-187
QuiUen 71, 76, 105, 131, 133, 220
Ftegensburg j
30-43
X ,
HlO
v
NOTES
X K T iyrATM H i I V I i ^ T
E d i t e d by A. Dold and B. Eckmann
X X J R J E l
X
3
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1988.
1
Vol. 1355: J. Bokowski, B. Sturmfels Computational Synthetic Geometry. V 168 pages. 1989.
1
1
Vol. 1356: H. Volkmer Multiparameter Eigenvalue Problems and
Expansion Theorems. VI, 157 pages. 1988.
1
Vol. 1357: S. Hildebrandt, R. Leis (Eds.), Partial Differential Equations
and Calculus of Variations. VI, 423 pages. 1988.
Vol. 1328: J.L. Bueso, P. Jara B. Torrecillas (Eds.), Ring Theory.
Proceedings, 1986. IX 331 pages. 1988.
Vol. 1358: D. Mumford, The Red Book of Varieties and Schemes. V,
309 pages. 1988.
Vol. 1329: M. Alfaro J.S. Dehesa F.J. Marcellan, J.L. Rubio de
Francia, J. Vinuesa (Eds.): Orthogonal Polynomials and their Applications. Proceedings, 1986. XV, 334 pages. 1988.
Vol. 1359: P. Eymard J.-P. Pier (Eds.), Harmonic Analysis. Proceedings, 1987. Vlll 287 pages. 1988.
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1
1
1
Vol. 1330: A. Ambrosetti F. Gori, R. Lucchetti (Eds.), Mathematical
Economics. Montecatini Terme 1986. Seminar. Vll 137 pages. 1988.
1
1
Vol. 1331: R. Bamon, R. Labarca, J. Palis Jr. (Eds.), Dynamical
Systems, Valparaiso 1986. Proceedings. VI, 250 pages. 1988.
Vol. 1332: E. Odell, H. Rosenthal (Eds.), Functional Analysis. Proceedings, 1986-87. V, 202 pages. 1988.
Vol. 1333: A.S. Kechris D.A. Martin, J.R. Steel (Eds.), Cabal Seminar
81-85. Proceedings, 1981-85. V 224 pages. 1988.
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1
Vol. 1360: G. Anderson, C. Greengard (Eds.), Vortex Methods.
Proceedings, 1987. V, 141 pages. 1988.
Vol. 1361: T. torn Dieck (Ed.), Algebraic Topology and Transformation
Groups. Proceedings, 1987. VI 298 pages. 1988.
1
Vol. 1362: P. Diaconis, D. Elworthy, H. Folimer, E. Nelson, G.C.
Papanicolaou, S.R.S. Varadhan. Ecole d'Ete de Probabilites de SaintFlour XV-XVII, 1985-87. Editor: P.L. Hennequin. V, 459 pages.
1988.
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Vol. 1334: Yu. G. Borisovich Yu. E. Gliklikh (Eds.), Global Analysis
- Studies and Applications III. V, 331 pages. 1988.
Vol. 1363: P.G. Casazza, T.J. Shura. Tsirelson's Space. VIII 204
pages. 1988.
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1
Vol. 1335: F. Guillen, V. Navarro Aznar, P. Pascual-Gainza, F. Puerta,
Hyperr^solutions cubiques et descente cohomologique. XII, 192
pages. 1988.
Vol. 1364: R.R. Phelps, Convex Functions, Monotone Operators and
Differentiability. IX, 115 pages. 1989.
Vol. 1365: M. Giaquinta (Ed.), Topics in Calculus of Variations.
Seminar, 1987. X, 196 pages. 1989.
Vol. 1336: B. Helffer, Se mi-Classical Analysis for the Schrodinger
Operator and Applications. V, 107 pages. 1988.
Vol. 1366: N. Levitt, Grassmannians and Gauss Maps in PL-Topology.
V 203 pages. 1989.
Vol. 1337: E. Sernesi (Ed.), Theory of Moduli. Seminar, 1985. VIII 232
pages. 1988.
Vol. 1367: M. Knebusch, Weakly Semialgebraic Spaces. XX, 376
pages. 1989.
Vol. 1338: A.B. Mingarelli, S.G. Halvorsen, Non-Oscillation Domains
of Differential Equations with Two Parameters. XI, 109 pages. 1988.
Vol. 1368: R. Hubl, Traces of Differential Forms and Hochschild
Homology. Ill, 111 pages. 1989.
Vol. 1339: T Sunada (Ed.), Geometry and Analysis of Manifolds.
Procedings, 1987. IX, 277 pages. 1988.
Vol. 1369: B. Jiang, Ch.-K. Peng, Z. Hou (Eds.), Differential Geometry
and Topology. Proceedings, 1986-87. VI, 366 pages. 1989.
Vol. 1340: S. Hildebrandt, D.S. Kinderlehrer, M. Miranda (Eds.),
Calculus of Variations and Partial Differential Equations. Proceedings,
1986. IX 301 pages. 1988.
Vol. 1370: G. Carlsson, R.L. Cohen, H.R. Miller, D.C. Ravenel (Eds.),
Algebraic Topology. Proceedings, 1986. IX 456 pages. 1989.
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1
Vol. 1341: M. Dauge, Elliptic Boundary Value Problems on Corner
Domains. VIII, 259 pages. 1988.
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Vol. 1371: S. Glaz, Commutative Coherent Rings. XI, 347 pages.
1989.
Vol. 1372: J. Azema PA. Meyer, M. Yor (Eds.), Seminaire de Probabilites XXIII. Proceedings. IV, 583 pages. 1989.
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Vol. 1342: J . C Alexander (Ed.), Dynamical Systems. Proceedings,
1986-87. Vlll 726 pages. 1988.
1
Vol. 1343: H. Ulrich, Fixed Point Theory of Parametrized Equivariant
Maps. Vll 147 pages. 1988.
1
Vol. 1344: J. Kral J. Lukes, J. Netuka J. Vesely (Eds.), Potential
Theory - Surveys and Problems. Proceedings, 1987. Vlll 271 pages.
1988.
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1
Vol. 1345: X. Gomez-Mont, J. Seade, A. Verjovski (Eds.), Holomorphic
Dynamics. Proceedings, 1986. VII, 321 pages. 1988.
Vol. 1346: O. Ya. Viro (Ed.), Topology and Geometry - Rohlin
Seminar. XI, 581 pages. 1988.
Vol. 1347: C Preston, Iterates of Piecewise Monotone Mappings on
an Interval. V, 166 pages. 1988.
Vol. 1348: F. Borceux (Ed.), Categorical Algebra and its Applications.
Proceedings, 1987. Vlll 375 pages. 1988.
Vol. 1349: E. Novak, Deterministic and Stochastic Error Bounds in
Numerical Analysis. V, 113 pages. 1988.
Vol. 1373: G. Benkart, J.M. Osborn (Eds.), Lie Algebras, Madison
1987. Proceedings. V, 145 pages. 1989.
Vol. 1374: R.C. Kirby, The Topology of 4-Manifolds. VI, 108 pages.
1989.
Vol. 1375: K. Kawakubo (Ed.), Transformation Groups. Proceedings, 1987.
VIII, 394 pages, 1989.
Vol. 1376: J. Lindenstrauss, V.D. Milman (Eds.), Geometric Aspects of
Functional Analysis. Seminar (GAFA) 1987-88. VII, 288 pages. 1989.
Vol. 1377: J.F. Pierce, Singularity Theory, Rod Theory, and SymmetryBreaking Loads. IV, 177 pages. 1989.
Vol. 1378: R.S. Rumely Capacity Theory on Algebraic Curves. Ill, 437
pages. 1989.
1
1
Vol. 1379: H. Heyer (Ed.), Probability Measures on Groups IX.
Proceedings, 1988. VIII, 437 pages. 1989
Vol. 1380: H. P. Schlickewei. E. Wirsing (Eds.), Number Theory, Ulm
1987. Proceedings. V, 266 pages. 1989.
Vol. 1381: J.-O. Strdmberg, A. Torchinsky. Weighted Hardy Spaces.
V. 193 pages. 1989.
Vol. 1382: H. Reiter, Metaplectic Groups and Segal Algebras. XI, 128
pages. 1989.
Vo!. 1383: D. V. Chudnovsky, G . V. Chudnovsky, H. C o h n M. B. Nathanson
(Eds.), NumberTheory, NewYork 1 9 8 5 - 8 8 . Seminar. V, 256 pages. 1989.
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Vol.
1384:
J. Garcia-Cuerva
(Ed.), Harmonic
Analysis
and
Partial
Differential Equations. Proceedings, 1987. V l l 213 pages. 1989.
1
Vol. 1385: A . M . Anile, Y. Choquet-Bruhat (Eds.), Relativistic Fluid Dynamics. Seminar, 1987. V 3 0 8 pages. 1989.
1
Vol. 1386: A. Bellen, C W. Gear, E. Russo (Eds.), Numerical Methods for
OrdinaryDifferentiaI Equations. Proceedings, 1987. VII, 136 pages. 1989.
Vol. 1387: M. Petkovic, Iterative Methods for Simultaneous Inclusion of
Polynomial Zeros. X, 263 pages. 1989.
Vol. 1388: J. Shinoda T.A. Slaman T. Tugue (Eds.), Mathematical Logic
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1
and Applications. Proceedings, 1987. V, 223 pages. 1989.
Vol. 1000: Second Edition. H. Hopf, Differential Geometry in the Large. VII,
184 pages. 1989.
Vol. 1389: E. Ballico C. Ciliberto (Eds.), Algebraic Curves and Projective
Geometry. Proceedings, 1988. V 2 8 8 pages. 1989.
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1
Vol. 1390: G . Da Prato L. Tubaro (Eds.), Stochastic Partial Differential
Equations and Applications II. Proceedings, 1988. VI, 2 5 8 pages. 1989.
1
Vol. 1391: S. Cambanis A. Weron (Eds.), Probability Theory on Vector
Spaces IV. Proceedings, 1987. Vlll 4 2 4 pages. 1989.
1
1
Vol. 1392: R. Silhol, Real Algebraic Surfaces. X, 215 pages. 1989.
Vol. 1393: N. Bouleau, D. Feyel R Hirsch G . Mokobodzki (Eds.),
Seminaire de Theorie du Potentiel Paris, No. 9. Proceedings. VI, 265
pages. 1989.
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Vol. 1394: T. L. Gill W. W. Zachary (Eds.), Nonlinear Semigroups, Partial
1
Differential Equations and Attractors. Proceedings, 1987. IX, 233 pages.
1989.
Vol. 1395: K. Alladi (Ed.), NumberTheory Madras 1987. Proceedings. VII,
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234 pages. 1989.
Vol. 1396: L. Accardi, W. von Waldenfels (Eds.), Quantum Probability and
Applications IV. Proceedings, 1987. V l 3 5 5 pages. 1989.
1
Vol. 1397: P R . Turner (Ed.), Numerical Analysis and Paratle) Processing.
Seminar, 1987. VI, 264 pages. 1989.
Vol. 1398: A . C . Kim, B . H . Neumann (Eds.), Groups - Korea 1988. Proceedings. V, 189 pages. 1989.
Vol. 1399: W.-P. Barth, H. Lange (Eds.), Arithmetic of Complex Manifolds.
Proceedings, 1988. V, 171 pages. 1989.
Vol. 1400: U. Jannsen. Mixed Motives and Algebraic K-Theory. XIII, 246
pages. 1990.

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