Doc Math J DMV
Chow Groups with Coefficients
Markus Rost
Received Octob er
Communicated by Alexander S Merkurjev
Abstract We develop a generalization of the classical Chow groups in
order to have available some standard prop erties for homology theories long
exact sequences sp ectral sequences for brations homotopy invariance and
intersections The basis for our constructions is Milnor s K theory
Mathematics Sub ject Classi cation Primary C
Contents
Summary
Conventions and Notations
Introduction
Cycle Premo dules
Cycle Mo dules
The Four Basic Maps
Compatibilities
Cycle Complexes and Chow Groups
Acyclicity for Smo oth Lo cal Rings
The Cycle Mo dules A M
q
Fibrations
Homotopy
Deformation to the Normal Cone
The Basic Construction
The Pull back Map
Intersection Theory for Fibrations
Pro ducts
References
Documenta Mathematica
Markus Rost
Summary
The pap er considers generalities for lo calization complexes for varieties Examples of
these complexes are given by the Gersten resolutions in various contexts in particular
in K theory and in etale cohomology The pap er gives a general notion of co e cient
systems for such complexes the so called cycle mo dules There are the corresp onding
complexes of cycles with co e cients and their homology groups the Chow groups
with co e cients For these some general constructions are developed prop er push
forward at pull back sp ectral sequences for brations homotopy invariance and
intersection theory
If one sp ecializes the material to the case of Milnor s K theory as co e cient
system one obtains in particular an elementary development of intersections for the
classical Chow groups This treatment is somewhat di erent to former approaches
The main to ol is still the deformation to the normal cone The ma jor di erence is
that homotopy invariance is not established alone for the Chow groups but for the
cycle complex with co e cients in Milnor s K Theory This enables one to keep
control in b ered situations The pro of of asso ciativity of intersections is based on a
doubled version of the deformation to the normal cone
Conventions and Notations
We work over a ground eld k and a base scheme B Sp ec k The word scheme means
a lo calization of a separated scheme of nite type over k This includes schemes of
nite type over a eld nitely generated over k From Section on all schemes are
of nite type over a eld Moreover all schemes and morphisms are de ned over B
with exceptions in Section The letter M stands from Section on for a cycle
mo dule If not mentioned otherwise it is de ned over B in Sections or over X
in Sections
fxg of x in X For x X we denote by dim x X the dimension of the closure
and by co dim x X the dimension of the lo calization X The set of p oints of X of
x
p
dimension resp co dimension p is denoted by X resp X We make free use
p
of some basic facts from commutative algebra and refer for this to Hartshorne
Matsumura and in particular to Fulton App A App B
p
In Sections and we use the sp ecial notation X Y for certain
maps b etween the cycle complexes This is explained in
Documenta Mathematica
Chow Groups with Coefficients
Introduction
The classical Chow groups CH X of p dimensional cycles on a variety X may b e
p
de ned as the cokernel of the divisor map
a a
d
x Z
x X x X
(p+1) (p)
Here X is the set of p oints of X of dimension p and x is the residue class eld
p
of x This pap er studies complexes C X M of the following type
a a a
d d d d
M x M x M x
x X x X x X
(p+1) (p) (p 1)
Here M is what we call a cycle mo dule This is a functor F M F on elds to
ab elian groups equipp ed with four structural data the even ones restriction and
corestriction the o dd ones multiplication with K and residue maps for discrete
valuations Moreover there is imp osed a list of certain rules and axioms A particular
example of a cycle mo dule is M K given by Milnor s or Quillen s K ring
K F Z F K F
Other examples are provided by Galois cohomology sp eci cally
a
n n
M F H F D
r
n
with D a Galois mo dule over a ground eld k with char k prime to r
The complex C X M is called the chain complex of cycles on X and its ho
mology groups A X M are called the Chow groups of X with co e cients in M
p
The Chow groups A X M enclose various familiar ob jects The classical Chow
p
group CH X is a direct summand of A X K The E terms of the lo cal
p p
global sp ectral sequences in etale cohomology and in Quillen s K theory are of type
A X M For prop er smo oth X of dimension d the group A X M is a birational
p d
invariant the M valued analogue of unrami ed Galois cohomology
The pap er develops some basic constructions for the cycle complexes
C X M and the Chow groups A X M for schemes X of nite type over a eld
There are prop er push forward at pull back and homotopy invariance Moreover
intersection theory is available for regular imbeddings and morphisms to smo oth
varieties there is a pull back map Finally for a morphism X Z there is a
sp ectral sequence
E A Z A M A X M
p q p q
p q
Here the A M are certain cycle mo dules obtained from taking homology in the
q
b ers All the mentioned functorial b ehavior extends for appropriate b er diagrams
to the cycle mo dules A M and the sp ectral sequences
q
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Markus Rost
The constructions are carried out on complex level in a p ointwise manner
The treatment has some parallels to a standard development of homology of CW
complexes This analogy should not b e taken to o serious but may give a rst impres
sion ab out the sort of technicalities In this picture our cells are just all p oints of
the variety in question The patching data for the cells are given by the geomet
ric valuations on the residue class eld of one p oint having center in another p oint
The appropriate lo cal co e cient systems are the cycle mo dules However the nature
of these co e cient systems is more complicated than in top ology First of all their
ground ring is provided by Milnor s K theory of elds Moreover b esides the usual
functorial b ehavior there is need for transfer maps basically b ecause one has to deal
with non algebraically closed elds and there are residue maps for valuations to give
passage from one p oint of a variety to its sp ecializations
The material of this pap er grew out from considerations concerning the bijectivity
of the norm residue homomorphism and Hilb ert s Satz for Milnor s K There the
n
computation of the Chow groups of certain norm varieties and quadrics plays an
imp ortant role As a general technique see also Karp enko and Merkurjev we
used a sp ectral sequence for morphisms X Z relating the Chow groups of the
total space to something like the Chow groups of the base with co e cients in the
Chow groups of the b ers moreover these sp ectral sequences should b e compatible
with intersection op erations The goal of the pap er was to present an appropriate
framework in a fairly direct manner
With the remarks following we have tried to draw the line of development of
the pap er In the discussion of intersection theory we restrict for simplicity to typi
cal situations and with Milnor s K theory as co e cient system although the actual
treatment is more general
Even if one is interested in classical Chow groups alone one is led to consider
some more general versions of Chow groups To start with a simple situation let
Y X b e a closed subvariety Then there is an exact sequence
CH Y CH X CH X n Y
p p p
For concrete computations as well as for general considerations there app ears the
problem to extend this sequence to the left in a reasonable way by a sort of higher
variants of Chow groups Similarly let X Z b e a morphism of varieties and
try to relate the Chow groups of X with the Chow groups of Z and of the b ers
When working within the classical Chow groups alone there will b e no go o d answer
in general
In this pap er the approach to these problems is provided by Milnor s K theory
For a variety X one forms for n Z the complex C X n with
a
C X n K x
p n p
x X
(p)
}
The complex of cycles with co e cients in Milnor s K theory to b e considered later splits up as
C X n according to the grading of Milnor s K ring a direct sum C X K
n
Documenta Mathematica
Chow Groups with Coefficients
where K F is Milnor s n th K group of a eld F The homology groups of the
n
complex C X n are denoted by A X n For n p it ends up with
p
a a a
d d d
K x K x K x
x X x X x X
(p+2) (p+1) (p)
and one has CH X A X p
p p
Then for a subvariety Y X there is a long exact sequence
A X n Y n A Y n A X n A X n Y n
p p p p
Moreover let X Z b e a morphism The ltration of the set X given by
p
the dimension of the image gives rise to a ltration of the complex C X n The
corresp onding E sp ectral sequence lo oks like
a
E A X n p A X n
q z p q
p q
z Z
(p)
with X X Sp ec z
z Z
A ma jor problem in intersection theory is to pro duce for a regular imbedding
q
f X X a pull back map f on the Chow groups having the geometric meaning of
intersecting cycles on X with X For a general account on intersections we refer to
Fulton
These maps are in the actual context of type
q
f A X n A X n d
p p d
with d co dim f
q
In the pap er the maps f are de ned by rst constructing homomorphisms of
complexes
I f C X n C X n d
d
and then passing to homology In a b ered situation that is f lies over some map
Z Z with appropriate smo othness conditions the maps I f can b e chosen
to resp ect the ltrations thereby inducing homomorphisms on the corresp onding
sp ectral sequences
As the reader might guess the maps I f cannot b e de ned canonically in
terms of f Namely I f gives in particular a lift of the classical pull back map
q
f CH X CH X to the cycle groups But if a cycle W on X do es not
p p d
meet X prop erly there is in general no way to de ne W X by a canonical cycle
It may b e surprising that one can handle with such pull back maps I f on
complex level in a reasonable way Therefore we will discuss here the nature of these
maps in some detail
When working with the complexes C X n it turns out that the necessary
constructions can b e describ ed in terms of four basic op erations These called the
four basic maps are of the following type
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Markus Rost
For a morphism f X Y there is a push forward map
f C X n C Y n
p p
For a morphism g X Y with b er dimension s there is a pull back map
g C Y n C X n s
p p s
Moreover there is multiplication with K for a global unit a on X there is a
map
fag C X n C X n
p p
given by p ointwise multiplication with a x x K x
Finally for a closed immersion there is a canonical b oundary map
C X n Y n C Y n
p p
All these maps are de ned in a p ointwise manner If f is prop er and g is at the
maps f and g commute with the di erentials of the complexes One uses f also
for op en immersions f and g also for closed immersions g then f and g are just
the corresp onding pro jections and don t commute with the di erentials The maps
fag and anti commute with the di erentials
In fact the four basic maps are enough to de ne intersections on complex level
by their very de nition the maps I f are sums of comp ositions of the four basic
maps For the construction of the I f the rst ma jor to ol is the deformation to the
normal cone This yields a canonical deformation map
C X n C N n
where N is the normal cone of f The next step is to de ne for a vector bundle
V X of dimension d a homotopy inverse
C V n C X n d
d
to the pull back map It is at this place where one needs some extra noncanonical
choices The choice to b e made is at most that of what we call a co ordination
of This is a strati cation of X together with bundle trivializations on the strata
In the end there is a canonical pro cedure which starts from the choice of a co
ordination of the normal bundle of f and yields a map I f as desired de ned in
terms of the four basic maps Di erent choices lead to homotopic maps I f with
the homotopies again expressible in terms of the four basic maps In a b ered situ
ation one may arrange things to end up with ltration preserving maps I f Once
having made the necessary choices the construction is quite functorial For example
it is compatible with resp ect to base change and lo calization In order to establish
functoriality namely I f f should b e homotopic to I f I f if necessary under
a ltration preserving homotopy we use a kind of doubled deformation space
Documenta Mathematica
Chow Groups with Coefficients
The viewp oint of the pap er is to put the four basic maps in the center In
particular the maps fag are treated as if they were a kind of morphisms in their own
right of equal rank as the more familiar push forward and pull back maps This has
at least technical advantages For example in order to check various compatibilities
q
concerning the maps f it is very convenient to reduce to a separate treatment of the
four basic maps
The reader may ask why we insist to stay on complex level although one is
interested mainly in the Chow groups Over some range this is quite natural from the
material However the pro of of homotopy invariance with resp ect to vector bundles
is much simpler for the Chow groups using the sp ectral sequences than for the cycle
complexes themselves where one has to construct explicit homotopy inverses
The ma jor motive for keeping the complex level throughout was to keep control
on the ltrations in b ered situations
Besides this we hop e that our metho d is of some interest concerning questions
e
for corresp ondences b etween arbitrary varieties To give an example let f X X
b e a prop er birational morphism with X smo oth Then there are pull back maps
e
I f C X n C X n
similar to the I f ab ove The I f are unique up to homotopy and have the standard
push forward map f as left inverse In particular I f identi es C X n as a
e
sub complex of C X n In the case of a blow up in a p oint x the choice to b e made
in the construction of I f is at most that of a system of parameters around x
We think of the maps I f as a sort of generalized corresp ondences One can make
this more precise in a further development which we call bivariant theory of cycles
There the four basic maps nd their place as morphisms of varieties in an appropriate
di erential category and the homotopy classes of the maps I f app ear rather as
morphisms in a category of varieties admitting pro ducts than just as homomorphisms
of complexes as in this pap er
The motive of introducing a general notion of co e cient systems for cycles ap
p ears when lo oking at the sp ectral sequence Its E terms are the homology
groups of complexes of type
a a
d d d
A X n A X n
q z q z
z Z z Z
(p) (p 1)
We interpret this by saying that the collection of functors with n Z
A n F A X Sp ec F n
q q Z
de ned on elds F over Z app ear as new co e cient systems This pro cess of creating
co e cient systems may in fact b e iterated
Therefore it seems convenient to have available some appropriate general notion
of co e cient systems The class considered in this pap er is provided by the notion
of what we call cycle mo dules Its de nition is formal and somewhat ad ho c The
imp ortant thing for us is that it contains standard functors like Milnor s or Quillen s
K theory and Galois cohomology as indicated ab ove that it is closed under pro cesses
like M A M and that it allows intersection theory Anyway it might b e of at
q
least heuristic interest that many general constructions intersections also the pro of
of acyclicity for smo oth lo cal rings can b e based on pure formal prop erties at least
if one starts from Milnor s K theory of elds
Documenta Mathematica
Markus Rost
Milnor s K theory is the fundamental base of the whole pap er This was at rst
suggested by our original problem Hilb ert s Satz for Milnor s K Besides this
n
Milnor s K theory seems to give the minimal framework needed in order to express
the considerations on intersections discussed ab ove By the way it seems likely that
the general metho d works also with Milnor s K theory replaced by the Witt ring of
quadratic forms of elds of characteristic di erent from
Milnor s K theory has a simple de nition in terms of generators and relations
Despite this fact it is by no means a simple and well understo o d functor Already
to de ne the norm homomorphisms takes some e ort An even more serious and in
general an unsolved problem is for example the computation of the torsion in Mil
nor s K groups These problems are related with Hilb ert s Satz Merkurjev and
Suslin and are part of a broader picture Beilinson conjectures motivic co
homology In this context there app ear other and more general higher versions of the
classical Chow groups than the groups A X n based on Milnor s K theory namely
p
motivic cohomology Blo ch s higher Chow groups and Suslin s singular homology and
also K cohomology Blo ch Quillen Suslin and Voevodsky Milnor s
K theory forms a central part of motivic cohomology and of Quillen s K theory In
fact in the smo oth case there are natural maps from the motivic cohomology of X
to the groups A X n and from A X n to the K cohomology of X b oth of which
are isomorphisms in some low degrees On the other hand motivic cohomology and
Quillen s K theory give rise to cycle mo dules in our sense In the case of Quillen s
K theory this is made more precise in Sections and These functors are def
initely necessary for a full understanding of Milnor s K theory For the purp ose of
this pap er however it turned out to b e enough to rely on elementary prop erties of
Milnor s K theory
The pap er may b e roughly divided in four parts In Sections the notion
of cycle mo dules is de ned Here we have lent some weight to a discussion of the
axioms In Sections the cycle complexes the Chow groups and their basic func
torial b ehavior are established Section is a side remark concerning the acyclicity
of Gersten type resolutions Sections treat the sp ectral sequences Sections
are concerned with intersection theory
I am indebted to Inge Meier for typesetting a rst version of this pap er
Documenta Mathematica
Chow Groups with Coefficients
Cycle Premodules
Cycle premo dules are roughly said functors on elds which have transfer are mo dules
over Milnor s K theory are equipp ed with residue maps for discrete valuations and
satisfy the usual rules The de nition is quite formal It forms the lo cal dimension
part of the notion of cycle mo dules A ma jor di erence to cycle mo dules is that cycle
premo dules do not have to ob ey laws involving an in nite number of valuations like
the sum formula for P
Cycle premo dules are de ned by a list of data and rules These are just usual
prop erties quite familiar to standard examples Equivalently one may de ne cycle
premo dules as the additive functors on a certain category which has an explicit de
scription in terms of Milnor s K theory and valuations see Remark This p oint
of view is p erhaps more satisfying It tells that our list of data and rules is in a sense
a complete list However it would take some e ort to establish the comp osition rule
in the category and we omit therefore a detailed discussion Moreover in order to
establish certain functors as cycle premo dules it is more convenient to refer to the
explicit lists of prop erties
The viewp oint of the four basic maps mentioned in the introduction would at
rst lead to functors F M F such that each M F is a mo dule over the tensor
algebra TF However the existence of norm maps and the homotopy prop erty leads
one to pass to mo dules over Milnor s K ring see Remark
We rst recall basic facts from Milnor s K theory Let F b e a eld By de nition
Milnor s K ring Milnor of F is
K F TF J
where F is the multiplicative group of F TF is the tensor algebra of F as ab elian
group and J is the two sided ideal of TF generated by the set
f a b j a b F a b g
The standard grading on TF induces a grading
a
K F K F
n
n
K F is the n th Milnor s K group of F By de nition K F Z and K F F
n
The elements of K F represented by tensors a a a F are called
n n i
symbols and denoted by fa a g The group law in K F is written additively
n n
e g fabg fag fbg There are the rules fa ag and fa bg fb ag see
Milnor In particular K F is an anti commutative ring with resp ect to the
natural Z grading
For a homomorphism of elds F E there is the ring homomorphism
K F K E
fa a g f a a g
n n
} M
In the literature one often uses the notation K F for Milnor s K ring while K F stands for
Quillen s K ring
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Markus Rost
If is nite there is the norm homomorphism
K E K F
preserves the Z grading Its comp onent Z Z in degree is multiplication with
deg E F In degree it is the usual norm map N E F for the nite eld
extensions has b een de ned by Bass and Tate with resp ect to a choice of
generators of E over F it is in fact indep endent of such a choice Kato For a
characterization of see the remark after Theorem
For a valuation v F Z we denote by O m v its ring maximal ideal and
v v
residue class eld resp ectively For nontrivial v there is the residue homomorphism
K F K v see Milnor is of degree It has the characterizing
v v
prop erties
f u u g fu u g
v n n
fu u g
v n
for a prime of v and for v units u with residue classes u v De ne
i i
s K F K v
v
s x f g x
v
v
is a ring homomorphism and is characterized by s
v
s fu u g fu u g
n n
v
s f u u g
n
v
Rules b etween the maps and the multiplicative structure of K are com
v
prised b elow in Theorem
Let B b e a scheme over a eld k recall our conventions In the following we
mean by a eld over B a eld F together with a morphism Sp ec F B such that F
is nitely generated over k By a valuation over B we mean a discrete valuation v
of rank together with a morphism Sp ec O B such that v is of geometric type
v
over k The latter means that O is the lo calization of an integral domain of nite
v
type over k in a regular p oint of co dimension Alternatively valuations of geometric
type may b e characterized by k O the quotient eld F and the residue class eld
v
v are nitely generated over k and tr deg F jk tr deg v jk
This geometric setting is convenient for our later purp oses We imp ose its re
strictive conditions from the b eginning in order to keep things straight For some
purp oses one may consider also arbitrary elds and valuations discrete of rank
and eventually not equicharacteristic over an arbitrary scheme B
In the following the letters stand for homomorphisms of elds over B and
all maps b etween various M F M E are understo o d as homomorphisms of
graded ab elian groups
Definition Let F B be the class of elds over B A cycle premodule M
consists of an object function M F B A to the class of abelian groups together
with a Z grading M M M or a Z grading M M and with the fol lowing
n
n
data D D and rules R a R e
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Chow Groups with Coefficients
D For each F E there is M F M E of degree
D For each nite F E there is M E M F of degree
D For each F the group M F is equipp ed with a left K F mo dule structure
denoted by x for x K F and M F The pro duct resp ects the
gradings K F M F M F
n m n m
D For a valuation v on F there is M F M v of degree
v
For a prime of v on F we put
s M F M v
v
s f g
v
v
R a For F E E L one has
R b For nite F E E L one has
R c Let F E F L with nite Put R L E For p Sp ec R
F
let L R p E R p b e the natural maps Moreover let l b e the
p p p
length of the lo calized ring R Then
p
X
l
p p p
p
R For F E x K F y K E M F M E one has with
nite in the pro jection formulae R b and R c
R a x x
R b x x
R c y y
R a Let E F and let v b e a valuation on F which restricts to a nontrivial
valuation w on E with rami cation index e Let w v b e the induced
map Then
e
v w
R b Let F E b e nite and let v b e a valuation on F For the extensions w
of v to E let v w b e the induced maps Then
w
X
v w
w
w
R c Let E F and let v b e a valuation on F which is trivial on E Then
v
R d Let v b e as in R c let E v b e the induced map and let b e a
prime of v Then
s
v
R e For a valuation v on F a v unit u and M F one has
fug fu g
v v
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Markus Rost
The maps are called the restriction and corestriction homomorphisms resp ec
tively We use the notations r c if there is no ambiguity
E jF E jF
Note that R c with y K E gives
R d For nite F E one has
deg id
Moreover R c implies
R e For nite totally inseparable F E one has
deg id
We consider M F also as a right K F mo dule via
nm
x x
for x K F and M F
n m
The maps are called the residue homomorphisms and the maps s are called the
v
v
sp ecialization homomorphisms It is easy to check that R e implies
R f For a valuation v on F x K F M F and a prime of v one has
n
n
x x s s x f g x
v v v v v
v v
s x s x s
v v v
If is another prime and u is the v unit with u then
0
s x s x fu g x
v v
From this and the rule R c it follows in particular that the rule R d holds for every
prime
More remarks concerning these formulae and the residue homomorphisms in general
are given b elow
All relevant cycle premo dules M known to us are Z graded with M for
n
n Within the general theory however there is need only for a Z grading and
we will understand this case if not mentioned otherwise
A morphism f B B de nes a transformation F B F B and the restric
tion of a cycle premo dule M over B to F B is a cycle premo dule over B It will
b e sometimes denoted by f M but mostly by M as well If B Sp ec R is a ne we
call a cycle premo dule over B a cycle premo dule over R If R is a eld we sp eak of
a constant cycle premo dule The reference to the base B will b e often dropp ed
Definition A pairing M M M of cycle premodules over B is given by
bilinear maps for each F in F B
M F M F M F
which respect the gradings and which have the properties P P stated below
A ring structure on a cycle premodule M is a pairing M M M which induces
on each M F an associative and anti commutative ring structure
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Chow Groups with Coefficients
P For x K F M F M F one has
P a x x
P b x x
P For F E M F M E M F M E one has with
nite in P b P c
P a
P b
P c
P For a valuation v on F M F M F and a prime of v one has
n
n
s s f g
v v v v v
v v
Note that P implies
s s s
v v v
Definition A homomorphism M M of cycle premodules over B of even
resp odd type is given by homomorphisms
M F M F
F
which are even resp odd and which satisfy with the signs corresponding to even resp
odd type
F E
E F
fag fag
F F
v F v
v
A unit a on B provides a simple example of a homomorphism of o dd type namely
fag M M given by fag fa g where a F is the restriction of a
F F F
The cycle premo dules over B together with the notion of homomorphism of
De nition form an Z graded ab elian category
Theorem Milnor s K theory K together with the data
multiplication
v
is a Z graded cycle premodule over any eld k With its multiplication K is a cycle
premodule with ring structure
This statement is a compact form of results in Bass and Tate Kato
Milnor we omit a detailed deduction
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Markus Rost
Theorem holds also in the setting of arbitrary elds and valuations discrete
of rank and with a restriction in R b see Remark b elow
Given the rings K F for each F in F Sp ec k the maps and are
v
uniquely determined by R b R c P P and
fag f a g
deg
fag N a
v
fag v a
v
v a v b v a v b
fa bg b a mo d m
v v
Here v denotes a normalized valuation v
This statement is trivial for the maps and for the uniqueness of the maps
v
see in particular Bass and Tate p
The multiplication maps of the K F mo dule structures on M F for each F give
rise to a pairing of cycle premo dules
K M M
Here the axioms P P and P follow from D R and R f
In order to establish a cycle premo dule it is convenient to use the following
reduction
Lemma For the validity of R d it su ces under presence of the other rules
of De nition to require R d for the case E v
Proof By R a the rule R d holds for E if it holds for some extension E of E with
E O Moreover by R a we may replace O by any unrami ed extension O with
v v
v
since the same residue class eld we don t want to pass to the henselization lim O
v
our elds should b e nitely generated over k Now by lifting a transcendence base of
v over E to O we may assume that v is nite over E Moreover we may assume
v
as ab ove Then v is totally inseparable that E is algebraically closed in any O
v
over E Supp ose p char F We argue by induction on v E Let a E
p
p
Then the extension v a is contained in v but not in any O such that E E
v
p
p
a has rami cation index p has the same residue class eld and of v to F F
v E v E Using R c R b R c and R e it is now easy to see that R d
holds for the pair v E if it holds for the pair v E use the fact that the norm of
a prime for v is a prime for v
The rest of this section will not b e used later within the general theory However
the following remarks may b e of at least heuristic interest and we will refer to them
partially in later side remarks
Remark There is the following p oint of view concerning R f See also Bass
and Tate Milnor remark at the end of p
For a valuation v F Z let
K v K F f m g K F
v
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Consider the ring homomorphisms
p K F K v
i K v K v
given by pro jection resp by the formula
i fu u g p fu u g
n n
for v units u There is an exact sequence
i
i
K v K v K v
with p Any prime gives rise to a section y p f g i y of
v
We put
M v K v M v
K v
Then there is an exact sequence
i
M v M M v v
and the splittings ab ove give for every a decomp osition of K v mo dules
M v M v M v
We de ne
p M F M v
p s p f g
v
v
Note that p is indep endent of the choice of One has p
v
Now R f may b e reformulated by saying that p is a mo dule homomorphism over
the ring homomorphism p Similarly one may understand P via pairings
M v M v M v
K v
Remark A particular consequence of R e is the fact that the subgroup
f m g M F
v
is killed whenever one passes to M v This seems to b e a reasonable condition
from a geometric p oint of view However note that the continuous Steinberg symbol
K Q Z corresp onding to the adic valuation on Q Milnor x maps
f g to the nontrivial element
Remark If one wants to consider arbitrary valuations discrete and of rank
one has to require in R b that the integral closure of O in E is nite over O This
v v
condition holds for geometric and for complete valuation rings see Serre By
lo oking at completions and using R c and R a one may then derive for arbitrary
valuations a formula
X
l
v w w
w
w
with certain integers l This remark applies in particular to Milnor s K theory
w
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Markus Rost
Lemma In the situation of R a let be a prime of v let be a prime of w
e
and let u be the v unit with u Then
s s fu g
w
v w
Proof First note that the validity of the statement do es not dep end on the choices
of and Moreover if E K F is an intermediate eld we may restrict to
consider the extensions K jE and F jK
If F jE is unrami ed e we may take and the claim follows from R a
and R a
After lifting a transcendence base of v over w to O we may therefore
v
assume that F is nite over E
If e F E case of total rami cation see Serre Chap I x we may
take N then u and the claim follows from R c and R b We have
now already covered totally inseparable extensions
For a separable nite extension F jE let LjE b e a Galois extension containing F
x some extension of v to L and let D LjF D LjE Gal LjE b e the decomp o
sition groups
D LjF
Then F L is unrami ed over F with the same residue class eld by R a
D LjE
we are reduced to consider the extension F jE The eld E L is contained
in F since it is unrami ed over E we know the claim for E jE and we are reduced
U
to consider F jE Let K L where
U f g Gal LjE j g acts trivially on v g
is the inertia group Then K jE is unrami ed and F jK is totally rami ed
Remark The rules and Lemma show that every comp osite of maps b e
tween various M F given by the data D D is a sum of comp osites of the form
x y
v v
1 r
This kind of normal form for comp osites can b e made more precise as follows There
e
is a category F with ob jects the class of arbitrary elds and with morphism groups
a a
b
Hom F E K H K v
K v
v
H
r
Here v runs through the valuations on F with value groups Z with lexicographical
order and with r The groups K v are de ned exactly as ab ove for r
Moreover H runs through those comp osites of v and E which are nite over E
b
and denotes the graded tensor pro duct
Restricting to the class F B and to geometric higher rank valuations one ob
e
tains a category F B The cycle premo dules over B may b e then characterized as
e
the additive functors on F B In this alternative de nition all the rules including
e
Lemma are hidden in the comp osition law of F
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Remark Galois cohomology as cycle premodule Any torsion etale sheaf
on B with the torsion prime to char k gives rise via Galois cohomology to a cycle
premo dule over B For simplicity we restrict here to the case B Sp ec k with k a
eld and to nite Galois mo dules over k For generalities of Galois cohomology we
refer to Serre Shatz
Let k b e a separable closure of k let r b e prime to char k let k b e the group
r
of r th ro ots of unity and let D b e a nite continous Gal k jk mo dule of exp onent r
For a eld F over k let F b e a separable closure containing k Then and D are
r
Gal F jF mo dules via Gal F jF Gal k jk Put
a
n n
e
H F D H F D
r
n
Here we use for a nite Galois mo dule C the notation
n n n
H F C H Gal F jF C lim H Gal LjF C
where L runs through the nite Galois sub extensions of F jF such that Gal F jL acts
trivially on C
e e e
H F Z r is a ring and H F D is a mo dule over H F Z r via cup pro ducts
e
The ob ject function H D on F k given by H D F H F D is in a
natural way a Z graded cycle premo dule over k This statement is just a collection
of well known prop erties of Galois cohomology In the following we restrict ourselves
to a description of the data D D The rules follow from standard prop erties of the
cohomology of nite groups and from standard rami cation theory
D and D For F E let F E b e some extension over k and let
Gal E jE Gal F jF b e the induced map De ne as the usual restriction
homomorphism induced from For nite de ne as the usual transfer homo
morphism induced from times the degree of inseparability E E F cf Serre
e
D The K F mo dule structure on H F D is given by cup pro ducts and the
norm residue homomorphism
e
h K F r K F H F Z r
F
h is the Z graded ring homomorphism which in degree is given by the Kummer
F
r
isomorphism F F H F For the rule h fag h f ag see for
r F F
example Tate or Remark
D Let E b e the completion of F with resp ect to v Then there is a natural
exact sequence
I Gal E jE Gal j
n
where I is the inertia group Put D D and consider the corresp onding
n
r
Ho chschild Serre sp ectral sequences
p q
p q p q
H H I D H E D E
n n
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Markus Rost
The cohomology of the inertia group I is given by H I D D H I D
n n n
q
Hom D D and H I D for q Serre Chap IV Hence
r n n n
the sp ectral sequences give rise to homomorphisms
n n
H E D H D
v n n
n n
Comp osing with H F D H E D de nes the desired maps
n n
n n
H D F H D
v
Remark Quil len s K theory as cycle premodule We denote by K F
K F Quillen s K ring of a eld F Hereby we understand the de nition K F
n n
n
BQ Mod F of Quillen with the pro duct as de ned in Grayson
n
Here Mod F is the category of nite dimensional F mo dules For generalities of
Quillen s K theory see also Grayson Srinivas
The ob ject function F K F de nes a Z graded cycle premo dule with ring
structure over any eld k Its data are given as follows
D and D One takes the pull back map resp the push forward map of
Quillen x where Sp ec E Sp ec F is the morphism corresp onding to
D One uses the natural homomorphism K F K F from Milnor s to
F BGL F and the com Quillen s K theory To de ne one may refer to K
n
n
putations BGL F H GL F Z K F BGL F H E F Z
K F Matsumoto s theorem see Milnor Another p ossibility is to de ne directly
a homomorphism F K F and then to check the rule fag f ag
using the arguments of Remark
D One uses the connecting map of the long exact lo calization sequence for O
v
Quillen x
The veri cation of the rules is omitted It is a lengthy but straightforward exercise
to deduce them from Grayson Quillen
}
According to the conventions made for the cup pro duct and the sp ectral sequence one may
have di erent signs in the pro duct rules for the di erentials This a ects rule R e so if necessary
one should replace by an appropriate sign dep ending alone on n
v
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Chow Groups with Coefficients
Cycle Modules
In this section we de ne the notion of a cycle mo dule and derive imp ortant prop erties
the homotopy prop erty for A and the sum formula for prop er curves Moreover we
give a simpli cation of the axioms for a constant cycle mo dule over a p erfect eld
The axioms of a cycle mo dule are basic for all further considerations Therefore we
have included discussions on various related prop erties to a much larger extent than
is actually needed in the following sections
Throughout the section M denotes a cycle premo dule over some scheme B recall
our conventions
For a scheme X over B we write M x M x for x X The generic
p oint of an irreducible scheme X is denoted by or If X is normal then for
X
x X the lo cal ring of X at x is a valuation ring let M M x b e the
x X
corresp onding residue homomorphism
For x y X we de ne
x
M x M y
y
x
as follows Let Z fxg If y Z then Otherwise let Z Z b e the
y
normalization and put
X
x
c
z
z j y
y
z jy
with z running through the nitely many p oints of Z lying over y
Definition A cycle module M over B is a cycle premodule M over B which
satis es the fol lowing conditions FD and C
FD Finite support of divisors Let X b e a normal scheme and M
X
Then for all but nitely many x X
x
C Closedness Let X b e integral and lo cal of dimension Then
X
x
M M x
X
x x
0
(1)
x X
where is the generic and x is the closed p oint of X
X
Many remarks and de nitions of Section are understo o d accordingly for cycle mo d
ules For example a homomorphism of cycle mo dules is a homomorphism of the
underlying cycle premo dules
Of course C has sense only under presence of FD which guarantees niteness in
the sum More generally note that if FD holds then for any X x X and M x
x
one has for all but nitely many y X
y
If X is integral and FD holds for X we put
a
d M M x
(1)
X
x X
x
(1)
x X
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Markus Rost
Sp ec F u is the a ne line In the following F denotes a eld over B and A
F
over F with function eld F u Pro ofs of Prop osition and Theorem are given
after Remark
Proposition Let M be a cycle module over B Then the fol lowing properties
H and RC hold for al l elds F over B
H Homotopy property for A The sequence
a
r d
M F M F u M x
1
x A
F
(0)
is an exact complex with r r
F u jF
RC Reciprocity for curves Let X b e a prop er curve over F Then
a
d c
M M x M F
X
x X
(0)
P
c is a complex c d with c
x jF
The prop erties FD C H RC are all what we need in further sections Axiom
FD enables one to write down the di erentials d of the complexes C X M axiom
C guarantees that d d prop erty H yields the homotopy invariance of the
Chow groups A X M and nally prop erty RC is needed to establish prop er push
forward For the material from Section on the reader may take H and RC just
as additional axioms of cycle mo dules and skip without much harm everything after
Remark b elow
For another example of the fundamental role of axioms like RC in formal de
nitions of functors on elds see also Somekawa
For integral X we put
ker M A X M ker d
X
x
(1)
x X
One may think of A X M as the group of unrami ed M valued functions on X
Theorem Let M be a cycle premodule over a perfect eld k Then M is a
cycle module over k if and only if the fol lowing properties FDL and WR hold for
al l elds F over k
FDL Finite support of divisors on the line Let M F u Then
for all but nitely many valuations v of F u over F
v
WR Weak reciprocity Let b e the residue map for the valuation of
F u jF at in nity Then
A A M
F
One implication here is obvious from Prop osition since trivially FD FDL
and RC WR Conditions FDL and WR are comparatively weak they deal
only with the a ne line and involve no corestriction maps For nonconstant cycle
mo dules i e B is not the sp ectrum of a eld we don t know any similar simpli cation
of the axioms FD and C
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Chow Groups with Coefficients
Further prop erties of cycle mo dules are
Co Continuity Let X b e smo oth and lo cal and let Y X b e the blow up
in the unique closed p oint x Then
A X M A Y M
In other words if v is the valuation corresp onding to the exceptional b er
over x then
A X M
v
E Evaluation In the situation of Co there is a unique homomorphism
ev A X M M x
evaluation at x such that
r ev s j A X M
v j x
v
0
for any prime of v
The validity of these two prop erties will follow from the construction of the pull back
q
maps f A X M A Z M for morphisms f Z X in Section Namely
q
the inclusion of Co is given by f with f Y X the blow up Moreover in E one
q
has ev f with f Sp ec x X the inclusion See also Remark b elow
Remark A basic example of a cycle mo dule over any eld k is Milnor s K
ring K Axiom FD follows as for classical divisors For H see Milnor The
validity of RC for X P is intrinsic to the de nition of the norm homomorphisms
in Bass and Tate Kato has used RC to prove C by passing to
completions
of Remarks and are Remark The cycle premo dules H D and K
cycle mo dules Axioms FD and C are contained in Blo ch and Ogus and in
Quillen x Sect resp ectively
For H D one may use here alternatively Theorem and Tsen s Theorem as
follows see also Serre FDL follows from the fact that every nite exten
sion of F u is rami ed only in nitely many places of F u jF One has trivially
H WR If F is separably closed then H follows from Tsen s Theorem i e
q
H F u for q see for example Shatz and the Kummer isomor
r
phism K F u r H F u To deduce H for arbitrary F one applies the
r
Ho chschild Serre sp ectral sequence for the extension F u jF u
Remark Probably the considerations of this section and of the whole pap er
may b e developed in characteristic also for a version of cycle mo dules which
are mo dules over the Witt ring of quadratic forms instead over Milnor s K ring A
transferring would b e not at all formal b ecause the residue maps for the Witt ring
dep end on choices of parameters
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Markus Rost
In the following pro ofs of Prop osition and Theorem we use the notations
A Sp ec F u A Sp ec F s t and Z A the lo calization of A at
s t
Moreover y z Z A denote the p oints with parameters s t resp ectively
We pro ceed in several steps
Step FD C WR Given A A M put
ftg t s M F s t
or more precisely ftg with F u F s t u t s Using R
and R one nds
for x Z n fy z g
x
ftg r
y
y jF
fug fsg
z z
r fug fsg r
z jF z jF
C and give
X
y
x
x y
(1)
x Z
Step FDL WR H Note that d r by R c Moreover any sp ecial
ization map for an F rational p oint on P is a left inverse to r by R d
Surjectivity of d For a closed p oint x A let
x
M x M F u
x
c fu u x g r
x u jF u x u j x
and let
X a
x
M x M F u
1
x
x A
( 0)
Then d id by R b R e
Exactness at M F u Given A A M put
ftg u t u M F u t
More precisely Let E F u let i E E t b e the homomorphisms over F with
i u u u u t and put ftg i
We compute for the valuations w of E t over E One nds easily
w w
for w But also by R d since the valuation at t restricts trivially
under i and and since the induced homomorphisms E coincide Hence WR
tells On the other hand one has
ftg i
ft u t g fu tg ftg
fug t u t g f
r fug
E jF
by making particular use of R d and R e note that t u t has residue class
So
r fug r M F
E jF E jF
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Chow Groups with Coefficients
Step FD H RC There is a nite morphism X P over F Using
this and R b one reduces to the case X P Then it su ces to check
X
x
c
u
u jF
1
u P
(0)
x
for as in Step This equation follows from the computation d id and
x
c
x jF
a consequence of R b and R d
The pro of of Prop osition is now complete We next consider the nontrivial
implication of Theorem We will refer at some places to Sections and but
only in a mild way Note that H is available by Step
n n n
Step FDL FD for X A Let p A A b e the n standard pro
i
jections Then
n
p A
i
i
n
where is the generic p oint of A FD follows from FDL applied to F
Step FD for X A WR C for X Z As in Step we have
X
x
M M F s M x
Z
1
x A
F (s)
(0)
with
x
c ft t x g r
x t jF s t x t j x
x
C holds obviously on M F s Let us verify C on the image of for xed x
By d id in Step we are reduced to check
y
x x
y
Let v run through the valuations on x t which restrict on F s t to the valuation
with parameter s and corresp onding to y Let v b e the restriction of v to x and let
fxg b e the center of v If c v then t x is a v unit with residue t c v c v
Supp ose fxg and let R b e the residue class ring of x lo calized at The
x
valuations v with c v restrict in a one to one manner to the valuations w of x
with R O For these one has v t x since t x is nilp otent in R sR
x w x x
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Markus Rost
In the following u runs through fxg fy g n f g One nds
X
x
ft t x g r c
y v
x t j x v j y
v
X
c ft t c v g r
v
v j y v j v
v c v
X
c r ft y g
v
v j y v j v
v c v
X
x
ft t u g r c
u t j u u t j y
u
u
x
ft y g r
y jF
y
Since t u the sum vanishes under and we are done by R d
Step Reduction of C to the case x O Let X b e as in C and
X
write X Sp ec R Lift a transcendence base of x over k to elements t R
i
and put K k t t R Then x is a nite extension of K Since k is
n
p erfect we may take here a transcendence base such that x is separable over K
Let X Sp ec R x let u X b e the canonical lift of x and let X b e the
K
lo calization of X in u We assume that FD holds for X and X Consider the pull
back along the at map X X see Section The induced map M x M u
is injective since x and u have the same residue class elds An application of R a
and R b see Prop osition shows that C holds for X if C holds for X
We know now in particular that C holds for every lo calization of A in some
closed p oint
Step Proof of FD There exists a generically nite separable rational map
n n
X A All but nitely many x X corresp ond to p oints of A The argument
k
of Step yields a reduction to a plane curve X over some eld K So consider the case
We may assume that X maps dominantly to Sp ec F s fxg for some x A X
x x
so that as in Step is de ned Put We have Moreover
x
only for nitely many u Step The closure of u x meets X only
u
in nitely many p oints Now since C holds for every lo calization of A we have
for all but nitely many w X A
w
Step Proof of C By Step we may assume that F x is contained in O
X
n
Cho ose a closed dimensional subscheme Y P such that X is the lo calization
F
n
of Y in a F rational p oint y We consider the generic pro jection from P to P
F F
n
More precisely let T b e the Grassmannian of co dimensional linear subspaces of P
F
n n
let E F T let H P b e the tautological subspace and let P n H P b e
E E E
a linear pro jection Then H Y and restricts to a prop er map p Y P
E E
E
Let D p p y Then D is the intersection of Y with a generic co dimensional
E
linear subspace passing through y Hence
D n fy g Y n Y
E
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In particular D Y fy g Now we consider at pull back along the base change
q Y Y followed by the push forward along p see Section One nds see
E
and of Prop osition for M
X
X X
u x
r p q
u
E jF
x x
p u
0
(1) (1)
u U x X
where U is the lo calization of P in p y The right hand side vanishes by Step
E
and r is injective since E jF is a rational extension
E jF
We conclude with some more considerations concerning the axioms of cycle mo d
ules These have b een included here more for illustration than for application In
order not to b e to o tiring we have taken here the freedom to b e a bit vague ab out
our actual assumptions
Remark In the datum D of cycle premo dules there is hidden a strong rule
namely the relation fa ag of Milnor s K theory The main justi cation within
this pap er for using Milnor s K theory is that it works well Asking naively one may
try to weaken D by requiring only the existence of bilinear pairings
K F M F M F
fag fag
and restricting to x K E y K F in R Then M F would b e a TF mo dule
However if one wants to develop a geometric theory one is in the end led to
pass to mo dules over Milnor s K theory A reasoning for this is given by the following
little game It refers in a mild way to the rules of cycle premo dules and to a part of
the homotopy prop erty H
Let M F let L b e an over eld of F let u L n f g and consider
u fug f ug r M L
LjF
Our aim is to conclude u for the case L F Assuming reasonable sp ecial
ization maps this follows from the generic case L F u with u a variable To treat
this case our strategy is to argue that u is unrami ed on the whole a ne line
Then by homotopy invariance u is constant An extra argument nally shows
u
To b e sp eci c rst a little calculation which provides by the way already the
divisibility of u referring only to the existence of norm maps and the pro jection
F u b e the function eld in the variable u and let L F u L formula Let L
n
0
with u u Then u N u and the pro jection formulae R b and R c
L jL
give
0
u fug fN u g r
L jL LjF
0 0
fug c f u g r
L jL L jF
n
0 0
c fu g f u g r
L jL L jF
0
n c u
L jL
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Markus Rost
u for all nite places v of LjF This is We want to conclude that
v
quite natural to assume as long as u and u are v units For the place at u
and similarly at u one may argue as follows
Let u M F b e the residue for the valuation of LjF at u
Similarly let u now with resp ect to the valuation of L jF at u A
change of variables u u shows But the ab ove computation and rule R b
yields n Taking n gives
Now H tells that u comes from M F that is u r for some
LjF
M F Naturality with resp ect to the homomorphisms L L over F with
u u and u u gives
u u u
On the other hand one has
u u u
just by linearity One concludes u But then u as well and the ab ove
computation for n tells u
Remark As already mentioned the prop erties Co and E of cycle mo dules
follow from the material in Section The considerations there use the deformation
to the normal cone and homotopy inverses But things simplify considerably if one
may pass to the limits X Sp ec k t t In the following we consider the
n
case X Sp ec k s t tacitly assuming that our cycle mo dules are de ned on an
appropriate category of schemes In fact we could have taken also more general
schemes as basis for our notions say excellent schemes over a p erfect eld however
one should then b e careful with Theorem
C Co Let E F u and T Sp ec E s t Given A X M put
ft usg r M
T
E s t jF s t
One may then calculate
X
z
z v
(1)
z T
Hence by C
v
C E By the last argument we may use Co It follows that for A X M
the value of s is indep endent of the choice of the prime Since v F t s is
v
rational one is by H reduced to check
s
w
v
for all valuations w of F t s over F except the one with w t s Every w
de nes a p oint in the exceptional b er of the blow up Y X One calculates for
the w in question
X
y
fsg fsg
y w v
w
(1)
y Y
Finally note s fsg for the choice s
v
v
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Remark In the case of a constant cycle premo dule one may derive C from
Co under presence of FD This tells that axiom C app ears naturally in our
q
framework if we require the existence of pull back maps f As in Remark we
consider here the case X Sp ec k s t
Co C To derive C from Co for constant M and for X Sp ec k s t
one argues rst similarly as for Step ab ove as follows Let y b e the p oint with
parameter s and de ne for x X n fy g
x
M x M
X
x
r c ft t x g r
F s t jF s t x t jF s t x t j x
x x
As in Step one has id and for z x y moreover one nds
x z
x x x
ftg r This shows that C holds on the image of the
y
F t jF
In order to verify C for M we may arrange things such that
X y
We are reduced to check C for
X
x
x
x y
Using the ab ove computations one nds
for x y
x
ftg r for x y
y jF
for some M F
We must show Put E F r and written in a somewhat sloppy form
r s r t s t fs r g M E s t
One computes A Sp ec E s t M and
fr g M E s t
v
Co gives fr g in M F r s t Applying appropriate sp ecialization and
residue maps shows
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The Four Basic Maps
The purp ose of this section is to introduce the cycle complexes and all the types of
op erations on them needed further on except the cross pro ducts to b e de ned in
Section
Let M b e a cycle mo dule over X let N b e a cycle mo dule over Y and let U X
V Y b e subsets For a homomorphism
a a
M x N y
x U y V
x
we write M x N y for the comp onents of
y
Change of coefficients Let M N b e a homomorphism of cycle mo d
ules over X and let U X b e a subset We put
a a
M x N x
x U x U
x x
where and for x y
x
x y
Cycle complexes For a cycle mo dule M over X and an integer p let
a
C X M M x
p
x X
(p)
We de ne
d d C X M C X M
X p p
x x x
by d with as in This de nition has sense by axiom FD
y y y
Lemma d d
X X
x
Proof One has to check d d for x X z X This is trivial
p p
z
fxg Otherwise let Y b e the lo calization of fxg in z Since our schemes are if z
catenary we have X Y Y and dim Y Now apply axiom C to Y
p
The complex C X M C X M d is called the complex of cycles
p X p
on X with coe cients in M
When developing a theory of cycles rst natural questions are the following
Given a prop er morphism f X Y what is the push forward map f on cycles
Or given a at morphism g Y X what is the pull back map g on cycles In
fact we will de ne such maps However these questions are not our guiding p oint of
view We rather x schemes X Y and numbers p q and then ask what is the class
of maps
C X M C Y M
p q
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which we should consider Our answer is then motivated by what we want to do with
the complexes namely developing intersection theory etc This leads to the four
basic maps as de ned in
The de nitions of the basic maps multiplication with K and b oundary maps
in and are easy to understand However our way of introducing push
forward and pull back maps as in and deserves some words of comment It
turns out that these maps denoted by f and A g s are sums of comp ositions of
maps of simpler type namely push forward maps f for prop er morphisms f pull back
maps g for at morphisms g and the pro jections i and inclusions j corresp onding
to closed or op en subvarietes see This fact which we will not prove seems
to b e however only of heuristic interest In fact it would b e a nuisance if we had to
consider at each step such a reduction of the language expressing the maps b etween
the cycle complexes
Push forward For a morphism f X Y of schemes of nite type over a
eld we de ne
f C X M C Y M
p p
x
c Otherwise as follows If y f x and if x is nite over y then f
x j y
y
x
f
y
Pull back Our main interest is to de ne the particular types of pull back
maps as considered in b elow In our general de nition in we de ne
pull back maps C X M C Y M asso ciated to any morphism g Y X of
p q
relative dimension q p Moreover we use coherent sheaves A on Y as mo di ers
of the arising multiplicities This construction gives great technical exibility and is
useful in Section
For a morphism g Y X let
s g max dim y Y dim g y X j y Y
Moreover let Y Y Sp ec x for x X
x X
Note that if x X y Y g y x and s g q p then necessarily
p q
y Y
x
Let g Y X b e a morphism and let A b e a coherent sheaf on Y For x X
and y Y we de ne an integer
x
x
Z A g
y
as follows The lo calization Y of Y in y is the sp ectrum of an artinian ring R
x
x y
with only residue class eld y Let A b e the pull back of A via Y Y Y
x
x y
x
l A as the length of A considered as R mo dule for the notion and de ne A g
R
y
of length and further prop erties we refer to Fulton App A
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Fix s Z Let g Y X b e a morphism with s g s and let A b e a coherent
sheaf on Y We de ne homomorphisms
A g s C X M C Y M
p p s
by
x
A g r if g y x
y j x
y
x
A g s
y
otherwise
Here x is considered as a sub eld of y via g
Let F b e a eld let g Y Sp ec F b e a morphism and let
A A A
b e an exact sequence of coherent sheaves over Y Then
A g s A g s A g s
This follows from the additivity of length with resp ect to short exact sequences
For some particularly interesting cases we use the following notations Let F E
b e a homomorphism of elds let X b e of nite type over F and let g Y X
Sp ec F
Sp ec E X b e the base change Then we put g O g
Y
A morphism g Y X of schemes of nite type over a eld is said to have
constant relative dimension s if all b ers are either empty or equidimensional of
dimension s In this case we write dim g s and put
g O g dim g
Y
Particular cases are here op en and closed immersions with s
Multiplication with units For global units a a O we de ne
n
X
homomorphisms
fa a g C X M C X M
n p p
by
fa x a x g for x y
n
x
fa a g
n
y
otherwise
This de nition turns C X M into a mo dule over the tensor algebra of O If X is
p
X
de ned over some eld F then C X M b ecomes via F O a mo dule over K F
p
X
Boundary maps Let X b e of nite type over a eld let i Y X b e a closed
immersion and let j U X n Y X b e the inclusion of the op en complement We
will refer to Y i X j U as a b oundary triple and de ne
U
C U M C Y M
p p
Y
x U
by taking for the de nition in with resp ect to X The map is called the
y
Y
b oundary map asso ciated to the b oundary triple or just the b oundary map for the
closed immersion i Y X
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We conclude this section with a few notations and remarks concerning the four
basic maps
Generalized correspondences We introduce the notation
p
X Y
to denote homomorphisms
C X M C Y M
which are sums of comp osites of the four basic maps f g fag and for schemes of
nite type over a eld
This notation is made for the sake of simpli cation It also stresses the fact that
we think of the maps in question rather as a sort of morphisms of varieties than just
maps of complexes asso ciated to every M As mentioned in the introduction this can
b e made more precise in a further development The di erential d is not sub ject
X
to this notation convention we rather think of d as a part of the inner structure
X
of X Similarly for homomorphisms induced by a change of co e cients
Gradings The Z gradings on M induces a Z grading on C X M by
a
C X M n M x
p n p
x X
(p)
p
with n Z Supp ose X Y resp ects this grading in the sense that
C X M n C Y M n r
r
for some r Z In this case we write sgn One has sgn f sgn g
n
sgn fa a g and sgn Moreover we put
n
d sgn d
Then
sgn sgn
sgn
All the maps to b e considered will resp ect the Z grading Moreover if M is
Z graded then the will resp ect the corresp onding Z gradings on the complexes
Additionally they resp ect the natural Z gradings given by dimension So if M is
Z graded there is an underlying Z Z grading see also Section In the general
treatment however there is need only for the Z grading
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Boundary triples Let Y i X j U b e a b oundary triple The set theoretic
union X Y U yields a natural decomp osition
p p p
C X M C Y M C U M
p p p
of ab elian groups Here the complex C Y M is a sub complex of the complex
C X M with C U M as quotient complex The maps i j and i j are the
corresp onding inclusions and pro jections resp ectively In a formal way the situation
is describ ed by the following formulae
i d j
X
i i id j j id
Y U
i j j i
i i j j id
X
j i i j
i j i j
i j
Later on we will make free use of these simple rules in particular in Sections
and The canonical decomp osition is the source of our formal treatment of
intersection theory on complex level
Compatibilities
In this section we establish the basic compatibilities for the maps considered in the
last section All arguments are simple in nature or at least familiar to cycle theories
They are basically of lo cal nature As usual the treatment of at pull back causes
most of the technicalities
Rules among the maps of are formulated in Prop osi
tion is concerned with the compatibility with the di erentials The compatibilities
with change of co e cients are obvious and we don t make a p oint of them here and
further
Proposition
For f X Y f Y Z as in one has f f f f
Let g Y X and g Z Y be morphisms Let s s g and s s g
and let A A be coherent sheaves on Y Z respectively with A at over Y Then
s s s g g and
A g g s s A g s A g s g A
O
Z
In particular
g g g g
for g g as in with g at
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Chow Groups with Coefficients
Consider a pul l back diagram
0
g
U Z
0
f
f
y y
g
Y X
with f and f as in Let s s g s g and let A be a coherent sheaf on Y
Then
A g s f f f A g s
In particular
g f f g
for g as in
Proof is immediate from the de nitions and R a
Pro of of The inequality is obvious Let x X y Y z Z with
x y
dim y Y dim x X s dim z Z dim y Y s We have to check
x y x
A g g A g A g g A
O
Z
z z y
We may assume X Sp ec x and Y Sp ec R with R as in By devisage
using the atness of A over R and we may reduce to the case A y Now
the claim is trivial
z
Pro of of Let A g s f f f A g s We have to show for
y
z Z and y Y
p p s
This obvious if f z g y Otherwise let x f z g y Our assumptions
give dim x X dim z Z and dim x X dim y Y s g p hence dim x X
dim z Z p and z is nite over x
Let u U b e a maximal p oint of the b er over z Our assumptions give
z
dim u U dim y Y p s and dim u U dim z Z s g p s hence
z
remains unchanged if we replace X by Sp ec x Z u U This shows that
p s
y
by Sp ec z Y by Y Sp ec R see and p s by Then f is nite and at
x y
Hence f is at and by devisage using we may assume A y as R mo dule
But then it su ces to consider the case Y Sp ec y and the claim follows from
rule R c
Lemma Let f Y X be as in
If a is a unit on X then
f ff a g fag f
Let f be nite and at and let a be a unit on Y Then
f fag f ff a g
Here f O O is the standard transfer map
Y X
Proof is immediate from R b For we may assume X Sp ec F with F a
eld Then for y Y let l b e the length of O By R c we have
y y Y
X
f fag f l c fa y g
y
y jF
y
and the claim follows
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Lemma Let a be a unit on X
For g Y X as in one has
g fag fg ag g
For a boundary triple Y i X j U one has
U U
fj a g fi a g
Y Y
Proof follows from R a and from R b and R e
Let h X X b e a morphism of schemes of nite type over a eld and let
Y X b e a closed immersion Consider the induced diagram given by U X n Y
and pull back
Y X U
h
h
y y y
h
Y X U
Proposition
If h is proper then
0
U U
h h
0
Y Y
If h is at of constant relative dimension then
0
U U
h h
0
Y Y
Proof Immediate from Prop osition and b elow
Lemma Let g Y X be a smooth morphism of schemes of nite type over a
eld of constant ber dimension let X Y be a section to g and let t O be
Y
a global parameter de ning the subscheme X Moreover let g Y n X X be
the restriction of g and let be the boundary map associated to Then
g and ftg g id
X
Proof One reduces to X Sp ec E and applies R c and R d
Proposition
For proper f X Y as in one has
d f f d
Y X
Let g Y X be a morphism and let A be a coherent sheaf on Y at over X
Then
d A g s A g s d
Y X
for s s g In particular
g d d g
X Y
for at g as in
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Chow Groups with Coefficients
For a unit a on X one has
d fag fag d
X X
For a boundary triple Y i X j U one has
U U
d d
Y U
Y Y
Proof follows as Lemma and follows from Lemma
x
Pro of of Let f d f f d We have to show f for
Y X
y
fz g the claim is x X and y Y Let z f x and q dim z Y If y
p p
obvious If y z we rst replace Y by Sp ec y and then X by fxg This is the
case of a prop er curve over a eld considered in RC of Section If y fz g and
fz g and y z we must have q p and x is nite over z We may assume Y
X fxg Consider the diagram
g
X X
f
f
y y
h
Y Y
where g and h are the normalizations Let x X and z Y b e the generic p oints
lying over x and z resp ectively We have g j M x by the very de nition of
the di erentials similarly h j M z This and show
f g j M x d h f f g d j M x
Y
X
h f j M x
x
Since g j M x is an isomorphism onto M x we are reduced to show f for
y
x
y Y Let u X b e a p oint over y We have u X Now f follows
p
y
from rule R b the prop erness of f and the fact that the lo cal rings of y and of all
the preimages u are valuation rings
x
Pro of of Let d A g s A g s d We have to show for
Y X
y
x X y Y Put z g y If z fxg the claim is obvious If z x then
p p s
for u Y all valuations on u with center y are trivial on x the claim follows
x
fxg z x Then z X from rule R c We are now reduced to the case z
p
since dim z X dim y Y s p We may assume X fxg Moreover
by Prop ositions and we may additionally assume that X is normal Let
U u Y j y fug Then
x
X
x x z x u
A g s A g s
y z y u y
u U
We may replace X and Y by its lo calizations in z and y resp ectively Then X
Sp ec R with R a valuation ring Y Sp ec S with S lo cal of dimension and
U Y
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Markus Rost
In this case we have by de nition
X
x u x
A r l A A r l
S S
u u j x y j z
y y z
(u)
u U
where S A are the lo calizations at u and where is a prime element of R
u u
For u U let S b e the normalization of S u For a S mo dule H of nite length
u u
we de ne
X
L H l H w y
u
w
S
(w )
w
where w runs through the maximal prime ideals of S and where S resp H are
u
w w
the lo calizations of S resp H at w We claim that L H is the length of H as
u u
S mo dule
L H l H
u S
To prove this use devisage to reduce to the trivial case H w for some w
Moreover we have
l S S l S u S
S u u S
This follows from the fact that the cokernel and the trivial kernel of S u S
u
have nite S length and is a nonzero divisor of S u and S see Fulton
u
Lemma A
We have for xed u
X
u u
r c r
u j x w j y u j x
y w
w
X
x
l S S c r
w w w j y w j z
z
S
(w )
w
x
L S S r
u u u
y j z
z
u
Here we used the de nition of and R a R b R d
y
x
Putting things together one nds that follows from
y
X
l A A A l S u S l
S S S
u
(u)
u U
This formula is exactly the formula of Fulton Lemma A b ecause the map
A A a a is injective by the atness of A over R
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Cycle Complexes and Chow Groups
This section contains notations and a few remarks and examples In Section we
have introduced for a cycle mo dule M over X the complexes
a
C X M M x
p
x X
(p)
with di erentials
d d C X M C X M
X p p
Sometimes it is convenient to use the co dimension index instead of the dimension
index We put
a
p
M x C X M
(p)
x X
and de ne
p p
d d C X M C X M
X
x x x
again by d with as in Similar as in Lemma one nds d d
y y y
The choice b etween the dimension and co dimension index dep ends on the matter
Our basic interest is in schemes X of nite type over a eld F In this case the
dimension setting is in general appropriate since then the dimension of a p oint x is
an absolute notion indep endent of the ambient space dim x X tr deg x jF
p
If X is in addition equidimensional of dimension d then X X and
d p
p
C X M C X M Then we will freely switch b etween the two notions if
d p
it is convenient in particular if we consider intersections in the smo oth case The
co dimension setting will also b e used for certain schemes not necessarily of nite type
over a eld e g for sp ectra of lo cal rings In this case we understand the material of
Section to b e transferred from the dimension to the co dimension setting via nite
type mo dels
In practice all M have a Z grading and one likes to keep track on it We put for
Z graded M
a
C X M n M x
p n p
x X
(p)
a
p
C X M n M x
n p
(p)
x X
with n Z Then there are decomp ositions of complexes
a
C X M C X M n
n
a
C X M C X M n
n
In the introduction we have used the notation C X n C X K n
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The Chow group of p dimensional cycles with coe cients in M is de ned as the
p th homology group of the complex C X M and denoted by A X M Similarly
p
p p
we de ne A X M A X M n and A X M n according to the notations used
p
for the complexes
U
The homomorphisms f for prop er f g for at g fa a g and of
n
Y
Section anti commute with the di erentials see Prop osition The induced
maps on the co homology groups will b e denoted by the same letters The compat
ibilities of carry over for prop er f f and at g
It is obvious from that for a b oundary triple Y i X j U there is the long
exact sequence
j
i i
A X M A Y M A U M A Y M
p p p p
We conclude by mentioning a few examples H D and K denote the cycle
mo dules given by Galois cohomology and Quillen s K theory as considered in Sec
tions
Remark Classical Chow groups We understand here CH X as the group
p
of p cycles mo dulo rational equivalence as de ned in Fulton Sect denoted
by A X From this de nition it is obvious that
p
A X K p CH X
p p
p
For the Chow group CH X of p co dimensional cycles for smo oth irreducible X say
our notations give
p
p
A X K p CH X
Remark Unrami ed cohomology For a prop er smo oth variety X over a
eld k and a cycle mo dule M over k the group
A X M M k X
is a birational invariant of the eld extension k X jk see Corollary A well
known example here is the unrami ed Brauer group of k X jk Its n torsion subgroup
is in our notations given by A X H
n
Remark Relations with local global spectral sequences In Quillen s K
theory as well as in etale cohomology there are sp ectral sequences given by co di
mension of supp ort see Quillen Sect Blo ch and Ogus The corre
sp onding E terms together with the d di erentials may b e identi ed with the com
plexes C X K n and C X H D n The corresp onding E terms are of the
p q p q
p p q
form E A X K q and E A X H D q resp ectively where
r
r D
}
Namely the de nition of CH X mentioned in the rst sentence of the introduction
p
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Chow Groups with Coefficients
Remark The map from Milnor s to Quil len s K theory The natural ho
momorphisms K F K F form a homomorphism of cycle mo dules It is an isomor
phism in degrees Moreover for a valuation v on F one has K F K F
v v
see Merkurjev and Suslin It follows that the induced homomorphisms
A X K n A X K n
p p
are bijective for n p
Acyclicity for Smooth Local Rings
The following observations have b een included to underpin the notion of cycle mo d
ules They are not needed in further sections M is a cycle mo dule over a eld k
Theorem Let X be smooth and semi local Then
p
A X M for p
This theorem is known in Quillen s K theory Quillen x Theorem and
in etale cohomology Blo ch and Ogus and has b een proved by O Gabb er for
Milnor s K theory The main step in these pro ofs is sometimes called Quillen s trick
and carries over to cycle mo dules as well Here we follow essentially this metho d but
with a simpli cation due to I Panin
Let V b e a vector space over k and let A V b e the asso ciated a ne space For
a linear subspace W of V let
A V A V W
W
v v W
W
b e the pro jection
Lemma Let X A V be an equidimensional closed subvariety with dim X d
and let Y X be a closed subvariety with dim Y d Moreover let S Y be a nite
subset such that X is smooth in S Then for a generic d codimensional linear
subspace W of V the fol lowing conditions hold
The restriction
j Y Y A V W
W
is nite
The restriction
j X X A V W
W
is locally around S smooth of relative dimension
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Proof Panin We extend the situation to the pro jective closure A V P V k
with P V P V k as hyperplane at in nity Let
P V k n P W P V W k
W
v t v W t
W
b e the pro jection is an a ne bundle over P V W k which extends the a ne
W
bundle over A V W
W
P V k b e the closure of Y and let Y Y P V Then dim Y Let Y
d Hence for generic d co dimensional W we have Y P W Therefore
there is the map
j Y Y P V W k
W
This map is nite since it is prop er and since is an a ne bundle This shows
W
b ecause j Y is the pull back of j Y along A V P V k
W W
For condition one just needs to guarantee that maps for each s S the
W
tangent space T X V of X in s epimorphically onto the tangent space V W of
s
A V W in s This is again an op en condition for W
W
This lemma is very close to Quillen x Lemma and su ces for all
applications I know The existence of such a space W is not clear over nite ground
elds and needs some extra discussion However if one is in the end interested in
co homology groups there is usually no problem with replacing the ground eld k
by a rational extension k t t In this case one may take for W for example
r
the tautological subspace of V de ned over the function eld of the Grassmannian of
d co dimensional subspaces of V In our situation we refer here to the following
remark
Lemma Let X be a variety over k and let g X X be the base change
k t
Then
g A X M A X M
q q
k t
is injective
This lemma will b e b ecome obvious in the next section where we show that F
A X M is a cycle mo dule Then g is just the restriction map r for this cycle
q F
k t jk
at a rational p oint of P yields a left inverse What mo dule and any sp ecialization s
v
one really uses here is Lemma with Y X P and Prop osition
Proposition Let X be a smooth variety over a eld and let Y X be a closed
subscheme of codimension Then for any nite subset S Y there is an open
neighborhood X of S in X such that the map
i A Y X M A X M
is the trivial map Here i Y X X is the inclusion
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Chow Groups with Coefficients
Proof We may assume that X is a ne By Lemma we nd a diagram at least
after replacing k by a rational extension
i
Y X
A
with Y A nite and with X A smo oth of relative dimension in S
Put Z Y X and consider the diagram
A
Z
g
i
Y X
where g and are the pro jections and is the diagonal Note that is nite g is
smo oth of relative dimension in S and that is a section to g and a lift of the immer
sion i Moreover after a lo calization to an op en subset X X containing S we may
assume that there is a global parameter t O de ning the closed subscheme Y
Z
Let Y Z j Q b e the b oundary triple given by with Q Z n Y and let
g Q Y b e the restriction of g Now consider the comp osite
ftg
g j
p p p p
X Z H Y Q Q
One nds
Q
ftg g i H
Y
by Lemma Therefore i is nullhomotopic
Pro of of Theorem We may assume that X is connected Put d dim X
Consider pairs U S where U is a smo oth d dimensional variety of nite type over k
and S U is a nite subset such that X is the lo calization of U in S Then
p p
C X M lim C U M
U S
Moreover
p
C U M C U M lim C Y M
d p d p
Y
where Y runs over the closed p co dimensional subsets of U Hence
p p
A X M lim A U M lim lim A Y M
d p
Y
U S U S
p p
But Prop osition tells that A Y M A U M A U M is the trivial
d p
map for small enough U U
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In the smo oth case we shea fy cycle mo dules as follows For a smo oth variety X
let M b e the Zariski sheaf on X given by
X
M U A U M M
X X
for op en subsets U of X
Corollary For a smooth variety X over k there are natural isomorphisms
p
p
A X M H X M
X
Zar
p
Proof De ne the Zariski sheaves C on X by
p p
C U C U M
Then there is a complex of sheaves
d d
C M C
X
The complex is exact This holds at M and at C by the very de nitions Theo
X
p
rem implies exactness at p ositive dimensions The corollary follows since the C
are asque
The resolution of M considered in this pro of has nice functorial prop erties
X
Namely we will de ne for morphisms f Y X maps of complexes Section
p p
I f C X M C Y M
and under presence of a ring structure for M a pairing of complexes Section
C X M C X M C X M
These are functorial with resp ect to lo calizations Therefore the isomorphisms of
Corollary are compatible with pull backs and with pro ducts
The following example is a nice illustration of Corollary Let X b e smo oth
and de ne the Zariski sheaf K on X by
n
K U A U K n K k X
n n
The sheaf K has a comparatively simple de nition it just refers to the de nition
n
of Milnor s K groups for elds and of the residue maps for valuations Corollary
yields the following interpretation of the classical Chow groups on a smo oth variety
p
p
X K CH X H
p
Zar
The same result holds with Milnor s K theory replaced by Quillen s K theory
The corresp onding sheaf K coincides with the sheaf induced from the presheaf U
n
Q Q
K U where K U denotes the n th Quillen s K group of the category of vector
n n
bundles on U In this context is known as Blo ch s formula see Quillen
Thm Grayson
Another sp ecial case of Corollary for M K is
d
A X K n H X K
n d
Zar
with d dim X This interpretation of the Chow groups of zero cycles on X with
co e cients in K was obtained already in Kato
n
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Chow Groups with Coefficients
The Cycle Modules A M
q
In this section we show that new cycle mo dules can b e obtained from the Chow groups
of the b ers of a morphism It was in fact this pro cess of forming lo cal co e cient
systems for cycles which motivated the notion of cycle mo dules
Let Q B b e a morphism of nite type and let M b e a cycle mo dule over Q
For elds F over B let Q Q Sp ec F We de ne an ob ject function A M
F B q
on F B by
A M F A Q M
q q F
Our aim is to show that A M is in a natural way a cycle mo dule over B
q
All the prop erties of cycle mo dules except axiom C hold already on complex
level i e for the groups C Q M It is appropriate to establish rst the corre
q F
sp onding ob ject function as a cycle premo dule
c
So let M b e the ob ject function on F B de ned by
c
M F C Q M
q F
b
We rst describ e its data as a cycle premo dule These will b e denoted by b b
v
rb bc etc in order to distinguish them from the data etc of M
v
E jF E jF
For a homomorphism of elds F E let Q Q b e the induced mor
E F
phism We de ne the data D and D by
b C Q M C Q M
q F q E
b C Q M C Q M
q E q F
For D we take the K F mo dule structure on C Q M describ ed in To
q F
e
establish D put Q Q Sp ec O It has over Sp ec O the generic b er Q and
v B v v F
the sp ecial b er Q De ne
v
b
C Q M C Q M
v q F q
v
x x x
b
e
by with as in with resp ect to the scheme Q
v v
y y y
c
Theorem Together with these data M is a cycle premodule over B
Proof All the required prop erties follow from the rules and axioms for M and from
Section
Below we consider R a in detail Here is a sketch for the other less complicated
cases
for R a use for R b use
for R c use and a length consideration
for R a use for R b use for R c use R c and R c
for R b use for R c use R c
for R d use and R d for R e use R b and R e
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Markus Rost
Pro of of R a Let g Q Q and g Q Q b e the pro jections We
F E
v w
have to show that the following diagram is commutative
b
v
C Q M C Q M
q q F
v
x x
g
e g
0
b
w
C Q M C Q M
q q E
w
e e
We want to apply Prop osition to the pro jection g Q Q The pull back
v w
of g along Sp ec E Sp ec O is g Let
w
e
Sp ec w Q g Q Q
v Sp ec O
w v
w
b e the pull back of g along Sp ec w Sp ec O Note that Q and Q have
w
v v
the same reductions and therefore the same cycle groups
w Note that g g are the pull We claim g e g Let R O
v O
w
backs along Q Sp ec w of the morphisms Sp ec v Sp ec w Sp ec R
w
Sp ec w resp ectively The claim follows from e l R and a standard length
R
consideration
It remains to show that the diagram commutes with e g replaced by g This
follows cum grano salis see the following remark from Prop osition
Remark When applying here Prop osition in a formal way there app ears
an arti cial problem caused by the fact that the dimension index do es not b ehave
e
p erfectly well for schemes over lo cal rings like Q However note that to check a
v
it su ces to restrict to the comp onents cor g commutativity like g
w v
fxg Q For these p oints one resp onding to p oints x Q with
w q E q
e
has x Q by the dimension inequality Matsumura p A similar
w
q
e
remark applies to Q Therefore the desired identity follows from d g g d on
v
e
C Q M One may avoid these considerations by lo oking more closely to the
q w
pro of of Prop osition
c
We have to relate the di erentials for the cycle premo dule M to the di erentials
for the cycle mo dule M
e
Let X B b e a scheme over B and let X Q X Then for x y X there
B
is the map
x
b
c c
M x M y
y
according to By de nition this is a map
x
b
C Q M C Q M
q q
x y
y
b etween cycle groups with co e cients in M
e
Proposition Let x y X be points lying over x y X respectively and
x x x
b b
suppose x Q and y Q Denote by the component of with
x q y q
y y
y
respect to x and y Then
x x x
b
M x M y
y y y
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Proof We may assume y fx g since otherwise b oth sides are trivial The dimen
fxg Let v run through sion inequality Matsumura p shows then y
the valuations of x with center y in X Moreover let w run through the valuations
on x with center y in X The restriction of any w to x is one of the valuations v
e
Let w Q b e the center of w in X Sp ec O Now the claim follows from
X v
v
X
x x x
b b
bc
v
v j y
y y y
v
X X
w x
b
bc
v
v j y
y w
v
w jv
X X
c c
w
w j y w j w
v
w jv
X
x
c
w
w j y
y
w
c
It follows from Prop osition that the data of the cycle premo dules M for
various q commute resp anti commute with the di erentials of the complexes
C Q M Passing to homology we obtain data D D for the ob ject functions
F
A M
q
Theorem Together with these data A M is a cycle module over B
q
Proof The rules for the data of the cycle premo dule A M are immediate from
q
c
the rules for M Moreover axiom FD for M and Prop osition show that FD
c
holds for M consequently also for A M It remains to verify axiom C
q
Consider the maps
a
C Q C Q C Q C Q C Q
q q q q q
x x x
0 0
(1)
x X
z z z
de ned by with as in with resp ect to the scheme Q X
B
y y y
By Prop osition we are reduced to show It su ces to check
z
for z Q and y Q with y fz g here fz g is the
q x q
y
0
e
closure of z in X The dimension inequality Matsumura p shows
Q Q Z Q
x q x q q
0
x
with Z fz g We are done by axiom C for M
y
In the following prop osition we formulate some functorial prop erties of the con
struction A M Let
q
h
Y X
B
}
Here the indication of the cycle mo dule M has b een dropp ed
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Markus Rost
b e a commutative diagram with and of nite type and let M b e a cycle mo dule
over X For a eld F over B let
h Y X
F F F
b e the morphism induced by h
Proposition The fol lowing transformations are homomorphisms of cycle
modules over B
For proper h let
h A M A M
q q
with h h
F F
For at h of relative dimension s let
h A M A M
q q s
with h h
F
F
For a global unit a on X let
fag A M A M
q q
with fag fa j X g
F F
For a boundary triple Y i X j U let
U
A j M A i M
q q
Y
with the boundary map for Y X
F F F
Proof One has to check the compatibility with D D This follows for from
and for from and
for from the anti commutativity of K and for from
and C
Let Q B b e at and not necessarily of nite type One may then de ne
q
cycle mo dules A M with
q q
A M F A Q M
F
To establish these cycle mo dules one pro ceeds analogous to the A M ab ove Al
q
ternatively one may reduce to the consideration of the A M as follows If is of
q
nite type one may assume that it is of constant dimension s In this case one has
q
A M A M For the general case note that at least lo cally with resp ect
s q
q q
to B one has A Q M lim A Q M where Q B runs through the at
F
F
nite type mo dels of
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Fibrations
In this section we consider the sp ectral sequence asso ciated to a morphism and formu
late some basic functorial prop erties A rst application yields the homotopy prop erty
for vector bundles
From now on all schemes are assumed to b e of nite type over a eld and M is
with exceptions in Section a cycle mo dule over X
For a morphism X X we put
a
C M x C X M
p l l
x X
(p l)
where
p X x X j dim x X
p l l
Then
C C C X M
p p
is a nite ltration of C X M by sub complexes This ltration has the sub quotients
a
C X M
u
0
u X
(p)
n
b e the asso ciated sp ectral sequence see e g Hilton and Stamm Let E
p q
n
bach The di erential for X restricts on C X M to the di erential
u
for X Therefore
u
a
E A X M
q
u
p q
0
u X
(p)
of this spectral sequence equals the di er Proposition The di erential d
p q
0
for the cycle module A M ential d
q X
we have to check equality of the corresp onding y X Proof For u X
p p
0
comp onents of d and d
X
p q
0
0
u
u
0
0 0
d A X M A X M d
0
q q X
u y
0
p q
y
y
0
u
The map d is by de nition induced from the map
0
p q
y
0 0
C X M C X M
q q
u y
u u
where for u y X lying over u y resp ectively
y y
The claim follows from Prop osition
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Markus Rost
Corollary There is a convergent spectral sequence
E A X A M A X M
p q p q
p q
If X is equidimensional and is at then there is a convergent spectral sequence
p q
p q p q
E A X A M A X M
Here the second statement follows from the rst by a formal switch to co dimension
index In this co dimension setting one may drop the nite type hypotheses
Remark We will use the following dictions Let X X Y Y b e
p
morphisms Then X Y is called ltration preserving with resp ect to of
degree r t if
C C
p l p r l t
If see Sec for the de nition of then is homomorphism of ltered
complexes and induces maps denoted by the same letter
n n
E E
p q p r q t r
p
Two ltration preserving maps X Y of degree r t are called homotopic
p
if there is a ltration preserving H X Y of degree r t such that
H If and then the induced maps on the E
terms coincide If the homotopy H can b e chosen of degree r t then already
the induced maps on the E terms coincide This follows from a little calculation
working for arbitrary ltered complexes
Let
f
Y X
y y
0
f
X Y
b e a commutative diagram of morphisms The following statement is trivial
Lemma
One has
C C f
p l p l
Suppose f has relative dimension t and let s s f see Then
C C f
p l p s l t
If a is a unit on X then
fag C C
p l p l
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Chow Groups with Coefficients
Let Y i X j U be a boundary triple Then
U
C j C i
p l p l
Y
For the diagram one has
U
h C h C
p l p l
Y
Let
f f
b
Y Y X
y y y
0
f
Y Y X
b
0
X and f f f b e the natural decomp osition of diagram with Y Y
X
We call the diagram a at square if f and f are at of some constant relative
dimensions This holds then also for f
We use the natural identi cation
A f M f A M
q q
of cycle mo dules over Y
Proposition
If f and f are proper then the map
f E E
p q p q
corresponding to equals the composite
0
f
f
#
A Y A M A Y A M A X A M
p q p q p q
Suppose the square is at and put r dim f s dim f Then the map
f E E
p q p r q s
corresponding to equals the composite
0
f
f
#
A X A M A Y A M A Y A M
p q p r q p r q s
For a global unit a on X the map
fag E E
p q p q
corresponding to equals fag
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The map
E j E i
p q p q
U
corresponding to equals
Y
The map
h E h E
p q p q
corresponding to equals the map
0
U
A U A h M A Y A h M
0
p q p q
Y
Proof is trivial follows from Prop osition In and one may supp ose
either f f or f f
Pro of of for f f Here X Y and the map
f C Y M C X M
l l
is the family of maps
f C Y M C X M
q q
u u u
with u X and p q l On the other hand
p
f C X A M C X A M
p q p q
is comp onentwise induced by the maps f
u
0
X We Pro of of for f f Here we have a pull back diagram Y Y
X
consider the maps induced by
f C Y M C X M
l l
on the E terms These are maps with p q l
a a
0 0
C X M C X M
q q
y x
0 0
0 0
y Y x X
(p) (p)
0
Their comp onents are the corestrictions c with y X y Y and
y j x y q
p
0
f y X x f y X Here y is necessarily nite over x since
x q p
b oth elds have the same transcendence degree Therefore the maps on the E terms
are given by the maps
0 0 0
f C X M C X M
0
q q
y f y
y
with y Y such that y j f y is nite and where
p
0 0 0
f X X
0
y f y
y
is the asso ciated nite morphism On the other hand
f C Y A M C X A M
p q p q
is induced exactly by the maps f
0
y
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Chow Groups with Coefficients
Pro of of for f f One argues as for and notes that
f C X M C Y M
l l s
is the family of maps
f C X M C Y M
q q s
u u u
with u X and p q l
p
Pro of of for f f The map
f C X M C Y M
l l r
is the family of maps
x
O f r M x M y
Y
y j x
y
with y Y x X and f y x
l r l
The map
f C X A M C Y A M
p q p r q
is the family of maps induced by the maps
0
x
0
0 0
f O f C X M C X M
0 0
Y q q
x y
y y
0 0
f y x and where f x X with y Y X X is the natural
0
y x
y
p p r
map Moreover f is the family of maps
0
y
x
O y f r M x M y
0
X O
x j y
0
y y
X
with y Y lying over x X and over y
p q r p q
The claim amounts to show for such p oints y x y x the equality
0 0
0
l x y l x O l x O
O O Y O Y
y 0 y y 0 0
X X X
For this see Fulton A
Pro of of The map
U
C U M C Y M
l l
Y
is on the sub quotients of the ltrations given by the family of maps
C U M C Y M
u q q
u u
with u X p q l and where is the b oundary map for the closed immersion
u
p
Y X On the other hand
u u
U
C X A j M C X A i M
p q p q
Y
is comp onentwise induced by the maps
u
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Markus Rost
By an a ne bundle of dimension n we mean a bundle V X which is lo cally
n
on X isomorphic to X A X with a ne transition maps In applications we
are mainly interested in the sp ecial case of vector bundles
A rst application of the sp ectral sequence is
Proposition Let V X be an a ne bund le of dimension n Then
A X M A V M
p p n
is bijective for al l p If X is equidimensional then
p p
A X M A V M
is bijective for al l p
Here again the second statement follows from the rst and one may drop in the
co dimension setting the nite type hypothesis
Proof By Corollary and Prop osition applied to Y X X Y V
f f all we need to show is
A M for q n
q
and that
M A id M A M
X n
is an isomorphism of cycle mo dules over X
Therefore we are reduced to the case X Sp ec F Then V is a trivial bundle
n
V A For n the claim is H of Section So we know Prop osition for
F
n
line bundles over an arbitrary base But then the case V A follows by induction
F
on n
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Chow Groups with Coefficients
Homotopy
We have just observed the homotopy prop erty for a ne bundles In this section we
show that this fact can b e made more precise on cycle level by means of a homotopy
inverse
p
A homomorphism X Y with is called a strong homotopy equiva
p p
lence if there is r Y X and H Y Y such that
r
r id
H
H id r
The pair r H will b e called h data for
p
Let V X b e an a ne bundle We will show that X V is a strong
homotopy equivalence A crucial p oint here is the treatment of the case V X A
The general case is then more or less clear in view of the decomp osition of the cycle
complexes corresp onding to b oundary triples We give here explicit formulas in order
to make clear compatibility with base change and ltrations
By a coordination X of an a ne bundle V X of dimension n we
i i
mean a sequence X X X X of closed subsets of X together with
k
trivializations
n
V j X n X X n X A
i i i i i
We use the notation V j U V U for U W and a scheme V W over W
W
Co ordinations clearly exist since X is no etherian For a morphism f Y X we
denote by f the induced co ordination on the pull back bundle f V
In the following we construct in several steps h data
r C V M C X M
p p n
H C V M C V M
p p
for dep ending on a co ordination
p
The case V X A h data r H for X X A are given by the
comp osites
f tg
j
1
p p p
X r X A X A nf g X A nf g
p
p
fs tg
1
2
p p p
H X A X A A n X A A n X A
Here t is the co ordinate of A Sp ec Z t and s t are the co ordinates of A A
Sp ec Z s Sp ec Z t Moreover fs t g is the diagonal j is the standard
inclusion p and p are given by the standard pro jections and is induced by
X X X P n f g with op en complement X A n f g
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We have to verify for r H the de ning prop erties of h data and
are immediate and follows from Lemma To check consider the
decomp osition
p
q
1
p p
X A P X A p X A A n
where q is the inclusion and p is the pro jection p is prop er and therefore
H p q fs tg p
Moreover
q i i
where i X X A P i X A X A P are the inclusions
and are the b oundary maps for X X A A X A
X A P n resp ectively
Since s t is a parameter for one nds
p i fs tg p id
by Lemma
Let W A P n A Moreover let p b e the restriction of p to
U X W n A and let b e the b oundary map corresp onding to the
inclusion X A X W Then
fs tg p fs tg p
Since s t t is a unit on W with constant value on X A one has
fs tg p f tg p
The compatibility of the b oundary maps with at pull back now gives
p i fs tg p r
Putting things together yields
n n
The case V X A Let X A X b e the pro jection and put
n
n n n n n
p
X X A By induction on n we de ne h data r H for
n
X X X X
n
n n n
Let Y X A so that Y A X A Note that where
X
Y
Y X is the pro jection Put
n
n
r r r
X
Y
n n n
n
H H r H
X
Y Y Y
Here r H are the h data for from moreover r id and H
Y
Y Y Y
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Chow Groups with Coefficients
The prop erties and can b e easily veri ed For note that
n
n n n
H H
X X X
Y
n n n n
n
H H r
Y
Y Y Y Y
n
H
Y
Finally follows from
n n n
n
H H H r
X
Y Y Y
n n n n
r r r
Y Y Y Y
n n
r
X Y
Glueing Let V X b e an a ne bundle let Y X b e closed let
U X n Y and put V V j Y V V j U For given h data r H for
p p
j V Y V and r H for j V U V we de ne h data r H for
p
X V by the formulae
H H H r r H
H r
H r
Here the matrix notation refers to the natural decomp ositions
C X M C Y M C U M
C V M C V M C V M
p
Moreover V V is the b oundary map corresp onding to V V The veri
cation of is straightforward and omitted
The general case Given a co ordination one uses iteratively the recip e of
to construct h data r H for
It turns out that the glueing pro cess of is asso ciative in the sense that
r H do es not dep end on the ordering in which the di erent pieces are glued
together However this is not at all imp ortant for us one should just decide oneself
for some xed standard ordering
Functoriality The construction of r H is compatible with manip
ulations on the base given by the four types of maps f g fag and We omit a
formulation since this will b e used only in the trivial case of op en immersions g
Proposition
Let V X be an a ne bund le of dimension n with coordination and let
X X be a morphism Then
r C C
p l p l n
H C C
p l p l
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Markus Rost
Let
V X
y y
0
V X
be a pul l back diagram with an a ne bund le of dimension n and let be a coor
dination for Then
r C C
p l p n l n
H C C
p l p l
Proof This is straightforward but nevertheless tedious by following the construc
tions
In order to de ne h data as ab ove one needs less than the choice of a co ordination
For example in one refers alone to trivializations of the one dimensional bundles
m m
X A X A
We have not tried to describ e the precise amount of information of a co ordination
needed in order to p erform the ab ove construction
Deformation to the Normal Cone
This section describ es three general constructions asso ciated to closed imbeddings
the normal cone the deformation space and the double deformation space
For the general role of the deformation space in intersection theory we refer to
Fulton The double deformation space will b e our to ol to verify asso ciativity
of the intersection op erations
We rst x notations and describ e some signi cant prop erties Explicit descrip
tions are given in b elow
Let Z Y X b e closed imbeddings
The normal cone to Y in X is denoted by N N X N X Y There is the
Y
pro jection N X Y and the inclusion Y N X If Y X is a regular imbedding
Y Y
then N X is a vector bundle over Y with the inclusion as zero section
Y
The deformation space D D X Y is a scheme over X A It is at over A
Over A n f g A one has
D j A n f g X A n f g
Furthermore the pro jection D j f g X f g factors through Y X f g and one
has
D j f g N X
Y
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as schemes over Y Our D is in Fulton Chap denoted by M moreover we
have taken instead of as the basep oint of the sp ecial b er
D X Y Z is a scheme over X A It is The double deformation space D
at over A and one has the following canonical identi cations of schemes over A
assuming in that Z Y X are regular imbeddings
D j A A n f g D X Y A n f g
j A n f g A A n f g D X Z D
D j A f g D N X N Y
Z Z
D j f g A D N X N X j Z
Y Y
j f g X f g factors through Z X and one Moreover the pro jection D
has
D j f g N N X N Y N N X N X j Z
Z Z Y Y
as schemes over Z
There is a more symmetric but less general version of the double deformation
space Namely let Y Y b e closed subschemes of X and let Z b e the intersection
e
of Y and Y i e Z Y Y Then there is a double deformation space D
X
e
D X Y Y X A relating in the transversal case all ve inclusions induced
e
from Z Y Y X The deformation space D is at over A and symmetric with
resp ect to a simultaneous interchange of Y Y and of the factors of A A A
Supp ose that Y and Y meet transversally Then
e
D j A n f g A D X Y A
e
D j f g A D N X N X j Z
Y Y
e
Moreover one has D j L D X Z for any line L A through the origin as long as
L is di erent from the two axes
X Y Z is the pull back of the space In the case Z Y Y the space D
X
e e
D X Y Y along the a ne blow up A A t s ts s We don t need D but
we have included b elow its de nition since it might b e a bit simpler to understand
than D
We have to recall facts from lo cal algebra Remark is a sp ecial version of
the lo cal criterion of atness Matsumura G p Remark may b e
deduced by considering lo cally regular sequences for J containing regular sequences
for I see Serre For a compact account of other facts needed in the following
we refer to Fulton App A App B
Remark A morphism U V A is at if and only if the morphisms
1 1
U A n f g V A n f g U f g V f g and U A are at
A A
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Remark If Z Y and Y X are regular imbeddings then Z X is a
regular imbedding If X is a ne and if X Sp ec A and I J A are the ideals
corresp onding to Y and Z resp ectively then
n m m n n m
I J J I J
n m n n m
I J I I J
n n
Here we understand n m Z with I J A for n
e
and D for a ne X From the naturality We next give the de nitions of N D D
of the a ne constructions it will b e obvious that they extend to global ones
We keep the notations of Remark Moreover we use the co ordinates A
Sp ec k t and A Sp ec k t s The indices n m always run in Z
The normal cone N N X is de ned as the sp ectrum of the ring
Y
a
n n
O I I
N
n
O is a ring over O A I and pro jection to the degree zero summand gives a
N Y
homomorphism O O
N Y
The deformation space D D X Y is de ned as the sp ectrum of the
subring
X
n n
O I t A t t
D
n
O is a nitely generated ring over A t with generators x t if x are generators
D i i
of I After inverting t one has
O t A t t
D
Since t is not a zero divisor it follows that O is at over k t Moreover
D
a
n n
O t O I I O
D D N
n
n
For later purp oses we are very precise ab out this identi cation for x I the residue
n n n n n
of x t mo d I t corresp onds to x mo d I This sign convention
will avoid some other signs later on
The double deformation space D D X Y Z is the sp ectrum of the
subring
X
n m n n m
O I J t s A t s t s
D
n m
is nitely generated over A s t After inverting s or t one has with D O
D
D X Z
X
n n m
O s I t s O s s
D
D
n m
X
m n m
0
t t t O J t s O
D
D
n m
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Chow Groups with Coefficients
This shows and For note rst
a
n m n n m n n m
s O I J I J t s O
D D
n m
In order to make clear the ring structures in particular as ring over k t s we keep
n m
here the terms t s having now merely the meaning of symbols
Moreover N X and N Y are the sp ectra of
Z Z
a
m m m
R J J s
m
a
m m m
R J I J I s
m
The pro jection R R yields an inclusion N Y N X
Z Z
m m
e
Let I ker R R By Remark one has J I I J and therefore
a
m m m m
e
I I J J J s
m
and
a
n n m n m m m
e
I I J J J s
m
Hence D N X N Y is the sp ectrum of
Z Z
a a
n n m n m m n m
e
I I J J J t s
n n m
follows now from Remark
For note rst
a
n m n n m n n m
t O I J I J t s O
D D
n m
For the ring of N X we write now
Y
a
n n n
O I I u
N
n
e
Let J O b e the ideal corresp onding to the closed subscheme N X j Z Its p owers
N Y
are
a
m m n n n n
e
J J I I I u
n
Hence D N Z N X j Z is the sp ectrum of
Y Y
a
m n n n n m
R J I I I u s
n m
De ne
O t O R
D D
n m n n m n n m n
x mo d I J t s x mo d I u s
It is easy to see that is a surjective ring homomorphism over k s Moreover
is injective by Remark The map gives the identi cation of Now
is obvious The atness over A not needed in the following may b e deduced
from Remark
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Markus Rost
e
The double deformation space D We just give the de nition Let I A
b e the ideal corresp onding to Y X One puts
X
n m n m
O I I v s A v s v s
e
D
n m
e
in In the transversal case one has One may handle with D similar as with D
O k t s O
k v s
D
e
D
where k v s k t s via v ts s s and with J I I
The Basic Construction
For a closed immersion i Y X we de ne
p
J i J X Y X N X
Y
as the comp osite of
ftg
p p p
X X A n f g X A n f g N X
Y
Here X A n f g X is the pro jection A Sp ec k t and is the b oundary
map for N X D X Y One has J X Y so that J X Y is a homomor
Y
phism of complexes C X M C N X M
Y
If M K then the restriction of J X Y to the classical cycle groups coincides
with the sp ecialization homomorphisms of Fulton Chap this may b e
deduced from the description of in Fulton Prop via Cartier divisors
As for classical cycles one may think of J X Y as the pull back along tubular
neighborho o ds followed by a linearization pro cess In the following we have collected
the remarks on J X Y which are needed in further sections We have not tried to
give a detailed geometrical description
The construction of J X Y is lo cal in the sense that
J U Y U j J X Y
where j U X is an op en immersion and N X j Y U N X is the induced
Y Y
inclusion
Lemma Let Y N X be the inclusion Then
Y
J X Y i
Proof The statement follows from Lemma and the fact that the closure of Y
A n f g in D X Y is Y A
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Chow Groups with Coefficients
Let X b e normal y X and Y fy g Moreover let F and E b e the function
elds of X and N X resp ectively We want to compute the co dimension comp onent
Y
J M F M E
of J X Y The problem is purely lo cal in y Let v b e the valuation on F corre
sp onding to y and let y v Moreover let m b e the ideal of y let m
b e a prime and let m m b e its image The normal cone N X is the sp ectrum of
Y
a
n n
m m
n
and one has E
The following lemma shows that there is a factorization
p
J M F M v M E
where p is from Remark
Lemma
J r s f g r
v
E j E j
v
Proof We may supp ose X Sp ec A and that the ideal I corresp onding to y is
generated by Then D X Y is the sp ectrum of
A t t A t t
By de nition we have
J ftg r
w
F t jF
where w is the valuation on F t corresp onding to the principal ideal t A t t
Note that E w and that is the residue of the w unit t by the sign
convention in The claim follows now from
ftg r f g r f t g r
w w w
F t jF F t jF F t jF
r f g f g r
w w
F t jF F t jF
and the fact that w restricts on F to v
The preceding remarks yield a complete description of J X Y for smo oth
curves X
The rest of the section contains a series of technical lemmata
Lemma Let Y X be a closed immersion let g V X be at of constant
relative dimension and let
N g N V Y V N X Y V N X Y
X X
be the projection Then
J V Y V g N g J X Y
X
Proof This follows from the atness of D V Y V D X Y
X
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Markus Rost
Lemma Let U V be a closed immersion and let p V W be at Suppose
that the composite
q N V U V W
U
is at of the same relative dimension as p Then
p
J V U p q W N V
U
Proof Let W A n f g W b e the pro jection and let f b e the comp osite
p id
f D V U V A W A
Then by de nition
J V U p ftg f j V A n f g
Now f is at by Remark and f j N V q Hence
U
J V U p q ftg
where is the b oundary map corresp onding to W f g W A But ftg
id by Lemma
Lemma Let U V be a regular imbedding let p V W be smooth of con
stant relative dimension and suppose p i is a regular imbedding Then the projection
q N V N W
U U
is an epimorphism of vector bund les and
J V U p q J W U
Proof Use the atness of D V U D W U
Lemma Let T T be a morphism let T T T be closed subschemes
0
T for i and let T T
i T
i
e e
Put T T n T T T T T T T n T T T n T and let
be the boundary maps for the closed immersions
e e
T T n T T T T T n T T T
respectively Then
p
T T
under a ltration preserving homotopy of degree
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Chow Groups with Coefficients
Proof Corresp onding to the set theoretic decomp osition of T we have
e e
C T M C T M C T M C T M C T M
Let
p
T T
b e the corresp onding comp onent of d Then d d gives
T T T
Hence is a homotopy as required
D X Y Z T D j f g A T D j A f g We keep the Let T D
notations of Lemma Then T X A n f g A n and T D j f g
Let T X b e the pro jection and let t s b e the co ordinates of A as in
so that T ft g T fs g
Lemma Let Z Y X be regular imbeddings Then
ft sg J N X N X j Z J X Y
Y Y
fs tg J N X N Y J X Z
Z Z
Proof Let
i
X A n f g A n f g X A n f g X A n f g X
i
b e the pro jections with
x t s x s x s x
x t s x t x t x
One nds using in particular Lemma
ft sg fsg ftg
fsg J X A n Y A n f g
fsg N X A n f g N X J X Y
Y Y
J N X N X j Z J X Y
Y Y
fsg ftg fs tg
ftg J X A n f g Z A n f g
J X Z ftg N X A n f g N X
Z Z
J N X N Y J X Z
Z Z
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Markus Rost
The Pull back Map
In this section we de ne the pull back maps for morphisms to smo oth varieties Some
prop erties are formulated in particular the functoriality of the sp ectral sequences
We conclude with applications and discussions concerning birational questions The
pro ofs of Theorems and are given in the next section
In the following all schemes X Y X are at over B of some constant relative
dimension denoted by dim X All pro ducts Y X Y X are taken over B
B
and cycle mo dules will b e induced via pro jection to the second factor pro jection to
the rst factor do es not exist for us We use the notations T X N X X and
S X S
TX T X We are primarily interested in the case B Sp ec k but we don t have
B
to pay much for considering arbitrary B M is a cycle mo dule over X
Let X b e smo oth over B Then TX is a vector bundle on X For a morphism
f Y X let
p
i
f Y Y X X
b e the factorization with i y y f y and p y x x Then i is a regular
imbedding and N Y X f TX
Y
We choose a co ordination on TX and de ne
p
I f I f r f J Y X Y p X Y
Note that the construction is lo cal in the sense that for an op en immersion j U X
one has
I f j j I f
where f f U U is the restriction of f and f U Y is the inclusion
One has I f and
I f C X M C Y M
p p r
where r dim Y dim X If B is equidimensional then
B B
p p
I f C X M C Y M
We de ne
q
f A X M A Y M
p p r
and
q
p p
f A X M A Y M
q
as the induced maps on co homology f do es not dep end on the choice of One
has the following prop erties
Theorem For g Z Y and f Y X with X and Y smooth over B one
q q q
has f g g f
For the pro of see the next section
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Chow Groups with Coefficients
Proposition If f is at then I f f
Proof It su ces to show
J Y X Y p f
where f TX Y is the pro jection For this apply Lemma with U Y
V Y X and W X
Proposition If i Y X is a regular imbedding and X is smooth over B
p
then I i is homotopic to r J i where r is any retraction to Y N X
Y
Proof Apply Lemma with U Y V Y X and W X
The following corollary applied to the blow up at x implies together with The
orem prop erty E of Section
p
Corollary Let X be smooth over B Sp ec k let x X and let
fxg X i
x
be the inclusion Moreover let be any regular sequence at x and let
p
v v be the induced sequence of valuations with the fraction elds k X
p
v v and with the residue classes of as primes Then
p p
q
i A X M A fxg M
x
is the restriction of
1 p
M k X M x s s
v v
1 p
Proof Let X X X X b e the sequence of smo oth schemes lo cally
p
around x with X de ned by h i Using Theorem one reduces to p
i i
This case follows from Prop osition and Lemma
Proposition Pro jection formula Consider a pul l back square
f
X Y
h
y y
h
f
Y X
with h smooth and proper and with X smooth over B Then
q q
h f f h
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Markus Rost
Proof One considers the diagram
Y X X Y
y
h
Y Y X X
h
h
y y y
Y Y X X
Here the b ottom diagram is the pull back along h and h h id We have three
X
p
Y the rst is constructed along D Y maps X X Y and is given by pull back
X and sp ecialization to Y followed by push forward the second go es along to Y
D Y X Y and is given by pull back to Y X and sp ecialization to Y followed by
push forward the third go es along D Y X Y and is given by push forward pull
back to Y X and sp ecialization to Y The rst two may b e related using Lemma
V Y X and W Y X the last two by the compatibility of the with U Y
constructions with prop er push forward
Consider the triangle and assume that is smo oth and is at De ne
p p
h B A M A M
q q
by h B h Here we understand B Sp ec F in the de nition of h
F F F
Proposition h B is a homomorphism of cycle modules over B
Proof We apply Prop osition Since the pro jection N Y X Y Y is a
vector bundle we know that
q q
A A
is an isomorphism of cycle mo dules Moreover
h B ftg p
where p Y X A n f g X is the pro jection and is the b oundary for
N Y X Y D Y X Y
Theorem Consider the square and its decomposition Suppose
that B is equidimensional is at is smooth and X hence also X is smooth
over B Then the spectral sequences
p q
p q p q
E A X A M A X M
p q
p q p q
E A Y A M A Y M
commute with the maps
q
0
0
f Y
f
#
p q p q p q
A X A M A Y A M A Y A M
q
p p
f A X M A Y M
For the pro of see the next section Switching to dimension indices this theorem holds
without the equidimensionality assumption on B
Documenta Mathematica
Chow Groups with Coefficients
For the rest of the section we assume B Sp ec k
Lemma Let X be smooth let Y be integral let f Y X be a dominant
morphism and let k X k Y be the induced homomorphism of the function
elds Then
I f j M k X M k X M k Y
Proof After replacing Y by an op en subset we may assume that f is at The claim
follows from Prop osition
Lemma Assume in additional ly that f is proper and that is an iso
morphism Then f I f id
Proof Let
f D Y X Y D X X X
b e the prop er map induced from f There is the commutative diagram
ftg r f
p p p
n f g TX X Y X A f Y
p p p p
f
f
f
y y y y
ftg r
p p p
X X X A n f g TX X
where f f are the restrictions of f The diagram shows f I f I id But
I id id by Prop osition
Lemma shows in particular that for any blow up Y X the complex
C X M is a direct summand of C Y M This splitting via I f dep ends alone
on the choice of a co ordination of TX near the singular lo cus and is unique up to
homotopy
Corollary Let X be a proper smooth variety over k and let M be a cycle
module over k Then the group A X M is a birational invariant of X
Proof If X X are prop er and birational isomorphic there exist a prop er Y and
birational morphisms Y X take for Y the closure in X X of a common
i
M one has the trivial op en subset of the X Then as subgroups of M
Y i X
i
inclusions A Y M A X M Lemma shows A X M A Y M
i i
For an illustration let X b e a smo oth and prop er variety over k with function
eld F Then for any geometric valuation v on F of rank there is a birational
morphism f Y X such that v has center y in Y with v y The map I f
yields a formula
X
x
x v
v
(1)
x X
where
x
M x M v
v
x
equals the comp onent I f
y
Documenta Mathematica
Markus Rost
This formula is a sort of higher dimensional analogue of the sum formula for
one dimensional function elds It has the following prop erties
where z is the center of v in X in it is lo cal that is x runs only through X
z
x
other words for x X
v
z
it is not unique but dep ends only on the choice of a co ordination of the tangent
bundle of X restricted to X
z
x
it is universal in the sense that the can b e written as sums of comp ositions of the
v
data of cycle mo dules indep endent of M This is quite obvious from the construction
x
of I f One can make this more precise by interpreting the as morphisms in the
v
e e
category F of Remark In this way the category F app ears as the natural place
x
for the co e cients of formulas like
y
x
Exercise Describ e the for dim X and v the valuation corresp onding to the
v
exceptional b er of the blow up in a closed p oint see Remark
Birational invariants like A X M have b een considered in various contexts
like etale cohomology and K theory see Colliot Th el ene for a survey The
advantage of the metho d of pro of of Corollary lies in its general and essentially
simple nature after having established the functors in question as cycle mo dules
moreover the formula makes things p erhaps more enlightening A similar
metho d works probably for functors related with the Witt ring of quadratic forms
To mention a particular example let Z Sp ec k b e prop er and let M
resp N b e the Z graded cycle mo dule over k given as the cokernel resp image of
A K A id K K
Sp ec k
By Corollary the group A X M which is a sub quotient of k X is a
birational invariant for prop er smo oth X over k The pro of of this fact was the main
aim of Rost There it was achieved by a di erent metho d using the triviality of
e e
A X N for smo oth lo cal X proved in this pap er in more generality in Section
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Chow Groups with Coefficients
Intersection Theory for Fibrations
The purp ose of this section is to prove Theorems and We de ne pull back
maps on complex level for regular imbeddings and for morphisms to smo oth varieties
in b ered situations Moreover we establish functoriality of the constructions Most
of the work has b een done already in Sections
Consider a commutative square
f
Y X
y y
0
f
Y X
The square is called a regular imbedding if f and f are regular imbeddings of
some constant co dimensions and if the induced map
0
X p N X N
Y Y
is an epimorphism of vector bundles over Y The kernel bundle of p is denoted by N
0
We consider p also as vector bundle and identify it with q N where q N X Y
Y
is the pro jection
Let b e a regular imbedding and let and b e co ordinations of N Y and
0
X Y resp ectively We de ne of N
Y
p
X Y J
by
J r r q J X Y J
The following is clear from Sections One has J and
J C C
p l p s l t
with s co dim f and t co dim f Moreover the homotopy class of J
with resp ect to the degree s t do es not dep end on the choice of and
we wanted to b e as canonical as p ossible If one is In the de nition of J
interested only in the homotopy class one may put J r J X Y for any
ltration preserving retraction r to N X X ltration preserving means always
Y
with resp ect to the natural degrees
Next consider a diagram
f f
1 2
Z Y X
y y y
0 0
f f
1 2
Z Y X
We denote the left square by the right square by and the comp osed square
by
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Markus Rost
Theorem If and are regular imbeddings then is a regular imbed
ding and
J J J
under a ltration preserving homotopy
Proof The rst statement is straightforward In the following deduction of the second
statement the letters r stand for some ltration preserving retractions We make use
i
of Lemma for the homotopy of Lemmata and for the homotopy
and of Lemma for the equality
J r J Y Z r J X Y J
r N N X N X j Z N Y Z J Y Z r J X Y
Y Y
r J N X N X j Z N Y X Y r J X Y
Y Y
r J N X N X j Z J X Y
Y Y
r J N X N Y J X Z
Z Z
r J N X N Y N X Z r J X Z
Z Z Z
r N N X N Y Z r J X Z
Z Z
r J X Z
J
The square is called admissible if is at is smo oth and X is smo oth
over B Consider the diagram
i
Y Y X X
y y y
0
0
i
X Y X Y
and denote the left square by and the right square by If is admissible then
i p
is a regular imbedding and is a at square see Moreover the normal
i p
is given bundles of i and i are given by f TX and f TX resp ectively and N
i
0
by f T X
X
0
X and TX resp ec Let b e admissible and let and b e co ordinations of T
X
tively We de ne
p
I X Y
by
I I J f f p
i
One has I and
C C I
p l p s l t
with s dim Y dim X and t dim Y dim X Moreover the homotopy
B B B B
class of I with resp ect to the degree s t do es not dep end on the choice of
and
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Chow Groups with Coefficients
Theorem If in the squares and are admissible then the
square is admissible and
I I I
under a ltration preserving homotopy
Proof The rst statement is trivial For the second we consider the diagrams
i i
4 1
Z Y X Z Y Z
p p
4 1
y y
i
2
Y X Y
p
2
y
X
p
i i
5
5 3
Z Y X Z X X Z
The regular imbeddings i lie over accordingly de ned regular imbeddings i the
j
j
corresp onding squares are denoted by
j
The are regular imbeddings An application of Lemma and Theorem
j
shows that by noting i i i i and p p p
I J J p I
By de nition we have
I J p i
Finally Lemma shows
p p i J
Theorem implies Theorem by passing to homology For a pro of of
Theorem we consider six squares with the top arrows
i i
2 1
0
Y X X Y Y
X
p p
2 1
y y
p
i
3
3
0
Y X Y X X
X
lying over the b ottom arrows
0
i
Y Y Y X
0
0
p
i
Y Y X X
Documenta Mathematica
Markus Rost
Let b e the square corresp onding to i for j The are regular imbed
j j j
q
By the de nition of I and Theorem dings The map f is induced from I
we have
I J J p p
Lemma shows that
p p J J
q
Therefore f is the comp osition of the maps induced by J p and by J p
Next note that p i f An application of Prop osition shows that J
p induces on the E terms the map f Y Finally note that p i f and that
the squares under i and p are pull back squares An application of Prop osition
q
shows that J p induces on the E terms the map f
Products
In this section M is a cycle mo dule over B and N is a cycle mo dule over k We
assume that either N K or that M N is a cycle mo dule with ring structure over
B Sp ec k So in any case we are given a pairing N M M of cycle mo dules
over B
The restriction to these sp ecial cases are made for simpli cation For example
in forming intersections of cycles with co e cients in a cycle mo dule M with ring
structure one needs to know that its pairing factors through a cycle mo dule over
B B However the existence of a corresp onding appropriate notion of tensor pro duct
of cycle mo dules is not clear to me and a settling of this question would lead to far here
anyway The problem could b e avoided in the following by assuming the necessary
factorizations but this is somewhat tiring
Cross products Let Y b e a scheme over k and let Z b e a scheme over B
all of nite type over k We de ne the cross pro duct
C Y N C Z M C Y Z M
p q p q
as follows For y Y let Z Sp ec y Z let Z Z b e the pro jection and let
y y y
i Z Y Z b e the inclusion For z Z we understand similarly Y Y Y
y y z z z
and i Y Y Z We give the following two equivalent de nitions
z z
X
i
y y
y
y Y
(p)
X
i
z z
z
z Z
(q )
Here N y is the y comp onent of and the pro duct is understo o d after p ointwise
y
restriction of The map
y
i C Z M C Y Z M
y q y p q
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Chow Groups with Coefficients
is the inclusion corresp onding to Z Y Z Similarly we understand
y q p q
M z and
z
i C Y M C Y Z M
z p z p q
To check equality of the two de nitions consider the u comp onents for u Y Z Let
y z b e the images of u under the pro jections Y Z Y Z and let R y z
k
Then the u comp onents are either trivial or u is a minimal prime of R In the latter
case the u comp onents are given by
r r l R
u y z R
u j y u j z u
Associativity Additionally let X b e of nite type over k and let
C X N Then
r
For a pro of consider the u comp onents for u X Y Z Let x y z b e the
images of u in X Y Z resp ectively and let R x y z Then the
k k
u comp onents are either trivial or u is a minimal prime of R In the latter case it
follows from standard rules for length that the u comp onents are given by
r r r l R
u x y z R
u j x u j y u j z u
Commutativity Supp ose M N is a cycle mo dule with ring structure over
B Sp ec k Let Y Z Z Y b e the interchange of factors For C Y M n
p
and C Z M m one has
q
nm
C Z Y M n m
p q
This is immediate from the de nitions
Chain rule For C Y N n and C Z M m one has
p q
n
d d d
For a pro of we may assume M y M z for some y Y and z Z
p q
Consider for u Y Z the u comp onents of the three terms If one of them is
nontrivial we must have dim u Y Z p q and the images y z of u must b e
in the closures of y z resp ectively Dimension reasons show y y or z z Now the
claim follows from one of the two de nitions of the cross pro duct and Prop osition
Compatibility The cross pro duct is compatible with the four basic types
of maps f g fag and acting on one of the two factors This follows from the
compatibility with at pull back and De nition We omit a detailed formulation
We conclude with a consideration of the intersection pairing for cycles on a
smo oth variety Let X b e smo oth over k and let b e a co ordination of TX We
de ne
I C X N C X M C X M
X
I r J X X X
X
By this is a pairing of complexes Let
A X N A X M A X M
b e the induced pairing
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Markus Rost
The next theorem follows from the preceding remarks and in particular from
Theorem It holds accordingly on chain level up to homotopy
Theorem If M N is a cycle module with ring structure over B Sp ec k
the pairing turns A X M into an anti commutative associative ring If N K
the pairing turns A X M into a module over A X K
We have de ned in particular a ring structure on the classical Chow groups
a
p
CH X A X K p
p
of a smo oth variety This ring structure coincides with the classical one This
may b e deduced from the remark at the b eginning of Section and Fulton
Chaps
References
Bass H Tate J The Milnor ring of a global eld in Algebraic K Theory I I
Classical Algebraic K Theory and Connections with Arithmetic Pro ceedings
Battelle Institute Lecture Notes in Math
Blo ch S Algebraic cycles and higher algebraic K theory Adv Math
Blo ch S Ogus A Gersten s conjecture and the homology of schemes Ann
scient Ec Norm Sup
Colliot Th el ene J L Birational invariants purity and the Gersten conjecture
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