of $(\pi^{{\rm adiv}}_{1}(X,D)_{0})^{\vee}$ which vanishes on $C(X,D)_{0}$ is
zero. We fix $\chi\in(\pi^{{\rm adiv}}_{1}(X,D)_{0})^{\vee}$ which vanishes on
$C(X,D)_{0}$.

By [16, Corollary 8.5], there is a commutative diagram

(5.19)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widehat{{\mathbb{Z}}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{(\pi^{{\rm
adiv}}_{1}(X,D))^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\vee}_{X|D}}$$\textstyle{(\pi^{{\rm
adiv}}_{1}(X,D)_{0})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\vee}_{X|D}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(X,D)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(C(X,D)_{0})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the left vertical arrow is an isomorphism.

This diagram shows that $\chi$ lifts to an element $\chi^{\prime}\in(\pi^{{\rm
adiv}}_{1}(X,D))^{\vee}$. Since $\rho^{\vee}_{X|D}(\chi)=0$, a diagram chase
shows that there exists an element
$\chi^{\prime\prime}\in(\widehat{{\mathbb{Z}}})^{\vee}$ such that
$\rho^{\vee}_{X|D}(\chi^{\prime}-\chi^{\prime\prime})=0$. On the other hand,
since the image of the map $\rho_{X|D}$ is dense, it follows from [16, Lemma
7.11] that the middle vertical arrow in (5.19) is injective. It follows that
$\chi^{\prime}=\chi^{\prime\prime}$. Equivalently, $\chi=0$. The proof of
Theorem 1.1 is now complete. ∎

######  Corollary 5.15.

Let $X$ and $D$ be as in Theorem 1.1 with $X$ projective. Then for every
$m\in{\mathbb{N}}$, the reciprocity map $\rho_{X|D}$ induces an isomorphism of
finite groups

$\rho_{X|D}\colon{C(X,D)}/m\xrightarrow{\cong}{\pi^{{\rm adiv}}_{1}(X,D)}/m.$

###### Proof.

Use [16, Lemma 8.4] and Theorem 1.1. ∎

##  6\. Reciprocity theorem for $\pi^{\rm ab}_{1}(X,D)$

In this section, we shall prove a reciprocity theorem for $\pi^{\rm
ab}_{1}(X,D)$ in the pro-setting as $D$ varies over a set of closed subschemes
of pure codimension one $X$ all of which have the same support. The main
application for us will be in the proof of Theorem 1.4.

Let $k$ be a finite field and $X$ a projective integral normal scheme of
dimension $d\geq 1$ over $k$. We let $C\subset X$ be a reduced closed
subscheme of pure codimension one with complement $U$. We assume that $U$ is
regular. We shall need the following results.

######  Lemma 6.1.

Let $f\colon X^{\prime}\to X$ be a morphism of integral normal schemes whose
image is not contained in $C$. Let $D^{\prime}\subset X^{\prime}$ be an
effective Weil divisor. Assume that $f^{*}(D)$ is an effective Weil divisor on
$X^{\prime}$ such that $f^{*}(D)\leq D^{\prime}$. Then there is a push-forward
map

$f_{*}\colon\pi^{\rm ab}_{1}(X^{\prime},D^{\prime})\to\pi^{\rm ab}_{1}(X,D).$

###### Proof.

Obvious from Definition 5.3. ∎

######  Lemma 6.2.

Assume that $k$ is finite and $X$ is projective over $k$. Then the group
$\pi^{\rm ab}_{1}(X,D)_{0}$ is finite. In particular, the map $\pi^{\rm
ab}_{1}(X,D)_{0}\to{\pi^{\rm ab}_{1}(X,D)_{0}}/{\infty}$ is an isomorphism.

###### Proof.

This is shown in [26, Corollary 1.2] when $D\subset X$ is an effective Cartier
divisor. But one does not need the latter assumption. The reason is that since
$D$ is anyway a closed subscheme which defines a Weil divisor, its inverse
image $f^{*}(D)$ becomes an effective Cartier divisor on $X^{\prime}$ where
$f\colon X^{\prime}\to X$ is any smooth alteration. We can replace $X$ by any
such alteration using Lemma 6.1 because the map $f_{*}\colon\pi^{\rm
ab}(f^{-1}(U))\to\pi^{\rm ab}_{1}(U)$ has finite cokernel. ∎

The following is the key step for proving the main result of this section.

######  Lemma 6.3.

For every $D\in{\rm Div}_{C}(X)$, there exists $D^{\prime}\in{\rm Div}_{C}(X)$
such that the composite map
$\widetilde{C}_{U/X}\xrightarrow{\widetilde{\rho}_{U/X}}\pi^{\rm
ab}_{1}(U)\xrightarrow{q^{\prime}_{X|D}}\pi^{\rm ab}_{1}(X,D)$ factors through

$\rho^{c}_{X|D}\colon C(X,D^{\prime})\to\pi^{\rm ab}_{1}(X,D).$

###### Proof.

We fix a closed subscheme $D\in{\rm Div}_{C}(X)$. For any $D^{\prime}\in{\rm
Div}_{C}(X)$, let $F_{D^{\prime}}={\rm
Ker}(\widetilde{C}_{U/X}\twoheadrightarrow C(X,D^{\prime}))$. It follows from
[16, Proposition 4.8] that $F_{D^{\prime}}={\rm
Ker}((\widetilde{C}_{U/X})_{0}\twoheadrightarrow C(X,D^{\prime})_{0})$. Hence,
it suffices to show that the composite map
$(\widetilde{C}_{U/X})_{0}\xrightarrow{\widetilde{\rho}_{U/X}}\pi^{\rm
ab}_{1}(U)\twoheadrightarrow\pi^{\rm ab}_{1}(X,D)$ annihilates
$F_{D^{\prime}}$ for some $D^{\prime}$. Using [16, Lemma 8.4], we see that
$\widetilde{\rho}_{U/X}$ induces the maps
$(\widetilde{C}_{U/X})_{0}\xrightarrow{\widetilde{\rho}_{U/X}}\pi^{\rm
ab}_{1}(U)_{0}\twoheadrightarrow\pi^{\rm ab}_{1}(X,D)_{0}$. Hence, we need to
show that this composite map annihilates $F_{D^{\prime}}$ for some
$D^{\prime}$.

Now, we know from Theorem 4.8 that each $C(X,D^{\prime})_{0}$ is finite. It
follows from [16, Corollary 4.9] that
$(\widetilde{C}_{U/X})_{0}\xrightarrow{\cong}{\underset{D^{\prime}\in{\rm
Div}_{C}(X)}{\varprojlim}}C(X,D^{\prime})_{0}$. In particular,
$(\widetilde{C}_{U/X})_{0}$ is profinite. Since the maps
$(\widetilde{C}_{U/X})_{0}\xrightarrow{\widetilde{\rho}_{U/X}}\pi^{\rm
ab}_{1}(U)_{0}\twoheadrightarrow\pi^{\rm ab}_{1}(X,D)_{0}$ are continuous, it
follows that the images of $F_{D^{\prime}}$ under these maps are closed
subgroups. We let $E_{D^{\prime}}$ be the image of $F_{D^{\prime}}$ in
$\pi^{\rm ab}_{1}(X,D)_{0}$ under the composite map.

We now let $\chi\in(\pi^{\rm ab}_{1}(X,D)_{0})^{\vee}$ be a continuous
character. Then the composite $\chi^{\prime}:=\chi\circ
q^{\prime}_{X|D}\circ\widetilde{\rho}_{U/X}$ is a continuous character of
$(\widetilde{C}_{U/X})_{0}$. Since $(\widetilde{C}_{U/X})_{0}$ is profinite as
we just saw, $\chi^{\prime}$ factors through some $C(X,D_{\chi})_{0}$. In
other words, $\chi(E_{D_{\chi}})=0$. Since $\pi^{\rm ab}_{1}(X,D)_{0}$ is
finite by Lemma 6.2, $(\pi^{\rm ab}_{1}(X,D)_{0})^{\vee}$ is also finite (see
[38, Example 2.9.5]). We can therefore choose $D^{\prime}\in{\rm Div}_{C}(X)$
which dominates $D_{\chi}$ for all $\chi\in(\pi^{\rm
ab}_{1}(X,D)_{0})^{\vee}$. It is then clear that $\chi$ annihilates
$E_{D^{\prime}}$ for all $\chi\in(\pi^{\rm ab}_{1}(X,D)_{0})^{\vee}$. We have
thus shown that $E_{D^{\prime}}$ is a closed subgroup of $\pi^{\rm
ab}_{1}(X,D)_{0}$ which is annihilated by all characters of the latter group.
But then the Pontryagin duality theorem says that $E_{D^{\prime}}$ must be
zero. Equivalently, $F_{D^{\prime}}$ lies in the kernel of the composite map
$(\widetilde{C}_{U/X})_{0}\xrightarrow{\widetilde{\rho}_{U/X}}\pi^{\rm
ab}_{1}(U)_{0}\twoheadrightarrow\pi^{\rm ab}_{1}(X,D)_{0}$. This proves the
lemma. ∎

Let $D\in{\rm Div}_{C}(X)$ and $nD\subset X$ the closed subscheme defined by
${\mathcal{I}}^{n}_{D}$, where ${\mathcal{I}}_{D}$ is the sheaf of ideals
defining $D$. For every $n\in{\mathbb{N}}$, the set of closed subschemes
$n^{\prime}D\in{\rm Div}_{C}(X)$ which have the property described in Lemma
6.3 is linearly ordered by inclusion. Since $X$ is Noetherian, there exists
smallest integer $\lambda(n)\in{\mathbb{N}}$ such that
$\lambda(n)\geq\lambda(n-1)$ and $\lambda(n)D$ satisfies the property asserted
in Lemma 6.3. That is, the composite map
$\widetilde{C}_{U/X}\xrightarrow{\widetilde{\rho}_{U/X}}\pi^{\rm
ab}_{1}(U)\xrightarrow{q^{\prime}_{X|D}}\pi^{\rm ab}_{1}(X,nD)$ factors
through

$\rho^{c}_{X|nD}\colon C(X,\lambda(n)D)\to\pi^{\rm ab}_{1}(X,nD).$

We let $\lambda\colon{\mathbb{N}}\to{\mathbb{N}}$ be the above function. We
write $\lambda(1)D=D_{\rho}$. As a consequence of Lemma 6.3, we conclude the
following.

######  Theorem 6.4.

The reciprocity map ${\rho}_{U/X}\colon{C}_{U/X}\to\pi^{\rm ab}_{1}(U)$ of
Theorem 5.7 induces a continuous homomorphism between the topological pro-
abelian groups

$\rho^{\bullet}_{X|D}\colon\\{C(X,nD)\\}_{n\in{\mathbb{N}}}\to\\{\pi^{\rm
ab}_{1}(X,nD)\\}_{n\in{\mathbb{N}}}$

such that ${\underset{n\in{\mathbb{N}}}{\varprojlim}}\
\rho^{\bullet}_{X|D}=\widetilde{\rho}_{U/X}$.

From the construction of $\widetilde{\rho}_{U/X}$, we actually get the
following commutative diagram of short exact sequences of topological pro-
abelian groups. We ignore to write the indexing set ${\mathbb{N}}$.

(6.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{C(X,nD)_{0}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\bullet}_{X|D}}$$\textstyle{\\{C(X,nD)\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg}/{n_{0}}}$$\scriptstyle{\rho^{\bullet}_{X|D}}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{\pi^{\rm
ab}_{1}(X,nD)_{0}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{\pi^{\rm
ab}_{1}(X,nD)\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg^{\prime}}/{n_{0}}}$$\textstyle{\widehat{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the right vertical arrow is the pro-finite completion map and
$n_{0}\in{\mathbb{N}}$ depends only on $U$ and not on ${\rm Div}_{C}(X)$.
Moreover, taking the limits of the vertical arrows, we get the commutative
diagram of short exact sequences of topological abelian groups

(6.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widetilde{C}_{U/X})_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\rho}_{U/X}}$$\textstyle{\widetilde{C}_{U/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg}/{n_{0}}}$$\scriptstyle{\widetilde{\rho}_{U/X}}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{\rm
ab}_{1}(U)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{\rm
ab}_{1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg^{\prime}}/{n_{0}}}$$\textstyle{\widehat{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

by Proposition 5.4 and Corollary 5.5. Note that the projective limits over
${\rm Div}_{C}(X)$ and $\\{nD\\}_{n\in{\mathbb{N}}}$ coincide.

Our reciprocity theorem for the 1-skeleton étale fundamental group with
modulus is the following.

######  Theorem 6.5.

For each $D\in{\rm Div}_{C}(X)$, the reciprocity map of pro-abelian groups

$\rho^{\bullet}_{X|D}\colon\\{C(X,nD)\\}_{n\in{\mathbb{N}}}\to\\{\pi^{\rm
ab}_{1}(X,nD)\\}_{n\in{\mathbb{N}}}$

is injective and has dense image.

###### Proof.

The proof of the density assertion is same as in Theorem 1.1. We shall prove
pro-injectivity.

Using the commutative diagram (6.1), it suffices to show the injectivity at
the level of the degree zero subgroups. Before we do this, we claim that
$\rho^{c}_{X|nD}\colon C(X,\lambda(n)D)_{0}\to\pi^{\rm ab}_{1}(X,nD)_{0}$ is
surjective. Indeed, as the latter group is finite by Lemma 6.2, it suffices to
show that this map has dense image. But the proof of this is identical to that
of the surjectivity of the map (5.17) in Theorem 1.1, which only used that
$\rho_{X|D}$ has dense image and not its injectivity.

Let $n\in{\mathbb{N}}$ and write $n^{\prime}=\lambda(n)$. We let $F_{n}$
denote the kernel of $C(X,n^{\prime}D)_{0}\twoheadrightarrow\pi^{\rm
ab}_{1}(X,nD)_{0}$. Using the Pontryagin duality for finite groups, it
suffices to show that the ind-abelian group $\\{F^{\vee}_{n}\\}$ is zero.

Choose a character $\chi\in F^{\vee}_{n}$ and lift it to a character of
$C(X,n^{\prime}D)_{0}$. We denote this lift also by $\chi$. We choose
$m\in{\mathbb{N}}$ large enough such that
$C(X,n^{\prime}D)_{0}\cong{C(X,n^{\prime}D)_{0}}/m$ using Theorem 4.8. Using
[16, Proposition 4.8, Lemma 7.10], $\chi$ lifts to a character of
${C(X,n^{\prime}D)}/m$. Choose one such lift and denote its image in
$C(X,n^{\prime}D)^{\vee}$ also by $\chi$. We let
$\widetilde{\chi}=p^{\vee}_{X|n^{\prime}D}(\chi)$ and consider the commutative
diagram (see (5.18))

(6.3) $\textstyle{{\rm
fil}^{c}_{nD}H^{1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\rho^{c}_{X|nD})^{\vee}}$$\scriptstyle{{q^{\prime}}^{\vee}_{X|nD}}$$\textstyle{C(X,n^{\prime}D)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\vee}_{X|{n^{\prime}D}}}$$\textstyle{({C(X,n^{\prime}D)}/m)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\rho}^{\vee}_{U/X}}$$\scriptstyle{(\rho^{\infty}_{U/X})^{\vee}}$$\scriptstyle{\cong}$$\textstyle{(\widetilde{C}_{U/X})^{\vee}}$$\textstyle{({\widetilde{C}_{U/X}}/{\infty})^{\vee}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Since $\chi\in({C(X,n^{\prime}D)}/m)^{\vee}$, it follows from Corollary 5.10
that $\widetilde{\chi}\in({\widetilde{C}_{U/X}}/{\infty})^{\vee}$. We conclude
from Corollary 5.11 that there exists $\chi^{\prime}\in H^{1}(U)$ such that
$\widetilde{\rho}^{\vee}_{U/X}(\chi^{\prime})=\widetilde{\chi}$. It follows
from Proposition 5.4 that ${\chi}^{\prime}\in{\rm fil}^{c}_{n_{1}D}H^{1}(K)$
for some $n_{1}\gg n$.

We let $\chi_{\rho}$ be the image of ${\chi}^{\prime}$ under the map
$(\pi^{\rm ab}_{1}(X,n_{1}D))^{\vee}\to(\pi^{\rm
ab}_{1}(X,n_{1}D)_{0})^{\vee}$. Then it follows that the image of
$\chi_{\rho}$ in $(C(X,\lambda(n_{1})D)_{0})^{\vee}$ lies in the image of
$(\rho^{c}_{X|n_{1}D})^{\vee}$. Since $F^{\vee}_{n}={\rm
Coker}((\rho^{c}_{X|nD})^{\vee})$ by [16, Lemma 7.10], it follows that $\chi$
dies under the canonical map $F^{\vee}_{n}\to F^{\vee}_{n_{1}}$. Since
$F^{\vee}_{n}$ is finite, we can find $n_{1}\gg n$ such that the map
$F^{\vee}_{n}\to F^{\vee}_{n_{1}}$ is zero. We have thus shown that the ind-
abelian group $\\{F^{\vee}_{n}\\}$ is zero. This finishes the proof. ∎

######  Corollary 6.6.

The morphism of pro-abelian groups

$\rho^{\bullet}_{X|D}\colon\\{C(X,nD)_{0}\\}_{n\in{\mathbb{N}}}\to\\{\pi^{\rm
ab}_{1}(X,nD)_{0}\\}_{n\in{\mathbb{N}}}$

is an isomorphism.

###### Proof.

The map $\rho^{\bullet}_{X|D}$ is injective by Theorem 6.5. Moreover, we also
showed in the proof of this theorem that $\rho^{\bullet}_{X|D}$ is surjective.
∎

######  Corollary 6.7.

The morphism of pro-abelian groups

$\rho^{\bullet}_{X|D}\colon\\{{C(X,nD)}/m\\}_{n\in{\mathbb{N}}}\to\\{{\pi^{\rm
ab}_{1}(X,nD)}/m\\}_{n\in{\mathbb{N}}}$

is an isomorphism for every $m\in{\mathbb{N}}$.

###### Proof.

The proof is identical to that of Corollary 5.15 using Corollary 6.6. ∎

The following result will be improved in § 9.2 when $X$ is regular. We
nevertheless need this weaker version to prove Bloch’s formula for normal
varieties.

######  Theorem 6.8.

For every $D\in{\rm Div}_{C}(D)$, the identity map of $\pi^{\rm ab}_{1}(U)$
induces an isomorphism of topological pro-abelian groups

$\theta^{\bullet}_{X|D}\colon\\{\pi^{{\rm
adiv}}_{1}(X,nD)\\}_{n\in{\mathbb{N}}}\xrightarrow{\cong}\\{\pi^{\rm
ab}_{1}(X,nD)\\}_{n\in{\mathbb{N}}}$

making the diagram

$\textstyle{\widetilde{C}_{U/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\rho}_{U/X}}$$\textstyle{\\{C(X,nD)\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\\{\rho^{{\rm
div}}_{X|nD}\\}}$$\scriptstyle{\rho^{\bullet}_{X|D}}$$\textstyle{\pi^{\rm
ab}_{1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{\pi^{{\rm
adiv}}_{1}(X,nD)\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\bullet}_{X|D}}$$\textstyle{\\{\pi^{\rm
ab}_{1}(X,nD)\\}}$

commute.

###### Proof.

The commutativity of the diagram is clear from the construction of various
reciprocity maps. We fix $n$ and let $n^{\prime}=\lambda(n)$. Let $F$ denote
the kernel of the map $\pi^{\rm ab}_{1}(U)\to\pi^{{\rm
adiv}}_{1}(X,n^{\prime}D)$. Then $F$ is same as the kernel of the map
$\pi^{\rm ab}_{1}(U)_{0}\to\pi^{{\rm
adiv}}_{1}(X,n^{\prime}D)_{0}\cong{\pi^{{\rm adiv}}_{1}(X,n^{\prime}D)_{0}}/m$
for all $m\gg 0$ (see Theorem 1.1).

By Corollary 5.11, $\rho^{\infty}_{U/X}$ is an isomorphism and it induces an
isomorphism between $(\widetilde{C}_{U/X})_{0}$ and $\pi^{\rm ab}_{1}(U)_{0}$
by Corollary 5.10. Hence, we conclude from Theorem 1.1 that
$\rho^{\infty}_{X|D}$ induces an isomorphism ${\rm
Ker}((\widetilde{C}_{U/X})_{0}\to C(X,n^{\prime}D)_{0})\xrightarrow{\cong}F$.
However, the kernel on the left hand side dies in $\pi^{\rm
ab}_{1}(X,nD)_{0}$. It follows that $F$ dies in $\pi^{\rm ab}_{1}(X,nD)_{0}$.

To prove that $\theta^{\bullet}_{X|D}$ is an isomorphism, we only need to show
it is injective. To prove this, it is equivalent to show that the map
$\\{\pi^{{\rm adiv}}_{1}(X,nD)_{0}\\}\to\\{\pi^{\rm ab}_{1}(X,nD)_{0}\\}$ is
injective. But this follows from Theorem 1.1 and Corollary 6.6. ∎

We conclude this section by proving the following property of ${\rm
fil}^{c}_{D}H^{1}(K)$ which will be used in the proof of Theorem 1.2 in § 9.2.

######  Proposition 6.9.

Let $k$ be any field. Assume that $\text{\rm dim}(X)\geq 2$ and $A\subset X$
is a closed subscheme such that $A_{\rm red}\subset C$ and $\text{\rm
dim}(A)\leq\text{\rm dim}(X)-2$. Let $X^{\prime}=X\setminus A,\
C^{\prime}=C\setminus A$ and $D^{\prime}=D\setminus A$. Then there is an
inclusion ${\rm fil}^{c}_{D}H^{1}(K)\subset{\rm fil}^{c}_{D^{\prime}}H^{1}(K)$
of subgroups of $H^{1}(K)$.

###### Proof.

Suppose $\chi\in{\rm fil}^{c}_{D}H^{1}(K)$ and let $Y^{\prime}\subset
X^{\prime}$ be an integral curve not contained in $D^{\prime}$. Let $Y\subset
X$ be the scheme-theoretic closure of $Y^{\prime}$ in $X$. Then $Y$ is an
integral curve in $X$ not contained in $D$. Let $\nu\colon Y_{n}\to X$ be
induced map from the normalization of $Y$. We let $D_{Y_{n}}=D\times_{X}Y_{n}$
and $D^{\prime}_{Y^{\prime}_{n}}=D\times_{X}Y^{\prime}_{n}$. We then get a
commutative diagram

(6.4)
$\textstyle{D^{\prime}_{Y^{\prime}_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{\prime}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{\prime}}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{Y_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{X}$

in which the two squares are Cartesian, vertical arrows are open immersions
and horizontal arrows are finite.

We write $D_{Y_{n}}={\underset{x\in\Sigma}{\sum}}m_{x}[x]$, where $\Sigma$ is
the support of $\nu^{-1}(C)$. Then the above diagram says that
$D^{\prime}_{Y^{\prime}_{n}}={\underset{x\in\Sigma^{\prime}}{\sum}}m_{x}[x]$,
where $\Sigma^{\prime}=\nu^{-1}(C^{\prime})$. In particular,
$D^{\prime}_{Y^{\prime}_{n}}$ is an effective Weil divisor on $Y_{n}$ such
that $D^{\prime}_{Y^{\prime}_{n}}\leq D_{Y_{n}}$ and every
$x\in\Sigma^{\prime}$ (note that $D^{\prime}_{Y^{\prime}_{n}}$ will be zero if
$Y\cap C\subset A$) has the property that the multiplicity of
$D^{\prime}_{Y^{\prime}_{n}}$ at $x$ is same is that of $D_{Y_{n}}$. But this
implies that $\nu^{\prime*}(\chi)\in
H^{1}(\nu^{\prime-1}(U))=H^{1}(\nu^{-1}(U))$ has the property that its image
in $H^{1}(k(Y)_{x})$ lies in ${\rm fil}^{{\rm ms}}_{m_{x}}H^{1}(k(Y)_{x})$,
for all $x\in\Sigma^{\prime}$. This proves the proposition. ∎

##  7\. A moving lemma for 0-cycles with modulus

Our next goal is to prove a moving lemma for the Chow group of 0-cycles with
modulus which will be the key ingredient in the proof of Theorem 1.2. In this
section, we shall recall the Chow group of 0-cycles with modulus and prove
some intermediate results.

###  7.1. Chow group with modulus

We recall the definition of the Chow group of 0-cycles with modulus from [2]
and [27]. Let $k$ be any field and $X$ a reduced quasi-projective scheme over
$k$ of dimension $d\geq 1$. Let $D\subset X$ be an effective Cartier divisor
and $U$ its complement. Let $D^{\prime}=D_{\rm red}$. Let
${\mathcal{Z}}_{0}(U)$ denote the free abelian group on the set of closed
points in $U$. Suppose that $Y\subset X$ is an integral curve not contained in
$D^{\prime}$ and $\nu\colon Y_{n}\to X$ is the projection map from the
normalization of $Y$. We let $D^{\prime}_{Y}=\nu^{-1}(D^{\prime})$. Let
$I_{D_{Y}}$ denote the ideal of $\nu^{*}(D)$ in the semilocal ring
${\mathcal{O}}_{Y_{n},D^{\prime}_{Y}}$. We have the divisor map ${\rm
div}\colon
K^{M}_{1}({\mathcal{O}}_{Y_{n},D^{\prime}_{Y}},I_{D_{Y}})\to{\mathcal{Z}}_{0}(\nu^{-1}(U))$.
We let ${\mathcal{R}}_{0}(Y|D)$ denote the image of the composite map

(7.1)
$K^{M}_{1}({\mathcal{O}}_{Y_{n},D^{\prime}_{Y}},I_{D_{Y}})\xrightarrow{{\rm
div}}{\mathcal{Z}}_{0}(\nu^{-1}(U))\xrightarrow{\nu_{*}}{\mathcal{Z}}_{0}(U).$

We let ${\mathcal{R}}_{0}(X|D)$ be the image of the map
${\underset{Y}{\bigoplus}}\ {\mathcal{R}}_{0}(Y|D)\to{\mathcal{Z}}_{0}(U)$,
where $Y$ runs through the set of curves as above. An element
$\alpha\in{\mathcal{R}}_{0}(X|D)$ is said to be a 0-cycle rationally
equivalent to zero. The Chow group of 0-cycles with modulus is the quotient

(7.2) ${\rm CH}_{0}(X|D)=\frac{{\mathcal{Z}}_{0}(U)}{{\mathcal{R}}_{0}(X|D)}.$

The functor $(X,D)\mapsto{\rm CH}_{0}(X|D)$ has appropriate covariant
functorial property for proper maps and contravariant functorial property for
flat maps. It is also clear from the definition that for $D_{1}\leq D_{2}$,
there is a canonical map ${\rm CH}_{0}(X|D_{2})\to{\rm CH}_{0}(X|D_{1})$. In
particular, there is a canonical map ${\rm CH}_{0}(X|D)\to{\rm
CH}^{F}_{0}(X)$. If $X$ is projective over $k$, composing with the classical
degree map ${\rm CH}^{F}_{0}(X)\to{\mathbb{Z}}$, we get the degree map

(7.3) $\deg\colon{\rm CH}_{0}(X|D)\to{\mathbb{Z}},$

whose image coincides with the image of the composite map
${\mathcal{Z}}_{0}(U)\hookrightarrow{\mathcal{Z}}_{0}(X)\to{\mathbb{Z}}$.
Hence, there exists an integer $n\geq 1$ which depends only on $U$ such that
the sequence

(7.4) $0\to{\rm CH}_{0}(X|D)_{0}\to{\rm
CH}_{0}(X|D)\xrightarrow{{\deg}/n}{\mathbb{Z}}\to 0$

is exact. Furthermore, we have $n=1$ if either $k$ is finite and $X$ is
geometrically irreducible or $k$ is separably closed. We shall consider ${\rm
CH}_{0}(X|D)$ a topological group with discrete topology.

We let $C(U)={\underset{E}{\varprojlim}}\ {\rm CH}_{0}(X|E)$ and
$C(U)_{0}={\rm Ker}(C(U)\xrightarrow{\deg}{\mathbb{Z}})$, where the limit is
over all effective Cartier divisors $E$ with support $D^{\prime}$. Since we
have ${\rm CH}_{0}(X|D_{2})_{0}\twoheadrightarrow{\rm CH}_{0}(X|D_{1})_{0}$
for every $D_{1}\leq D_{2}$, it follows that

(7.5) $C(U)_{0}\xrightarrow{\cong}{\underset{E}{\varprojlim}}\ {\rm
CH}_{0}(X|E)_{0}.$

We shall consider $C(U)$ and $C(U)_{0}$ as topological groups with their
inverse limit topology. It is then clear that $C(U)_{0}$ is a closed subgroup
of $C(U)$.

The moving lemma we wish to prove for ${\rm CH}_{0}(X|D)$ is the following.
This result is of independent interest in the theory of cycles with modulus
aside from its current application.

######  Theorem 7.1.

Let $X$ be a smooth quasi-projective scheme of dimension $d\geq 2$ over a
perfect field $k$ and let $D\subset X$ be an effective Cartier divisor. Let
$A\subset D^{\prime}$ be a closed subscheme such that $\text{\rm dim}(A)\leq
d-2$. Then ${\mathcal{R}}_{0}(X|D)$ is generated by the images of
${\mathcal{R}}_{0}(Y|D)$ as in (7.1), where $Y\subset X$ has the additional
property that $Y\cap A=\emptyset$.

###  7.2. The Levine-Weibel Chow group of the double

The proof of Theorem 7.1 is similar to that of the fundamental exact sequence
[2, (6.3)], which relates the Chow group with modulus with an improved version
of the Levine-Weibel Chow group of a singular variety. However, we need some
modifications to prove Theorem 7.1 and shall therefore present a complete
proof.

Throughout the rest of § 7, we shall follow the notations and assumptions of
Theorem 7.1. Recall from [2, § 2.1] that the double of $X$ along $D$ is the
quasi-projective scheme $S_{X}$ such that the square

(7.6)
---
$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{\iota}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{+}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{-}}$$\textstyle{S_{X}}$

is co-Cartesian. That is, $S_{X}=X_{+}\amalg_{D}X_{-}$. The arrows
$\iota_{\pm}$ are closed immersions and $\iota\colon D\hookrightarrow X$ is
the inclusion of $D$ in $X$. There is a projection map $\Delta\colon S_{X}\to
X$ which is finite and flat. Moreover, $\Delta\circ\iota_{\pm}={\rm Id}_{X}$.
We also have $D^{\prime}=(S_{X})_{\rm sing}$ via the inclusion
$\iota_{\pm}\circ\iota$. Let us denote this by $\iota^{\prime}$. The double
$S_{X}$ is a reduced scheme with two irreducible components $X_{\pm}$, each a
copy of $X$.

Let ${\rm CH}^{LW}_{0}(S_{X})$ denote the Levine-Weibel Chow group of $S_{X}$
(see [2, § 3.4]). Recall that this is the quotient of
${\mathcal{Z}}_{0}(S_{X}\setminus D)$ by the subgroup of rational
equivalences, denoted by ${{\mathcal{R}}^{LW}_{0}(S_{X})}$. This subgroup is
generated by the divisors of rational functions on certain ‘Cartier curves on
$X$ relative to $D$’. We refer to [2, § 3.4] for these terms.

In this subsection, we shall prove some lemmas which will allow us to choose
some refined Cartier curves in order to define the rational equivalence of
0-cycles on $S_{X}$. We shall consider $A$ as a closed subscheme of $S_{X}$
via the inclusions $A\hookrightarrow D\hookrightarrow S_{X}$. We shall assume
in this subsection that $k$ is infinite and perfect.

######  Lemma 7.2.

Assume $d\geq 3$. Then ${\mathcal{R}}^{LW}_{0}(S_{X})$ is generated by the
divisors of functions on (possibly non-reduced) Cartier curves
$C\hookrightarrow S_{X}$ relative to $D$, where $C$ satisfies the following:

  1. (1)

There is a locally closed embedding $S_{X}\hookrightarrow{\mathbb{P}}^{N}_{k}$
and distinct hypersurfaces

$H_{1},\cdots,H_{d-2}\hookrightarrow{\mathbb{P}}^{N}_{k}$

such that $Y=S_{X}\cap H_{1}\cap\cdots\cap H_{d-2}$ is a complete intersection
which is reduced;

  2. (2)

$X_{\pm}\cap Y=X_{\pm}\cap H_{1}\cap\cdots\cap H_{d-2}:=Y_{\pm}$ are integral;

  3. (3)

No component of $Y$ is contained in $D$;

  4. (4)

$Y\cap A$ is finite;

  5. (5)

$C\subset Y$;

  6. (6)

$C$ is a Cartier divisor on $Y$;

  7. (7)

$Y_{\pm}$ are smooth away from $C$.

###### Proof.

The proof is identical to that of [2, Lemma 5.4]: we obtain $Y$ and $C$ using
the Bertini theorems of Altman-Kleiman [1, Theorem 1] and Jouanolou [17].
Since $k$ is infinite, these Bertini theorems also allow us to ensure that a
general hypersurface of a given degree in ${\mathbb{P}}^{n}_{k}$ will
intersect $A$ properly under any chosen locally closed embedding
$X\hookrightarrow{\mathbb{P}}^{N}_{k}$. Since (4) is an open condition on the
linear system of hypersurfaces of a given degree in ${\mathbb{P}}^{N}_{k}$, we
can also include it in the proof of [2, Lemma 5.4] because $\text{\rm
dim}(A)\leq d-2$. ∎

######  Lemma 7.3.

Let $d\geq 2$. Let $\nu\colon C\hookrightarrow S_{X}$ be a (possibly non-
reduced) Cartier curve relative to $D\subset S_{X}$. Assume that either $d=2$
or there are inclusions $C\subset Y\subset S_{X}$, where $Y$ is a reduced
complete intersection surface and $C$ is a Cartier divisor on $Y$, as in Lemma
7.2. Let $f\in\mathcal{O}_{C,\nu^{*}D}^{\times}\subset k(C)^{\times}$, where
$k(C)$ is the total quotient ring of ${\mathcal{O}}_{C,\nu^{*}D}$.

We can then find two Cartier curves $\nu^{\prime}\colon
C^{\prime}\hookrightarrow S_{X}$ and $\nu^{\prime\prime}\colon
C^{\prime\prime}\hookrightarrow S_{X}$ relative to $D$ satisfying the
following:

  1. (1)

There are very ample line bundles
${\mathcal{L}}^{\prime},{\mathcal{L}}^{\prime\prime}$ on $S_{X}$ and sections
$t^{\prime}\in H^{0}(S_{X},{\mathcal{L}}^{\prime}),\ t^{\prime\prime}\in
H^{0}(S_{X},{\mathcal{L}}^{\prime\prime})$ such that
$C^{\prime}=Y\cap(t^{\prime})$ and $C^{\prime\prime}=Y\cap(t^{\prime\prime})$
(with the convention $Y=S_{X}$ if $d=2$);

  2. (2)

$C^{\prime}$ and $C^{\prime\prime}$ are reduced;

  3. (3)

$C^{\prime}\cap A=C^{\prime\prime}\cap A=\emptyset$;

  4. (4)

The restrictions of both $C^{\prime}$ and $C^{\prime\prime}$ to $X$ via the
two closed immersions $\iota_{\pm}$ are integral curves in $X$, which are
Cartier and smooth along $D$;

  5. (5)

There are functions
$f^{\prime}\in{\mathcal{O}}_{C^{\prime},(\nu^{\prime})^{*}D}^{\times}$ and
$f^{\prime\prime}\in{\mathcal{O}}_{C^{\prime\prime},(\nu^{\prime\prime})^{*}D}^{\times}$
such that $\nu^{\prime}_{*}({{\rm
div}}(f^{\prime}))+\nu^{\prime\prime}_{*}({{\rm
div}}(f^{\prime\prime}))=\nu_{*}({{\rm div}}(f))$ in
${\mathcal{Z}}_{0}(S_{X}\setminus D)$.

###### Proof.

In this proof, we shall assume that $Y=S_{X}$ if $d=2$. In the latter case, a
Cartier curve on $S_{X}$ along $D^{\prime}=(S_{X})_{{\rm sing}}$ must be an
effective Cartier divisor on $S_{X}$. Hence, we can assume that $C$ is an
effective Cartier divisor on $Y$ for any $d\geq 2$. Note that
$Y=Y_{+}\amalg_{D}Y_{-}$.

Since $Y$ is quasi-projective over $k$ and $C$ is an effective Cartier divisor
on $Y$, we can add some very ample effective divisor to $C$ to get another
effective Cartier divisor $C^{\prime}$ (see the proof of [33, Lemma 1.4]) on
$Y$ such that

  1. (1)

$C\subset C^{\prime}$;

  2. (2)

$\overline{C^{\prime}\setminus C}\cap(C\cap D)=\emptyset$;

  3. (3)

${\mathcal{O}}_{Y}(C^{\prime})$ is very ample on $Y$.

Setting $g=f$ on $C$ and $g=1$ on $C^{\prime}\setminus C$, we then see that
$g$ defines an element in ${\mathcal{O}}^{\times}_{C^{\prime},C^{\prime}\cap
D}$ such that ${\rm div}(g)={\rm div}(f)$. We can thus assume that
${\mathcal{O}}_{Y}(C)$ is very ample on $Y$.

We now choose $t_{0}\in H^{0}(Y,{\mathcal{O}}_{Y}(C))$ such that $C=(t_{0})$,
where for a line bundle ${\mathcal{L}}$ and a meromorphic section $h$ of
${\mathcal{L}}$, $(h)$ denotes the (effective) divisor of zeros of $h$. Since
${\mathcal{O}}_{Y}(C)$ is very ample, and $Y$ is reduced, we can find a new
section $t_{\infty}\in H^{0}(Y,{\mathcal{O}}_{Y}(C))$, sufficiently general so
that:

  1. (1)

$(t_{\infty})$ is reduced;

  2. (2)

$(t_{\infty})\cap(t_{0})\cap D=\emptyset$;

  3. (3)

$(t_{\infty})\cap A=\emptyset$;

  4. (4)

$(t_{\infty})$ contains no component of $(t_{0})$;

  5. (5)

$D$ contains no component of $(t_{\infty})$.

Note that we can achieve (3) because $Y\cap A$ is finite by Lemma 7.2. Denote
by $C_{\infty}$ the divisor $(t_{\infty})$. Notice that the function $h=(1,f)$
is meromorphic on $C_{\infty}\cup C$ and regular invertible in a neighborhood
of $(C_{\infty}\cup C)\cap D$ by (2). Since $Y\cap A$ is a finite closed set
of $S_{X}$ which does not meet $C_{\infty}$, we can write $Y\cap A=S_{1}\amalg
S_{2}$, where $S_{1}\subset C\cap A$ and $S_{2}\cap(C_{\infty}\cup
C)=\emptyset$. By setting $h=1$ on $S_{2}$, we can extend the function $h$ to
a meromorphic function on $T:=(Y\cap A)\cup C_{\infty}\cup C$ which is regular
invertible in a neighborhood of $T^{\prime}:=(Y\cap A)\bigcup((C_{\infty}\cup
C)\cap D)$. Let $S$ denote the finite set of closed points on $C_{\infty}\cup
C$, which either lie on $C_{\infty}\cap C$ or where $h$ is not regular. In
particular, $S\supset\\{\text{poles of $f$}\\}\cup(C\cap C_{\infty})$.

Since $S_{X}$ is reduced quasi-projective and $X_{\pm}$ (as well as $Y_{\pm}$)
are integral closed subschemes of $S_{X}$, we can find a very ample line
bundle ${\mathcal{L}}$ on $S_{X}$ and a section $s_{\infty}\in
H^{0}(S_{X},{\mathcal{L}})$ (see again [33, Lemma 1.4]) so that:

  * a)

$(s_{\infty})$ and $(s_{\infty})\cap Y$ are reduced;

  * b)

$Y\not\subset(s_{\infty})$;

  * c)

$(s_{\infty})\cap X_{\pm}$ and $(s_{\infty})\cap Y_{\pm}$ are integral;

  * d)

$(s_{\infty})\cap T^{\prime}=\emptyset$;

  * e)

$(s_{\infty})\supset S$;

  * f)

$C_{\infty}\cup C$ contains no component of $(s_{\infty})\cap Y$.

If $\overline{S_{X}}$ is the scheme-theoretic closure of $S_{X}$ in the
projective embedding given by ${\mathcal{L}}={\mathcal{O}}_{S_{X}}(1)$ and if
${\mathcal{I}}$ is the ideal sheaf of $\overline{S_{X}}\setminus S_{X}$ in the
ambient projective space, then we can find a section $s^{\prime}_{\infty}$ of
the sheaf ${\mathcal{I}}\otimes{\mathcal{O}}_{{\mathbb{P}}^{N}_{k}}(m)$ for
some $m\gg 0$, which restricts to a section $s_{\infty}$ on $S_{X}$ satisfying
the properties (a) - (f) on $S_{X}=\overline{S_{X}}\setminus
V({\mathcal{I}})$. This implies in particular that
$S_{X}\setminus(s_{\infty})=\overline{S_{X}}\setminus(s^{\prime}_{\infty})$ is
affine. Set ${\mathcal{L}}^{\prime}={\mathcal{L}}^{m}$.

We now know that $Y_{+}$ is smooth away from $C$, and (d) tells us that
$(s_{\infty})$ intersects $Y_{+}$ along $D$ at only those points which are
away from $C_{\infty}\cup C$. It follows that $(s_{\infty})\cap Y_{+}\cap
D\subset(s_{\infty})\cap(Y_{+})_{\rm reg}$, where $(Y_{+})_{\rm reg}$ denotes
the smooth locus of $Y_{+}$. On the other hand, we can use the Bertini theorem
of Altman and Kleiman [1, Theorem 1] to ensure that
$(s_{\infty})\cap(Y_{+})_{\rm reg}$ is smooth. The same holds for $Y_{-}$ as
well. We conclude that we can choose ${\mathcal{L}}^{\prime}$ and
$s_{\infty}\in H^{0}(S_{X},{\mathcal{L}}^{\prime})$ such that (a) - (f) above
as well as the following hold.
g) $(s_{\infty})\cap Y_{\pm}$ are smooth along $D$.
h) $S_{X}\setminus(s_{\infty})$ is affine.

Now, because $h$ is a meromorphic function on $T$ which is regular outside
$S$, and the latter is contained in $(s_{\infty})$, the property (h) implies
that $h|_{T\setminus(s_{\infty})}$ extends to a regular function $H$ on
$U=S_{X}\setminus(s_{\infty})$. Since $H$ is a meromorphic function on $S_{X}$
which has poles only along $(s_{\infty})$, it follows that $Hs^{N}_{\infty}$
is an element of $H^{0}(U,({\mathcal{L}}^{\prime})^{N})$ which extends to a
section $s_{0}$ of $({\mathcal{L}}^{\prime})^{N}$ on all of $S_{X}$, if we
choose $N\gg 0$.

Since $s_{\infty}$ and $h$ are both invertible on $T$ in a neighborhood of
$T^{\prime}$, we see that $s_{0}$ is invertible on $T$ in a neighborhood of
$T^{\prime}$. In particular, $s_{0}\notin
H^{0}(S_{X},({\mathcal{L}}^{\prime})^{N}\otimes{\mathcal{I}}_{T})$. Applying
[33, Lemma 1.4] and [1, Theorem 1] (see their proofs), we can thus find
$\alpha\in
H^{0}(S_{X},({\mathcal{L}}^{\prime})^{N}\otimes{\mathcal{I}}_{T})\subset
H^{0}(S_{X},({\mathcal{L}}^{\prime})^{N})$ such that
$s^{\prime}_{0}:=s_{0}+\alpha$ has the following properties:

  * a’)

$(s^{\prime}_{0})$ and $(s^{\prime}_{0})\cap Y$ are reduced;

  * b’)

$Y\not\subset(s^{\prime}_{0})$;

  * c’)

$(s^{\prime}_{0})\cap X_{\pm}$ and $(s^{\prime}_{0})\cap Y_{\pm}$ are
integral;

  * d’)

$(s^{\prime}_{0})\cap T^{\prime}=\emptyset$;

  * e’)

$C_{\infty}\cup C$ contains no component of $(s^{\prime}_{0})\cap Y$;

  * f’)

$(s^{\prime}_{0})\cap Y_{\pm}$ are smooth along $D$.

We then have

$\frac{s^{\prime}_{0}}{s^{N}_{\infty}}=\frac{Hs^{N}_{\infty}+(\alpha
s^{-N}_{\infty})s^{N}_{\infty}}{s^{N}_{\infty}}=H+\alpha
s^{-N}_{\infty}=H^{\prime},\ (\mbox{say}).$

Since $\alpha$ vanishes along $C_{\infty}\cup C$ and $s_{\infty}$ is
invertible along $U$, it follows that $H^{\prime}_{|{(C_{\infty}\cup C)\cap
U}}=H_{|{(C\cup C_{\infty})\cap U}}=h_{|U}$. In other words, we have
${s^{\prime}_{0}}/{s^{N}_{\infty}}=h$ as rational functions on $C_{\infty}\cup
C$. We can now compute:

$\nu_{*}({\rm div}(f))=(s^{\prime}_{0})\cdot C-N(s_{\infty})\cdot C$ $0={\rm
div}(1)=(s^{\prime}_{0})\cdot C_{\infty}-N(s_{\infty})\cdot C_{\infty}.$

Setting $(s^{Y}_{\infty})=(s_{\infty})\cap Y$ and $(s^{\prime
Y}_{0})=(s^{\prime}_{0})\cap Y$, we get

$\begin{array}[]{lll}\nu_{*}({\rm
div}(f))&=&(s^{\prime}_{0})\cdot(C-C_{\infty})-N(s_{\infty})(C-C_{\infty})\\\
&=&(s^{\prime Y}_{0})\cdot({\rm
div}({t_{0}}/{t_{\infty}}))-N(s^{Y}_{\infty})\cdot({\rm
div}({t_{0}}/{t_{\infty}}))\\\ &=&\iota_{{s^{\prime Y}_{0}},*}({\rm
div}(f^{\prime}))-N\iota_{{s^{Y}_{\infty}},*}({\rm
div}(f^{\prime\prime})),\end{array}$

where $f^{\prime}=({t_{0}}/{t_{\infty}})|_{(s^{\prime
Y}_{0})}\in{\mathcal{O}}^{\times}_{(s^{\prime Y}_{0}),D\cap(s^{\prime
Y}_{0})}$ (by (d’)) and
$f^{\prime\prime}=(t_{0}/t_{\infty})|_{(s^{Y}_{\infty})}\in{\mathcal{O}}_{(s^{Y}_{\infty}),D\cap(s^{Y}_{\infty})}^{\times}$
(by (d)).

It follows from (g) and (f’) that $(s^{\prime Y}_{0})_{|X_{+}},\ (s^{\prime
Y}_{0})_{|X_{-}},\ (s^{Y}_{\infty})_{|X_{+}}$ and $(s^{Y}_{\infty})_{|X_{-}}$
are all smooth along $D$. Setting
${\mathcal{L}}^{\prime\prime}=({\mathcal{L}}^{\prime})^{N},\
t^{\prime\prime}=s^{\prime}_{0}$ and $t^{\prime}=s_{\infty}$, we see that the
curves $C^{\prime}=(t^{\prime})\cap Y$ and
$C^{\prime\prime}=(t^{\prime\prime})\cap Y$ together with the functions
$f^{\prime}$ and $f^{\prime\prime}$ satisfy the conditions of the Lemma. ∎

###  7.3. The map $\tau^{*}_{X}$

Let ${\rm CH}^{A}_{0}(X|D)$ be the quotient of the free abelian group
${\mathcal{Z}}_{0}(X\setminus D)$ on the closed points on $X\setminus D$ by
the subgroup of rational equivalences ${\mathcal{R}}^{A}_{0}(X|D)$ generated
by the images of ${\mathcal{R}}_{0}(Y|D)$ as in (7.1), where $Y\subset X$ has
an additional property that $Y\cap A=\emptyset$. It is clear that there is a
canonical surjection ${\rm CH}^{A}_{0}(X|D)\twoheadrightarrow{\rm
CH}_{0}(X|D)$. Our goal is to prove that this is an isomorphism.

We let $\tau^{*}_{X}\colon{\mathcal{Z}}_{0}(S_{X}\setminus
D)\to{\mathcal{Z}}_{0}(X\setminus D)$ be the map
$\tau^{*}_{X}=\iota^{*}_{+}-\iota^{*}_{-}$ under the embeddings
$X\xrightarrow{\iota_{\pm}}S_{X}$. We want to show that $\tau^{*}_{X}$
preserves the subgroups of rational equivalences. We continue to assume that
$k$ is infinite and perfect.

######  Lemma 7.4.

Assume $d=2$. Then the map $\tau_{X}^{*}$ descends to a group homomorphism
${\rm CH}^{LW}_{0}(S_{X})\to{\rm CH}^{A}_{0}(X|D)$.

###### Proof.

The proof is essentially identical to that of [2, Proposition 5.7], but subtle
changes are required at several places. So we provide the details.

We have shown in Lemma 7.3 that in order to prove that $\tau^{*}_{X}$
preserves the subgroups of rational equivalences, it suffices to show that
$\tau^{*}_{X}({\rm div}(f))\in{\mathcal{R}}^{A}_{0}(X|D)$, where $f$ is a
rational function (which is regular and invertible along $D$) on a Cartier
curve $\nu\colon C\hookrightarrow S_{X}$ that we can choose in the following
way.

  1. (1)

There is a very ample line bundle ${\mathcal{L}}$ on $S_{X}$ and sections
$t\in H^{0}(S_{X},{\mathcal{L}}),t_{\pm}=\iota^{*}_{\pm}(t)\in
H^{0}(X,\iota^{*}_{\pm}({\mathcal{L}}))$ such that $C=(t)$ and
$C_{\pm}=(t_{\pm})$.

  2. (2)

$C$ is a reduced Cartier curve of the form $C=C_{+}\amalg_{E}C_{-}$, where
$E=\nu^{*}(D)$ such that $C_{\pm}$ are integral curves on $X$, none of which
is contained in $D$, none of which meets $A$ and each of which is smooth along
$D$ (see [2, Remark 5.6]).

If $E=\emptyset$, then $C_{\pm}$ are two integral curves on $X$ away from $D$
and $\tau^{*}_{X}({\rm div}(f))={\rm div}(f|_{C_{+}})-{\rm
div}(f|_{C_{-}})\in{\mathcal{R}}^{A}_{0}(X|D)$. We can thus assume that
$E\neq\emptyset$. Then (1) implies that

(7.7) $(t_{+})_{|D}=\iota^{\prime*}(t)=(t_{-})_{|D},$

where recall that
$\iota^{\prime}=\iota_{+}\circ\iota=\iota_{-}\circ\iota\colon D\hookrightarrow
S_{X}$ denotes the inclusion map.

Let $(f_{+},f_{-})$ be the image of $f$ in
${\mathcal{O}}^{\times}_{C_{+},E}\times{\mathcal{O}}^{\times}_{C_{-},E}\hookrightarrow
k(C_{+})\times k(C_{-})$. It follows from [2, Lemma 2.2] that there is an
exact sequence

$0\to{\mathcal{O}}_{C,E}\to{\mathcal{O}}_{C_{+},E}\times{\mathcal{O}}_{C_{-},E}\to{\mathcal{O}}_{E}\to
0.$

In particular, we have

(7.8) $(f_{+})_{|E}=(f_{-})_{|E}\in{\mathcal{O}}^{\times}_{E}.$

Let us first assume that $C_{+}=C_{-}$ as curves on $X$. Let $C$ denote this
curve and let $C_{n}$ denote its normalization. Let $\pi\colon C_{n}\to
C\hookrightarrow X$ denote the composite map. Since $C$ is regular along $E$
by (2), we get $f_{+},f_{-}\in{\mathcal{O}}^{\times}_{C_{n},E}$. Setting
$g:=f_{+}f^{-1}_{-}\in{\mathcal{O}}^{\times}_{C_{n},E}$, it follows from (7.8)
that $g\in{\rm
Ker}({\mathcal{O}}^{\times}_{C_{n},E}\to{\mathcal{O}}^{\times}_{E})$.
Moreover, $\tau^{*}_{X}({\rm div}(f))=\iota^{*}_{+}({\rm
div}(f))-\iota^{*}_{-}({\rm div}(f))={\rm div}(f_{+})-{\rm
div}(f_{-})=\pi_{*}({\rm div}(g))$. Since $C\cap A=\emptyset$ by (2), we
conclude that $\tau^{*}_{X}({\rm div}(f))$ dies in ${\rm CH}^{A}_{0}(X|D)$.

We now assume that $C_{+}\neq C_{-}$. Since $C_{+}\cap D=C_{-}\cap D=E$ as
closed subschemes, we see that the support of $(C_{+}\cup C_{-})\cap D$ is
same as $E_{\rm red}$, where $C_{+}\cup C_{-}\subset X$ is the closed
subscheme defined by the ideal sheaf
${\mathcal{I}}_{C_{+}}\cap{\mathcal{I}}_{C_{-}}$. Note that since $C_{\pm}$
are integral, $C_{+}\cup C_{-}$ is a reduced closed subscheme of $X$ with
irreducible components $C_{\pm}$.

Since $A$ is a finite closed subscheme of $X$ and $(C_{+}\cup C_{-})\cap
A=\emptyset$, the functions $f_{\pm}$ extend to meromorphic functions on
$T_{\pm}=A\cup C_{\pm}$ which are regular invertible in a neighborhood of
$T^{\prime}=A\cup E$ by letting $f_{\pm}=1$ on $A$. Let $S_{\pm}$ denote the
set of closed points on $C_{\pm}$, where $f_{\pm}$ have poles. We let
$T=T_{+}\cup T_{-}$ and $S=S_{+}\cup S_{-}$. It is clear that $S\cap
D=\emptyset$.

We now repeat the constructions in the proof of Lemma 7.3 to find a very ample
line bundle ${\mathcal{L}}$ on $X$ and a section $s_{\infty}\in
H^{0}(X,{\mathcal{L}})$ (see [33, Lemma 1.4] and [1, Theorem 1]) such that:

  * a)

$(s_{\infty})$ is integral (because $X$ is integral);

  * b)

$(s_{\infty})\cap T^{\prime}=\emptyset$;

  * c)

$(s_{\infty})\supset S$;

  * d)

$(s_{\infty})\not\subset C_{+}\cup C_{-}$;

  * e)

$(s_{\infty})$ is smooth away from $S$;

  * f)

$X\setminus(s_{\infty})$ is affine.

It follows that $f_{\pm}|_{T_{\pm}\setminus(s_{\infty})}$ extend to regular
functions $F_{\pm}$ on $X\setminus(s_{\infty})$. Since $F_{\pm}$ are
meromorphic functions on $X$ which have poles only along $(s_{\infty})$, it
follows that $F_{\pm}s^{N}_{\infty}$ are elements of
$H^{0}(X\setminus(s_{\infty}),{\mathcal{L}}^{N})$ which extend to sections
$(s_{0})_{\pm}$ of ${\mathcal{L}}^{N}$ on all of $X$, if we choose $N\gg 0$.

Since the functions $s_{\infty}$ and $F_{\pm}$ are all meromorphic functions
on $X$ which are regular on $X\setminus(s_{\infty})$, it follows that each of
them restricts to a meromorphic function on $T$ which is regular on
$T\setminus(s_{\infty})$ and
$F_{\pm}|_{T\pm\setminus(s_{\infty})}=f_{\pm}|_{T_{\pm}\setminus(s_{\infty})}$.
Since $s_{\infty}$ and $F_{\pm}$ are invertible on $T_{\pm}$ in some
neighborhood of $T^{\prime}$, we see that $(s_{0})_{\pm}$ are invertible on
$T_{\pm}$ in some neighborhood of $T^{\prime}$. In particular,
$(s_{0})_{\pm}\notin H^{0}(X,{\mathcal{L}}^{N}\otimes{\mathcal{I}}_{T})$. As
before, using [33, Lemma 1.4] and [1, Theorem 1], we can moreover find
$\alpha_{\pm}\in H^{0}(X,{\mathcal{L}}^{N}\otimes{\mathcal{I}}_{T})\subset
H^{0}(X,{\mathcal{L}}^{N})$ such that
$(s^{\prime}_{0})_{\pm}:=(s_{0})_{\pm}+\alpha_{\pm}$ have the following
properties:

  * a’)

$((s^{\prime}_{0})_{\pm})$ are integral;

  * b’)

$((s^{\prime}_{0})_{\pm})\not\subset C_{+}\cup C_{-}$;

  * c’)

$((s^{\prime}_{0})_{\pm})\cap T^{\prime}=\emptyset$;

  * d’)

$((s^{\prime}_{0})_{\pm})$ are smooth away from $S$.

We then have

(7.9)
$\frac{(s^{\prime}_{0})_{\pm}}{s^{N}_{\infty}}=\frac{F_{\pm}s^{N}_{\infty}+(\alpha_{\pm}s^{-N}_{\infty})s^{N}_{\infty}}{s^{N}_{\infty}}=F_{\pm}+\alpha_{\pm}s^{-N}_{\infty}=H_{\pm},\
(\mbox{say}).$

We can now find a dense open subscheme $U^{\prime}\subset
X\setminus((s_{\infty})\cup((s^{\prime}_{0})_{+})\cup((s^{\prime}_{0})_{-}))$
which contains $T^{\prime}$ and where $F_{\pm},\alpha_{\pm},s_{\infty}$ and
$(s^{\prime}_{0})_{\pm}$ are all regular. In particular, $H_{\pm}$ are
rational functions on $X$ which are regular on $U^{\prime}$.

Since $T^{\prime}\subset U^{\prime}$ and $T^{\prime}\neq\emptyset$ (because
$E\neq\emptyset$), it follows that $T_{\pm}\cap U^{\prime}$ are dense open in
$T_{\pm}$. It follows that $s_{\infty}$ and $(s^{\prime}_{0})_{\pm}$ restrict
to regular functions on $T_{\pm}\cap U^{\prime}$ which are invertible in a
neighborhood of $T^{\prime}$. Since $\alpha_{\pm}$ vanish along $T$ and
$s_{\infty}$ is invertible on $U^{\prime}$, it follows that
${H_{\pm}}_{|{T_{\pm}\cap U^{\prime}}}={F_{\pm}}_{|{T_{\pm}\cap U^{\prime}}}$.
Since $F_{\pm}$ restrict to regular functions on $T_{\pm}\cap U^{\prime}$ and
are invertible along $T^{\prime}$, it follows that $H_{\pm}$ restrict to
regular functions on $T_{\pm}\cap U^{\prime}$ and are invertible along
$T^{\prime}$.

As $H_{+}$ (resp. $H_{-}$) is regular on $U^{\prime}$, it restricts to a
regular function on the dense open subset $C_{-}\cap U^{\prime}$ (resp.
$C_{+}\cap U^{\prime}$) of $C_{-}$ (resp. $C_{+}$). Furthermore, we have

(7.10) ${H_{+}}_{|E}={F_{+}}_{|E}={f_{+}}_{|E}\ =^{\dagger}\
{f_{-}}_{|E}={F_{-}}_{|E}={H_{-}}_{|E},$

where $\dagger$ follows from (7.8).

We thus saw above that $H_{+}$ and $H_{-}$ are both regular functions on
$C_{-}\cap U^{\prime}$ such that $H_{-}\neq 0$. In particular,
${H_{+}}/{H_{-}}$ is a rational function on $C_{-}$. Since
$(s^{\prime}_{0})_{+}$ and $(s^{\prime}_{0})_{-}$ are both invertible
functions on $C_{-}\cap U^{\prime}$, it follows that the restriction of
${(s^{\prime}_{0})_{+}}/{(s^{\prime}_{0})_{-}}$ on $C_{-}$ is a rational
function on $C_{-}$, which is regular and invertible on the dense open
$C_{-}\cap U^{\prime}$. On the other hand, we have

(7.11)
$\frac{(s^{\prime}_{0})_{+}}{(s^{\prime}_{0})_{-}}=\frac{{(s^{\prime}_{0})_{+}}/{s^{N}_{\infty}}}{{(s^{\prime}_{0})_{-}}/{s^{N}_{\infty}}}=\frac{H_{+}}{H_{-}},$

as rational functions on $X$. In particular,
$(s^{\prime}_{0})_{+}\cdot{H_{-}}={(s^{\prime}_{0})_{-}}\cdot{H_{+}}$ as
regular functions on $U^{\prime}$. In particular, this identity holds after
restricting these regular functions to $C_{-}\cap U^{\prime}$. We thus get

(7.12)
$\frac{(s^{\prime}_{0})_{+}}{(s^{\prime}_{0})_{-}}=\frac{{(s^{\prime}_{0})_{+}}/{s^{N}_{\infty}}}{{(s^{\prime}_{0})_{-}}/{s^{N}_{\infty}}}=\frac{H_{+}}{H_{-}},$

as rational functions on $C_{-}$. Note that $H_{-}$ is non-zero on $C_{-}$.
Since $\frac{(s^{\prime}_{0})_{+}}{(s^{\prime}_{0})_{-}}$ restricts to a
rational function on $C_{-}$ which is regular and invertible in the dense open
$C_{-}\cap U^{\prime}$, we conclude that ${H_{+}}/{H_{-}}$ restricts to an
identical rational function on $C_{-}$ which is regular and invertible on
$C_{-}\cap U^{\prime}$.

We now compute

$\begin{array}[]{lll}\tau^{*}_{X}({\rm div}(f))&=&\iota^{*}_{+}({\rm
div}(f))-\iota^{*}_{-}({\rm div}(f))\\\ &=&{\rm div}(f_{+})-{\rm
div}(f_{-})\\\ &=&\left[((s^{\prime}_{0})_{+})\cdot
C_{+}-(s^{N}_{\infty})\cdot C_{+}\right]-\left[((s^{\prime}_{0})_{-})\cdot
C_{-}-(s^{N}_{\infty})\cdot C_{-}\right]\\\
&=&\left[((s^{\prime}_{0})_{+})\cdot C_{+}-((s^{\prime}_{0})_{+})\cdot
C_{-}\right]+\left[((s^{\prime}_{0})_{+})\cdot
C_{-}-((s^{\prime}_{0})_{-})\cdot C_{-}\right]\\\
&&-\left[(s^{N}_{\infty})\cdot C_{+}-(s^{N}_{\infty})\cdot C_{-}\right]\\\
&=&\left[((s^{\prime}_{0})_{+})\cdot(C_{+}-C_{-})\right]+\left[C_{-}\cdot(((s^{\prime}_{0})_{+})-((s^{\prime}_{0})_{-}))\right]-\left[(s^{N}_{\infty})\cdot(C_{+}-C_{-})\right]\\\
&=&((s^{\prime}_{0})_{+})\cdot({\rm div}({t_{+}}/{t_{-}}))+C_{-}\cdot({\rm
div}({(s^{\prime}_{0})_{+}}/{(s^{\prime}_{0})_{-}}))-N(s_{\infty})\cdot({\rm
div}({t_{+}}/{t_{-}}))\\\ &=&((s^{\prime}_{0})_{+})\cdot({\rm
div}({t_{+}}/{t_{-}}))+C_{-}\cdot({\rm
div}({H_{+}}/{H_{-}}))-N(s_{\infty})\cdot({\rm
div}({t_{+}}/{t_{-}})).\end{array}$

It follows from (b) and (c’) that $t_{\pm}$ restrict to regular invertible
functions on $((s^{\prime}_{0})_{+})$ and $(s_{\infty})$ along $D$. We set
$h_{1}={(\frac{t_{+}}{t_{-}})}_{|((s^{\prime}_{0})_{+})},\
h_{2}={(\frac{H_{+}}{H_{-}})}_{|C_{-}}$ and
$h_{3}={(\frac{t_{+}}{t_{-}})}_{|s_{\infty}}$. Let
$((s^{\prime}_{0})_{+})_{n}\to((s^{\prime}_{0})_{+})$, $(C_{-})_{n}\to C_{-}$
and $(s_{\infty})_{n}\to(s_{\infty})$ denote the normalization maps. Let
$\nu_{1}\colon((s^{\prime}_{0})_{+})_{n}\to X$, $\nu_{2}\colon(C_{-})_{n}\to
X$ and $\nu_{3}\colon(s_{\infty})_{n}\to X$ denote the composite maps. We now
note that $((s^{\prime}_{0})_{+}),\ C_{-}$ and $(s_{\infty})$ are all regular
along $D$ by (2), (e) and (d’). Furthermore, none of these meets $A$ by (2),
(b) and (c’). It follows from (7.7) and (7.10) that $(\nu_{1})_{*}(h_{1}),\
(\nu_{2})_{*}(h_{2})$ and $(\nu_{3})_{*}(h_{3})$ die in ${\rm
CH}^{A}_{0}(X|D)$. We conclude that $\tau^{*}_{X}({\rm div}(f))$ dies in ${\rm
CH}^{A}_{0}(X|D)$. In particular, $\tau^{*}_{X}$ descends to a map
$\tau^{*}_{X}\colon{\rm CH}^{LW}_{0}(S_{X})\to{\rm CH}^{A}_{0}(X|D)$. This
finishes the proof. ∎

The following result generalizes Lemma 7.4 to higher dimensions.

######  Proposition 7.5.

Assume $d\geq 2$. Then the map $\tau_{X}^{*}$ descends to a group homomorphism
$\tau_{X}^{*}\colon{\rm CH}^{LW}_{0}(S_{X})\to{\rm CH}^{A}_{0}(X|D)$.

###### Proof.

We can assume $d\geq 3$ by Lemma 7.4. Let $\nu\colon C\hookrightarrow S_{X}$
be a reduced Cartier curve relative to $D$ and let
$f\in{\mathcal{O}}^{\times}_{C,E}$, where $E=\nu^{*}(D)$. By Lemma 7.2, we can
assume that there are inclusions $C\hookrightarrow Y\hookrightarrow S_{X}$
satisfying the conditions (1) - (7) of Lemma 7.2. The only price we pay by
doing so is that $C$ may no longer be reduced. We achieve its reducedness
using Lemma 7.3 as follows.

We replace $C$ by a reduced Cartier curve (which we also denote by $C$) that
is of the form given in Lemma 7.3. We shall now continue with the notations of
proof of Lemma 7.3.

We write $C=(t)\cap Y$, where $t\in H^{0}(S_{X},{\mathcal{L}})$ such that
${\mathcal{L}}$ is a very ample line bundle on $S_{X}$. Let
$t_{\pm}=\iota^{*}_{\pm}(t)\in H^{0}(X,\iota^{*}_{\pm}({\mathcal{L}}))$ and
let $C_{\pm}=(t_{\pm})\cap Y=(t_{\pm})\cap Y_{\pm}$. Let $\nu_{\pm}\colon
C_{\pm}\hookrightarrow X$ denote the inclusions. It follows from our choice of
the section that $(t_{\pm})$ are integral. If $C_{+}=C_{-}$, exactly the same
argument as in the case of surfaces applies to show that $\tau^{*}_{X}({\rm
div}(f))$ dies in ${\rm CH}^{A}_{0}(X|D)$. So we assume $C_{+}\neq C_{-}$. We
can also assume $E\neq\emptyset$.

Let $\Delta(C)=C_{+}\cup C_{-}$ denote the scheme-theoretic image in $X$ under
the finite map $\Delta$. Since $X$ is smooth and integral, we can find a
complete intersection integral surface $Z\subset X$ satisfying the following:

  1. (1)

$Z\supset\Delta(C)$;

  2. (2)

$Z\cap A$ is finite;

  3. (3)

$Z\cap(t_{\pm})$ are integral curves;

  4. (4)

$Z$ is smooth away from $\Delta(C)$.

Set $t^{Z}_{\pm}=(t_{\pm})_{|Z}$. Since $C_{\pm}$ are integral and contained
in $Z\cap(t_{\pm})$, it follows that

(7.13) $(t^{Z}_{\pm})=C_{\pm}.$

We let $f_{\pm}=\nu_{\pm}^{*}(f)$. As in the proof of Lemma 7.4, the functions
$f_{\pm}$ extend to meromorphic functions on $T_{\pm}=(Z\cap A)\cup
C_{\pm}=(Z\cap A)\amalg C_{\pm}$ which are regular invertible in a
neighborhood of $T^{\prime}=(Z\cap A)\cup E$ by letting $f_{\pm}=1$ on $Z\cap
A$. Moreover, $f_{+}|_{E}=f_{-}|_{E}$, where $E=\nu^{*}(D)$. Let $S_{\pm}$
denote the set of closed points on $C_{\pm}$, where $f_{\pm}$ have poles. We
let $T=T_{+}\cup T_{-}$ and $S=S_{+}\cup S_{-}$. It is clear that $S\cap
D=\emptyset$.

We now choose another very ample line bundle ${\mathcal{M}}$ on $X$ and
$s_{\infty}\in H^{0}(X,{\mathcal{M}})$ (see the proof of Lemma 7.3) such that:

  * i)

$(s_{\infty})$ is integral;

  * ii)

$(s_{\infty})\cap Z$ and $(s_{\infty})\cap(t_{\pm})$ are proper and integral;

  * iii)

$(s_{\infty})\supset S$;

  * iv)

$(s_{\infty})\cap T^{\prime}=\emptyset$;

  * v)

$X\setminus(s_{\infty})$ is affine;

  * vi)

$(s_{\infty})$ is smooth away from $S$;

  * vii)

$(s_{\infty})\cap Z$ is smooth away from $\Delta(C)$;

  * viii)

$(s_{\infty})\cap Z\not\subset\Delta(C)$.

As shown in the proof of Lemma 7.3, it follows from (3), (iv) and (vii) above
that $(s^{Z}_{\infty}):=(s_{\infty})\cap Z$ is smooth along $D$. Using (v), we
can lift $f_{\pm}\in k(T_{\pm})^{\times}$ to regular functions $F_{\pm}$ on
$X\setminus(s_{\infty})$. Using an argument identical to that given in the
proof of Lemma 7.4, we can extend $(s_{0})_{\pm}=s_{\infty}^{N}F_{\pm}$ (for
some $N\gg 0$) to global sections $(s_{0})_{\pm}$ of $\mathcal{M}^{N}$ on $X$
so that:

  * a)

$((s_{0})_{\pm})$ and $((s_{0})_{\pm})\cap Z$ are integral;

  * b)

$((s_{0})_{\pm})\cap T^{\prime}=\emptyset$;

  * c)

$((s_{0})_{\pm})\cap Z\not\subset\Delta(C)$;

  * d)

$((s_{0})_{\pm})\cap Z$ are smooth away from $\Delta(C)$.

As we argued in the proof of Lemma 7.3, it follows from (iv), (vii), (viii),
(c) and (d) that $(s^{Z}_{\infty})$ and
$((s^{Z}_{0})_{\pm}):=((s_{0})_{\pm})\cap Z$ are smooth along $D$.

Setting $H_{\pm}={(s_{0})_{\pm}}/{s^{N}_{\infty}}$ and using the argument of
the proof of Lemma 7.4, we get $H_{\pm}\in k(X)^{\times}$ and they restrict to
rational functions on $C_{\pm}$ which are regular and invertible along $D$.
Moreover, ${H_{+}}/{H_{-}}$ restricts to a rational function on $C_{-}$ which
is regular and invertible along $D$. Since

(7.14) ${H_{+}}_{|E}={F_{+}}_{|E}={f_{+}}_{|E}\ =^{\dagger}\
{f_{-}}_{|E}={F_{-}}_{|E}={H_{-}}_{|E},$

where $\dagger$ follows from (7.8), we have ${H_{+}}/{H_{-}}=1$ on $E$.

We now compute

$\begin{array}[]{lll}\tau^{*}_{X}({\rm div}(f))&=&\iota^{*}_{+}({\rm
div}(f))-\iota^{*}_{-}({\rm div}(f))\\\ &=&{\rm div}(f_{+})-{\rm
div}(f_{-})\\\ &=&\left[((s^{Z}_{0})_{+})\cdot C_{+}-N(s^{Z}_{\infty})\cdot
C_{+}\right]-\left[((s^{Z}_{0})_{-})\cdot C_{-}-N(s^{Z}_{\infty})\cdot
C_{-}\right]\\\ &=&\left[((s^{Z}_{0})_{+})\cdot C_{+}-((s^{Z}_{0})_{+})\cdot
C_{-}\right]+\left[((s^{Z}_{0})_{+})\cdot C_{-}-((s^{Z}_{0})_{-})\cdot
C_{-}\right]\\\ &&-N\left[(s^{Z}_{\infty})\cdot C_{+}-(s^{Z}_{\infty})\cdot
C_{-}\right]\\\
&=&\left[((s^{Z}_{0})_{+})\cdot(C_{+}-C_{-})\right]+\left[C_{-}\cdot(((s^{Z}_{0})_{+})-((s^{Z}_{0})_{-}))\right]-N\left[(s^{Z}_{\infty})\cdot(C_{+}-C_{-})\right]\\\
&{=}^{\dagger}&\left[((s^{Z}_{0})_{+})\cdot((t^{Z}_{+})-(t^{Z}_{-}))\right]+\left[C_{-}\cdot(((s^{Z}_{0})_{+})-((s^{Z}_{0})_{-}))\right]\\\
&&-N\left[(s^{Z}_{\infty})\cdot((t^{Z}_{+})-(t^{Z}_{-}))\right]\\\
&=&((s^{Z}_{0})_{+})\cdot({\rm div}({t^{Z}_{+}}/{t^{Z}_{-}}))-C_{-}\cdot({\rm
div}({(s_{0})_{+}}/{(s_{0})_{-}}))-N(s^{Z}_{\infty})\cdot({\rm
div}({t^{Z}_{+}}/{t^{Z}_{-}}))\\\ &=&((s^{Z}_{0})_{+})\cdot({\rm
div}({t^{Z}_{+}}/{t^{Z}_{-}}))-C_{-}\cdot({\rm
div}({H_{+}}/{H_{-}}))-N(s^{Z}_{\infty})\cdot({\rm
div}({t^{Z}_{+}}/{t^{Z}_{-}})),\end{array}$

where ${=}^{\dagger}$ follows from (7.13).

It follows from (iv) and (b) that ${t^{Z}_{+}}/{t^{Z}_{-}}$ restricts to
regular and invertible functions on $((s^{Z}_{0})_{+})$ and $(s^{Z}_{\infty})$
along $D$. Since $t\in H^{0}(S_{X},{\mathcal{L}})$ and
$t_{\pm}=\iota^{*}_{\pm}(t)\in H^{0}(X,\iota^{*}_{\pm}({\mathcal{L}}))$, it
follows that $(t_{+})_{|D}=\iota^{\prime*}(t)=(t_{-})_{|D}$. In particular,
${({t^{Z}_{+}}/{t^{Z}_{-}})}_{|E}=1$. We have seen before that
${(\frac{H_{-}}{H_{+}})}_{|C_{-}}$ is a regular and invertible function on
$C_{-}$ along $D$ and ${(\frac{H_{+}}{H_{-}})}_{|E}=1$.

We set $h_{1}={(\frac{t^{Z}_{+}}{t^{Z}_{-}})}_{|((s^{Z}_{0})_{+})},\
h_{2}={(\frac{H_{+}}{H_{-}})}_{|C_{-}}$ and
$h_{3}={(\frac{t^{Z}_{+}}{t^{Z}_{-}})}_{|s^{Z}_{\infty}}$. Let
$((s^{Z}_{0})_{+})_{n}\to((s^{Z}_{0})_{+})$, $(C_{-})_{n}\to C_{-}$ and
$(s^{Z}_{\infty})_{n}\to(s^{Z}_{\infty})$ denote the normalization maps. Let
$\nu_{1}\colon((s^{Z}_{0})_{+})_{n}\to X$, $\nu_{2}\colon(C_{-})_{n}\to X$ and
$\nu_{3}\colon(s^{Z}_{\infty})_{n}\to X$ denote the composite maps. The curves
$((s^{Z}_{0})_{+})$ and $(s^{Z}_{\infty})$ are all smooth along $D$, and
$C_{-}$ is smooth along $D$ by Lemma 7.3. Since none of these curves meets $A$
by (iv), (b) and Lemma 7.3, we see that $(\nu_{1})_{*}(h_{1}),\
(\nu_{2})_{*}(h_{2})$ and $(\nu_{3})_{*}(h_{3})$ all die in ${\rm
CH}^{A}_{0}(X|D)$. It follows that $\tau^{*}_{X}({\rm div}(f))$ dies in ${\rm
CH}^{A}_{0}(X|D)$. This finishes the proof. ∎

##  8\. Proof of the moving lemma

In this section, we shall finish the proof of Theorem 7.1. We need to recall
the definition of the Chow group of 0-cycles on singular varieties introduced
by Binda-Krishna [2]. This is an improved version of the Levine-Weibel Chow
group.

Let $k$ be a field and $X$ a reduced quasi-projective scheme of dimension
$d\geq 2$ over $k$. A good curve over $X$ is a reduced scheme $C$ of pure
dimension one together with a finite map $\nu\colon C\to X$ whose image is not
contained in $X_{\rm sing}$ and which is a local complete intersection
(l.c.i.) morphism (see [2, § 2.3] for the definition of such a morphism) over
a neighborhood of $\nu(C)\cap X_{\rm sing}$ in $X$. We let
${\mathcal{R}}_{0}(X)$ be the subgroup of ${\mathcal{Z}}_{0}(X_{\rm reg})$
generated by $\nu_{*}({\rm div}(f))$, where $\nu\colon C\to X$ is a good curve
over $X$ and $f\in{\mathcal{O}}^{\times}_{C,\nu^{-1}(X_{\rm sing})}$. We let
${\rm CH}_{0}(X)={{\mathcal{Z}}_{0}(X)}/{{\mathcal{R}}_{0}(X)}$. One can in
fact consider only those good curves $C$ in this definition which are regular
away from $\nu^{-1}(X_{\rm sing})$ (see [2, Lemma 3.5]). It is known that the
identity map of ${\mathcal{Z}}_{0}(X_{\rm reg})$ induces a surjection ${\rm
CH}^{LW}_{0}(X)\twoheadrightarrow{\rm CH}_{0}(X)$. In certain cases, this map
is known to be an isomorphism if $k$ is algebraically closed. Otherwise, ${\rm
CH}_{0}(X)$ has better behavior than ${\rm CH}^{LW}_{0}(X)$. Later in this
section, we shall also prove a moving lemma for ${\rm CH}_{0}(X)$.

###  8.1. Factorization of $\tau^{*}_{X}$ through ${\rm CH}_{0}(S_{X})$

Let $k$ be a field and $X$ a smooth quasi-projective scheme of pure dimension
$d\geq 2$ over $k$. Let $D\subset X$ be an effective Cartier divisor with
$D^{\prime}=D_{\rm red}$. Let $A\subset D$ be a closed subscheme such that
$\text{\rm dim}(A)\leq d-2$. We shall now show that the map $\tau^{*}_{X}$
that we constructed in Proposition 7.5 actually factors through ${\rm
CH}_{0}(S_{X})$.

Before we do this, recall that if $f\colon{X_{1}}\to{X_{2}}$ is a proper map
and $D_{1}$ (resp. $D_{2}$) is an effective Cartier divisor on ${X_{1}}$
(resp. ${X_{2}}$) such that $f^{*}(D_{2})\leq D_{1}$, then there is a push-
forward map $f_{*}\colon{\rm CH}_{0}({X_{1}}|D_{1})\to{\rm
CH}_{0}({X_{2}}|D_{2})$ (see [4, Lemma 2.7] or [31, Proposition 2.10]). But
the existing proofs of this use a different (but equivalent) definition of the
Chow group of 0-cycles with modulus from the one presented in § 7.1. In
particular, if we are given a closed subscheme $A_{2}\subset D_{2}$ such that
$\text{\rm dim}(A_{2})\leq\text{\rm dim}(X_{2})-2$ and $A_{1}=f^{*}(A_{2})$,
then it is not immediately clear from our definition that the push-forward map
$f_{*}\colon{\rm CH}^{A_{1}}_{0}(X_{1}|D_{1})\to{\rm
CH}^{A_{2}}_{0}(X_{2}|D_{2})$ exists. The next lemma shows that this map
actually exists.

######  Lemma 8.1.

The map $f_{*}\colon{\mathcal{Z}}_{0}(X_{1}\setminus
D_{1})\to{\mathcal{Z}}_{0}(X_{2}\setminus D_{2})$ induces a push-forward map

$f_{*}\colon{\rm CH}^{A_{1}}_{0}(X_{1}|D_{1})\to{\rm
CH}^{A_{2}}_{0}(X_{2}|D_{2}).$

###### Proof.

It suffices to consider the case when $D_{1}=f^{*}(D_{2})$. Let $Y_{1}\subset
X_{1}$ be an integral curve not contained in $D_{1}$ and not meeting $A_{1}$.
Let $\nu_{1}\colon(Y_{1})_{n}\to X_{1}$ be the induced finite map from the
normalization of $Y_{1}$. Let $Y_{2}=f(Y_{1})$. As $f$ is proper,
$Y_{2}\subset X_{2}$ is closed and $Y_{2}\cap A=\emptyset$. If $Y_{2}$ is a
closed point, the proof is straightforward. So we assume $Y_{2}$ is an
integral curve. Then it can not be contained in $D_{2}$. We let
$\nu_{2}\colon(Y_{2})_{n}\to X_{2}$ be the induced map from the normalization
of $Y_{2}$. This gives rise to a finite dominant map
$f^{\prime}\colon(Y_{1})_{n}\to(Y_{2})_{n}$.

We let $g\in
K^{M}_{1}({\mathcal{O}}_{(Y_{1})_{n},\nu^{-1}_{1}(D_{1})},I_{D_{1}})$. We know
from [10, Chapter 1] that $f_{*}({\rm div}(g))={\rm div}(N(g))$, where
$N\colon K^{M}_{1}(k(Y_{1}))\to K^{M}_{1}(k(Y_{2}))$ is the norm map. So all
we need to show to finish the proof is that

$N(K^{M}_{1}({\mathcal{O}}_{(Y_{1})_{n},\nu^{-1}_{1}(D_{1})},I_{D_{1}}))\subset
K^{M}_{1}({\mathcal{O}}_{(Y_{2})_{n},\nu^{-1}_{1}(D_{2})},I_{D_{2}}).$

But this is well known (e.g., see [37, Lemma 6.19]) since
$I_{D_{1}}=f^{*}(I_{D_{2}})$. ∎

Using Lemma 8.1, we can prove the following result which will be used later in
this section. This shows that ${\rm CH}^{A}_{0}(X|D)$ can also be defined in
the style of the definition of ${\rm CH}_{0}(X|D)$ given in [4]. For an
integral curve $C\subset{\mathbb{P}}^{1}_{X}$ and a point
$t\in{\mathbb{P}}^{1}_{k}(k)$ such that $C\not\subset(X\times\\{t\\})$, we let
$[C_{t}]=\pi_{*}([C]\cdot(X\times\\{t\\}))$, where
$\pi\colon{\mathbb{P}}^{1}_{X}\to X$ is the projection and
$[C]\cdot(X\times\\{t\\})$ is the 0-cycle associated to the scheme theoretic
intersection of $C$ and $X\times\\{t\\}$.

######  Lemma 8.2.

Let $X$ and $D$ be as above. Let
${\mathcal{R}}^{\prime}_{0}(X|D)\subset{\mathcal{Z}}_{0}(X\setminus D)$ be the
subgroup generated by the 0-cycles $[C_{0}]-[C_{\infty}]$, where
$C\subset{\mathbb{P}}^{1}_{X}$ is an integral curve satisfying the following
properties:

  1. (1)

$C\cap{\mathbb{P}}^{1}_{D}$ is finite;

  2. (2)

$C\cap(D\times_{k}\\{0,\infty\\})=\emptyset$;

  3. (3)

$C\cap{\mathbb{P}}^{1}_{A}=\emptyset$;

  4. (4)

The Weil divisor $\nu^{*}(X\times\\{1\\})-\nu^{*}({\mathbb{P}}^{1}_{D})$ is
effective, where $\nu:C_{n}\to C\hookrightarrow{\mathbb{P}}^{1}_{X}$ is the
composite finite map.

Then ${\rm CH}^{A}_{0}(X|D)={\rm
Coker}({\mathcal{R}}^{\prime}_{0}(X|D)\to{\mathcal{Z}}_{0}(X\setminus D))$.

###### Proof.

Let ${\mathcal{R}}^{A}_{0}(X|D)={\rm Ker}({\mathcal{Z}}_{0}(X\setminus
D)\twoheadrightarrow{\rm CH}^{A}(X|D))$. Let $C\subset X$ be an integral curve
not contained in $D$ and not meeting $A$. Let $\nu\colon C_{n}\to X$ be the
induced map and let $f\in K^{M}_{1}({\mathcal{O}}_{C_{n},\nu^{-1}(D)},I_{D})$.
Taking the closure $\Gamma_{f}$ of the graph of $f\colon
C\dasharrow{\mathbb{P}}^{1}_{k}$ in ${\mathbb{P}}^{1}_{X}$, one easily sees
that ${\rm div}(f)\in{\mathcal{R}}^{\prime}_{0}(X|D)$.

Conversely, suppose $C\hookrightarrow{\mathbb{P}}^{1}_{X}$ is an integral
curve defining an element in ${\mathcal{R}}^{\prime}_{0}(X|D)$. Let $\pi\colon
C_{n}\to{\mathbb{P}}^{1}_{X}\to X$ be the composite map. The projection
$C\to{\mathbb{P}}^{1}_{k}$ defines an element $g\in k(C)^{\times}$ which lies
in $K^{M}_{1}({\mathcal{O}}_{C_{n},\pi^{-1}(D)},I_{D})$ by (1) - (4). In
particular, ${\rm
div}(g)\in{\mathcal{R}}_{0}(C_{n}|\pi^{*}(D))={\mathcal{R}}^{\pi^{-1}(A)}_{0}(C_{n}|\pi^{*}(D))$.

We let $C^{\prime}$ be the image of $C$ under the projection to $X$. If
$C^{\prime}$ is a closed point, the proof is straightforward. So we can assume
that $C^{\prime}$ is a curve, not contained in $D$ and not meeting $A$ by (1)
and (3). Moreover, projection to $X$ defines a finite dominant map $C\to
C^{\prime}$. This in turn induces a finite dominant map $f^{\prime}\colon
C_{n}\to C^{\prime}_{n}$. Furthermore, $\pi$ is same as the composition
$C_{n}\xrightarrow{f^{\prime}}C^{\prime}_{n}\xrightarrow{\nu}X$. Since
$[C_{0}]-[C_{\infty}]=\nu_{*}\circ f^{\prime}_{*}({\rm div}(g))=(\nu\circ
f^{\prime})_{*}({\rm div}(g))$, it suffices therefore to show that
$\pi_{*}({\rm div}(g))=(\nu\circ f^{\prime})_{*}({\rm
div}(g))\in{\mathcal{R}}^{A}_{0}(X|D)$. But this follows directly from Lemma
8.1. ∎

######  Lemma 8.3.

Assume that for every integer $n\geq 0$, the map
$\tau^{*}_{{\mathbb{P}}^{n}_{X}}$ descends to a well defined homomorphism

$\tau^{*}_{{\mathbb{P}}^{n}_{X}}\colon{\rm
CH}_{0}^{LW}(S_{{\mathbb{P}}^{n}_{X}})\to{\rm
CH}^{{\mathbb{P}}^{n}_{A}}_{0}({\mathbb{P}}^{n}_{X}|{\mathbb{P}}^{n}_{D}).$

Then the map $\tau^{*}_{X}\colon{\rm CH}_{0}^{LW}(S_{X})\to{\rm
CH}^{A}_{0}(X|D)$ factors through ${\rm CH}_{0}(S_{X})$, giving a well defined
homomorphism

$\tau_{X}^{*}\colon{\rm CH}_{0}(S_{X})\to{\rm CH}^{A}_{0}(X|D).$

###### Proof.

Let $\delta\colon{\mathcal{Z}}_{0}(S_{X}\setminus D)\to{\rm CH}^{A}_{0}(X|D)$
be the composition ${\mathcal{Z}}_{0}(S_{X}\setminus D)\to{\rm
CH}_{0}^{LW}(S_{X})\xrightarrow{\tau^{*}_{X}}{\rm CH}^{A}_{0}(X|D)$. We have
to show that $\delta$ factors through ${\rm CH}_{0}(S_{X})$. Using [2, Lemma
3.5], we have to show more precisely that $\delta(\nu_{*}({\rm div}(f)))=0$
for every finite l.c.i. morphism $\nu\colon C\to S_{X}$ from a reduced curve
$C$ whose image is not contained in $D$ and for every rational function $f$ on
$C$ that is regular and invertible along $\nu^{-1}(D)$.

Since $\nu$ is a finite l.c.i. morphism, we can factor it as a composition
$\nu=\pi\circ\mu$, where $\mu\colon
C\hookrightarrow{\mathbb{P}}^{n}_{S_{X}}=S_{{\mathbb{P}}^{n}_{X}}$ (using [2,
Proposition 2.3]) is a regular embedding (see [7, Lemma 37.55.3]) and
$\pi\colon{\mathbb{P}}^{n}_{S_{X}}\to S_{X}$ is the projection. In particular,
$\mu(C)=C$ is a Cartier curve on the double $S_{{\mathbb{P}}^{n}_{X}}$. We let
$Y=S_{{\mathbb{P}}^{n}_{X}}$ and $E={\mathbb{P}}^{n}_{D}$. It is then clear
that $\pi^{-1}(A)={\mathbb{P}}^{n}_{A}$ is a closed subscheme of $Y_{\rm
sing}$ and $\text{\rm dim}({\mathbb{P}}^{n}_{A})\leq\text{\rm dim}(Y)-2$.

It follows from the commutative diagram

(8.1) $\textstyle{{\mathcal{Z}}_{0}(Y\setminus
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\pm}^{*}}$$\scriptstyle{\pi_{*}}$$\textstyle{{\mathcal{Z}}_{0}({\mathbb{P}}^{n}_{X}\setminus
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\textstyle{{\mathcal{Z}}_{0}(S_{X}\setminus
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{*}_{\pm}}$$\textstyle{{\mathcal{Z}}_{0}(X\setminus
D)}$

and the formula $\delta=\iota^{*}_{+}-\iota^{*}_{-}$ that the square

$\textstyle{{\mathcal{Z}}_{0}(Y\setminus
E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{Y}}$$\scriptstyle{\pi_{*}}$$\textstyle{{\rm
CH}^{{\mathbb{P}}^{n}_{A}}_{0}({\mathbb{P}}^{n}_{X}|E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}}$$\textstyle{{\mathcal{Z}}_{0}(S_{X}\setminus
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{{\rm
CH}^{A}_{0}(X|D)}$

commutes. That is, $\delta(\nu_{*}({\rm div}(f)))=\delta(\pi_{*}(\mu_{*}({\rm
div}(f))))=\pi_{*}(\delta_{Y}(\mu_{*}({\rm div}(f))))$. Note that the push-
forward map $\pi_{*}$ on the right exists by Lemma 8.1.

By assumption, we have $\delta_{Y}(\mu_{*}({\rm div}(f)))=0\in{\rm
CH}^{{\mathbb{P}}^{n}_{A}}_{0}({\mathbb{P}}^{n}_{X}|E)$, where $\delta_{Y}$ is
the composition ${\mathcal{Z}}_{0}(Y\setminus E)\to{\rm
CH}_{0}^{LW}(Y)\xrightarrow{\tau^{*}_{{\mathbb{P}}^{n}_{X}}}{\rm
CH}^{{\mathbb{P}}^{n}_{A}}_{0}({\mathbb{P}}^{n}_{X}|E)$. Since the push-
forward map $\pi_{*}\colon{\rm
CH}^{{\mathbb{P}}^{n}_{A}}_{0}({\mathbb{P}}^{n}_{X}|E)\to{\rm
CH}^{A}_{0}(X|D)$ is well defined as we saw above, we are done. ∎

######  Lemma 8.4.

Let $k$ be an infinite perfect field. Then $\tau^{*}_{X}$ factorizes through a
homomorphism

$\tau^{*}_{X}\colon{\rm CH}_{0}(S_{X})\to{\rm CH}^{A}_{0}(X|D).$

###### Proof.

Combine Proposition 7.5 and Lemma 8.3. ∎

###  8.2. Behavior of ${\rm CH}^{A}_{0}(X|D)$ under the change of base field

Let $k$ now be any field and $X$ a smooth quasi-projective scheme of pure
dimension $d\geq 2$ over $k$. Let $D\subset X$ be an effective Cartier
divisor. Let $A\subset D$ be a closed subscheme such that $\text{\rm
dim}(A)\leq d-2$. We shall need the following base change property of ${\rm
CH}^{A}_{0}(X|D)$.

######  Proposition 8.5.

Let $k\hookrightarrow k^{\prime}$ be a separable algebraic (possibly infinite)
extension of fields. Let $X^{\prime}=X_{k^{\prime}},\
D^{\prime}=D_{k^{\prime}}$ and $A^{\prime}=A_{k^{\prime}}$ denote the base
change of $X,D$ and $A$, respectively. Let ${\rm
pr}_{{k^{\prime}}/{k}}:X^{\prime}\to X$ be the projection map. Then the
following hold.

  1. (1)

There exists a pull-back ${\rm pr}^{*}_{{k^{\prime}}/{k}}:{\rm
CH}^{A}_{0}(X|D)\to{\rm CH}^{A^{\prime}}_{0}(X^{\prime}|D^{\prime})$.

  2. (2)

If there exists a sequence of separable field extensions $k=k_{0}\subset
k_{1}\subset\cdots\subset k^{\prime}$ with $k^{\prime}=\cup_{i}k_{i}$, then we
have ${\underset{i}{\varinjlim}}\ {\rm
CH}^{A_{k_{i}}}_{0}(X_{k_{i}}|D_{k_{i}})\xrightarrow{\simeq}{\rm
CH}^{A^{\prime}}_{0}(X^{\prime}|D^{\prime})$.

  3. (3)

If $k\hookrightarrow k^{\prime}$ is finite, then there exists a push-forward
${\rm pr}_{{k^{\prime}}/{k}\ *}:{\rm
CH}^{A^{\prime}}_{0}(X^{\prime}|D^{\prime})\to{\rm CH}^{A}_{0}(X|D)$ such that
${\rm pr}_{{k^{\prime}}/{k}\ *}\circ{\rm pr}^{*}_{{k^{\prime}}/{k}}$ is
multiplication by $[k^{\prime}:k]$.

###### Proof.

Let $x\in X\setminus D$ be a closed point. Since ${\rm pr}_{{k^{\prime}}/{k}}$
is smooth (ind-smooth to be precise), it follows from our hypothesis that
${\rm pr}^{*}_{{k^{\prime}}/{k}}([x])$ is a well defined 0-cycle in
${\mathcal{Z}}_{0}(X^{\prime}|D^{\prime})$. We thus have a pull-back map ${\rm
pr}^{*}_{{k^{\prime}}/{k}}:{\mathcal{Z}}_{0}(X\setminus
D)\to{\mathcal{Z}}_{0}(X^{\prime}\setminus D^{\prime})$. Let
${\mathcal{R}}^{\prime}_{0}(X|D)\subset{\mathcal{Z}}_{0}(X\setminus D)$ be as
in Lemma 8.2. To show that ${\rm pr}^{*}_{{k^{\prime}}/{k}}$ preserves
rational equivalence, it suffices to show that it takes
${\mathcal{R}}^{\prime}_{0}(X|D)$ to
${\mathcal{R}}^{\prime}_{0}(X^{\prime}|D^{\prime})$ by Lemma 8.2.

Let $C\subset X\times_{k}{\mathbb{P}}^{1}_{k}$ be an integral curve as in
Lemma 8.2. It follows from the smoothness of ${\rm pr}_{{k^{\prime}}/{k}}$
that $C^{\prime}={\rm
pr}^{*}_{{k^{\prime}}/{k}}(C)=C_{k^{\prime}}\hookrightarrow(X\times_{k}{\mathbb{P}}^{1}_{k})_{k^{\prime}}=X^{\prime}\times_{k^{\prime}}{\mathbb{P}}^{1}_{k^{\prime}}$
is a reduced curve whose irreducible components
$C^{\prime}_{1},\ldots,C^{\prime}_{r}$ satisfy conditions (1)-(4) of Lemma
8.2. Furthermore, the flat pull-back property of Bloch’s cycle complex (see
[6, Proposition 1.3]) says that

${\rm
pr}^{*}_{{k^{\prime}}/{k}}([C_{0}]-[C_{\infty}])=[C^{\prime}_{0}]-[C^{\prime}_{\infty}]=\
\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}([(C^{\prime}_{i})_{0}]-[(C^{\prime}_{i})_{\infty}]).$

In particular, ${\rm pr}^{*}_{{k^{\prime}}/{k}}([C_{0}]-[C_{\infty}])$ dies in
${\rm CH}^{A^{\prime}}_{0}(X^{\prime}|D^{\prime})$. This proves (1).

It is clear that the map ${\underset{i}{\varinjlim}}\ {\rm
CH}^{A_{i}}_{0}(X_{i}|D_{i})\to{\rm
CH}^{A^{\prime}}_{0}(X^{\prime}|D^{\prime})$ is surjective. To show
injectivity, suppose there is some $i\geq 0$ and
$\alpha\in{\mathcal{Z}}_{0}(X_{i}|D_{i})$ such that ${\rm
pr}^{*}_{{k^{\prime}}/{k_{i}}}(\alpha)\in{\mathcal{R}}^{\prime}_{0}(X^{\prime}|D^{\prime})$.
We can replace $k$ by $k_{i}$ and assume $i=0$.

Let $C^{j}\hookrightarrow
X^{\prime}\times_{k^{\prime}}{\mathbb{P}}^{1}_{k^{\prime}}=(X\times_{k}{\mathbb{P}}^{1}_{k})_{k^{\prime}}$
for $j=1,\cdots,r$ be a collection of integral curves as in the proof of (1)
so that ${\rm pr}^{*}_{{k^{\prime}}/{k}}(\alpha)=(\stackrel{{\scriptstyle
r}}{{\underset{j=1}{\sum}}}\ [C^{j}_{0}])-(\stackrel{{\scriptstyle
r}}{{\underset{j=1}{\sum}}}\ [C^{j}_{\infty}])$. Let $\nu^{j}:C^{j}_{n}\to
X^{\prime}\times_{k^{\prime}}{\mathbb{P}}^{1}_{k^{\prime}}$ denote the maps
from the normalizations of the above curves.

We can then find some $i\gg 0$ and integral curves $W^{j}\hookrightarrow
X_{i}\times_{k_{i}}{\mathbb{P}}^{1}_{k_{i}}$ such that
$C^{j}=W^{j}\times_{k_{i}}k^{\prime}$ for each $j=1,\cdots,r$. In particular,
we have $C^{j}_{0}={\rm pr}^{*}_{{k^{\prime}}/{k_{i}}}(W^{j}_{0})$ and
$C^{j}_{\infty}={\rm pr}^{*}_{{k^{\prime}}/{k_{i}}}(W^{j}_{\infty})$ for
$j=1,\cdots,r$. Since ${\rm pr}_{{k^{\prime}}/{k_{i}}}$ is smooth, it follows
that $C^{j}_{n}=(W^{j}_{n})_{k^{\prime}}$ for each $j$. Moreover, it follows
from [31, Lemma 2.2] that condition (4) above holds on each $W^{j}_{n}$. It
follows that each $W^{j}$ defines a rational equivalence for 0-cycles with
modulus $D_{i}$ on $X_{i}$.

We now set $\alpha_{i}={\rm pr}^{*}_{{k_{i}}/{k}}(\alpha)$ and let
$\beta=\alpha_{i}-(\stackrel{{\scriptstyle
r}}{{\underset{j=1}{\sum}}}([W^{j}_{0}]-[W^{j}_{\infty}]))\in{\mathcal{Z}}_{0}(X_{i}\setminus
D_{i})$. It then follows that ${\rm pr}^{*}_{{k^{\prime}}/{k_{i}}}(\beta)={\rm
pr}^{*}_{{k^{\prime}}/{k_{i}}}(\alpha_{i})-\stackrel{{\scriptstyle
r}}{{\underset{j=1}{\sum}}}([C^{j}_{0}]-[C^{j}_{\infty}])={\rm
pr}^{*}_{{k^{\prime}}/{k}}(\alpha)-\stackrel{{\scriptstyle
r}}{{\underset{j=1}{\sum}}}([C^{j}_{0}]-[C^{j}_{\infty}])=0$ in
${\mathcal{Z}}_{0}(X^{\prime}\setminus D^{\prime})$. Since the map ${\rm
pr}^{*}_{{k^{\prime}}/{k_{i}}}:{\mathcal{Z}}_{0}(X_{i}\setminus
D_{i})\to{\mathcal{Z}}_{0}(X^{\prime}\setminus D^{\prime})$ of free abelian
groups is clearly injective, we get $\beta=0$, which means that
$\alpha_{i}\in{\mathcal{R}}_{0}(X_{i}|D_{i})$. This proves (2). The existence
of push-forward follows from Lemma 8.1 and the formula ${\rm
pr}_{{k^{\prime}}/{k}\ *}\circ{\rm pr}^{*}_{{k^{\prime}}/{k}}=[k^{\prime}:k]$
is obvious from the definitions. This proves (3). ∎

###  8.3. Proof of Theorem 7.1

We shall now prove Theorem 7.1. This is equivalent to showing that the
canonical surjection ${\rm CH}^{A}_{0}(X|D)\twoheadrightarrow{\rm
CH}_{0}(X|D)$ is also a monomorphism. Using Proposition 8.5 and the standard
pro-$\ell$ extension trick, we easily reduce to the case when $k$ is infinite
and perfect. We assume this to be the case in the rest of the proof.

By Lemma 8.4, the map
$(\iota^{*}_{+}-\iota^{*}_{-})\colon{\mathcal{Z}}_{0}(S_{X}\setminus
D)\to{\mathcal{Z}}_{0}(X\setminus D)$ defines a homomorphism
$\tau^{*}_{X}\colon{\rm CH}_{0}(S_{X})\to{\rm CH}^{A}_{0}(X|D)$. We now define
two maps in opposite direction,

$p_{\pm,*}\colon{\mathcal{Z}}_{0}(X\setminus
D)\rightrightarrows{\mathcal{Z}}_{0}(S_{X}\setminus D)$

by $p_{+,*}([x])=\iota_{+,*}([x])$ (resp. by $p_{-,*}([x])=\iota_{-,*}([x])$)
for a closed point $x\in X\setminus D$. Concretely, the two maps $p_{+,*}$ and
$p_{-,*}$ copy a cycle $\alpha$ in one of the two components of the double
$S_{X}$ (the $X_{+}$ or the $X_{-}$ copy). Since $\alpha$ is supported outside
$D$ (by definition of ${\mathcal{R}}_{0}(X|D)$), the cycles $p_{+,*}(\alpha)$
and $p_{-,*}(\alpha)$ give classes in ${\rm CH}_{0}(S_{X})$. By [2,
Proposition 5.9], the maps $p_{\pm,*}$ descend to group homomorphisms
$p_{\pm,*}\colon{\rm CH}_{0}(X|D)\to{\rm CH}_{0}(S_{X})$. Composing with the
canonical surjection ${\rm CH}^{A}_{0}(X|D)\twoheadrightarrow{\rm
CH}_{0}(X|D)$, we get maps $p^{A}_{\pm,*}\colon{\rm CH}^{A}_{0}(X|D)\to{\rm
CH}_{0}(S_{X})$. Furthermore, it is clear from the definitions of
$p^{A}_{\pm,*}$ and $\tau^{*}_{X}$ that $\tau^{*}_{X}\circ p^{A}_{\pm,*}={\rm
Id}$. We thus get maps

${\rm CH}^{A}_{0}(X|D)\twoheadrightarrow{\rm
CH}_{0}(X|D)\xrightarrow{p_{\pm,*}}{\rm
CH}_{0}(S_{X})\xrightarrow{\tau^{*}_{X}}{\rm CH}^{A}_{0}(X|D),$

whose composition is identity. It follows that the first arrow from the left
is injective. This finishes the proof. ∎

###  8.4. Moving lemma for ${\rm CH}_{0}(X)$

We shall end this section with a moving lemma for the Chow group of 0-cycles
${\rm CH}_{0}(X)$ for a singular scheme $X$. This result is of independent
interest in the study of 0-cycles and algebraic $K$-theory on singular
varieties. We remark that this moving lemma for ${\rm CH}^{LW}_{0}(X)$ is yet
unknown over finite fields.

Let $k$ be a field and $X$ a reduced quasi-projective scheme of dimension
$d\geq 2$ over $k$. Suppose that $A\subset X_{\rm sing}$ is a closed subscheme
such that $\text{\rm dim}(A)\leq d-2$. We can then define ${\rm
CH}^{A}_{0}(X)$ by repeating the definition of ${\rm CH}_{0}(X)$, except that
we allow only those good curves $\nu\colon C\to X$ which satisfy the
additional condition that $\nu(C)\cap A=\emptyset$. We clearly have a
canonical surjection ${\rm CH}^{A}_{0}(X)\twoheadrightarrow{\rm CH}_{0}(X)$.

######  Proposition 8.6.

Let $k$ be any perfect field. Then the map ${\rm CH}^{A}_{0}(X)\to{\rm
CH}_{0}(X)$ is an isomorphism.

###### Proof.

We only need to show injectivity. To reduce it to the case when $k$ is
infinite, we note that an analogue of Proposition 8.5 is proven for ${\rm
CH}_{0}(X)$ in [2, Proposition 6.1]. The reader may check that this proof
works verbatim for ${\rm CH}^{A}_{0}(X)$. This allows us to use the pro-$\ell$
extension trick as we did in Theorem 7.1 to reduce the proof of the
proposition when $k$ is infinite and perfect.

Let $\nu\colon C\to X$ be a good curve over $X$ and let $f$ be a rational
function on $C$ which is regular and invertible in a neighborhood of
$\nu^{-1}(X_{\rm sing})$. It suffices to show that $\nu_{*}({\rm div}(f))$
belongs to the subgroup ${\mathcal{R}}^{A}_{0}(X)$ of rational equivalences
that define ${\rm CH}^{A}_{0}(X)$.

First of all, we can assume by [2, Lemma 3.5] that $\nu\colon C\to X$ is an
l.c.i. morphism. As a consequence, we have a factorization $\nu=\pi\circ\mu$,
where $\mu\colon C\hookrightarrow{\mathbb{P}}^{n}_{X}$ is a regular embedding
and $\pi\colon{\mathbb{P}}^{n}_{X}\to X$ is the projection. In particular,
$\mu(C)=C$ is a Cartier curve on ${\mathbb{P}}^{n}_{X}$. The smoothness of
$\pi$ implies that $({\mathbb{P}}^{n}_{X})_{\rm sing}={\mathbb{P}}^{n}_{X_{\rm
sing}}$. Furthermore, $\pi^{-1}(A)={\mathbb{P}}^{n}_{A}$ is a closed subscheme
of $({\mathbb{P}}^{n}_{X})_{\rm sing}$ and $\text{\rm
dim}({\mathbb{P}}^{n}_{A})\leq\text{\rm dim}({\mathbb{P}}^{n}_{X})-2$. It is
also clear that $\nu_{*}({\rm div}(f))=\pi_{*}(\mu_{*}({\rm div}(f)))$.

We can now apply [8, Lemma 1.3] to find a reduced Cartier curve
$\mu^{\prime}\colon C^{\prime}\hookrightarrow{\mathbb{P}}^{n}_{X}$ such that
$C^{\prime}\cap{\mathbb{P}}^{n}_{A}=\emptyset$, and a rational function
$f^{\prime}$ on $C^{\prime}$ which is regular and invertible along
$C^{\prime}\cap{\mathbb{P}}^{n}_{X_{\rm sing}}$ and $\mu_{*}({\rm
div}(f))={\rm div}(f^{\prime})$. This can also be easily deduced from the
proof of Proposition 7.5. We let $\nu^{\prime}\colon
C^{\prime}\xrightarrow{\mu^{\prime}}{\mathbb{P}}^{n}_{X}\xrightarrow{\pi}X$
denote the composite map. Since $\mu^{\prime}$ is a regular immersion along
${\mathbb{P}}^{n}_{X_{\rm sing}}$, it follows that $\nu^{\prime}$ is an l.c.i.
morphism along $X_{\rm sing}$. In particular, $\nu^{\prime}\colon
C^{\prime}\to X$ is a good curve over $X$. Furthermore, we have
$\nu^{\prime}(C^{\prime})\cap
A=\pi(C^{\prime}\cap{\mathbb{P}}^{n}_{A})=\emptyset$. Since
$\nu^{\prime}_{*}({\rm div}(f^{\prime}))=\pi_{*}({\rm
div}(f^{\prime}))=\pi_{*}\circ\mu_{*}({\rm div}(f))=\nu_{*}({\rm div}(f))$, it
follows that ${\rm div}(f)$ dies in ${\rm CH}^{A}_{0}(X)$. This finishes the
proof. ∎

##  9\. Comparison of the idele class groups

We shall work under the following assumptions in this section. We let $k$ be a
finite field and $X$ a normal projective integral scheme of dimension $d\geq
1$ over $k$. Let $C\subset X$ be a reduced closed subscheme of pure
codimension one whose complement $U$ is regular. We let $A=C_{\rm sing}$ with
reduced induced closed subscheme structure. Note that $\text{\rm dim}(A)\leq
d-2$. In particular, $A=\emptyset$ when $d=1$. Recall that ${\rm Div}_{C}(X)$
is the directed set of closed subschemes of $X$ with support $C$. Let $K$
denote the function field of $X$. We fix a separable closure $\overline{K}$ of
$K$ and let $K^{\rm ab}\subset\overline{K}$ be the maximal abelian extension
of $K$.

We fix a closed subscheme $D\in{\rm Div}_{C}(X)$. If $x\in U$ is a closed
point, then it defines a Parshin chain on $(U\subset X)$ of length zero.
Hence, we have the injection ${\mathbb{Z}}=K^{M}_{0}(k(x))\hookrightarrow
I_{U/X}$. Extending it linearly over $U_{(0)}$, we see that there is canonical
inclusion ${\mathcal{Z}}_{0}(U)\hookrightarrow I_{U/X}$. Composing with the
canonical maps $I_{U/X}\twoheadrightarrow C_{U/X}\to\widetilde{C}_{U/X}$, we
get a map

(9.1) ${\operatorname{\rm
cyc}}_{U/X}\colon{\mathcal{Z}}_{0}(U)\to\widetilde{C}_{U/X}.$

Composing with the surjection $\widetilde{C}_{U/X}\twoheadrightarrow C(X,D)$,
we get our cycle class map

(9.2) ${\operatorname{\rm cyc}}_{X|D}\colon{\mathcal{Z}}_{0}(U)\to C(X,D).$

Recall that Bloch’s formula for the 0-cycles with modulus asks whether the
following hold if $D$ is an effective Cartier divisor.

  1. (1)

${\operatorname{\rm cyc}}_{X|D}$ annihilates the subgroup of rational
equivalences.

  2. (2)

The resulting map ${\operatorname{\rm cyc}}_{X|D}\colon{\rm CH}_{0}(X|D)\to
C(X,D)$ is an isomorphism.

These conditions together are equivalent to asking whether the idele class
groups ${\rm CH}_{0}(X|D)$ and $C_{KS}(X,D)$ are isomorphic. The goal of this
section is to prove this to be the case when $X$ is regular.

###  9.1. Reciprocity maps for 0-cycles

At any rate, we see using (7.4) and [16, Proposition 4.8, Lemma 8.4] that we
have a commutative diagram of short exact sequences of topological abelian
groups

(9.3)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}_{0}(U)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}_{0}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg}/n}$$\scriptstyle{{\operatorname{\rm
cyc}}_{X|D}}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(X,D)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg}/n}$$\scriptstyle{\rho_{X|D}}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{{\rm
adiv}}_{1}(X,D)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{{\rm
adiv}}_{1}(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg}/n}$$\textstyle{\widehat{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $n\geq 1$ is an integer depending only on $U$.

We also have a commutative diagram of short exact sequences of topological
abelian groups (see § 6 for the definition of $D_{\rho}$)

(9.4)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}_{0}(U)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}_{0}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg}/n}$$\scriptstyle{{\operatorname{\rm
cyc}}_{X|D_{\rho}}}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(X,D_{\rho})_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(X,D_{\rho})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg}/n}$$\scriptstyle{\rho^{c}_{X|D}}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{\rm
ab}_{1}(X,D)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{\rm
ab}_{1}(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg}/n}$$\textstyle{\widehat{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Let $\vartheta_{U/X}=\widetilde{\rho}_{U/X}\circ{\operatorname{\rm
cyc}}_{U/X},\ \vartheta_{X|D}=\rho_{X|D}\circ{\operatorname{\rm cyc}}_{X|D}$
and $\vartheta^{c}_{X|D}=\rho^{c}_{X|D}\circ{\operatorname{\rm
cyc}}_{X|{D_{\rho}}}$. It is then clear from the definition of the reciprocity
map $\rho_{U/X}$ (see § 5.2) and the cycle class map ${\operatorname{\rm
cyc}}_{U/X}$ that the composition $\vartheta_{U/X}$ is the Frobenius
substitution which takes a closed point $x\in U$ to the image of the Frobenius
automorphism under the canonical map ${\rm
Gal}({\overline{k(x)}}/{k(x)})\to\pi^{\rm ab}_{1}(U)$. Hence, the same holds
for $\rho_{X|D}$ and $\vartheta^{c}_{X|D}$.

It follows from the classical ramified class field theory for curves that for
every effective Cartier divisor $D\in{\rm Div}_{C}(X)$, the map
$\vartheta^{c}_{X|D}$ factors through rational equivalences and defines a map
(obviously continuous) $\vartheta^{c}_{X|D}\colon{\rm CH}_{0}(X|D)\to\pi^{\rm
ab}_{1}(X,D)$ (see [27, Proposition 3.2] and [3, § 8]). Taking the limit and
using Proposition 5.4, we see that $\vartheta_{U/X}$ descends to a continuous
homomorphism

(9.5) $\vartheta_{U/X}\colon C(U)\to\pi^{\rm ab}_{1}(U).$

It follows from the generalized Chebotarev-Lang density theorem (e.g., see
[41, Theorem 5.8.16]) that this map has dense image.

###  9.2. Bloch’s formula for regular schemes

Throughout this subsection, we shall assume that $X$ is regular. Note that
every $D\in{\rm Div}_{C}(X)$ is then an effective Cartier divisor on $X$.
Under the extra assumption, we would like to work with a new subgroup of
$H^{1}(K)$ which we introduce below.

######  Definition 9.1.

Let ${\rm fil}^{A}_{D}H^{1}(K)$ be the subgroup of characters $\chi\in
H^{1}(U)$ such that for every integral curve $Y\subset X$ not contained in $C$
and not meeting $A$, the finite map $\nu\colon Y_{n}\to X$ has the property
that the image of $\chi$ under $\nu^{*}\colon H^{1}(U)\to H^{1}(\nu^{-1}(U))$
lies in ${\rm fil}_{\nu^{*}(D)}H^{1}(k(Y))$, where $Y_{n}$ is the
normalization of $Y$ (see Definition 5.1). We let $\pi^{A}_{1}(X,D)$ denote
the Pontryagin dual of ${\rm fil}^{A}_{D}H^{1}(K)$ and let
$\pi^{A}_{1}(X,D)_{0}$ be the kernel of the map $\pi^{A}_{1}(X,D)\to{\rm
Gal}({\overline{k}}/k)$.

We have the canonical inclusions

(9.6) ${\rm fil}^{c}_{D}H^{1}(K)\subset{\rm fil}^{A}_{D}H^{1}(K)\subset
H^{1}(U)\subset H^{1}(K).$

These give rise to the surjective continuous homomorphisms of profinite groups

(9.7) ${\rm Gal}({K^{\rm ab}}/K)\twoheadrightarrow\pi^{\rm
ab}_{1}(U)\twoheadrightarrow\pi^{A}_{1}(X,D)\twoheadrightarrow\pi^{\rm
ab}_{1}(X,D).$

######  Lemma 9.2.

The composite map $C(U)\xrightarrow{\vartheta_{U/X}}\pi^{\rm
ab}_{1}(U)\twoheadrightarrow\pi^{A}_{1}(X,D)$ factors through a continuous
homomorphism

$\vartheta^{A}_{X|D}\colon{\rm CH}^{A}(X|D)\to\pi^{A}_{1}(X,D).$

###### Proof.

Since $C(U)$ is the limit of the pro-abelian group $\\{{\rm
CH}^{A}_{0}(X|D)\\}_{D\in{\rm Div}_{C}(X)}$ by Theorem 7.1, it suffices to
show that the composite map ${\mathcal{Z}}_{0}(U)\to\pi^{\rm
ab}_{1}(U)\twoheadrightarrow\pi^{A}_{1}(X,D)$ kills
${\mathcal{R}}^{A}_{0}(X|D)$. Equivalently, we need to show that for every
character $\chi\in{\rm fil}^{A}_{D}H^{1}(K)$, the induced character
$\vartheta^{*}_{U/X}(\chi)$ annihilates ${\mathcal{R}}^{A}_{0}(X|D)$.

Let $Y\subset X$ be an integral curve not contained in $D$ and not meeting $A$
and let $\nu\colon Y_{n}\to X$ be the induced finite map. Let $V=\nu^{-1}(U)$
and $E=\nu^{*}(D)$. Then we get a commutative diagram

(9.8) $\textstyle{{\rm
fil}^{A}_{D}H^{1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta^{*}_{U/X}}$$\scriptstyle{\nu^{*}}$$\textstyle{{\rm
Hom}_{{\rm
cont}}({\mathcal{Z}}_{0}(U),{{\mathbb{Q}}}/{{\mathbb{Z}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{*}}$$\textstyle{{\rm
fil}_{E}H^{1}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta^{*}_{V/{Y_{n}}}}$$\textstyle{{\rm
Hom}_{{\rm cont}}({\mathcal{Z}}_{0}(V),{{\mathbb{Q}}}/{{\mathbb{Z}}}),}$

where the right vertical arrow is induced by the push-forward map
$\nu_{*}\colon{\mathcal{Z}}_{0}(V)\to{\mathcal{Z}}_{0}(U)$ and the left
vertical arrow is induced by the definition of ${\rm fil}^{A}_{D}H^{1}(U)$. We
need to show that $\vartheta^{*}_{V/{Y_{n}}}\circ\nu^{*}(\chi)$ annihilates
${\mathcal{R}}_{0}(Y_{n}|E)$. But this follows from the classical ramified
class field theory for curves (e.g., see [40]). ∎

It is clear from the above definitions that there is a commutative diagram

(9.9)
$\textstyle{C(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta_{U/X}}$$\textstyle{{\rm
CH}^{A}_{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta^{A}_{X|D}}$$\textstyle{{\rm
CH}_{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta^{c}_{X|D}}$$\textstyle{\pi^{\rm
ab}_{1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{A}_{1}(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{\rm
ab}_{1}(X,D).}$

######  Lemma 9.3.

The map $\vartheta^{A}_{X|D}\colon{\rm CH}^{A}_{0}(X|D)\to\pi^{A}_{1}(X,D)$ is
injective with dense image. Moreover, the induced map
$\vartheta^{A}_{X|D}\colon{\rm CH}^{A}_{0}(X|D)_{0}\to\pi^{A}_{1}(X,D)_{0}$ is
an isomorphism.

###### Proof.

The density of the image follows from (9.5). We show the other assertions. We
first show that $\vartheta^{A}_{X|D}\colon{\rm
CH}^{A}_{0}(X|D)_{0}\to\pi^{A}_{1}(X,D)_{0}$ is surjective. It follows from
[3, Lemma 8.4, Theorem 8.5] that the map $\vartheta^{c}_{X|D}\colon{\rm
CH}_{0}(X|D)_{0}\to\pi^{\rm ab}_{1}(X,D)_{0}$ is an isomorphism for all
$D\in{\rm Div}_{C}(X)$. Taking the limit and noting that
$C(U)_{0}\cong{\underset{D}{\varprojlim}}\ {\rm CH}_{0}(X|D)_{0}$ by (7.5), it
follows that $\vartheta_{U/X}\colon C(U)_{0}\to\pi^{\rm ab}_{1}(U)_{0}$ is an
isomorphism. We now use the commutative diagram

(9.10)
$\textstyle{C(U)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta_{U/X}}$$\textstyle{\pi^{\rm
ab}_{1}(U)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}^{A}_{0}(X|D)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta^{A}_{X|D}}$$\textstyle{\pi^{A}_{1}(X,D)_{0}.}$

We have shown above that the top horizontal arrow is an isomorphism. The right
vertical arrow is surjective. The surjectivity of $\vartheta^{A}_{X|D}$ on
degree zero subgroups follows.

To show injectivity, we consider the diagram

(9.11) $\textstyle{{\rm
CH}^{A}_{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta^{A}_{X|D}}$$\textstyle{\pi^{A}_{1}(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta^{c}_{X|D}}$$\textstyle{\pi^{\rm
ab}_{1}(X,D).}$

The left vertical arrow is an isomorphism by Theorem 7.1 and the bottom
horizontal arrow is injective by [3, Lemma 8.4, Theorem 8.5]. It follows that
the top horizontal arrow is injective. ∎

######  Corollary 9.4.

The canonical map $\pi^{A}_{1}(X,D)\to\pi^{\rm ab}_{1}(X,D)$ is an
isomorphism. In particular, the canonical inclusion ${\rm
fil}^{c}_{D}H^{1}(K)\hookrightarrow{\rm fil}^{A}_{D}H^{1}(K)$ is a bijection.

###### Proof.

We only need to show that $\pi^{A}_{1}(X,D)_{0}\to\pi^{\rm ab}_{1}(X,D)_{0}$
is injective. But this follows immediately by restricting the diagram (9.9) to
degree zero subgroups and noting that all arrows in the right square except
the bottom horizontal arrow are isomorphisms by [3, Lemma 8.4, Theorem 8.5],
Theorem 7.1 and Lemma 9.3. ∎

Proof of Theorem 1.2. By Proposition 5.2 and Definition 5.3, the theorem is
equivalent to the statement that ${\rm fil}_{D}H^{1}(K)={\rm
fil}^{c}_{D}H^{1}(K)$ as subgroups of $H^{1}(K)$.

We first show that ${\rm fil}_{D}H^{1}(K)\subset{\rm fil}^{c}_{D}H^{1}(K)$. In
order to show this, we can replace ${\rm fil}^{c}_{D}H^{1}(K)$ by ${\rm
fil}^{A}_{D}H^{1}(K)$ using Corollary 9.4. We now let $\chi\in{\rm
fil}_{D}H^{1}(K)$ and let $Y\subset X$ be an integral curve not contained in
$D$ and not meeting $A$. Let $\nu\colon Y_{n}\to X$ be the induced map from
the normalization of $Y$. We need to show that $\nu^{*}(\chi)\in{\rm
fil}_{\nu^{*}(D)}H^{1}(k(Y))$.

Since $Y\cap C_{\rm sing}=Y\cap A=\emptyset$, we can replace $X$ (resp. $C$)
by $X\setminus A$ (resp. $C_{\rm reg}$) to show the above assertion. Since
$C_{\rm reg}$ is a smooth divisor inside the smooth variety $X\setminus A$,
[27, Corollary 2.8] (which does not require $X\setminus A$ to be projective)
applies. This yields $\nu^{*}(\chi)\in{\rm fil}_{\nu^{*}(D)}H^{1}(k(Y))$.

We now show the reverse inclusion. Let $\chi\in{\rm
fil}^{c}_{D}H^{1}(K)\subset H^{1}(U)$. We let $X^{\prime}=X\setminus A,\
D^{\prime}=D\cap X^{\prime}$ and $C^{\prime}=C\cap X^{\prime}$. Note that
$U=X^{\prime}\setminus C^{\prime}$. By [16, Theorem 7.19], it suffices to show
that $\chi\in{\rm fil}_{D^{\prime}}H^{1}(K)$. To prove this, we first deduce
from Proposition 6.9 that $\chi\in{\rm fil}^{c}_{D^{\prime}}H^{1}(K)$. Since
$X^{\prime}$ is regular and $C^{\prime}$ is a regular divisor on $X^{\prime}$
with complement $U$, it follows from [27, Corollary 2.8] that $\chi\in{\rm
fil}_{D^{\prime}}H^{1}(K)$. This finishes the proof. ∎

######  Remark 9.5.

With $(X,D,U)$ as in Theorem 1.2 and $r\geq 1$ an integer, let
${\mathcal{R}}_{r}(U)$ be the set of isomorphism classes of lisse
$\overline{{\mathbb{Q}}}_{\ell}$-Weil sheaves on $U$ of rank $r$ up to semi-
simplification. For a given $V\in{\mathcal{R}}_{r}(U)$, one can define the
Swan conductor ${\rm Sw}(V)$ as ${\rm Sw}(V)={\underset{E}{\sum}}{\rm
Sw}_{E}[E]\in{\rm Div}(X)$, where $E$ runs through integral divisors on $X$
and ${\rm Sw}(E)$ is defined as in [9, definition 3.1] but with a modification
that we use the Abbes-Saito filtration $G^{(\bullet)}_{K_{E}}$ of $G(K_{E})$,
where $K_{E}$ is the Henselization of $K$ at the generic point of $E$. One
then says that $V\in{\mathcal{R}}^{{\rm
div}}_{r}(X,D)\subset{\mathcal{R}}_{r}(U)$ if ${\rm Sw}(V)\leq D$.

We can now ask if ${\mathcal{R}}^{{\rm div}}_{r}(X,D)={\mathcal{R}}_{r}(X,D)$,
where the latter group is as in [9, Definition 3.6]. Theorem 1.2 can be viewed
as an answer to the $r=1$ case of this question. An attempt to answer this
question may also lead one to a proof of Theorem 1.2 using only ramification
theory.

Proof of Theorem 1.3. We need to show that ${\operatorname{\rm
cyc}}_{X|D}\colon{\mathcal{Z}}_{0}(U)\to C_{KS}(X,D)$ induces an isomorphism
${\operatorname{\rm cyc}}_{X|D}\colon{\rm
CH}_{0}(X|D)\xrightarrow{\cong}C_{KS}(X,D)$.

The theorem is already known when $d\leq 2$ by [3]. We shall therefore assume
that $d\geq 3$. We can also assume that $X$ is integral. Since the composite
map

(9.12) ${\mathcal{Z}}_{0}(U)\xrightarrow{{\operatorname{\rm
cyc}}_{X|D}}C(X,D)\xrightarrow{\psi_{X|D}}C_{KS}(X,D)$

is the cycle class map of (1.1), we can replace $C_{KS}(X,D)$ by $C(X,D)$
using [16, Theorem 3.8].

Let $s_{X|D}\colon{\mathcal{Z}}_{0}(U)\twoheadrightarrow{\rm CH}_{0}(X|D)$ be
the canonical quotient map. We denote the quotient maps
$C(U)\twoheadrightarrow{\rm CH}_{0}(X|D)$ and $\pi^{\rm
ab}_{1}(U)\twoheadrightarrow\pi^{\rm ab}_{1}(X,D)$ by $t_{X|D}$ and
$q^{\prime}_{X|D}$, respectively. Let
$\widehat{\rho}_{U/X}\colon{\mathcal{Z}}_{0}(U)\to C(U)$ be the canonical map
induced by taking the limit of the maps $s_{X|D^{\prime}}$ as $D^{\prime}$
runs through ${\rm Div}_{C}(X)$.

We now consider the commutative diagram

(9.13)
$\textstyle{{\mathcal{Z}}_{0}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{X|D}}$$\scriptstyle{{\operatorname{\rm
cyc}}_{U/X}}$$\scriptstyle{\widehat{\rho}_{U/X}}$$\textstyle{C(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta_{U/X}}$$\scriptstyle{t_{X|D}}$$\textstyle{\pi^{\rm
ab}_{1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\prime}_{X|D}}$$\textstyle{{\rm
CH}_{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta^{c}_{X|D}}$$\scriptstyle{{\operatorname{\rm
cyc}}_{X|D}}$$\textstyle{\pi^{\rm
ab}_{1}(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{X|D}}$$\textstyle{C(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X|D}}$$\textstyle{\pi^{{\rm
adiv}}_{1}(X,D)}$$\textstyle{\widetilde{C}_{U/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{X|D}}$$\scriptstyle{\widetilde{\rho}_{U/X}}$$\textstyle{\pi^{\rm
ab}_{1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{q_{X|D}}$

The triangle on the top left commutes by the definitions its various arrows.
The square on the top right commutes by (9.9). The bottom trapezium commutes
by (5.8). The big outer square is commutative because we have seen before that
for any closed point $x\in U$, both the maps $r_{X|D}\circ
q^{\prime}_{X|D}\circ\vartheta_{X|D}\circ\widehat{\rho}_{U/X}$ and
$q_{X|D}\circ\widetilde{\rho}_{X|D}\circ{\operatorname{\rm cyc}}_{U/X}$
coincide with the Frobenius substitution at $x$. The map $\vartheta^{c}_{X|D}$
is injective by Theorem 7.1, Lemma 9.3 and Corollary 9.4. The map $r_{X|D}$ is
an isomorphism by Theorem 1.2 and $\rho_{X|D}$ is injective by Theorem 1.1. It
follows by a diagram chase that the composite map
$p_{X|D}\circ{\operatorname{\rm cyc}}_{U/X}$ annihilates
${\mathcal{R}}_{0}(X|D)$ and induces an honest map

(9.14) ${\operatorname{\rm cyc}}_{X|D}\colon{\rm CH}_{0}(X|D)\to C(X,D)$

such that the middle square in (9.13) commutes. Moreover, this map is
injective. It is surjective by (2.10) and [21, Theorem 2.5]. This proves the
theorem. ∎

###  9.3. Bloch’s formula without regularity

In this subsection, we shall prove Theorem 1.4, Bloch’s formula in the pro-
setting when $X$ is not regular. We remark that it is not even clear if
Bloch’s formula would hold for each $D$ when $X$ is singular. Our assumption
now is the following. We let $X$ be a normal projective integral scheme of
dimension $d\geq 1$ over a finite field $k$ and let $D$ be an effective
Cartier divisor on $X$. We assume that $U=X\setminus D$ is regular.

We shall prove Theorem 1.4 after the following lemma. We shall ignore to write
the indexing set ${\mathbb{N}}$ of the pro-abelian groups used in the theorem
in what follows.

######  Lemma 9.6.

The cycle class map ${\operatorname{\rm
cyc}}_{U/X}\colon{\mathcal{Z}}_{0}(U)\to\widetilde{C}_{U/X}$ descends to a
continuous homomorphism of pro-abelian groups

${\operatorname{\rm cyc}}^{\bullet}_{X|D}\colon\\{{\rm
CH}_{0}(X|nD)\\}\to\\{C(X,nD)\\}.$

###### Proof.

We need to show that the map of pro-abelian groups $\\{{\operatorname{\rm
cyc}}_{X|D}\\}\colon\\{{\mathcal{R}}_{0}(X|nD)\\}\to\\{C(X,nD)\\}$ is zero.
Since ${\mathcal{R}}_{0}(X|nD)\subset{\mathcal{Z}}_{0}(U)_{0}$, it suffices to
show using (9.3) that the map of pro-abelian groups $\\{{\operatorname{\rm
cyc}}_{X|nD}\\}\colon\\{{\mathcal{R}}_{0}(X|nD)\\}\to\\{C(X,nD)_{0}\\}$ is
zero. To show this, we look at the sequence of maps

(9.15) $\\{{\mathcal{R}}_{0}(X|nD)\\}\xrightarrow{\\{{\operatorname{\rm
cyc}}_{X|nD}\\}}\\{C(X,nD)_{0}\\}\xrightarrow{\\{\rho_{X|nD}\\}}\\{\pi^{\rm
ab}_{1}(X,nD)_{0}\\},$

where the second map $\\{\rho_{X|nD}\\}=\rho^{\bullet}_{X|D}$ exists by
Theorem 6.4.

We have seen just above (9.5) that the map ${\mathcal{Z}}_{0}(U)\to\pi^{\rm
ab}_{1}(X,nD)$ factors through $\vartheta^{c}_{X|nD}\colon{\rm
CH}_{0}(X|nD)\to\pi^{\rm ab}_{1}(X,nD)$ for every $n$. It follows that the
composite arrow in (9.15) is zero. The lemma now follows from Theorem 6.5. ∎

Using Lemma 9.6, we get commutative diagrams

(9.16)
$\textstyle{C(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
cyc}}_{U/X}}$$\scriptstyle{\vartheta_{U/X}}$$\textstyle{\widetilde{C}_{U/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\rho}_{U/X}}$$\textstyle{\\{{\rm
CH}_{0}(X|nD)\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
cyc}}^{\bullet}_{X|D}}$$\scriptstyle{\vartheta^{\bullet}_{X|D}}$$\textstyle{\\{C(X,nD)\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\bullet}_{X|D}}$$\textstyle{\pi^{\rm
ab}_{1}(U)}$$\textstyle{\\{\pi^{\rm ab}_{1}(X,nD)\\},}$

where the triangle on the left is the limit of the one on the right.

Proof of Theorem 1.4. The composite map ${\mathcal{Z}}_{0}(U)\to\\{{\rm
CH}_{0}(X|nD)\\}\to\\{C_{KS}(X,nD)\\}$ is surjective by [21, Theorem 2.5]. So
we only need to show that ${\operatorname{\rm cyc}}^{\bullet}_{X|D}$ is
injective. For this, we can replace $\\{C_{KS}(X,nD)\\}$ by $\\{C(X,nD)\\}$
using (2.10).

Using the commutative diagram of exact sequences of pro-abelian groups

(9.17)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{{\rm
CH}_{0}(X|nD)_{0}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
cyc}}^{\bullet}_{X|D}}$$\textstyle{\\{{\rm
CH}_{0}(X|nD)\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg}}$$\scriptstyle{{\operatorname{\rm
cyc}}^{\bullet}_{X|D}}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{C(X,nD)_{0}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{C(X,nD)\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\deg}}$$\textstyle{{\mathbb{Z}},}$

it suffices to show that the left vertical arrow is injective.

For this, we look at the sequence of maps (see (9.16))

(9.18) $\\{{\rm CH}_{0}(X|nD)_{0}\\}\xrightarrow{{\operatorname{\rm
cyc}}^{\bullet}_{X|D}}\\{C(X,nD)_{0}\\}\xrightarrow{\rho^{\bullet}_{X|D}}\\{\pi^{\rm
ab}_{1}(X,nD)_{0}\\}.$

We have seen in the proof of Theorem 1.3 that the composite map is an (in fact
levelwise) isomorphism (see [3, Lemma 8.4, Theorem 8.5]). The map
$\rho^{\bullet}_{X|D}$ is an isomorphism by Corollary 6.6. We conclude that
${\operatorname{\rm cyc}}^{\bullet}_{X|D}$ is an isomorphism. This finishes
the proof. ∎

As a consequence of Theorem 1.4, we have the following.

######  Corollary 9.7.

The inverse limit idele class group $\widetilde{C}_{U/X}$ is independent of
the normal compactification $X$ of $U$.

###### Proof.

Follows from Theorem 1.4 and [27, Lemma 3.1]. ∎

_Acknowledgements._ Gupta was supported by the SFB 1085 _Higher Invariants_
(Universität Regensburg). He would also like to thank TIFR, Mumbai for
invitation in March 2020 and extending the invitation during the tough times
of the Covid-19 pandemic. The authors would like to thank Shuji Saito for
telling them about his positive expectation of Theorem 1.2 when they were
working on its proof. The authors would like to thank Moritz Kerz for some
fruitful discussion with Gupta on the contents of this paper.

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