Doc Math J DMV
Classical Motivic Polylogarithm
According to Beilinson and Deligne
Annette Huber Jorg Wildeshaus
Received February
Revised September
Communicated by Peter Schneider
Abstract The main purp ose of this pap er is the construction in motivic
cohomology of the cyclotomic or classical polylogarithm on the pro jective
line minus three p oints and the identi cation of its image under the regula
tor to absolute Deligne or l adic cohomology By sp ecialization to ro ots of
unity one obtains a compatibility statement on cyclotomic elements in mo
tivic and absolute cohomology of ab elian number elds As shown in BlK
this compatibility completes the pro of of the Tamagawa number conjecture
on sp ecial values of the Riemann zeta function
The main constructions and ideas are contained in Beilinson s and Deligne s
unpublished preprint Motivic Polylogarithm and Zagier Conjecture
BD We work out the details of the pro of setting up the foundational
material which was missing from the original source the pap er contains
an app endix on absolute Ho dge cohomology with co e cients and its inter
pretation in terms of Saito s Ho dge mo dules The second app endix treats
K theory and regulators for simplicial schemes
Mathematics Sub ject Classi cation Primary F Secondary R
R R D F
Keywords Polylogarithm motivic and absolute cohomology regulators cy
clotomic elements
Introduction
The aim of this work is to present the construction of the class of the cyclotomic or
classical p olylogarithm in motivic cohomology It maps to the elements in Deligne and
l adic cohomology de ned and studied in Beilinson s Polylogarithm and cyclotomic
elements B The latter elements can b e seen as b eing represented by a pro
variation of Ho dge structure or a pro l adic sheaf on the pro jective line minus three
p oints
1
See correction on page of this volume
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Our main interest lies in the sp ecialization of these sheaves to ro ots of unity
they represent the cyclotomic one extensions of Tate twists already studied by
Soul e Sou Deligne D and Beilinson B
Let us b e more precise denote by the set of primitive d th ro ots of unity in
d
Q Q T T d We get an alternative pro of of the following theorem of
d d
Beilinson s
Corollary Assume n and denote by r the regulator map
D
M
n
C i R H Sp ec Q Q n
d
M
Q C
d
There is a map of sets
H Sp ec Q Q n
n d
d M
such that
M
n
C i R r
D n
d
Q C
d
P
k
maps a ro ot of unity to Li
n
n
k
k
Now x a d th primitive ro ot of unity in Q This choice allows to identify
Sp ec Q Q n with a Q subspace of continuous etale cohomology H
d l l
cont
1
Gal Q Q
l
r
n
l
A
1 1
Q Q Q lim
r
Z l l l
l
l
r
Note that there is a distinguished ro ot of unity T in Q As was observed already
d
in B the study of the cyclotomic p olylogarithm gives a pro of of BlK Conjecture
cf Sou Th eor eme for the case n Gr Th eor eme IV for the lo cal
version if l d
Corollary Let b e the map constructed in Under the ab ove
n
inclusion the l adic regulator
r H Sp ec Q Q n H Sp ec Q Q n
l d d l
M cont
b
maps T to
n
X
d n
A
n
d n
r
l b
r
This result implies in particular that Soul e s cyclotomic elements in the group
K F Z for an ab elian number eld F and a prime l are induced by el
n Z l
ements in K theory itself Corollary Furthermore the case d of forms a
central ingredient of the pro of of the Tamagawa number conjecture mo dulo p owers of
for o dd Tate twists Q n n BlK x Finally as shown in KNF Theorem
the general case of implies the mo di ed version of the Lichtenbaum conjecture
for ab elian number elds
Documenta Mathematica
Classical Motivic Polylogarithm
The main ideas necessary for b oth the construction of the motivic p olylogarithm
and the identi cation of the realization classes together with a sketch of pro of are
contained in the unpublished preprint Motivic Polylogarithm and Zagier Conjecture
BD and its predecessors B BD p Our aim in this pap er is to work out the
details of the pro ofs To do this we have to set up a lot of foundational material which
was missing from the original sources K theory of simplicial schemes regulators to
absolute Ho dge and l adic cohomology of simplicial schemes and an interpretation
of the latter as Ext groups of Ho dge mo dules and l adic sheaves resp ectively This
material is contained in the two app endices which we regard as our main contribution
to the sub ject We hop e they prove to b e useful in other contexts than that treated
in the main text
Other parts of BD deal with the weak version of the Zagier conjecture We
do not treat this since a complete pro of has b een given by de Jeu Jeu although
by somewhat di erent means from those used in BD
We see two main groups of pap ers related to p olylogarithms
The rst deals with mixed sheaves i e variations of Ho dge structure or l adic
mixed lisse sheaves Maybe the nucleus of these pap ers is Deligne s observation that
the analytic and top ological prop erties of the dilogarithm Li viewed as a multivalued
holomorphic function on P C nf g can b e co ded by saying that Li is an entry
of the p erio d matrix of a certain rank three variation of Q Tate Hodge structure on
P n f g
C
We refer to Rm section for a nice survey of the construction of a pro
variation on P n f g containing all Li The etale analogue is constructed in
k
C
Beilinson s Polylogarithm and Cyclotomic elements B where he de ned pro
ob jects in the categories of l adic sheaves on P nf g In b oth settings the bres
at ro ots of unity di erent from coincide with the cyclotomic extensions mentioned
ab ove
The hop e and indeed the motivation underlying these pap ers is that once a
satisfactory formalism of motivic sheaves is developed the de nition of p olylogarithms
should basically carry over We would thus obtain p olylogarithmic classes in Ext
groups of motives these groups b eing supp osedly closely connected to K theory of
which everything already de ned on the level of realizations would turn out to b e the
resp ective regulator
Nowhere is this hop e do cumented more manifestly than in Beilinson s and
Deligne s Interpr etation motivique de la conjecture de Zagier reliant p olylogarithmes
et r egulateurs BD if there is such a motivic formalism then the weak version of
Zagier s conjecture necessarily holds not only the values at ro ots of unity of higher
logarithms but also appropriate linear combinations of arbitrary values must lie in
the image of the regulator
For the time b eing and in each case separately honest work is needed to p erform
the K theoretic constructions and calculate their images under the regulators
The second class of pap ers is concerned with precisely that task In analogy with
the ab ove one should rst mention Blo ch s Application of the dilogarithm function
in algebraic K theory and algebraic geometry Bl
Beilinson s Higher regulators and values of L functions B provided the
K theoretic construction of cyclotomic elements together with the computation of
their images in Deligne cohomology lo c cit Theorem Neu E
Documenta Mathematica
Annette Huber Jorg Wildeshaus
As for Zagier s conjecture we mention Goncharov s Polylogarithms and Motivic
Galois Groups Go where Zagier s conjecture including the surjectivity statement
is proved for K of a number eld and de Jeu s Zagier s Conjecture and Wedge
Complexes in Algebraic K theory Jeu which contains the pro of of the weak
version of Zagier s conjecture indep endently of motivic considerations for K of
n
a number eld and arbitrary n
Typically the ob jects of interest in this class of pap ers are complexes co cy
cles and symbols i e ob jects which do not constantly a ord a geometric or sheaf
theoretic interpretation It is by no means easy to see say how a concrete element
in some Deligne cohomology group can b e interpreted as an extension of variations
of R Ho dge structure These and similar di culties present themselves to the reader
willing to translate from one class to the other
The authors like to think of the present article as an attempt to bridge the gap
b etween the two disciplines
In a sense the coarse structure of the article follows the ab ove scheme sections
are entirely sheaf theoretic Anything we say there is therefore a priori restricted
to the level of realizations i e non motivic In sections K theory enters The
app endices provide the foundations necessary to connect the two p oints of view
Given that quite a lot has b een said ab out the l adic and Ho dge theoretic incar
nations of the classical p olylogarithm B BD WiIV the reader may wonder
why sheaf theoretic considerations still take up one third of this work
Indeed the construction of the motivic p olylog could b e achieved much more eas
ily if a satisfactory formalism of mixed motivic sheaves were available The necessity
to replace a simple geometric situation by a rather complicated one in order to replace
complicated co e cients like Log by Tate twists should b e seen as the main source of
di culty in any attempt to the construction of motivic versions of p olylogarithms
We now turn to the description of the ner structure of the main text sections
In section we normalize the sheaf theoretic notations used throughout the
whole article
Section gives a quick axiomatic description of the logarithmic sheaf Log and
the small p olylogarithmic extension pol The universal prop erty is needed
only to connect the general de nition of the logarithmic sheaf as a solution of a
representability problem to the somewhat ad ho c but much more geometric de nition
of section A reader prepared to accept the results on the shap e of the Ho dge
theoretic and l adic incarnation of the p olylogarithm may therefore take
the constructions in sections and as a de nition of b oth Log and pol and view
section as an extended introduction providing background material
In section we establish the geometric situation used thereafter As section
it is mainly intended for easier reference
In section we construct a pro unip otent sheaf G on U P n f g as
pro jective limit of relative cohomology ob jects of p owers of G over U relative to
m
certain singular subschemes The transition maps are given by the b oundary maps in
the relative residue sequence The universal prop erty then allows to identify
G with the restriction of Log to U
Section contains a geometric pro of of the splitting principle the bres
of Log at ro ots of unity have split weight ltration Since we need a pro of which
Documenta Mathematica
Classical Motivic Polylogarithm
translates easily to the motivic situation we return to Beilinson s original approach
to the splitting principle B which consists of an analysis of the action of the
multiplication by natural numbers on our absolute cohomology groups
The main ob jective of section is the description of pol in terms of geometric
data The Leray sp ectral sequence suggests that one extensions of Q by Log
should b e describ ed as elements of the pro jective limit of cohomology groups with
Tate co e cients of p owers of G relative to certain subschemes The main result
m
allows to identify pol under this corresp ondence
In Section our main to ol the residue sequence is constructed in the setting of
motivic cohomology Prop osition and Lemma The arguments are very much
parallel to those used for absolute cohomology of realizations in section However
we have to replace the singular schemes by explicit simplicial schemes with regular
comp onents This is where the material of App endix B enters
Section is the K theoretic analogue of section We consider a certain pro
jective system of motivic cohomology groups In order to identify its pro jective limit
Corollary we use bijectivity or at least controlled injectivity of the regulator
to Deligne cohomology and the results of section We are then able to de ne the
universal motivic p olylog
In the nal section the motivic version of the splitting principle is shown
Again we strongly use the known b ehaviour of the regulator to show that the action
of multiplication by natural numbers splits into eigenspaces Applied to the universal
motivic p olylogarithm this induces the cyclotomic elements in motivic cohomology In
the light of section it is clear from their very construction that they induce the right
elements not only in Deligne but also in continuous etale cohomology We conclude
by drawing the corollaries which are the main results announced at the b eginning
The App endices can b e read indep endently of the main text and of each other
They are meant to b e used as a reference but a careful reader might actually want
to read them rst We refer to the resp ective introductions for an account of their
content
The reader might nd it useful to consult HW for an overview of the strategy
of the pro of of the main results
Acknowledgments It is a pleasure for us to thank b oth Beilinson and Deligne
for not only letting us work with their ideas but explicitly encouraging us to bring
mathematics into a state in which the constructions of BD can b e p erformed
We thank C Deninger for suggesting to us that the metho ds developed and
results obtained in our resp ective PhD theses H Wi might form a sound basis
of a successful treatment of this theme
We are grateful to G Banaszak H Esnault W Ga jda T Geisser U Jannsen
M Joachim K K unnemann F Lecomte A J Scholl and J Sch urmann for useful
discussions We thank R de Jeu for his remarks on App endix B and C Soul e for
making GSo available and explaining some critical p oints We are very grateful to
the referee for her or his helpful and detailed comments on App endix B
Part of this work was written up during a stay at the Fields Institute at Toronto
in Octob er We wish to thank M Kolster and V Snaith for the invitation
A
Finally we are obliged to Mrs Gabi Weckermann for L T Xing part of this pap er
E
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Contents
Mixed sheaves
The Logarithmic Sheaf and the Polylogarithmic Extension
The Geometric Set Up
Geometric Origin of the Logarithmic Sheaf
The Splitting Principle Revisited
Polylogs in Absolute Cohomology Theories
Calculations in K theory
Universal Motivic Polylogarithm
The Cyclotomic Case
A Absolute Hodge Cohomology with Coefficients
A Algebraic Mixed Ho dge Mo dules
A Algebraic Mixed Ho dge Mo dules over R
B K Theory of Simplicial Schemes and Regulators
B Generalized Cohomology Theories
B K theory
B Cohomology of Ab elian Sheaves
B Continuous Etale Cohomology
B Absolute Ho dge Cohomology
B A Combinatorial Lemma
Mixed sheaves
We start by de ning the sheaf categories which will b e relevant for us For our
purp oses it will b e necessary to work in the settings of mixed l adic p erverse sheaves
H and of algebraic mixed Ho dge mo dules over R A Since the pro cedures are
entirely analogous we introduce for economical reasons the following rules whenever
an area of pap er is divided by a vertical bar
the text on the left of it will concern the Ho dge theoretic setting while the text on
the right will deal with the l adic setting Of course we hop e that b efore long there
will b e a satisfactory formalism of mixed motivic sheaves providing a third setting to
Documenta Mathematica
Classical Motivic Polylogarithm
which our constructions can b e applied We let
l a xed prime number
A R
A Z
l
F Q
F Q
l
and set B Sp ec A
For any reduced separated and at scheme X of nite type over B we let
X X C as a top ol space
X X Q
top
top A
Sh X Perv X Q
Sh X Perv X Q
top top
top top l
the latter categories denoting the resp ective categories of p erverse sheaves on X
top
BBD
Next we de ne the category Sh X in the l adic setting we x a pair S L con
sisting of a horizontal strati cation S of X H x and a collection L fL S j S
Sg where each L S is a set of irreducible lisse l adic sheaves on S For all S S
and F L S we require that for the inclusion j S X all higher direct images
n
R j F are S L constructible i e have lisse restrictions to all S S which are
extensions of ob jects of L S We assume that all F L S are pure
We can make this more explicit in our computations X will always b e a lo
n
cally closed subscheme of some A the strati cation is by the number of vanishing
n
co ordinates in A L S is the set of all Tate sheaves on S
b b
Following H x we de ne D X Q as the full sub category of D X Q
l l
c
S L
of complexes with S L constructible cohomology ob jects Note that all ob jects will
b
b e mixed By H x D X Q admits a p erverse t structure whose heart we
l
S L
denote by Perv X Q
l
S L
Sh X Perv X Q Sh X MHM X R
l Q
S L
see A
Because of the horizontality requirement in the l adic situation we have the full
b
U Q of the formalism of Grothendieck s functors only on the direct limit D
X l
m
b
D X Q for U op en in B and S L as ab ove see H x However for a
U l
S L
xed morphism
X Y
we have a notion of e g admissibility for a pair S L this is the case if
b b b
D U Q D X Q D U Q
Y l l X l
m m
S L
b
factors through some D Y Q Our computations will show at least a p osteriori
l
T K
that for our choice of S L all functors which app ear are admissible We will not
stress these technical problems and even suppress S L from our notation
Documenta Mathematica
Annette Huber Jorg Wildeshaus
As in BBD we denote by Hom etc the resp ective functors on the cate
gories
b b
b b
D Sh X D X Q
D Sh X D MHM X R
l
Q
S L
q
and H for the p erverse cohomology functors
We refer to ob jects of Sh X as sheaves and to ob jects of Sh X as top ological
top
sheaves Let us denote by
V V
top
the forgetful functor from Sh X to Sh X If we use the symbol W it will always
top
refer to the weight ltration
If X is smo oth we let
s
l m
s
Sh X Var X R Sh X
Sh X Et X Sh X
Q
Q
l
see A
the category of lisse
s
mixed Q sheaves on X
Sh X the category of
l
top
s
Q lo cal systems on X
top
Sh X the category of
top
lisse Q sheaves on X
l top
s s
We refer to ob jects of Sh X as smo oth sheaves and to ob jects of Sh X as
top
s
smo oth top ological sheaves Denote by U Sh X the category of unip otent ob jects
s
of Sh X i e those smo oth sheaves admitting a ltration whose graded parts are
s
pullbacks of smo oth sheaves of Sh B via the structure morphism Similarly one
s
de nes U Sh X
top
Remark Note that in the l adic situation the existence of a weight ltration i e
W
an ascending ltration W by subsheaves indexed by the integers such that Gr is
m
s
of weight m is not incorp orated in the de nition of Sh compare the warnings in
H x In the Ho dge theoretic setting the existence of a weight ltration is part of
the data
Remark We have to deal with a shift of the index when viewing e g a variation as
a Ho dge mo dule which o ccurs either in the normalization of the embedding
b
Var X R D MHM X R
Q Q
or in the numbering of cohomology ob jects of functors induced by morphisms b etween
schemes of di erent dimension In order to conform with the conventions laid down
in app endix A and WiI chapter we chose the second p ossibility a variation is
a Ho dge mo dule not just a shift of one such Similarly a lisse mixed Q sheaf is a
l
p erverse mixed sheaf Therefore if X is of pure relative dimension d over B then the
embedding
l m
b
U Q Et X D
X l
m
Q
l
asso ciates to V the complex concentrated in degree d whose only non trivial coho
mology ob ject is V
As a consequence the numbering of cohomology ob jects of the direct image say
will di er from what the reader might b e used to e g the cohomology of a curve
Documenta Mathematica
Classical Motivic Polylogarithm
is concentrated in degrees and instead of and Similarly one has to
s
distinguish b etween the naive pullback of a smo oth sheaf and the pullback
b s
on the level of D Sh X lands in the category of smo oth sheaves while
of a smo oth sheaf yields only a smo oth sheaf up to a shift
In the sp ecial situation of pullbacks we allow ourselves one notational inconsis
tency if there is no danger of confusion e g in Theorem we use the notation
also for the naive pullback of smo oth sheaves Similar remarks apply for smo oth
top ological sheaves
For a scheme a X B we de ne
b
F n a F n D Sh X
X
where F n is the usual Tate twist on B
If X is smo oth we also have the naive Tate twist
s
F n Sh X Sh X
on X If X is of pure dimension d then we have the equality
F n F n d
X
In order to keep our notation transparent we have the following
Definition For any morphism X S of reduced separated and at
B schemes we let
b b
R X D Sh X D Sh S
S
i i b
H X H D Sh X Sh S
S
Definition For a closed reduced subscheme Z of a separated reduced at B
b
scheme X of nite type with complement j U X and an ob ject M of D Sh X
de ne
a R X M R Hom b F M
abs X
D Sh X
i i
H X M H R X M
abs
abs
the absolute complex and absolute cohomology groups of X with co e cients in M
b R X n R X F n
abs abs X
i i
H X n H X F n
X
abs abs
c R X rel Z n R X j F n
abs abs U
i i
H X rel Z n H X j F n
U
abs abs
the relative absolute complex and relative absolute cohomology with Tate co e cients
Documenta Mathematica
Annette Huber Jorg Wildeshaus
In the Ho dge setting absolute cohomology with Tate co e cients coincides with
Beilinson s absolute Ho dge cohomology over R Theorem A In the l adic set
ting it yields continuous etale cohomology see the remark following De nition B
Remark If X is a scheme over S then we have the formulae
R X R S R X
abs abs S
i i
H X H S R X
S
abs abs
The Logarithmic Sheaf and the Polylogarithmic Extension
We aim at a sheaf theoretic description of the small classical p olylogarithm on
P n f g The rst step is an axiomatic de nition of the logarithmic pro sheaf
We need the following result
Theorem Let X b e the complement in a smo oth prop er B scheme of an NC
divisor relative to B SGA Exp XI I I all of whose irreducible comp onents
are smo oth over B Let x X B and write a X B The functor
s s
x U Sh X Sh B
is representable in the following sense
a There is a pro ob ject
s
G en pro U Sh X
x
the generic pro unip otent sheaf with basep oint x on X which has a weight
ltration satisfying
s
G en W G en U Sh X for all n
x n x
Note that this implies that the direct system
R a Hom G en W G en V
x n x
n N
s
of smo oth sheaves on B b ecomes constant for any V U Sh X
This constant value is denoted by
a Hom R G en V
x
b There is a section
B x G en
x
s s
c The natural transformation of functors from U Sh X to Sh B
G en x ev R a Hom
x
x
Documenta Mathematica
Classical Motivic Polylogarithm
s
is an isomorphism Similarly for the transformation of functors from U Sh X
top
s
to Sh B
top
G en x ev R a Hom
x top
x
Consequently the pairs G en and G en are unique up to unique
x x top
isomorphism
d The natural transformations of functors
s s
Hom G en Hom F x and
x
U Sh X Sh B
s
Hom G en B x
x top top
U Sh X
top
s s
from U Sh X and U Sh X resp ectively are isomorphisms
top
Pro of For a c we refer to
WiI Remark d after Theorem WiI Theorem i
s
and B and lo c cit Theorem ii Apply the functors Hom F
top
Sh B
to the result in c in order to obtain d
Remark In the Ho dge setting and for the constant base B Theorem is
equivalent to the classi cation theorem for admissible unip otent variations of Ho dge
structure HZ Theorem In this case G en is the canonical variation with base
x
p oint x of lo c cit section
Now let
G G U P n f g
m m B B
B
j U G
m
p G B pe p j U B
m
We may form the generic pro unip otent sheaf with basep oint on G
m
s
Definition Log G en pro U Sh G is called the logarithmic pro sheaf
m
As we shall see b elow there is an isomorphism
Y
W
Gr Log F k
k
Assuming this for the moment we now describ e the higher direct images
H U j Log
B
q
Theorem a H U j Log for q
B
H U j Log is split b H U j Log has a weight ltration and W
B B
More precisely any isomorphism as ab ove induces an isomorphism
Y
W H U j Log F k
B
k
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Remark By these statements on the higher direct images of the pro sheaf j Log
we mean the following
a For q the pro jective system
q
H U j Log W Log
n
B n
is ML zero
b induces a morphism of pro jective systems
m
Y
H U j Log W Log F k
m
B
m
k
m
of sheaves with a weight ltration such that the weight parts of the
pro jective systems of kernels and co kernels are ML zero
Pro of One uses the exact triangle
id
G
m
j j
of it and the fact that H G Log is easily computable For or rather H G
m m
B B
the details see WiI I I Theorem Or use and whose pro of is indep endent
of
A xed choice of
Y
W
Gr Log F k
k
W
induces in particular an isomorphism of Gr Log and F The theorem then en
ables one to de ne the small p olylogarithmic extension as the extension
W
Gr Log j Log j pol Ext
s
U U
U Sh U
Q
F k under the isomorphism mapping to the natural inclusion F
k
F Log j Hom F Log j Ext
b
U U U
D Sh U
Sh U
Hom b pe F j Log
D Sh U
Y
A
Hom F F k
Sh B
k
induced by the pro jective limit of the edge homomorphisms in the Leray sp ectral
sequence for pe and the isomorphism of b Note that the de nition of pol is
indep endent of the choice of For the details we refer to WiI I I Theorem as
there we de ne
Ext F Log W Log j Ext F Log j lim
n U U
Sh U Sh U
n
Documenta Mathematica
Classical Motivic Polylogarithm
A description of Log and pol in b oth incarnations was given by Beilinson and Deligne
see B and BD x for the Ho dge version and B for the l adic
setting The reader may nd it useful to also consult WiIV chapters and
setting N in the notation of lo c cit
We recall the values of pol at sp ectra of cyclotomic elds let d and
T T where T is the d th cyclotomic C Sp ec R where R A
d d
d
p olynomial
C is canonically a closed reduced subscheme of G A For any integer b
m A
d
prime to d there is an embedding
i C C G A
b m A
d
b
Since d is invertible on C the image of i is actually contained in U and hence we
b
may form the pullback of pol via i
b
pol Ext F Log
s
b b
Sh C
where Log denotes the pullback of Log
b
Now we have the following
Theorem Splitting Principle Log splits uniquely into a direct pro duct
b
Y
W
Log Gr Log
b b
k
k
W
and Gr Log is isomorphic to F k for any k
b
k
Pro of B or BD or WiIV Lemma Or use and whose
pro of is indep endent of
In order to identify pol with an element of
b
Y
Ext F F k
s
Sh C
k
we need to x an isomorphism
Y
W
Gr Log F k
b b
k
By de nition is the pullback via i of the isomorphism
b b
Y
W
Gr Log F k
k
of pro sheaves on G of WiIV chapters and which we brie y describ e now
m
By d there is a canonical pro jection
Log F
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Furthermore there is a canonical isomorphism
W
Gr Log p H G F
m
B
given by the fact that b oth sides are equal to p of the mixed structure on the
ab elianized fundamental group G see WiI chapter
m top
Observe that there is an isomorphism
res H G F F
m
B
given by the map residue at
Q
W
Finally b oth Gr Log and F k carry a canonical multiplicative structure
k
W
for Gr Log this is a formal consequence of
WiI Corollary ii WiI Corollary ii
see Remark b at the end of chapter of lo c cit
Our isomorphism
Y
W
Gr Log F k
k
is the unique isomorphism compatible with res and the multiplicative struc
ture of b oth sides
Using the framing of Log given by we may identify pol with an element of
b b b
Y
Ext s F F k
Sh C
k
or after twisting and forgetting the comp onent k as an element of
Y
F F k Ext
s
Sh C
k
Note that in the Ho dge setting we do not lose any information by forgetting the
s
comp onent k as there are no non trivial extensions in Sh C of F by itself
This latter statement fails to hold in the l adic context It is however true that the
zero comp onent of pol is trivial One way to see this is via WiI I I Corollary
b
where it is proved that there is in fact a mixed realization pol of which the ab ove
b
extensions are merely the Ho dge and l adic comp onents In the category of mixed
realizations there is a go o d concept of p olarization which ensures that there are
no non trivial extensions of pure realizations of the same weight Alternatively one
uses Theorem where it is proved that our pol lie in the image of the resp ective
b
regulators The claim then follows from the vanishing of H C
M
Theorem Beilinson Under the isomorphism of A we have in the Ho dge
setting
Y M
k b k
A
pol Li C i Q
b k
k
k
C C
P
n
z
for jz j and z where Li z
k
k
n
n
Documenta Mathematica
Classical Motivic Polylogarithm
Pro of B or BD i or WiIV Theorem
Note that one may identify C C with f Q C g by asso ciating to the
d
unique embedding mapping T Q Q T T to
d d
Q It allows to identify In the l adic situation choose a geometric p oint C
C and
Sp ec Z
l d
and furthermore the category of continuous Q mo dules under the Galois group of
l
l m
s
Q that are mixed and unrami ed outside l d and the category Sh C Et C
Q
l
Given this we think of Ext Q Q k as sitting inside
s
l l
Sh C
H Q Q k
l
cont
Together with the natural map of Lemma B we thus have an inclusion of
Ext Q Q k into
s
l l
Sh C
1
Gal Q Q
l
r
k
l
A
1 1
Q Q Q lim
r
Z l l l
l
l
r
Theorem Beilinson Under the ab ove inclusion we have in the
l adic setting
X
d k k
A
pol
b
k
d k
r
l b
r k
Pro of B or BD ii or WiIV Theorem
Remarks a Using the de ning prop erty of pol one can show see B or
BD pro of of that it coincides with a sp eci c sub quotient of the generic
pro unip otent sheaf on U The sp ecializations to sp ectra of cyclotomic elds of this
sub quotient were already studied in D section In particular Theorems and
are equivalent to the Ho dge and l adic versions of D Th eor eme
b One of the main results of this work will b e Theorem that the elements in
and for xed b and d are the resp ective regulators of one and the same element
in motivic cohomology This implies that Soul e s construction of cyclotomic elements
in the K theory with Z co e cients of an ab elian number eld Sou Lemma
l
Sou actually factors over the image of K theory prop er Corollary As shown
in BlK x Theorem also implies that the Tamagawa number conjecture mo dulo
p owers of is also true for o dd Tate twists see our Corollary Finally is used
in KNF Theorem to prove the mo di ed version of the Lichtenbaum conjecture
for ab elian number elds
c There are relative versions of and for schemes over a base scheme S smo oth
over B They allow to directly de ne the small p olylogarithmic extension pol on
S
U S which however turns out to b e the base change to S of pol
B
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Remark In our de nition of pol we chose not to follow BD The approach via
the universal prop erty of Log and the computation of its cohomology rather imitates
that of Beilinson and Levin in the elliptic case BL In fact one of the
predecessors of lo c cit contains a uni ed de nition of Log and pol for relative curves
of arbitrary genus BLp
The Geometric Set Up
For easier reference we assemble the notation used in the next sections
As b efore we let
l a xed prime number
A R
A Z
l
B Sp ec A
G G U P n f g
m m B B
B
denote a smo oth separated scheme over B of pure relative Furthermore we let S
dimension d S
G S
m
the op en subscheme of S where and are disjoint We assume S to b e dense S S
in S
j S S
n S S i S
n S is equipp ed with the reduced scheme structure where S
Z S S
with the reduced scheme structure
G n Z V
m S
For n de ne
n n
S G p
m S
n
n n
V v G
m S
n n
n n n
G G Z n V z
m S m S
n
v and Z where Z carries the reduced scheme structure So p id
S
Documenta Mathematica
Classical Motivic Polylogarithm
The base change of the ab ove ob jects and morphisms to S is denoted by the same
letters not underlined
S G
m S
Z S q S
V G n Z
m S
n n
p G S
m S
n n n
v V G
m S
n n n
z Z G
m S
n
Also we de ne partial compacti cations of p
n n n
g G A
m S S
n n n n n
h H A n G A
S m S S
n
where again H has the reduced structure
n n
p A S
S
V A n Z
S
n
n n
v V A
S
n n
n n n
z Z A n V A
S S
n
Z is equipp ed with the reduced structure So Z Z Z where
Remarks a The under lined ob jects should remind the reader that the partial com
overlined ob jects pacti cation comes from the compacti cation j of the base S The
n
refer to compacti cation upstairs induced from g
b For xed n we have a natural action of the symmetric group S on our geometric
n
situation
For the purp oses of K theory in section we will have to replace the singular
n
scheme Z by some smo oth simplicial scheme Put
n
n n n
Z Z G q G Z G q q G Z
S m S S S
m S m S m S
n
n
is a prop er covering of Z This is the easiest case of a morphism Note that Z
of schemes with cohomological descent meaning that for any reasonable cohomology
n
theory the cohomology of Z will agree with the cohomology of the smo oth simplicial
scheme
n
n n
Z cosk Z G
m S
i e
n n n
n n
k fold pro duct Z Z Z
G G
k
m S m S
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Put Z corresp onding to the empty scheme We will also use the simplicial
n n
n
Z Z sitting in A in the same way Finally let which is attached to scheme
S
n n n
G Cone Z G
m S m S
n
n n
A Cone Z A
S S
where the cone is taken in the category of p ointed simplicial sheaves on the big Zariski
site cf the discussion in app endix B
Geometric Origin of the Logarithmic Sheaf
In section we de ned a pro sheaf
s
Log pro U Sh G
m
and an element
F Log j pol Ext
U
Sh U
F Log W Log j lim Ext
n U
Sh U
n
The aim of this section is to identify Log j or rather its No etherian quotients
U
as relative cohomology ob jects with co e cients in Tate twists of certain schemes over
U Theorem
Recall that according to our conventions we have
F F
U
and hence we may view pol as an element of
Hom b F Log j H U Log j
U U U
D Sh U
abs
where we have used the notation introduced in De nition
For the schemes of section we have the following
Definition For n
sgn sgn
n d S
n n n n n
n
G v F n H G v F n G H
V
m S m S S
S
where the sup erscript sgn refers to the sign eigenspace under the natural action of
n n
the symmetric group S on G and V
n
m S
Observe in particular that G F
The following is an immediate consequence of the K unneth formula
Lemma There is a canonical isomorphism
n
n
Sym G G
Documenta Mathematica
Classical Motivic Polylogarithm
n
We want to compute G and simultaneously construct for each n a
pro jection
n n
G G
via the residue at whose pro jective limit over n we shall then identify for sp ecial
and and S U with the restriction Log j of the logarithmic pro sheaf to U
U
n n n
n n
the smo oth part b e the singular part of H and H H n H Let H
reg
sing sing
n
n
We the subscript reg will mean the complement of H For any subscheme of A
S
sing
work with the following geometric arrangement
n n
n
n
V H V V
reg
reg reg
n n
n
v v
v
y y y
H reg reg
n n
h g
reg reg
n
n n
A H G
reg
S reg m S
Both squares are cartesian All maps are either op en or closed immersions and each
line gives in fact a smo oth pair of S schemes
n
b n
v M is a shift of a Lemma For any complex M D Sh A such that
reg
S reg
n
V smo oth sheaf on there is an exact triangle
reg
n n
n n n n
h v v v v M h M
reg reg
reg reg
H reg H reg
n n n n
M v g v g
reg reg
n
v applied to the exact triangle obtained from purity for the closed Pro of This is
reg
immersion
n n
n
V H V
reg reg reg
of smo oth schemes
n
We apply this lemma to M F n and evaluate the cohomological functors
A
S r eg
n sgn i
A on the triangle Following c we write everything as relative H
S reg
abs
cohomology with Tate co e cients
n
n i n sgn n sgn i
rel A H rel Z n G Z n H
S reg abs m S abs reg
n
i
n n sgn
H H rel Z H n
reg reg
abs
n
i
sgn n
Z n H A rel
reg S reg
abs
We refer to this as the absolute residue sequence
i n sgn
Application of the cohomological functors H A to the same exact
S S reg
triangle yields a long exact sequence of sheaves on S that we call the relative residue
Documenta Mathematica
Annette Huber Jorg Wildeshaus
sequence
sgn
sgn
n n i n i n
n
n
G v F n H v F n H A
V
m S S reg S S reg
V
reg
sgn
i
n n
H H n v F n
n
reg H reg
S
V H
reg
reg
sgn
i
n n
n
A H v F n
S reg reg
S
V
reg
sgn
n d S
n n n
n
o ccurs in this sequence F n v G Note that G H
V
m S
S
We are now going to further analyse and reshap e these sequences The nal
form will b e achieved in Prop osition and Theorem
First we need to identify the terms
n
i
n sgn n
Z H n n H H rel
reg reg
abs
sgn
i
n n
H H v F n n n
n
reg H reg
S
H V
reg
reg
n
n n
Z H in H is given by The complement of
reg reg
n
n n n
v V H H
H reg reg reg reg
n
n
n
n
V H V under the identi cation Since
reg
reg
k
n
a
n
n
H G
reg
m S
k
and these comp onents are p ermuted transitively by S we conclude
n
Lemma
n
a
n n
`
n
n
v F n a F n n v
n
V
H reg
V H
k
reg
reg
k
n
M
n
i n i
n n n
b rel Z n Z H n H G H H rel
reg reg
m S
abs abs
k
n
n n
i
sgn
and hence the sign eigenspace H H rel Z H n is isomorphic to
reg reg
abs
i n
n sgn
H G rel Z n
m S
abs
where the last sgn refers to the action of S The isomorphism is given by pro jection
n
onto the comp onents unequal to k for some choice k f ng It is indep endent
of the choice of k
n
M
n n
n n
n
F n v c G v F n H R R n
n
S S
V
H reg reg
m S
V H
reg
reg
k
Documenta Mathematica
Classical Motivic Polylogarithm
sgn
n
i
n
v F n As in b the sign eigenspace H H is canoni n
n
reg
H reg
S
V H
reg
reg
cally isomorphic to
sgn
i n n
n
H G v F n
V
S m S
n
the latter equals G For i n d S
Pro of The only p oint that remains to b e shown is the indep endence of the isomor
phisms in b and c of the choice of k Recall the identity
sgn
sgn
n n n n
n
G rel Z n G v F n S R R R
V S abs abs
m S m S
q
sgn n n
n
for q n d S We are going to prove in d that H G v F n
V
m S
S
So the asso ciated sp ectral sequence degenerates and shows that the indep endence of
the map in b follows from that of the map in c
n d S
n sgn n n
n
For c we only need to consider G H There our F n G v
V
m S
S
claim follows from Lemma and the graded compatibility of the cup pro duct with
b oundary morphisms GH Prop osition and Corollary
Remark The arguments of this section would b ecome simpler if we could use an
sgn
in c However we do not know whether it is p ossible to make a decom ob ject R
S
p osition into eigenspaces in our triangulated categories
By the identi cation of the lemma the residue sequences de ne canonical residue
maps
i n
i n n sgn n sgn
res H G rel Z n rel Z n H G
abs m S
m S
abs
sgn
sgn
i n n
n n i
n
n
G v H F n res H v F n G
V
V
m S S
S m S
tting into the relative and absolute residue sequences In particular observe that
we have a residue map
n n
res G G
Now we concern ourselves with the identi cation of the remaining terms
n
sgn i n
Z n n H A rel
reg abs S reg
sgn
i n n
n
n H v A F n
S reg S reg
V
reg
of the residue sequences
n
We use the following ltration of A by op en subschemes
S
n n
j at most k co ordinates vanishg f x x A F A
n k
S S
n n n n
So we have F A A and F A G
n
S S S m S
The graded pieces of this ltration are
n n n
n F A F A G A
k k k
S S S
n
f x x A j precisely k co ordinates vanishg
n
S
Documenta Mathematica
Annette Huber Jorg Wildeshaus
n
is equipp ed with the reduced scheme structure Note that it splits into several G A
k
S
disjoint pieces For k and any such piece there is a transp osition of S acting
n
trivially By using triangles similar to for the inclusions
n n n
G A F A F A
k k k
S S S
n
is trivial we conclude inductively that the sign eigenpart of the cohomology of H
sing
Lemma The adjunction morphism induces isomorphisms
n n
i n sgn i n sgn
H A rel Z n H A rel Z n
abs S abs S reg reg
sgn
sgn
i n i n n n
n n
H A H A v F n v F n
S S reg S S reg
V V
reg
By b and the absolute residue sequence takes the form
n
sgn i n n sgn i n
Z n H G rel Z n H A rel
abs m S abs S
res
i n
n sgn
H G rel Z n
m S
abs
n
i
n sgn
H A rel Z n
S
abs
Similarly the relative residue sequence lo oks as follows
sgn
sgn
i n n n i n
n
n
H v F n G v F n H A
V
S m S S S
V
sgn
res
n n i
n
G v F n H
V
m S S
sgn
i
n n
n
H A v F n
S
S V
For the computation of the term
sgn
n i n
n
v F n H A
S S
V
we use the K unneth formula
n n
n n
Lemma a R A v v F n H A F n and the K unneth formula
S
S S S
gives an isomorphism
sgn
n
n n n n
A v v Sym H v F n H F n F A H A
S S S S S S
b The choice of an ordering of the sections and gives an isomorphism
A v v F H F F R A
S
S S S
Up to sign it is canonical
c The isomorphisms of a and b induce an isomorphism
n d S
n n n n
n
A H v F n v F n F n H A
S S S
S
V
n
It dep ends on the choice made in b only up to the sign The group S acts
n
on these ob jects via the sign character
d For i we have
sgn sgn
i n d S
i n n n n
n
H G v F n G v F n H
V
S m S m S
S
Documenta Mathematica
Classical Motivic Polylogarithm
Pro of For b consider the long exact cohomology sequence asso ciated to the triangle
v F F
z F
We have
F i
i
H A F
S S
i
and
M
F i
i
A H z F
S S
i
The long exact cohomology sequence thus reads
M
H A A v v F F F F H
S S S
S
If we let f g fs s g then we identify the cokernel of
M M
F F F
i
L
f F to f f with F by mapping f
s s s s
i
N
n
n
a follows from b since F Sym F
c is a consequence of a and b
d follows from a and the relative residue sequence by induction on n
On the level of absolute cohomology the isomorphism of c induces an iso
morphism
sgn
n n
i n i n
n n i
H A rel Z n H A Z n rel H S n
S S abs
abs abs
This gives the nal shap e of the absolute residue sequence
i n
i n n sgn
H S n H G rel Z n
abs m S
abs
res
i n n
n sgn
n G rel Z H
m S
abs
i
H S n
abs
By d the relative residue sequence collapses into the short exact sequence
of sheaves on S
res
n n
F n G G
In order to identify the long exact absolute cohomology sequence asso ciated to
this sequence with the absolute residue sequence we need the following
Documenta Mathematica
Annette Huber Jorg Wildeshaus
b
Lemma Let K D Sh X b e a complex of sheaves on a separated reduced and
at B scheme X Supp ose there is an action of a nite group G on K Let b e
the character of an absolutely irreducible representation of G over F For any ob ject
V with a G action of an F linear ab elian category denote by V the isotypical
comp onent of V i e the image under the pro jector
X
g g e
G
g G
i
Supp ose that H K vanishes for all i Then
Hom b F K i Hom b F H K i
D D
Pro of By applying e and e one checks the statement for a complex of the
H K For the general case consider the sp ectral sequence for sp ecial form K
Hom b F i induced by the truncation functors It degenerates after applying
n
D
e
Now that we know that formation of absolute cohomology commutes with for
mation of sign eigenspaces we have
Proposition The absolute residue sequence is the long exact sequence in abso
lute cohomology attached to the short exact sequence
res
n n
G F n G
We conclude the computational part of this section by collecting our results
Theorem a For n we have
sgn
n n n
G v F n G H
m S S
sgn
i n n
and H G v F n for i
S m S
b The residue at i e the b oundary map of gives an epimorphism
n n
res G G
for n
c The K unneth formula gives an isomorphism
sgn
n n n n
H A v v ker res F n H F n A
S S S S
for n A choice of an ordering of the sections and induces an isomorphism
F n ker res
n
which dep ends on this choice only up to the sign
n
n
d Let G Sym G b e the canonical isomorphism of and
n
Sym F F
n
Sym F F n
Documenta Mathematica
Classical Motivic Polylogarithm
the isomorphisms given by multiplication Then the diagrams
n
G F
o o
n n
Sym G Sym F
and
n
F n G
o o
n n
Sym F Sym G
commute Here the horizontal maps are given by the successive residue maps and
by c resp ectively
n
e Let W G
n
n n n k
W G W G ker G G f or k n
k k
n n
and W G G The choice in c induces isomorphisms
n
M
W
n
F i Gr G
i
which by their construction t into commutative diagrams
L
n
W
n
F i Gr G
i
W
Gr res can
L
n
W
n
Gr F i G
i
n
The ltration W is therefore the weight ltration of G
Pro of a b and c follow from the previous results The commutativity of the rst
diagram in d follows from the de nition of the residue map For the second diagram
we use the fact that the K unneth formula of is compatible with the K unneth
formula of the pro of of a For e apply induction on n
Recall that S is the op en subscheme of S where the sections and of G
m S
are disjoint For sp ecial S and the following is the main step towards the
n
identi cation of the pro jective limit of the G with the restriction Log j of the
U
logarithmic sheaf
n
n
Lemma a There is a unique smo oth sheaf G on S It has a extending G
weight ltration
b There is a canonical isomorphism
n n
Sym G G
and a unique isomorphism
n
M
W n
n
Gr G F i
i
Documenta Mathematica
Annette Huber Jorg Wildeshaus
which is compatible with the isomorphism of e
n
is split there is a canonical isomorphism c The weight ltration of i G
n
M
W n n
i G F i Gr i G
b
i
n S into S Here i denotes the inclusion of S
d There is an exact sequence
v F G G i F H
m S
S
of sheaves on S
Pro of If there is any smo oth sheaf as in a then it will automatically b e unique and
hence b follows from a and d e Also it will su ce b ecause of d to show
the lemma for the case n
There we have the following diagram
y
i F
y
H F K H F
y
L
F
y
where
M
K H Cone d S i F d S F
S
S nS
with v v v v in terms of constructible sheaves this is just Ker shifted in
the appropriate degree to de ne a p erverse sheaf The horizontal sequence is as in
the pro of of b the long exact cohomology sequence on S asso ciated to the short
exact sequence on G
m S
z F v F F
where we have set
i i
F G H H v
m S
S
We thus get the equality
v F H v F R G G
S m S m S
S
Documenta Mathematica
Classical Motivic Polylogarithm
and an exact sequence of sheaves on S
K F H v F F G
m S
S
whose restriction to S is isomorphic via the choice of an ordering of and to
F G F
Push out of the ab ove via the morphism
M
A
K F F F
whose kernel is i F recall again that we use p erverse indices gives the desired
extension G By construction b and d hold Applying i to the pushout diagram
is the pushout of F via and taking cohomology we see that the sheaf i G
F
and we get c
We now sp ecialize our geometric situation we let
S G
m B
G B G
m B m B
id G G
m B m B
n S the closed subscheme of G given by the So we have S U and S
B m B
immersion of B into G
m B
After having made precise which choice of normalization we have and in how far
it a ects our identi cations we now x it we let
s and s id in c
n
We thus get a pro jective system G of smo oth Tate sheaves on G with
n m B
n
M
n
G F i j
B
i
By the universal prop erty of Log Theorem d there is a unique morphism
n
Log G G lim
n
to such that j sends B Log j
B
B
Y
F i G F j
B
i
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Theorem is an isomorphism
Pro of The claim can b e shown on the level of the underlying top ological sheaves The
l adic statement follows from the statement for the top ological spaces of C valued
p oints by comparison recall that we are dealing with lo cally constant sheaves
Over C the bre at of the pro lo cal system Log equals the completion of the
top
Z with resp ect to the augmentation ideal a group ring Q of G C
m
The representation of is given by multiplication compare the general construction
in WiI In particular we have
n
Log lim Sym Log
top top
n
where Log Log a is of dimension two Now in the category of unip otent
top top
lo cal systems on G C the pro sheaf Log has the universal prop erty of Theorem
m top
d
The resulting map factors over G We apply this universal prop erty to G
top top
Since G is two dimensional the representation of Q is necessarily
top
top
trivial on a and we get a morphism of lo cal systems
Log G
top top
top
giving rise to a morphism
n
Sym Log G lim
top top
top
n
Again b ecause of the universal prop erty of Log this morphism is identical to
top top
It therefore su ces to show that is bijective which amounts to saying that
top
the coinvariants of G under the action of are one dimensional But taking
top
coinvariants under of a unip otent variation V amounts to computing singular
cohomology
G V H G C V H
m R m
Sp ec R
Firstly we claim that
F i
i
G G v H F
m R m R
Sp ec R
i
e g identify the left hand side with
i
H G C G C G C f g G C F
m m m m
i
H G C G C G C f g f g G C F
m m m m
and apply the K unneth formula From the pro of of we recall remember that
G we have S
m
G G v F H F G G v R
m R m R m R m R G
G
m R
m R
from which we conclude
F i
i
G H G G v H F
m R m R m R
G
Sp ec R
m R
i
Documenta Mathematica
Classical Motivic Polylogarithm
The long exact sequence obtained by applying R G to the exact se
m R
Sp ec R
quence of d
F G F H G G v
m R m R
G
m R
then shows that
F G G H
m R
Sp ec R
Remark The geometric situation used in this section is identical to the one of
BD see in particular lo c cit The comparison statement of our
Prop osition is implicit in lo c cit We mention that basically the same
geometric arrangement was used in Jeu More precisely writing down the iterated
cone construction of lo c cit one arrives at a simplicial ob ject which is homotopy
equivalent to Beilinson s and Deligne s construction used here
The Splitting Principle Revisited
In order to b e able to translate easily to the motivic context we recall Beilinson s
original pro of B of the splitting of the logarithmic pro sheaf over sp ectra of
cyclotomic elds Theorem
First we return to the general situation considered at the b eginning of section
For N we have the morphism of S schemes
G G
m S m S
N
x x
and for each n the induced morphism
n n n
G G
m S m S
We work under the additional assumption
A
n n n
If this is the case we have V V and hence get a morphism
n n n
v F n v F n
and hence a morphism
n n n
v F n F n v
n
which after application of p and pro jection onto the sign eigenpart induces
n n n
G G
n n
We need to understand the action of on G and on absolute cohomology
n
First we establish in how far is compatible with the residue at
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Lemma a Under any isomorphism
n
M
W
n
F i Gr G
i
W
n n i
the map Gr is multiplication by N on F i
b For any n the diagram
n
n n
G G
res res
n n
y y
n
N
n n
G G
commutes
Pro of Since the morphisms in b are strict with resp ect to the weight ltration it
su ces to check that
W W W W
n n
Gr res Gr N Gr Gr res
n n
W
But if we choose the isomorphism of e then Gr res is simply the canonical
n
pro jection
n n
M M
F i F i
i i
and therefore b follows from a For a we note rst that it su ces to show the
statement for one choice of isomorphism
n
M
W
n
F i G Gr
i
This time we use the isomorphism on graded ob jects induced by thereby reducing
ourselves to the case n There we consider the long exact cohomology sequence
asso ciated to the exact sequence
z F v F F
i
and the cohomological functors H G We know the cohomology of G
m S m
S
F i
i
F i
H G F
m S
S
i f g
Of course we know the cohomology of two p oints
M
F i
i
H F G z
m S
S
i
Documenta Mathematica
Classical Motivic Polylogarithm
We get an exact sequence
M
G v F F F F G H
m S
S
n
and b ecause of assumption A it carries an action of But this action can b e
i i
G z F it is trivial on the F and G F and H identi ed on H
m S m S
S S
multiplication by N on F
Certainly A is only satis ed in very sp ecial situations namely if and are
supp orted in the schemes of N torsion of G
m S
Let again d C Sp ec R where R A T T as in section For
d
d
b prime to d consider
i C C G
b m
b
The pullback Log of the pro sheaf Log j on U via i is identical to the pro jective
b U b
n
limit of the sheaves G obtained by setting
b
C S
C B G
m
i
b
Since A is satis ed with N d we may apply and conclude
n
Corollary G splits into a direct sum
b
n
M
n n
W
Gr G G
i
b b
i
Therefore there is a unique isomorphism
n
M
n n
F i G
b b
i
n
which is compatible with the isomorphism of b
n
n i n
Pro of F i G is the eigenspace of d under the morphism
b
We conclude with the implications of and for absolute cohomology with
co e cients For this recall the absolute residue sequence for n
res
n n n
sgn sgn n
H G n n C n H G H
m C abs
m C
abs abs
introduced after where we have set
n n
n sgn sgn n n
rel Z n n H G H G
m C m C
abs abs
thus saving enough space to get the ab ove sequence into a single line
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Corollary a For n the absolute residue sequence splits into short exact
sequences
n n n
n sgn sgn
H C n H G n n H G
abs m C
m C
abs abs
n
b For N d the map acts on the short exact sequences of a there is a
commutative diagram
n n n
sgn sgn n
n H G n C n H G H
m C abs
m C
abs abs
n n
id d
y y y
n n n
n sgn sgn
H C n H G n H G n
abs m C
m C
abs abs
Pro of By the absolute residue sequence is the absolute cohomology sequence for
the exact sequence of sheaves on C
res
n n
b
G F n G
b b
Therefore a follows from while b follows from b and the fact that under
the identi cation of a
n
n n sgn
H C G H G n
abs m C
abs
the map induced by
n n
n
G G
b b
n
is the map of the absolute cohomology groups
n
n n sgn
It follows that the eigenvalues of on H G n are
m C
abs
n
d d The eigenspace decomp osition yields
n
M
n
n n
n sgn n n sgn
H C i H G n H G rel Z n
abs m C m C
b abs abs
i
which in sheaf theoretic terms corresp onds to the decomp osition
n
M
n n
Ext F F i Ext F G
Sh C Sh C
b b
i
given by Corollary
The pullback pol of the small p olylogarithmic extension pol on U is an element
b
of
n
n
n sgn n
rel Z n H G Ext F G lim lim
m C Sh C
abs b
n n
n
sgn n
n H G lim
m C
abs
n
Documenta Mathematica
Classical Motivic Polylogarithm
n
We have shown that using the eigenspace decomp osition for the action of the
these groups are isomorphic to
Y Y
Ext F F k H C k
Sh C abs
k k
and describ e pol as an element in this group
b
Actually in order to relate the ab ove decomp osition to the one used for and
we shall need to compare the isomorphism
Y
W
n
F k Gr G lim
k
n
of b to the isomorphism
Y
W W
Gr Log F k Gr G
k
of section
A priori we know that the isomorphisms
W
Gr G F k
k k
k
satisfy an identity of the type
q
k k k
for a constant q F
k
We remark that in order to prove the main results announced in the introduction
all one needs to know is that q is a rational number which is indep endent of
k
whether we work in the Ho dge or the l adic setting
In order to exhibit the precise relation of the motivic analogue of pol see section
to the cyclotomic elements in K theory see Corollary b we need to identify
q
k
Proposition We have the equality
k
k k
Pro of Because of the compatibility of with the canonical pro jection
G F
we have In order to show we compare the classes of G in
Ext F F
Sh G
m
induced by and resp ectively Let
K C K Q
Documenta Mathematica
Annette Huber Jorg Wildeshaus
and choose any K valued p oint t of U Of course the value of q can still b e detected
from the extensions of
mixed Q Ho dge structures
Galois mo dules
of G via t In b oth settings there is a natural morphism given by the pullback t G
of K F into the resp ective Ext F F see e g
Z
WiIV Theorem WiIV Theorem
By
WiIV Prop osition a WiIV Prop osition a
the class of t G calculated in the framing given by equals the image of t K
under this morphism By Sch the same holds for the framing given by
note that here it is vital to choose the ordering of the sections and in the way we
did b efore For k let
k
k k
G Sym G
b e the isomorphism of b By d the diagram
k
F G
k
o o
k k
Sym F Sym G
commutes By WiIV Theorem a the commutativity of this diagram char
k
acterizes uniquely From lo c cit Theorem b and c we know that the
diagram
k
k
F k G
k
o o
k
Sym
k k
Sym F Sym G
commutes So our identity
k
k k
follows from d
Polylogs in Absolute Cohomology Theories
In section we showed that the logarithmic pro sheaf is the pro jective limit of
relative cohomology ob jects with co e cients in Tate twists of certain schemes over
U The Leray sp ectral sequence suggests that is should b e p ossible to recover pol as
a pro jective limit of elements in absolute cohomology with Tate co e cients of these
schemes and indeed this is what we do in Theorem That the co e cients are Tate
is of course the central p oint it allows us in section to imitate the construction
Documenta Mathematica
Classical Motivic Polylogarithm
of this section and thus to de ne a motivic version of pol This detour is necessary
b ecause we know up to date of no satisfactory formalism of mixed motivic sheaves
whose absolute cohomology with Tate co e cients would give back motivic cohomology
de ned via K theory
We return to the geometric situation set up b efore and start by computing
the higher direct images of the restriction of Log to U
and the pro jection Lemma a The inclusion F G
G F induce natural isomorphisms
H F G G
B m
B
H G F G G H
m m
B B
and the latter group is isomorphic to F via the map residue at
B
n n
b The inclusion F n G and the pro jection G F induce natural identi
cations
F n i
B
n
i
F i
H G G
B
m
B
i f g
Pro of The statements need only b e checked on the level of lo cal systems Part a
is shown in the pro of of From there we also recall that we have to compute
the invariants and coinvariants under the action of the group G C or
m
equivalently of a generator of Using b we may deduce b from a
Corollary
F n i
B
i n
H U G
B
i f g
n
L
n
F k and an ob ject For i the sheaf H U G is the direct sum of
B
B
k
which is an extension of F by itself
B
i n
Pro of By WiI Theorem there is a weight ltration on H U G Now use
B
the exact triangle
id
G
m B
j j
purity and c
Remark In the setting of Ho dge mo dules where a concept of p olarization is avail
able any extension of pure ob jects of the same weight is necessarily split
n
The map H U G F of the corollary yields in particular a map residue
B
at for n
B
n n
res H U G H B R U G H B
B
abs abs abs
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Definition Let n The map
n
n n n sgn
res H U G H G rel Z n H B
abs m U abs
abs
is called the total residue map
For later reference we note
Corollary H G rel Z
abs m U
Pro of We have
H G rel Z H U G
abs m U
abs
which b ecause of equals H B F
abs
Next we have
Lemma i The transition morphism
n n
res G G
satis es
U res F n F n H
B B
B
n n
H U res H U G H U G
B B B
is surjective with kernel F n
B
In particular the total residue for n factors over the total residue for n
there is a commutative diagram
res
n
H B H U G
abs abs
res res
n
U G H
abs
ii The Leray sp ectral sequences for n give exact sequences
res
n
H B n H U G H B
abs abs abs
The map
n
H B n H U G
abs abs
is the comp osition of H B n H U n H U F n and the map induced
abs abs abs
n
by the inclusion of F n into G in other words the same noted map of the residue
sequence
The pro jective limit of the ab ove sequences identi es
n
U G H U Log j lim H
U
abs abs
n
B and H
abs
Documenta Mathematica
Classical Motivic Polylogarithm
iii There are unique splittings
n
s H B H U G
n
abs abs
of the sequences in ii for any n such that for any n we have a commutative
diagram
s
n
n
H U G H B
abs abs
s res
n
n
H U G
abs
Pro of i The rst statement is clear For the second either go through the construc
tion or observe that the direct image of the morphism U B has cohomological
top top
U is right exact on smo oth sheaves dimension one hence H
B
ii We have the Leray sp ectral sequence
p q p q p q
n n
E H B H U G H U G
B
abs abs
whose low term sequence reads
n
d
n
H B n H U G H B H B n
abs abs abs abs
By i the Mittag Le er condition is satis ed for the pro jective system
H B n and therefore
n
abs
n
ker d H B H U Log j lim
U
abs abs
n
n
B n is ML zero since the pro jective system im d H
n n
abs
But then any of the
n
H U G H B
abs abs
must b e surjective as well
iii Apply ii
n
Denote by pol the image of the small p olylogarithmic extension pol under
n
H U Log j H U G
U
abs abs
Theorem a Under the isomorphism
B U Log j H H
U
abs abs
of ii the small p olylogarithmic extension pol is mapp ed to
b For each n the map
n
s H B H U G
n
abs abs
n
maps to pol
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Pro of This is the de nition of pol and the s
n
Recall a that we may identify
n n n sgn
H U G H G v F n
abs abs m U
n
sgn n n
n
H G v F n
V
m U
abs
n
n n sgn
H G rel Z n
m U
abs
In section we are going to prove a motivic analogue of ii and then de ne
pol as the element in
n
n n sgn
lim H G rel Z n
m U
M
n
B mapping to under the isomorphism to H
M
In order to prove a motivic version of ii we shall frequently use injectivity
of the Beilinson regulator on certain motivic cohomology groups and two technical
results on H that will o ccupy the rest of this section
abs
While this may app ear arti cial at rst sight we remind the reader that in the motivic
setting we cannot make use of any sheaf theoretic means like Leray sp ectral sequences
An imp ortant means will b e the lo calization sequence asso ciated to the geometric
situation
f g A U
B
B
It is the result of the degeneration of the Leray sp ectral sequence and reads
H A p H U p H f g p
B
abs B abs
abs
H A p
B
abs
Lemma a The structure morphism is an isomorphism
H B p H A p
abs abs B
b The b oundary map is trivial i e we have short exact sequences
M
H B p H B p H U p
abs abs
abs
i
F p A F p Pro of For a note that R
B B
A
B
B
b follows from the fact that there are B valued p oints of U
In particular for p we have the exact sequence
M
B H U B H H
abs abs abs
i
The last map equals the map of Ext groups
Ext F F Hom F H U F
Sh B
Sh U B
Documenta Mathematica
Classical Motivic Polylogarithm
obtained from the Leray sp ectral sequence observe that the residues at and
B B
provide an isomorphism
M
F H U F
B
i
We have a natural map
O U H U
abs
Its comp osition with
M
H B H U
abs abs
i
asso ciates to a function on U its orders at and resp ectively
We need to understand the comp osition
res H U Ext F F
abs Sh U
U G Hom F H
Sh B
B
Observe that due to the last group is equal to H B Furthermore we recall
abs
from the pro of of and the de nition of res that the comp osition
M
res
F H U F H U G F
B B
i
L
F We have thus proved is given by pro jection onto the comp onent of
i
Lemma Consider the non vanishing functions t and t on U We have
res t res t
In particular the map
G rel Z H U H U G H
m U abs abs abs
do es not map t O U to zero
Pro of Observe that res factorizes through
Remark The main technical result of this section ii corresp onds to BD
ii Observe that pol and the p olylogarithmic class of lo c cit do not quite
agree in our notation
H U Log j
U
abs
while pol H U Log j The connection is as follows there is a canonical
U
abs
monomorphism
Log Log
identifying Log with W Log and pol is the push out of via The present
de nition of the p olylog seems more natural since it is an element of an H B
abs
mo dule of rank one which is canonically trivialized By contrast H U Log j
U
abs
is of rank two
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Calculations in K theory
The next step is to do the constructions of section with K groups or more precisely
with relative K cohomology as introduced in app endix B For technical reasons we
will have to use simplicial schemes to replace the singular schemes that app eared b e
fore All constructions will b e compatible with the regulator maps to absolute Ho dge
cohomology app endix A and B and to continuous etale cohomology app endix
B
A priori these regulators have values in absolute cohomology groups for the same
simplicial ob ject cf B and B Using B and B these absolute cohomol
ogy groups are then identi ed with relative cohomology of singular schemes This
identi cation is made tacitly
Let B Sp ec Z and S a smo oth a ne B scheme We will work in the category
of smo oth S schemes K cohomology is taken on the Zariski site over B
Before returning to the geometric situation introduced in section we have to
check a technical lemma Let us consider the following general construction Let X
b e a smo oth quasi pro jective S scheme and Y a closed subscheme of X which is itself
also smo oth over S Put
n
n n n
Y Y X q X Y X q q X Y
S S S
n
Note that Y is a prop er covering of the singular scheme
n n n
Y X r X r Y
This is the easiest case of a morphism of schemes with cohomological descent meaning
n
that for any reasonable cohomology theory the cohomology of Y will agree with
the cohomology of the smo oth simplicial scheme
n
n n
X Y cosk Y
i e
n n n
n n
k fold pro duct Y Y Y
X X
k
For etale cohomology and absolute Ho dge cohomology the corresp onding results are
B and B resp ectively
We will work in the setting of spaces i e p ointed simplicial sheaves of sets
on the Zariski site of smo oth B schemes We refer to app endix B for details and
terminology We use the notation
n n n
X Cone Y X
for the space that computes relative cohomology for the closed embedding cf B
n
The space Y do es not b ecome degenerate ab ove any simplicial degree How
ever we have
n
Lemma a Y is isomorphic in Ho sT to a simplicial scheme which is degen
erate ab ove degree n
Documenta Mathematica
Classical Motivic Polylogarithm
n n
b In particular Y and X are K coherent
n n
c X is a space constructed from schemes in a nite diagram over X in the sense
of B
d If T is another closed subscheme of X which is smo oth over S and disjoint of Y
then the inclusions
i n i n
T X X
S
are tor indep endent of all morphisms in the diagram in c
Pro of By de nition
n
Y q q Y Y
n
n
where Y is the reduced closed subscheme of X of those p oints whose i th co ordinate
i
n
lies in Y This induces a decomp osition of Y into disjoint subschemes of the form
k
n n
Y Actually this subscheme is canonically isomorphic to Y
i X X i
k
n
Y for j k g Y f x x X j x Y
i n i i
j
k
We get the following more familiar form of the simplicial scheme
a
n
Y Y
i
k
k
i I
I f ng
Let n b e the simplicial set with
n f i i j i i ng
k k k
n
We de ne the simplicial scheme Y by
a
n
Y Y
i
k
i I
I n
k
It is degenerate ab ove the simplicial degree n and from our previous considerations
n
We consider these simplicial schemes as we see that it is a natural subspace of Y
spaces in the sense of app endix B by adding a disjoint base p oint
For a scheme U in the big Zariski site over B we consider the morphism of simplicial
sets
n n
U U Y Y
By the combinatorial Lemma B it induces an isomorphism of homotopy sets
Hence the inclusion is a weak homotopy equivalence of spaces
b is an immediate consequence of a and B b Recall that Y and X were assumed
n
smo oth over B We already have seen that all comp onents of X are disjoint unions
n
Y and a disjoint base p oint All morphisms of X schemes of the form Y
i i
k
b etween the scheme comp onents are given by the natural closed immersions b etween
Documenta Mathematica
Annette Huber Jorg Wildeshaus
them The condition on the tor dimension required in B follows b ecause are
schemes are regular T Y and X are all at over S hence the maps in the diagram
X Y
S
y
T X X X
S S
are easily seen to b e tor indep endent The inclusions of T and Y into X are trivially
tor indep endent b ecause this is a lo cal condition
Basically this lemma tells us that all conditions hold that are needed to apply
the machinery of app endix B We have a well behaved relative motivic cohomology
theory cf B
Now we return to the geometric situation set up in section We consider
n n
Z G
m S
y y
n
n
Z A
S
where Z Z S q S with disjoint S rational p oints and of G There
m S
n
is a simplicial op eration of S on the situation which induces an op eration on relative
K cohomology and on motivic cohomology
Proposition There is a natural residue map
res
n
i n
n sgn i n n sgn
H G rel Z j H G rel Z j
M m S
M m S
where sgn means the sign eigen space under the op eration of the resp ective symmetric
group
Moreover there is a long exact sequence
n
n i
sgn n sgn i n
j Z rel Z j H A rel G H
M S
m S M
i n n sgn
H G rel Z j
M m S
n i
n sgn
rel Z G j H
m S M
Under the regulators the long exact sequences are compatible with the ones in abso
lute cohomology after
k k
Remark Recall that Z and hence H G rel Z j H S j by
M m S M
de nition
n n
Pro of We lter A by the op en subschemes F A de ned just b efore Lemma
k
S S
n n n n n
In particular F A G Again G A F A r F A We use the notation
k k k
S m S S S S
n n n
F A and G A for the induced op en resp ectively lo cally closed subspaces of A
k k
Note that the situation is still symmetric under p ermutation of co ordinates Hence
there is a compatible op eration of the symmetric group on the space constructed from
n
schemes F A
k
Documenta Mathematica
Classical Motivic Polylogarithm
n n
The closed immersion G A F A satis es the rst condition in TC in B
k k
The maps we have to consider for the rest of TC are lo cally of the form considered
in d Hence B applies i e we can use the lo calization sequences for motivic
n n n
cohomology induced by the triples F A F A G A We get
k k k
i i
i n n n
H G A j H F A j H F A j
k k k
M
M M
i
n
H G A j
k
M
The sequence remains exact when we take sign eigenspaces Now let us compute one
of the groups involved
M
i n i n
n
G a a j H A H G A j
m S k A k
M M
f a a a ng
k
where
G a a f x x j x if i a for some j x else g
m S k n i j i
n
The decomp osition corresp onds to the decomp osition of G A into its connected
k
n
n
G a a means the op en subspace lying comp onents The notation A
m S k A
over the lo cally closed scheme Now consider the op eration of the symmetric group
If k then there is for each comp onent some transp osition which acts trivially
namely one that interchanges two vanishing co ordinates Hence the sign eigenspace
vanishes altogether For k the decomp osition has the form
M
a n a
i n i n
n
H G A j G f g G j H A
A
M M
m S m S
a n
The op eration of the symmetric group p ermutes the factors transitively The stabilizer
n
of one summand is the symmetric group S We get
n
sgn i i n sgn
j H G H G A j
M M
m S
where the sign eigenspace on the right hand side is taken with resp ect to the smaller
n
symmetric group S We have a choice of isomorphism here and use the one that
n n
identi es G with G f g Putting these results in the long exact sequences
m S m S
we get iteratively
n
sgn i n sgn i n sgn i n
Z j H F A j H F A j H A rel
n
M M M
So the ab ove sequence for k gives the desired residue sequence We can do the
same construction for absolute cohomology Ho dge or l adic considered as general
ized cohomology theories By B B and B the long exact sequences for
motivic cohomology will b e compatible via the regulator with the ones in generalized
cohomology The next step is to pass from generalized cohomology to cohomology of
ab elian sheaves By B and B this can b e done In fact we get precisely the
residue sequence for absolute cohomology constructed in section
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Remark a By B we have the same maps and long exact sequences for the
K cohomology groups themselves However note that there is a Riemann Ro ch hid
den in the compatibility of the lo calization sequence in K cohomology and absolute
cohomology
b We shall show injectivity of the Beilinson regulator on
n
n n sgn
H G rel Z n
m S
M
in Prop osition Together with Lemma b it shows that the residue map on
i n n sgn
H G rel Z j
M m S
n
in do es not dep end on the choice of embedding of G
m S
a n a
G f g G
m S m S
a n
of the ab ove pro of if i j n n Since we are only interested in these sp ecial
indices we chose to exclude from the statement of the dep endence of res in the
n
general case from the ab ove choice
Lemma Let j k Then
n
k n
k n
S j H Z j H A rel
M S
M
where the isomorphism is induced by a choice of ordering of the sections and It
is compatible with the identi cation in under the regulator map S op erates by
n
sign on the left hand side
i
Remark Here and in the sequel we put H S j if j i This makes sense
M
as S is regular and the corresp onding K group vanishes see B
Pro of Fix j We consider the skeletal sp ectral sequence B We have
p q q
n
A E H j
p
S
M
We will show that the only non trivial E terms are concentrated in one vertical line
n q q
S j E H
M
This means that the sp ectral sequence converges in the strongest p ossible way This
yields isomorphisms as stated Before we can check this we need some preparation
If X is a space constructed from schemes we denote by C p X the simplicial set of
n n
has the same singular cohomology as C p Z Z its connected comp onents C p
cf pro of of which is the simplicial set attached to a CW complex dual to the
Z has two disjoint comp onents b oundary of the n dimensional hypercub e note that
n
This means that C p Z has a vertex for every n cell of the cub e etc In
n
particular we see that it has the homotopy type of an n sphere C p A is of
S
n
course contractible It follows that C p A has singular cohomology concentrated
in degree n where it is one dimensional
Documenta Mathematica
Classical Motivic Polylogarithm
Let us make this more explicit
In order to compute the cohomology of a cosimplicial group it su ces to consider
n
Z is completely the sub complex corresp onding to nondegenerate simplices C p
degenerate from cosimplicial degree n on In degree n there is one nondegenerate
n
simplex for each vertex of the hypercub e They are indexed by f g Hence any
n
n n n
element of H C p A H C p Z is represented by an element of
M
n
Q K
n
f g
n
Z do es not b ecome degenerate Let g b e a generator of the cohomology group C p
n
The nondegenerate part in degree n is given by one copy of f g for each p ossible
sgn
p ermutation of the numbers n It is easy to see that g is in
n
Z the kernel of the di erential It represents the generator of cohomology of C p
We see that S op erates by the sign of the p ermutation
n
We choose the generator g of cohomology given by the tuple
s i s i
n
Q
i i
n
where i f g and s s This choice of generator amounts to
k
picking the ordering and extending it by the K unneth formula Now let us
analyze our E term For xed q we have the complex attached to the cosimplicial
q
n n
are isomorphic All connected comp onents of A ab elian group H A j
p p N
S
M
to a copy of some p ower of A By the homotopy prop erty of K theory we have
S
q q
n n
S j C H j A H
Q p N S
p
M M
n
where C is the cosimplicial vector space computing singular cohomology of
n
C p A By the previous considerations we already know its cohomology It also
follows that the op eration of S on our motivic cohomology is by the sign
n
Now compare our isomorphism to the one constructed in the realization We have the
same sp ectral sequence there attached to the weight ltration The identi cation of
the E term also uses K unneth formula and choice of an ordering of the sections
Using this identi cation we obtain the motivic residue sequence
k n k n
k n sgn sgn
H S j H G j H G j
M m S
M M m S
k n
H S j
M
for j k By construction we have the following
Theorem Under the regulator the motivic residue sequence maps to the abso
lute residue sequence of section
Note that the residue sequences for all indices k and n organize into a sp ectral
sequence connecting the relative motivic cohomology of A and the relative motivic
cohomology of G In particular for each n there is the converging cohomological
m
sp ectral sequence
p q p q pq p q n
n n n
G rel Z n G n H S p H E H
m S m S
M M M
Documenta Mathematica
Annette Huber Jorg Wildeshaus
This is the motivic version of the weight sp ectral sequence in absolute cohomology
We refer to it as the motivic residue sp ectral sequence
Remark As in section the residue sequence or equivalently the residue sp ectral
sequence turns out to b e the central technical to ol in the construction of the motivic
p olylog see De nition The sp ectral sequence is identical to the one constructed
in BD The de nition and basic prop erties of motivic cohomology of simplicial
schemes B B allow to justify the construction
At this p oint we should stress that the pro of of the inno cent lo oking Theorem
requires the whole of the theory covered in the app endices
Universal Motivic Polylogarithm
We now return to the sp ecial situation used in section Let B Sp ec Z We
consider now the case S U Let and the diagonal section of U G
B m B
First we compute the motivic cohomology of U We use the embedding of U into
A to do so The long exact lo calization sequence B reads
n
n n
H B q B j H A j H U j
M B M
M
n
H B q B j
M
By the homotopy prop erty of K theory we get
n n
n n
H B j H U j H B j H B j
M M
M M
n
B j H
M
The Gysin map for the inclusion of a p oint in the a ne line vanishes by Q Thm
ii Hence we are actually dealing with a system of short exact sequences As all
motivic cohomology groups of B vanish for n this sequence only gives non trivial
cohomology of U for n
Lemma For B Sp ec Z we have
Q if i
H U i
M
else
for j
H U j
Q Q for j
M
H B j for j
M
H U j H B j H B j
M M M
n
H U j if n
M
Pro of Clear from the ab ove using B
By Borel s Theorem B the Beilinson regulator
i i
X j R H H X R j
p
Q R
H M
is injective for X Sp ec Z even an isomorphism but in the one case H B
M
where the co dimension is one We call Beilinson regulator what strictly sp eaking is
Documenta Mathematica
Classical Motivic Polylogarithm
i
U k its tensor pro duct with R This implies that it is also an isomorphism for H
M
with the exception of the indices and where the co dimension is resp
This means that many of the residue maps are actually isomorphisms The
following computations are carried out in the case B Sp ec Z With a little more
e ort they generalize to the case of the ring of integers of a number eld
Consider the residue sequence for n j and S U
H U H G H U
M M m U M
H U H G H U
M M m U M
The Beilinson regulator induces a map b etween the ab ove sequence and the residue
sequence in section On H U R the regulator is an isomorphism and on
M
H U R it is injective of co dimension one By the absolute Ho dge coho
M
mology group H G R vanishes Hence the map from the rst to the second
p
H m U
R
line is injective and the regulator is injective of co dimension one on H G
M m U
Furthermore this last group is one dimensional
The image of under the Beilinson regulator is the map o ccurring in for
n
Definition Let s b e the comp osition of the maps
M
i
H B H U H G Q H B
M M M m U M
i
where i is the inclusion of the summand and is the map of the residue sequence
Lemma s is an isomorphism
Pro of Because of dimension reasons we only have to check that do es not vanish on
the image of i This follows from
Definition Let res b e the inverse of s We de ne the total residue map
n
n sgn
res H G n Q
m U
M
by comp osition of the residue maps in our long exact sequence with res
We now have to check that the total residue map deserves its name By de nition
and i it su ces to consider res
Lemma The regulators map the motivic res to res in absolute cohomology
Pro of Let us consider the situation of The morphism
O U H U R
p
R
H
factors through H U K U There is a commutative diagram
Q
M
L
H H B R U R
p
R R
p
H H
H
i
x x
L
H B O U H U
M M
i
Documenta Mathematica
Annette Huber Jorg Wildeshaus
hence the functions t and t on U corresp ond to the canonical generators of the two
summands We consider the commutative diagram for absolute Ho dge cohomology
res
R R
H U R H G R H B R
p p p
R R
H H m U H
R
x x
r
L
G B H H
m U M M
i
By the comp osition from the b ottom left to the top right corner is given by the
pro jection to the comp onent tensored by R It follows that res r R is an
R
isomorphism In turn vanishes on the comp onent and is an isomorphism on the
comp onent But then by de nition res is also the pro jection to the summand
As is surjective this su ces The same argument works in the etale situation
Lemma There is a short exact sequence
r es
sgn
Q H B H G
M M m U
and the Beilinson regulator is an isomorphism on the middle term
Pro of This is nothing but the residue sequence using our computation of
H G The zero es on b oth sides come from vanishing cohomology groups
M m U
Comparison with the short exact sequence ii shows that the regulator is an
isomorphism
Proposition There are short exact sequences
res
n
n
n sgn
Q G n H H B n
m U M
M
n
sgn n
n It is even an isomorphism The Beilinson regulator is injective on all H G
m U
M
for n
Pro of The n and n cases are the previous lemmas By induction one checks
n n sgn
that all H G n vanish for n Hence the residue sequence reads
M m U
n
n n
n sgn n sgn
G H n H n H G U n H B n
m U M M M
M m U
By the ve lemma and inductive hypothesis we see that the regulator is an isomor
phism on the middle term for n We need the previous lemma to get started
Now consider the sequences of the prop osition All maps are well de ned It follows
from ii that the sequence is exact
n
n sgn
Corollary There are canonical splittings s Q H G n such
n
m U
M
that the diagram
s
n
n
n sgn
H B H G n
M m U
M
s res
n
n
n sgn
H G n
M
m U
commutes They are compatible with the ones in iii Furthermore the group
n
sgn n
n is canonically isomorphic to Q G lim H
m U
M
Documenta Mathematica
Classical Motivic Polylogarithm
n
sgn n
n This Pro of Im r es is isomorphic to Q by the total residue on H G
n
m U
M
induces the same splitting as in
Definition For n N the system pol s de nes the universal motivic
n n
p olylogarithm
By construction pol is mapp ed to the p olylogarithmic system in absolute Ho dge
n
cohomology and continuous etale cohomology
Remark The main result of this section is identical to BD Although
part of the argument involves only constructions within K theory the pro of of
relies heavily on a detailed analysis of the b ehaviour of the regulator b etween the
motivic and absolute residue sequences
The Cyclotomic Case
Let d As b efore let R A d T T the ring of d integers of the cyclotomic
d
eld of d th ro ots of unity Put C Sp ec R Let b e a primitive d th ro ot of unity
Q and b an integer prime to d We work in the situation S C G C in
m
and i G C as in section
b m
Lemma a For n we have
n n sgn n n n sgn
H G n H G rel Z n Q
M m C M m C
The Beilinson and the l adic regulators are isomorphisms
b For n the residue sequence induces short exact sequences
n n
n sgn n sgn
H C n H G n H G n
M m C M
M m C
n
n sgn
G n for n The l adic regulator is injective on the group H
m C
M
Pro of For n we have H G H C which is canonically isomorphic
M m C M
to Q by B In particular b oth regulator are isomorphisms
H G and its counterpart in absolute cohomology vanish
M m C
Consider the following bit of the residue sequence for n
n n
sgn n n sgn
n H G G H n H C n
M m C M
m C M
The rst map is injective since H C n The l adic regulator is always
M
injective on the last term by B By inductive hypothesis it is an isomorphism on
the middle term By Cor the last map vanishes in absolute cohomology This
implies a for n In the next bit of the long exact sequence
n n
n sgn n sgn
H C n H G n H G n
M m C M
M m C
the rst map is injective by a For n we have H C n while for
M
n the term
n
n
H G n H C
M M
m C
vanishes Hence in any case we end up with the short exact sequence in b The
regulator maps it to the short exact sequence By induction and B we can
control the injectivity of the l adic regulator
Documenta Mathematica
Annette Huber Jorg Wildeshaus
C b ecause d is inverted Remark The Beilinson regulator is not injective on H
M
in C
Consider the morphism G G that raises p oints to the d th p ower
m C m C
n n n
By contravariance A As in section it induces a morphism of spaces A
C C
it induces an op eration on motivic cohomology
Lemma BD Remark i i on page
n
op erates on the short exact sequence of the previous lemma as follows
n n
n sgn n sgn
H C n H G n H G n
M m C M
M m C
n
n
d
id
y y y
n n
n sgn n sgn
H C n H G n H G n
M m C M
M m C
Pro of This description follows immediately from the injectivity of the l adic regulator
and Cor b
n
Remark The op eration on H C n is given by the op eration on
M
n
n
H A n It is easy to check that it is trivial by considering the op eration
C
M
on the starting terms of the degenerating skeletal sp ectral sequence To understand
the compatibility with the residue map in terms of K theory is a lot harder The
factor d is induced by a push forward from a non reduced scheme to its reduction
The theory in App endix B is not even set up to handle such schemes
n
As in the case of absolute cohomology it follows that the eigenvalues of on
n
n sgn n
G rel Z n are d d H
m C
M
Lemma The eigenspace decomp osition yields a splitting
M
n
n
n sgn
H G n H C i
m C M
M
b
i n
n
which is compatible with the splitting after Cor There is a canonical
b
isomorphism
Y
n
n sgn
lim H G n H C i
b
m C M
M
i
Pro of The rst assertion is clear by construction The second follows b ecause the
eigenspace decomp osition is compatible with the residue map
Definition Let i C U b e as b efore Let pol b e the pullback of the univer
b b
n
n sgn
sal p olylogarithm system pol de ned in to the inverse limit lim H G n
m C
M
n
n n sgn
lim H G rel Z n Via the isomorphism of we have constructed
b
m C
M
Q
H C i an element in
M
i
Theorem Under the regulators the element
Y
n
n sgn
pol lim H G n H C i
b
m C M
M
i
Documenta Mathematica
Classical Motivic Polylogarithm
is mapp ed to the elements
Y
n
n sgn
pol lim H G n H C i
b
m C abs
abs
i
constructed at the end of section
Pro of This follows from the construction
We list the consequences of this result denote by the set of primitive d th ro ots
d
of unity in Q
d
Firstly the description of the regulator to absolute Ho dge cohomology yields an
alternative pro of of the following
Corollary Assume n
a B Neu I I E
There is a map of sets
H C n
n
d M
H Sp ec Q n for n
d
M
such that
r H Sp ec Q R n
p
D n d R
d H
M
n
A
C i R
A
Q C
d
maps a ro ot of unity to Li For n this prop erty characterizes the
n
map uniquely
n
b
b For a ro ot of unity T Q Q T T the element
d d
b
T H C n
n
M
is given by
b n
T n comp onent of pol
n b
n
Pro of Note that a really is Beilinson s formulation of the result his normalization
of the isomorphism
M
n
H Sp ec Q R n C i R
p
d R
H
di ers from ours by the factor The unicity assertion is a direct consequence of
the injectivity of the regulator So our claim follows from and from
Documenta Mathematica
Annette Huber Jorg Wildeshaus
In B the ab ove compatibility statement is used to prove Gross s conjecture
ab out sp ecial values of Dirichlet L functions An alternative pro of of this conjecture
using an entirely di erent geometric construction is given in section of Den
Recall that the l adic regulator r factorizes as follows
l
K C Q H C n H C n
n Z
l
M M
H C n
l
cont
H Sp ec Q n
d
cont
where we let C C Z
Z
l
l
For the rest of this section we x C Q As was observed already in B
the study of the cyclotomic p olylog yields a pro of of the following result
Corollary Assume n
a Sou Th eor eme for the case n Gr Th eor eme IV for the lo cal version
if l d
Let d and b e as in Let l b e a prime Under the embedding of the l adic
n
regulator
r H C n H Sp ec Q n
l d
M cont
b
maps T to
n
X
d n
A
n
d n
r
l b
r
b Conjecture of BlK holds
Pro of a is and As for b it remains to check the comparison statement of
BlK Conjecture for the ro ot of unity For this observe the relations
n
c c
n n
n
n
c c
n n
n
in the notation of lo c cit if n D Prop osition i
Soul e has constructed maps
K C Z
l n Z l
l
d
for any prime l see end of App endix B for more details
The l adic regulator
r K C Q H Sp ec Q n Prop B
l n Z l d
l
cont
b
takes T to the cyclotomic element in continuous Galois cohomology
l
X
d n
A
r
l b
r
de ned by Soul e and Deligne cf Sou page D
Documenta Mathematica
Classical Motivic Polylogarithm
Corollary For each d and n there is a unique map
K Sp ec Q
n d
d
such that for each prime number l the map
K C Z
l n Z l
l
d
K Sp ec Q Z
n d Z l
equals the comp osition of and the natural map
K Sp ec Q K Sp ec Q Z
n d n d Z l
Furthermore the map Q agrees with
Z
H Sp ec Q n
d
n d M
n
given by d n
n
Pro of The uniqueness assertion is a formal consequence of the nite generation of
K Sp ec Q to give an element in a nitely generated ab elian group M is the
n d
same as giving elements in M Q and all M Z which coincide in M Q By
Z Z l Z l
the maps r and r agree for all l From Theorem B we conclude
l l l
n
that and agree as maps to K Q
l n Z l
n
As shown by Blo ch and Kato Corollary implies the validity of the following
also for even n
Corollary Let n
Then the Tamagawa number conjecture BlK Conjecture is true mo dulo a
p ower of for the motif Q n
Pro of BlK Theorem i gives the complete pro of for o dd n which is indep endent
of anything said in the present article In lo c cit Theorem ii it is shown that
the conjecture holds for even n if BlK holds But the latter is the content of
Finally the compatibility statement of forms a central ingredient in the
pro of of the mo di ed version of the Lichtenbaum conjecture for ab elian number elds
KNF Theorem
A Absolute Hodge Cohomology with Coefficients
The aim of this app endix is to provide a natural interpretation of absolute Ho dge
cohomology as extension groups in the category of algebraic Ho dge mo dules over R
A That such a sheaf theoretic interpretation should b e p ossible was already
anticipated by Beilinson B long b efore Ho dge mo dules were de ned
The app endix is divided into two subsections The rst A starts with a
summary of those parts of Saito s theory relevant to us The central result is A
where we prove that for a smo oth scheme a U Sp ec C the p olarizable Ho dge
Documenta Mathematica
Annette Huber Jorg Wildeshaus
complex R U F of D and B x is a representative for a F the
U
ob ject in the derived category of p olarizable F Ho dge structures de ned via Saito s
formalism S As a consequence we are able A to identify absolute
Ho dge cohomology of a smo oth scheme U over C as de ned in B x it equals
the Ext groups of Tate twists in the category of algebraic Ho dge mo dules on U The
compatibility b etween the approaches of Deligne Beilinson and of Saito will come as
no surprise to the exp erts see e g S However we were unable to nd a
quotable reference
In A we turn to the variant of the theory we really need algebraic Ho dge
mo dules over R These live on the complexi cation of separated reduced schemes of
nite type over R and are basically the ob jects xed by the natural involution on
the category of mixed Ho dge mo dules given by complex conjugation The compari
son statement for absolute Ho dge cohomology over R Theorem A then follows
formally from A
A Algebraic Mixed Hodge Modules
In S x the category MHM X of algebraic mixed A Ho dge mo dules is de ned
A
where A is a eld contained in R and X a separated reduced scheme of nite type
over C
Saito s construction admits the full formalism of Grothendieck s functors
b
D on the level of b ounded derived categories D MHM S Hom
A
and a forgetful functor
rat MHM X Perv X
A A
to the category of p erverse sheaves on the top ological space X underlying X C which
have algebraic strati cations such that the restrictions of their cohomology sheaves
to the strata are lo cal systems By the de nition of MHM which we shall partly
A
sketch in a moment rat is faithful and exact The functor rat on the level of derived
categories is compatible with Grothendieck s functors S
For smo oth X one constructs MHM X as an ab elian sub category S Prop o
A
sition of the category MF W D A whose ob jects are
h X
M F W K W
where M F is an ob ject of the category MF D i e a regular holonomic alge
h X
X W is a braic D mo dule M together with a go o d ltration F and K Perv
X A
lo cally nite ascending ltration and is an isomorphism
DR M K C
A
resp ecting W Here DR denotes the de Rham functor from the category of D
X
mo dules to the category of p erverse sheaves
We note that by de nition the weight graded ob jects of all algebraic Ho dge
mo dules satisfy a certain p olarizability condition see S
Call an algebraic Ho dge mo dule on a smo oth variety smo oth if the underlying
p erverse sheaf is a lo cal system up to a shift
Documenta Mathematica
Classical Motivic Polylogarithm
Theorem A Saito Let X b e smo oth and separated Then there is an equiv
alence
s
Var X MHM X
A A
b etween the category of admissible variations of mixed A Ho dge structure Ks and
the category of smo oth algebraic A Ho dge mo dules on X
Pro of This is the remark following S Theorem
In particular we see that MHM Sp ec C is the category MHS of p olarizable
A A
mixed A Ho dge structures
If V is a variation on X with underlying lo cal system For V then the p erverse
sheaf underlying the Ho dge mo dule V under the corresp ondence of A is
For V d
if X is of pure dimension d
It turns out that the de nition of Tate twists in MHM X is compatible with
A
the ab ove equivalence only up to shift
Definition A S Let n Z and A n MHS the usual Tate
A
twist For a separated reduced scheme a X Sp ec C de ne
b
A n a A n D MHM X
X A
If X is smo oth and of pure dimension d then A n d is the variation of Ho dge
X
structure which one denotes A n
For arbitrary X the complex A n will not even b e the shift of a Ho dge mo dule
X
b p
but a prop er element of D MHM X whose cohomology ob jects H A n are a
A X
priori trivial only for p dim X S
We note again that we follow Saito s convention and write e g for the functor
on derived categories
b b
D MHM X D MHM Y
A A
induced by a morphism X Y
In order to compare the Ho dge structures on Betti cohomology given by Saito s
and Deligne s constructions we need to go into the details of S
Theorem A Saito Let j U X b e an op en immersion of smo oth sepa
rated schemes over C with Y X n U a divisor with normal crossings If X is of
pure dimension d then
d
j A d H j A MHM X MF W D A
U U A h X
equals the ob ject
w Y j A d
X top U
d
where w Y denotes the D mo dule log Y and j the direct image for
X X top
X
the derived category of p erverse sheaves
Documenta Mathematica
Annette Huber Jorg Wildeshaus
The de Rham complex with logarithmic singularities is quasi isomorphic to
L
O d DR w Y d hence w Y
X X X D
X
DR w Y log Y d
X
X
compare Bo VI I I and
log Y d j C d
top
X
is the usual quasi isomorphism
log Y j j C
top top
X U
compare D shifted by d
The Ho dge ltration F on w Y is induced from the stupid ltration while
X
the weight ltrations W on w Y and j C d are those induced from the
X top
canonical ltration on j shifted by d
top U
d
Pro of The equation j A d H j A follows from the faithfulness of rat
U U
and the fact that the corresp onding statement for j is true since j is a ne In
top
our geometric situation the explicit construction of j of any admissible variation of
A Ho dge structure is carried out in the pro of of S Theorem For A it
U
sp ecializes to our claim
In B Beilinson extends Deligne s notion of Ho dge complexes D
to the p olarizable situation
Definition A Beilinson A mixed A Ho dge complex
K K F W K W
C
is called p olarizable if the cohomology ob jects of the weight n Ho dge complexes
W
Gr K are p olarizable A Ho dge structures
n
Remark The weight ltration W of a mixed Ho dge complex K induces mixed
W
i
Ho dge structures on its cohomology Observe however that Gr H K is of weight
n
n i
As in the non p olarizable situation Beilinson proves
Theorem A B Lemma There is an equivalence of categories b e
b
tween D MHS and the derived category of p olarizable A Ho dge complexes
A
Let X b e smo oth and separated over C Forgetting part of the structure of a
Ho dge mo dule yields a functor
b
For C MHM X T X
A
Here T X is the category of triples
M M F W K W
Documenta Mathematica
Classical Motivic Polylogarithm
b
where M F W is a class in the ltered derived category D W MF D of
h X
MF D and K W a class in the ltered derived category of sheaves of A
h X
b
vector spaces on X C denoted by D W X C A Furthermore the map is an
isomorphism
DR M K C
A
resp ecting W
b
Recall that in order to obtain a class in D W X C A from a complex of p erverse
sheaves one applies the realization functor of BBD
b
The global section functor can b e derived on D W X C A By S we
b
have a functor R on D W MF D if X is prop er and the two constructions are
h X
compatible with the comparison isomorphism of any ob ject in T X S
We end up with an ob ject
R M R M F W R K W R
of T Sp ec C The functor
b
R R For C MHM X T Sp ec C
A
b
factorizes through D MHM X
A
Our second comparison result is the following
Theorem A Let a X Sp ec C b e smo oth and prop er and M an ob ject of
b
D MHM X Write
A
For M M F W K W T X
a
M R M F W R K W R R
is a mixed p olarizable A Ho dge complex
b The class of R M in the derived category of p olarizable Ho dge complexes is
canonically isomorphic under the identi cation of A to
b
a M D MHS
A
c Let f Y X b e a prop er morphism of smo oth and prop er schemes over C
and let b denote the structure morphism of Y such that
b a f
b
For any N D MHM Y together with a morphism M f N in
A
b
D MHM X the morphism
A
M R N b N a f N a a M R
equals under the isomorphism of a the morphism
R R R
of A Ho dge complexes
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Pro of a We may assume that M is pure of some weight Using S we are
reduced to the case where M M is a Ho dge mo dule of weight n and we have to
M is a p olarizable Ho dge complex of the same weight Axiom CH show that R
of D follows from S Prop osition in particular applied to
pr M where
pr X A X
C
C
Furthermore by the remark following S and by lo c cit we have
isomorphisms in MF W D A
h
Sp ec C
i i i i i
M R M F W i R K W i R H a M R
Since the right hand side is a p olarizable Ho dge structure of weight i n S
we have CH and in addition p olarizability
b In the pro of of a we constructed a functor
b b
a R D MHM X D MHS
A A
such that
i i b
H a H a MHM X D MHM X MHS
A A A
b b
for all i Comp osition with j D MHM U D MHM X for op en immersions
A A
j U X de nes
b b
a j a j D MHM U D MHS
A A
But for a ne U a j is the left derived functor of
H a j MHM U MHS
A A
S pro of of Theorem If U is a ne then so is j U X and hence j is
exact Therefore
j
H a
H a j H a j MHM U MHS MHM X
A A A
and we get a natural transformation coincides with H a j
a j a j
which is an isomorphism since this is true on the level of cohomology ob jects as
one checks on the level of vector spaces Observe that this natural transformation is
compatible with restriction to smaller a ne subschemes of X Now recall S pro of
of that the functor a is constructed using the Cech complex asso ciated to an
a ne covering of X for details see B In the same way the functor a is
recoverable from the a j We end up with an isomorphism of a and a which is
indep endent of the covering
c In the pro of of b we constructed a natural isomorphism
a a
Documenta Mathematica
Classical Motivic Polylogarithm
b b
of functors from D MHM X to D MHS For f id our claim is therefore
A A
proved For the general situation we use the same techniques as in the pro of of b to
rst construct a natural isomorphism
a f b
b b
of functors from D MHM Y to D MHS and then to see that the triangle
A A
b a f
y
a f
commutes
Corollary A cf S Let j U X b e a smo oth compacti cation
of a smo oth and separated scheme a U Sp ec C such that Y X n U is a divisor
with normal crossings
b
a a A D MHS is isomorphic under the identi cation of A to the
U A
class of the mixed p olarizable A Ho dge complex
U A R DR log Y j A R
top U
X
of D and B x with the same notation
b If f X X is a morphism of compacti cations j U X and j U X
of U as in a then f induces an isomorphism
DR DR log Y j A log Y j A R R
0
top U U
X X top
D remark preceding so R U A dep ends only on U
The isomorphism in a also dep ends only on U
c In particular the Ho dge structures on
n n n
rat H a A n H U C i A
U
B
given by Deligne s and Saito s constructions coincide
Pro of a Combine A and A b
b Use A c
c follows from a and b
Actually the statement A c implies the functoriality prop erty we were after
we have two functors
b
S m C D MHS
A
where S m C denotes the category of smo oth separated schemes over C
A U R U A R
A a U Sp ec C a A
U
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Corollary A The isomorphism of A a is functorial in U S m C In
other words there is a natural isomorphism
A R A
b
of functors from S m C to D MHS
A
Pro of Let
0
j
U X
f f
y y
j
U X
b e a commutative diagram of smo oth and separated schemes over C where X and
X are prop er and Y X n U and Y X n U are divisors with normal crossings
We have a morphism
0
j A f j A
U U
Application of a gives the morphism
X
0 0
A a A a
U U U U
b elonging to the functoriality requirement for A Our claim follows from A c
applied to a shift of the morphism
Definition A Let X C b e separated reduced and of nite type and M an
b
ob ject of D MHM X
A
a The absolute Ho dge complex of X with co e cients in M is
p
R X M R Hom b A M
H X
D MHM X
A
b Its cohomology groups
i i
p
H X M X M H R
p
H
H
are called absolute Ho dge cohomology groups of X with co e cients in M
c We denote absolute Ho dge cohomology with co e cients in Tate twists by
i i
H X n H X A n
p p
X
H H
d For a closed reduced subscheme Z of X with complement j U X we de ne
relative absolute Ho dge cohomology with co e cients in Tate twists as
i
p
H X j A n X rel Z n H
p
U H
H
Documenta Mathematica
Classical Motivic Polylogarithm
Note that if X is smo oth and of pure dimension d and if
M M MHM X
A
then the right hand side of A b b eing equal to
Hom b A d M d i
X
D MHM X
A
admits an interpretation as the group of d i extensions of Ho dge mo dules mo dulo
Yoneda equivalence
Corollary A If X is smo oth and separated over C and n Z then
p p
X A n and H X n H X n R X A n R
p p
X H X H
H H
coincide functorially with the same noted ob jects of B x
Pro of This follows from A and the adjunction formula
A a M R Hom b A M R Hom b
X
D MHM X D MHS
A A
Remark The Leray sp ectral sequence for a X Sp ec C yields exact sequences
i i i
A H Ext A H H X A n Hom
p
X MHS
MHS A H
A
k k n
with H H X C i A since MHS has cohomological dimension one B
A
B
i
Corollary Comparing them with the analogous sequences for H we see that
H
i i
H X A n H X A n
p
X X
H H
i
n
in the notation of B x if H X C i A has weights smaller than zero
B
which is the case if i n i n if X is prop er
Observe that this is the same range of indices where Deligne cohomology coincides
i
X R n N we have natural morphisms with H
X
H
i i i
H X R n H X R n H X R n
p
X X X
H H D
b oth of which are isomorphisms if i n i n if X is prop er
A Algebraic Mixed Hodge Modules over R
Algebraic Ho dge mo dules over R are de ned as the category of Ho dge mo dules xed
under a certain involution given by complex conjugation We start by constructing
this involution
Let X C b e smo oth and let X denote the complex conjugate scheme We have
an equivalence
Var X Var X
A A
Documenta Mathematica
Annette Huber Jorg Wildeshaus
of the categories of admissible variations induced by complex conjugation
X C X C
and de ned as follows
The lo cal system and the weight ltration on X C are the pullbacks via of the
lo cal system and the weight ltration on X C and the Ho dge ltration on X C is
the pullback of the conjugate of the Ho dge ltration on X C
preserves admissibility and b ehaves in an obvious sense involutively
In particular if X is de ned over R we get an involution on Var X C
A R
Definition A Let X R b e smo oth and separated
a The category Var X R consists of pairs V F where V is an ob ject of
A
Var X C and F is an isomorphism
A R
V V
of variations such that F F
In the category Var X R we may de ne Tate twists A n F acts via
A
n
multiplication by
b Var X R the category of admissible variations of mixed A Ho dge structure
A
over R is the full sub category of Var X R of pairs V F which are graded
A
p olarizable for n Z there is a morphism
W W
Gr V F Gr V F A n
A
n n
in Var X R such that the induced morphism
A
W W
Gr V Gr V A n
A
n n
is a p olarization in the usual sense
Remark We note that implicit in our de nition is a descent datum over R of the
bi ltered at vector bundle on X C underlying any admissible variation V F
R
of mixed A Ho dge structure over R
For this claim to make sense recall rst D I I Th eor eme that any at
analytic vector bundle on X C carries a canonical algebraic structure If the vector
bundle underlies an admissible variation then the Ho dge ltration is a ltration by
algebraic subbundles Ks Prop osition
Now the descent datum is given by the anti linear isomorphism
c F F c c F F F V F V
DR di di di di
of the C bundles underlying V and V Here c denotes the anti linear involutions
given by complex conjugation of co e cients and F is the forgetful functor to C
di
bundles
Lemma A The category Var Sp ec R R equals the category MHS of mixed
A
A
p olarizable A Ho dge structures over R B x
Documenta Mathematica
Classical Motivic Polylogarithm
Pro of Straightforward
Our aim is to generalize our de nition of sheaves over R to algebraic Ho dge
mo dules
For smo oth and separated X C recall that MHM X is an ab elian sub category
A
of MF W D A Ob jects of the latter are
h X
M F W K W
where M F is an ob ject of the category MF D of regular holonomic algebraic
h X
X W is a lo cally nite ascend D mo dules with a go o d ltration and K Perv
X A
ing ltration and is an isomorphism
DR M K C
A
resp ecting W
The equivalence
MF W D A MF W D A
h X h X
is constructed comp onentwise
The p erverse sheaf and the weight ltration on X C are the pullbacks via
X C X C of the p erverse sheaf and the weight ltration on X C
The equivalence
Mo d Mo d
D D
X
X
which by construction will resp ect holonomicity comes ab out as follows
Given a D mo dule N we may form the inverse image in the sense of sheaves
X
of ab elian groups N which is a D mo dule All we therefore need is an
X
D of sheaves of rings extending the isomorphism isomorphism c D
X X
O given by complex conjugation of co e cients we then de ne c O
X X
D N N
X
D
X
Of course the map c is itself given by conjugation of co e cients in lo cal co ordinates
x x we have
n
X X
c f f c
x x
Altogether we get
MF W D A MF W D A
h X h X
which again b ehaves involutively
Going through the de nition one checks that induces
MHM X MHM X
A A
Using lo cal embeddings as in S we can de ne for any scheme X which is
separated reduced and of nite type over C Furthermore if X is de ned over R we
get an involution on MHM X C
A R
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Theorem A Let X and Y b e separated and reduced schemes of nite type over
C
b
a is compatible with Hom and D e g for M N D MHM X we
A
have
M N M N Hom Hom
X X
b If X Y is a morphism then is compatible with e g for
b
M D MHM X we have
A
b
M M D MHM Y
A
Pro of This follows from the de nitions
Definition A a Let a X Sp ec R b e smo oth and separated The
category MHM X R consists of pairs M F where M is an ob ject of
A
MHM X C and F is an isomorphism
A R
M M
such that F F
By A b we have a A n MHM X R
A
b Let a X Sp ec R b e smo oth and separated MHM X R the category
A
of algebraic mixed A Ho dge mo dules over R on X is the full sub category of
MHM X R of pairs M F which are graded p olarizable for any n Z
A
there is a morphism
W W
Gr M F Gr M F a A n
A
n n
in MHM X R such that the induced morphism
A
W W
Gr M Gr M a A n
A
n n
is a p olarization in the sense of S
As in A we identify the category of smo oth ob jects in MHM X R with
A
Var X R
A
c For an arbitrary separated and reduced scheme X of nite type over R one
de nes the category MHM X R using lo cal embeddings as in S
A
Remark a As in the case of variations over R we get a descent datum over R for
the bi ltered D mo dule underlying any Ho dge mo dule over R on a smo oth and
X C
R
separated scheme X over R
b As in S the category MHM Z R for any closed reduced subscheme
A
Z of X is equivalent to the category of Ho dge mo dules over R on X with supp ort in
Z
Documenta Mathematica
Classical Motivic Polylogarithm
Theorem A There is a formalism of Grothendieck s functors
b
Hom D on D MHM R It is compatible with the forgetful functor
A
b b
D MHM R D MHM C
A A R
Pro of By A we may e g de ne
M F M F
Definition A Let X R b e separated reduced and of nite type and M an
b
ob ject of D MHM X R
A
a The absolute Ho dge complex of X R with co e cients in M is
p
X R M R Hom b A M R
X H
D MHM X R
A
b Its cohomology groups
i i
p
H X R M X R M H R
p
H
H
are called absolute Ho dge cohomology groups of X R with co e cients in M
c We denote absolute Ho dge cohomology with co e cients in Tate twists by
i i
H X R n H X R A n
p p
X
H H
d For a closed reduced subscheme Z of X with complement j U X we de ne
relative absolute Ho dge cohomology with co e cients in Tate twists as
i
p
H X R j A n X rel Z R n H
p
U H
H
Again if X is smo oth and of pure dimension d and M M MHM X we have
A
d i
i
H X R M Ext A d M
p
X
H
MHM X R
A
We have statements analogous to A A for the situation over R For
reference we note explicitly
Theorem A If X is smo oth and separated over R and n Z then
p
X R n and H R X R n
p
H
H
coincide functorially with the absolute Ho dge complex and cohomology groups of B
x
Next we have
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Lemma A Let X R b e separated reduced and of nite type and M an ob ject
b
of D MHM X R Then the forgetful functor
A
b b
D MHM X R D MHM X C
A A R
induces functorial isomorphisms
p p
X C M X R M R R
R H H
H X R M H X C M
p p
R
H H
Here the sup erscript denotes the xed part of the action of the involution on
R Hom b A M
X C
D MHM X C
R
A
R
In particular the category MHS has cohomological dimension one since this is
A
p
X true for MHS Furthermore observe that the ab ove action of Z Z on R
R A H
C A n is precisely that of B x
X C
R
Corollary A Let X R b e separated reduced and of nite type The forgetful
functor
X C rat MHM X R Perv
R A A
is faithful and exact
Remark Again we have
i i
H X R A n H X R A n
p
X X
H H
if i n i n if X is prop er We have natural morphisms
i i i
H X R R n H X R R n H X R R n
p
X X X
H H D
which are isomorphisms in the same range of indices
We conclude with an explicit formula for Ext in MHM X R of a nite scheme
A
X R
there is a canonical isomorphism Theorem A For any H MHS
A
A H Ext W H W H W F H
C A C
MHS
A
H Sp ec R R H
p
H
where the sup erscript on the left hand side denotes the xed part of the de Rham
conjugation
c
1
F H W H W H W W H W H W F H
C C A C A C
W H W H W F H
C A C
F
1
W H W H W F H
C A C
Documenta Mathematica
Classical Motivic Polylogarithm
The isomorphism is given by sending the class of h W H to the extension describ ed
C
by the matrix
h id
H
This means that we equip C H with the diagonal weight and Ho dge ltrations and
C
the A rational structure extending the A rational structure H of H by the vector
A C
h C H
C
thereby obtaining an extension E of A by H in the category MHS
A
A H is given with the same The conjugate extension E Ext
MHS
A
notation by the matrix
F h id
H
and the extension of F to an isomorphism
F E E
sends h to F h Thus
F id F C H C H
C C C C
Pro of Using B x or Jn Lemma and Remark a we see that there is
an isomorphism
A H W H W H W F H Ext
C A C
MHS
A
Note that our normalization follows that of Jannsen and therefore di ers from that
of Beilinson by the factor
In general if h W H corresp onds to an extension E in MHS then c h W H
C A C
corresp onds to E and its pullback via
F H H
A H therefore is describ ed by F c h The action of the involution on Ext
MHS
A
corresp onds to F c on the left hand side of the ab ove isomorphism
Corollary A Let X R b e nite and reduced and M MHM X R Then
A
there is a canonical isomorphism
M
A
W M W M W F M
x C x A x C
x X C
M
A
M A Ext
x
MHS
A
A
x X C
H X R M
p
H
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Pro of The last isomorphism is given by the observation that we have
M
MHS MHM X
A A
x X C
Corollary A For X R nite and reduced and n we have
M
n
A
C i A Ext A A n
X X
MHM X R
A
x X C
H X R n
p
H
Here the sup erscript denotes the xed part with resp ect to the conjugation on b oth
n
X C and C i A and the isomorphism asso ciates to z the extension
x
x X C
whose stalk at x X C is given by the matrix
z
n x
i
n n
if e and e are the base vectors F C and i i A C then the
n
Ho dge structure is sp eci ed by
F he i W C he i
C n A n C
and the A rational structure is generated by e and
n
z e e
x n
n
i
Pro of This is A and A using the basis e of A n
n
B K Theory of Simplicial Schemes and Regulators
We start with a presentation of K theory B for simplicial schemes in terms
of generalized cohomology Applied to a regular scheme we get back its K groups
cf B a Next we de ne op erations on K cohomology cf B Motivic
cohomology of simplicial schemes in particular relative motivic cohomology B
is introduced as graded pieces of the ltration with resp ect to these op erations
This discussion is based on the extremely useful unfortunately unpublished pap er
GSo by Gillet and Soul e More often than not the results in B and B will b e due
to them The wish for a complete published reference made us go over the material
again Meanwhile an alternative approach to K theory of simplicial schemes and
op erations was also worked out by Levine Le De Jeu was the rst to use the setting
of GSo to de ne motivic cohomology of simplicial ob jects In his article Jeu he
proves Riemann Ro ch in this setting We give a more general version in B
We then construct regulators i e Chern classes from K cohomology to con
tinuous etale cohomology B and to absolute Ho dge cohomology B in this sit
uation Our main interest is the construction of a long exact sequence for relative
Documenta Mathematica
Classical Motivic Polylogarithm
K cohomology of simplicial schemes as well as for their motivic cohomology which is
mapp ed to the corresp onding long exact sequences in sheaf cohomology B
We would like to thank the referee for her or his comp etent and detailed comments
and corrections
B Generalized Cohomology Theories
We need a framework which is general enough to treat K theory and the usual coho
mology theories in parallel It turns out such a framework is given by homotopical
algebra as axiomatized by Quillen in Q
We de ne cohomology of spaces simplicial sheaves of sets with co e cients in
another space B We then construct a long exact sequence for relative cohomol
ogy in this context B Finally we deduce the sp ectral sequence relating general
ized cohomology of a space to generalized cohomology of its comp onents B
A systematic investigation of generalized cohomology for Grothendieck top ologies
was carried out by Jardine in particular Jr We recapitulate the de nitions for the
convenience of the reader A rst introduction to the necessary simplicial metho ds is
M
We x a regular a ne irreducible base scheme B of nite Krull dimension In
our applications B is either a eld or an op en subscheme of the ring of integers of
a number eld We x a small category of no etherian nite dimensional B schemes
which is closed under nite disjoint unions and contains all op en subschemes of all
its ob jects We turn it into a site using the Zariski top ology Typically this will b e a
sub category of all smo oth schemes over the base B
Let T b e the top os of sheaves of sets on our Zariski site over B Let sT b e
the category of p ointed simplicial T ob jects Its ob jects will b e called spaces in the
sequel We denote the nal and initial ob ject of sT by
Remark A space is given by a simplicial sheaf of sets X and a simplicial map from
the constant simplicial sheaf all of whose comp onents are given by the constant
sheaf attached to the set with one element to X Equivalently we can consider it
as a simplicial ob ject in the category of sheaves p ointed by
Let X b e a scheme We can also see it as an ob ject of T The corresp onding
constant simplicial ob ject p ointed by a disjoint base p oint
U Mor U X f g for connected U T
B
will also b e denoted X
Definition B A space is said to b e constructed from schemes if all comp onents
are representable by a scheme in the site plus a disjoint base p oint
Note that any simplicial scheme whose comp onents are schemes in the site
gives rise to a space constructed from schemes but there are many spaces constructed
from schemes which do not come from simplicial schemes The main example is the
mapping cone of a map of schemes taken in sT cf B b elow
If P is a prop erty of schemes and if the space X is constructed from schemes we
say X has P if the scheme parts of the comp onents have P
The easiest way to de ne the homotopy sets X x of a simplicial set X with
n
basep oint x X is to take the homotopy sets of its geometric realization X x
n
Documenta Mathematica
Annette Huber Jorg Wildeshaus
is a group for n even ab elian for n If X is a space and K a nite simplicial
set i e all K are nite then we de ne the space X K comp onentwise as the sum
n
of p ointed sheaves
n X
n
K
n
Definition B Brown Gersten Gillet Soule Let X b e a space and f
X Y b e a map of spaces
a f is called a weak equivalence if all stalks f X Y are weak equivalences of
P P P
simplicial sets i e if f induces an isomorphism on all homotopy sets for all
P
choices of base p oint
b f is called a co bration if for all schemes U in T the induced map f U X U
Y U is injective
c f is called a bration if it has the following lifting prop erty given a commutative
diagram
A X
f
i
y y
B Y
where i is a co bration and a weak equivalence there exists a map B X that
makes the diagram commute
d For two spaces X and Y let Hom X Y b e the p ointed simplicial set
n Hom X n Y
sT
where n is the standard simplicial n simplex e g M p ointed by zero
This is the p ointed version of the global theory discussed in Jr x
Quillen s notion of a closed mo del category axiomatizes the prop erties which
are needed in order to pass to a homotopy category which b ehaves similar to the
homotopy category of CW spaces
Proposition B Brown Gersten Joyal sT is a p ointed closed simplicial
mo del category in the sense of Quillen Q
Pro of For a mo del category we need brations co brations and weak equivalences
satisfying a set of axioms Q I Def This is GSo Theorem Gillet and
Soul e attribute this theorem to Joyal letter to Grothendieck For simplicial sheaves
a published pro of of all prop erties can b e found in Jr Cor It is an abstract
non sense fact that with the category of simplicial sheaves the category of p ointed
simplicial sheaves is also a mo del category It is p ointed by The simplicial structure
Q II Def is given by B d
Documenta Mathematica
Classical Motivic Polylogarithm
Technical Remark Note that the unique map X is always a co bration
i e all spaces are co brant A space will b e called brant if the unique map X
is a bration If a space is brant than its sections X U over a scheme U form
a simplicial set satisfying Kan s extension condition cf M However this
prop erty do es not su ce to make X brant Part of the pro of of the prop osition is
the existence of brant resolutions In fact the construction in Jr Lemma is
even functorial
Let Ho sT b e the homotopy category asso ciated to the mo del category sT by
lo calizing at the class of weak equivalences As usual we will write X Y for the
morphisms from X to Y in the homotopy category If Y is brant then this set is
given by the set of morphisms from X to Y in sT up to simplicial homotopy For
general Y we compute X Y by X Y where Y is a brant resolution of Y
Remark The category of p ointed presheaves with the same notions as in B is
also a p ointed mo del category By Jr Lemma the map from a presheaf to its
shea cation is a weak equivalence and we get the same homotopy category from
presheaves or sheaves
If X is a space then its susp ension SX is given by X where is
the usual equivalence relation generated by x x By Q Ch I the lo op
space functor is right adjoint to S on the homotopy category
There are two natural ways of thinking ab out Ho sT From the p oint of view of
algebraic top ology it corresp onds to the category of CW complexes with morphisms
up to homotopy From the p oint of view of homology theory it corresp onds to the
category of homological complexes which are concentrated in p ositive degrees with
morphisms up to homotopy S and shift the complexes This second p oint of view
is not quite precise note that in general morphisms in Ho sT form p ointed sets
rather than groups
Definition B For any space A we de ne cohomology of spaces with co e cients
in A by setting
m
m
H X A S X A for m
sT
This is a p ointed set for m a group for m and even an ab elian group for
m If A b elongs to an in nite lo op sp ectrum i e if there are spaces A for i
i
with A A and weak equivalences A A then we also de ne cohomology
i i
groups with p ositive indices by setting
n m
m
H X A S X A for m n
n
sT
Note that the set only dep ends on n m b ecause the susp ension S and the lo op
functor are adjoint
Definition B Let f X Y b e a map of spaces Then the mapping cone of f
is the space
C f X q Y
where is the usual equivalence relation of the mapping cone i e x f x
x For any map of spaces f X Y we de ne relative cohomology by
m m
H Y rel X A H C f A
sT sT
Documenta Mathematica
Annette Huber Jorg Wildeshaus
C f is the standard construction of the homotopy co bre of a map
Proposition B For any morphism f X Y of spaces there is a long exact
cohomology sequence
m m m m
H Y A H X A H Y rel X A H Y A
sT sT sT sT
Pro of By Q Ch I we have the ab ove long exact sequence attached to the triple
of spaces
i
X Y Y
X
if i is a co bration The mapping cylinder of f is de ned as X Y It is weakly
X
equivalent to Y and the induced mapping X X Y is a co bration The
X
mapping cone of f is nothing but the co bre of this inclusion Hence the long exact
sequence of the lemma is a sp ecial case of Quillen s with Y X Y
X
If A is only a space then the sequence will end at the index zero There is no
reason for the last arrow to b e right exact The H are only p ointed sets The
sT
H are groups all others are even ab elian groups However if A is an in nite lo op
sT
sp ectrum then all cohomology groups will b e ab elian groups and the sequence is
unbounded in b oth directions
We will consider a couple of sp ectral sequences which are constructed by means
of homotopical algebra Their di erentials are
p q p r q r
d E E
r
r r
We refer to this b ehaviour as homological sp ectral sequence as opp osed to a cohomo
logical sp ectral sequences with di erentials
p q p r q r
d E E
r
r r
In the same way as with the long exact sequences which involve p ointed sets we
also have to b e careful ab out our sp ectral sequences They will b e constructed by the
metho d of Bous eld Kan cf BouK Ch IX xx We refer to them as sp ectral
sequences of Bous eld Kan type We give an overview over their prop erties They
lo ok like this
p q q p
E L q p r
r
with homological di erentials
are ab elian groups if q p
q p p q
L E
are groups if q p
r
are p ointed sets if q p
p q
p q p r q r p q
We have E Ker d im d Treat non existing E as zero for this
r r
r
formation By BouK IX iv this makes also sense for p q Let
p q p q p q
lim E E E
r
r r
r p
Documenta Mathematica
Classical Motivic Polylogarithm
n n
There is a descending co ltration Q on the limit term L i e Q L is a quotient of
i
n
L Let
p q q p q p
e Ker Q L Q L
p p
p q p q
In general there will b e an injection e E Convergence is a more complicated
p q
question The sp ectral sequence stabilizes if all pro jective systems E b ecome
r p
r
eventually stable Then we have complete convergence BouK IX Hence the
n n
co ltration on the limit term is exhaustive lim Q L L and we have isomor
s
phisms
p q p q
e E for p q
Note that even then the case p q has to b e discussed separately We refer to this
problem and more generally the fact that p ointed sets rather then groups app ear as
the fringe e ect
Proposition B a Let X and A b e spaces The ltration of X by its skeletons
sq X induces a sp ectral sequence of Bous eld Kan type for its A cohomology
n
q p
p q q
E H X A H X A for q p
p
sT sT
It converges completely if X is degenerate ab ove some degree i e if there is N
such that for n N X is covered by the image of the degeneracy maps
n
b If A is an in nite lo op sp ectrum and X as in a then we have a converging
homological sp ectral sequence
q p
p q q
E H X A H X A for p
p
sT sT
Pro of This is the hypercohomology sp ectral sequence of GSo We sketch
their pro of We can assume A to b e brant We can construct a weak equivalence
p
S X The Hom sk X A form a tower of X X such that sk X sk X
p p p p
brations of simplicial sets converging to Hom X A The attached Bous eld Kan
sp ectral sequence BouK x x has starting terms
p q
E Hom sk X sk X A
q n p p
q
p
Hom S X A H X A
q p p p
sT
This nishes the construction of the sp ectral sequence In order to discuss convergence
we consider the same sp ectral sequence attached to X itself It stabilizes by the
assumption on degeneracy see BouK x Both sp ectral sequences agree from r
on
For b we consider the sp ectral sequence in a for each space in the sp ectrum By
shifting q accordingly we get a direct system of sp ectral sequences whose limit is the
one we are interested in
Remark It would b e much nicer to work with sp ectra and their homotopy category
throughout It would b e a triangulated category It would help to get rid of the fringe
e ects However the question of convergence of the sp ectral sequences do es not get
Documenta Mathematica
Annette Huber Jorg Wildeshaus
easier the reason b ehind this b eing that all these sp ectral sequences are constructed
for some kind of homotopy limit and pro jective limits are not exact However the
literature we want to use is in the setting of spaces The reason is that we want to
use the ring structure in order to de ne motivic cohomology and the op erators
do not delo op
B K theory
We now introduce higher algebraic K theory of spaces as a generalized cohomology
theory It gives back usual K theory in the case of regular schemes B We then
de ne op erators on these K cohomology groups B This allows de nition of
motivic cohomology of spaces as graded parts of the ltration B We then
prove a Grothendieck Riemann Roch type theorem B As a consequence we
get a long exact lo calization sequence for motivic cohomology B
Recall that all schemes in the site underlying T are assumed to b e no etherian
and nite dimensional
Let K b e the space Z Z B Gl where Z B Gl is the simplicial sheaf asso ciated
to the simplicial presheaf U Z B Gl U lim Z B Gl U K is p ointed by
n
lim B Gl E It is in fact part of an in nite lo op sp ectrum We also need the
n n
N N
unstable spaces K Z Z B Gl There are natural transition maps K
N
N
K K As K groups commute with direct limits the stalk of K in a p oint P
on U T is weakly equivalent to
Z Z B Gl O K
P P
where O is the stalk of the structural sheaf
P
Remark Even though it is well known that K theory is de ned by a sp ectrum it
is not completely trivial to de ne it as a functor from schemes to sp ectra rather
than just a functor up to homotopy We refer to GSo for the details of this
construction For a di erent account of K theory as a presheaf and its prop erties
including the pro duct structure we also refer to Jardine s b o ok Jr
Definition B Gillet Soule For any space X in sT we de ne its K
cohomology
m
m
X K S X K for m Z H
sT
m
N
X K for m Following GSo we call a space and the unstable K groups H
sT
m m
N
X K for m is an isomorphism X K H K coherent if lim H
sT sT
Proposition B Brown Let K b e the shea cation of the presheaf Y
q
q
H Y K Let X b e a scheme in T There is a homological sp ectral sequence
sT
q p
pq
E H X K
sT
with
p pq
H X K E
q
ZAR
It converges completely
Documenta Mathematica
Classical Motivic Polylogarithm
Pro of For q p this is the sp ectral sequence GSo Prop The basic version
for the small Zariski site was constructed in BrG Theorem Our generalization
follows from the pro of of Jr and which deals with the etale top ology The
key is to construct a Postnikov tower for K This is done as in in the pro of of BrG
Thm We then have to check that the homotopy sheaves of K are isomorphic to
the homotopy sheaves of the limit of its Postnikov tower It su ces to check this
for the small Zariski site Zar Y for all schemes Y in T Hence we are reduced to
the situation considered in lo c cit Note that Y was assumed to b e no etherian and
nite dimensional We extend to arbitrary p q using the full K theory sp ectrum
Convergence follows b ecause X has nite cohomological dimension
Remark We could generalize the sp ectral sequence to arbitrary spaces X
p
H X K would have to b e understo o d as in B Convergence would not b e
q
ZAR
guaranteed anymore
The most imp ortant application of this prop osition is that it allows to transp ort
prop erties which are well known for cohomology with co e cients in an ab elian sheaf
to cohomology with co e cients in a space One such prop erty is the comparison
b etween di erent Zariski sites
Proposition B Gillet Soule de Jeu a Let X b e a no etherian regular
m
nite dimensional scheme in the site Then one has the equality H X K
sT
K X where the right hand side means Quillen K theory of the scheme X
m
m
X K for m In particular H
sT
b Let X b e a space constructed from schemes Assume that all comp onents are
regular No etherian nite dimensional schemes and that X is degenerate ab ove
some simplicial degree Then X is K coherent
Pro of The constant case is proved in GSo Prop We sketch a slightly
di erent argument We use the converging Brown sp ectral sequence and comparison
theorems for sheaf cohomology to show that it su ces to prove the prop osition in
the case of T Zar X Note that the existence of the whole sp ectrum means we
do not have to worry ab out fringe e ects In this case we have a Mayer Vietoris se
quence for K theory Q Rem and hence the presheaf de ning K cohomology
is pseudo asque in the sense of Brown and Gersten BrG p By lo c cit
Thm this implies a for the site Zar X
The vanishing follows b ecause the K theory sp ectrum is connective The general
ization to spaces constructed from schemes using the skeletal sp ectral sequence was
carried out in Jeu and Lemma
Corollary B If X is a space meeting the conditions of part b of the prop osi
tion then its K cohomology do es not dep end on the category of schemes underlying
the top os
Pro of If X is constant then we always get its K theory For more general X we
have to use the converging skeletal sp ectral sequence There are no fringe problems
b ecause K is an in nite lo op sp ectrum
Documenta Mathematica
Annette Huber Jorg Wildeshaus
The direct sum of matrices cf Lo together with addition on Z induces
a compatible system of maps
N N
K K K
Our aim is to show that its direct limit de nes an H group structure on K It will
b e used to de ne addition on K cohomology
Lemma B Let G G b e algebraic groups over Z E a subgroup of G with E
E E Let f f G G b e homomorphisms which di er by conjugation by a
global section of E Then the induced maps
f f
Z BG Z BG
agree in the homotopy category of spaces
Pro of The construction in Lo A is functorial Hence it yields a free homotopy
b etween B f and B f By construction we get a commutative diagram
j
Z BE
i
y y
Z BG Z BG
The comp osition of with d Z BG C i is a homotopy b etween df and df
Now it su ces to show that d is a weak equivalence i e that Z BE is contractible
This can b e checked on stalks As homotopy groups commute with direct limits it is
enough to show that Z BE U is contractible for all a ne schemes U We consider
the diagram
BE U BE U
y y
Z
1
Z BE U Z BE U
By de nition of Quillen s construction see Lo ch induces an isomorphism
on homology Hence Z is a weak equivalence BlK Ch I BE U is con
tractible b ecause E U E U E U Lo Prop osition Hence Z BE U
is also contractible
The standard application of this lemma is with G Gl G Gl and E the
n
subgroup generated by elementary matrices which contains all even p ermutation
matrices see Lo
Proposition B The direct sum of matrices induces an H group structure on
K
Pro of The same pro of as in Lo Theorem allows to check the identities of an
H space On nite level they hold up to conjugation with a p ermutation matrix
Documenta Mathematica
Classical Motivic Polylogarithm
By the previous lemma this implies that they hold in the homotopy category We
N N
use that the transition maps K K are co brations in order to show that the
maps on nite level de ne one on K For the existence of a homotopy inverse we
argue di erently An H space is an H group if and only if the shear map
K K K K k k k k k
is a weak equivalence This can b e checked on stalks But the stalks of K are the
simplicial sets computing K theory of lo cal rings They are H groups with the same
addition by the a ne case Lo
Remark We now have two H group structures on K the explicit one we just have
constructed and one b ecause K is a lo op space as part of a sp ectrum We exp ect
them to b e equal but have not b een able to prove it They certainly induce the same
X K they agree at least if X is addition on higher K cohomology groups On H
sT
represented by a scheme b ecause they do for K theory of schemes This is enough for
our needs In the sequel the addition on K cohomology is the one of the prop osition
The next aim is the de nition of a multiplicative structure on K We start with
the op eration of Z on K The H group structure on K allows to de ne a map of
spaces
Z K K
Z
It vanishes on Z K and hence factors over Z K
The construction of the Lo day pro duct Lo
Z B Gl U Z B Gl U Z B Gl U
N N
is functorial in U Together with the pro duct on the factor Z it de nes a system
Z
of maps
N N
K K K
K
compatible up to homotopy which de nes a pro duct
Y K Y K Y K
n
for all K coherent spaces Y It turns all H Y K for n into a ring p ossibly
sT
without unity
n
Remark Note that this pro duct on Y K is zero on H Y K for n cf Kr
sT
Ex p The same map of spaces also induces a non trivial pro duct
K
n m n m
S Y K S Y K S Y K
This is the one which is usually called Lo day pro duct We do not need it in the sequel
Let S b e the simplicial version of the sphere i e the constant simplicial sheaf
asso ciated to f g p ointed by We will use the notation K sT for H S K
sT
It is a ring with unity where the ring structure is induced by the ring structure on Z
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Lemma B If the site underlying T has a nal ob ject X then
K X K sT
Pro of If X is the nal ob ject of the site then the space we denote by X is equal to
S
The following lemma generalizes an op eration of K X which was explained to
us by de Jeu in the case where Y is constructed from X schemes
n
Lemma B Let Y b e a space in sT Then the ring K sT op erates on H Y K
sT
for n and makes it into an K sT algebra
Pro of If Y is a space in sT then there is canonical isomorphism Y S Y The
n
pro duct K sT with H Y K is de ned by the comp osition
sT
K
Y S Y K K K
Lemma B Gillet Soule Let G b e a group over Z Let R G b e the
Z
Grothendieck group of representations of G on free Z mo dules of nite type
a Let A b e an N dimensional representation of G There is a canonical class in
Z Z B G K which dep ends only on the equivalence class of A The direct
sum of representations is mapp ed to the sum of classes
b The map in a induces an algebra homomorphism
r R G Z Z B G K
Z
Pro of We follow GSo or the a ne case Kr By choice of a basis of an
N dimensional representation A induces a map of sheaves
A G Gl
N
and hence by functoriality a map
N
r A Z BG fN g Z B Gl K
N
For di erent choices of basis the maps di er by conjugation with an element of
Gl The matrix is in the p erfect subgroup E Gl Gl hence by Lemma
N
N
B the image of r A in Z B G K do es not dep end on the choice of matrix
Viewed as map to K this r A extends to the factor Z using the ab ove pro duct
Z
The last statement of a follows by de nition of the H group structure on K
For b we have to check that the relations of the Grothendieck group are mapp ed to
zero and that the multiplicative structure is well behaved We rst prove the analogue
of Kr Theorem The canonical maps
Gl Gl
induce weak equivalences of simplicial sheaves after applying Z B This can b e
checked on stalks and is hence reduced to the a ne case From now the pro of works
precisely as in the a ne case see Kr Cor
Documenta Mathematica
Classical Motivic Polylogarithm
K sT is a ring i e the axioms in Kr Def are satis ed If R is a
K sT algebra then it is called a K sT algebra if it is equipp ed with op erators
i
for i such that K sT R is a ring cf Kr Note that has to have
the constant value If R itself do es not have a unity then it cannot b e a ring
Theorem B Gillet Soule Let Y b e a K coherent space For k and
m there are maps
m m
k
H Y K H Y K
sT sT
m
They turn H Y K into a K sT algebra
sT
Pro of This is essentially GSo Prop Put G Gl in the previous lemma Let
n
n n n
is the canonical representation of Gl on n R Gl where Z Z Z
n Z n
id id
n k k n
Z and is the trivial representation We de ne r Z By comp osition it
n
m m
k n
induces a map H Y K H Y K These form a pro jective system and
n
sT sT
hence de ne an op eration on K cohomology of a K coherent space Well de nedness
n
and all prop erties of a ring are checked on the universal level i e on K for varying
n and hence as in the a ne case Kr Thm For example we want to show
k
X
i j k
x y x y
i
n
Assume that x y are represented by elements in Y K On R Gl Gl we have
Z n n
the ring identity
k
X M
i j k
i
n
We evaluate this identity in Z and get an equality of elements in R Gl Gl
Z n n
n n
By the previous lemma it induces the same equality of elements in K K K
Comp osed with x y this is the required equality
Remark A more conceptual pro of was suggested to us by Soul e and the referee
One should use the integral completion functor constructed by Go erss and Jardine
Go eJr It has a universal prop erty similar to the one of the construction and
hence allows to copy directly Kratzer s arguments
Technical Remark When we try to de ne in the same way then we still get a
map
Z B Gl Z Z B Gl
N
It do es not extend to the factor Z b ecause Z Z do es not resp ect the base
p oint in fact it maps to This re ects the fact that the ring K Y do es not
have a unity for a general space Y The most striking example is Y C i where
i Z X is a morphism b etween regular schemes cf Sou Then K Y
Ker K X K Z do es not contain
Gillet and Soul e GSo Prop consider the structure as a H Y K algebra
sT
Y K happ ens to have a unity However we can check This only makes sense if H
sT
Documenta Mathematica
Annette Huber Jorg Wildeshaus
m
Y K on H Y K is compatible with the in general that the op eration of H
sT
sT
K sT algebra structure of b oth groups
Note that the structure is compatible with the contravariant functoriality of
K cohomology This means that the long exact sequences for relative K theory are
compatible with the op eration where it is de ned
Once we have op erations we get as usual a ltration and Adams op erators
n
Y K for n If the ltration is lo cally nite then we on the mo dule H
sT
have in particular the Chern character
M
j
n n
ch H Y K Gr H Y K for n
Q Q
sT sT
j N
which is an isomorphism For a quick survey cf T pp
Definition B Let Y b e a K coherent space Supp ose that the ltration is
lo cally nite and hence that rationally K cohomology splits into Adams eigenspaces
Then we put for j n
n j
j
n
H Y K H Y j Gr
Q
M
sT
the motivic cohomology of the space Y If i X Y is a morphism of spaces then
we de ne relative motivic cohomology by
n n
H Y rel X j H Cone i j
M M
Remark We restrict to this range of indices b ecause we did not de ne Adams
eigenspaces for K cohomology with p ositive indices K theory with negative in
dices However if these K groups vanish we can simply de ne the corresp onding
motivic cohomology groups to b e zero This is the case if X is a regular scheme
The long exact sequence for relative cohomology B together with the ab ove
remarks on the op eration give a long exact sequence for relative motivic cohomology
m m m m
H Y A H X A H Y rel X A H Y A
M M M M
Lemma B Let X b e a space degenerate ab ove some simplicial degree We
assume the conditions of the previous de nition Fix an integer j There is a coho
mological sp ectral sequence with starting terms
t
H X j for s j t
s
s t
M
E
else
s t
It converges to H X j for j s t
M
Pro of Consider the skeletal sp ectral sequence B a with co e cients in the space
K It reads
q p
p q q
X K H X K E H
p
sT sT
for p By carefully checking the construction of the sp ectral sequence we see that
p q
are induced by functoriality in the rst argument Hence they all di erentials d
r
Documenta Mathematica
Classical Motivic Polylogarithm
are morphisms of mo dules For q p the limit terms are also mo dules and
p q p q
by construction the morphisms e E are compatible with this structure They
are isomorphisms for q p Note however that we do not get enough information
p p
on the limit terms on the p q line Convergence only implies that e injects into
p p
E We want to show that it is even a bijection In order to see this we consider the
skeletal sp ectral sequence with co e cients in the sp ectrum K The sp ectral sequences
agree where the rst is de ned in particular convergence of the second sp ectral se
quence implies our isomorphism There is an issue here with the H group structure
A priori the two sp ectral sequences use di erent group laws But on all initial terms
they give the same addition and hence also on all higher terms
Now we take Adams eigenspaces By re indexing s p t q j we get a coho
mological sp ectral sequence as stated Note that we use the terms b elow the p q
diagonal to compute the terms on it but we do not consider their limit terms
The same sp ectral sequence also shows that the conditions in the de nition of
motivic cohomology hold if X is a space constructed from schemes and degenerate
ab ove some degree
The next thing we need is pushout at least for certain closed immersions and a
Riemann Ro ch theorem Over a eld push forward was de ned by de Jeu in Jeu
We adapt his metho d to more general bases and formalize the geometric situation
Definition B Let S b e a regular irreducible No etherian a ne scheme Let X
b e smo oth and quasi pro jective over S A nite diagram D over X is a category
X
of nitely many smo oth quasi pro jective S schemes with nal ob ject X such that all
Y Y are nite sets and such that all morphisms in D are of nite Tor Mor
X D
X
dimension
we mean the category of all nite disjoint unions of By the small Zariski site Zar
D
X
op en subschemes of ob jects in D with the induced morphisms b etween them It is
X
equipp ed with the Zariski top ology The corresp onding top os will b e denoted T
X
An easy case of such a diagram is a single morphism Y X that meets the
conditions
We consider the following situation Let i Z X b e a closed immersion of
smo oth quasi pro jective S schemes and D a nite diagram over X We assume the
X
following conditions corresp onding to the ones formulated by de Jeu in Jeu
TC For all X in D the pullback X Z is S smo oth If f X X is a
X X
morphism in D then in the cartesian diagram
X
Z X Z X
X
f f Z
X
y y
i
Z X Z X
X
the maps f and i are tor indep endent i e
k
O O Tor
X Z
O
X
k
for k Tor denotes the sheaf of tor groups
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Lemma B The pullback D of D by Z satis es the conditions for a nite
Z X
diagram over Z
Pro of Finite Tor dimension in D follows from Tor independence and the same prop
Z
erty in D
X
Let Y b e a space in sT Let j U X b e the op en complement of Z in X
X
Let Y U b e the p ointed version of j j Y i e the sheaf asso ciated to the presheaf
X
Y V if V U X
V
else
It is a space in sT Let Y Z i Y a space in sT If Y is constructed from
X X Z
schemes then so are Y U and Y Z The scheme comp onents are given by
X X
the base change with U or Z resp ectively Note that i Y U is empty i e only
X
consists of the base p oint
Proposition B de Jeu Let i Z X b e a closed immersion with op en
complement U Let D b e a nite diagram over X such that TC holds with resp ect
X
to i Then for Y sT
a There is a natural pushout map
k k
Y K Y Z K H H
X
sT sT
X Z
b Let Y b e a space in sT which is constructed from schemes We assume that it
X
is degenerate ab ove some simplicial degree Then
Y Z C Y U Y Z
X X X
and the pushout
k k
Y rel Y U K Y Z K H H
X X
sT sT
X Z
is an isomorphism
let M V b e the category of all coherent Pro of For an ob ject V of the site Zar
D
X
sheaves on V In it let P V D b e the sub category of those sheaves F satisfying
X
j
0
F O Tor
V
O
V
for all j and all V V in D Note that there are only nitely many conditions
X
as our diagram is nite The nice thing ab out P V D is that it is contravariantly
X
functorial Hence Quillen s B QP D lo op space of the classifying space of the
X
Q construction de nes a presheaf of simplicial sets on the site by Q x It is
here where we use the fact that all schemes are quasi pro jective Let B QP b e the
X
space in sT de ned by its shea cation By Quillen s Resolution Theorem Q
X
Thm Cor p there is a weak equivalence of spaces B QP K Basically
X
X
this is the fact that K theory and K theory agree for regular schemes
We also have the space B QP in sT For the closed immersion i V Z V
Z X
Z
the pushout i is exact on the category of coherent sheaves Because of TC it maps
Documenta Mathematica
Classical Motivic Polylogarithm
the the sub category P V Z D to P V D In fact we get a morphism of spaces
Z X
in sT
X
i
B QP i B QP
X Z
Using the weak equivalences to K this de nes a map in the homotopy category
i
K i K
X Z
If Y is a space in sT then we get the map in a as
X
k k k
Y K i i Y i K H i Y K H H
X Z Z
sT sT sT
X X Z
In the sp ecial case of a scheme Y part b is nothing but Quillen s pushout isomorphism
K i Y K Y rel Y U
n n X
for regular schemes Q x Prop recall that all schemes in the site are regular
This generalizes to the case of spaces constructed from schemes by the skeletal sp ectral
sequence
Lemma B Consider a cartesian diagram of smo oth quasi pro jective S schemes
0
i
X Z
f f
X Z
y y
i
Z X
where i is a closed immersion Let D b e a nite diagram on X Assume that the
X
0
de nes a nite diagram over X and that b oth i and i satisfy TC pullback D
X
We also assume that for all V in D the maps
X
V X V
X
and
V Z V
X
are tor indep endent
Then for all spaces Y in sT there is a commutative diagram
X
0
i
k k
H f i Y K H f Y K
sT Z sT X
0 0
Z X
x x
f f
X Z
i
k k
H H i Y K Y K
sT sT
Z X
Pro of We have to re ne the categories P V D used in the pro of of B further
Z
Let P V D b e the sub category of P V D of those coherent sheaves F satisfying
Z Z
j
0
O F Tor
Z
O
Z
Documenta Mathematica
Annette Huber Jorg Wildeshaus
is again weakly equivalent to K By Q x there The induced space B QP
Z
Z
is a commutative diagram of spaces in sT
X
i f B QP f P
0 0
X X
Z X
x x
i B QP B QP
Z X
This proves the lemma
We also need the following lemma from algebraic geometry
Lemma B Supp ose we are given a cartesian diagram
0
i
Z X
f
y y
i
Z X
of smo oth S schemes where i is a closed embedding then the blow up of X in Z is
the base change by f of the blow up of X in Z provided i and f are tor indep endent
n
Pro of In order to see this note that by EGAI I we have to check that f I
n n
0
is isomorphic to J where I is the sheaf of ideals of Z in X and J the O I
X O
X
one of Z in X This follows from tor indep endence in the case n Note that in
n n
general we have a surjection f I J Let K b e the kernel Pull back by f is
n
right exact i e we have an exact sequence
f I J f I I
Together with the ab ove surjectivity this implies f I I J J As X resp ec
tively X are regular and Z resp ectively Z are lo cally given by regular sequences the
structural theorem Ha II Theorem A e implies
n n n n
f I I J J
n n k n n k
By the snake lemma K K is surjective and hence f I I J J
n n
for all k But then
n n k n n n k n n k n
f I lim f I J f I lim f I im f I lim J J J
Push forward is not a ring morphism but it do es resp ect the ltration up to a
shift at least under go o d conditions This is made precise in the following Riemann
Ro ch Theorem which is a slight generalization of de Jeu s in Jeu He considers
a sp ecial type of diagram and restricts to a base eld De Jeu imitates the pro of in
T Theorem which is over a eld However his arguments work for our base as
well Indeed the original article Sou Thm treated the more general case
Documenta Mathematica
Classical Motivic Polylogarithm
Theorem B Grothendieck Riemann Roch Let S b e a regular irre
ducible No etherian a ne scheme S Let i Z X b e a closed immersion of
constant co dimension d of quasi pro jective smo oth S schemes For X Z let
td Gr K b e the usual Todd classes e g T p Let a nite diagram
Q
D b e given that satis es the conditions TC with resp ect to i Finally let Y b e a
X
space constructed from schemes in sT
X
a The homomorphism i K i Y K Y has degree d with resp ect to
n Q n Q
the ltration i e
i
j d j
F K Y F K i Y
n Q n Q
b The following diagram commutes
td Z ch
K i Y Gr K i Y
n Q n Q
i i
y y
td X ch
K Y Gr K Y
n Q n Q
Remark td is a unit with augmentation Hence the horizontal maps in b are
isomorphisms
Pro of We essentially have to prove classical Riemann Ro ch for the inclusion Z X
The conditions on our situation are chosen in a way that the diagrams we drag along do
not make any di culties Note also that we can replace Y by the cone of Y U Y
i e we can assume that all pushout maps are isomorphisms Having observed this
we can follow de Jeu s arguments in Jeu
The rst step is to prove the analogue of T Theorem or Jeu Prop osition
Riemann Ro ch without denominators We only sketch the idea Because
of functoriality B and the homotopy prop erty of K theory we can make the
transformation to the normal cone Hence we can assume without loss of generality
that i is a section of a pro jective bundle over Z The existence of the pro jection
p which is a left inverse of i allows to make explicit calculations All details of the
argument can b e found in Jeu when replacing K Y K X there by
K X K sT The necessary compatibility of blow up and base change is
X
guaranteed by the previous lemma
We then show that up to multiplication with the appropriate Todd class i has
the required b ehaviour with resp ect to Adams eigenspaces The argument is the same
as in Jeu Prop osition or T Lemma Now the theorem follows by the same
formal manipulations as in the pro of of T Lemma
Corollary B Let i Z X closed immersion of constant co dimension d
and Y b e as in the theorem Let U X r Z Then there is a natural lo calization
sequence
K Z Y K Y K U Y
m X Q m Q m X Q
K Z Y
m X Q
Documenta Mathematica
Annette Huber Jorg Wildeshaus
or in terms of motivic cohomology
i d
i i
H Z Y j d H Y j H U Y j
X
M M
M
i d
H Z Y j d
X
M
Pro of Part b of Theorem B implies that
M M
j j d
i Gr K Y rel Y U Gr K Y Z
m m
j N j N
i d
i
Y rel Y U j is an isomorphism i e H H Z Y j d
X
M
M
We consider the long exact sequence of relative K cohomology or relative motivic
cohomology for the op en embedding U Y Y We can use i to identify the
relative cohomology with cohomology of the closed complement
Only a few K groups are known However the ranks of the K groups of number
elds are understo o d
Theorem B Borel Let K b e a number eld with ring of S integers o
S
where S is a nite set of primes of K Let B Sp ec o As usual r is the number of
S
real places of K and r the number of complex places Then the motivic cohomology
has the following ranks
H B
M
H S r r B
M
H B n r n even
M
H B n r r n o dd
M
i
else H B j
M
Pro of The computation of K B and K B is classical Ba Ch IX Prop and
Ch X Cor The higher K groups for the ring of integers o were calculated by
K
Borel Bo Prop It follows from Quillen s computation of the K groups of
nite elds that the ranks are not changed by lo calizing at nite primes
B Cohomology of Abelian Sheaves
We now show how the usual cohomology theories t in the set up of generalized
cohomology This is well do cumented in the literature BrG G Jeu In the
case of a cohomology theory de ned by a pseudo asque complex of presheaves F
we compare the di erent p ossible p oints of view These are Zariski cohomology of
the asso ciated complex of sheaves generalized cohomology of the asso ciated space or
simply cohomology of the sections We always get the same cohomology groups B
and B If the complex of presheaves F is part of a twisted duality theory B
we de ne Chern classes from K cohomology of spaces to cohomology with co e cients
in F Finally we check compatibility of the lo calization sequence in K cohomology
with the one for cohomology of spaces with co e cients in F B
By a complex we always mean a cohomological complex Of course it can also
b e considered as a homological complex by inverting the signs of the indices
The Dold Pupp e functor M Thm attaches to a complex of ab elian groups G
which is concentrated in non p ositive degrees a simplicial ab elian group K G p ointed
Documenta Mathematica
Classical Motivic Polylogarithm
i
by whose homotopy groups K G agree with the cohomology groups h G
i
It induces an equivalence b etween the homotopy category of simplicial ab elian groups
and the homotopy category of complexes of ab elian groups concentrated in non
p ositive degrees By construction of the functor K there is a natural weak equivalence
of spaces
Cone K G K Cone G K G
and hence a natural map K G K G in the homotopy category of p ointed
simplicial sets which is a homotopy equivalence If G is an arbitrary complex of
ab elian groups let G b e the canonical sub complex in degrees less or equal to N
N
We put
K G K G N
N N
The natural map G N G N induces
N N
K G N K G K G
N N N
which is a weak equivalence This means the K G form an in nite lo op sp ectrum
N
whose homotopy groups re ect all cohomology groups of the complex
Definition B Let G b e a cohomological complex of sheaves of ab elian groups
on the big Zariski site The shea ed version of the ab ove construction yields an
in nite lo op sp ectrum of spaces K G with
i
h G K G
i
where the right hand side is the shea cation of the presheaf
U K G U
i
As a sp ectrum K G de nes generalized cohomology groups with indices in Z for
any space X
Proposition B Let G b e a b ounded b elow complex of sheaves on the big Zariski
site Let X b e a scheme Then
i i
H X K G H X G
sT ZAR
Pro of As G is b ounded b elow it has a b ounded b elow resolution by asque sheaves
Now the pro of pro ceeds as in BrG Prop The main ingredient is that K I is a
brant space if I is a asque sheaf
Definition B a Following BrG Sect a complex F of ab elian presheaves
on the big Zariski site is called pseudo asque if it has the Mayer Vietoris prop
erty i e for op en subschemes U and V of some scheme X we have a long exact
sequence of ab elian groups
i i i
h F U V h F U F V h F U V
i
h F U V
Documenta Mathematica
Annette Huber Jorg Wildeshaus
More precisely the square
F U V F U
y y
F V F U V
is homotopically cartesian
b Let F b e a complex of ab elian presheaves For the ob ject q U in T where U
is a scheme we put
F q U F U
Let X b e a space constructed from schemes Then we put
F X Tot F X
i i
the total complex of the cosimplicial complex F X
i i N
Taking the total complex of a bicomplex as in b of course involves a choice of
signs which we x once and for all Di erent choices of signs di er by a canonical
isomorphism of the total complex
Lemma B Let F b e a b ounded b elow pseudo asque complex of ab elian
presheaves Let F b e its shea cation Then
i i
H X K F h F X
sT
for all spaces X constructed from schemes
Pro of Let I b e a b ounded b elow asque resolution of F This is in particular a
pseudo asque complex of presheaves that is quasi isomorphic to F as a complex of
presheaves b ecause b oth compute Zariski cohomology of F As in the pro of of BrG
Theorem the simplicial sheaf K I is a brant resolution of K F Hence we can
assume without loss of generality that F itself is a complex of asque sheaves
For the case of a scheme X the lemma is the reformulation of BrG Theorem in the
easier case of simplicial presheaves that come from a complex of ab elian presheaves
In the general case
i
H X K F Hom X K F
i
sT
Hom ho colim X K F
i j
holim Hom X K F BouK XI I Prop
i j
i i
h Tot F X h F X
i
This means if we de ne a cohomology theory by a pseudo asque complex of
presheaves on the big Zariski site we can freely change from the p oint of view of
generalized cohomology to ordinary Zariski cohomology or cohomology of the sections
of the presheaf
Documenta Mathematica
Classical Motivic Polylogarithm
If X Y is a morphism of schemes we consider as usual its Cech nerve
cosk X Y i e the simplicial Y scheme given by
cosk X Y X X n fold pro duct
n Y Y
with the natural b oundary and degeneracy morphisms
Definition B We say that a morphism X Y of schemes has cohomological
descent for the cohomology theory given by the complex of ab elian Zariski sheaves G
if the natural morphisms
i i
H Y K G H cosk X Y K G
sT sT
are isomorphisms for all i Z
This is of course a very sp ecial case of the general notion of cohomological descent
Lemma B Let j U X b e an op en immersion with closed complement Y
Let F b e a pseudo asque complex of presheaves on ZAR with shea cation F
X
a There are natural isomorphisms
i i
H X rel Y K F H X j j F
sT ZAR
b If Y Y is a morphism with cohomological descent for F then we get a natural
isomorphism
i i
H X rel cosk Y Y K F H X j j F
sT ZAR
Pro of By B the left hand side of a is canonically isomorphic to the cohomology
of
F i
i
F C Y X Cone F X F Y
where the right hand side is the cone in the category of cohomological complexes We
assume without loss of generality that F is a asque complex The key p oint is the
short exact sequence of complexes of sheaves on X
j j F F i i F
It induces a canonical quasi isomorphism of complexes
j j F Cone F i i F
We now take R X of the right hand side Because F was assumed to b e
Zar
pseudo asque the morphism
Cone F X F Y Cone F X F Y
is a quasi isomorphism This last fact follows from B and B Of course it can
also b e proved even more easily in terms of complexes of ab elian groups rather than
Documenta Mathematica
Annette Huber Jorg Wildeshaus
simplicial ab elian groups In the case of a morphism Y Y with cohomological
descent the left hand side of the statement is by B given by the cohomology of
Cone F X F cosk Y Y
The natural morphism F Y F cosk Y Y is a quasi isomorphism by de nition
and Lemma B
L
Theorem B Gillet de Jeu Let F F i b e a pseudo asque com
i Z
plex of ab elian presheaves on the big Zariski site Assume that F de nes a twisted
duality theory i e the extra data of G Def exist and all conditions of lo c cit
Def are ful lled Then
There are Chern class maps of spaces
c K K F j j
j
They induce morphisms
i j
i
c H Y K F j Y K H
j
sT
sT
for all spaces Y in sT
If Y is a K coherent space then the total Chern class c is a morphism of
algebras on K cohomology of Y
Let i Z X a closed immersion of smo oth S schemes with op en complement
U The map i i F r j F r d j d required in G Def induces
Z X
push forward on generalized cohomology If Y is a space over X as in B
then the diagram
i
j d j
Gr K Y Gr K Y Z
n Q n X Q
c c
j
j d
y y
i
j n j d n
H Y Z K F j H Y K F j d
X Q Q
sT sT
is commutative
Pro of The construction of the Chern classes is G Thm Gillet s formulation is
for schemes but he constructs in fact a morphism of spaces lo c cit p so the
results hold for more general spaces see also GSo The assertion on the ring
structure is GSo Thm We sketch the idea Everything is de ned on the level
of co e cients so it do es not dep end on Y Compatibility with multiplication is G
Compatibility with op erators can b e checked on the level of universal Chern
i
classes i e for elements C H B Gl F i Now use the splitting principle
i N n
sT
G
The last part of the prop osition is a generalization of Gillet s Riemann Ro ch Theorem
G to spaces of our sp ecial type The pro of carries over by the same metho d as in
the pro of of Riemann Ro ch for K cohomology B Mutis mutanda the statement
can b e found in Jeu Lemma
Documenta Mathematica
Classical Motivic Polylogarithm
Remark This will allow to de ne regulator maps from K cohomology to the coho
mology theories we are interested in
Corollary B Let X Z d Y and F b e as in the theorem In addition assume
that F is pseudo asque Let U b e the complement of Y in X We abbreviate
Y Y U Y Y Z and F K F j Then there is a natural morphism
U X Z X j
of long exact sequences
i i d
i i
H Y j H Y j d H Y j H Y j
U Z U
M M
M M
y y y y
i i d
i i
Y F Y F H H Y F H Y F H
U j j U j Z j d
sT sT
sT sT
y y y y
i i d i i
h F j Y h F j d Y h F j Y h F j Y
U Z U
Pro of We start with the long exact sequences for relative cohomology B with
co e cients in the sp ectrum K and in the sp ectrum K F Their compatibility is
nothing but functoriality Relative cohomology is replaced by cohomology of Y Z
X
using B Finally we pass to graded pieces of the ltration Note that the indices
in the de nition of motivic cohomology are chosen in a way that they agree with the
indices of other cohomology theories under Chern class maps Equality of the last
two lines is B
Note that the last line has nothing to do with generalized cohomology or spaces
B Continuous Etale Cohomology
There are di erent ways of de ning continuous etale cohomology We will see that
they all give the same thing
Fix a number eld K and a prime l Let B b e an op en subscheme of Sp ec o l
K
where o is the ring of integers of K
K
Proposition B Deligne Ekedahl Let f Y X b e a morphism of B
b
schemes of nite type Then there are triangulated categories D X Z and D Y
l c
c
Z admitting the following there is a t structure whose heart are the constructible
l
l adic systems There are functors
b b
f f D Y Z D X Z
l l
c c
and
b b
f f D X Z D Y Z
l l
c c
having all the usual prop erties of Grothendieck functors
Pro of This is Ek Thm In the case B Sp ec o l the category was already
K
constructed in D
Documenta Mathematica
Annette Huber Jorg Wildeshaus
b
Remark D X Z should b e thought of as the b ounded derived categories of
l
c
b
constructible l adic sheaves on X By Ekedahl s construction D X Z is a sub
et l
c
category of a lo calization of a sub category of the derived category of the ab elian
N
category X Z By this notation Ekedahl means the category of pro jective sys
et l
n
tems of etale sheaves on X ringed by the pro jective system Z l The four functors
are de ned on the level of this last derived category Ekedahl then shows that they
b
induce well de ned functors on D X Z In the case B op en in Sp ec o l we
l K
c
get away with Deligne s more straightforward construction
Definition B Version a For k Z let Z k b e the constructible l adic
l
k
sheaf on B given by the pro jective system
n
l
b We de ne continuous etale cohomology of s X B by
i
H X k Hom b s Z s Z k i
l l
D X Z
cont
l
c
c If j U X is an op en immersion with complement Y we de ne relative contin
uous etale cohomology by
i
H X rel Y k Hom b s Z j s j Z k i
l l
D X Z
cont
l
c
b
X Z We de ne continuous etale d More generally let M b e an ob ject of D
l
c
cohomology of X with co e cients in M as
i
H X M Hom b s Z M i
l
D X Z
cont
l
c
This de nition allows to derive all the usual sp ectral sequences from the calculus
of the Grothendieck functors
Remark As checked in H x this de nition coincides with Jannsen s original one
in Jn sect In our case continuous etale cohomology with co e cients in a
n
X F b ecause all constructible l adic sheaf F is nothing but the naive lim H
n n n
et
n
X F are nite H
n
et
Let us now de ne continuous etale cohomology in a way that ts in with the
setting of the previous section
Definition B Version Consider the pro jective system of sheaves
k
on the big etale site over B Let I b e an injective resolution in the
n
l
n N
category of pro jective systems It is given by a pro jective system I of injective res
n
k
on the big etale site with split surjective transition morphisms Jn olutions of
n
l
By taking sections we get a pro jective system of complexes of Zariski presheaves
k
The functor R lim turns it into a complex F k of Zariski presheaves R
n
l
l
n N
For any space X put
i i
H X k H X K F k
l
cont sT
In particular if Y X is a morphism of spaces then we put
i i
H X rel Y k H C K F k
l
cont sT
Documenta Mathematica
Classical Motivic Polylogarithm
Lemma B If X is a B scheme then b oth versions of the de nition of continuous
etale cohomology agree canonically If Z X is a closed immersion then the same
is true for b oth de nitions of relative continuous etale cohomology
Pro of F X is nothing but an explicit version of the derived functor R lim R X
l
from the derived category of pro jective systems of etale sheaves to the derived category
of ab elian groups Hence the complex F X computes the rst version of continuous
l
etale cohomology In particular it has the Mayer Vietoris prop erty Hence we can
apply the lemmas of the previous section B and get
i i
H X K F h F X
l l
sT
To extend the result to relative etale cohomology we use essentially the same
argument as in B b
Remark When we say that the isomorphism is canonical we think in particular of
the following situation The cartesian diagram of schemes
0
0
j
i
Y U X
f
y y y
j
i
U X Y
x x x
g
0 0
0 0
j
i
Y X U
f j op en g i closed complements induces a map
f
i i
H X rel Y n H X rel Y n
cont cont
which is compatible with the identi cation If all schemes are smo oth and X inter
sects Y transversally then we also get the same long exact sequence
i d
i i
H X rel Y n d H X rel Y n H X rel Y n
cont cont cont
i d
H X rel Y n d
cont
using either de nition of relative cohomology
Lemma B If Y Y is a prop er covering i e a prop er and surjective map
then it has cohomological descent for continuous etale cohomology In particular if
Y X is a closed embedding and Y a prop er covering of Y then there is a natural
isomorphism
i i
H X rel Y j H X rel cosk Y Y j
cont cont
where the right hand side is taken in the sense of spaces
Pro of Cohomological descent is a consequence of the same descent for etale coho
mology with torsion co e cients prime to the characteristic of the schemes SGA I I
Exp Vbis By B b the second part follows
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Proposition B On the Zariski site of smo oth schemes over B the presheaf
F has the prop erties of a twisted duality theory There are regulator maps from
l
K cohomology to continuous etale cohomology
i i
H Y j H Y j
M cont
for all K coherent spaces Y They are compatible with pullback i e if f Y Y
is a map of K coherent spaces we get commutative diagrams
f
i i
H Y j H Y j
M M
c c
j j
y y
f
i i
H Y j H Y j
cont cont
If i Z X is a closed immersion of smo oth schemes constant co dimension d with
op en complement U and Y a space constructed form schemes over X as in B
then the regulator is compatible with pushout i e the diagram
i
n d
n
H Y j H Y Z j d
X
M
M
c
c
j d j
y y
i
n d
n
Y j H Y Z j d H
X
cont cont
is commutative
Pro of We restrict to smo oth schemes for simplicity We have to de ne the extra
structure from G and We put
d i
H X j H X d j
i
cont
for a d dimensional smo oth connected scheme Pull back on cohomology and pushout
on homology are induced from the functors on sheaves on the etale site We do not
n
this is actually one of Gillet s work out the details For a single etale sheaf
l
examples iii
There is really only one case when this regulator is understo o d
Lemma B Let K b e a number eld o b e its ring of integers and l a prime
K
Assume i k then Soul e s l adic regulator
k
Sp ec o l i K o l Z H
K i k K l
cont
agrees with the one obtained from Prop B
Pro of Put A o l Soul e s de nition in Sou is the comp osition
K
c lim
i k
k
H A Z l i lim K A Z l K A lim
i k i k
et
Documenta Mathematica
Classical Motivic Polylogarithm
where c is as in Sou I I There is a natural map of presheaves F i
i k l
R Z l i Hence in Gillet s de nition of Chern classes we get a commutative
diagram
c
i
k
H Sp ec A i K A
i k
cont
y
k
H Sp ec A Z l i
et
Hence we only have to consider nite co e cients Furthermore in this simple case of
a regular commutative ring we do not really need to consider the shea ed versions
and generalized cohomology Gillet s construction b oils down to a comp osition of the
Hurewicz map with universal Chern classes
c is de ned by the same type of comp osition Sou II For i k the map
i k
with the same universal Chern classes
By the de nition of K theory with co e cients we have a commutative diagram lo c
cit I I with X Z B Gl A
l
X X X Z l
n n n
h
q h h
y y y
l
H X Z H X Z H X Z q
n n n
For the prime compare also We
Theorem B Soule Let K b e a number eld o b e its ring of integers and l
K
any prime Let S b e a nite set of prime ideals of o and S S fl g Let o b e
K S
the lo calization of o at S The regulator map
K
i i
0
j Q H Sp ec o j c H Sp ec o
Q l S Q j S
cont M
is always injective and an isomorphism for i and j We have the following
b ehaviour for pairs of indices i j
j j Z isomorphism
j j mot coh vanishes l adic do es not in general
injective of nite co dimension
isomorphism j j
j j conjectured to b e isom i e etale coh to vanish
injective of nite co dimension
isomorphism i e b oth vanish j j
i j else b oth vanish
Pro of We have
i i
Sp ec o j H G Q j H
S Q S l
cont
where G is the Galois group of the maximal extension of K that is unrami ed
S
outside of S We rst check that these groups vanish for i By Mi I Cor
j
i
all H G are nite This means that the pro jective systems for varying n are
n
S
l
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Artin Rees We do not get a lim contribution to continuous cohomology Moreover
j
i
by lo c cit I c the H G for i are torsion This implies that their
n
S
l
i
pro jective limit is torsion In total we have vanishing cohomology H G Q j
S l
for i
The case i is trivial H G Q E Q where E are the S units while
S l S l S
0
o Sp ec o H Q For H G Q the S Brauer group the co dimension
0
S l S l
S M
is the same as in the case by Euler Poincar e duality cf the discussion in
Jn Lemma and Cor In the remaining cases neither motivic B nor
continuous etale cohomology Jn Lemma is changed by the inversion of S at
least up to torsion We assume S For o dd l the cases j and j for
j are Soul e s result in Sou Theorem Note that we are in the range where
the previous lemma applies
For l we have to re ne the argument On the level of Q co e cients we may by
p
note that the only prime which could Galois descent assume that K contains
p ossibly ramify in this quadratic extension has b een inverted and hence we get an
etale extension of rings By DwF Theorem and the succeeding remark we have
surjectivity even for l
To conclude we need to show that the Q vector spaces have the right dimension
Let j By Jn pro of of Lemma the dimension of
i
H Sp ec o j
K Q
cont
equals the corank of
i
H Sp ec o Q Z j
K
cont
By Sou and Prop osition this corank for i equals the rank of the
K group if and only if
H Sp ec o Q Z j
K
cont
is torsion This in turn follows from We Theorem
Finally we want to discuss Soul e s elements in K theory with co e cients Every
thing is in the setting of simplicial sets and sp ectra in the usual sense Generalized
r
cohomology do es not enter Let b e the sphere sp ectrum and l a prime p ower By
de nition of the Mo ore sp ectrum there is a co bration sequence
r
r
r
j
i
l
l
l
r
M S
l
Recall that for the ring of integers in a number eld A
K A Z lim K A Z l lim K M
n l n n l
The Mo ore sp ectrum has a unique pro duct for l For l r there are two
r
O Theorem a b pro jective systems of regular pro duct structures on M
l
and Lemma Together with the pro duct structure on K this de nes a pro duct on
K A Z for l
l
Documenta Mathematica
Classical Motivic Polylogarithm
For d we de ne R Z dl Recall Sou Lemma Sou
d
Soul e s construction of maps
primitive elements of K R Z K R Z
l d n l n Z l
The original statement is for o dd primes l but using the ab ove adic pro duct the
construction works without any changes for l For a primitive d th ro ot of unity
l r
r
satisfying Let lim K R Z l choose some lim
r r r r dl
r
K R Using the b e the pro jective system of Bott elements with j
r r
formalism of norm compatible units developed in Sou one lets denote the
l
pro jective system
r d n
lim K R Z l Z N
n r r
r
r
Remark It is not clear to the authors whether the adic Soul e elements dep end on
the choice of pro duct on the Mo ore sp ectrum By O pp the di erence
r
b etween the two regular pro ducts and on M is given by
r r
r
j j
i
r r r
M M S S M
Lemma B Let b e a ro ot of unity and n The restriction map from
Q Q n into H
l
cont
1
Gal Q Q
l
1
Q n H Q
l l
cont
1
Gal Q Q
l
n
A
r 1
lim Q H Q
r
Z l l l
l cont
l
r
1
Gal Q Q
l
r
n
l
A
1 1
Q Q Q lim
r
Z l l l
l
l
r
is injective
Pro of Note that the argument given in the discussion preceding WiIV Theorem
is incorrect since the transition maps
n n
m
m
H Q H Q
r r
l
l
cont cont
l l
are in general not injective The kernel of the restriction map is given by
1
Q Q n Q H
l l
cont
Since Q Q is prime to l we have to show that
l
1
Q Z n Q H
l l l
cont
1
Q is isomorphic to Z and is torsion But the Galois group G of Q
l l l
hence its rst cohomology equals the functor of coinvariants Our claim follows since
n
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Proposition B Let b e a xed d th ro ot of unity The l adic regulator
r K R Q H Q Q n
l n Z l d l
cont
b
takes to the cyclotomic element in continuous Galois cohomology
l
X
d n
A
r
l b
r
in the description of the last lemma de ned by Soul e and Deligne cf Sou page
D
Pro of If l is o dd then this is Sou Th eor emes and For l the same is true
using the prop erties of the adic regulator see We
B Absolute Hodge Cohomology
Let B Sp ec C or B Sp ec R in this section
In A a de nition of absolute Ho dge cohomology and relative cohomology for
general varieties over C was given The variant over R was A
By A resp A absolute Ho dge cohomology of smo oth varieties is given
p
B n functorially by Beilinson s complexes R
H
Lemma B These form a pseudo asque complex of presheaves on the Zariski site
of smo oth B schemes
U where U is a com Pro of By construction B they form a presheaf on pairs U
pacti cation with complement an NC divisor For more details cf H Prop
U we get the desired presheaf To say it is pseudo Taking the limit over all choices of
asque means that absolute Ho dge cohomology has the Mayer Vietoris prop erty In
the context of A and A it is a formal consequence of the existence of triangles
i i id j j for op en immersions j with closed complement i In the context of B
it follows from the Mayer Vietoris prop erty of De Rham cohomology and singular
cohomology
We now consider the corresp onding generalized cohomology
Definition B Version If X is a space over B then we de ne absolute
Ho dge cohomology by
i i
f
p
H B n X B n H X K R
p
H
H sT
If f Z X is a morphism of spaces then we de ne relative cohomology
i i
f
p
B n H X rel Z B n H Cone f K R
p
H
H sT
Lemma B There is a functorial isomorphism b etween b oth de nitions of absolute
Ho dge cohomology for a smo oth variety X If Y X is a closed immersion of smo oth
schemes then the same is true for relative cohomology
Documenta Mathematica
Classical Motivic Polylogarithm
Pro of Lemma B and Lemma B a
In order to get the same equalities at least for some singular varieties we have
to check a descent prop erty for Ho dge mo dules For this we need functoriality of
i i with values in complexes of Ho dge mo dules rather than ob jects in the derived
category
Lemma B Let X C b e smo oth and i Y X a closed reduced subscheme of
S
n
pure co dimension Let Y Y For I f ng and M MHM X let
i F
i
i Y Y X
I I i
i I
j U X r Y X
I I i
i I
M j j M MHM X
I I F
I
All Y are equipp ed with the reduced structure Then i i M de nes a functor
I I
I
b
fsubsets of f ngg C MHM X
F
Pro of As j is a ne b oth j and j map Ho dge mo dules to such Note that lo cally
I I
I
each Y is given by a function f on X The functor i i has an explicit description
i i I
I
for closed subschemes of the type Y given in the pro of of S Prop In fact
I
M M
0 0
M M M i i M
I I I
I
0 0 0 0
I I jI j I I jI j
where the complex sits in degrees less or equal to zero
Proposition B Let X C b e smo oth and i Y X a closed subscheme as in
the lemma Let Y Y q q Y and
n
s
Y cosk Y Y Y
i e
Y k factors Y Y
k Y Y
Then the functor s s de ned by the total complex of the cosimplicial complex
is isomorphic to i i s s
n n N
n
Pro of Note that
a
Y Y
k I
k
I f ng
where Y Y in the notation of the previous lemma Let M b e in MHM X
I F
fi i g
k
By the previous lemma we get indeed a cosimplicial complex hence s s M is a well
de ned complex of Ho dge mo dules Let Y b e the simplicial subscheme given by
a
s
k
Y Y Y
I
k
I i i i
k
Documenta Mathematica
Annette Huber Jorg Wildeshaus
By the Ho dge mo dule version of the combinatorial Lemma B the morphism
s s M s s M is a quasi isomorphism By de nition S
i i M M M
f ng
and this complex is canonically quasi isomorphic to the total complex of the constant
cosimplicial complex i i M It is easy to see that the natural morphism
Tot i i M s s M
is a quasi isomorphism
Corollary B Let X B b e smo oth Supp ose Y X is an NC divisor over B
i
all of whose irreducible comp onents are smo oth over B Then the group H Y B j
p
H
as de ned in A resp A is isomorphic to the generalized cohomology group
i
H Y B j and to the same noted group in B
p
H
Pro of The condition on Y ensures that Y is indeed a smo oth simplicial scheme It
gives rise to a space over B Cohomological descent for the co e cients as in B
implies cohomological descent for their global sections in the sense of B We can
use Y as the smo oth prop er hyper covering needed in Beilinson s de nition Equality
to the generalized cohomology version is again B
This is of course cohomological descent for a closed Cech covering We have
restricted to this case which is built into the very de nition of Ho dge mo dules for
simplicity There is no reason why there should not b e cohomological descent in the
same generality as for constructible sheaves
Lemma B Let X B b e smo oth and Z X a closed immersion of an NC
divisor all of whose irreducible comp onents are smo oth over B Let Z b e the smo oth
simplicial scheme of B then there is a canonical isomorphism
i i
H X rel Z B n H X rel Z B n
p p
H H
where we use the original de nition on the left and the second on the right
Pro of This follows by the general metho d of B b from the descent prop erty that
we have just established
Remark If we had checked cohomological descent in general then we would get
B for arbitrary varieties and B for arbitrary closed immersions
Theorem B On the site of smo oth schemes over B the presheaves
p
B n have the prop erties of a twisted duality theory There are regu R
H
lator maps from K cohomology to absolute Ho dge cohomology
i i
H Y j H Y B j
p
M H
for all K coherent spaces Y They are compatible with pullback i e if f Y Y
is a map of K coherent spaces we get commutative diagrams
f
i i
H Y j H Y j
M M
c c
j j
y y
f
i i
H Y B j H Y B j
p p
H H
Documenta Mathematica
Classical Motivic Polylogarithm
If i Z X is a closed immersion of smo oth schemes constant co dimension d with
op en complement U and Y a space constructed form schemes over X as in B
then the regulator is compatible with pushout i e the diagram
i
n d
n
Y j H H Y Z j d
X
M
M
c c
j
j d
y y
i
n d
n
H H Y Z B j d Y B j
p
p
X
H
H
is commutative
Pro of We use Gillet s metho d B All axioms of a twisted duality theory hold e g
H Ch Granted this the pro of pro ceeds as in the l adic case B
Remark Recall N that there is a natural transformation from absolute
Ho dge to Deligne cohomology The comp osition of the ab ove regulator with this
transformation was already constructed in Jeu
Theorem B Borel Let K b e a number eld with r real and r pairs of
complex embeddings into C We consider the ring of integers o as a scheme over Z
K
Then the Beilinson regulator
i i
H Sp ec o j R H Sp ec o R j
p
K Q K R
M H
is an isomorphism for all pairs i j It is injective of co dimension
r r for i j and injective of co dimension one in the case i j
Pro of Note that the cohomological dimension of the category of Ho dge structures is
The case i is trivial and the case is Dirichlet s classical result In Bo
the claim and much more is proved for the Borel regulator instead of the Beilinson
regulator By Rp Corollary the two regulators coincide up to a non vanishing
rational factor
B A Combinatorial Lemma
This section gives a purely combinatorial pro of why two conceivable de nitions of the
Cech nerve of a covering are homotopically equivalent This is well known at least
for op en coverings and Cech cohomology and probably in general But for lack of
nding an appropriate reference we work out the combinatorics here
Let C n b e the following simplicial set
k
C n f ng
k
b e the simplicial subset of with the obvious face and degeneracy maps Let C n
simplices whose entries are ordered by In fact this is the simplicial version of the
n simplex
Supp ose we are given a covariant functor from the category of subsets of
f ng to the category of sets We get simplicial sets by setting
A n A
k I
I C n
k
A n A
I
k
I C n
k
Documenta Mathematica
Annette Huber Jorg Wildeshaus
where A is the value of our functor on the set I fi i g Note that the elements
I k
of C n are ordered tuples but the value of A do es not dep end on the ordering
k I
Lemma B If the functor has constant value A then b oth simplicial sets have
the homotopy
A if i
A n
i
else
Pro of Obviously it is enough to consider the case A i e of the simplicial sets
C n C n themselves Both simplicial sets satisfy the extension condition M
rather trivially Hence we can use the combinatorial computation of the homotopy
groups given in M Def We immediately get the result
Proposition B For a general functor A the injection A n A n of simpli
cial sets is a weak homotopy equivalence
i
Pro of We lter the simplicial sets C n by the simplicial subsets F C n of sim
plices in which at most i di erent integers o ccur This induces a ltration of the simpli
i i i
cial sets A n Let G A n b e the co bre of the co bration F A n F A n
It consists of simplices in which precisely i di erent integers o ccur We argue by
induction on i for all functors A at the same time There is a long exact homotopy
sequence attached to the co bration sequence
i i i
F A n F A n G A n
i i
By induction it su ces to show that all co bres G A n G A n are weakly equiv
alent to the nal ob ject The co bre decomp oses into a union of simplicial sets
corresp onding to a di erent choice of i elements in f ng each If su ces to
prove acyclicity for one choice e g for the subset f ig Hence we only have to
i i i i
consider G A i G A i But this last co bre is isomorphic to G B i G B i
where B is the functor with constant value A For i it is easy to see that
f ig
i i
G B i G B i By B the quotients B n B n are acyclic for all n Us
ing the same co bration sequence as for A and the inductive hypothesis this implies
i i
that all G B i G B i are acyclic
Note that A could also b e a functor to the category of ab elian groups or to the
dual of the category of ab elian groups
References
Ba H Bass Algebraic K theory Benjamin Inc
B A A Beilinson Notes on absolute Ho dge cohomology in Applications
of Algebraic K theory to Algebraic Geometry and Number Theory Pro
ceedings of a Summer Research Conference held June in
Boulder Colorado Contemp Math vol Part I AMS Providence
pp
B A A Beilinson Higher regulators and values of L functions Jour So
viet Math pp
Documenta Mathematica
Classical Motivic Polylogarithm
B A A Beilinson On the derived category of p erverse sheaves in
Yu I Manin ed K Theory Arithmetic and Geometry LNM
Springer Verlag
B A A Beilinson Polylogarithm and Cyclotomic Elements typewritten
preprint MIT
BBD A A Beilinson J Bernstein P Deligne Faisceaux p ervers in B
Teissier J L Verdier Analyse et Topologie sur les Espaces singuliers
I Ast erisque So c Math France
BD A A Beilinson P Deligne Motivic Polylogarithm and Zagier Conjec
ture preprint
BD p A A Beilinson P Deligne Polylogarithms and Regulators the very
preliminary version handwritten preprint
BD A A Beilinson P Deligne Interpr etation motivique de la conjecture
de Zagier reliant p olylogarithmes et r egulateurs in U Jannsen S L
Kleiman J P Serre eds Motives Pro ceedings of the Research Con
ference on Motives held July August in Seattle Washington
Pro c of Symp in Pure Math Part I I AMS pp
BL A A Beilinson A Levin The Elliptic Polylogarithm in U Jannsen
S L Kleiman J P Serre Motives Pro ceedings of the Research Con
ference on Motives held July August in Seattle Washington
Pro c of Symp in Pure Math Part I I AMS pp
BLp A A Beilinson A Levin Elliptic Polylogarithm typewritten preprint
MIT
Bl S Blo ch Application of the dilogarithm function in algebraic K theory
and algebraic geometry in Pro c Int Symp Alg Geometry Kyoto
pp
BlK S Blo ch K Kato L functions and Tamagawa Numbers of Motives
in P Cartier et al eds The Grothendieck Festschrift Volume I
Birkh auser pp
Bo A Borel Stable real cohomology of arithmetic groups Ann Sc Ec
Norm Sup pp
Bo A Borel Cohomologie de SL et valeurs de fonctions z eta Ann Scuola
n
Normale Sup eriore di Pisa pp
Bo A Borel et al Algebraic D mo dules Perspectives in Mathematics
Academic Press
BouK A K Bous eld D M Kan Homotopy Limits Completions and Lo cal
izations LNM Springer
Br K S Brown Abstract homotopy theory and generalized sheaf cohomolo
gy Trans AMS pp
BrG K S Brown S M Gersten Algebraic K theory as generalized sheaf co
homology in Alg K Theory I LNM pp Springer
Brw W Browder Algebraic K theory with co e cients Z p in M G Bar
ratt M E Mahowald eds Geometric Applications of Homotopy The
ory I Pro ceedings Evanston LNM Springer Verlag pp
Documenta Mathematica
Annette Huber Jorg Wildeshaus
D P Deligne Equations Di erentielles a Points Singuliers R eguliers LNM
Springer Verlag
D P Deligne Th eorie de Ho dge I I Publ Math IHES pp
D P Deligne Th eorie de Ho dge I I I Publ Math IHES pp
D P Deligne La Conjecture de Weil I I Publ Math IHES pp
D P Deligne Le Group e Fondamental de la Droite Pro jective Moins Trois
Points in Y Ihara K Rib et J P Serre Galois Groups over Q Pro
ceedings of a Workshop held March at the MSRI Berkeley
California Springer pp
Den C Deninger Higher regulators and Hecke L series of imaginary quadratic
elds I I Ann of Math pp
DwF W G Dwyer E M Friedlander Algebraic etale K theory Trans Am
Math So c no pp
E H Esnault On the Lo day Symbol in the Deligne Beilinson Cohomolo
gy K Theory pp
EGAI I A Grothendieck J Dieudonn e El ements de g eom etrie alg ebrique I I
Publ Math IHES
Ek T Ekedahl On the Adic Formalism in P Cartier et al e ds The
Grothendieck Festschrift Volume I I Birkh auser pp
GH J Gamst K Ho echsmann Pro ducts in sheaf cohomology T ohoku
Math J pp
G H Gillet Riemann Ro ch theorems for higher algebraic K theory Adv
in Math pp
GSo H Gillet C Soul e Filtrations on higher algebraic K theory unpub
lished revised version in preparation
GSo H Gillet C Soul e Descent Motives and K Theory Preprint
Go eJr P G Go erss J F Jardine Lo calization theories for simplicial
presheaves preprint
Go A B Goncharov Polylogarithms and Motivic Galois Group in U
Jannsen S L Kleiman J P Serre Motives Pro ceedings of the Re
search Conference on Motives held July August in Seattle
Washington Pro c of Symp in Pure Math Part I I AMS pp
Gr M Gros R egulateurs syntomiques et valeurs de fonctions L p adiques
I I Invent Math pp
HZ R M Hain S Zucker Unip otent variations of mixed Ho dge structure
Inv math pp
Ha R Hartshorne Algebraic Geometry Graduate Texts in Mathematics
Springer Verlag
H A Hub er Mixed Motives and Their Realization in Derived Categories
LNM Springer
H A Hub er Mixed Perverse Sheaves for Schemes over Number Fields
Comp Math pp
Documenta Mathematica
Classical Motivic Polylogarithm
HW A Hub er J Wildeshaus The Classical Polylogarithm Abstract of a
series of lectures given at the workshop on p olylogs in Essen May
available under http www math uiuc edu K theory
Jeu R de Jeu Zagier s Conjecture and Wedge Complexes in Algebraic K
theory Comp Math pp
Jn U Jannsen Continuous etale cohomology Math Ann pp
Jn U Jannsen On the l adic Cohomology of Varieties over Number Fields
and its Galois Cohomology in Y Ihara K Rib et J P Serre Galois
Groups over Q Pro ceedings of a Workshop held March at
the MSRI Berkeley California Springer pp
Jn U Jannsen Mixed Motives and Algebraic K Theory LNM
Springer Verlag
Jr J F Jardine Simplicial ob jects in a Grothendieck top os in Applica
tions of Algebraic K theory to Algebraic Geometry and Number Theory
Pro ceedings of a Summer Research Conference held June in
Boulder Colorado Contemp Math vol Part I AMS Providence
pp
Jr J F Jardine Simplicial presheaves Journal of Pure and Appl Algebra
pp
Jr J F Jardine Stable homotopy theory of simplicial presheaves Can J
Math pp
Jr J F Jardine Generalized Etale Cohomology Theories Progress in
Mathematics Birkh auser Verlag
Kn D Kan On c s s complexes Am Journ Math pp
KNF M Kolster T Nguyen Quang Do V Fleckinger Twisted S units p adic
class number formulas and the Lichtenbaum conjectures Duke Math J
pp
Kr C Kratzer strucure en K th eorie alg ebrique Comm Math Helv
no pp
Ks M Kashiwara A study of variation of mixed Ho dge structure Publ
RIMS Kyoto Univ pp
Le M Levine Lambda op erations K theory and motivic cohomology in
V P Snaith ed Algebraic K theory pap ers from the nd Great Lakes
conference Canada March in memory of R W Thomason Provi
dence RI AMS Fields Inst Comm pp
Lo J L Lo day K th eorie alg ebrique et repr esentations de group es Ann
scient Ec Norm Sup e s erie t pp
M P May Simplicial ob jects in algebraic top ology Van Nostrand
Mi J S Milne Arithmetic Duality theorems Academic Press
N J Nekov a r Beilinson s Conjectures in U Jannsen S L Kleiman J P
Serre Motives Pro ceedings of the Research Conference on Motives held
July August in Seattle Washington Pro c of Symp in Pure
Math Part I AMS pp
Documenta Mathematica
Annette Huber Jorg Wildeshaus
Neu J Neukirch The Beilinson Conjecture for Algebraic Number Fields in
M Rap op ort N Schappacher P Schneider eds Beilinson s Conjec
tures on Sp ecial Values of L Functions Perspectives in Mathematics
Academic Press pp
O S Oka Multiplications on the Mo ore Sp ectrum Memoirs of the Fac of
Sciences Kyushu Univ Ser A Vol No pp
Q D Quillen Homotopical Algebra LNM Springer
Q D Quillen Higher Algebraic K Theory in Alg K Theory I LNM
Springer pp
Rm D Ramakrishnan Regulators algebraic cycles and values of L
functions in M R Stein R Keith Dennis eds Algebraic K Theory
and Algebraic Number Theory Pro c of a Seminar held at the East West
Center in Honolulu Hawaii on Jan Cont Math
pp
Rp M Rap op ort Comparison of the regulators of Beilinson and of Borel
in M Rap op ort N Schappacher P Schneider eds Beilinson s Con
jectures on Sp ecial Values of L Functions Perspectives in Mathematics
Academic Press pp
S Morihiko Saito Mo dules de Ho dge Polarisables Publ RIMS Kyoto
Univ pp
S Morihiko Saito Mixed Ho dge Mo dules Publ RIMS Kyoto Univ
pp
S Morihiko Saito Ho dge Conjecture and Mixed Motives I in J A Carl
son C H Clemens D R Morrison Complex Geometry and Lie Theory
Pro c of Symp in Pure Math AMS pp
Sch A J Scholl Height pairings and Sp ecial Values of L functions in U
Jannsen S L Kleiman J P Serre Motives Pro ceedings of the Re
search Conference on Motives held July August in Seattle
Washington Pro c of Symp in Pure Math Part I AMS pp
SGA A Grothendieck et al Rev etements Etales et Group e Fondamental
LNM Springer Verlag
SGA I I M Artin A Grothendieck J L Verdier et al Th eorie des Topos et
Cohomologie Etale des Sch emas T ome LNM Springer Verlag
SGA I I I M Artin A Grothendieck J L Verdier et al Th eorie des Topos et
Cohomologie Etale des Sch emas T ome LNM Springer Verlag
Sou C Soul e K th eorie des anneaux d entiers de corps de nombres et coho
mologie etale Inv Math pp
Sou C Soul e On higher p adic regulators in E M Friedlander M R
Stein eds Algebraic K theory Pro ceedings Evanston LNM
Springer Verlag
Sou C Soul e The rank of etale cohomology of varieties over p adic and
number elds Comp Math pp
Documenta Mathematica
Classical Motivic Polylogarithm
Sou C Soul e Op erations en K th eorie alg ebrique Can Jour Math
no pp
Sou C Soul e El ements cyclotomiques en K th eorie Ast erisque
pp
T G Tamme The Theorem of Riemann Ro ch in M Rap op ort N Schap
pacher P Schneider eds Beilinson s Conjectures on Sp ecial Values of
L Functions Perspectives in Mathematics Academic Press pp
We C Weibel Etale Chern classes at the prime in P G Go erss and J F
Jardine eds Algebraic K Theory and Algebraic Topology Pro c Lake
Louise NATO Adv Sci Inst Ser C Math Phys Sci Kluwer
Acad Publ pp
Wi J Wildeshaus Realizations of Polylogarithms LNM Springer
Verlag
WiI J Wildeshaus Mixed structures on fundamental groups in Wi pp
WiI I I J Wildeshaus Polylogarithmic Extensions on Mixed Shimura varieties
Part I Construction and basic prop erties in Wi pp
WiIV J Wildeshaus Polylogarithmic Extensions on Mixed Shimura varieties
Part I I The classical p olylogarithm in Wi pp
Annette Hub er J org Wildeshaus
Math Institut Math Institut
Einsteinstr Einsteinstr
M unster M unster
Germany Germany
huber math uni muenster de wildesh math uni muenster de
Documenta Mathematica
Documenta Mathematica