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Computations for bivariant cycle cohomology

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Computations for bivariant cycle cohomology

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Content COMPUTATIONS FOR BIV ARIANT CYCLE COHOMOLOGY
by
Taylan Bilal
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
August 2013
Copyright 2013 Taylan Bilal
Dedication
To Gulsevin, Haluk and Melis.
ii
Acknowledgments
First of all, I would like to thank my advisor, Eric Friedlander, for his guidance and support.
Without his patience, insight and generosity, my thesis would not exist. Thank you!
I would also like to thank professors Ko Honda, Aravind Asok, Francis Bonahon,
Thomas Geisser, Robert Guralnick, Susan Montgomery and Edmond Jonckhere for the
knowledge and support they have so generously offered.
I want to thank Andrei Pavelescu for all his support through the years. Adam Ericksen,
Changlong Zhong and Joe Timmer; thank you so much for valuable math discussions and
your friendship. Mihaela Ignatova, Rado Marinov, Ibrahim Ekren, Ozlem Ejder, Bahar
Acu, Burcin Oztuna, Can Hankendi; thank you all so much.
A special thank you goes to my parents. Also, a huge thank you to my cousin Murat
Karakas and his lovely wife Charlotte Jane Thomas. I thank you all for providing me with
all your love and support.
Last but not least, I want to thank my spouse, Melis. Through these years you were a
constant source of love and inspiration; You are truly amazing.
iii
Table of Contents
Dedication ii
Acknowledgments iii
Abstract v
Chapter 1 Introduction 1
Chapter 2 Background 4
2.1 Chow Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Algebraic simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Naïve Bivariant Cycle Cohomology . . . . . . . . . . . . . . . . . . . . . 7
2.4 cdh-topology onSch=k . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Pretheories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Bivariant Cycle Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter 3 Some Computations Using Theorem 2.6.3 19
3.1 Blow-ups of planar curves . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Boundary of the algebraicn-simplex in the first variable . . . . . . . . . . 24
Chapter 4 Continuous Algebraic Maps 31
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Continuous Algebraic Maps in Codimension 1 . . . . . . . . . . . . . . . . 33
Chapter 5 Bivariant Cycle Cohomology in Codimension 1 44
Chapter 6 Pseudo-Flasqueness and Conjectures 49
6.1 Generalizing Theorem 5.0.14 . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Naïve Version onSch=k . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References 54
iv
Abstract
E. Friedlander and V . V oevodsky introduced Bivariant Cycle Cohomology in 1995. The
theory relates to many other known cycle theoretic functors, such as Chow groups and
motivic cohomology. I provide various computations for Bivariant Cycle Cohomology,
first explicitly, then using some properties of the theory, and then by comparing it to other
theories. I give a description of the Bivariant Cycle Cohomology groups in codimension 1,
using special rational functions, units and the Chow group.
v
Chapter 1
Introduction
Bivariant cycle cohomology was introduced by E. Friedlander and V . V oevodsky in
1995 [3] as a successor to the graded Chow group A

(X). Recall that the graded Chow
group is an invariant for schemes X of finite type over a field k, defined as the rational
equivalence classes of algebraic cycles onX. The grading comes from the dimension of
the cycle. S. Bloch then introduced the higher Chow groups, CH
q
(X;p) in [1]. This
generalized Grothendieck’s relation between K
0
(X) and A

(X) where X is nonsingular
(i.e. the fact that the two theories are isomorphic after tensoring withQ) to the existence of
a spectral sequence converging to higher Algebraic K-Theory ofX.
In his paper [10] published in 1989, B. Lawson introduced the Lawson homology, a
theory defined for algebraic varieties overC. He considered topological spaces of algebraic
cycles, and his theory related to algebraic equivalence rather than rational equivalence.
This could be thought of as a topological analogue of higher Chow groups. In [5], E.
Friedlander and H.B. Lawson developed a bivariant extension of Lawson homology, which
they denoted asL

H

(Y;X).
In [6], E. Friedlander and O. Gabber briefly introduced a theory called “algebraic bivari-
ant cycle theory”. This was defined in terms of continuous algebraic maps, which we will
go into detail in Section 4.1.1. The Bivariant Cycle Cohomology is a more sophisticated
version of this theory, having numerous good properties such as localization, cdh-descent
and Gysin. See Theorem 2.6.3 for more details. The bivariance, duality and pairings for
this theory should be useful in various contexts.
1
The Bivariant Cycle Cohomology is defined using a complex of presheavesC

z
equi
(X;r)
defined in terms of equidimensional cycles. Specializing toC

z
equi
(A
i
; 0)[2i], one recov-
ers the Suslin-Friedlander motivic complexes calledZ
SF
(i), i 0. When the base field
k is perfect, these complexes are quasi-isomorphic to V . V oevodsky’s motivic complexes
Z(i), and in fact the Bivariant Cycle Cohomology groups agree with motivic cohomology
groups for a smooth varietyU over a fieldk (see [3, 9.2]), that is:
H
m
(U;Z(s)) =A
0;2sm
(U;A
s
)
Another use of the presheavesz
equi
(X; 0) is in the definition of motives with compact sup-
port, specifically for any scheme X of finite type over k, the motive of X with compact
support isz
equi
(X; 0) considered as an object ofDM
eff;
Nis
(k). Motives with compact sup-
ports are used to prove the cancellation theorem[11, 16.25], which allows one to embed
effective motives into the triangulated category of all motives.
This paper aims to carry out computations for Bivariant Cycle Cohomology in a few
accessible cases. The Bivariant Cycle Cohomology for a pair (X;Y ) of schemes of finite
type over a fieldk is defined to be
A
r;i
(Y;X) :=H
i
cdh

Y; [C

(z
equi
(X;r))]
cdh

:
Some of the tools I use within the computations are the exact sequences listed in Theorem
2.6.3. The imposed cdh-descent in the definition of Bivariant Cycle Cohomology allows
one to compute the theory in terms of other Bivariant Cycle Cohomology groups for a
simpler pair of schemes. Moreover, the close relationship between equidimensional cycles
and the groups that the Chow variety represents allows us to compare the Bivariant Cycle
Cohomology to another theory defined in terms of “continuous algebraic maps”.
2
Chapter 2 establishes the conventions, and incorporates the framework on which I base
the computations in the paper. It also summarizes the nice properties of the Bivariant Cycle
Cohomology proven in [3].
Chapter 3 uses some of those properties to carry out explicit computations of Bivariant
Cycle Cohomology in some specific cases. The computations are interesting in the sense
that they involve nonsingular varieties as theY variable, given that the theory is much bet-
ter understood when that variable is smooth. To be precise, when Y is smooth, there is
another formulation of Bivariant Cycle Cohomology that Friedlander and V oevodsky call
the “naïve” version, which is much simpler to understand. The main result of those com-
putations is the fact that we may get nonzero Bivariant Cycle Cohomology in the negative
degrees whenY is nonsingular, whereas the “naïve” version vanishes in those degrees.
In [4], Friedlander computes the cohomology of a chain complex, defined in terms of
the Chow variety and continuous algebraic maps. This complex is very similar to the naïve
Bivariant Cycle Cohomology; the relation between the two stems from the correspondence
between effective cycles in YX that are equidimensional and of relative dimension r
over Y , and morphisms Y ! C
r
(X) where C
r
(X) is the Chow variety of effective r-
cycles on a projective varietyX. Chapter 4.1.1 generalizes the results in [4], and concludes
in Theorem 4.2.9 and Corollary 4.2.10. These results compute the theory in codimension 1
for a pair (X;Y ) whereY is smooth and quasi-projective. Chapter 5 then ties the results in
Section 4.1.1 to the modern language of Bivariant Cycle Cohomology. Theorem 5.0.14 is
the main result, and it is the analogue of Corollary 4.2.10 for the world of Bivariant Cycle
Cohomology.
Chapter 6 contains unanswered questions and possible future research topics, related to
the link between continuous algebraic maps and Bivariant Cycle Cohomology.
3
Chapter 2
Background
Throughout this paper, we will work with separated schemes of finite type over a field that
we will denote byk, that is, the categorySch=k. Often, we will restrict our attention to the
categorySm=k of smooth schemes overk. Most of the time, we will assume that the field
k has characteristic 0.
Recall that, anS-scheme of finite typeZ is said to be “equidimensional" overS of rela-
tive dimensionr, if every irreducible component ofZ dominates a connected component of
S and for every points ofS, the fiberZ
s
is either empty or each of its components have di-
mensionr. We say a cycle is equidimensional overU if the elementary cycles constituting
the cycle are all equidimensional overU.
LetX be a scheme of finite type overk, a field, and letr 0 be an integer. Friedlander
and V oevodsky consider the following definition:
Definition 2.0.1. For every smooth schemeU overk, letz
equi
(X;r)(U) be the free abelian
group generated by the closed irreducible reduced subschemes ofXU, which are equidi-
mensional and of relative dimensionr overU. Let alsoz
eff
equi
(X;r)(U) be the sub-monoid
ofz
equi
(X;r)(U), consisting of effective cycles, andz
eff
equi
(X;r;d)(U) be the sub-monoid
consisting of effective cycles of degreed.
So for example ifrdim(X) + 1,z
equi
(X;r)() is the zero group.
If U
0
! U is any map of smooth schemes over k, the pullback of relative cycles
induces a natural map
z
equi
(X;r)(U)!z
equi
(X;r)(U
0
):
4
This givesz
equi
(X;r) a structure of a presheaf on the categorySm=k [11, 16.1]
In order to better understand the presheaf structure, let’s consider another approach to
equidimensional cycles. Under some hypotheses, the functor z
equi
(X;r) turns out to be
represented by the Chow variety. Let us now recall the classical construction.
2.1 Chow Variety
Let X be a projective scheme over k, and fix a closed embedding X P
m
k
= P
m
.
The dual (P
m
)

of the projectivem-space parametrizes hyperplanes inP
m
, so ((P
m
)

)
r+1
parametrizes families of (r + 1)-uples of hyperplanes (L
0
;:::;L
r
) inP
m
. One can then
consider the incidence correspondence
=f(x; (L
0
;:::;L
r
)) :x2L
0
\\L
r
gP
m
((P
m
)

)
r+1
It is easy to see that the projection onto the first coordinate
pr
1
: !P
m
is smooth. Therefore, for any cycleZ of dimension r in P
m
, there is a defined cycle
Chow(Z) = (pr
2
)

(pr
1
)

(Z) on (P
m
)
r+1
. One can see thatChow(Z) is of codimension
1, and ifdeg(Z) = d thenChow(Z) is of multi-degree (d;:::;d). It is known that effec-
tive cycles of codimension 1 and given multi-degree on a product of projective spaces are
parametrized by another projective space, therefore we get an injection
z
eff
equi
(P
m
;r;d)(Spec

k

),!P
N
(

k)
5
(See Definition 2.0.1) for some integerN depending onm;r andd. The image of this injec-
tion coincides with the

k-points of a closed reduced subscheme, which is called the Chow
variety of effective cycles of degreed and dimensionr inP
m
, denotedC
r;d
(P
m
). Moreover,
the set of

k-points ofC
r;d
(P
m
) corresponding to cycles supported onX corresponds to

k-
points of another closed reduced subscheme, called the Chow variety of effective cycles of
degreed and dimensionr inX, and denotedC
r;d
(X).
The Chow varietyC
r
(X) of effectiver-cycles onX is then defined to be:
C
r
(X) :=
a
d0
C
r;d
(X)
The following is the well-known result signifying the importance of Chow varieties.
Theorem 2.1.1. Let k be a field of characteristic 0, X be a projective scheme, U be a
smooth quasi-projective scheme over k. Then there exists a one-to-one correspondence
between effective equidimensional cycles onUX that are equidimensional of relative
dimensionr overU, and morphisms fromU to the Chow varietyC
r
(X).
The correspondence actually goes deeper, but this is the version we will use in this
paper. For a proof of this Theorem and more, see [2, 1.4]. The reader can refer to [12] for
more details.
Under the hypotheses of Theorem 2.1.1, we can now reinterpret the presheaf struc-
ture of z
equi
(X;r). Let U
0
! U be a map of smooth schemes over k, and let Z 2
z
equi
(X;r)(U) be an elementary cycle inXU that is equidimensional overU, of relative
dimensionr. We know now thatZ corresponds to a morphism

Z
:U!C
r
(X):
This morphism can be composed to get a morphismU
0
!C
r
(X), which in turn gives
6
us a cycleZ
0
inU
0
X, equidimensional overU
0
of relative dimensionr. This is the same
cycle obtained by pullback of relative cycles.
2.2 Algebraic simplices
Recall the algebro-geometrical analogue of then-simplex
n
is

n
= Spec

k[t
0
;:::;t
n
]=(
X
t
i
1)

:
Considering the natural face and degeneracy maps, one obtains a co-simplicial object in
Sch=k, orSm=k. This allows one to get a simplicial presheaf out of a presheaf on these
categories, via
F ()!F (

)
Let C

be the functor that takes a presheaf (on Sch=k or Sm=k) and sends it to the
chain complex of presheaves associated to the simplicial presheaf above, i.e. for a presheaf
G, then
th
term in the complex is
C
n
(G)(Y ) =G(Y
n
)
and the boundary maps are given by alternating sums of face maps.
2.3 Naïve Bivariant Cycle Cohomology
One thing one can do at this point is to applyC

toz
equi
(X;r) and take homology. Fried-
lander and V oevodsky called this the “naïve” version of Bivariant Cycle Cohomology. The
7
i
th
naïve Bivariant Cycle Cohomology group is then
A
naive
r;i
(U;X) =h
i
(C

(z
equi
(X;r)) (U))
By doing so, it is very hard to prove a lot of properties that would be desirable in a coho-
mology theory, such as localization, cdh-descent etc. In fact, the next construction in [3] is
first extending the theory fromSm=k toSch=k and imposing cdh-descent by doing so.
2.4 cdh-topology onSch=k
Recall that an abstract blow-up is a pair (i
Z
:Z!X; :X
0
!X) such that i
Z
is a
closed immersion of positive codimension and is a proper surjective map such that the
restriction

1
(XnZ)
red
! (XnZ)
red
:
is an isomorphism.Z is called the “center” of the blow-up.
Thecdh-topology onSch=k is then the minimal Grothendieck topology admitting the
following as coverings: Nisnevich coverings together with the maps;
Z
`
X
0
X
i
Z
`

where the pair (i
Z
;) is an abtract blow-up.
Remark 2.4.1. For any schemeX, the closed embeddingX
red
! X is a cdh-covering.
So there is no difference betweenX andX
red
when working with cdh-topology. Also, it is
useful to observe that cdh-topology permits abstract blow-ups, blow-ups, disjoint union of
the irreducible components of a reducible scheme as coverings.
From now on, we will work with fields k that have resolution of singularities in the
8
following sense:
1. For any scheme of finite typeX overk, there is a proper surjective morphismY!
X with Y being a smooth scheme overk.
2. For any smooth scheme X over k, and an abstract blow-up q : X
0
! X, there
exists a sequence of blow-ups
p :X
n
!:::!X
1
=X
with smooth centers, such thatp factors throughq.
It is well known that any field of characteristic zero admits resolution of singularities in
the above sense. See [9] for more on that.
A cdh-sheaf on (Sch=k)
cdh
is then a presheaf satisfying the sheaf axiom with respect
to cdh-coverings.
Observe that there is a natural morphism of sites
: (Sch=k)
cdh
! (Sm=k)
Zar
Then we have the functor of inverse image of sheaves

:Shv((Sm=k)
Zar
)!Shv((Sch=k)
cdh
)
which sends a Zariski sheafF onSm=k to the cdh-sheafification of the following presheaf
onSch=k:
Y7! lim
!
Y!U
F (U)
whereU runs over smooth schemes overk.
9
Lemma 2.4.2. Letk be a field admitting resolution of singularities. Then the functor of
inverse image

:Shv((Sm=k)
Zar
)!Shv((Sch=k)
cdh
)
is exact.
Proof. See Lemma 3.6 in [3].
In an abuse of notation, for any presheafF onSm=k, we will denote

(F
Zar
) byF
cdh
.
For any schemeY of finite type, one can now define cdh-cohomology onY of a cdh-
sheafG by
H
i
cdh
(Y;F ) :=Ext
i
(Z(Y )
cdh
;F ):
Here,Z(Y ) is the presheaf of abelian groups on Sch=k freely generated by the sheaf of
sets represented by Y . More generally, for any cochain complex K of cdh-sheaves, one
can define
H
i
cdh
(Y;K) :=Hom(Z(Y )
cdh
;K[i])
2.5 Pretheories
The presheaves z
equi
(X;r) fall into a special subclass of presheaves called “pretheories”
defined below, which behave very nicely and allow one to prove various technical lem-
mas that end up in theorems such as existence of Mayer-Vietoris and localization exact
sequences for Bivariant Cycle Cohomology.
LetU be a smooth scheme overk andC!U a smooth curve overU. Letc(C=U; 0)
be the free abelian group generated by integral closed subschemes inC that are finite over
U and dominant over an irreducible component ofU. Observe, for any morphism of smooth
10
schemesf :U
0
!U, there is a homomorphism
c(C=U; 0)!c(C
U
U
0
=U
0
; 0)
For any sections : U! C, the image is obviously inc(C=U; 0). Let’s denote this
image bys in an abuse of notation.
Definition 2.5.1. A “pretheory onSm=k” is a presheaf of abelian groups onSm=k together
with a family of homomorphisms

C=U
:c(C=U)!Hom(F (C);F (U))
satisfying
1. IfU
1
,U
2
are smooth schemes overk, the canonical morphism
F (U
1
a
U
2
)!F (U
1
)F (U
2
)
is an isomorphism.
2. For any section s : U ! C, the map F (s) : F (C)! F (U) equals
C=U
(s) :
F (C)!F (U).
3. For any morphism f : U
0
! U, any a 2 F (C) and anyZ 2 c(C=U; 0), the
11
following diagram commutes:
F (C) F (C
U
U
0
)
F (U) F (U
0
)

C=U
(Z)
F (C
U
f)
F (f)

C
U
U
0
=U
0(Z)
A morphism of pretheories is a morphism of presheaves which respects the 3 condi-
tions above.
Note that the category of pretheories is abelian and the forgetful functor from prethe-
ories to presheaves of abelian groups is exact [3]. The following is the most important
technical result about pretheories.
Theorem 2.5.2. Let k be a field that admits resolution of singularities, and let F be a
pretheory onSm=k. IfF
cdh
= 0, then [C

(F )]
Zar
is an acyclic chain complex of sheaves
onSm=k.
Proof. This is Theorem 5.5.2 in [3].
Corollary 2.5.3. Let
0!F!G!H! 0
be a sequence of pretheories onSm=k such that the corresponding sequence of cdh-sheaves
onSch=k
0!F
cdh
!G
cdh
!H
cdh
! 0
is exact. Then,
12
1. we have an exact triangle of complexes of Zariski sheaves onSm=k of the form
[C

(F )]
Zar
! [C

(G)]
Zar
! [C

(H)]
Zar
! [C

(F )]
Zar
[1]
2. there is a long exact sequence of abelian groups
:::!h
i
(C

(F ))(Spec(k))!h
i
(C

(G))(Spec(k))!h
i
(C

(H))(Spec(k))
!h
i1
(C

(F ))(Spec(k))!:::
whereh
i
() are the homology presheaves.
3. we have the following exact triangle of complexes of cdh-sheaves onSch=k of the
form
[C

(F )]
cdh
! [C

(G)]
cdh
! [C

(H)]
cdh
! [C

(F )]
cdh
[1]
Proof. First assertion follows from Theorem 2.5.2. Second assertion follows from any
presheaf on Spec(k) being a Zariski sheaf. Third assertion follows from Theorem 2.4.2.
The relevance of pretheories comes from the fact the we can define a structure of prethe-
ory on the presheavesz
equi
(X;r). See the discussion after Lemma 5.6 in [3].
2.6 Bivariant Cycle Cohomology
We are now ready for the definition:
Definition 2.6.1. For any schemesX andY of finite type over a fieldk, the Bivariant Cycle
13
Cohomology groups are defined to be
A
r;i
(Y;X) :=H
i
cdh

Y; [C

(z
equi
(X;r))]
cdh

The negative sign in the definition is just a convention, which makes the chain complex
C

(z
equi
(X;r)) into a cochain complex.
From the definition of both the naïve version and the actual version, it’s easy to check
that in the first variable X, the theories are covariantly functorial with respect to proper
maps (via proper push-forward of cycles), and contravariantly functorial with respect to flat
equidimensional maps (via flat pull-back of cycles) with a shift inr. They are automatically
contravariantly functorial inY .
It’s important to note that while the naïve Bivariant Cycle Cohomology is defined for
pairs (Y;X) whereY is a smooth scheme overk, the definition above works for anyY in
the categorySch=k. This is because the groupsz
equi
(X;r)(Y ) do not carry the structure of
a presheaf for all schemesY of finite type overk,Y needs to be smooth for the restriction
maps to exist. Whereas in the definition above, thecdh-sheafification functor provides us
with a sheaf onSch=k. Another contrast between the two versions is the relative difficulty
of computing the sophisticated version directly. The trade-off is, it’s now easier to prove
nice properties such as Mayer-Vietoris and localization for Bivariant Cycle Cohomology,
using techniques from Homological Algebra.
The result below relates the two versions of Bivariant Cycle Cohomology. As one might
expect, the sophisticated version is actually generalization of the naïve case.
Theorem 2.6.2. Let k have resolution of singularities and let Y be smooth and quasi-
projective. Then, we have
A
naive
r;i
(Y;X) A
r;i
(Y;X)

14
Proof. Theorem 8.1 in [3]
For convenience, here is a list some of the properties of Bivariant Cycle Cohomology.
Theorem 2.6.3. AssumeX,Y as before, and the fieldk has resolution of singularities. Let
r 0.
1. For any schemeX overk, the groupA
naive
r;0
(Spec(k);X) is canonically isomorphic
toA
r
(X), the Chow group ofr-cycles onX.
2. (Mayer-Vietoris inY ) Given a proper map : Y
0
! Y and a closed subscheme
ZY such that the morphism

1
(YnZ)!YnZ
is an isomorphism, we have a long exact sequence of Bivariant Cycle Cohomology
groups;
:::!A
r;i
(Y;X)!A
r;i
(Y
0
;X)A
r;i
(Z;X)!A
r;i
(
1
(Z);X)
!A
r;i1
(Y;X)!:::
Also, iffU
1
;U
2
g is a Zariski open covering ofY , we get the long exact sequence
:::!A
r;i
(Y;X)!A
r;i
(U
1
;X)A
r;i
(U
2
;X)!A
r;i
(U
1
\U
2
;X)
!A
r;i1
(Y;X)!:::
3. (Mayer-Vietoris inX) Let : X
0
! X be a proper map andZ X be a closed
15
subscheme such that the morphism

1
(XnZ)!XnZ
is an isomorphism. Then, there is a canonical long exact sequence
:::!A
r;i
(Y;
1
(Z))!A
r;i
(Y;X
0
)A
r;i
(Y;Z)!A
r;i
(Y;X)
!A
r;i1
(Y;
1
(Z))!:::
Also, iffV
1
;V
2
g is a Zariski open covering ofX, then we have the following exact
sequence
:::!A
r;i
(Y;X)!A
r;i
(V
1
;X)A
r;i
(V
2
;X)!A
r;i
(V
1
\V
2
;X)
!A
r;i1
(Z;X)!:::
4. (Localization inX) LetZX be a closed subscheme ofX, andY be any scheme.
Then,
:::!A
r;i
(Y;Z)!A
r;i
(Y;X)!A
r;i
(Y;XnY )
!A
r;i1
(Y;Z)!:::
5. (Homotopy invariance inY ) The projectionYA
1
!Y induces an isomorphism
A
r;i
(Y;X) A
r;i
(YA
1
;X)

for anyi2Z.
16
6. (Homotopy invariance inX) The pull-back morphism
z
equi
(X;r)!z
equi
(XA
1
;r + 1)
induces an isomorphism
A
r;i
(Y;X) A
r+1;i
(Y;XA
1
)

for anyi2Z.
7. (Suspension) For anyX,Y , the natural homomorphism
i

:z
equi
(X;r + 1)z
equi
(X;r)!z
equi
(XP
1
;r + 1)
where
i :X!XP
1
is a closed embedding and
:XP
1
!X
is the natural projection, induces an isomorphism
A
r+1;i
(Y;X)A
r;i
(Y;X) A
r+1;i
(Y;XP
1
)

8. (Cosuspension) There are canonical isomorphisms:
A
r;i
(YP
1
;X) A
r+1;i
(Y;X)A
r;i
(Y;X)

9. (Gysin) Let Z be a closed immersion of smooth schemes of codimension c in Y .
17
Then, there is a canonical long exact sequence
:::!A
r+c;i
(Z;X)!A
r;i
(Y;X)!A
r;i
(YnZ;X)
!A
r+c;i1
(Z;X)!:::
10. (Motivic Cohomology) IfX is smooth, then
H
m
(X;Z(s)) =A
0;2sm
(X;A
s
)
where the left hand-side is motivic cohomology.
Proof. Assertion 1 follows from the definition.
Assertion 2 is proved in [3, 4.4] and [3, 4.5].
Assertions 3 and 4 are proved in [3, 5.12] and [3, 5.14]
Assertion 5 is proved in [3, 5.9].
Assertions 6, 7, 8 and 9 are proved in [3, 8.3].
For assertion 10, see [3, 9.2].
18
Chapter 3
Some Computations Using Theorem 2.6.3
In this section, we will compute the Bivariant Cycle Cohomology using various properties
listed in Theorem 2.6.3.
Because Bivariant Cycle Cohomology is functorial with respect to the first variableY ,
we will have a splitting for anyk-schemeY with a rational point given by the structure map
Y! Spec(k) and the inclusion of any rational point. We call the complement
~
A
r;i
(Y;X),
so we have
A
r;i
(Y;X) =A
r;i
(Spec(k);X)
~
A
r;i
(Y;X)
3.1 Blow-ups of planar curves
Example 3.1.1. Let’s begin by a simple example. LetC be the nodey
2
=x
2
(x + 1). Then
we have an abstract blow-up
A
1
!C
The center of this abstract blow-up is the point (0; 0), and its inverse image consists of
two points. Therefore, for anyX, using Theorem 2.6.3 Assertion 2, we get
::: ! A
r;i
(C;X)!A
r;i
(A
1
;X)A
r;i
(Spec(k);X)! 2A
r;i
(Spec(k);X)
! A
r;i1
(C;X)!:::
By homotopy invariance inY ,A
r;i
(A
1
;X) =A
r;i
(Spec(k);X) and thus the exact sequence
19
becomes
::: ! A
r;i
(C;X)! 2A
r;i
(Spec(k);X)! 2A
r;i
(Spec(k);X)
! A
r;i1
(C;X)!:::
It’s easy to see that the map 2A
r;i
(Spec(k);X)! 2A
r;i
(Spec(k);X) sends a pair (;)
to (;), so we can split off the long exact sequence
::: ! A
r;i
(Spec(k);X)! 2A
r;i
(Spec(k);X)!A
r;i
(Spec(k);X)
! A
r;i1
(Spec(k);X)!:::
to obtain
::: !
~
A
r;i
(C;X)! 0!A
r;i
(Spec(k);X)
!
~
A
r;i1
(C;X)! 0!:::
Hence, we obtain an isomorphism between
~
A
r;i1
(C;X) andA
r;i
(Spec(k);X). There-
fore, the Bivariant Cycle Cohomology forC is
A
r;i
(C;X) =A
r;i
(Spec(k);X)A
r;i+1
(Spec(k);X):
In fact, for any integern 2, we can consider the planar curveC given byy
n
= (x+1)x
n
.
It is irreducible by Eisenstein’s criterion withp = x + 1 ink[x][y]. Taking a quick look
at the partial derivatives, we see that the only singularity of this curve is at (0; 0). The
20
morphism
A
1
! C
t 7! (t
n
1; t(t
n
1))
gives us a blow-up at (0; 0), withn points (n
th
roots of unity) lying above (0; 0). Therefore,
we have the following long exact sequence.
::: ! A
r;i
(C;X)!A
r;i
(A
1
;X)A
r;i
(Spec(k);X)!A
r;i
(Spec(k);X)
n
! A
r;i1
(C;X)!:::
By homotopy invariance inY ,A
r;i
(A
1
;X) =A
r;i
(Spec(k);X) and thus the exact sequence
becomes
:::!A
r;i
(C;X)!A
r;i
(Spec(k);X)
2
!A
r;i
(Spec(k);X)
n
!A
r;i1
(C;X)!:::
It’s again easy to see that the map
A
r;i
(Spec(k);X)
2
!A
r;i
(Spec(k);X)
n
sends a pair (;) to (;:::;), so we can split off the long exact sequence
::: ! A
r;i
(Spec(k);X)!A
r;i
(Spec(k);X)
2
!A
r;i
(Spec(k);X)
! A
r;i1
(Spec(k);X)!:::
21
to obtain the exactness of
::: !
~
A
r;i
(C;X)! 0!A
r;i
(Spec(k);X)
n1
!
~
A
r;i1
(C;X)! 0!:::
Hence, we obtain an isomorphism between
~
A
r;i1
(C;X) and
A
r;i
(Spec(k);X)
n1
:
Therefore, the Bivariant Cycle Cohomology forC is
A
r;i
(C;X) =A
r;i
(Spec(k);X)A
r;i+1
(Spec(k);X)
n1
:
Example 3.1.2. Let’s consider the case where C is n + 1 lines in a plane, meeting at a
point, say, given by the vanishing locus of
x(x +y)::: (x +ny):
There is an abstract blow up
p :
~
C = Spec

k[x;y;t]
xyt; t(1 +t)::: (n +t)

!C
It’s easy to see that
~
C is n + 1 skew lines in A
3
, the center of the abstract blow-up is
Z = Spec(k) corresponding to (0; 0), andp
1
(Z) is (n + 1)k-points. Then, we obtain the
22
exact sequence:
::: ! A
r;i
(C;X)!A
r;i
(A
1
;X)
n+1
A
r;i
(Spec(k);X)!A
r;i
(Spec(k);X)
n+1
! A
r;i1
(C;X)!:::
By homotopy invariance inY ,A
r;i
(A
1
;X) =A
r;i
(Spec(k);X) and thus the exact sequence
becomes
::: ! A
r;i
(C;X)!A
r;i
(Spec(k);X)
n+2
!A
r;i
(Spec(k);X)
n+1
! A
r;i1
(C;X)!:::
It’s easy to see that the map
A
r;i
(Spec(k);X)
n+2
!A
r;i
(Spec(k);X)
n+1
sends (x
1
;:::;x
n+1
;y) to (x
1
y;:::;x
n+1
y). So, splitting off the exact sequence
::: ! A
r;i
(Spec(k);X)!A
r;i
(Spec(k);X)
n+2
!A
r;i
(Spec(k);X)
n+1
! A
r;i1
(Spec(k);X)!:::
we see that
~
A
r;i
(C;X) = 0, so the Bivariant Cycle Cohomology is
A
r;i
(C;X) =A
r;i
(Spec(k);X):
23
3.2 Boundary of the algebraicn-simplex in the first variable
Letn be a nonnegative integer. FixX2 Obj(Sch=k), and defineS
n
as the boundary of
the well known algebraic (n + 1)-simplex, that is
S
n
= Spec
0
@
k[x
0
;:::;x
n+1
]
X
x
i
= 1;
Y
x
i
= 0
1
A
Let’s write
S
n
=
n+1
[
j=0
fx
0
+::: +x
n+1
= 1; x
j
= 0g:
SetX
0
to be the disjoint union of
fx
0
+::: +x
n+1
= 1; x
0
= 0g
and
n+1
[
j=1
fx
0
+::: +x
n+1
= 1; x
j
= 0g:
The idea is that, the first piece is isomorphic to the affinen-spaceA
n
, and the second piece,
let’s call it “M
n
", is “contractible” so it should behave like a point, because of homotopy
invariance inY .
Lemma 3.2.1. For alln2 N andX2 Obj(Sch=k), the structure mapM
n
! Spec(k)
induces an isomorphism
A
r;i
(M
n
;X) A
r;i
(Spec(k);X):

24
Proof. To deal with the combinatorics in the computation, let
M
n;j
:=fx
0
+::: +x
n+1
= 1; x
1
= 0g[[fx
0
+::: +x
n+1
= 1; x
j
= 0g
forj = 1;:::;n + 1.
So,M
n;n+1
=M
n
, andM
n;1
wA
n
. We’ll prove that for allk-schemesX of finite type,
for allj = 1;:::;n, for allr andi, the structure map induces isomorphisms
A
r;i
(Spec(k);X)!A
r;i
(M
n;j
;X):
Now let’s proceed by induction onn.
Forn = 0, we have
S
0
= Spec

k[x
0
;x
1
]
x
0
+x
1
= 1,x
0
x
1
= 0

= Spec

k[x]
x(1x) = 0

:
SoS
0
is just two points,M
0
is one of those points, and the statement forM
0
is trivial.
Now assume the result forn. Forn + 1, let’s do another induction now onj. Forj = 1,
the statement is just the homotopy invariance of the theory with respect to the first variable,
see Theorem 2.6.3, claim 5. Assume true forj. Observe,
M
n+1;j+1
= M
n+1;j
[

X
x
i
= 1,x
j+1
= 0

The second piece in the union above is isomorphic toA
n
, and thus we could consider the
abstract blow-up ofM
n+1;j+1
by the disjoint union ofM
n+1;j
andA
n
mapping toM
n+1;j+1
.
25
The centerZ of this abstract blow up is the intersection
Z = M
n+1;j
\

X
x
i
= 1,x
j+1
= 0

=
j
[
l=1

X
x
l
= 1,x
j+1
= 0,x
l
= 0

Also, the inverse image ofZ via this abstract blow-up map is
Z
a
ZM
n+1;j
a
A
n
:
It’s easy to see thatZ is isomorphic toM
n;j
via relabeling of the indices. Hence, by
induction onn, we know the result forZ. Thus, the long exact sequence looks like
::: ! A
r;i
(M
n+1;j+1
;X)!A
r;i
(M
n+1;j
;X)A
r;i
(A
n
;X)A
r;i
(Z;X)
! A
r;i
(Z;X)
2
!A
r;i1
(M
n+1;j+1
;X)!:::
And by the induction we’re running onn andk and by homotopy invariance, we can rewrite
the sequence above as;
::: ! A
r;i
(M
n+1;j+1
;X)!A
r;i
(Spec(k);X)
3
!A
r;i
(Spec(k);X)
2
! A
r;i1
(M
n+1;j+1
;X)!:::
and the mapA
r;i
(Spec(k);X)
3
!A
r;i
(Spec(k);X)
2
is given by
(a;b;c)7! (ac;bc)
26
This map is obviously surjective for allr,i, and its kernel is isomorphic to the diagonal
inA
r;i
(Spec(k);X)
3
, which is the image ofA
r;i
(Spec(k);X) inA
r;i
(M
n+1;j+1
;X) under
the map
A
r;i
(M
n+1;j+1
;X)!A
r;i
(Spec(k);X)
3
:
Since the sequence is exact, this proves that
A
r;i
(M
n+1;j+1
;X) =A
r;i
(Spec(k);X)
This ends the induction ink, and that immediately ends the induction onn.
Now we can compute the theory forS
n
.
Proposition 3.2.2. For alln2N andX2Obj(Sch=k),
A
r;i
(S
n
;X) =A
r;i
(Spec(k);X)A
r;i+n
(Spec(k);X)
Proof. As we described above, consider the covering ofS
n
:A
n
a
M
n
!S
n
27
This is an abstract blow-up ofS
n
with centerZ, where
Z =

n+1
X
i=0
x
i
= 1,x
0
= 0

\

n+1
[
j=1
f
n+1
X
i=0
x
i
= 1; x
j
= 0g
!
=
n+1
[
j=1

n+1
X
i=0
x
i
= 1,x
0
= 0,x
j
= 0

= Spec
0
B
B
B
B
B
@
k[x
0
;:::;x
n+1
]
X
x
j
= 1,x
0
= 0,
n+1
Y
j=1
x
j
= 0
1
C
C
C
C
C
A
= Spec
0
B
B
B
B
B
@
k[x
1
;:::;x
n+1
]
X
x
j
= 1,
n+1
Y
j=1
x
j
= 0
1
C
C
C
C
C
A
w S
n1
It is also clear that
1
(Z)wZ
`
Z.
This suggests that we would get the result by induction onn. Forn = 0, the result is
immediate becauseS
0
is just two rational points. Assume nown 1 and the statement is
true forn 1.
The long exact sequence stemming from the abstract blow-up reads as follows:
::: ! A
r;i
(S
n
;X)!A
r;i
(A
n
;X)A
r;i
(M
n
;X)A
r;i
(S
n1
;X)!A
r;i
(S
n1
;X)
2
! A
r;i1
(S
n
;X)!:::
Now we can split off the long exact sequence
28
::: ! A
r;i
(Spec(k);X)!A
r;i
(Spec(k);X)
3
!A
r;i
(Spec(k);X)
2
! A
r;i1
(Spec(k);X)!:::
to obtain
::: !
~
A
r;i
(S
n
;X)!
~
A
r;i
(S
n1
;X)!
~
A
r;i
(S
n1
;X)
2
!
~
A
r;i1
(S
n
;X)!:::
The map
~
A
r;i
(S
n1
;X)!
~
A
r;i
(S
n1
;X)
2
is given byx7! (x;x), hence we have
found
~
A
r;i
(S
n
;X) =
~
A
r;i+1
(S
n1
;X)
By induction onn, we have
~
A
r;i+1
(S
n1
;X) =A
r;i+1+n1
(Spec(k);X)
which yields
~
A
r;i
(S
n
;X) =A
r;i+n
(Spec(k);X)
which completes the proof.
Remark 3.2.3. In the naïve version of the theory, the chain complex doesn’t have any non-
zero terms in the negative degrees. Therefore, we only get zero if we look at the cohomol-
ogy in the negative degrees. Example 3.1.1 and Proposition 3.2.2 show that, computing the
theory with a singular scheme in theY variable, we may get non-zero groups in negative
29
degrees, as a result of cdh-descent. In the decomposition
A
r;i
(S
n
;X) =A
r;i
(Spec(k);X)A
r;i+n
(Spec(k);X);
the termA
r;i+n
(Spec(k);X) is in general non-zero ifin.
30
Chapter 4
Continuous Algebraic Maps
We will now move on to carry out a computation involving cycles in codimension 1. As
usual, this is the most accesible case. This section first reviews the concepts present in
[4], and then generalizes its results. Theorem 2.1.1 will allow us to connect the results we
obtain to the world of Bivariant Cycle Cohomology.
4.1 Background
Definition 4.1.1. For X, Y reduced schemes of finite type over a field k, a continuous
algebraic mapf :Y !X is a closed subvariety
f
YX with the property that the
projection ontoY is finite and bijective on geometric points.
Remark 4.1.2. As pointed out in [7], a continuous algebraic mapf :Y!X is equivalent
to a morphismY
w
!X whereY
w
is the weak normalization ofX.
Let X, Y be reduced schemes of finite type over a field k which has resolution of
singularities, andX!P
N
k
an embedding. For anyr 0, we consider the Chow variety
of the projective closure ofX,
C
r
(

X) =
a
C
r;d
(

X):
Leta;b : Spec(

k)! C
r
(

X) be two geometric points ofC
r
(

X). By the correspondence
stated in Theorem 2.1.1, we obtain two 0-cycles in

X
k
. We can add them to get another
31
0-cycle, which in turn corresponds to another geometric point
a +b : Spec(

k)!C
r
(

X):
This way we get an abelian monoid structure onC
r
(

X)(

k).
To get a meaningful group completion out of this, letR be the equivalence relation on
C
r
(

X)
2
(

k) consisting of pairs of pairs of points ((a
1
;a
2
); (b
1
;b
2
)) in C
r
(

X)(

k) with the
property thata
1
+b
2
restricts toa
2
+b
1
onYX. Note the subtlety here, for example
if (a;b) corresponds to a cycle that lies outside of X in

X, then it is R-equivalent to 0.
Let nowZ
r
(X) denote the set theoretic quotientC
r
(

X)
2
(

k)=R. Friedlander proposed the
following definition in [4].
Definition 4.1.3. A “continuous algebraic map :Y !Z
r
(X)” is a set-theoretic func-
tion
:Y (

k)!Z
r
(X) =C
r
(

X)
2
(

k)=R
induced by an algebraic correspondenceC

YC
r
(

X)
2
.
Definition 4.1.4. With the hypotheses above, letZ
X;r
(Y ) be the set of all continuous alge-
braic maps :Y (

k)!C
r
(

X)
2
(

k)=R.
The definitions above are independent (up to natural isomorphism) of a choice of pro-
jective closure forX [6, 4.5].
Remark 4.1.5. The associationY 7! Z
X;r
(Y ) is contravariantly functorial inY . To see
this, observe that a morphism :Y
0
!Y of reducedk-schemes induces
7! :Y
0
(

k)!Y (

k)!C
r
(X)
2
(

k)=R:
Obviously, is a continuous algebraic map because it is induced by the correspondence
32

Y
C

Y
0
C
r
(

X)
2
.
For a non-reducedk-schemeY of finite type, declare
Z
X;r
(Y ) :=Z
X;r
(Y
red
)
This means we can regardZ
X;r
() as a presheaf onSch=k. The monoid structure on the
geometric points ofC
r
(

X) now makesZ
X;r
(Y ) into a presheaf of abelian groups.
4.2 Continuous Algebraic Maps in Codimension 1
From now on, assume that the fieldk is of characteristic 0.
Definition 4.2.1. For anyk-schemeX of finite type, letk(X)

=
Q

k()

where runs
over the generic points ofX andk() is the residue field at. Elements ofk(X)

are called
invertible rational functions.
Definition 4.2.2. For varietiesX,Y , letR
X
(Y ) k(YX)

consist of those invertible
rational functions f for which there exists a blow-up (i.e. a proper birational surjective
morphism)Y
0
!Y some line bundleL onY
0

X, and non-zero global sectionsF;G2
L(Y
0
X), such that
1. f =F=G2k(Y
0
X)

2. the zero lociZ
F
,Z
G
YX are equidimensional overY
0
.
3. for every geometric point y ofY ,F
E
=G
E
2k(EX)

lies in the image ofk(X
y
)

,
whereE = yY
0
,F
E
andG
E
are restrictions ofF ,G toE.
Proposition 4.2.3. f2 R
X
(Y ) iff there exists a blow-upY
0
! Y , an affine open cover
fV
i
g ofY
0
, and some affine open subsetU
i
= Spec (A
i
)2 V
i
X, dense in each fiber of
33
U
i
! V
i
, such that the restriction off to eachk(U
i
) is a regular functiong
i
2 A
i
with
the property that for every geometric point y ofY and everyV
i
admitting a lifting of y, the
restriction ofg
i
to yY
0
X is an invertible rational function ink( yY
0
X)

, lying
in the image ofk( yX)

Proof. Prop 2.2 in [4]
It’s easy to check thatR
X
(Y ) is a subgroup ofk(YX)

and the association
Y7!R
X
(Y )
is contravariantly functorial. See [4] the discussion after Prop 2.2.
From now on X will be a quasi-projective variety of pure dimension m 1. Now
comes the computation for the homology of the complexC

(Z
X;m1
(Y )).
Theorem 4.2.4. There exists a natural cycle map
:R
X
(Y )!Z
X;m1
(Y ); f7!
f
with the property that for all geometric point y ofY ,
f
( y)2 is the Chow coordinate of
the principal divisor off
y
2 k( yX). Moreover, ifY is of pure dimensionn, then the
following sequence of abelian groups is exact:
R
X
(Y )!Z
X;m1
(Y )!A
m+n1
(YX)
Proof. Theorem 2.4 in [4]
Proposition 4.2.5. For varietiesX,Y where the connected components ofX has dimen-
sion 1, the simplicial abelian group R
X
(Y

) is acyclic, i.e. all of its homotopy
groups are zero.
34
Proof. The strategy here is the same as in Proposition 3.1 in [4], namely to prove that any
finitely generated simplicial subcomplexKR
X
(Y

) is contained in a contractible
subcomplex ofR
X
(Y

).
Let f
i
2 R
X
(Y
n
i
), i = 1;:::;s. Choose an affine open U = Spec (A) YX
such that there existsP
i
,Q
i
2A[t
1
;:::;t
n
i
] withf
i
=P
i
=Q
i
. Also choose data forf
i
as in
Proposition 4.2.3, i.e. letp
i
:W
i
!Y
n
i
be a blow-up,fW
i;
g be a finite affine cover
ofW
i
, andU
i;
= Spec (A
i;
)W
i;
X dense in each fiber ofW
i;
X!W
i;
, and
f
i;
2A
i;
. We can visualize the data by the diagram below.
U
i;
W
i;
X
X W
i;
W
i
Y
n
i

X
p
i
Now, using the projections
X
:U
i;
!X, letV be the finite dimensionalk-vector sub-
space of the ring of rational functions onX, spanned by the coefficients off
i;
2A
i;
. Pick
an invertible rational functionh onX, which is not a quotient of two nonzero functions in
V

k

k.
Then let
g
i
(t
1
;:::;t
n
i
;t
n
i
+1
) =ht
n
i
+1
+f
i
(t
1
;:::;t
n
i
) (1t
n
i
+1
)
Theg
i
,i = 1;:::;s are rational functions onY
n
i
+1
X. We would like to say
g
i
2R
X
(Y
n
i
+1
). First, It is obvious thatg
i
are invertible rational functions. We may
35
now find data as in Proposition 4.2.3. We will do so by modifying the data forf
i
.
Observe,
p
i
id :W
i

1
!Y
n
i
+1
is a blow-up,
fW
i;

1
g is an affine cover ofW
i

1
,
U
i;

1
= Spec (A
i;
[t
n
i
+1
]) W
i;

1
X are affine open subsets, dense in
each fibre ofW
i;

1
X!W
i;

1
,
The restriction ofg
i
tok(U
i;

1
) isg
i;
(t
1
;:::;t
n
i
;t
n
i
+1
) =ht
n
i
+1
+f
i;
(1
t
n
i
+1
). But we don’t know ifh is regular onU
i;

1
, so we cannot conclude the
regularity ofg
i;
onU
i;

1
.
To remedy this, first we note that sinceh is a function onX, it is regular on an open
subsetV ofW
i;

1
X if and only if it is regular on its projection ontoX,
X
(V ). So
letO be a dense open set inX on whichh is a regular function, and replace the affine opens
U
i;
byU
0
i;
=

U
i;
\
1
X
(O)

. These are affine by separatedness, and they are dense in
each fiber ofW
i;

1
X!W
i;
becauseU
i;
and
1
X
(O) both are.
Let’s then consider the following data forg
i
,i = 1;:::;s:
p
i
id :W
i

1
!Y
n
i
+1
is a blow-up,
fW
i;

1
g is an affine cover ofW
i

1
,
U
0
i;

1
= Spec (A
i;
[t
n
i
+1
]) W
i;

1
X are affine open subsets, dense in
each fibre ofW
i;

1
X!W
i;

1
,
The restriction ofg
i
tok(U
0
i;

1
) isg
i;
(t
1
;:::;t
n
i
;t
n
i
+1
) =ht
n
i
+1
+f
i;
j
U
0
i;

(1t
n
i
+1
). This is now regular onU
0
i;

1
.
36
Here is the diagram for the above:
U
0
i;

1
W
i;

1
X
X W
i;

1
W
i

1
Y
n
i
+1

X
p
i
id
It remains to check if theseg
i;
satisfy the lifting property in Proposition 4.2.3.
Let y be a geometric point ofY
n
i
+1
that lifts toW
i;

1
. We want the restriction
ofg
i;
to (W
i;

1
)
y
X to live in the image ofk(X
y
)

. Observe that y gives a geometric
point ofY that lifts toW
i;
, andf
i;
restricted to (W
i;
)
y
X lives in the image ofk(X
y
)

.
Sinceh is already ink(X)

, this guarantees thatg
i;
satisfies the lifting property.
Finally, let
~
K be the simplicial subcomplex of R
X
(Y

) generated by g
i
, i =
1;:::;s. This new subcomplex is the cone on the subcomplexK with vertexh. So,
~
K is
contractible, containingK, and that completes the proof.
Following the definition in [4], for varietiesY ,X withX of pure dimensionm 1, let
U
X
(Y ) = kerf :R
X
(Y )!Z
X;m1
(Y )g
Proposition 4.2.6. For varietiesY ,X withX of pure dimensionm 1,
1. U
X
(Y ) =ff2k(YX)

: (f) =;g
2. U
X
(Y
n
) =U
X
(Y )
Proof. The first statement is proved in [4]. The key argument is, any rational function
whose divisor is empty is necessarily an element ofR
X
(Y ). For the second statement, by
37
induction reduce to the case wheren = 1. By the first statement in the proposition, it is
enough to prove a function ink(YX)(T ) has zero divisor if and only if it is ink(YX).
The inclusionU
X
(Y ) U
X
(Y
1
) is trivial, because a function with empty divisor in
YX is still going to have empty divisor inY
1
X. Letf2 k(YX)(t) with
empty divisor. Seeingf as a function on the affine line over the fieldk(YX), its divisor
is still zero, because
div(f;Y
1
X) =
X

n

+
X

n

where runs over the codimension 1 generic points ofY
1
X whose image in
YX includes the generic point ofYX, and runs over the codimension 1 generic
points ofY
1
X whose image inYX is again a codimension 1 point.
Then, we may write f = g(t)=h(t) with g, h relatively prime elements in the UFD
k(YX)[t]. Ifg(t) = 2k(YX), then by Hilbert’s Nullstellensatz, we can find a geometric
point! A
1
k(YX)
whereg vanishes andh does not. This contradicts the assumption
that the divisor off is zero.
We need one more result. The following lemma is Theorem 7.4 in [3].
Lemma 4.2.7. Letk have resolution of singularities. LetX andY be varieties overk,Y
being smooth, quasi-projective and of pure dimensionn. Then, for allr 0, the embed-
ding of presheaves
z
equi
(X;r)(Y) z
equi
(XY;r +n)()
38
induces a quasi-isomorphism of complexes of abelian groups
C

z
equi
(X;r)(Y ) C

z
equi
(XY;r +n)(Spec(k))
In other words, the embedding induces an isomorphism
A
naive
r;i
(Y;X) A
naive
r+n;i
(Spec(k);XY )

for alli.
Proof. See Theorem 7.4 in [3]
Corollary 4.2.8. Letk have resolution of singularities. LetX andY be varieties overk,
Y being smooth, quasi-projective and of pure dimensionn. Then, for anyr n and for
anyr-cycle inXY , there exists anotherr-cycle in the same rational equivalence class,
which is equidimensional overY .
Proof. By Assertion 1 in Theorem 2.6.3, we have
A
r
(XY )'A
naive
r;0
(Spec(k);XY )
By Lemma 4.2.7, we know that
A
naive
rn;0
(Y;X) A
naive
r;0
(Spec(k);XY )

So, if is anr-cycle inXY , its rational equivalence class [] corresponds to, under
the inverse of the isomorphism induced by inclusion, a class represented by an r-cycle

0
which is equidimensional over Y . If we apply the inclusion map now, we get [
0
]2
39
A
r
(XY ), which is then [].
Finally, we are ready to state and prove the theorem that gives us the computation about
the homotopy groups of the simplicial abelian groupZ
X;r
(Y

).
Theorem 4.2.9. Let k have characteristic 0. Let X, Y be varieties of pure dimension
m 1,n respectively. Then, there exists an exact sequence of simplicial abelian groups
0!U
X
(Y )!R
X
(Y

)!Z
X;m1
(Y

)!A
1
(YX)
whereU
X
(Y ) andA
1
(YX) are seen as constant simplicial abelian groups. Moreover, if
X is projective,Y is smooth and quasi-projective, the sequence above is also exact on the
right.
Proof. By Theorem 4.2.4 and the definition ofU
X
(Y ), there exists an exact sequence of
abelian groups
0!U
X
(Y
i
)!R
X
(Y
i
)!Z
X;m1
(Y
i
)!A
1
(Y
i
X)
for alli2N. This collection of exact sequences are compatible with face and degeneracy
maps (by functoriality), and consequently we get an exact sequence of simplicial abelian
groups.
0!U
X
(Y

)!R
X
(Y

)!Z
X;m1
(Y

)!A
1
(Y

X)
By Proposition 4.2.6, the simplicial abelian groupU
X
(Y
1
) is degenerate, isomor-
phic to the simplicial abelian group generated byU
X
(Y ) at degree 0. On the other hand,
it is well known that the Chow group of cycles modulo rational equivalence is homotopy
40
invariant, hence
A
1
(YX) =A
1
(Y
k
X)
and this similarly implies that the simplicial abelian group
A
1
(Y

X)
is generated byA
1
(YX) at degree 0.
Assume now thatX is projective, Y is smooth and quasi-projective. Observe that by
the assumptions on the dimensions of the schemesX andY , we have
A
m+n1
(YX) =A
1
(YX)
Therefore, to prove that the exactness on the right, it is sufficient to show that the
morphism of abelian groups
Z
X;m1
(Y )!A
1
(YX)
is surjective.
By assertion 1 in Theorem 2.6.3, the naive version of Bivariant Cycle Cohomology
satisfies
h
0
(C

z
equi
(YX;m +n 1))(Spec(k)) =A
m+n1
(YX)
and by the assumptions on the dimensions ofX andY , observe that the two groupsA
1
(Y
X) andA
m+n1
(YX) are the same.
SinceY is smooth and quasi-projective, Corollary 4.2.8 applies. Let []2A
m+n1
(Y
41
X) be a rational equivalence class, and let be equidimensional overY . The positive and
negative parts of are effective cycles that are equidimensional over Y , and since Y is
smooth, there exists morphismsY ! C
m1
(X) corresponding to those effective cycles
(see Theorem 2.1.1). Combining the two, we get a morphism Y ! C
m1
(X)
2
, which
gives us a continuous algebraic map 2Z
X;r
(Y ). It is easy to see that maps to
under
the mapZ
X;r
(Y )!A
m+n1
(YX).
Corollary 4.2.10. Let k have characteristic 0. Let X, Y be varieties of pure dimension
m 1,n respectively, such thatY is smooth and quasi-projective. Then,
h
0
(C

Z
X;m1
(Y )) = A
m+n1
(YX)
h
1
(C

Z
X;m1
(Y )) = U
X
(Y )
h
j
(C

Z
X;m1
(Y )) = 0 forj 2
Proof. Taking the associated chain complexes of the simplicial abelian groups in Theorem
4.2.9, we get an exact sequence of chain complexes of abelian groups
0!C

U
X
(Y )!C

R
X
(Y )!C

Z
X;m1
(Y )!C

A
m+n1
(YX)! 0
Out of this exact sequence, we get two short exact sequences of chain complexes of
abelian groups
0!C

U
X
(Y )!C

R
X
(Y )!K

! 0
and
0!K

!C

Z
X;m1
(Y )!C

A
m+n1
(YX)! 0
We know, by the previous results, that the complexesC

U
X
(Y ) andC

A
m+n1
(YX)
are quasi-isomorphic to the complexesU
X
(Y ) andA
m+n1
(YX) at degree 0, respec-
42
tively. Also, the complex C

R
X
(Y ) is acyclic by Proposition 4.2.5. Hence we get the
following long exact sequences of abelian groups
0!H
i
(K

)! 0!::: 0!H
1
(K

)!U
X
(Y )! 0!H
0
(K

)! 0
and
:::! 0!H
j
(K

)!h
j
(C

Z
X;m1
(Y ))! 0!:::
0!H
0
(K

)!h
0
(C

Z
X;m1
(Y ))!A
m+n1
(YX)! 0
wherei 2 andj 1. Combining the two, we get the desired result.
43
Chapter 5
Bivariant Cycle Cohomology in Codimension 1
We would like to carry the Corollary 4.2.10 to the language of Bivariant Cycle Cohomol-
ogy.
Proposition 5.0.11. LetX be a projective variety. Then, the presheafZ
X;r
() is the sheafi-
fication for the h-topology of the presheaf
Y7!Hom(Y;C
r
(X))
+
where + denotes the group completion. In particular, Z
X;r
() is a cdh-sheaf, since cdh
topology is included in h topology.
Proof. proposition 4.1 in [4]
The following result establishes a relation between the two theories.
Theorem 5.0.12. Assume that the field k is of characteristic 0, X is projective and U is
smooth overk. Then, for allr 0,
Z
X;r
(U) = (z
equi
(X;r))
cdh
(U):
In particular, the restriction of the presheafZ
X;r
() toSm=k coincides with the restriction
of (z
equi
(X;r))
cdh
().
44
Proof. By Proposition 5.0.11,
Z
X;r
(U) =

Hom(;C
r
(X))
+

h
(U)
SinceU is smooth, Hom(U;C
r
(X)) is the monoid of effective cycles inUX that are
equidimensional overU. That is,
Z
X;r
(U) = (z
equi
(X;r))
h
(U)
By Remark 4.6 in [3], ifk is of characteristic 0, then the map
z
equi
(X;r)
cdh
!z
equi
(X;r)
h
is an isomorphism of presheaves onSm=k. Therefore, we obtain
Z
X;r
(U) = (z
equi
(X;r))
cdh
(U):
Note that it’s obvious that the restriction maps agree as well, because the pullback of rel-
ative cycles applied to equidimensional cycles corresponds to composition of continuous
algebraic maps.
Observe that the assumption of characteristic 0 guarantees resolution of singularities.
So, every cdh-cover will have a refinement consisting of smooth schemes. Therefore, every
cdh-sheaf onSch=k will be completely determined by its restriction toSm=k [8]. Hence,
the lemma above implies;
Corollary 5.0.13. IfX is a projective scheme of finite type over a fieldk of characteristic
0, the cdh-sheaves (z
equi
(X;r))
cdh
() andZ
X;r
() are equal.
45
By the Remark 4.6 in [3], the natural morphism
z
equi
(X;r)! (z
equi
(X;r))
cdh
of presheaves onSm=k is a monomorphism. Therefore, under the hypotheses of Theorem
5.0.12, we may seez
equi
(X;r)(U) as a subgroup ofZ
X;r
(U).
Recall that by Theorem 2.6.2, the Bivariant Cycle Cohomology groupsA
r;i
(U;X) equal
h
i
(C

(z
equi
(X;r)(U))). So now in the codimension 1 case, we have a way to compute the
Bivariant Cycle Cohomology via the embedding
i :z
equi
(X;r)(U) Z
X;r
(U):
Theorem 5.0.14. Assume that the fieldk is of characteristic 0,X,Y schemes of finite type
overk such thatX is projective of pure dimensionm 1,Y is smooth and quasi-projective
of pure dimensionn. Then, the embedding
C

z
equi
(X;m 1)(Y ) C

Z
X;m1
(Y )
is a quasi-isomorphism, and consequently
A
m1;0
(Y;X) = A
1
(YX)
A
m1;1
(Y;X) = U
X
(Y )
A
m1;j
(Y;X) = 0 forj 2
Proof. Recall the map : R
X
(Y )! Z
X;m1
(Y ) from Theorem 4.2.4. By abuse of
46
notation, let denote the induced map of simplicial abelian groups
R
X
(Y

)!Z
X;m1
(Y

)
as well. For simplicity, let
B

:=
1
(z
equi
(X;m 1)(Y

))
By definition ofB

, we have
kerf :B

!z
equi
(X;m 1)(Y

)g =U
X
(Y

):
The discussion above implies, assumingr =m 1, we have the commutative diagram of
simplicial abelian groups
0 U
X
(Y ) R
X
(Y

) Z
X;r
(Y

) A
1
(YX) 0
0 U
X
(Y ) B

z
equi
(X;r)(Y

) A
1
(YX) 0

k

i
k
Here,U
X
(Y ) andA
1
(YX) denote again the simplicial abelian groups generated by the
respective abelian groups at degree 0. The top row is exact by Theorem 4.2.9. On the other
hand,z
equi
(X;r)(Y

) surjects ontoA
m+n1
(YX) by Corollary 4.2.8 and the fact
that for allk 0, A
1
(YX) = A
1
(Y
k
X). Together with the definition ofB

we obtain the exactness of the bottom sequence. Therefore, to conclude the proof, it will
suffice to show that the associated chain complexB

is acyclic (see Corollary 4.2.10).
47
To that end, first observe that we have the following exact sequence describingB
i
;
0!U
X
(Y
i
)!B
i
!fW2z
equi
(X;r)(Y
i
) : [W ] = 02A
1
(Y
i
X)g! 0
Hence,
B
i
=ff2k(Y
i
X)

: (f) is equidimensional overY
n
g
Note that this description allows functions with empty divisors. Following the same strat-
egy as in Proposition 4.2.5, letK B

be a finitely generated simplicial subcomplex. It
will be enough to show thatK is contained in a contractible subcomplex ofB

.
Letf
i
2B
n
i
fori = 1;:::;s, generatingK. Seeing these as elements ofR
X
(Y

),
we can apply the same procedure as in Proposition 4.2.5 to obtain an elementh2k(X)

.
Let then
g
i
(t
1
;:::;t
n
i
;t
n
i
+1
) =ht
n
i
+1
+f
i
(t
1
;:::;t
n
i
) (1t
n
i
+1
):
It’s easy to see thatg
i
2 B
n
i
+1
by the choice ofh; for any point (y;t)2 Y
n
i
+1
, the
functiong
i
j
(y;t)
will not vanish, so the fiber will be of codimension 1 or empty.
Just as in Proposition 4.2.5, let
~
K be the simplicial subcomplex ofB

generated by the
g
i
,i = 1;:::;s. This subcomplex is the cone on the subcomplexK with vertexh. So,
~
K is
contractible, containingK, and that shows thatB

is an acyclic chain complex of abelian
groups, which in turn completes the proof.
48
Chapter 6
Pseudo-Flasqueness and Conjectures
6.1 Generalizing Theorem 5.0.14
We will attempt to enlarge the result of Theorem 5.0.14 for higher codimension, and also
to a larger class of pairs of varieties. Throughout this section, assume thatX is a projective
variety, the base fieldk has characteristic 0.
For any schemeX of finite type over a fieldk, the natural monomorphism of presheaves
z
equi
(X;r)(),! (z
equi
(X;r))
cdh
()
is a morphism of pretheories (See section 2.5). LetK be the cokernel of this morphism.
Then, applying Corollary 2.5.3 to the exact sequence
0!z
equi
(X;r)()! (z
equi
(X;r))
cdh
()!K()! 0
gives us an isomorphism of cohomology groups
h
i
(C

z
equi
(X;r))(Spec (k)) h
i
(C

((z
equi
(X;r)
cdh
)))(Spec (k))

We can do somewhat better than that. LetY be a smooth scheme of finite type overk, and
consider the following presheaves onSm=k;
U7!z
equi
(X;r)(YU)
49
and
U7! (z
equi
(X;r))
cdh
(YU):
We may denote these presheaves byz
equi
(X;r)(Y) and (z
equi
(X;r))
cdh
(Y)
respectively. Now injectivity of
z
equi
(X;r)(),! (z
equi
(X;r))
cdh
()
clearly implies the injectivity of
z
equi
(X;r)(Y),! (z
equi
(X;r))
cdh
(Y):
LetF denote the presheaf cokernel of this injection. Applying Corollary 2.5.3 to the exact
sequence of presheaves onSm=k
0!z
equi
(X;r)(Y)! (z
equi
(X;r))
cdh
(Y)!F ()! 0
we have a long exact sequence of cohomology groups
:::!h
i
(C

z
equi
(X;r))(Y )!h
i
(C

(z
equi
(X;r))
cdh
)(Y )!h
i
(C

(F ))(Spec(k))
!h
i1
(C

z
equi
(X;r))(Y )!:::
In light of Corollary 5.0.13, if we had that the groupsh
i
(C

(F ))(Spec(k)) vanish, we
would be able to compare the Bivariant Cycle Cohomology withh
i
(Z
X;r
(Y )). Another for-
mulation of this obstruction is as follows; considering thecdh-sheafification of the presheaf
z
equi
(X;r)(Y), we have a commutative diagram of maps of presheaves
50
z
equi
(X;r)(Y) (z
equi
(X;r)(Y))
cdh
z
equi
(X;r)
cdh
(Y)
i
j
h
We know that the map i is a monomorphism. The map j is the canonical map from
the presheaf to its sheafification, and the maph comes from the universal property of the
sheafification. Applying Corollary 2.5.3 to the cokernel of the mapj, we get a quasi iso-
morphism of complexes
C

z
equi
(X;r)(Y )
and
C

(z
equi
(X;r)(Y)
cdh
)(Spec(k)):
Hence, we get the following result:
Proposition 6.1.1. Let the base field k have characteristic 0, let X be projective, Y be
smooth and quasi-projective. Assume also that the map
h : [z
equi
(X;r)(Y)]
cdh
!z
equi
(X;r)
cdh
(Y)
induces a quasi isomorphism of complexes of abelian groups
C

[(z
equi
(X;r)) (Y)
cdh
] (Spec(k)) C

[z
equi
(X;r)
cdh
] (Y ):

Then, we have
A
r;i
(Y;X) =h
i
(C

(Z
X;r
(Y ))):
Proof. By the discussion above, the complex
C

[(z
equi
(X;r)) (Y)
cdh
] (Spec(k))
51
is quasi isomorphic to C

z
equi
(X;r)(Y ), and by Theorem 2.6.2, the homology gives the
Bivariant Cycle Cohomology A
r;i
(Y;X). Again by the discussion above and Corollary
5.0.13, the complex
C

[z
equi
(X;r)
cdh
] (Y )
givesC

(Z
X;r
(Y )). This concludes the proof.
The obvious question is when are the hypotheses of Proposition 6.1.1 satisfied.
6.2 Naïve Version onSch=k
We know that for any scheme X of finite type over k, the abelian groups z
equi
(X;r)(U)
have the structure of a presheaf onSm=k, which does not extend toSch=k. However, for
any schemeY of finite type overk, we can consider the abelian groupz
equi
(X;r)(Y ):
Although we don’t have functoriality any more, with respect to face and degeneracy
maps of the algebraic simplex, we do have functoriality. This is because the degeneracy
maps are smooth projections, and the face maps are complete intersection morphisms [13].
Therefore, it makes sense to talk about the simplicial abelian group
z
equi
(X;r)(Y

);
the associated chain complex
z
equi
(X;r)(Y

);
and then take cohomology of this complex.
This way, to every pair (Y;X) of schemes of finite type, we can associate a sequence
of groups. This association is not functorial, but when we restrict our attention to the case
whereY is smooth, we recover the naïve Bivariant Cycle Cohomology.
52
It is obvious that the groups h
i
(z
equi
(X;r)(Y

)) are zero for i < 0. In Chapter
3, we saw that it was not the case for Bivariant Cycle Cohomology. It is an open prob-
lem now to determine whether the Bivariant Cycle Cohomology groups and the groups
h
i
(z
equi
(X;r)(Y

)) are different fori 0 as well.
53
References
[1] S. Bloch, “Algebraic Cycles and Higher K-theory”, 1986
[2] E. Friedlander, “Algebraic Cycles, Chow varieties and Lawson Homology”, 1991
[3] E. Friedlander, V . V oevodsky, “Bivariant Cycle Cohomology”, 1996
[4] E. Friedlander, “Some Computations of Algebraic Cycle Homology”, 1994
[5] E. Friedlander, H. B. Lawson, “A theory of algebraic cocycles”, 1992
[6] E. Friedlander, O. Gabber, “Cycle spaces and intersection theory”, Topological Meth-
ods in Modern Mathematics, 1993
[7] E. Friedlander, M. Walker, “Function Spaces and Continuous Algebraic Pairings for
Varieties”, 1991
[8] B. Gordon, J. Lewis, S. Müller-Stach, S. Saito, N. Yui, ed.,“Bloch-Kato conjecture
and motivic cohomology with finite coefficients, The Arithmetic and Geometry of
Algebraic Cycles” , Nato ASI series C, vol. 548, Kluwer, 2000, pp. 117Ð189.
[9] H. Hironaka, “Resolution of singularities of an algebraic variety over a field of char-
acteristic zero”, 1964
[10] H. B. Lawson, “Algebraic Cycles and Homotopy Theory”, 1989
[11] C. Mazza, V . V oevodsky, C. Weibel, “Lecture Notes on Motivic Cohomology”, 2005
[12] P. Samuel, “Methodes d’algebre abstraite en geometrie algebrique”, Ergebnisse der
Mathematik, N. F. 4 Springer-Verlag, Berlin, 1955
[13] W. Fulton, “Intersection Theory”, Ergeb. Math. Grenzgeb. (3), vol. 2, Springer, Berlin,
1984
54

Abstract (if available)

Abstract E. Friedlander and V. Voevodsky introduced Bivariant Cycle Cohomology in 1995. The theory relates to many other known cycle theoretic functors, such as Chow groups and motivic cohomology. I provide various computations for Bivariant Cycle Cohomology, first explicitly, then using some properties of the theory, and then by comparing it to other theories. I give a description of the Bivariant Cycle Cohomology groups in codimension 1, using special rational functions, units and the Chow group.

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Creator Bilal, Taylan (author)

Core Title Computations for bivariant cycle cohomology

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 07/03/2013

Defense Date 05/24/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag algebraic geometry,algebraic K-theory,bivariant,cohomology,cycles,equidimensional,motives,OAI-PMH Harvest

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Contributor Electronically uploaded by the author (provenance)

Advisor Friedlander, Eric M. (committee chair), Asok, Aravind (committee member), Jonckheere, Edmond A. (committee member)

Creator Email taylanbil@gmail.com,tbilal@usc.edu

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