# Zero-cycle groups on algebraic varieties

Federico Binda and Amalendu Krishna  Dipartimento di Matematica “Federigo
Enriques”, Università degli Studi di Milano
Via Cesare Saldini 50, 20133 Milano, Italy<EMAIL_ADDRESS>Department
of Mathematics, Indian Institute of Science, Bangalore, 560012, India.
<EMAIL_ADDRESS>
###### Abstract.

We compare various groups of 0-cycles on quasi-projective varieties over a
field. As applications, we show that for certain singular projective
varieties, the Levine-Weibel Chow group of 0-cycles coincides with the
corresponding Friedlander-Voevodsky motivic cohomology. We also show that over
an algebraically closed field of positive characteristic, the Chow group of
0-cycles with modulus on a smooth projective variety with respect to a reduced
divisor coincides with the Suslin homology of the complement of the divisor.
We prove several generalizations of the finiteness theorem of Saito and Sato
for the Chow group of 0-cycles over $p$-adic fields. We also use these results
to deduce a torsion theorem for Suslin homology which extends a result of
Bloch to open varieties.

###### Key words and phrases:

Cycles with modulus, Cycles on singular varieties, Motivic cohomology

###### 2020 Mathematics Subject Classification:

Primary 14C25; Secondary 14F42, 19E15

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries: cycles groups and motivic cohomology
  3. 3 Chow group vs. motivic cohomology
  4. 4 Proofs of Theorems 1.1 and 1.2
  5. 5 Bloch’s torsion theorem for Suslin homology
  6. 6 The finiteness theorems
  7. 7 Chow groups of normal crossing surfaces
  8. 8 Proof of Theorem 1.6

##  1\. Introduction

The goal of this work is to clarify the relation between various groups of
0-cycles associated to algebraic varieties and prove some new properties of
these groups. One of the questions we want to address is whether one can
isolate a class of algebraic varieties for which one could describe the a
priori non-${\mathbb{A}}^{1}$-invariant motivic cohomology groups (such as the
Levine-Weibel Chow group and the Chow group with modulus) in terms of the —
more classical — ${\mathbb{A}}^{1}$-invariant motivic cohomology groups such
as the ones defined by Friedlander, Suslin and Voevodsky. We answer this
question for a projective variety which has either normal crossing
singularities or it is obtained as the double of a smooth variety along a
reduced divisor. The interest in the latter case originates from the fact that
it allows us to describe the Chow group of 0-cycles with modulus on smooth
varieties with reduced modulus in terms of the singular homology à la
Suslin–Voevodsky. In the following subsections, we describe our main results
and their applications.

###  1.1. Cycles with modulus vs. Suslin homology

Let $k$ be a field and $X$ a smooth projective variety over $k$. Let $D\subset
X$ be an effective Cartier divisor and let $U=X\setminus D$. Let ${\rm
CH}_{0}(X|D)$ denote the Chow group of 0-cycles with modulus (see [32] or
[7]). These groups are supposed to provide the cycle theoretic description of
the relative algebraic $K$-theory in the same way as the classical Chow groups
do for the non-relative $K$-theory: for the case of 0-cycles this relationship
has now been established in many important cases (see e.g., [6], [23], [24]).
The relative $K$-theory is an important tool to study the ordinary $K$-theory
of singular varieties by embedding them inside smooth varieties. It is
therefore important to know the structure of the Chow groups with modulus
under various special conditions on the base field and the variety.

It is not hard to see that there is a natural surjection
$\lambda_{X|D}\colon{\rm CH}_{0}(X|D)\twoheadrightarrow H^{sus}_{0}(U)$, where
the latter is the 0-th Suslin homology of $U$ (also known as the Suslin-
Voevodsky singular homology). It acts, in many ways, as a replacement of the
classical Chow group of 0-cycles for non-proper varieties over $k$ (see [52]
for an early example of this philosophy).

Since $H^{sus}_{0}(U)$ is part of the well developed theory of motivic
cohomology with compact support, there are many tools to study and compute it.
On the other hand, the nature of the _relations_ defining ${\rm CH}_{0}(X|D)$
in some cases can be used to grasp new information about $H^{sus}_{0}(U)$
using $\lambda_{X|D}$. It is therefore important to get conditions under which
the above surjection is an isomorphism. We give a complete answer to this when
$k$ is algebraically closed and a partial answer in general.

######  Theorem 1.1.

Assume that $k$ is algebraically closed and $D$ is reduced111$\lambda_{X|D}$
is almost never an isomorphism with integral coefficients if $D$ is not
reduced.. Then the map

$\lambda_{X|D}\colon{\rm CH}_{0}(X|D)\to H^{sus}_{0}(U)$

has the following properties.

  1. (1)

It is an isomorphism if ${\rm char}(k)>0$.

  2. (2)

It is not always an isomorphism if ${\rm char}(k)=0$.

######  Theorem 1.2.

Assume that $k$ is perfect and the irreducible components of $D_{\rm red}$ are
regular. Then the following hold.

  1. (1)

If ${\rm char}(k)=p>0$, then

$\lambda_{X|D}\colon{\rm CH}_{0}(X|D)[\tfrac{1}{p}]\to
H^{sus}_{0}(U)[\tfrac{1}{p}]$

is an isomorphism.

  2. (2)

If ${\rm char}(k)=0$, then

$\lambda_{X|D}\colon{{\rm CH}_{0}(X|D)}/m\to{H^{sus}_{0}(U)}/m$

is an isomorphism for every integer $m\neq 0$.

Part (2) Theorem 1.1 may appear surprising, because one would expect ${\rm
CH}_{0}(X|D)$ to behave like an ${\mathbb{A}}^{1}$-invariant theory if $D$ is
reduced. One application of this is that it implies that the Chow groups with
modulus defined by Russell [47] does not in general coincide with the above
Chow group with modulus. It also implies that Russell’s Chow groups can not
satisfy Bloch’s formula in terms of the top Zariski/Nisnevich cohomology of
Milnor or Quillen $K$-sheaves.

###  1.2. Bloch’s torsion theorem for Suslin homology

Let $U$ be a quasi-projective variety of dimension $d\geq 0$ over an
algebraically closed field $k$ which admits a smooth compactification. Let
$\ell\neq{\rm char}(k)$ be a prime number. If $U$ itself is complete, then
Bloch’s torsion theorem [8] (established independently from Roitman’s theorem)
affirms the existence of an isomorphism

${\rm
CH}_{0}(U)\\{\ell\\}\xrightarrow{\cong}H^{2d-1}_{{\acute{e}t}}(U,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(d)).$

This provides a simple description of the prime-to-$p$ torsion in the Chow
group of zero cycles in terms of an étale cohomology group. This isomorphism
is very subtle because there is no obvious map in either direction and its
construction is one of the main challenges.

When $U$ is no longer complete, but admits a smooth compactification as
indicated above, we can replace the Chow group ${\rm CH}_{0}(U)$ by
$H_{0}^{sus}(U)$, as explained in the previous paragraph. An extension of
Roitman’s torsion theorem for $H_{0}^{sus}(U)$, providing an isomorphism
between the prime-to-$p$ torsion part of $H_{0}^{sus}(U)$ and the
corresponding torsion in the semi-abelian Albanese variety of $U$, has been
proved by Spieß and Szamuely [52]. As a consequence of our main result, we
also get the following theorem (see Remark 5.5 for a different approach). Let
$H^{*}_{c,{\acute{e}t}}(U,{\mathcal{F}})$ denote the étale cohomology of $U$
with compact support (e.g., see [45]).

######  Theorem 1.3.

Let $U$ be a quasi-projective variety of dimension $d\geq 0$ over an
algebraically closed field such that $U$ admits a smooth compactification.
Then there is an isomorphism

$H^{sus}_{0}(U)\\{\ell\\}\xrightarrow{\cong}H^{2d-1}_{c,{\acute{e}t}}(U,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(d)).$

######  Remark 1.4.

When $U$ is complete, the $p$-adic analogue of Theorem 1.3 can be deduced from
[21, Cor. 3.18] using [17, Thm. 8.1]. We hope to address this question in the
non-complete case in a future project. It is also an interesting question if
it is possible to eliminate the condition in Theorem 1.3 that $U$ admits a
smooth compactification. We expect this to be the case.

###  1.3. Finiteness of 0-cycle groups

To motivate our next set of results, suppose that $k$ is a $p$-adic field and
$m\geq 1$ is an integer such that $p\nmid m$. A conjecture of Colliot-Thélène
predicted that if $X$ is a smooth projective variety over $k$, then ${{\rm
CH}_{0}(X)}/m$ is finite. Such a finiteness result was previously known over
finite and algebraically closed fields (without any restriction on $m$).

The conjecture of Colliot-Thélène was affirmatively answered by Saito and Sato
[48] (see also Bloch’s appendix to [11] and [12] for simplified proofs). It is
natural to expect that a similar finiteness result would hold even if $X$ is
not complete provided we replace ${{\rm CH}_{0}(X)}/m$ by
${H^{sus}_{0}(X)}/m$. In the same spirit, one would expect that ${{\rm
CH}_{0}(X|D)}/m$ is finite for $X$ projective, if we interpret ${{\rm
CH}_{0}(X|D)}$ as a Chow group with compact support for $X\setminus|D|$.

We give positive answers to both questions. Furthermore, we show that the
finiteness result of Saito and Sato also holds for singular schemes. The
question of finiteness of Suslin homology with finite coefficients arose
during a conversation of authors with Colliot-Thélène on his conjecture.

For a reduced quasi-projective scheme $X$ over a field, let ${\rm
CH}^{LW}_{0}(X)$ denote the Levine-Weibel Chow group of $X$ (see § 2).

######  Theorem 1.5.

Let $X$ be an integral quasi-projective scheme over $k$ which is regular in
codimension one. Let $D\subset X$ be an effective Cartier divisor. We then
have the following.

  1. (1)

${H^{sus}_{0}(X)}/m$ is finite if $X$ is smooth.

  2. (2)

${{\rm CH}_{0}(X|D)}/m$ is finite if $X$ is smooth and complete.

  3. (3)

${{\rm CH}^{LW}_{0}(X)}/m$ is finite if $X$ is complete.

We show that the above finiteness result also holds over finite and
algebraically closed fields (see § 6).

###  1.4. Levine-Weibel Chow group vs. motivic cohomology

Let $k$ be a field of exponential characteristic $p\geq 1$. Let $\Lambda$ be a
commutative ring, and assume that $p$ is invertible in $\Lambda$ or that $k$
admits resolution of singularities. Let $X$ be a reduced quasi-projective
$k$-scheme of dimension $d\geq 0$.

Let $H^{2d}(X,\Lambda(d))$ denote the Friedlander-Voevodsky motivic cohomology
of $X$ (see § 2 for a reminder of the definition). It was shown in [37] that
there is a canonical map

(1.1) $\lambda_{X}\colon{\rm CH}^{LW}_{0}(X)\otimes_{{\mathbb{Z}}}\Lambda\to
H^{2d}(X,\Lambda(d)),$

where $\Lambda$ is as above. Note that, a priori, the source and the target of
$\lambda_{X}$ are very different objects: on one side we have a group of
cycles, defined by generators and relations (in particular, no topology is
involved), while on the other side we have a group defined either as a Hom-
group in a motivic category, or as the cdh-hypercohomology of a certain
complex of sheaves.

It is well known that this map can not be an isomorphism for arbitrary
varieties: for instance, $H^{2d}(X,\Lambda(d))$ does not see the difference
between $X$ and its semi-normalization. On the other hand, one can easily show
for a curve $X$ with a cusp singularity that ${\rm
CH}^{LW}_{0}(X)\otimes_{{\mathbb{Z}}}\Lambda$ changes after passing to the
semi-normalization of $X$.

On the other hand, it is a result of Voevodsky that for regular $X$, both
groups are isomorphic, as a (very) special case of the identification of
motivic cohomology with Bloch’s higher Chow groups. It is therefore reasonable
to ask if there is a specific class of singular varieties or, rather, a
specific class of singularities, for which $\lambda_{X}$ is an isomorphism
(for instance, we have already observed that semi-normality is a necessary
condition).

In this paper, we prove the following positive result when $X$ has normal
crossings singularities (see Definition 2.12).

######  Theorem 1.6.

Assume that $k$ is an infinite perfect field and $X$ is a normal crossing
projective variety of dimension $d\geq 0$ over $k$. Then $\lambda_{X}$ is an
isomorphism in the following cases.

  1. (1)

$p>1$ and $\Lambda={\mathbb{Z}}[\tfrac{1}{p}]$.

  2. (2)

$m\in k^{\times}$ and $\Lambda={{\mathbb{Z}}}/m$.

We prove a similar isomorphism when $X$ is the join of a smooth projective
variety along a reduced divisor if $k$ is algebraically closed, see Theorem
3.5.

Note that $H^{2d}(X,\Lambda(d))$ is not known to have a presentation in terms
of generators and relations in general. In [12, Thm. 7.1], a variant (denoted
by ${\rm CH}^{EKW}_{0}(X)$ in this paper) of the Levine-Weibel Chow group was
defined for $X$ as in Theorem 1.6 in terms of generators and relations. It was
shown that ${\rm CH}^{EKW}_{0}(X)\otimes_{{\mathbb{Z}}}\Lambda$ is isomorphic
to $H^{2d}(X,\Lambda(d))$. Theorem 1.6 implies that under its hypothesis,
there is an isomorphism

${\rm CH}^{EKW}_{0}(X)\otimes_{{\mathbb{Z}}}\Lambda\xrightarrow{\cong}{\rm
CH}^{LW}_{0}(X)\otimes_{{\mathbb{Z}}}\Lambda.$

As a consequence, it follows that ${\rm
CH}^{EKW}_{0}(X)\otimes_{{\mathbb{Z}}}\Lambda$ can be defined in terms of only
one type of relations instead of two types of relations used in [12].

We end the discussion of our main results with yet another consequence. Let
$X$ be a smooth projective variety of dimension $d\geq 2$ over an infinite
perfect field and let $D\subset X$ be a simple normal crossing divisor. Let
$\iota\colon D\hookrightarrow X$ be the inclusion. By the classical moving
lemma for Chow groups, one knows that ${\rm CH}_{1}(X)$ is the quotient of the
free abelian group ${\mathcal{Z}}_{1}(X)_{D}$ of integral 1-dimensional cycles
on $X$ which do not meet $D_{\rm sing}$ by a suitable rational equivalence
relation. We have the obvious pull-back map
$\iota^{*}\colon{\mathcal{Z}}_{1}(X)_{D}\to{\mathcal{Z}}_{1}(D,D_{\rm sing})$
(see § 2.1) defined by taking the scheme theoretic intersection with $D$. But
it is not clear if it descends to a map between the Chow groups. As a
consequence of Theorem 1.6, we can conclude the following.

######  Corollary 1.7.

Let $X$ and $D$ be as above, and let $\Lambda={\mathbb{Z}}[\tfrac{1}{p}]$ if
$p>1$ or $\Lambda={{\mathbb{Z}}}/m$ with $m\in k^{\times}$. Then the map
$\iota^{*}\colon{\mathcal{Z}}_{1}(X)_{D}\to{\mathcal{Z}}_{1}(D,D_{\rm sing})$
descends to a homomorphism

$\iota^{*}\colon{\rm CH}_{1}(X)\otimes_{{\mathbb{Z}}}\Lambda\to{\rm
CH}^{LW}_{0}(D)\otimes_{{\mathbb{Z}}}\Lambda.$

###  1.5. Overview of proofs

To prove the first part of Theorem 1.1, the key ingredients are the Roitman
torsion theorems of [35] and [52], and the Lefschetz hyperplane section
theorem for Serre’s Albanese variety from [20]. Using these, the theorem is
deduced from Proposition 4.2. The latter is a general result of independent
interest and estimates the kernel of the map from the Chow group with modulus
to Suslin homology. We prove the second part of Theorem 1.1 by constructing a
specific example.

The proof of Theorem 1.2 uses three main tricks. The first is that in order to
compare Chow group with modulus with Suslin homology, we need to replace the
integral normal curves used in defining the Suslin homology by some embedded
curves. To take care of this, we pass from $X$ to a projective space over it
where the curve is automatically embedded and then use the covariant
functoriality of the Chow group with modulus. The second trick is to perform a
series of monoidal transformations on $X$ in order to make the given integral
curve transverse with respect to the modulus divisor $D$. We achieve this
using a result of Jannsen (see the appendix of [48]). The last trick is to
apply a result of Miyazaki [46] which allows us to assume that the pull-back
of the divisor under blow-ups is reduced.

We prove Theorem 1.3 by first reducing it to the case of surfaces. We solve
the latter case by translating the problem to a torsion theorem for the Chow
group of a singular surface, namely, the double of $X$ along $D$. We solve
this problem using the methods of the torsion theorems of [35].

We deduce part (1) of Theorem 1.5 from a general finiteness result (see
Proposition 6.2) that we prove in this paper. This result applies to all
projective schemes over certain fields. Using this, the finiteness of the
Suslin homology follows using the localization sequence for motivic cohomology
with compact support. The part (2) of Theorem 1.5 follows from Theorem 1.2 and
part (1) of Theorem 1.5 after we make $D$ reduced using a theorem of Miyazaki
[46], and a divisor with normal crossings using a sequence of blow ups with
smooth centers. We deduce part (3) of Theorem 1.5 from its part (1) and an
independent result Theorem 6.5 which compares the Levine-Weibel Chow group and
the Suslin homology.

The strategy to prove Theorem 1.6 is to construct a surjective map ${\rm
CH}^{EKW}_{0}(X)\to{\rm CH}^{LW}_{0}(X)$ such that its composite with
$\lambda_{X}$ is the one shown to be an isomorphism by Esnault, Kerz and
Wittenberg. In order to construct such a map, our key step is Theorem 8.3,
which in turn relies on a moving lemma from [4].

We recollect various cycle and motivic cohomology groups in § 2. We recall and
prove a result about the Albanese maps in § 3. These are used in proving
Theorem 1.1. The main result of this section is Theorem 3.5 which proves an
isomorphism between the Levine-Weibel Chow group and motivic cohomology of a
special kind of singular variety. The results of § 1.1 are proven in § 4.
Theorem 1.5 is shown in § 6. In § 7, we prove Theorem 1.6 for surfaces and in
§ 8, we complete the proof of the general case.

### Notations and conventions

If $A$ is a Noetherian ring, we shall write
${\operatorname{\mathbf{Sch}}}_{A}$ for the category of schemes which are
separated and of finite type over ${\rm Spec\,}(A)$. We let
${\mathbf{Sm}}_{A}$ denote the full subcategory of
${\operatorname{\mathbf{Sch}}}_{A}$ whose objects are smooth schemes over
${\rm Spec\,}(A)$. For $X\in{\operatorname{\mathbf{Sch}}}_{A}$, we shall write
$X_{\rm reg}$ (resp. $X_{\rm sm}$) for the regular locus of $X$ (resp. for the
smooth locus of $X\to{\rm Spec\,}(A)$). By a regular (or smooth) closed point
of $X$ we will mean a closed point lying in $X_{\rm reg}$ (resp. in $X_{\rm
sm}$). We will write $X_{\rm sing}$ for the singular locus of $X$ (i.e., the
complement $X\setminus X_{\rm reg}$) with reduced closed subscheme structure.
We let $X_{(i)}$ denote the set of points $x\in X$ such that $\text{\rm
dim}(\overline{\\{x\\}})=i$.

We will be mostly interested in the case where $A=k$ is a perfect field. If
$p$ is the exponential characteristic of $k$, we shall let $\Lambda$ be the
ring ${\mathbb{Z}}$ if $k$ admits resolution of singularities, or the ring
${\mathbb{Z}}[\tfrac{1}{p}]$ otherwise ([15, Def. 3.4]). For any abelian group
$M$, we shall write $M\\{p^{\prime}\\}$ (resp. $M\\{p\\}$) for the subgroup of
elements in $M$ which are torsion of order prime to $p$ (resp. a power of
$p$). For a commutative ring $R$, we let
$M_{R}=M{\underset{{\mathbb{Z}}}{\otimes}}R$.

In this paper, we shall describe the setting of a subsection in the beginning
of the subsection. Further assumptions will be explicitly stated whenever they
are used in the middle of a subsection.

##  2\. Preliminaries: cycles groups and motivic cohomology

We begin by fixing some notations and clarifying our terminology. Throughout
this paper, we shall encounter several different groups of 0-cycles: the
groups will often share the same set of generators, but will be subject to
different notions of rational equivalence. In a nutshell, our main results
will be a comparison between the different notions, by means of suitable
moving lemmas, and to deduce new properties of these 0-cycle group. We shall
draw some consequences. In this preliminary section, we introduce the main
players, recalling some definitions already existing in the literature. New
groups of cycles, tailored on the specific geometry of the varieties we are
interested in, will be defined later in the paper.

###  2.1. Cycle groups on singular schemes

Let $k$ be a field of exponential characteristic $p\geq 1$. Let
$X\in{\operatorname{\mathbf{Sch}}}_{k}$ be a reduced $k$-scheme and $Y\subset
X$ a closed subscheme. For any $r\geq 0$, we write ${\mathcal{Z}}_{r}(X,Y)$
for the free abelian group on the set of integral $r$-dimensional cycles in
$X$ (i.e., integral closed subschemes of dimension $r$) which do not meet $Y$.
Let ${\mathcal{R}}_{r}(X)$ be the subgroup of
${\mathcal{Z}}_{r}(X):={\mathcal{Z}}_{r}(X,\emptyset)$ generated by the cycles
which are rationally equivalent to zero (see for example [14, §1]). We write
${\rm CH}_{r}(X)$ for the quotient ${\mathcal{Z}}_{r}(X)/{\mathcal{R}}_{r}(X)$
and refer to it as the (Fulton’s) Chow group of $r$-cycles on $X$ modulo
rational equivalence. We shall also use the notation ${\rm CH}^{d-r}(X)$ for
${\rm CH}_{r}(X)$ if $X$ has pure dimension $d$.

We now recall the definition of the Levine-Weibel Chow group of 0-cycles on
$X$ from [42] and its modified version from [4]. Let $Y\subset X$ be a nowhere
dense closed subscheme containing $X_{\rm sing}$.

######  Definition 2.1.

Let $C$ be a reduced $k$-scheme which is of pure dimension one (from now on,
such $C$ will be simply called _a curve_). We shall say that a pair $(C,Z)$ is
_a good curve relative to $Y$_ if there exists a finite morphism $\nu\colon
C\to X$ and a closed proper subscheme $Z\subsetneq C$ such that the following
hold.

  1. (1)

No component of $C$ is contained in $Z$.

  2. (2)

$\nu^{-1}(Y)\cup C_{\rm sing}\subseteq Z$.

  3. (3)

$\nu$ is local complete intersection at every point $x\in C$ such that
$\nu(x)\in Y$.

Let $(C,Z)$ be a good curve relative to $Y$ and let
$\\{\eta_{1},\cdots,\eta_{r}\\}$ be the set of generic points of $C$. Let
${\mathcal{O}}_{C,Z}$ denote the semilocal ring of $C$ at
$S=Z\cup\\{\eta_{1},\cdots,\eta_{r}\\}$. Let $k(C)$ denote the ring of total
quotients of $C$ and write ${\mathcal{O}}_{C,Z}^{\times}$ for the group of
units in ${\mathcal{O}}_{C,Z}$. Notice that ${\mathcal{O}}_{C,Z}$ coincides
with $k(C)$ if $|Z|=\emptyset$. As $C$ is Cohen-Macaulay,
${\mathcal{O}}_{C,Z}^{\times}$ is the subgroup of group of units in the ring
of total quotients $k(C)^{\times}$ consisting of those
$f\in{\mathcal{O}}_{C,x}$ which are regular and invertible for every $x\in Z$.

Given any $f\in{\mathcal{O}}^{\times}_{C,Z}\hookrightarrow k(C)^{\times}$, we
denote by ${\rm div}_{C}(f)$ (or ${\rm div}(f)$ in short) the divisor of zeros
and poles of $f$ on $C$, which is defined as follows. If $C_{1},\ldots,C_{r}$
are the irreducible components of $C$, and $f_{i}$ is the factor of $f$ in
$k(C_{i})$, we set ${\rm div}(f)$ to be the $0$-cycle $\sum_{i=1}^{r}{\rm
div}(f_{i})$, where ${\rm div}(f_{i})$ is the usual divisor of a rational
function on an integral curve in the sense of [14]. As $f$ is an invertible
regular function on $C$ along $Z$, ${\rm div}(f)\in{\mathcal{Z}}_{0}(C,Z)$.

By definition, given any good curve $(C,Z)$ relative to $Y$, we have a push-
forward map
${\mathcal{Z}}_{0}(C,Z)\xrightarrow{\nu_{*}}{\mathcal{Z}}_{0}(X,Y)$. We shall
write ${\mathcal{R}}_{0}(C|Z,X|Y)$ for the subgroup of
${\mathcal{Z}}_{0}(X,Y)$ generated by the set $\\{\nu_{*}({\rm
div}(f))|f\in{\mathcal{O}}^{\times}_{C,Z}\\}$.

######  Definition 2.2.

(1) Let ${\mathcal{R}}_{0}(X,Y)$ denote the subgroup of
${\mathcal{Z}}_{0}(X,Y)$ generated by the image of the map
${\mathcal{R}}_{0}(C|Z,X|Y)\to{\mathcal{Z}}_{0}(X,Y)$, where $(C,Z)$ runs
through all good curves relative to $Y$. We let ${\rm
CH}^{BK}_{0}(X,Y)=\frac{{\mathcal{Z}}_{0}(X,Y)}{{\mathcal{R}}_{0}(X,Y)}$ and
call it the _lci Chow group_ of zero cycles of $X$. We shall write
${\mathcal{R}}_{0}(X,X_{\rm sing})$ as ${\mathcal{R}}_{0}(X)$ and ${\rm
CH}^{BK}_{0}(X,X_{\rm sing})$ as ${\rm CH}^{BK}_{0}(X)$.

(2) Let ${\mathcal{R}}^{LW}_{0}(X)$ denote the subgroup of
${\mathcal{Z}}_{0}(X)$ generated by the divisors of rational functions on good
curves as above, where we further assume that the map $\nu\colon
C\hookrightarrow X$ is a closed immersion. We let ${\rm
CH}^{LW}_{0}(X)=\frac{{\mathcal{Z}}_{0}(X,X_{\rm
sing})}{{\mathcal{R}}^{LW}_{0}(X)}$ and call it the _Levine-Weibel Chow group_
of zero cycles on $X$. We shall refer to good curves relative to $X_{\rm
sing}$ as lci curves and embedded good curves on $X$ as Cartier curves.

By definition, there is a canonical surjection ${\rm
CH}^{LW}_{0}(X)\twoheadrightarrow{\rm CH}^{BK}_{0}(X)$.

######  Remark 2.3.

The Chow group ${\rm CH}^{LW}_{0}(X)$ was discovered by Levine and Weibel [42]
in an attempt to describe the Grothendieck group of a singular scheme in terms
of algebraic cycles. The modified version ${\rm CH}^{BK}_{0}(X)$ was
introduced in [4]. $\square$

Note that there is always a map

(2.1) ${\rm CH}^{BK}_{0}(X)\to{\rm CH}_{0}(X)$

which is far, in general, from being an isomorphism if $X_{\rm
sing}\neq\emptyset$. This is already clear when $\text{\rm dim}(X)=1$, see
[42] or [4, Lem. 3.12].

Assume now that $X$ is complete of pure dimension $d\geq 0$ and let
$X_{1},\ldots,X_{r}$ be the irreducible components of $X$. For any 0-cycle
$\alpha\in{\mathcal{Z}}_{0}(X,X_{\rm sing})$, we let $\alpha_{i}$ be the
restriction of $\alpha$ on $X_{i}$ and let
$\deg(\alpha)=(\deg(\alpha_{1}),\ldots,\deg(\alpha_{r}))\in{\mathbb{Z}}^{r}$.
This defines a homomorphism $\deg\colon{\mathcal{Z}}_{0}(X,X_{\rm
sing})\to{\mathbb{Z}}^{r}$. It is well known and easy to check from the
definition of ${\mathcal{R}}^{LW}_{0}(X)$ that
$\deg({\mathcal{R}}^{LW}_{0}(X))=0$. One therefore has a degree homomorphism
$\deg\colon{\rm CH}^{LW}_{0}(X)\to{\mathbb{Z}}^{r}$. Using a projective bundle
trick (e.g., see the proof of Theorem 4.5), one deduces that $\deg$ actually
factors through ${\rm CH}^{BK}_{0}(X)$. We shall let ${\rm
CH}^{LW}_{0}(X)_{\deg 0}$ and ${\rm CH}^{BK}_{0}(X)_{\deg 0}$ denote the
kernels of these degree maps. Note that $\deg$ does not factor through ${\rm
CH}_{0}(X)$ unless $X$ is integral.

###  2.2. Singular homology

Let $k$ be a field and $U$ a smooth $k$-scheme. Recall that the Suslin-
Voevodsky singular homology of $U$ (also called Suslin homology) is defined as
follows. Let $\Delta^{n}$ be the algebraic $n$-simplex and denote by
$C_{n}(U)$ the free abelian group generated by those integral closed
subschemes $Z$ of $U\times\Delta^{n}$ for which the projection
$Z\to\Delta^{n}$ is finite and surjective. By pulling back via the structure
maps of the cosimplicial scheme $\Delta^{\bullet}_{k}$, one equips
$C_{\bullet}(U)$ with the structure of a simplicial abelian group. We denote
the associated chain complex (via the Dold-Kan correspondence) also by
$C_{\bullet}(U)$. The $n$-th singular homology group $H^{sus}_{n}(U)$ of $U$
is the $n$-th homology of this complex.

When $n=0$, another, explicit, description is available (and will be taken as
a definition for us). For $i\colon C\hookrightarrow U$ an integral curve, let
$\phi\colon C^{N}\to C$ be its normalization and let $\overline{C}^{N}$ be the
unique projective normal compactification of $C^{N}$. We denote by
$C_{\infty}$ the complement $\overline{C}^{N}\setminus C^{N}$, and consider it
as a _reduced_ Cartier divisor on $\overline{C}$. Write
$G(\overline{C}^{N},C_{\infty})$ for ${\rm
Ker}(\mathcal{O}_{\overline{C}^{N},C_{\infty}}^{\times}\to\mathcal{O}_{C_{\infty}}^{\times})$,
where $\mathcal{O}_{\overline{C}^{N},C_{\infty}}$ denotes as before the
semilocal ring of $\overline{C}^{N}$ at the union of the closed points
supporting $C_{\infty}$. Note that $G(\overline{C}^{N},C_{\infty})\subset
k(C)^{\times}=k(\overline{C}^{N})^{\times}$.

For $f\in G(\overline{C}^{N},C_{\infty})$, we can consider its divisor ${\rm
div}_{\overline{C}^{N}}(f)\in{\mathcal{Z}}_{0}(\overline{C}^{N},C_{\infty})$.
Since $\phi$ is finite and $i$ is a closed immersion, we can consider the
composite

${\mathcal{Z}}_{0}(\overline{C}^{N},C_{\infty})={\mathcal{Z}}_{0}(C^{N})\xrightarrow{\phi_{*}}{\mathcal{Z}}_{0}(C)\xrightarrow{i_{*}}{\mathcal{Z}}_{0}(U).$

We write ${\mathcal{R}}^{S}_{0}(C,U)$ for the subgroup of
${\mathcal{Z}}_{0}(U)$ generated by the set $\\{i_{*}\phi_{*}({\rm
div}(f))\,|\,f\in G(\overline{C}^{N},C_{\infty})\\}$. Let
${\mathcal{R}}^{S}_{0}(U)$ denote the subgroup of ${\mathcal{Z}}_{0}(U)$
generated by ${\mathcal{R}}^{S}_{0}(C,U)$ where $C$ runs through all integral
curves in $X$.

######  Theorem 2.4.

$($[49, Thm. 5.1]$)$ There is a canonical isomorphism

$\frac{{\mathcal{Z}}_{0}(U)}{{\mathcal{R}}_{0}^{S}(U)}\xrightarrow{\cong}H^{sus}_{0}(U).$

In the sequel, we shall use the left hand side of this isomorphism as the
definition of $H^{sus}_{0}(U)$. There is a degree homomorphism $\deg\colon
H^{sus}_{0}(U)\to{\mathbb{Z}}^{r}$, where $r$ is the number of connected
components of $U$. This is the product of maps on Suslin homology induced by
the projection of each connected component of $U$ to ${\rm Spec\,}(k)$. We let
$H^{sus}_{0}(U)_{\deg 0}$ be the kernel of $\deg$.

######  Remark 2.5.

The $j$-th Suslin homology group of $U$ with coefficients in $\Lambda$ has the
following motivic interpretation which will be explored further later in the
text.

$H_{j}^{sus}(U)_{\Lambda}={\rm Hom}_{\mathbf{DM}^{\rm
eff}_{{\operatorname{Nis}}}(k,\Lambda)}(\Lambda[j],M(U)),$

where $\mathbf{DM}^{\rm eff}_{{\operatorname{Nis}}}(k,\Lambda)$ is Voevodsky’s
category of effective motives with $\Lambda$-coefficients and $M(U)$ is the
motive of $U$.

(2) It is clear from the above description that $H_{0}^{sus}(U)\cong{\rm
CH}_{0}(U)$ if $U$ is complete.

###  2.3. Motivic cohomology

Let $k$ be a perfect field of exponential characteristic $p\geq 1$. Let $X$ be
a quasi-projective scheme of dimension $d\geq 0$ over $k$. Recall from [44,
Lecture 16] that given $T\in{\operatorname{\mathbf{Sch}}}_{k}$ and integer
$r\geq 0$, the presheaf $z_{equi}(T,r)$ on ${\mathbf{Sm}}_{k}$ is defined by
letting $z_{equi}(T,r)(U)$ be the free abelian group generated by integral
closed subschemes $Z\subset U\times T$ such that $Z$ is dominant and
equidimensional of relative dimension $r$ (any fiber is either empty or all
its components have dimension $r$) over a component of $U$. It is known that
$z_{equi}(T,r)$ is a sheaf on the big étale site of ${\mathbf{Sm}}_{k}$.

Let $C_{\ast}z_{equi}(T,r)$ be the chain complex of presheaves of abelian
groups associated to the simplicial presheaf on ${\mathbf{Sm}}_{k}$ given by
$C_{n}z_{equi}(T,r)(U)=z_{equi}(T,r)(U\times\Delta^{n}_{k})$. The simplicial
structure on $C_{\ast}z_{equi}(T,r)$ is though the cosimplicial scheme
$\Delta^{\bullet}_{k}$, as before. Recall the following definition of motivic
cohomology of singular schemes from [15, Defn. 9.2].

######  Definition 2.6.

The motivic cohomology groups of $X$ are defined as the hypercohomology
$H^{m}(X,{\mathbb{Z}}(n))={\mathbb{H}}^{m-2n}_{cdh}(X,C_{\ast}z_{equi}({\mathbb{A}}^{n}_{k},0)_{cdh})$.
Here, $C_{\ast}z_{equi}({\mathbb{A}}^{n}_{k},0)_{cdh}$ is the complex obtained
by applying term wise the change of site functor from ${\mathbf{Sm}}_{k}$ to
${\operatorname{\mathbf{Sch}}}_{k}$.

This definition is rather untouchable. However, if one is willing to work with
$\Lambda$-coefficients (or under the assumption of resolution of
singularities), motivic cohomology groups have a description in terms of Hom-
groups in Voevodsky’s category of motives, and even on the complexes
$C_{\ast}z_{equi}({\mathbb{A}}^{n}_{k},0)_{cdh}$ we have a better
understanding. More precisely, we have the following result (see [37, Thm.
3.10]).

######  Theorem 2.7.

There is a functorial isomorphism

$H^{m}(X,\Lambda(n))\simeq{\rm
Hom}_{\mathbf{DM}(k,\Lambda)}(M(X),\Lambda(n)[m]),$

where $\mathbf{DM}(k,\Lambda)$ is Voevodsky’s non-effective category of
motives for the cdh-topology (also known as the ‘big’ category of motives)
with $\Lambda$-coefficients.

######  Remark 2.8.

When $X$ is smooth, the motivic cohomology groups are isomorphic to Bloch’s
higher Chow groups. More precisely, for any $m,n\in{\mathbb{Z}}$, there is a
natural isomorphism $H^{m}(X,{\mathbb{Z}}(n))={\rm CH}^{n}(X,2n-m)$. See [56,
Cor. 2]. $\square$

In this paper, we are specifically interested in the bi-degree $(2d,d)$. By
[37, Lem. 7.12], there is a canonical group homomorphism

(2.2) $\gamma_{X}\colon{\rm CH}_{0}^{BK}(X)_{\Lambda}\to
H^{2d}(X,\Lambda(d)).$

We remark here that the construction of $\gamma_{X}$ uses [37, Thm. 5.1] whose
proof holds verbatim with integral coefficients when $k$ admits resolution of
singularities. The map $\gamma_{X}$ has the property that for a closed point
$x\in X_{\rm reg}$, the diagram

(2.3)
$\textstyle{\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{{\rm
CH}_{0}(k(x))_{\Lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{H^{0}(k(x),\Lambda(0))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}^{BK}(X)_{\Lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{X}}$$\textstyle{H^{2d}(X,\Lambda(d))}$

commutes, where the vertical arrows are the push-forward maps (see [37, § 7],
and [5, §5.1] where a similar construction is performed with motivic
cohomology replaced by étale cohomology with finite coefficients). We let

$\lambda^{LW}_{X}\colon{\rm CH}^{LW}_{0}(X)_{\Lambda}\to{\rm
CH}_{0}^{BK}(X)_{\Lambda}\to H^{2d}(X,\Lambda(d))$

denote the composite map.

We let $M^{c}(X)\in\mathbf{DM}(k,\Lambda)$ denote the motive of $X$ with
compact support. One knows that $M^{c}(X)\simeq M(X)$ if $X$ is complete.
Under our assumption on $\Lambda$, it follows from [55] and [30, Chap. 5] that
for a closed immersion $Z_{1}\subset Z_{2}$ of schemes, there is a
distinguished triangle (called the localization sequence) in
$\mathbf{DM}(k,\Lambda)$:

(2.4) $M^{c}(Z_{1})\to M^{c}(Z_{2})\to M^{c}(Z_{2}\setminus Z_{1})\to
M^{c}(Z_{1})[1].$

Using the duality between the motivic homology and the motivic cohomology with
compact support, one gets a functorial isomorphism
$H^{sus}_{n}(Z)_{\Lambda}\simeq H^{2d-n}_{c}(Z,\Lambda(d))$ if $Z$ is a smooth
scheme of pure dimension $d$, see [15] and [30, Chap. 5], where the motivic
cohomology groups with compact support is defined as

$H^{m}_{c}(Z,\Lambda(n))\simeq{\rm
Hom}_{\mathbf{DM}(k,\Lambda)}(M^{c}(Z),\Lambda(n)[m]).$

In what follows, we will frequently use these identifications. We shall
sometimes write $H^{m}(Z,{\mathbb{Z}}(n))$ as $H^{m,n}(Z)$.

###  2.4. Cycles with modulus

Let $k$ be a field. Given any integral normal curve $C$ over $k$ and an
effective Cartier (or Weil) divisor $E$ on $C$, we say that a rational
function $f\in k(C)^{\times}$ on $C$ has modulus $E$ if $f\in{\rm
Ker}(\mathcal{O}_{C,E}^{\times}\to\mathcal{O}_{E}^{\times})$. As in § 2.2, we
let $G(C,E)$ denote the subgroup of $k(C)^{\times}$ given by such rational
functions. Note that $E$ is not necessarily reduced.

Let $X\in{\operatorname{\mathbf{Sch}}}_{k}$ be an integral scheme, and let $D$
be an effective Cartier divisor on $X$. Write as before
${\mathcal{Z}}_{0}(X,D)$ for the free abelian group on the set of closed
points in $X\setminus D$. If $C$ is an integral normal curve over $k$ equipped
with a finite morphism $\phi_{C}\colon C\to D$ such that
$\phi_{C}(C)\not\subset D$, we have a well defined group homomorphism

$\tau_{C}\colon G(C,\phi_{C}^{*}(D))\to{\mathcal{Z}}_{0}(X,D),\quad
f\mapsto(\phi_{C})_{*}({\rm div}(f)),$

where $\phi_{C}^{*}(D)$ denote the pullback of $D$ to $C$. Note that
$\phi_{C}^{*}(D)$ is not necessarily reduced, even when $D$ is. Recall the
following definition, that originally appeared in [32].

######  Definition 2.9 (Kerz-Saito).

We define the Chow group ${\rm CH}_{0}(X|D)$ of $0$-cycles of $X$ with modulus
$D$ as the cokernel of the homomorphism

${\rm div}\colon\bigoplus_{\phi_{C}\colon C\to
X}G(C,\phi_{C}^{*}D)\to{\mathcal{Z}}_{0}(X,D),$

where the sum runs over the set of finite morphisms $\phi_{C}\colon C\to X$
from integral normal curves such that $\phi_{C}(C)\not\subset D$.

The Chow groups of zero cycles with modulus are known to be covariantly
functorial (pushforward) for proper maps $f\colon X^{\prime}\to X$ with
respect to Cartier divisors $D^{\prime}$ on $X^{\prime}$ and $D$ on $X$ such
that $f^{*}(D)$ is defined and satisfies $f^{*}(D)\leq D^{\prime}$ as
effective Cartier divisors. See [7, Lem. 2.4] or [36, Prop. 2.10]. Similarly,
it is easy to construct a pullback along finite flat maps. See again [7, Lem.
2.4] or [36, Prop. 2.12].

######  Lemma 2.10.

Assume that $X$ is complete and $U=X\setminus D$ is a smooth $k$-scheme. Then
the canonical bijection
${\mathcal{Z}}_{0}(X,D)\xrightarrow{\cong}{\mathcal{Z}}_{0}(U)$ induces a
surjective homomorphism

(2.5) $\lambda_{X|D}\colon{\rm CH}_{0}(X|D)\twoheadrightarrow H_{0}^{sus}(U).$

###### Proof.

This is an easy exercise using Theorem 2.4. ∎

The map (2.5) is not an isomorphism if $D$ is not reduced. If $\text{\rm
dim}(X)=1$ and $X$ is a complete normal curve over $k$, the group ${\rm
CH}_{0}(X|D)$ agrees with the relative Picard group ${\rm Pic}(X,D)$ of line
bundles $\mathcal{L}$ on $X$ equipped with a trivialization
$\mathcal{L}_{|D}\cong\mathcal{O}_{D}$ along $D$ (see, e.g., [7, §4]), while
$H_{0}^{sus}(U)$ agrees with the relative Picard group ${\rm Pic}(X,D_{\rm
red})$, by [44, Thm. 7.16]. So, the two groups do not agree when $D$ is not
reduced. Theorem 1.1 of this paper describes the nature of $\lambda_{X|D}$
when $D$ is reduced.

###  2.5. Normal crossing varieties

Let $X$ be an integral Noetherian scheme of finite Krull dimension $d\geq 0$.
Let $D\subset X$ be an effective Cartier divisor and let $D_{i}\subset D,\
i\in I$ be the irreducible components of $D$. Note that $D=\emptyset$ if
$d=0$. Recall the following definition (e.g., see [10, 2.4]).

######  Definition 2.11.

(1) We say that $D$ is a _simple normal crossing divisor_ on $X$ if the local
ring $\mathcal{O}_{X,x}$ is regular for each point $x\in D$ and for each
$J\subset I$ nonempty, the scheme theoretic intersection $D_{J}=\bigcap_{j\in
J}D_{j}$ is a regular scheme which is either empty or of pure dimension
$d+1-|J|$.

(2) We say that $D$ is a _normal crossing divisor_ on $X$ if for every $x\in
D$, there exists an étale morphism $U\to X$ with $x$ in its image such that
$D\times_{X}U$ a simple normal crossing divisor on $U$.

######  Definition 2.12.

Let $k$ be a field and $X$ a reduced quasi-projective $k$-scheme. Let
$\\{X_{1},\ldots,X_{n}\\}$ be the set of irreducible components of $X$. We
shall say that $X$ is a normal crossing variety of dimension $d\geq 0$ if for
every nonempty subset $J\subset[1,n]$, the scheme theoretic intersection
$X_{J}:={\underset{i\in J}{\bigcap}}X_{i}$ is a smooth $k$-scheme which is
either empty or of pure dimension $d+1-|J|$.

###  2.6. The double construction

Let $k$ be any field. Let $X$ be an integral regular quasi-projective scheme
of dimension $d$ over $k$ and let $D\subset X$ be an effective Cartier
divisor. Recall from [4, § 2.1] that the double (or join) of $X$ along $D$ is
a quasi-projective scheme $S(X,D)=X\amalg_{D}X$ so that

(2.6) $\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
8.47755pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
21.9008pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\iota}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 45.00067pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\iota}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-16.58334pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
45.00067pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
52.53539pt\raise-13.20833pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.75696pt\hbox{$\scriptstyle{\iota_{+}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
52.53539pt\raise-15.91666pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.62927pt\raise-31.35416pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.07639pt\hbox{$\scriptstyle{\iota_{-}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
31.53471pt\raise-26.41666pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.53471pt\raise-26.41666pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{S(X,D)}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{array}$

is a co-Cartesian square in ${\operatorname{\mathbf{Sch}}}_{k}$. In
particular, the identity map of $X$ induces a finite flat map
$\Delta_{X}\colon S(X,D)\to X$ such that $\Delta_{X}\circ\iota_{\pm}={\rm
id}_{X}$ and $\pi=\iota_{+}\amalg\iota_{-}:X\amalg X\to S(X,D)$ is the
normalization map. We let $X_{\pm}=\iota_{\pm}(X)\subset S(X,D)$ denote the
two irreducible components of $S(X,D)$. We use $S_{X}$ as a shorthand for
$S(X,D)$ when the divisor $D$ is understood. $S_{X}$ is a reduced quasi-
projective scheme whose singular locus is $D\subset S_{X}$. It is projective
(resp. affine) whenever $X$ is so. It follows from [35, Lem. 2.2] that (2.6)
is a bi-Cartesian square.

We have a pullback isomorphism
${\mathcal{Z}}_{0}(S_{X},D)\xrightarrow{(\iota^{*}_{+},\iota^{*}_{-})}{\mathcal{Z}}_{0}(X_{+},D)\oplus{\mathcal{Z}}_{0}(X_{-},D)$,
as well as pushforward inclusion maps
${p_{\pm}}_{*}\colon{\mathcal{Z}}_{0}(X,D)\to{\mathcal{Z}}_{0}(S_{X},D)$ such
that $\iota^{*}_{+}\circ{p_{+}}_{*}={\operatorname{id}}$ and
$\iota^{*}_{+}\circ{p_{-}}_{*}=0$. The main result of [4] connects the lci
Chow group of zero cycles on $S_{X}$ with Fulton’s Chow group of $X$ and the
Kerz-Saito Chow group with modulus ${\rm CH}_{0}(X|D)$ as follows.

If $k$ is perfect222The perfectness condition has been removed recently in
[25, Thm. 1.1]., there are push-forward maps $p_{\pm,*}\colon{\rm
CH}_{0}(X|D)\to{\rm CH}_{0}^{BK}(S_{X})$ and pull-back maps
$\iota^{*}_{\pm}\colon{\rm CH}_{0}^{BK}(S_{X})\to{\rm CH}_{0}(X)$, where the
latter are split by $\Delta^{*}_{X}\colon{\rm CH}_{0}(X)\to{\rm
CH}_{0}^{BK}(S_{X})$. By [4, Thm. 7.1], there is a split short exact sequence

(2.7) $0\to{\rm CH}_{0}(X|D)\xrightarrow{p_{+,*}}{\rm
CH}_{0}^{BK}(S_{X})\xrightarrow{\iota^{*}_{-}}{\rm CH}_{0}(X)\to 0$

and a similar split exact sequence corresponding to the maps
$(p_{-,*},\iota^{*}_{+})$.

##  3\. Chow group vs. motivic cohomology

The goal of this section is to prove that the composition of the map (2.2)
with the Levine-Wiebel Chow group is an isomorphism for $S_{X}$. In
combination with (2.7), this will be the key step in the proof of Theorem 1.1.
We begin by recalling Serre’s Albanese varieties and Albanese maps and state
Theorem 3.2 which we shall use in the next section.

###  3.1. The semi-abelian Albanese variety for the double

We fix an algebraically closed field $k$ of any characteristic. We fix a pair
$(X,D)$ consisting of an integral smooth projective $k$-scheme $X$ of
dimension $d\geq 1$ and an effective Cartier divisor $D$ on it. We write
$S_{X}$ for the double $S(X,D)$ of § 2.6 and $U=X\setminus D$.

Let $\pi:\overline{S}_{X}=X_{+}\amalg X_{-}\to S_{X}$ denote again the
normalization map. $NS(\overline{S}_{X})$ denote the Neron-Severi group of
$\overline{S}_{X}$. Recall that this is the quotient of the class group of
$\overline{S}_{X}$ by the subgroup of Weil divisors which are algebraically
equivalent to zero. Let ${\rm Div}(S_{X})$ denote the free abelian group of
Weil divisors on $S_{X}$. Let $\Pi_{1}(S_{X})$ denote the subgroup of ${{\rm
Div}}(\overline{S}_{X})$ generated by the Weil divisors which are supported on
$\pi^{-1}(D)$. There is a canonical map $\iota_{S_{X}}\colon\Pi_{1}(S_{X})\to
NS(\overline{S}_{X})$. Let $\Pi_{\rm Serre}(S_{X})$ denote the kernel this map
and let $\Pi(S_{X})$ be the kernel of the canonical map

(3.1)
$\Pi_{1}(S_{X})\xrightarrow{(\iota_{S_{X}},\pi_{*})}NS(\overline{S}_{X})\oplus{\rm
Div}(S_{X}).$

Let $J^{d}(S_{X})$ and $J^{d}_{\rm Serre}(S_{X})$ denote the Cartier duals of
the 1-motives

$[\Pi(S_{X})\to{\rm Pic}^{0}(\overline{S}_{X})],\text{ and }[\Pi_{\rm
Serre}(S_{X})\to{\rm Pic}^{0}(\overline{S}_{X})],$

respectively. There is a surjective group homomorphism

$\rho_{S_{X}}\colon{\rm CH}_{0}^{LW}(S_{X})_{\deg 0}\twoheadrightarrow
J^{d}(S_{X})$

which is universal among regular homomorphisms from ${\rm
CH}_{0}^{LW}(S_{X})_{\deg 0}$ to semi-abelian varieties. $J^{d}(S_{X})$ is
called the semi-abelian Albanese variety of $S_{X}$. By [4, Prop. 9.7], there
is a factorization of $\rho_{X}$:

(3.2) ${\rm CH}_{0}^{LW}(S_{X})_{\deg 0}\twoheadrightarrow{\rm
CH}_{0}^{BK}(X)_{\deg
0}\stackrel{{\scriptstyle\widetilde{\rho}_{S_{X}}}}{{\twoheadrightarrow}}J^{d}(S_{X})$

and $J^{d}(S_{X})$ is in fact the universal regular semi-abelian variety
quotient of ${\rm CH}_{0}^{BK}(S_{X})_{\deg 0}$.

One knows from [51] that $J^{d}_{\rm Serre}(S_{X})$ is Serre’s Albanese
variety of $(S_{X})_{\rm reg}=U\amalg U$, the universal object in the category
of morphisms from $U\amalg U$ to semi-abelian varieties. We let $J^{d}(U)$
denote Serre’s Albanese variety of $U$ so that $J^{d}_{\rm
Serre}(S_{X})=J^{d}(U)\times J^{d}(U)$. Using the above constructions, one
checks that there is an exact sequence of morphisms of semi-abelian varieties

(3.3) $0\to T\to J^{d}_{\rm Serre}(S_{X})\to J^{d}(S_{X})\to 0,$

where $T$ is an algebraic torus of rank bounded by the number of components of
$D$.

We also have a commutative diagram

(3.4) $\textstyle{{\rm
CH}^{0}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{-,*}}$$\textstyle{NS(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{-,*}}$$\textstyle{\Pi_{1}(S_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{NS(X_{+}\amalg
X_{-}).}$

If we let $\Pi_{\rm Serre}(X|D)={\rm Ker}({\rm CH}^{0}(D)\to NS(X))$, we
obtain a commutative diagram of 1-motives

(3.5)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[0\to{\rm
Pic}^{0}(X)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{X}}$$\scriptstyle{\iota_{-,*}}$$\textstyle{[\Pi_{\rm
Serre}(X|D)\to{\rm
Pic}^{0}(X)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{-,*}}$$\textstyle{[\Pi(S_{X})\to{\rm
Pic}^{0}(\overline{S}_{X})]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{X}}$$\scriptstyle{\iota^{*}_{+}}$$\textstyle{[\Pi_{\rm
Serre}(S_{X})\to{\rm
Pic}^{0}(\overline{S}_{X})]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}_{+}}$$\textstyle{[\Pi(S_{X})\to{\rm
Pic}^{0}(X)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{X}}$$\textstyle{[\Pi_{\rm
Serre}(X|D)\to{\rm
Pic}^{0}(X)]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

in which the two columns are exact.

We now consider the commutative diagram

(3.6) $\textstyle{{\rm
CH}^{0}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\Delta}^{*}_{X}}$$\textstyle{NS(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\Delta}^{*}_{X}}$$\textstyle{\Pi_{1}(S_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{NS(\overline{S}_{X}),}$

where we let $\widetilde{\Delta}^{*}_{X}(a)=(a,-a)$. Note that
$\Pi_{1}(S_{X})={\rm CH}^{0}(D)\oplus{\rm CH}^{0}(D)$ and
$NS(\overline{S}_{X})=NS(X)\oplus NS(X)$. It is clear then that
$\widetilde{\Delta}^{*}_{X}$ factors through ${\rm CH}^{0}(D)\to{\rm
Ker}(\Pi_{1}(S_{X})\xrightarrow{\pi_{*}}{\rm Div}(S_{X}))$. We thus get a
morphism of 1-motives $\widetilde{\Delta}^{*}_{X}:[\Pi_{\rm Serre}(X|D)\to{\rm
Pic}^{0}(X)]\to[\Pi(S_{X})\to{\rm Pic}^{0}(\overline{S}_{X})]$. Composing this
with $\iota^{*}_{+}$ in the left column of (3.5), we get a map $[\Pi_{\rm
Serre}(X|D)\to{\rm Pic}^{0}(X)]\to[\Pi(S_{X})\to{\rm Pic}^{0}(X)]$.

######  Lemma 3.1.

The morphism of 1-motives

$\iota^{*}_{+}\circ\widetilde{\Delta}^{*}_{X}:[\Pi_{\rm Serre}(X|D)\to{\rm
Pic}^{0}(X)]\to[\Pi(S_{X})\to{\rm Pic}^{0}(X)]$

is an isomorphism.

###### Proof.

To prove the lemma, we first note that at the level of the abelian varieties,
this map of 1-motives is given by $a\mapsto(a,-a)\mapsto a$, so it is actually
identity. Thus, we only need to show that the map
$\widetilde{\Delta}^{*}_{X}:\Pi_{\rm Serre}(X|D)\to\Pi(S_{X})$ is an
isomorphism.

To prove this, let $\Pi_{2}(S_{X})={\rm
Ker}(\Pi_{1}(S_{X})\xrightarrow{\pi_{*}}{\rm Div}(S_{X}))$. We then have the
commutative diagram

(3.7) $\textstyle{\Pi_{\rm
Serre}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}^{0}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl_{X}}$$\textstyle{NS(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi(S_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi_{2}(S_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl_{S_{X}}}$$\textstyle{NS(\overline{S}_{X}),}$

where the vertical arrows are $\widetilde{\Delta}^{*}_{X}$. Since $D\amalg
D\xrightarrow{\pi}D$ is just the collapse map, it is immediate from various
definitions that any element of $\Pi_{2}(S_{X})$ must be of the form
$\widetilde{\Delta}^{*}_{X}(a)$ for some $a\in{\rm CH}^{0}(D)$. Furthermore,
we have that $\iota^{*}_{+}\circ\widetilde{\Delta}^{*}_{X}(a)$ is identity. It
follows that the middle vertical arrow in (3.7) is an isomorphism. The right
vertical arrow is injective and is split by $\iota^{*}_{+}$.

If we let $\widetilde{NS(X)}$ and $\widetilde{NS(\overline{S}_{X})}$ be the
images of ${cl_{X}}$ and $cl_{S_{X}}$, respectively, we therefore get a
commutative diagram of short exact sequences

(3.8)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi_{\rm
Serre}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}^{0}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl_{X}}$$\textstyle{\widetilde{NS(X)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi(S_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Pi_{2}(S_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl_{S_{X}}}$$\textstyle{\widetilde{NS(\overline{S}_{X})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the middle and the right vertical arrows are isomorphisms. It follows
that the left vertical arrow is an isomorphism too. This proves the lemma. ∎

###  3.2. Albanese homomorphisms for 0-cycles

We continue with the setup of § 3.1. We let $J^{d}(X|D):={\rm
Ker}(J^{d}(S_{X})\xrightarrow{\iota^{*}_{-}}J^{d}(X))$, where the map
$\iota^{*}_{-}$ is surjective and is split by $\Delta^{*}_{X}:J^{d}(X)\to
J^{d}(S_{X})$. Using [4, Prop. 9.7, § 11.1], one shows that $J^{d}(X|D)$ is
the universal regular semi-abelian variety quotient of ${\rm
CH}_{0}(X|D)_{\deg 0}$. Furthermore, there is a commutative diagram of split
short exact sequences

(3.9)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}(X|D)_{\deg
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{+,*}}$$\scriptstyle{\rho_{X|D}}$$\textstyle{{\rm
CH}^{BK}_{0}(S_{X})_{\deg
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}_{-}}$$\scriptstyle{\widetilde{\rho}_{S_{X}}}$$\textstyle{{\rm
CH}_{0}(X)_{\deg
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{d}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{+,*}}$$\textstyle{J^{d}(S_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}_{-}}$$\textstyle{J^{d}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Next, recall from [52, §3] that the semi-abelian Albanese variety $J^{d}(U)$
of $U$ is the universal regular quotient of $H_{0}^{sus}(U)_{\deg 0}$. We
write $\rho_{U}$ for the induced surjection $\rho_{U}\colon
H_{0}^{sus}(U)_{\deg 0}\twoheadrightarrow J^{d}(U)$. Composing $\rho_{U}$ with
the canonical map $\lambda_{X|D}$ of (2.5), we get a (surjective) morphism

${\rm CH}_{0}(X|D)_{\deg 0}\to J^{d}(U),$

and the universality of $J^{d}(X|D)$ as a regular quotient of ${\rm
CH}_{0}(X|D)_{\deg 0}$ induces then a canonical map
$\lambda^{alb}_{X|D}:J^{d}(X|D)\to J^{d}(U)$. Using (3.5) and Lemma 3.1, we
easily obtain the following description of various semi-abelian Albanese
varieties.

######  Theorem 3.2.

The canonical map ${\rm CH}_{0}(X|D)\to H^{sus}_{0}(U)$ induces a commutative
diagram

(3.10) $\textstyle{{\rm CH}_{0}(X|D)_{\deg
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{X|D}}$$\scriptstyle{\rho_{X|D}}$$\textstyle{H^{sus}_{0}(U)_{\deg
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{U}}$$\textstyle{J^{d}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda^{alb}_{X|D}}$$\textstyle{J^{d}(U)}$

such that the following hold.

  1. (1)

$\lambda^{alb}_{X|D}$ is an isomorphism of semi-abelian varieties.

  2. (2)

$J^{d}(X|D)$ is the Cartier dual of the 1-motive $[\Pi(S_{X})\to{\rm
Pic}^{0}(X)]$.

  3. (3)

$J^{d}(U)$ is the Cartier dual of the 1-motive $[\Pi_{\rm Serre}(X|D)\to{\rm
Pic}^{0}(X)]$.

  4. (4)

There is a commutative diagram of short exact sequences of semi-abelian
varieties

(3.11)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{d}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{+,*}}$$\textstyle{J^{d}(U)\times
J^{d}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}_{-}}$$\textstyle{J^{d}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{d}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{+,*}}$$\textstyle{J^{d}(S_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}_{-}}$$\textstyle{J^{d}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the vertical arrows are surjective.

###  3.3. Chow group vs. motivic cohomology of the double

We let $k$ be an algebraically closed field of characteristic $p>0$. We let
$(X,D)$ be a pair consisting of an integral smooth projective $k$-scheme $X$
of dimension $d\geq 1$ and a reduced effective Cartier divisor $D$ on it.
Under this setup, we shall identify the Levine-Weibel Chow group of $S_{X}$ to
its motivic cohomology. We shall use the following consequence of [35, Thm.
6.5] which uses our assumption on $D$ in an essential way. But note that it
does not use our assumption on ${\rm char}(k)$.

######  Lemma 3.3.

$($[35, Thm. 6.5]$)$ The canonical map $\delta_{S_{X}}\colon{\rm
CH}^{LW}_{0}(S_{X})\to{\rm CH}_{0}^{BK}(S_{X})$ is an isomorphism.

######  Lemma 3.4.

The map $\lambda_{S_{X}}\colon{\rm CH}_{0}^{BK}(S_{X})_{\Lambda}\to
H^{2d}(S_{X},\Lambda(d))$ is surjective and the map $\lambda_{S_{X}}\colon{\rm
CH}_{0}^{BK}(S_{X})_{\Lambda}\\{p^{\prime}\\}\to
H^{2d}(S_{X},\Lambda(d))\\{p^{\prime}\\}$ is an isomorphism.

###### Proof.

In this proof, we shall assume all abelian groups to be tensored with
$\Lambda$. By construction, there is a commutative diagram (see [37, (8.13)])

(3.12)
---
$\textstyle{{\mathcal{Z}}_{0}(S_{X},D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}_{0}(S_{X},D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{sus}_{0}(S_{X}\setminus
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}^{BK}(S_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{S_{X}}}$$\textstyle{H^{2d}(S_{X},\Lambda(d)),}$

where the map $H^{sus}_{0}(S_{X}\setminus D)\to H^{2d}(S_{X},\Lambda(d))$ is
induced by the localization sequence (2.4) associated to the open embedding
$S_{X}\setminus D\to S_{X}$. It follows from [37, Thm. 5.1] that this map is
surjective. This in particular implies that the motivic cohomology group
$H^{2d}(S_{X},\Lambda(d))$ is generated by the free abelian group on the set
of closed points in $S_{X}\setminus D$. But this is equivalent to
$\lambda_{S_{X}}$ being surjective.

To show the isomorphism between the primed torsion subgroups (i.e., on the
prime-to-$p$-torsion part), we consider the commutative diagram of split short
exact sequences (note that the lines stay exact even after applying
$(-)\\{p^{\prime}\\}$, since (2.7) is split)

(3.13)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
CH}_{0}(X|D))\\{p^{\prime}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
CH}_{0}^{BK}(S_{X}))\\{p^{\prime}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
CH}_{0}(X))\\{p^{\prime}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H^{sus}_{0}(X\setminus
D))\\{p^{\prime}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H^{2d,d}(S_{X}))\\{p^{\prime}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H^{2d,d}(X))\\{p^{\prime}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The right vertical arrow is an isomorphism as $X$ is smooth. It suffices
therefore to show that the left vertical arrow is an isomorphism. For this, we
consider the commutative diagram (3.10). It follows from Theorem 3.2(1) that
the lower horizontal arrow in (3.10) is an isomorphism. The left (resp. right)
vertical arrow is an isomorphism on the primed torsion subgroups by [4, Thm.
11.5] (resp. [52, Thm. 1.1]). Our assertion now follows. ∎

We can now state and prove the main result of § 3.

######  Theorem 3.5.

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a
smooth projective $k$-scheme of pure dimension $d\geq 1$ and $D\subset X$ a
reduced effective Cartier divisor. Then the composition

$\lambda^{LW}_{S_{X}}\colon{\rm CH}^{LW}_{0}(S_{X})_{\Lambda}\to{\rm
CH}^{BK}_{0}(S_{X})_{\Lambda}\to H^{2d}(S_{X},\Lambda(d))$

is an isomorphism.

###### Proof.

We can assume $X$ is integral. The map $\lambda^{LW}_{S_{X}}$ is surjective by
Lemmas 3.3 and 3.4. To show it is injective, we first assume
$\Lambda={\mathbb{Z}}[\tfrac{1}{p}]$.

It follows from [37, Thm. 5.9, Lem. 8.1] that there is a commutative diagram

(3.14) $\textstyle{{\rm
CH}^{LW}_{0}(S_{X})_{\Lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{S_{X}}^{LW}}$$\textstyle{K_{0}(S_{X})_{\Lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2d,d}(S_{X})_{\Lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{KH_{0}(S_{X})_{\Lambda},}$

where $KH_{*}(S_{X})$ is Weibel’s homotopy $K$-theory. We remark here that
even as [37, Lem. 8.1] is proven over the field of complex numbers, its proof
holds over any algebraically closed field.

The right vertical arrow in (3.14) is an isomorphism by [57] (see also [54,
Exer. 9.11]) as $p>0$. The kernel of the top horizontal arrow is a torsion
group of exponent $(d-1)!$ by [41, Cor. 2.7]. It follows that the kernel of
$\lambda^{LW}_{S_{X}}$ is torsion of exponent $(d-1)!$. We therefore have a
short exact sequence

(3.15) $0\to{\rm Ker}(\lambda^{LW}_{S_{X}})_{\Lambda}\to{\rm
CH}^{LW}_{0}(S_{X})_{\Lambda}\to H^{2d,d}(S_{X})_{\Lambda}\to 0,$

where the group on the left is torsion of exponent $(d-1)!$. On the other
hand, since $k$ is algebraically closed and $S_{X}$ is projective, one knows
that the degree zero part of ${\rm CH}^{LW}_{0}(S_{X})_{\Lambda}$ is divisible
(see the last part in the proof of Theorem 6.4). Since ${\rm
Ker}(\lambda^{LW}_{S_{X}})$ lies in this subgroup, it follows from Lemma 3.4
that ${\rm Ker}(\lambda^{LW}_{S_{X}})_{\Lambda}$ is also divisible. The result
now follows since a divisible torsion group of bounded exponent must be zero.

We are now left with proving the injectivity of $\lambda^{LW}_{S_{X}}$ when
$k$ admits resolution of singularities (even if ${\rm char}(k)>0$) and
$\Lambda={\mathbb{Z}}$. In this case, the above argument shows that
$\lambda^{LW}_{S_{X}}$ is surjective whose kernel is a $p$-primary torsion
group. We now consider the commutative diagram

(3.16) $\textstyle{{\rm
CH}^{LW}_{0}(S_{X})\\{p\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\rho_{S_{X}}}$$\textstyle{{\rm
CH}_{0}(\overline{S}_{X})\\{p\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\overline{S}_{X}}}$$\textstyle{J^{d}(S_{X})\\{p\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{J^{d}(\overline{S}_{X})\\{p\\}.}$

It follows from [35, Thm. 6.4] that the vertical arrows are isomorphisms. On
the other hand, we know that $J^{d}(S_{X})$ is a semi-abelian variety whose
abelian variety quotient is $J^{d}(\overline{S}_{X})$. Since an algebraic
torus over $k$ has no $p$-primary torsion (note that ${\rm char}(k)=p>0$), it
follows that the lower horizontal arrow in (3.16) is an isomorphism. In
particular, the upper horizontal arrow is also an isomorphism.

In the final step, we consider the commutative diagram

(3.17) $\textstyle{{\rm
CH}^{LW}_{0}(S_{X})\\{p\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda^{LW}_{S_{X}}}$$\scriptstyle{\pi^{*}}$$\textstyle{H^{2d,d}(S_{X})\\{p\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{{\rm
CH}_{0}(\overline{S}_{X})\\{p\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{\overline{S}_{X}}}$$\textstyle{H^{2d,d}(\overline{S}_{X})\\{p\\}.}$

Since $\overline{S}_{X}$ is smooth, $\lambda_{\overline{S}_{X}}$ is an
isomorphism. We have shown above that the left vertical arrow in (3.17) is an
isomorphism. It follows that $\lambda^{LW}_{S_{X}}$ is injective on the
$p$-primary torsion subgroups. We conclude that ${\rm
Ker}(\lambda^{LW}_{S_{X}})$ is zero. This completes the proof. ∎

We end this section with the following question. Using the diagrams (3.4) and
(3.6), one shows that there is an 1-motive $[\Lambda_{\rm Serre}(X|D)\to{\rm
Pic}^{0}(\overline{S}_{X})]$ and a map of 1-motives $[\Lambda_{\rm
Serre}(X|D)\to{\rm Pic}^{0}(\overline{S}_{X})]\to[\Lambda(S_{X})\to{\rm
Pic}^{0}(\overline{S}_{X})]$. Let $J^{d}_{cdh}(S_{X})$ denote the Cartier dual
of $[\Lambda_{\rm Serre}(X|D)\to{\rm Pic}^{0}(\overline{S}_{X})]$.

######  Question 3.6.

Is there a surjective homomorphism $\rho^{cdh}_{S_{X}}:H^{2d,d}(S_{X})_{\deg
0}\twoheadrightarrow J^{d}_{cdh}(S_{X})$ and a commutative diagram

(3.18) $\textstyle{{\rm CH}^{LW}_{0}(S_{X})_{\deg
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{S_{X}}}$$\scriptstyle{\rho_{S_{X}}}$$\textstyle{H^{2d,d}(S_{X})_{\deg
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{cdh}_{S_{X}}}$$\textstyle{J^{d}(S_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J^{d}_{cdh}(S_{X})}$

such that the vertical arrows are the universal regular quotient maps?

##  4\. Proofs of Theorems 1.1 and 1.2

In this section, we shall prove Theorems 1.1 and 1.2. We shall prove the two
parts of Theorem 1.1 in separate subsections for the sake of clarity of
exposition. We begin with the proof of its part (1).

###  4.1. Positive characteristic case

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $(X,D)$
be a pair consisting of an integral smooth projective $k$-scheme $X$ of
dimension $d\geq 1$ and an effective (but not necessarily reduced) Cartier
divisor $D$ on it. We let $U=X\setminus D$. We need a lemma and a proposition.

######  Lemma 4.1.

Let $E\subset X$ be an effective Cartier divisor which is contained in $D$ and
whose support coincides with that of $D$. Then the kernel of the canonical
surjection ${\rm CH}_{0}(X|D)\twoheadrightarrow{\rm CH}_{0}(X|E)$ is a
$p$-primary torsion group.

###### Proof.

We shall prove the lemma by induction on $d$. If $d\leq 2$, the cycle class
map ${\rm CH}_{0}(X|D)\to H^{d}_{\rm zar}(X,{\mathcal{K}}^{M}_{d,(X,D)})$ is
an isomorphism by [4, Thm. 1.8], where ${\mathcal{K}}^{M}_{n,(X,D)}={\rm
Ker}({\mathcal{K}}^{M}_{n,X}\twoheadrightarrow{\mathcal{K}}^{M}_{n,D})$ is the
Zariski sheaf of relative Milnor $K$-groups. It suffices therefore to show
that the kernel of the canonical surjection $H^{d}_{\rm
zar}(X,{\mathcal{K}}^{M}_{d,(X,D)})\twoheadrightarrow H^{d}_{\rm
zar}(X,{\mathcal{K}}^{M}_{d,(X,E)})$ is a $p$-primary torsion group. Using the
long exact cohomology sequence associated to the exact sequence of Zariski
sheaves

$0\to{\mathcal{K}}^{M}_{d,(X,D)}\to{\mathcal{K}}^{M}_{d,(X,E)}\to{\mathcal{K}}^{M}_{d,(D,E)}\to
0,$

our assertion is implied by the claim that $H^{d-1}_{\rm
zar}(D,{\mathcal{K}}^{M}_{d,(D,E)})$ is a $p$-primary torsion group (in fact,
of bounded exponent). But this is well known (e.g., see [24, Lem. 2.6]).

We now assume $d\geq 3$. Let $\nu\colon C\to X$ be a finite morphism from a
regular integral projective curve whose image is not contained in $D$ and let
$f\in{\rm
Ker}({\mathcal{O}}^{\times}_{C,E^{\prime}}\twoheadrightarrow{\mathcal{O}}^{\times}_{E^{\prime}})$,
where $E^{\prime}=\nu^{-1}(E)$. We need to show that $\nu_{*}({\rm
div}(f))\in{\rm CH}_{0}(X|D)$ is killed by a power of $p$.

We can get a factorization
$C\stackrel{{\scriptstyle\nu^{\prime}}}{{\hookrightarrow}}{\mathbb{P}}^{n}_{X}\xrightarrow{\pi}X$
of $\nu$, where $\nu^{\prime}$ is a closed immersion and $\pi$ is the
canonical projection. It is clear that $\nu^{\prime}_{*}({\rm div}(f))$ dies
in ${\rm CH}_{0}({\mathbb{P}}^{n}_{X}|{\mathbb{P}}^{n}_{E})$ and $\nu_{*}({\rm
div}(f))=\pi_{*}(\nu^{\prime}_{*}({\rm div}(f)))$. Using the push-forward map
on the Chow groups with modulus, it suffices therefore to show that
$\nu^{\prime}_{*}({\rm div}(f))$ is killed by some power of $p$ in ${\rm
CH}_{0}({\mathbb{P}}^{n}_{X}|{\mathbb{P}}^{n}_{D})$. We can thus assume that
$\nu\colon C\to X$ is a closed immersion.

We now fix a closed embedding $X\hookrightarrow{\mathbb{P}}^{N}_{k}$. We can
use [2, Thm. 7] to find a hypersurface $H\subset{\mathbb{P}}^{N}_{k}$
containing $C$ and not containing $X$ such that the scheme theoretic
intersection $X\cap H$ has the property that it is smooth and contains no
irreducible component of $D$. By [19, Cor. 3.13], we can assume that $X\cap H$
is integral. We let $Y=X\cap H$, $D_{Y}=D\cap H$ and $E_{Y}=E\cap H$.

Let
$C\stackrel{{\scriptstyle\nu^{\prime}}}{{\hookrightarrow}}Y\stackrel{{\scriptstyle\iota}}{{\hookrightarrow}}X$
be the factorization of $\nu$. It follows from the choice of $Y$ that
$\nu^{\prime}_{*}({\rm div}(f))$ dies in ${\rm CH}_{0}(Y|E_{Y})$. This implies
by induction that $\nu^{\prime}_{*}({\rm div}(f))\in{\rm CH}_{0}(Y|D_{Y})$ is
killed by a power of $p$. It follows that $\nu_{*}({\rm
div}(f))=\iota_{*}(\nu^{\prime}_{*}({\rm div}(f)))$ is killed by a power of
$p$ in ${\rm CH}_{0}(X|D)$. This concludes the proof. ∎

######  Proposition 4.2.

The kernel of the map $\lambda_{X|D}\colon{\rm CH}_{0}(X|D)\twoheadrightarrow
H^{sus}_{0}(U)$ is a $p$-primary torsion group. Furthermore, the map
$\lambda_{X|D}\colon{\rm CH}_{0}(X|D)\\{\ell\\}\twoheadrightarrow
H^{sus}_{0}(U)\\{\ell\\}$ is an isomorphism for any prime $\ell\neq p$.

###### Proof.

By Lemma 4.1, we can assume that $D$ is reduced. Now, the second part of the
proposition is already shown in the proof of Lemma 3.4 (see the argument after
(3.13)). To prove the first part, we let $S_{X}$ be the double of $X$ along
$D$ and consider the commutative diagram (see (3.13) again)

(4.1) $\textstyle{{\rm
CH}_{0}(X|D)[\tfrac{1}{p}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{X|D}}$$\textstyle{{\rm
CH}_{0}^{BK}(S_{X})[\tfrac{1}{p}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{S_{X}}}$$\textstyle{H^{sus}_{0}(U)[\tfrac{1}{p}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2d,d}(S_{X})[\tfrac{1}{p}].}$

The horizontal arrows are split injective, and the right vertical arrow is an
isomorphism by Theorem 3.5. It follows that the left vertical arrow is also an
isomorphism. This implies our assertion. ∎

We can now prove Theorem 1.1(1). We restate it for convenience.

######  Theorem 4.3.

Let $X$ be a smooth projective scheme over an algebraically closed field $k$
of characteristic $p>0$. Let $D\subset X$ be a reduced effective Cartier
divisor with complement $U$. Then the canonical map

$\lambda_{X|D}\colon{\rm CH}_{0}(X|D)\to H^{sus}_{0}(U)$

is an isomorphism.

###### Proof.

We can assume $X$ to be integral. By Proposition 4.2, we only need to show
that $\lambda_{X|D}$ induces an isomorphism between the torsion subgroups. For
this, we use the commutative diagram (3.10). It follows from Theorem 3.2(1)
that the bottom horizontal arrow in this diagram is an isomorphism. The left
(resp. right) vertical arrow is an isomorphism on the torsion subgroups by
[35, Thm. 1.2] (resp. [20, Thm. 1.7]). This implies that the same is true for
the top horizontal arrow. ∎

###  4.2. Characteristic zero case

Let $k$ be any field (not necessarily algebraically closed) of characteristic
zero. We let $\Omega_{k}$ be the module of absolute Kähler differentials of
$k$. The second part of Theorem 1.1 is an easy consequence of the following
general result.

######  Theorem 4.4.

Let $\overline{C}\subset{\mathbb{P}}^{2}_{k}$ be the cuspidal curve given by
the homogeneous equation $x_{0}x^{2}_{1}-x^{3}_{2}=0$. Then ${\rm
CH}_{0}({\mathbb{P}}^{2}_{k}|\overline{C})\cong\Omega_{k}\oplus{\mathbb{Z}}$
and $H^{sus}_{0}({\mathbb{P}}^{2}_{k}\setminus\overline{C})\cong{\mathbb{Z}}$.

###### Proof.

We shall freely use the duality results from [55]. We drop the subscript $k$
from the notations of ${\mathbb{P}}^{n}_{k}$ and ${\mathbb{A}}^{n}_{k}$.

Let us first prove that
$H^{sus}_{0}({\mathbb{P}}^{2}_{k}\setminus\overline{C})\cong{\mathbb{Z}}$. Let
$C_{\infty}=(0:1:0)$ be the point at infinity and $P=(1:0:0)$ the singular
point of $\overline{C}$. We write $C=\overline{C}\setminus
C_{\infty}=({\mathbb{P}}^{2}_{k}\setminus H_{\infty})\cap\overline{C}$, where
$H_{\infty}\cong{\mathbb{P}}^{1}_{k}$ is the line $\\{X_{0}=0\\}$. We claim
that there is a split exact sequence

(4.2) $0\to k^{\times}\to H_{c}^{4}({\mathbb{A}}^{2}\setminus
C,{\mathbb{Z}}(2))\to{\mathbb{Z}}\to 0$

Indeed, letting $U={\mathbb{A}}^{2}\setminus C$, the localization triangle

$M^{c}(C)\to M^{c}({\mathbb{A}}^{2})\to M^{c}(U)\to M^{c}(C)[1]$

in ${\mathbf{DM}}^{\rm eff}(k,{\mathbb{Z}})$ gives rise to an exact sequence
of cohomology with compact support

(4.3) $\cdots\to H_{c}^{3}(C,{\mathbb{Z}}(2))\to
H_{c}^{4}(U,{\mathbb{Z}}(2))\to H_{c}^{4}({\mathbb{A}}^{2},{\mathbb{Z}}(2))\to
H_{c}^{4}(C,{\mathbb{Z}}(2)).$

By duality and homotopy invariance, we have

$H_{c}^{4}({\mathbb{A}}^{2},{\mathbb{Z}}(2))=H_{0}({\mathbb{A}}^{2},{\mathbb{Z}}(0))=H_{0}(k,{\mathbb{Z}}(0))={\mathbb{Z}}.$

Since $k$ is infinite and $U$ is a rational variety, we can choose $x\in
U(k)$. We then get a sequence of maps

$H^{0}(k(x),{\mathbb{Z}}(0))\to H_{c}^{4}(U,{\mathbb{Z}}(2))\to
H_{c}^{4}({\mathbb{A}}^{2},{\mathbb{Z}}(2))\to H^{0}(k,{\mathbb{Z}}(0)),$

where the first arrow is the Gysin homomorphism and the last arrow is the
push-forward map between cohomology with compact support induced by the flat
morphism ${\mathbb{A}}^{2}\to{\rm Spec\,}(k)$. Since the composition is known
to be an isomorphism, we get a split surjection
$H_{c}^{4}(U,{\mathbb{Z}}(2))\twoheadrightarrow{\mathbb{Z}}$.

We are left to compute $H_{c}^{3}(C,{\mathbb{Z}}(2))$. Comparing the exact
triangles for $C$ and for ${\mathbb{A}}^{1}$, again we get
$H_{c}^{3}(C,{\mathbb{Z}}(2))=H_{c}^{3}({\mathbb{A}}^{1},{\mathbb{Z}}(2))$. By
duality and homotopy invariance, the latter is isomorphic to
$H_{-1}(k,{\mathbb{Z}}(-1))=H^{1}(k,{\mathbb{Z}}(1))=k^{\times}$. The long
exact sequence (4.3) then gives

$H_{c}^{3}({\mathbb{A}}^{2},{\mathbb{Z}}(2))\to k^{\times}\to
H_{c}^{4}(U,{\mathbb{Z}}(2))\to{\mathbb{Z}}\to 0.$

Since
$H_{c}^{3}({\mathbb{A}}^{2},{\mathbb{Z}}(2))=H_{1}({\mathbb{A}}^{2},{\mathbb{Z}})=H_{1}(k,{\mathbb{Z}})=0$
and $H_{0}(U,{\mathbb{Z}}(0))\cong H_{c}^{4}(U,{\mathbb{Z}}(2))$ by duality,
we finally get (4.2).

Next, note that $H_{\infty}\setminus C_{\infty}={\mathbb{A}}^{1}$. Looking at
the triangle in ${\mathbf{DM}}^{\rm eff}(k,{\mathbb{Z}})$:

$M^{c}(H_{\infty}\setminus C_{\infty})\to
M^{c}({\mathbb{P}}^{2}\setminus\overline{C})\to M^{c}(U)\to
M^{c}(H_{\infty}\setminus C_{\infty})[1],$

we get

$k^{\times}=H_{c}^{3}({\mathbb{A}}^{1},{\mathbb{Z}}(2))\to
H_{c}^{4}(U,{\mathbb{Z}}(2))\to
H_{c}^{4}({\mathbb{P}}^{2}\setminus\overline{C},{\mathbb{Z}}(2))\to 0.$

Since $H_{c}^{4}(U,{\mathbb{Z}}(2))=k^{\times}\oplus{\mathbb{Z}}$ by (4.2),
it’s clear by construction that
$H_{c}^{4}({\mathbb{P}}^{2}\setminus\overline{C},{\mathbb{Z}}(2))=H_{0}^{sus}({\mathbb{P}}^{2}\setminus\overline{C},{\mathbb{Z}}(0))={\mathbb{Z}}$.

We now turn to the first statement of the theorem. By [6, Thm. 1.1], we have
an identification ${\rm CH}_{0}({\mathbb{P}}^{2}|\overline{C})\cong H^{2}_{\rm
Zar}({\mathbb{P}}^{2},\mathcal{K}^{M}_{2,({\mathbb{P}}^{2},\overline{C})})$.
Since the relative $K_{0}$-group $K_{0}({\mathbb{P}}^{2},\overline{C})$
decomposes as $H^{2}_{\rm
Zar}({\mathbb{P}}^{2},\mathcal{K}^{M}_{2,({\mathbb{P}}^{2},\overline{C})})\oplus{\rm
Pic}({\mathbb{P}}^{2},\overline{C})$ and the group ${\rm
Pic}({\mathbb{P}}^{2},\overline{C})$ vanishes by direct inspection, it is
enough to compute $K_{0}({\mathbb{P}}^{2},\overline{C})$.

This can be done explicitly by means of the long localization sequence for
$K$-theory with support. In particular, there is a short exact sequence

$0\to{\mathbb{Z}}\to K_{0}({\mathbb{P}}^{2},\overline{C})\to
K_{0}({\mathbb{A}}^{2},C)\cong SK_{1}(C)\to 0$

and we are left to show that $SK_{1}(C)\cong\Omega_{k}$. This is classical,
and can be done by descent along the normalization morphism
$\nu\colon{\mathbb{A}}^{1}\to C$ using the $K_{1}(A,B,I)$ computation of [18].
We leave the details to the interested reader. ∎

###  4.3. Kerz-Saito vs. Russell’s Chow groups with modulus

There exists another notion different from the one used in this paper (and
specifically used in the previous theorem) of Chow group of 0-cycles with
modulus. This Chow group was introduced by H. Russell [47] in the study of a
higher dimensional analogue of the generalized Jacobian of Rosenlicht-Serre.
Russell’s Chow group with modulus is defined as the cokernel of the map

${\rm div}\colon\bigoplus_{\phi_{C}\colon C\to X}G(C,\phi_{C}^{*}(D-D_{\rm
red})+(\phi_{C}^{*}(D))_{\rm red})\to{\mathcal{Z}}_{0}(X,D).$

In particular, in view of Theorem 2.4, it is clear that it agrees with
$H_{0}^{sus}(X\setminus D)$ when $D$ is reduced and $X$ is proper. Theorem 4.4
then exhibits explicitly an example (at least in characteristic zero) for
which Russell’s Chow group and the Kerz-Saito Chow group do not agree. Several
authors have enquired in the past about this comparison. Note that in view of
[6, Thm. 1.1], it follows that Russell’s Chow group does not satisfy Bloch’s
formula, even for $D$ reduced. In particular, it may not admit a cycle class
map to relative $K$-theory.

###  4.4. Finite coefficient case

We now prove Theorem 1.2. We restate it for convenience.

######  Theorem 4.5.

Let $k$ be a perfect field and $X$ a smooth projective $k$-scheme. Let
$D\subset X$ be an effective Cartier divisor with complement $U$. Assume that
the irreducible components of $D_{\rm red}$ are regular. Then the map

$\lambda_{X|D}\colon{\rm CH}_{0}(X|D)_{R}\to H^{sus}_{0}(U)_{R}$

is an isomorphism, where $R=[\tfrac{1}{p}]$ if ${\rm char}(k)=p>0$ and
${{\mathbb{Z}}}/m$ with $m\neq 0$ if ${\rm char}(k)=0$.

###### Proof.

When $\text{\rm dim}(X)=1$, we can use [46, Thm. 1.3(2)] to assume that $D$ is
reduced. In the latter case, the theorem holds integrally as one can directly
check. We therefore assume $d\geq 2$. We can also assume $X$ is integral.
Since both sides of (2.5) are quotients of the free abelian group
${\mathcal{Z}}_{0}(U)$, we need to show that any relation imposed on
${\mathcal{Z}}_{0}(X|D)$ to define $H^{sus}_{0}(U)$ lies in the image of
${\mathcal{R}}_{0}(X|D)\to{{\mathcal{Z}}_{0}(U)}_{R}$, where
${\mathcal{R}}_{0}(X|D)={\rm Ker}({\mathcal{Z}}_{0}(U)\twoheadrightarrow{\rm
CH}_{0}(X|D))$.

So let $C\subset X$ be an integral curve not contained in $D$ and let
$\nu\colon C^{N}\to C\hookrightarrow X$ be the composite map, where $C^{N}$ is
the normalization of $C$. Let $f\in k(C^{N})^{\times}$ be a rational function
on $C^{N}$ which is regular in an affine neighborhood of $\nu^{-1}(D)$ such
that $f(x)=1$ for every $x\in\nu^{-1}(D)$. We need to show that ${\rm div}(f)$
dies in ${{\rm CH}_{0}(X|D)}_{R}$.

Since $\nu$ is finite, we can find a factorization

(4.4)
$\textstyle{C^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\nu}$$\textstyle{{\mathbb{P}}^{n}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X,}$

where $\phi$ is a closed immersion and $\pi$ is the canonical projection.
Since $\pi$ is smooth, the irreducible components of
$D^{\prime}:={\mathbb{P}}^{n}_{D}$ are regular. Furthermore, $C^{N}\not\subset
D^{\prime}$. Since $\pi_{*}({\rm div}(f)_{C^{N}})={\rm div}(f)_{C}={\rm
div}(f)$ under the proper push-forward map
$\pi_{*}\colon{\mathcal{Z}}_{0}({\mathbb{P}}^{n}_{U})\to{\mathcal{Z}}_{0}(U)$
and since
$\pi_{*}({\mathcal{R}}_{0}({\mathbb{P}}^{n}_{X}|D^{\prime}))\subset{\mathcal{R}}_{0}(X|D)$
(see e.g., [36, § 2] or [7, Lem. 2.7]), it suffices to show that ${\rm
div}(f)_{C^{N}}$ dies in ${{\rm
CH}_{0}({\mathbb{P}}^{n}_{X}|{\mathbb{P}}^{n}_{D})}_{R}$. Up to replacing $X$
with ${\mathbb{P}}^{n}_{X}$ and $C$ with $C^{N}$, we can therefore assume that
$C\subset X$ is regular so that $\nu$ is a closed immersion of regular
schemes.

Since $D_{\rm red}$ is reduced with regular irreducible components, we can
apply [48, Prop. A.6] to find a finite sequence of blow-ups
$\phi\colon\widetilde{X}\to X$ over the closed points of $D\cap C$ such that
the scheme theoretic inverse image $\widetilde{D}:=\widetilde{X}\times_{X}D$
satisfies the following.

  1. (1)

$\widetilde{D}_{\rm red}$ is reduced and its irreducible components are
regular.

  2. (2)

The strict transform $\widetilde{C}$ of $C$ is regular and intersects
$\widetilde{D}_{\rm red}$ only in the regular locus of $\widetilde{D}_{\rm
red}$ and transversely.

Since $\phi$ is proper, we have a commutative diagram

(4.5)
$\textstyle{{\mathcal{Z}}_{0}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{*}}$$\scriptstyle{\cong}$$\textstyle{{\rm
CH}_{0}(\widetilde{X}|\widetilde{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{*}}$$\textstyle{{\mathcal{Z}}_{0}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}({X}|{D}),}$

where $\phi_{*}$ is the push-forward map of the 0-cycles groups. Since
$\phi\colon\widetilde{C}\to C$ is an isomorphism, we see that $f\in
k(\widetilde{C})^{\times}$ such that $\phi_{*}({\rm
div}(f)_{\widetilde{C}})={\rm div}(f)_{C}={\rm div}(f)$. Moreover, $f$ is
regular in a neighborhood of $\widetilde{D}\cap\widetilde{C}$ with $f(x)=1$
for every $x\in\widetilde{D}\cap\widetilde{C}\subset\phi^{-1}(D\cap C)$. In
particular, ${\rm div}(f)_{\widetilde{C}}\in{\rm
Ker}({\mathcal{Z}}_{0}(U)\twoheadrightarrow H^{sus}_{0}(U))$.

It follows from (4.5) that ${\rm div}(f)_{C}$ will die in ${{\rm
CH}_{0}(X|D)}_{R}$ if we can show that ${\rm div}(f)_{\widetilde{C}}$ dies in
${{\rm CH}_{0}(\widetilde{X}|\widetilde{D})}_{R}$. We can therefore assume
that our original curve $C\subset X$ has the property that it is regular and
it intersects $D_{\rm red}$ only in the regular locus of $D_{\rm red}$ and
transversely. But it is immediate from various definitions that in this case,
$f\in k(C)^{\times}$ is regular in a neighborhood of $E$ and $f(x)=1$ for all
$x\in E$ if and only if $f\in{\rm
Ker}({\mathcal{O}}^{\times}_{C,E}\to{\mathcal{O}}^{\times}_{E})$, where
$E=C\times_{X}D_{\rm red}$. The latter condition implies that ${\rm
div}(f)\in{\mathcal{R}}_{0}(X|D_{\rm red})$. That is, ${\rm div}(f)$ dies in
${\rm CH}_{0}(X|D_{\rm red})$. By [46, Thm. 1.3(2)], this implies that ${\rm
div}(f)$ dies in ${{\rm CH}_{0}({X}|{D})}_{R}$. This concludes the proof. ∎

##  5\. Bloch’s torsion theorem for Suslin homology

The goal of this section is to prove Theorem 1.3. We shall first establish
some preliminary results of independent interest.

###  5.1. A torsion theorem for the double

We fix an algebraically closed field $k$ of characteristic zero and a prime
$\ell$. Let $X$ be an integral smooth projective surface over $k$ and let
$D\subset X$ be a reduced effective Cartier divisor. We let $S_{X}$ be the
double of $X$ along $D$.

######  Lemma 5.1.

The canonical map

$\lambda_{S_{X}}\colon{\rm CH}^{LW}_{0}(S_{X})\\{\ell\\}\to
H^{4}(S_{X},{\mathbb{Z}}(2))\\{\ell\\}$

is an isomorphism.

###### Proof.

By a Lefschetz principle argument, we can assume that $k={\mathbb{C}}$. In
this case, the Albanese homomorphism $\rho_{S_{X}}\colon{\rm
CH}^{LW}_{0}(S_{X})_{\deg 0}\to J^{2}(S_{X})$ has a factorization

${\rm CH}^{LW}_{0}(S_{X})_{\deg
0}\xrightarrow{\lambda_{S_{X}}}H^{4}(S_{X},{\mathbb{Z}}(2))_{\deg
0}\xrightarrow{\rho^{\prime}_{S_{X}}}J^{2}(S_{X})$

by [37, Thm. 8.4] such that $\rho^{\prime}_{S_{X}}$ is an isomorphism on the
torsion subgroups. On the other hand, the main result of [3] asserts that
$\rho_{S_{X}}$ is an isomorphism on the torsion subgroups. This forces
$\lambda_{S_{X}}$ to be an isomorphism on the torsion subgroups. ∎

By [35, Thm. 3.6], there is a canonical isomorphism
$H^{3}_{\text{\'{e}t}}(S_{X},{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(2))\xrightarrow{\cong}{\rm
CH}^{LW}_{0}(S_{X})\\{\ell\\}$. One has a similar isomorphism for $X$ by
Bloch. Let $\gamma_{S_{X}}$ be the composition of this isomorphism with
$\lambda_{S_{X}}$. We denote the corresponding map for $X$ by $\gamma_{X}$.

######  Lemma 5.2.

There is a commutative diagram

(5.1)
$\textstyle{H^{3}_{\text{\'{e}t}}(S_{X},{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}_{-}}$$\scriptstyle{\gamma_{S_{X}}}$$\textstyle{H^{3}_{\text{\'{e}t}}(X,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{X}}$$\textstyle{H^{4}(S_{X},{\mathbb{Z}}(2))\\{\ell\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}_{-}}$$\textstyle{H^{4}(X,{\mathbb{Z}}(2))\\{\ell\\}}$

such that the vertical arrows are isomorphisms.

###### Proof.

By [35, Thm. 3.6], there is a canonical isomorphism $\gamma_{S_{X}}\colon
H^{3}_{\text{\'{e}t}}(S_{X},{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(2))\xrightarrow{\cong}{\rm
CH}^{LW}_{0}(S_{X})\\{\ell\\}$ (this is true for any complete surface).
Moreover, it was shown in [35, Thm. 5.4] (see diagram (5.13) of op. cit.) that
$\gamma_{S_{X}}$ and $\gamma_{X}$ are compatible with $\iota^{*}_{-}$. Since
$\lambda_{S_{X}}$ and $\lambda_{X}$ are clearly compatible with
$\iota^{*}_{-}$, our assertion follows by Lemma 5.1. ∎

######  Lemma 5.3.

Let $U=X\setminus D$. There is a canonical isomorphism

$\gamma_{U}\colon
H^{3}_{c,{\acute{e}t}}(U,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(2))\xrightarrow{\cong}H^{sus}_{0}(U)\\{\ell\\}.$

###### Proof.

We have a commutative diagram of (split) exact sequences

(5.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}_{c,{\acute{e}t}}(U,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{+,*}}$$\textstyle{H^{3}_{\text{\'{e}t}}(S_{X},{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}_{-}}$$\scriptstyle{\gamma_{S_{X}}}$$\textstyle{H^{3}_{\text{\'{e}t}}(X,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{X}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{sus}_{0}(U)\\{\ell\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{+,*}}$$\textstyle{H^{4}(S_{X},{\mathbb{Z}}(2))\\{\ell\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}_{-}}$$\textstyle{H^{4}(X,{\mathbb{Z}}(2))\\{\ell\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Using this diagram, we conclude the proof by Lemma 5.2. ∎

###  5.2. Proof of Theorem 1.3

Let $k$ be an algebraically closed field and $U$ a quasi-projective $k$-scheme
of pure dimension $d\geq 0$. Let $U\hookrightarrow X$ be a dense open
immersion such that $X$ is a smooth projective $k$-scheme. Let $\ell$ be a
prime different from ${\rm char}(k)$. We can now prove Theorem 1.3. We restate
it for convenience.

######  Theorem 5.4.

There is an isomorphism

(5.3) $\vartheta_{U}\colon
H^{sus}_{0}(U)\\{\ell\\}\xrightarrow{\cong}H^{2d-1}_{c,{\acute{e}t}}(U,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(d)).$

###### Proof.

We can assume $d\geq 1$ because the theorem is trivial otherwise. We can
assume $U$ to be integral. Let $D\subset X$ be the complement of $U$ with
reduced closed subscheme structure. Let $p\geq 1$ be the exponential
characteristic of $p$. We consider several cases.

Assume first that $p>1$ and $\text{\rm dim}(D)\leq d-2$. Let
$\Lambda={\mathbb{Z}}[\tfrac{1}{p}]$. Since
$A\\{\ell\\}\xrightarrow{\cong}A_{\Lambda}\\{\ell\\}$ for any abelian group
$A$, we can tensor both sides of (5.3) with $\Lambda$. We then have the
localization exact sequence (see (2.4) and the discussion below it)

$H^{2d-1}(D,\Lambda(d))\to H^{sus}_{0}(U)_{\Lambda}\to{\rm
CH}_{0}(X)_{\Lambda}\to H^{2d}(D,\Lambda(d)).$

The first and the last terms of this exact sequence vanish by [37, Thm. 5.1],
and hence we get $H^{sus}_{0}(U)_{\Lambda}\xrightarrow{\cong}{\rm
CH}_{0}(X)_{\Lambda}$. Using the localization sequence for étale cohomology
and repeating the same argument, we also get
$H^{2d-1}_{c,{\acute{e}t}}(U,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(d))\xrightarrow{\cong}H^{2d-1}_{{\acute{e}t}}(X,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(d))$.
We therefore reduce to the case when $U$ is smooth and projective, where
Bloch’s theorem applies.

We next assume that $p>1$ and $\text{\rm dim}(D)=d-1$. Let $D^{\prime}\subset
D$ be the union of irreducible components of $D$ which have dimension $d-1$
and let $U^{\prime}=X\setminus D^{\prime}$. We have an exact sequence

$H^{2d-1}_{c}(U^{\prime}\setminus U,\Lambda(d))\to H^{sus}_{0}(U)_{\Lambda}\to
H^{sus}_{0}(U^{\prime})_{\Lambda}\to H^{2d}_{c}(U^{\prime}\setminus
U,\Lambda(d)).$

The two end terms vanish by [38, Lem. 6.2] because $\text{\rm
dim}(U^{\prime}\setminus U)\leq d-2$. Using the localization sequence for
étale cohomology and repeating the same argument, we get
$H^{2d-1}_{c,{\acute{e}t}}(U,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(d))\xrightarrow{\cong}H^{2d-1}_{c,{\acute{e}t}}(U^{\prime},{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(d))$.
It suffices therefore to prove the theorem for $U^{\prime}$. We can therefore
assume that $D$ has pure dimension $d-1$. Since $X$ is smooth, $D$ is
necessarily a Cartier divisor on $X$. In this case, we can combine Theorem 4.3
and [35, Thm. 1.1] to conclude our result.

We now assume $p=1$. The proof in the positive characteristic case remains
valid in characteristic zero if $d\leq 1$. However, it breaks down when $d\geq
2$ because Theorem 4.3 is no more available with us. We shall adopt a
different strategy which involves revisiting the proof of [35, Thm. 1.1].

We have already argued above that $D$ can be assumed to be a Cartier divisor
on $X$. This part of the argument does not involve ${\rm char}(k)$. We shall
assume that $X$ is integral and prove the theorem by induction on $d$. When
$d=2$, we can apply Lemma 5.3.

We now assume $d\geq 3$. We fix a closed embedding
$X\hookrightarrow{\mathbb{P}}^{n}_{k}$. By [2, Thm. 1], we can find a
hypersurface $H\subset{\mathbb{P}}^{n}_{k}$ of sufficiently high degree such
that the scheme theoretic intersection $Y=X\cap H$ satisfies the following.

  1. (1)

$Y$ is smooth and irreducible of dimension $d-1$.

  2. (2)

$Y$ contains no component of $D$.

  3. (3)

$Y\cap U\neq\emptyset$.

It follows from (1) that $Y$ is integral. We let $V=Y\cap U$ and let
$\tau\colon V\hookrightarrow U$ be the inclusion. We consider the diagram

(5.4)
$\textstyle{H^{2d-3}_{c,{\acute{e}t}}(V,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(d-1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{V}}$$\scriptstyle{\tau_{*}}$$\textstyle{H^{sus}_{0}(V)\\{\ell\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{*}}$$\textstyle{H^{2d-1}_{c,{\acute{e}t}}(U,{{\mathbb{Q}}_{\ell}}/{{\mathbb{Z}}_{\ell}}(d))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{sus}_{0}(U)\\{\ell\\}}$

The left vertical arrow is an isomorphism by [20, Prop. 5.1]. Since
$\gamma_{V}$ exists and is an isomorphism by induction, it remains to show
that the right vertical arrow is an isomorphism to finish the proof.

We now look at the commutative diagram

(5.5)
$\textstyle{H^{sus}_{0}(V)\\{\ell\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{V}}$$\scriptstyle{\tau_{*}}$$\textstyle{J^{d-1}(V)\\{\ell\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{*}}$$\textstyle{H^{sus}_{0}(U)\\{\ell\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{U}}$$\textstyle{J^{d}(U)\\{\ell\\},}$

where the horizontal arrows are the Albanese maps of Theorem 3.2. The
horizontal arrows are isomorphisms by [52, Thm. 1.1] and the right vertical
arrow is an isomorphism by [20, Thm. 7.3]. This forces the left vertical arrow
to be an isomorphism. ∎

######  Remark 5.5.

One of the referees remarked that a proof of Theorem 5.4 can also be obtained
using the Roitman torsion theorem of Spieß and Szamuely [52] together with
Poincaré duality. From a different perspective, one could use Theorem 5.4 to
show that the torsion theorem of [52] in positive characteristic is a direct
consequence of Bloch’s theorem together with the analogous result for singular
varieties.

##  6\. The finiteness theorems

In this section, we shall derive Theorem 1.5 from some general finiteness
results for motivic cohomology groups. Let $k$ be a field. Let $m\geq 1$ be an
integer which is invertible in $k$, and $p\nmid m$ if $k$ is a $p$-adic field.
Set $\Lambda^{\prime}={{\mathbb{Z}}}/m$. Let $X$ be a smooth projective
$k$-scheme.

If $k$ is algebraically closed, then it is classically known that ${{\rm
CH}_{0}(X)}/m\cong\Lambda^{\prime}$. Otherwise, the structure of ${{\rm
CH}_{0}(X)}/m$ is not fully understood. If $k$ is finite, then the main result
of unramified class field theory in geometric case due to Kato-Saito [28]
shows that ${{\rm CH}_{0}(X)}/m$ is finite. If $k$ is a $p$-adic field, a
conjecture of Colliot-Thélène predicted that ${{\rm CH}_{0}(X)}/m$ is finite.
This conjecture was settled by Saito and Sato in [48] (see also [12, Prop.
9.3]).

Let us now recall that there is an isomorphism ${{\rm CH}_{0}(X)}\cong
H^{sus}_{0}(X)$. One therefore concludes that ${H^{sus}_{0}(X)}/m$ is finite
if $k$ is either of the above three types. Suppose now that $U$ is a smooth
quasi-projective $k$-scheme which is not necessarily complete. One can then
ask if ${H^{sus}_{0}(X)}/m$ is finite. One can similarly ask if ${{\rm
CH}_{0}(X|D)}/m$ is finite if $D$ is an effective Cartier divisor on a smooth
projective scheme $X$ over $k$. To our knowledge, no answer is known about
these questions yet. We shall answer these in this section.

###  6.1. A general finiteness result

In this subsection, we prove a general result about the finiteness of motivic
cohomology groups of possibly singular schemes.

######  Lemma 6.1.

Let $k$ be either a finite, or a $p$-adic or an algebraically closed field.
Let $X$ be a smooth projective $k$-scheme of dimension $d\geq 0$. Then
$H^{2d+i}(X,\Lambda^{\prime}(d+i))$ is finite for every $i\geq 0$.

###### Proof.

If $X$ is integral of dimension $d$, then
$H^{2a-b}(X,{\mathbb{Z}}(b))\cong{\rm CH}^{a}(X,b)=0$ for $a>d+b$. This allows
us to assume that $X$ has pure dimension $d$. Subsequently, we can assume that
$X$ is integral. Using the above discussion of $i=0$ case, we can assume that
$i\geq 1$.

We now note that over any field $k$, there is a canonical isomorphism
$H^{2d+i}(X,\Lambda^{\prime}(d+i))\cong{{\rm CH}^{d+i}(X,i)}/m$. Furthermore,
Bloch’s spectral sequence $H^{p}_{\rm
zar}(X,{{\mathcal{C}}}{{\mathcal{H}}}^{q}(X,i))\Rightarrow{\rm CH}^{p+q}(X,i)$
and the isomorphisms ${\rm CH}^{p}({\mathcal{O}}_{X,x},p)\cong
H^{p}({\mathcal{O}}_{X,x},{\mathbb{Z}}(p))\cong
K^{M}_{p}({\mathcal{O}}_{X,x})$ (the latter is Gabber-Kerz improved Milnor
$K$-theory, [31]) together yield an exact sequence

(6.1) ${\underset{y\in X_{(1)}}{\bigoplus}}\
K^{M}_{i+1}(k(y))\xrightarrow{\partial}{\underset{x\in X_{(0)}}{\bigoplus}}\
K^{M}_{i}(k(x))\to{\rm CH}^{d+i}(X,i)\to 0.$

We thus get an exact sequence

(6.2) ${\underset{y\in X_{(1)}}{\bigoplus}}\
{K^{M}_{i+1}(k(y))}/m\xrightarrow{\partial}{\underset{x\in
X_{(0)}}{\bigoplus}}\ {K^{M}_{i}(k(x))}/m\to{{\rm CH}^{d+i}(X,i)}/m\to 0.$

If $k$ is algebraically closed, then $K^{M}_{i}(k)$ is divisible for $i\geq 1$
and hence we conclude that ${{\rm CH}^{d+i}(X,i)}/m=0$ for $i\geq 1$. This
proves the lemma for $k$ algebraically closed. For $k$ finite, the lemma
follows from [1, Thm. 1.2, 1.3].

Suppose now that $k$ is a $p$-adic field with residue characteristic $p$. We
first assume $i=1$. In this case, we can apply Temkin’s strengthening of the
alteration theorems of de Jong and Gabber (see [53, Thm. 1.2.5]) to find a
projective and generically finite morphism $f\colon X^{\prime}\to X$ of degree
$p^{n}$ such that $X^{\prime}$ is a smooth projective scheme over $k$ which
admits a strict semi-stable reduction over $k$. Since the composition ${\rm
CH}^{d+1}(X,1)\xrightarrow{f^{*}}{\rm
CH}^{d+1}(X^{\prime},1)\xrightarrow{f_{*}}{\rm CH}^{d+1}(X,1)$ is
multiplication by $p^{n}$, it follows that ${{\rm
CH}^{d+1}(X,1)}/m\hookrightarrow{{\rm CH}^{d+1}(X,1)}/m$. We can therefore
assume that $X$ admits a strict semi-stable reduction over $k$. In this case,
the finiteness of ${{\rm CH}^{d+1}(X,1)}/m$ follows from [13, Thm. 3.1.4].

We now assume $i\geq 2$. Let ${\mathcal{H}}^{p}_{{\acute{e}t}}(q)$ be the
Zariski sheaf on $X$ associated to the presheaf $U\mapsto
H^{p}_{{\acute{e}t}}(U,\Lambda^{\prime}(q))$. Let $X_{(i)}$ be the set of
points $x\in X$ such that $\text{\rm dim}(\overline{\\{x\\}})=i$. We consider
the commutative diagram with exact rows:

(6.3) $\textstyle{{\underset{y\in X_{(1)}}{\bigoplus}}\
{K^{M}_{i+1}(k(y))}/m\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{{\underset{x\in
X_{(0)}}{\bigoplus}}\
{K^{M}_{i}(k(x))}/m\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\rm
CH}^{d+i}(X,i)}/m\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{{\underset{y\in
X_{(1)}}{\bigoplus}}\
H^{i+1}_{{\acute{e}t}}(k(y),\Lambda^{\prime}(i+1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{{\underset{x\in
X_{(0)}}{\bigoplus}}\
H^{i}_{{\acute{e}t}}(k(x),\Lambda^{\prime}(i))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\rm
zar}(X,{\mathcal{H}}^{d+i}_{{\acute{e}t}}(d+i))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the bottom row is part of the Bloch-Ogus sequence for the étale
cohomology sheaves and the first two vertical arrows are the Norm-residue
homomorphisms. The two vertical arrows from the left are isomorphisms by the
solution of the Bloch-Kato conjecture due to Rost and Voevodsky. It follows
that the right vertical arrow is also an isomorphism. It suffices therefore to
show that $H^{d}_{\rm zar}(X,{\mathcal{H}}^{d+i}_{{\acute{e}t}}(d+i))$ is
finite.

We now consider the Leray spectral sequence

$E^{p,q}_{2}=H^{p}_{\rm
zar}(X,{\mathcal{H}}^{q}_{{\acute{e}t}}(d+i))\Rightarrow
H^{p+q}_{{\acute{e}t}}(X,\Lambda^{\prime}(d+i)).$

Since the étale cohomological dimension of any affine in $X$ is bounded by
$d+2$, one easily checks that $E^{d,d+i}_{2}=E^{d,d+i}_{\infty}$ and there is
an isomorphism
$E^{d,d+i}_{\infty}\xrightarrow{\cong}H^{2d+i}_{{\acute{e}t}}(X,\Lambda^{\prime}(d+i))$.
But the latter group is finite by the proper base change theorem. This
concludes the proof. ∎

######  Proposition 6.2.

Let $k$ be as in Lemma 6.1 and let $X$ be any projective $k$-scheme of
dimension $d\geq 0$ over $k$. Then $H^{2d+i}(X,\Lambda^{\prime}(d+i))$ is
finite for every $i\geq 0$.

###### Proof.

Let $q$ denote the exponential characteristic of $k$. Since the localization
map $H^{2d+i}(X,\Lambda^{\prime}(d+i))\to
H^{2d+i}(X,\Lambda^{\prime}(d+i))[\tfrac{1}{q}]$ is injective, it suffices to
show that the latter group is finite. We can assume that $X$ is reduced. If
$X$ is smooth, we are done by Lemma 6.1.

If $X$ is not smooth, we argue as follows. By Temkin’s strengthening of the
alteration theorems of de Jong and Gabber (see [53, Thm. 1.2.5]), there exists
$W\in{\mathbf{Sm}}_{k}$ and a surjective proper map $h:W\rightarrow X$, which
is generically finite and étale of degree $q^{n}$, $n\geq 1$. Then by a
theorem of Raynaud-Gruson [22, Thm. 5.2.2], there exists a blow-up
$g:X^{\prime}\rightarrow X$ with center $Z$ such that the following diagram
commutes, where $h^{\prime}$ is finite, flat and surjective of degree $q^{n}$
and $g^{\prime}:W^{\prime}\rightarrow W$ is the blow-up of W with center
$h^{-1}(Z)$:

(6.9) $\displaystyle\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
10.68991pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.38196pt\hbox{$\scriptstyle{g^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.77777pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.79068pt\raise 5.98611pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.98611pt\hbox{$\scriptstyle{h^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.52779pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.52779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
41.83249pt\raise-15.38889pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
41.83249pt\raise-27.77777pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
15.78511pt\raise-43.04163pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{h}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
33.18666pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.18666pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X.}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{array}$

Thus we have a $cdh$-cover $\\{X^{\prime}\amalg Z\rightarrow X\\}$ of $X$,
such that $\text{\rm dim}_{k}(Z)<\text{\rm dim}_{k}(X)$ and $\text{\rm
dim}_{k}(E)<\text{\rm dim}_{k}(X)$, where we set $E=X^{\prime}\times_{X}Z$. By
$cdh$-excision of [30, Prop. 5.3.4] (see also Thm. 2.3.14 of the same
reference), the following sequence is exact:

$\displaystyle H^{2d+i-1}(E,\Lambda^{\prime}(d+i))[\tfrac{1}{q}]\rightarrow
H^{2d+i}(X,\Lambda^{\prime}(d+i))[\tfrac{1}{q}]\rightarrow\hskip 142.26378pt$
$\displaystyle\hskip
170.71652ptH^{2d+i}(X^{\prime},\Lambda^{\prime}(d+i))[\tfrac{1}{q}]\oplus
H^{2d+i}(Z,\Lambda^{\prime}(d+i))[\tfrac{1}{q}].$

By induction on dimension, this reduces to showing that
$H^{2d+i}(X^{\prime},\Lambda^{\prime}(d+i))[\tfrac{1}{q}]$ is finite.

If we let $Z^{\prime}=h^{-1}(Z)$ and
$E^{\prime}=W^{\prime}\times_{W}Z^{\prime}$, then the same reasoning as above
shows that there is an exact sequence

$\displaystyle H^{2d+i}(W,\Lambda^{\prime}(d+i))[\tfrac{1}{q}]\to
H^{2d+i}(W^{\prime},\Lambda^{\prime}(d+i))[\tfrac{1}{q}]\to\frac{H^{2d+i}(E^{\prime},\Lambda^{\prime}(d+i))}{H^{2d+i}(Z^{\prime},\Lambda^{\prime}(d+i))}[\tfrac{1}{q}].$

The left term is finite since $W$ is smooth and projective, and the right term
is finite by induction on the dimension. It follows that the middle term is
finite too. It suffices therefore to show that the map $h^{\prime\ast}\colon
H^{2d+i}(X^{\prime},\Lambda^{\prime}(d+i))[\tfrac{1}{q}]\rightarrow
H^{2d+i}(W^{\prime},\Lambda^{\prime}(d+i))[\tfrac{1}{q}]$ is injective. But
this follows from [9, Prop. 6.3] because $h^{\prime}$ is finite and flat (in
particular, surjective) of degree $q^{n}$. This completes the proof. ∎

###  6.2. Proofs of parts (1) and (2) of Theorem 1.5

The following result generalizes the finiteness theorems of Kato-Saito [28]
and Saito-Sato [48] to non-complete schemes. The question whether such a
finiteness result would hold originated out of a discussion between Colliot-
Thélène and the authors. In the two results below, $k$ is either a finite, or
a $p$-adic or an algebraically closed field.

######  Theorem 6.3.

Let $U$ be a smooth quasi-projective $k$-scheme of pure dimension $d\geq 0$.
Then ${H^{sus}_{0}(U)}/m$ is finite.

###### Proof.

We can assume that $U$ is integral. Let $p$ be the exponential characteristic
of $k$. It suffices to show that $({H^{sus}_{0}(U)}/m)[\tfrac{1}{p}]$ is
finite. We have seen in § 2.3 that $({H^{sus}_{0}(U)}/m)[\tfrac{1}{p}]$ agrees
with $({H^{2d}_{c}(U,{\mathbb{Z}}(d))}/m)[\tfrac{1}{p}]$. The theorem is thus
equivalent to the statement that
$({H^{2d}_{c}(U,{\mathbb{Z}}(d))}/m)[\tfrac{1}{p}]$ is finite. By the
universal coefficient theorem, we have an exact sequence

(6.10) $0\to({H^{2d}_{c}(U,{\mathbb{Z}}(d))}/m)[\tfrac{1}{p}]\to
H^{2d}_{c}(U,\Lambda^{\prime}(d))[\tfrac{1}{p}]\to~{}_{m}H^{2d+1}_{c}(U,{\mathbb{Z}}(d)))[\tfrac{1}{p}].$

We choose an open embedding $U\subset X$ such that $X$ is an integral
projective scheme of dimension $d$ over $k$. We let $D=X\setminus U$ with
reduced closed subscheme structure. We then have an exact sequence

$H^{2d}(D,{\mathbb{Z}}(d)))[\tfrac{1}{p}]\to
H^{2d+1}_{c}(U,{\mathbb{Z}}(d)))[\tfrac{1}{p}]\to
H^{2d+1}(X,{\mathbb{Z}}(d)))[\tfrac{1}{p}].$

The two end terms are zero by [37, Thm. 5.1]. It follows that so is the middle
term. Using (6.10), we are reduced to showing that
$H^{2d}_{c}(U,\Lambda^{\prime}(d))[\tfrac{1}{p}]$ is finite. But this follows
from the exact sequence

$H^{2d-1}(D,\Lambda^{\prime}(d))[\tfrac{1}{p}]\to
H^{2d}_{c}(U,\Lambda^{\prime}(d))[\tfrac{1}{p}]\to
H^{2d}(X,\Lambda^{\prime}(d))[\tfrac{1}{p}]$

and Proposition 6.2. ∎

The second main result of this section is the following generalization of the
finiteness theorems of Kato-Saito [28] and Saito-Sato [48] to Chow groups with
modulus.

######  Theorem 6.4.

Let $X$ be a smooth projective $k$-scheme and $D\subset X$ an effective
Cartier divisor. Then ${{\rm CH}_{0}(X|D)}/m$ is finite.

###### Proof.

If $k$ is finite, the theorem is already known by [24, Thms. 1.3, 4.8]. Next,
suppose that $k$ is a $p$-adic field. We can then find a proper birational
morphism $f\colon X^{\prime}\to X$ from a smooth projective scheme
$X^{\prime}$ such that $f^{-1}(D)_{\rm red}$ is a strict normal crossing
divisor on $X^{\prime}$ and $f$ is an isomorphism over the complement of $D$.
It is clear from the definition of the Chow group of 0-cycles with modulus
that the pull-back map $f^{*}\colon{\rm CH}_{0}(X|D)\to{\rm
CH}_{0}(X^{\prime}|D^{\prime})$ exists and is an isomorphism, where
$D^{\prime}=f^{*}(D)$. We can therefore assume that $D_{\rm red}$ is a simple
normal crossing divisor. We are now done by Theorems 1.2 and 6.3.

We now assume that $k$ is algebraically closed. Using (2.7), we only need to
show that ${\rm CH}^{BK}_{0}(S_{X})_{\deg 0}$ is $m$-divisible. Since every
0-cycle of degree zero on $S_{X}$ lies on a complete intersection reduced
curve $C\subset S_{X}$, it suffices to show that ${\rm Pic}^{0}(C)$ is
$m$-divisible. But this is classical. ∎

###  6.3. Proof of Theorem 1.5(3)

We shall now complete the proof of Theorem 1.5 by proving its part (3). If $k$
is algebraically closed, the last part of the proof of Theorem 6.4 shows that
${\rm CH}^{LW}_{0}(X)_{\deg 0}$ is $m$-divisible. If $k$ is finite, part (3)
of Theorem 1.5 follows from [20, Thm. 1.3]. The remaining case follows from
the following general result of independent interest.

Let $k$ be any field and let $m\neq 0$ be an integer. Let $X$ be an integral
projective $k$-scheme which is regular is codimension one. It is shown in [20,
Lem. 8.3] that the identity map of ${\mathcal{Z}}_{0}(U)$ defines a surjection
$\theta_{X}\colon{\rm CH}^{LW}_{0}(X)\twoheadrightarrow H^{sus}_{0}(U)$ if we
let $U=X_{\rm reg}$.

######  Theorem 6.5.

Assume that ${\rm char}(k)=0$. Then the map

${\theta_{X}}/m\colon{{\rm CH}^{LW}_{0}(X)}/m\to{H^{sus}_{0}(U)}/m$

is an isomorphism.

###### Proof.

There is nothing to prove if $d\leq 1$ and we can thus assume that $d\geq 2$.
We shall prove the statement by induction on $d$. We first consider the
starting case $d=2$.

Let $\pi\colon\widetilde{X}\to X$ be a resolution of singularities of $X$ such
that the reduced exceptional divisor $E\subset\widetilde{X}$ has strict normal
crossings. For any integer $n\geq 1$, let $nE\hookrightarrow\widetilde{X}$
denote the infinitesimal thickening of $E$ in $\widetilde{X}$ of order $n$.

It is clear from the definitions of ${\rm CH}^{LW}_{0}(X)$, ${\rm
CH}_{0}(\widetilde{X}|D)$ and $H^{sus}_{0}(U)$ that the identity map of
${\mathcal{Z}}_{0}(U)$ defines, by the pull-back via $\pi^{*}$, the canonical
surjective maps

(6.11) ${\rm
CH}^{LW}_{0}(X)\stackrel{{\scriptstyle\pi^{*}}}{{\twoheadrightarrow}}{\rm
CH}_{0}(\widetilde{X}|nE)\twoheadrightarrow{\rm
CH}_{0}(\widetilde{X}|E)\twoheadrightarrow H^{sus}_{0}(U)$

for every integer $n\geq 1$ such that the composite map is $\theta_{X}$. The
first arrow from the left is an isomorphism for all $n\gg 1$ by [20, Thm.
1.5], and the third arrow is an isomorphism modulo $m$ by Theorem 1.2. We thus
have to show that the kernel of the surjection ${\rm
CH}_{0}(\widetilde{X}|nE)\twoheadrightarrow{\rm CH}_{0}(\widetilde{X}|E)$ is
divisible. By [6, Thm. 8.2], this is equivalent to showing that $H^{1}_{\rm
Zar}(E,{\mathcal{K}}_{2,(nE,E)})$ is divisible. But this is an immediate
consequence of the well known Goodwillie’s theorem for the relative $K$-theory
of nilpotent ideals, which implies that ${\mathcal{K}}_{2,(mE,E)}$ is a sheaf
of ${\mathbb{Q}}$-vector spaces on $E$.

We now assume $d\geq 3$. By [20, Cor. 2.5], we can assume that $X$ is normal.
We let $J^{d}(X)$ denote the Albanese variety of $X$ (which coincides with the
classical Albanese variety of any desingularization of $X$) and consider the
commutative diagram (see Theorem 3.2)

(6.12) $\textstyle{{\rm CH}^{LW}_{0}(X)_{\deg
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{X}}$$\scriptstyle{\rho_{X}}$$\textstyle{H^{sus}_{0}(U)_{\deg
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{U}}$$\textstyle{J^{d}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\prime}_{X}}$$\textstyle{J^{d}(U).}$

The vertical arrows are isomorphisms on the torsion subgroups by [40, Thm. 1]
and [52, Thm. 1.1]. Since a surjective morphism of semi-abelian varieties
induces surjection on $m$-torsion subgroups for every $m\geq 1$, it follows
that the same happens for $\theta_{X}$. This implies that ${\rm
Ker}(\theta_{X})\to{\rm Ker}({\theta_{X}}/m)$ is surjective. Hence, it
suffices to show that the map ${\rm Ker}(\theta_{X})\to{{\rm
CH}^{LW}_{0}(X)}/m$ is zero.

Let $\nu\colon C\to X$ be a finite morphism from a regular integral projective
curve whose image is not contained in $X_{\rm sing}$ and let $f\in{\rm
Ker}({\mathcal{O}}^{\times}_{C,E}\twoheadrightarrow{\mathcal{O}}^{\times}_{E})$,
where $E=\nu^{-1}(X_{\rm sing})$ with the reduced closed subscheme structure.
It suffices to show that $\nu_{*}({\rm div}(f))$ dies in ${{\rm
CH}^{LW}_{0}(X)}/m$. Equivalently, there is a 0-cycle
$\alpha\in{\mathcal{Z}}_{0}(U)$ such that $\nu_{*}({\rm
div}(f))-m\alpha\in{\mathcal{R}}^{LW}_{0}(X)$.

We can get a factorization
$C\stackrel{{\scriptstyle\nu^{\prime}}}{{\hookrightarrow}}{\mathbb{P}}^{r}_{X}\xrightarrow{\pi}X$
of $\nu$, where $\nu^{\prime}$ is a closed immersion and $\pi$ is the
canonical projection. Since the singular locus of ${\mathbb{P}}^{r}_{X}$
coincides with ${\mathbb{P}}^{r}_{X_{\rm sing}}$, it is clear that
$\nu^{\prime}_{*}({\rm div}(f))\in{\mathcal{R}}^{S}_{0}({\mathbb{P}}^{r}_{U})$
and $\nu_{*}({\rm div}(f))=\pi_{*}(\nu^{\prime}_{*}({\rm div}(f)))$. Using
[20, Cor. 2.6], it suffices therefore to show that $\nu^{\prime}_{*}({\rm
div}(f))-m\alpha^{\prime}\in{\mathcal{R}}^{LW}_{0}({\mathbb{P}}^{r}_{X})$ for
some $\alpha^{\prime}\in{\mathcal{Z}}_{0}({\mathbb{P}}^{r}_{U})$. We can thus
assume that $\nu\colon C\to X$ is a closed immersion.

We now fix a closed embedding $X\hookrightarrow{\mathbb{P}}^{N}_{k}$. Since
$U$ and $C$ are smooth and $d\geq 3$, we can use [2, Thm. 7] or [19, Thm. 3.9,
Cor. 3.13] to find a hypersurface $H\subset{\mathbb{P}}^{N}_{k}$ containing
$C$ and not containing $X$ such that the scheme theoretic intersection $X\cap
H$ has the property that it contains no irreducible component of $X_{\rm
sing}$ and $H\cap U$ is smooth and integral.

We let $Y=H\cap U$ and $W=H\cap U$. Since $Y$ is a Cartier divisor on $X$, we
must have $Y_{\rm sing}=X_{\rm sing}\cap Y$. Furthermore, $\text{\rm
dim}(Y_{\rm sing})=\text{\rm dim}(Y\cap X_{\rm sing})\leq\text{\rm dim}(X_{\rm
sing})-1\leq d-3=\text{\rm dim}(Y)-2$. In particular, $Y$ is regular in
codimension one. Let
$C\stackrel{{\scriptstyle\nu^{\prime}}}{{\hookrightarrow}}Y\stackrel{{\scriptstyle\iota}}{{\hookrightarrow}}X$
be the factorization of $\nu$. It follows from the choice of $Y$ that
$\nu^{\prime}_{*}({\rm div}(f))\in{\mathcal{R}}^{S}_{0}(W)$. It follows by
induction on $d$ that $\nu^{\prime}_{*}({\rm div}(f))$ dies in ${{\rm
CH}^{LW}_{0}(Y)}/m$. The push-forward map
$\iota_{*}\colon{\mathcal{Z}}_{0}(W)\to{\mathcal{Z}}_{0}(U)$ and [20, Cor.
2.6] together imply that $\nu_{*}({\rm div}(f))$ dies in ${{\rm
CH}^{LW}_{0}(X)}/m$. This concludes the proof. ∎

##  7\. Chow groups of normal crossing surfaces

We are now left with proving Theorem 1.6, the last of the main results listed
in § 1. In order to do this, we need to use an induction procedure on the
dimension of normal crossing varieties. In this section, we shall prove some
results for surfaces which will allow this induction.

###  7.1. Chow group with modulus to Levine-Weibel Chow group

The goal of this subsection is to prove the $d=2$ case of Theorem 8.3. Let $k$
be a field and $X$ a normal crossing variety of dimension $d\geq 1$ over $k$
(see Definition 2.12). We shall use the following notations in this section.

Let $\iota\colon Y\hookrightarrow X$ be the inclusion of an irreducible
component of $X$ and let $E\subset Y$ be the scheme theoretic intersection
$\overline{X\setminus Y}\cap Y$. Then $Y$ is a regular and $E$ is a simple
normal crossing divisor on $Y$. Let $D\subset X$ denote the singular locus of
$X$. Let $\iota^{\prime}\colon Y^{\prime}=\overline{X\setminus
Y}\hookrightarrow X$ be the inclusion of the union of other components of $X$.
Let ${\mathcal{I}}\subset{\mathcal{O}}_{X}$ be the sheaf of ideals on $X$
defining $Y^{\prime}$ and let ${\mathcal{J}}\subset{\mathcal{O}}_{Y}$ be the
sheaf of ideals on $Y$ defining $E$.

We have the following canonical inclusion of the groups of 0-cycles:

(7.1)
$\iota_{*}\colon{\mathcal{Z}}_{0}(Y,E)\hookrightarrow{\mathcal{Z}}_{0}(X,D).$

We wish to show that this map preserves the subgroups of rational equivalences
when $d\leq 2$. This will be generalized to higher dimensions in the next
section. We first deal with the case $d=1$.

######  Lemma 7.1.

$\iota_{*}$ descends to a homomorphism $\iota_{*}\colon{\rm
CH}_{0}(Y|E)\to{\rm CH}^{LW}_{0}(X)$ when $d=1$.

###### Proof.

Let $f\in{\rm
Ker}({\mathcal{O}}^{\times}_{Y,E}\to{\mathcal{O}}^{\times}_{E})$. The scheme
$X$ being a normal crossing variety means that the sequence

(7.2)
$0\to{\mathcal{O}}_{X,E}\to{\mathcal{O}}_{Y,E}\times{\mathcal{O}}_{Y^{\prime},E}\to{\mathcal{O}}_{E}\to
0$

is exact. If we let
$f^{\prime}=(f,1)\in{\mathcal{O}}^{\times}_{Y,E}\times{\mathcal{O}}^{\times}_{Y^{\prime},E}$,
then the modulus condition for $f$ implies that
$f^{\prime}\in{\mathcal{O}}^{\times}_{X,E}$. Moreover, we have ${\rm
div}(f^{\prime})={\rm div}(f)$ in ${\mathcal{Z}}_{0}(X,D)$. ∎

In the rest of § 7.1, we shall assume $d=2$. For any closed immersion of
schemes $W_{2}\subset W_{1}$ of $k$-schemes, we let
${\mathcal{K}}_{i,(W_{1},W_{2})}$ denote the Zariski sheaf on $W_{1}$
associated to the presheaf $U\mapsto K_{i}(U,U\cap W_{2})$ of relative Quillen
$K$-theory.

######  Lemma 7.2.

The canonical map $\iota^{*}\colon{\mathcal{I}}\to\iota_{*}({\mathcal{J}})$ is
an isomorphism.

###### Proof.

Let $A$ be a reduced affine $k$-algebra of dimension $d\geq 1$ such that ${\rm
Spec\,}(A)$ is a normal crossing variety. Let
$\\{{\mathfrak{p}}_{1},\ldots,{\mathfrak{p}}_{r}\\}$ be the set of minimal
primes of $A$. We write $I=\stackrel{{\scriptstyle
r}}{{\underset{i=2}{\bigcap}}}\ {\mathfrak{p}}_{i}$ and
$B={A}/{{\mathfrak{p}}_{1}}$. Our problem reduces to showing that the
homomorphism $I\to IB$ is bijective. This homomorphism is clearly surjective.
It is also injective because the kernel of the composite map $I\to IB\to B$ is
${\mathfrak{p}}_{1}\cap I=\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\bigcap}}}{\mathfrak{p}}_{i}=(0)$. ∎

######  Lemma 7.3.

The restriction map $\iota^{*}\colon
H^{2}(X,{\mathcal{K}}_{2,(X,Y^{\prime})})\to H^{2}(Y,{\mathcal{K}}_{2,(Y,E)})$
is an isomorphism.

###### Proof.

We have a commutative diagram of Zariski sheaves on $X$ with exact rows:

(7.3)
$\textstyle{{\mathcal{K}}_{2,(X,Y^{\prime})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}}$$\textstyle{{\mathcal{K}}_{2,X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}}$$\textstyle{\iota^{\prime}_{*}({\mathcal{K}}_{2,Y^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}}$$\textstyle{0}$$\textstyle{{\mathcal{K}}_{2,(Y,E)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{K}}_{2,Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\iota^{\prime}_{*}({\mathcal{K}}_{2,E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The canonical map $\iota^{*}\colon{\mathcal{I}}\to\iota_{*}({\mathcal{J}})$ is
an isomorphism by Lemma 7.2. It follows from [33] (see also [18, Thm. 0.2])
that the map
${\mathcal{K}}_{2,(X,Y^{\prime})}\to\iota_{*}({\mathcal{K}}_{2,(Y,E)})$ is
surjective. It is also clear that the kernel of this map is supported on $D$.
In particular, the map $H^{2}(X,{\mathcal{K}}_{2,(X,Y^{\prime})})\to
H^{2}(X,\iota_{*}({\mathcal{K}}_{2,(Y,E)}))\simeq
H^{2}(Y,{\mathcal{K}}_{2,(Y,E)})$ is an isomorphism. ∎

For any reduced quasi-projective surface $Z$ over $k$, let

$cyc_{Z}\colon{\rm CH}^{LW}_{0}(Z)\to F^{2}K_{0}(Z)\subset K_{0}(Z)$

be the $K$-theoretic cycle class map (e.g., see [4, Lem. 3.12]). As we have
already observed, it factors through $H^{2}_{\rm Zar}(Z,{\mathcal{K}}_{2,Z})$,
and we get canonical maps

(7.4) ${\rm CH}^{LW}_{0}(Z)\to{\rm CH}_{0}^{BK}(Z)\to H^{2}_{\rm
Zar}(Z,{\mathcal{K}}_{2,Z})\to H^{2}_{\rm Nis}(Z,{\mathcal{K}}_{2,Z}).$

The second arrow is an isomorphism by [6, Thm. 8.1] (or by [39] when $k$ is
algebraically closed), the third arrow is an isomorphism by [29, Thm. 2.5]
(the cited result assumes $X_{\rm reg}$ to be smooth but [23, § 3.5] explains
that this is unnecessary). The first arrow is an isomorphism when $k$ is
infinite by [6, Cor. 7.8].

######  Lemma 7.4.

The map $\iota_{*}$ of (7.1) descends to a homomorphism

$\iota_{*}\colon{\rm CH}_{0}(Y|E)\to{\rm CH}^{BK}_{0}(X).$

###### Proof.

Since $d=2$ and
$H^{i}(X,\iota^{\prime}_{*}({\mathcal{K}}_{2,Y^{\prime}}))\xrightarrow{\simeq}H^{i}(Y^{\prime},{\mathcal{K}}_{2,Y^{\prime}})$
for $i\geq 0$, we get from (7.3), a commutative diagram of Zariski cohomology
groups:

(7.5)
$\textstyle{H^{2}(X,{\mathcal{K}}_{2,(X,Y^{\prime})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\iota^{*}}$$\textstyle{H^{2}(X,{\mathcal{K}}_{2,X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}}$$\textstyle{H^{2}(Y^{\prime},{\mathcal{K}}_{2,Y^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}}$$\textstyle{0}$$\textstyle{H^{2}(Y,{\mathcal{K}}_{2,(Y,E)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(Y,{\mathcal{K}}_{2,Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(E,{\mathcal{K}}_{2,E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the two rows are exact.

For a closed point $x\in Y\setminus E=X\setminus Y^{\prime}\subset X_{\rm
reg}$, we have a commutative diagram

(7.6)
$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{H^{2}_{\\{x\\}}(X,{\mathcal{K}}_{2,(X,Y^{\prime})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\textstyle{H^{2}(X,{\mathcal{K}}_{2,(X,Y^{\prime})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}_{\\{x\\}}(X,{\mathcal{K}}_{2,X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{H^{2}(X,{\mathcal{K}}_{2,X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}}$$\textstyle{H^{2}_{\\{x\\}}(Y,{\mathcal{K}}_{2,(Y,E)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{H^{2}(Y,{\mathcal{K}}_{2,(Y,E)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}_{\\{x\\}}(Y,{\mathcal{K}}_{2,Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}(Y,{\mathcal{K}}_{2,Y}).}$

Since the composite
${\mathbb{Z}}\xrightarrow{\simeq}H^{2}_{\\{x\\}}(X,{\mathcal{K}}_{2,X})\hookrightarrow
K_{0}(X)$ is the cycle class of $x\in X_{\rm reg}$, it follows from Lemma 7.3,
(7.5) and (7.6) that the inclusion $Y\setminus E=X\setminus Y^{\prime}\subset
X_{\rm reg}$ induces a commutative diagram of cycle class maps

(7.7)
$\textstyle{{\mathcal{Z}}_{0}(Y,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cyc_{Y|E}}$$\scriptstyle{\iota_{*}}$$\textstyle{H^{2}(Y,{\mathcal{K}}_{2,(Y,E)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{Y}}$$\textstyle{{\mathcal{Z}}_{0}(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
cyc}}_{X}}$$\textstyle{H^{2}_{\rm
Zar}(X,{\mathcal{K}}_{2,X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(X),}$

where $\phi_{Y}=(\iota^{*})^{-1}\colon H^{2}(Y,{\mathcal{K}}_{2,(Y,E)})\to
H^{2}(X,{\mathcal{K}}_{2,(X,Y^{\prime})})$.

It follows from [6, Thm. 3.6] that $cyc_{X}$ factors through the quotient
${\rm CH}^{BK}_{0}(X)$ and it follows from [34, Thm. 1.2, Lem. 2.1] that the
map $cyc_{Y|E}$ factors through the quotient ${\rm CH}_{0}(Y|E)$. We thus get
a commutative diagram with solid arrows

(7.8)
$\textstyle{{\mathcal{Z}}_{0}(Y,E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}}$$\textstyle{{\rm
CH}_{0}(Y|E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cyc_{Y|E}}$$\textstyle{H^{2}_{\rm
Zar}(Y,{\mathcal{K}}_{2,(Y,E)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{Y}}$$\textstyle{{\mathcal{Z}}_{0}(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}^{BK}_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cyc_{X}}$$\textstyle{H^{2}_{\rm
Zar}(X,{\mathcal{K}}_{2,X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(X).}$

Now, the map ${\operatorname{\rm cyc}}_{X}\colon{\rm CH}^{BK}_{0}(X)\to
H^{2}_{\rm Zar}(X,{\mathcal{K}}_{2,X})$ is an isomorphism as shown above, and
the map ${\rm CH}_{0}(Y|E)\to H^{2}_{\rm Zar}(Y,{\mathcal{K}}_{2,(Y,E)})$ is
an isomorphism by [6, Thm. 8.2]. It follows that the composite map
$\iota_{*}\colon{\mathcal{Z}}_{0}(Y,E)\hookrightarrow{\mathcal{Z}}_{0}(X,D)\twoheadrightarrow{\rm
CH}^{BK}_{0}(X)$ factors through ${\rm CH}_{0}(Y|E)$. This proves the lemma. ∎

###  7.2. 0-cycles on normal crossing surfaces

In this subsection, we shall prove the $d=2$ case of Theorem 8.4. We continue
with the setup of § 7.1. Let $\\{X_{1},\ldots,X_{r}\\}$ be the set of
irreducible components of $X$. Let $C\subset X$ be a reduced curve. The
following definition is borrowed from [12].

######  Definition 7.5.

We shall say that $C$ is a snc subcurve in $X$ if its scheme theoretic
intersection with each irreducible component of $X$ is either empty or smooth
and integral of dimension one, its intersection with each $X_{i}\cap X_{j}$
($i\neq j$) is either empty or reduced and 0-dimensional, and its intersection
with each $X_{i}\cap X_{j}\cap X_{\ell}$ ($i\neq j\neq\ell\neq i$) is empty.

Let $X$ be as above and let $C\subset X$ be a snc subcurve. The following is
easy to verify.

######  Lemma 7.6.

The following hold for $C$.

  1. (1)

The inclusion $C\hookrightarrow X$ is a regular embedding.

  2. (2)

The embedding dimension of $C$ is at most two.

  3. (3)

$C$ is regular and integral if and only if it is contained in one component
$X_{i}$ of $X$ and $C\cap X_{j}=\emptyset$ for every component $X_{j}\neq
X_{i}$.

Let $X$ be a normal crossing variety of dimension $d\geq 2$ over $k$. Let