For all $n\gg 0$, the map $\phi^{n}_{X|D}\colon U_{n}(X^{o})(k)\to{\rm
Pic}^{0}(X|D)$ is a continuous open map whose image is dense.

  2. (2)

${\rm Pic}^{0}(X)$ is a profinite abelian group.

  3. (3)

The map ${\rm Pic}^{0}(X|D)\to{\rm Pic}^{0}(X)$ is a topological quotient.

  4. (4)

If $D^{\prime}\geq D$, the map ${\rm Pic}^{0}(X|D^{\prime})\to{\rm
Pic}^{0}(X|D)$ is a topological quotient.

  5. (5)

The map ${\rm Pic}^{0}(X|D)\to{\rm Pic}^{0}(Y)$ is a topological quotient.

###### Proof.

The statement (1) follows directly from Theorem 6.6 and [4, Thm. 10.5.1]. To
prove (2), we note that ${\mathbf{Pic}}^{0}(X)$ is an abelian variety over
$k$. This implies that ${\rm Pic}^{0}(X)$ is compact as an adic space (see op.
cit.). Since it is also totally disconnected, ${\rm Pic}^{0}(X)$ must be
profinite. Since the maps ${\rm Pic}^{0}(X|D^{\prime})\to{\rm Pic}^{0}(X|D)$
and ${\rm Pic}^{0}(X|D)\to{\rm Pic}^{0}(Y)$ are surjective, the remaining
statements of the corollary follow directly from Corollary 6.10 and [4, Thm.
10.5.1]. ∎

##  7\. Pontryagin dual of the Chow group with modulus

The goal of this section is to strengthen Corollary 6.11. More precisely, we
shall show that all maps in the exact sequence (5.1) are continuous with
respect to the adic topology and the resulting dual complex is partially
exact. This will be a key step in the proofs of the main results. We begin by
recalling the Kato topology and its relation with the adic topology.

###  7.1. Kato topology

Let $R$ be an equicharacteristic excellent Henselian discrete valuation ring
with maximal ideal ${\mathfrak{m}}$ whose residue field ${\mathfrak{f}}$ is a
local field of exponential characteristic $p\geq 1$. Let $L$ denote the
quotient field of $R$. In this case, one can find a two dimensional excellent
normal local integral domain $A$ whose residue field is finite such that
$R\cong(A_{\mathfrak{p}})^{h}$ for some height one prime ideal
${\mathfrak{p}}\subset A$. For any ideal $I\subset A$ not contained in
${\mathfrak{p}}$ and $n\geq 1$, we let ${\rm fil}^{I}_{n}{K}^{M}_{1}(R)$ be
the subgroup of $R^{\times}$ generated by $(1+{\mathfrak{m}}^{n})$ and
$(1+I)\subset A^{\times}\subset{K}^{M}_{1}(R)$. We let ${\rm
fil}^{I}_{0}{K}^{M}_{1}(R)=R^{\times}$.

When $p>1$, Kato defined a subgroup topology on $R^{\times}$ (see [20, § 7]
and [46, § 2.3]) for which the fundamental system of open neighborhoods of the
identity is given by subgroups of the form ${\rm fil}^{I}_{n}{K}^{M}_{1}(R)$,
where $I$ and $n$ vary as above. It follows from [20, § 7, Lem. 1] (see also
[46, § 2.3]) that this topology of $R^{\times}$ does not depend on the choice
of $A$. We shall call this the Kato topology on $R^{\times}$. The Kato
topology on $L^{\times}$ is the unique topology which is compatible with its
group structure and for which $R^{\times}$ with its Kato topology is an open
subgroup.

It is easy to check that the Kato topology of $R^{\times}$ coincides with the
subspace topology induced from the Kato topology of $\widehat{R}^{\times}$ via
the inclusion $R^{\times}\subset\widehat{R}^{\times}$. In particular, the same
also holds for the inclusion $L^{\times}\subset\widehat{L}^{\times}$. When
$p=1$, we shall assume the Kato topology on $R^{\times}$ and $L^{\times}$ to
be discrete. By choosing a uniformizer $\pi\in{\mathfrak{m}}$, we get a
canonical isomorphism of ${\mathfrak{f}}$-algebras
$\phi\colon{\mathfrak{f}}[[T]]\xrightarrow{\cong}\widehat{R}$ which sends $T$
to $\pi$. In § 6.3, we recalled the adic topology of ${\mathfrak{f}}$ and
${\mathcal{O}}_{\mathfrak{f}}$. We shall use the following description of the
Kato topology in terms of the adic topology.

######  Lemma 7.1.

When $p>1$, the map
$\phi\colon({\mathfrak{f}}[[T]])^{\times}\to(\widehat{R})^{\times}$ is an
isomorphism of topological abelian groups if we endow
${\mathfrak{f}}[[T]]\cong{\mathfrak{f}}\times{\mathfrak{f}}^{{\mathbb{N}}}$
with the product topology of the adic topology of ${\mathfrak{f}}$ and
$({\mathfrak{f}}[[T]])^{\times}$ with the subspace topology. In particular,
the quotient topology of the Kato topology on
$({\widehat{R}}/{{\mathfrak{m}}^{n}})^{\times}$ coincides with the adic
topology of $({{\mathfrak{f}}[T]}/{(T^{n})})^{\times}$.

###### Proof.

This is a straightforward consequence of [20, § 7, Rem. 1, Lem. 3] . ∎

Recall (e.g., see [44, App. A]) that for any commutative ring $A$ and the set
$\\{1,\ldots,m\\}$ (where $m=\infty$ is allowed), the ring of big Witt-vectors
of length $m$ is the commutative ring ${\mathbb{W}}_{m}(A)$ which is $A^{m}$
as a set and whose ring structure is uniquely determined by the condition that
the ghost map ${\rm gh_{m}}\colon{\mathbb{W}}_{m}(A)\to A^{m}$ is a ring
homomorphism with respect to the product ring structure on $A^{m}$. The ghost
map is in fact a natural transformation of functors between commutative rings.
We write ${\mathbb{W}}_{\infty}(A)$ as ${\mathbb{W}}(A)$.

Let $\Gamma(A):=(1+TA[[T]])^{\times}\subset(A[[T]])^{\times}$ be the subgroup
of the group of units of the power series ring $A[[T]]$. Recall that every
element of $\Gamma(A)$ has a unique presentation of the form
$f(T)={\underset{n\geq 1}{\prod}}(1-a_{n}T^{n})$ with $a_{n}\in A$. One knows
(e.g., see [44, App. A]) that as an abelian group with respect to its addition
operation, ${\mathbb{W}}(A)$ and ${\mathbb{W}}_{m}(A)$ have a simple
description. That is, the maps

(7.1) $\nu_{A}\colon{\mathbb{W}}(A)\to\Gamma(A);\
\nu_{A}((a_{n}))={\underset{n\geq 1}{\prod}}(1-a_{n}T^{n}),\ \ and$ (7.2)
$\mu_{A}\colon\Gamma(A)\to{\mathbb{W}}(A),\ \ \mu_{A}({\underset{n\geq
1}{\prod}}(1-a_{n}T^{n}))=(a_{n}).$

are group homomorphisms which are inverses to each other. These maps induce
isomorphisms

(7.3)
$\nu_{A}\colon{\mathbb{W}}_{m}(A)\xrightarrow{\cong}\Gamma_{m}(A):=\frac{(1+TA[[T]])^{\times}}{(1+T^{m+1}A[[T]])^{\times}};\
\ \mu_{A}\colon\Gamma_{m}(A)\xrightarrow{\cong}{\mathbb{W}}_{m}(A).$

It follows from (7.2) that there are polynomials $p_{i}\in
A[T_{1},\ldots,T_{m}]$ such that $\mu_{A}$ in (7.3) is of the form
$\mu_{A}(1+a_{1}T+\cdots+a_{m}T^{m})=(p_{1}(a_{1},\ldots,a_{m}),\ldots,p_{m}(a_{1},\ldots,a_{m}))$.
We shall let ${\mathbb{W}}_{m}(A)=0$ for $m\leq 0$.

Recall from § 6.2 that for any field $F$, there is a unipotent linear
algebraic group ${{\mathbb{W}}}_{m,F}$ over $F$ such that
${\mathbb{W}}_{m}(A)={{\mathbb{W}}}_{m,F}(A)$ for any $F$-algebra $A$. The map
$\psi_{m,F}\colon{{\mathbb{W}}}_{m,F}\to{\mathbb{A}}^{m}_{F}$, given by
$\psi_{m,F}((a_{i}))=(a_{i})$ for $(a_{i})\in{\mathbb{W}}_{m}(A)$ ($A$ any
$F$-algebra) is an isomorphism between $F$-schemes (e.g., see [50, Chap. V, §
3.13]). Note however that this is not an isomorphism of group schemes over
$F$. One can in fact write ${{\mathbb{W}}}_{m,F}={\rm Spec\,}(A_{m,F})$, where

(7.4) $A_{m,F}=\frac{F[X_{ij}|1\leq i,j\leq m+1]}{(X_{ij}\ \mbox{for}\ i>j,\
X_{ij}-1\ \mbox{for}\ i=j,\ X_{ij}-X_{i+1j+1}\ \mbox{for}\ i<j)}.$

The isomorphism $\psi_{m,F}$ is easily deduced from this description.

Suppose now that $F$ is our local field $k$. For any finite field extension
${k^{\prime}}/k$, the group scheme ${\mathbb{W}}_{m,k^{\prime}}$ then induces
the adic topology on the abelian group
${\mathbb{W}}_{m}(k^{\prime})={\mathbb{W}}_{m,k^{\prime}}(k^{\prime})$.
Furthermore, the isomorphism
${\mathbb{W}}_{m,k^{\prime}}\cong{\mathbb{A}}^{m}_{k^{\prime}}$ as
$k^{\prime}$-schemes implies that ${\mathbb{W}}_{m}(k^{\prime})\cong k^{\prime
m}$ as adic spaces, where the latter has the product of the adic topology of
$k^{\prime}$. Note here that the adic topology of
${\mathbb{W}}_{m}(k^{\prime})$ does not depend on whether it has been induced
by the $k$-scheme structure or by the $k^{\prime}$-scheme structure of
${\mathbb{W}}_{m,k^{\prime}}$ (e.g., see [20, § 7.2]). We shall henceforth
consider ${\mathbb{W}}_{m,k^{\prime}}$ as a unipotent algebraic group over
$k^{\prime}$ and ${\mathbb{W}}_{m}(k^{\prime})$ as a topological abelian group
with its adic topology. Using the descriptions of Kato and adic topologies, we
get the following.

######  Lemma 7.2.

Assume $p>1$. Let $X$ be a regular curve over $k$ and $\pi_{x}$ a uniformizer
of the local ring ${\mathcal{O}}_{X,x}$ at a closed point $x\in X$, then the
map

$\mu_{k(x)}\circ\phi^{-1}\colon\left(\frac{\widehat{{\mathcal{O}}_{X,x}}}{(\pi^{m+1}_{x})}\right)^{\times}\to
k(x)^{\times}\times{\mathbb{W}}_{m}(k(x))$

is a continuous bijection between topological abelian groups for every $m\geq
0$, where the left hand side is endowed with the quotient of the Kato
topology.

###### Proof.

We can assume $m\geq 1$, else the statement is immediate from Lemma 7.1. Using
Lemma 7.1 and (7.2), we only need to show that
$\mu_{k(x)}\colon\Gamma_{m}(k(x))\to{\mathbb{W}}_{m}(k(x))$ is continuous if
$\Gamma_{m}(k(x))$ is endowed with the subspace topology from
$\left({k(x)[T]}/{(T^{m+1})}\right)^{\times}$. For this, we look at the
diagram

(7.5)
$\textstyle{\Gamma_{m}(k(x))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{k(x)}}$$\scriptstyle{\theta_{k(x)}}$$\textstyle{{\mathbb{W}}_{m}(k(x))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{k(x)}}$$\textstyle{k(x)^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{k(x)}}$$\textstyle{k(x)^{m},}$

where $\gamma_{k(x)}((a_{i}))=(p_{1}((a_{i})),\ldots,p_{m}((a_{i})))$ (cf.
(7.3)). The arrow $\theta_{k(x)}$ is the canonical bijection induced by the
homeomorphism of topological spaces
$\alpha_{k(x)}\colon(k(x)[[T]])^{\times}\xrightarrow{\cong}k(x)^{\times}\times
k(x)^{{\mathbb{N}}}$ as in Lemma 7.1. It is clear from the definition of
various maps that this diagram is commutative. Since $\psi_{k(x)}$ is a
homeomorphism, it is enough to show that $\psi_{k(x)}\circ\mu_{k(x)}$ is
continuous. Equivalently, we need to show that
$\gamma_{k(x)}\circ\theta_{k(x)}$ is continuous.

Since $\alpha_{k(x)}$ is a homeomorphism of topological spaces (cf. Lemma 7.1)
and $\theta_{k(x)}$ is the induced map on the quotients with the quotient
topologies (note that the projection $k(x)^{{\mathbb{N}}}\to k(x)^{m}$ is a
quotient map), it is a homeomorphism. Since each component of $\gamma_{k(x)}$
is a polynomial map, it is clearly continuous. In particular, $\gamma_{k(x)}$
is continuous. It follows that $\gamma_{k(x)}\circ\theta_{k(x)}$ is
continuous. ∎

###  7.2. Pontryagin duals of Chow groups

Let $k$ be a local field of exponential characteristic $p\geq 1$. Let $(X,D)$
be a modulus pair over $k$, where $X$ is a geometrically connected smooth
projective curve over $k$. We let $D^{\dagger}=D_{\rm red}$. We shall assume
that $\emptyset\neq D^{\dagger}\subset X(k)$. We let $X^{o}=X\setminus D$. We
let $j\colon X^{o}\hookrightarrow X$ and $t\colon D\hookrightarrow X$ be the
inclusions. We write $D=\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}n_{i}[x_{i}]\in{\rm Div}(X)$. If $U\subset X$ is
any Zariski dense open, then $U(k)\subset X(k)$ is an adically dense open
subset (e.g., see [4, Thm. 10.5.1]). Since $X(k)\neq\emptyset$, it easily
follows that $U(k)$ must be infinite. In particular, $X^{o}(k)$ is infinite.
We fix a point $P\in X^{o}(k)$ such that ${\rm alb}_{X|D}(P)=0$. We have noted
in § 6.3 that ${\rm CH}_{0}(X|D)$ and ${\rm CH}_{0}(X)$ are adic topological
abelian groups and the degree zero parts are their open subgroups.

Assume that $p>1$. Using the identification
${\mathcal{O}}^{\times}(D)\cong{\underset{x\in
D^{\dagger}}{\prod}}\left(\frac{\widehat{{\mathcal{O}}_{X,x}}}{(\pi^{n_{x}}_{x})}\right)^{\times}$
as an adic space, we conclude from Proposition 6.9 and Lemma 7.2 that in
(5.1), the map $\partial_{X|D}$ is continuous, for, it is the product of maps
$\mu_{k(x)}\circ\phi^{-1}_{x}$ over $x\in D^{\dagger}$ followed by the
topological quotient $(k^{\times})^{r}\times(\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\prod}}}{\mathbb{W}}_{n_{i}-1}(k))\twoheadrightarrow(k^{\times})^{r-1}\times(\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\prod}}}{\mathbb{W}}_{n_{i}-1}(k))$. Since the map
$\vartheta_{X|D}$ and the inclusion
$k^{\times}\hookrightarrow{\mathcal{O}}^{\times}(D)$ are clearly continuous,
(5.1) gives rise to a sequence of homomorphisms

(7.6) $0\to{\rm CH}_{0}(X)^{\star}\xrightarrow{(\vartheta_{X|D})^{\star}}{\rm
CH}_{0}(X|D)^{\star}\xrightarrow{(\partial_{X|D})^{\star}}({\mathcal{O}}^{\times}(D))^{\star}\to(k^{\times})^{\star}$

######  Lemma 7.3.

(7.6) is a chain complex of abelian groups which is exact at ${\rm
CH}_{0}(X)^{\star}$ and ${\rm CH}_{0}(X|D)^{\star}$.

###### Proof.

The lemma is obvious if $\deg(D)=1$ by Lemma 5.2. We shall therefore assume
that $\deg(D)\geq 2$. That (7.6) is a complex is a direct consequence of
(5.1). The injectivity of $(\vartheta_{X|D})^{\star}$ follows again from
(5.1). To show the exactness at ${\rm CH}_{0}(X|D)^{\star}$, we can clearly
replace ${\rm CH}_{0}(X|D)$ and ${\rm CH}_{0}(X)$ by their degree zero
subgroups. We let $G=(k^{\times})^{r-1}\times(\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\prod}}}{\mathbb{W}}_{n_{i}-1}(k))$. We need to show that
the sequence

(7.7) $0\to({\rm
CH}_{0}(X)_{0})^{\star}\xrightarrow{(\vartheta_{X|D})^{\star}}({\rm
CH}_{0}(X|D)_{0})^{\star}\to G^{\star}\to 0$

is exact at the middle term.

To show the above exactness, we note that the sequence

(7.8) $0\to({\rm
CH}_{0}(X)_{0})^{\vee}\xrightarrow{(\vartheta_{X|D})^{\vee}}({\rm
CH}_{0}(X|D)_{0})^{\vee}\to G^{\vee}\to 0$

is clearly exact. Hence, given a continuous character $\chi\in({\rm
CH}_{0}(X|D)_{0})^{\star}$ whose restriction to $G$ is zero, we get a unique
character $\chi^{\prime}\in({\rm CH}_{0}(X)_{0})^{\vee}$ such that
$\chi=\chi^{\prime}\circ\vartheta_{X|D}$. It remains to show that
$\chi^{\prime}$ is continuous with respect to the adic topology of ${\rm
CH}_{0}(X)_{0}$. But this is an easy consequence of Lemma 5.1 and Corollary
6.11. ∎

We now assume $p=1$ and prove the analogous result. We begin with the
following.

######  Lemma 7.4.

If $p=1$, we have ${\rm CH}_{0}(X|D^{\dagger})^{\star}\cong({\rm
CH}_{0}(X|D)^{\star})_{\rm tor}\cong({\rm CH}_{0}(X|D)^{\vee})_{\rm tor}$. We
also have

(7.9) $(({\mathcal{O}}^{\times}(D))^{\star})_{\rm
tor}\cong({\mathcal{O}}^{\times}(D))^{\vee})_{\rm
tor}\cong({\mathcal{O}}^{\times}(D^{\dagger}))^{\star}\cong({\mathcal{O}}^{\times}(D^{\dagger}))^{\vee})_{\rm
tor},$

where ${\mathcal{O}}^{\times}(D)$ and ${\mathcal{O}}^{\times}(D^{\dagger})$
are endowed with the adic topology.

###### Proof.

The isomorphisms in (7.9) are immediate consequences of [39, Prop. II.5.7]
because ${\rm
Ker}({\mathcal{O}}^{\times}(D)\twoheadrightarrow{\mathcal{O}}^{\times}(D^{\dagger}))$
is a divisible group. To prove the first statement, we first assume
$D=\emptyset$. Since ${\rm CH}_{0}(X)_{0}={\mathbf{Pic}}^{0}(X)(k)$ is a
profinite abelian group by Corollary 6.11, it follows that $({\rm
CH}_{0}(X)_{0})^{\star}$ is a torsion group. Furthermore, the assumption $p=1$
implies using the Kummer sequence that ${{\rm CH}_{0}(X)}/n$ is discrete (and
finite) for the quotient topology for every integer $n\geq 1$. Since
${\mathbf{Pic}}^{0}(X)\xrightarrow{n}{\mathbf{Pic}}^{0}(X)$ is a smooth
isogeny, it follows from [4, Thm. 10.5.1] that ${\rm
CH}_{0}(X)\xrightarrow{n}{\rm CH}_{0}(X)$ is an open map of adic spaces. In
particular, ${{\rm CH}_{0}(X)}\to{{\rm CH}_{0}(X)}/n$ is continuous.

Now, if $\chi\in{\rm CH}_{0}(X)^{\star}$, then $\chi({\rm CH}_{0}(X)_{0})$ is
a finite cyclic group of the type ${{\mathbb{Z}}}/n$. This yields a
commutative diagram of exact sequences

(7.10)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}(X)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{{\rm
CH}_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\chi}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathbb{Z}}}/n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Since the image of $\widetilde{\chi}$ must be finite, it follows that $\chi$
has a finite order. Conversely, if $\chi\in({\rm CH}_{0}(X)^{\vee})_{\rm
tor}$, then it factors through ${{\rm CH}_{0}(X)}/n$ for some $n\geq 1$ and it
follows from the previous paragraph that $\chi$ is continuous on ${\rm
CH}_{0}(X)$.

Suppose now that $D$ is nonempty but reduced and let $\chi\in{\rm
CH}_{0}(X|D)^{\star}$. We have seen above that the restriction of $\chi$ to
$(k^{\times})^{r-1}$ has finite order. We thus get an integer $n\geq 1$ and a
commutative diagram of exact sequences (cf. Proposition 6.9)

(7.11)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(k^{\times})^{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{{\rm
CH}_{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{{\rm
CH}_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\chi}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathbb{Z}}}/n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Since ${\rm CH}_{0}(X|D)\twoheadrightarrow{\rm CH}_{0}(X)$ is a topological
quotient by Corollary 6.11, we get that $\widetilde{\chi}$ is continuous, and
hence, its image is finite. We conclude that the image of the middle vertical
arrow is finite.

Conversely, if $\chi\in({\rm CH}_{0}(X|D)^{\vee})_{\rm tor}$, then it factors
through ${{\rm
CH}_{0}(X|D)}/n\xrightarrow{\widetilde{\chi}}{{\mathbb{Q}}}/{{\mathbb{Z}}}$
for some $n\geq 1$. We have shown above that ${{\rm CH}_{0}(X)}/n$ is finite.
It follows from the top row of (7.11) and [39, Prop. II.5.7] that ${{\rm
CH}_{0}(X|D)}/n$ is finite. On the other hand, as
${\mathbf{Pic}}^{0}(X|D)\xrightarrow{n}{\mathbf{Pic}}^{0}(X|D)$ is a smooth
isogeny, it follows that ${{\rm CH}_{0}(X|D)}/n$ is a finite and discrete
quotient of ${\rm CH}_{0}(X|D)$. We conclude that $\chi$ is continuous.

If $D$ is not necessarily reduced, we look at the commutative diagram

(7.12) $\textstyle{{\rm
CH}_{0}(X|D^{\dagger})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
CH}_{0}(X|D^{\dagger})^{\vee})_{\rm
tor}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
CH}_{0}(X|D)^{\star})_{\rm
tor}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\rm
CH}_{0}(X|D)^{\vee})_{\rm tor},}$

in which all arrows are the canonical inclusions. Since any element of $({\rm
CH}_{0}(X|D)^{\vee})_{\rm tor}$ must annihilate the kernel of ${\rm
CH}_{0}(X|D)\twoheadrightarrow{\rm CH}_{0}(X|D_{\rm red})$, it follows that
the right vertical arrow in (7.12) is a bijection. We have shown in the
previous paragraph that the top horizontal arrow is a bijection. A diagram
chase shows that all arrows are bijections. This concludes the proof. ∎

######  Lemma 7.5.

If $p=1$ and $D$ is reduced, then (5.1) induces a chain complex of abelian
groups

(7.13) $0\to{\rm CH}_{0}(X)^{\star}\xrightarrow{(\vartheta_{X|D})^{\star}}{\rm
CH}_{0}(X|D)^{\star}\xrightarrow{(\partial_{X|D})^{\star}}({\mathcal{O}}^{\times}(D))^{\star}\to(k^{\times})^{\star}$

which is exact at ${\rm CH}_{0}(X)^{\star}$ and ${\rm CH}_{0}(X|D)^{\star}$.

###### Proof.

At any rate, we do have an exact sequence

$0\to({\rm CH}_{0}(X)^{\vee})_{\rm
tor}\xrightarrow{(\vartheta_{X|D})^{\vee}}({\rm CH}_{0}(X|D)^{\vee})_{\rm
tor}\xrightarrow{(\partial_{X|D})^{\vee}}(({\mathcal{O}}^{\times}(D))^{\vee})_{\rm
tor}\to((k^{\times})^{\vee})_{\rm tor}.$

The desired claim now follows by Lemma 7.4. ∎

##  8\. Brauer group with modulus

The goal of this section is to define the Brauer group of a modulus pair and
prove some functorial properties. We begin by recalling Kato’s ramification
filtration which will play a fundamental role in our exposition.

###  8.1. Étale motivic cohomology

Let $k$ be a field of exponential characteristic $p\geq 1$ and let $X$ be a
Noetherian $k$-scheme. If $n\in k^{\times}$ is an integer and
$r\in{\mathbb{Z}}$, we let ${{\mathbb{Z}}}/n(r)$ be the étale sheaf on $X$
defined as the usual Tate twist of the constant sheaf ${{\mathbb{Z}}}/n$
(e.g., see [35, p. 163]). If $p>1$ and $n=p^{s}m$ with $s\geq 0$ and $p\nmid
m$, we let ${{\mathbb{Z}}}/n(r)$ be the object ${{\mathbb{Z}}}/m(r)\oplus
W_{s}\Omega^{r}_{X,{\operatorname{log}}}[-r]$ as an object of
${\mathcal{D}}_{\text{\'{e}t}}(X)$. We have the cup product pairing of the
(hyper)cohomology of the form $H^{i}(X,{{\mathbb{Z}}}/{n}(j))\times
H^{i^{\prime}}(X,{{\mathbb{Z}}}/{n}(j^{\prime}))\to
H^{i+i^{\prime}}(X,{{\mathbb{Z}}}/{n}(j+j^{\prime}))$. For $q\in{\mathbb{Z}}$,
we let $H^{q}_{n}(X)$ denote the étale cohomology group
$H^{q}(X,{{\mathbb{Z}}}/{n}(q-1))$. We let $H^{q}(X)={\varinjlim}_{n}\
H^{q}_{n}(X)$ with respect to the canonical transition maps
${{\mathbb{Z}}}/{n}(r)\xrightarrow{m}{{\mathbb{Z}}}/{mn}(r)$ (see (2.6) for
their definition in positive characteristic). If $X={\rm Spec\,}(A)$ is
affine, we write $H^{q}_{n}(X)$ (resp. $H^{q}(X)$) as $H^{q}_{n}(A)$ (resp.
$H^{q}(A)$).

For $m,n\in k^{\times}$ and $s\geq 1$, we have commutative diagrams of exact
sequences

(8.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{{\mathbb{Z}}}/{n}}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
can}}}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\scriptstyle{{\operatorname{id}}}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{s}}$$\scriptstyle{{\operatorname{id}}}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{{{\mathcal{O}}^{\times}_{X}}/{p^{s}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{{\mathbb{Z}}}/{mn}}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{mn}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{s+1}}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathcal{O}}^{\times}_{X}}/{p^{s+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$
(8.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{{\mathbb{Z}}}/{mn}}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{mn}$$\scriptstyle{m}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{id}}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{s+1}}$$\scriptstyle{p}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{id}}}$$\textstyle{{{\mathcal{O}}^{\times}_{X}}/{p^{s+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
can}}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{{\mathbb{Z}}}/{n}}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{s}}$$\textstyle{{\mathcal{O}}^{\times}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathcal{O}}^{\times}_{X}}/{p^{s}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

of étale sheaves, where the diagrams on the right make sense when $p>1$.

The commutative diagrams in (8.1) give rise to an exact sequence of ind-
abelian groups

(8.3)
$0\to\\{{{\operatorname{Br}}(X)}/n\\}\to\\{H^{3}(X,{{\mathbb{Z}}}/m(1))\\}\to\\{_{m}H^{3}(X,{\mathcal{O}}^{\times}_{X})\\}\to
0,$

which are indexed by ${\mathbb{N}}$ and whose transition maps are induced by
${{\mathbb{Z}}}/m\xrightarrow{n}{{\mathbb{Z}}}/{mn}$. Taking the limits and
noting that $H^{i}(X,{\mathcal{O}}^{\times}_{X})$ is a torsion group for
$i\geq 2$ (e.g., see [4, Lem. 3.5.3]) when $X$ is regular, we get the
following.

######  Lemma 8.1.

If $X$ is regular, then the canonical map

$H^{3}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}}(1))\to
H^{3}(X,{\mathcal{O}}^{\times}_{X})$

is an isomorphism.

###### Proof.

This is an easy application of the above discussion once we use an elementary
fact that $A\otimes B=0$ if $A$ is a torsion abelian group and $B$ is a
divisible abelian group. ∎

We also get the following folklore result which we shall use throughout this
paper without giving reference.

######  Lemma 8.2.

Let $A$ be a equicharacteristic regular local ring. Then one has a canonical
isomorphism $H^{2}(A)\xrightarrow{\cong}{\operatorname{Br}}(A)$.

For a local ring $A$ over $k$, we have the Norm residue map (e.g., see [13, §
5.1])

(8.4) ${\rm NR}_{A}\colon{K^{M}_{i}(A)}/{n}\to H^{i}(A,{{\mathbb{Z}}}/n(i)).$

Composing this with the cup product, we see that there is a canonical bilinear
pairing

(8.5) $\langle,\rangle\colon
H^{i}(A,{{\mathbb{Z}}}/{n}(j))\times{K^{M}_{i^{\prime}}(A)}/{n}\to
H^{i+i^{\prime}}(A,{{\mathbb{Z}}}/{n}(j+i^{\prime})).$

If $A$ is any equicharacteristic local integral domain with maximal ideal
${\mathfrak{m}}_{A}$ and quotient field $E$, we let ${\rm
fil}_{0}{K}^{M}_{1}(E)=A^{\times}$ and ${\rm
fil}_{n}{K}^{M}_{1}(E)=(1+{\mathfrak{m}}^{n}_{A})$ if $n\geq 1$. For $i\geq
2$, we let ${\rm fil}_{n}{K}^{M}_{i}(E)$ be the image of the cup product map
${\rm fil}_{n}{K}^{M}_{1}(E)\otimes{K}^{M}_{i-1}(E)\to{K}^{M}_{i}(E)$. We let
${\rm fil}_{n}{K}^{M}_{i}(E)={K}^{M}_{i}(E)$ for $n<0$. It is clear that ${\rm
fil}_{\bullet}{K}^{M}_{i}(E)$ is a decreasing filtration of ${K}^{M}_{i}(E)$.
We shall call this ‘the logarithmic filtration’ of ${K}^{M}_{i}(E)$.

###  8.2. Kato’s ramification filtration

In order to define the Brauer group with modulus, we need to recall Kato’s
ramification filtration. We let $L$ be a Henselian discrete valuation field
with the ring of integers ${\mathcal{O}}_{L}$, the maximal ideal
${\mathfrak{m}}_{L}=(\pi_{L})$ and the residue field ${\mathfrak{l}}$ such
that ${\rm char}(L)={\rm char}({\mathfrak{l}})=p\geq 0$. Let $\widehat{L}$
denote the completion of $L$. Recall the following from [23, Cor. 2.5, Prop.
6.3], where we have shifted Kato’s filtration one place to the right.

######  Definition 8.3.

Let $q\geq 1$ be an integer.

  1. (1)

If $p=0$, we let ${\rm fil}_{0}H^{q}(L)=H^{q}({\mathcal{O}}_{L})$ and ${\rm
fil}_{n}H^{q}(L)=H^{q}(L)$ if $n\geq 1$.

  2. (2)

If $p>0$ and $n\geq 0$, we let ${\rm fil}_{n}H^{q}(L)$ be the subgroup of
elements $\chi\in H^{q}(L)$ such that
$\langle\chi,1+\pi^{n}_{L}{\mathcal{O}}_{L^{\prime}}\rangle=0$ for all
Henselian discrete valuation fields $L^{\prime}$ such that
${\mathcal{O}}_{L}\subset{\mathcal{O}}_{L^{\prime}}$ and
${\mathfrak{m}}_{L^{\prime}}={\mathfrak{m}}_{L}{\mathcal{O}}_{L^{\prime}}$,
where
$1+\pi^{0}_{L}{\mathcal{O}}_{L^{\prime}}:={\mathcal{O}}^{\times}_{L^{\prime}}$.

  3. (3)

We let ${\rm fil}_{n}H^{q}(L)=0$ for $n<0$.

It follows from [23, Lem. 2.2] that

(8.6) $H^{q}(L)={\underset{n\geq 0}{\bigcup}}{\rm fil}_{n}H^{q}(L).$

We shall call ${\rm fil}_{\bullet}H^{q}(L):=\\{{\rm
fil}_{n}H^{q}(L)\\}_{n\in{\mathbb{Z}}}$, the ramification filtration of
$H^{q}(L)$. Recall that for any $\chi\in H^{q}(L)$, the Swan conductor ${\rm
sw}(\chi)$ is the smallest integer $n$ such that $\chi\in{\rm
fil}_{n}H^{q}(L)$. The following result shows that one can characterize ${\rm
fil}_{\bullet}H^{q}(L)$ purely in terms of characters of the Milnor $K$-groups
of $L$ in cases of our interest.

######  Lemma 8.4.

Assume that $p>0$ and ${\mathfrak{l}}$ is a local field. Let $1\leq q\leq 2$
and $n\geq 0$ be integers. Then the following hold.

  1. (1)

There are canonical isomorphisms

$H^{3}(L)\xrightarrow{\cong}H^{3}(\widehat{L})\xrightarrow{\cong}H^{2}({\mathfrak{l}})\xrightarrow{\cong}{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

  2. (2)

The canonical map $H^{q}({\mathcal{O}}_{L})\to H^{q}(L)$ induced by the
inclusion ${\mathcal{O}}_{L}\hookrightarrow L$ fits into a split short exact
sequence

$0\to H^{q}({\mathcal{O}}_{L})\to{\rm fil}_{1}H^{q}(L)\to
H^{q-1}({\mathfrak{l}})\to 0,$

which is canonical for a given choice of $\pi_{L}$.

  3. (3)

${\rm fil}_{0}H^{q}(L)=H^{q}({\mathcal{O}}_{L})$.

  4. (4)

For $n\geq 1$, an element $\chi\in H^{q}(L)$ lies in ${\rm fil}_{n}H^{q}(L)$
if and only if $\langle\chi,{\rm fil}_{n}{K}^{M}_{3-q}(L)\rangle=0$ under the
pairing

$H^{q}(L)\times{K}^{M}_{3-q}(L)\xrightarrow{\langle,\rangle}H^{3}(L)\xrightarrow{\cong}{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

###### Proof.

Part (1) of the lemma is a special case of the general isomorphism ($\forall\
q$)

(8.7) $H^{q}(L)\xrightarrow{\cong}H^{q}(\widehat{L})$

as shown in [24, Lem. 21], and the well known case of complete discrete
valuation fields (see [21, § 3.2, Prop. 1]). For $q=1$, the other parts of the
lemma follow from [13, Thm. 6.3]. For $q=2$, part (2) follows directly from
[23, Prop. 6.1].

To prove (3), suppose first that $\chi\in{\rm fil}_{0}H^{2}(L)$. We consider
the commutative diagram of exact sequences

(8.8)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}({\mathcal{O}}_{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}_{1}H^{2}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha}$$\textstyle{H^{1}({\mathfrak{l}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\mathbb{Z}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(L^{\times})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{({\mathcal{O}}^{\times}_{L})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the vertical arrows are induced by (8.5). Our hypothesis says that
$\beta\circ\alpha(\chi)=0$. Since the right vertical arrow is injective by the
class field theory of local fields, it follows that $\chi\in
H^{2}({\mathcal{O}}_{L})$. Conversely, suppose that $\chi\in
H^{2}({\mathcal{O}}_{L})$. Then a diagram chase of (8.8) tells us that
$\chi\in{\rm fil}_{1}H^{2}(L)$ such that $\beta\circ\alpha(\chi)=0$. But this
is equivalent to saying that $\chi\in{\rm fil}_{0}H^{2}(L)$. This proves (3).

We now assume $n\geq 1$ and $q=2$. In view of [23, Prop. 6.3], we only need to
show that if $\chi\in H^{2}(L)$ is a character such that $\langle\chi,{\rm
fil}_{n}L^{\times}\rangle=0$, then it lies in ${\rm fil}_{n}H^{2}(L)$. At any
rate, we know that $\chi\in{\rm fil}_{m}H^{2}(L)$ for some $m\gg 0$. We can
assume that $m>n$ (else we are done). Our hypothesis implies that as a
character of $L^{\times}$, $\chi$ factors through ${L^{\times}}/{{\rm
fil}_{n}L^{\times}}$.

We now note that the ring of integers ${\mathcal{O}}_{\widehat{L}}$ of
$\widehat{L}$ is the ${\mathfrak{m}}_{L}$-adic completion
$\widehat{{\mathcal{O}}_{L}}$ of ${\mathcal{O}}_{L}$. Since
${{\mathcal{O}}^{\times}_{L}}/{{\rm
fil}_{n}L^{\times}}\xrightarrow{\cong}{{\mathcal{O}}^{\times}_{\widehat{L}}}/{{\rm
fil}_{n}{\widehat{L}^{\times}}}$, it follows from the commutative diagram

(8.9)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}^{\times}_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{L}}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}^{\times}_{\widehat{L}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\widehat{L}}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{\widehat{L}}}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

of exact sequences that ${L^{\times}}/{{\rm
fil}_{n}L^{\times}}\xrightarrow{\cong}{{\widehat{L}^{\times}}}/{{\rm
fil}_{n}{\widehat{L}^{\times}}}$. Hence, $\chi$ factors through
${{\widehat{L}^{\times}}}/{{\rm fil}_{n}{\widehat{L}^{\times}}}$ as a
character of $\widehat{L}^{\times}$. We conclude from [23, Prop. 6.3, Rem.
6.6] that $\chi\in{\rm fil}_{n}H^{2}(\widehat{L})$.

By an iteration of the commutative diagram

(8.10)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}_{m-1}H^{2}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}_{m}H^{2}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{{\rm
fil}_{m}H^{2}(L)}{{\rm
fil}_{m-1}H^{2}(L)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}_{m-1}H^{2}(\widehat{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}_{m}H^{2}(\widehat{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{{\rm
fil}_{m}H^{2}(\widehat{L})}{{\rm
fil}_{m-1}H^{2}(\widehat{L})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

it remains to show that the right vertical arrow in this diagram is injective.

We now look at the diagram

(8.11) $\textstyle{\frac{{\rm fil}_{m}H^{2}(L)}{{\rm
fil}_{m-1}H^{2}(L)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
rsw}_{L}}$$\textstyle{\Omega^{2}_{{\mathfrak{l}}}\oplus\Omega^{1}_{{\mathfrak{l}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\frac{{\rm
fil}_{m}H^{2}(\widehat{L})}{{\rm
fil}_{m-1}H^{2}(\widehat{L})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
rsw}_{\widehat{L}}}$$\textstyle{\Omega^{2}_{{\mathfrak{l}}}\oplus\Omega^{1}_{{\mathfrak{l}}}.}$

Since $\pi_{L}$ is also a uniformizer of ${\mathcal{O}}_{\widehat{L}}$, it
follows from [23, Thm. 0.1] that this diagram is commutative and the
horizontal arrows are injective. It follows that the left vertical arrow is
injective. This concludes the proof. ∎

We shall use the following result of Kato to prove various functorial
properties of the Brauer and Picard groups of modulus pairs.

######  Lemma 8.5.

Let ${L^{\prime}}/L$ be a finite extension of Henselian discrete valuation
fields of characteristic $p>0$. Let $e$ denote the ramification index of
${L^{\prime}}/L$. Let ${\mathfrak{l}}$ (resp. ${\mathfrak{l}}^{\prime}$)
denote the residue field of $L$ (resp. $L^{\prime}$). Assume that
${\mathfrak{l}}$ and ${\mathfrak{l}}^{\prime}$ are local fields. Then we have
the following.

  1. (1)

There is a commutative diagram

(8.12) $\textstyle{H^{2}(L^{\prime})\times
L^{\prime\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{{\rm
Cores}}$$\textstyle{H^{3}(L^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{L^{\prime}}}$$\scriptstyle{{\rm
Cores}}$$\textstyle{H^{2}({\mathfrak{l}}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Cores}}$$\scriptstyle{{\rm
inv}_{{\mathfrak{l}}^{\prime}}}$$\textstyle{H^{2}(L)\times
L^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Res}}$$\scriptstyle{\cup}$$\textstyle{H^{3}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{L}}$$\textstyle{H^{2}({\mathfrak{l}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
inv}_{{\mathfrak{l}}}}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}.}$

The same holds if we interchange ${\rm Cores}$ and ${\rm Res}$ on the left
side of the cup product maps.

  2. (2)

${\rm Cores}({\rm fil}_{en}H^{2}(L^{\prime}))\subseteq{\rm fil}_{n}H^{2}(L)$
for any $n\geq 0$.

  3. (3)

${\rm Res}({\rm fil}_{n}H^{2}(L))\subseteq{\rm fil}_{en}H^{2}(L^{\prime})$ for
any $n\geq 0$.

###### Proof.

(1) follows from [21, § 3.2, Lem. 1, Prop. 1(2)], (2) follows from (1) using
that ${\rm Res}({\rm fil}_{n}L^{\times})\subset{\rm fil}_{en}L^{\prime\times}$
by definition, and (3) follows from (1) and [42, Lem. 6.19]. ∎

###  8.3. The Brauer group with modulus

Let $k$ be a local field with ring of integers ${\mathcal{O}}_{k}$, maximal
ideal ${\mathfrak{m}}=(\pi)$ and residue field
${\mathfrak{f}}={{\mathcal{O}}_{k}}/{{\mathfrak{m}}}\cong{\mathbb{F}}_{q}$,
where $q=p^{n}$ for some prime $p\geq 2$ and integer $n\geq 1$. Let $X$ be a
connected regular quasi-projective scheme over $k$ of dimension $d\geq 1$. Let
$D\subset X$ be an effective divisor (possibly empty) and $D^{\dagger}=D_{\rm
red}$. We write $D=\sum_{x\in X^{(1)}}n_{x}\overline{\\{x\\}}$. Clearly,
$n_{x}=0$ unless $x$ is a generic point of $D^{\dagger}$, and $D=0$ if
$D^{\dagger}$ is empty. We let ${\rm Div}_{D^{\dagger}}(X)$ be the set of all
effective divisors on $X$ whose support is $D^{\dagger}$. This is a filtered
set under inclusion. We let ${\rm Div}(X)$ denote the filtered set of all
effective Cartier (equivalently Weil) divisors on $X$. We let
$X^{o}=X\setminus D$. We let $j\colon X^{o}\hookrightarrow X$ and $t\colon
D\hookrightarrow X$ be the inclusions. We let $K$ denote the function field of
$X$.

For any $x\in X^{(1)}$, we let $K_{x}$ denote the quotient field of
${\mathcal{O}}^{h}_{X,x}$ and $\widehat{K}_{x}$ the quotient field of
$\widehat{{\mathcal{O}}_{X,x}}$. We shall say that an element $\chi\in
H^{i}(K)$ is unramified at a point $x\in X^{(1)}$ if the image of $\chi$ under
the canonical map $H^{i}(K)\to H^{i}(K_{x})$ lies in the image of
$H^{i}({\mathcal{O}}^{h}_{X,x})$. We shall say that $\chi$ is unramified on an
open subscheme $U\subseteq X$ if it lies in the image of the canonical map
$H^{i}(U)\to H^{i}(K)$.

######  Lemma 8.6.

Let $U$ be a connected Noetherian regular scheme with function field $K$. Let
$\chi\in{\operatorname{Br}}(K)$ be unramified at all codimension one points of
$U$. Then $\chi$ is unramified on $U$.

###### Proof.

This is an easy consequence of the purity theorem for Brauer group, see [4,
Thm. 3.7.7]. ∎

######  Definition 8.7.

We let ${\operatorname{Br}}^{\rm div}(X|D)$ denote the subgroup of
${\operatorname{Br}}(K)$ consisting of elements $\chi$ such that for every
$x\in X^{(1)}$, the image $\chi_{x}$ of $\chi$ under the canonical map
${\operatorname{Br}}(K)=H^{2}(K)\to H^{2}(K_{x})$ lies in ${\rm
fil}_{n_{x}}H^{2}(K_{x})$.

######  Lemma 8.8.

We have the following relations between the subgroups of
${\operatorname{Br}}(K)$.

  1. (1)

For every pair of effective Cartier divisors $D\leq D^{\prime}$ on $X$, one
has

${\operatorname{Br}}(X)\subseteq{\operatorname{Br}}^{\rm
div}(X|D)\subseteq{\operatorname{Br}}^{\rm
div}(X|D^{\prime})\subseteq{\operatorname{Br}}(X^{o})\subseteq{\operatorname{Br}}(K).$

  2. (2)

${\underset{n\geq 1}{\varinjlim}}\ {\operatorname{Br}}^{\rm div}(X|nD)\
\xrightarrow{\cong}{\underset{D^{\prime}\in{\rm
Div}_{D^{\dagger}}(X)}{\varinjlim}}{\operatorname{Br}}^{\rm
div}(X|D^{\prime})\xrightarrow{\cong}{\operatorname{Br}}(X^{o}).$

  3. (3)

${\underset{D^{\prime}\in{\rm Div}(X)}{\varinjlim}}{\operatorname{Br}}^{\rm
div}(X|D^{\prime})\xrightarrow{\cong}{\operatorname{Br}}(K).$

  4. (4)

If ${\rm char}(k)=0$, then ${\operatorname{Br}}^{\rm
div}(X|D)\xrightarrow{\cong}{\operatorname{Br}}(X^{o})$ for every $D\in{\rm
Div}_{D^{\dagger}}(X)$.

###### Proof.

Part (1) follows easily from Lemmas 8.4 and 8.6. To prove (2), let
$w\in{\operatorname{Br}}(X^{o})$. For every irreducible component
$D^{\dagger}_{i}$ of $D^{\dagger}$, (8.6) says that the image of $w$ in
${\operatorname{Br}}(K_{x_{i}})$ lies in ${\rm fil}_{m_{i}}H^{2}(K_{x_{i}})$
for some $m_{i}\geq 0$, where $D^{\dagger}_{i}=\overline{\\{x_{i}\\}}$. We now
take $D=\sum_{i}m_{i}D^{\dagger}_{i}\in{\rm Div}_{D^{\dagger}}(X)$. Then it is
clear that $w\in{\operatorname{Br}}^{\rm div}(X|D)$. Part (3) follows from (2)
and [35, Lem. III.1.16], and (4) follows from the definition of
${\operatorname{Br}}^{\rm div}(X|D)$. ∎

We shall now define the Brauer group of a modulus pair. We let
${\mathcal{C}}(X)$ denote the set of integral curves on $X$ and let
${\mathcal{C}}(X|D)$ denote the subset of ${\mathcal{C}}(X)$ consisting of
those curves which are not contained in $D$. For any $C\in{\mathcal{C}}(X)$,
we let $\nu\colon C_{n}\to X$ denote the canonical map from the normalization
of $C$ and let $\nu^{*}(D)$ denote the scheme theoretic Pull-back of $D$. We
let $C^{o}_{n}=\nu^{-1}(X^{o})$.

######  Definition 8.9.

We let ${\operatorname{Br}}(X|D)$ denote the subgroup of
${\operatorname{Br}}(X^{o})$ consisting of elements $\chi$ such that for every
$C\in{\mathcal{C}}(X|D)$, the Brauer class
$\nu^{*}(\chi)\in{\operatorname{Br}}(C^{o}_{n})$ lies in the subgroup
${\operatorname{Br}}^{\rm div}(C_{n}|\nu^{*}(D))$. The group
${\operatorname{Br}}(X|D)$ will be called the Brauer group of the modulus pair
$(X,D)$.

The Brauer group of modulus pairs has the following functorial properties
first of which is not clear for ${\operatorname{Br}}^{\rm div}(X|D)$. Recall
that an admissible (resp. coadmissible) morphism of modulus pairs
$f\colon(X^{\prime},D^{\prime})\to(X,D)$ is a morphism of schemes $f\colon
X^{\prime}\to X$ such that $D^{\prime}\leq f^{*}(D)$ (resp. $D^{\prime}\geq
f^{*}(D)$) as Cartier divisors on $X^{\prime}$. One says that $f$ is strict if
$D^{\prime}=f^{*}(D)$.

######  Proposition 8.10.

Let $f\colon(X^{\prime},D^{\prime})\to(X,D)$ be a coadmissible morphism of
modulus pairs over $k$. Then the pull-back on étale cohomology induces a
homomorphism

$f^{*}\colon{\operatorname{Br}}(X|D)\to{\operatorname{Br}}(X^{\prime}|D^{\prime})$

such that
$f^{*}(({\operatorname{Br}}(X|E))\subset{\operatorname{Br}}(X^{\prime}|E^{\prime})$
if $E\leq D$ and $D^{\prime}\geq E^{\prime}\geq f^{*}(E)$. If
$g\colon(X^{\prime\prime},D^{\prime\prime})\to(X^{\prime},D^{\prime})$ is
another coadmissible morphism of modulus pairs, then $(f\circ
g)^{*}=g^{*}\circ f^{*}$.

###### Proof.

We let $U^{\prime}=f^{-1}(X^{o})$ so that $X^{\prime o}\subset U^{\prime}$. We
then have the pull-back map
$f^{*}\colon{\operatorname{Br}}(X^{o})\to{\operatorname{Br}}(U^{\prime})\subset{\operatorname{Br}}(X^{\prime
o})$. Suppose now that $\chi\in{\operatorname{Br}}(X|D)$ and let
$C^{\prime}\in{\mathcal{C}}(X^{\prime}|D^{\prime})$. If the image of
$C^{\prime}$ under $f$ is a closed point, then this closed point must lie in
$X^{o}$. In the latter case, it is clear that the pull-back of $\chi$ under
the composite map $C^{\prime}_{n}\to U^{\prime}\to X^{o}$ lies in
${\operatorname{Br}}(C^{\prime}_{n})\subset{\operatorname{Br}}(C^{\prime}_{n}|\nu^{\prime*}(D^{\prime}))={\operatorname{Br}}^{\rm
div}(C^{\prime}_{n}|\nu^{\prime*}(D^{\prime}))$.

Otherwise, the scheme theoretic image of $C^{\prime}$ under $f$ is a curve $C$
which is necessarily integral and lies in ${\mathcal{C}}(X|D)$. Since $f\colon
C^{\prime}\to C$ is also dominant, we get a commutative diagram

(8.13)
$\textstyle{C^{\prime}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{\prime}}$$\scriptstyle{g}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{C_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{X.}$

If we let $E=\nu^{*}(D)$ and $E^{\prime}=\nu^{\prime*}(D^{\prime})$, then we
get $E^{\prime}\geq g^{*}(E)$ by our assumption. In other words,
$g\colon(C^{\prime}_{n},E^{\prime})\to(C_{n},E)$ is a dominant morphism of
one-dimensional modulus pairs. We let $L$ (resp. $L^{\prime}$) be the function
field of $C$ (resp. $C^{\prime}$).

We now let $\omega=\nu^{*}(\chi)$ and let $x\in E^{\prime}$ be a closed point.
If $x\notin g^{-1}(E)$, then $g^{*}(\chi)\in{\rm
fil}_{0}H^{2}(L^{\prime}_{x})$. If $x\in g^{-1}(E)$, we let $y=g(x)$. We then
get an inclusion of Henselian discrete valuation fields $L_{y}\hookrightarrow
L^{\prime}_{x}$. It follows from Definition 8.3 that this inclusion induces a
map ${\operatorname{Br}}(L_{y})\to{\operatorname{Br}}(L^{\prime}_{x})$ which
preserves the ramification filtrations. In particular, if $n_{y}$ (resp.
$n_{x}$) denotes the multiplicity of $E$ (resp. $E^{\prime}$) at $y$ (resp.
$x$), then we get $g^{*}(w_{y})\in{\rm
fil}_{n_{y}}{\operatorname{Br}}(L^{\prime}_{x})\subset{\rm
fil}_{n_{x}}{\operatorname{Br}}(L^{\prime}_{x})$, where the latter inclusion
holds because $E^{\prime}\geq g^{*}(E)$. It follows that
$g^{*}(\omega)\in{\operatorname{Br}}(C^{\prime}_{n}|E^{\prime})$. We have thus
shown that $f^{*}(\chi)\in{\operatorname{Br}}(X^{\prime}|D^{\prime})$. The
inclusion
$f^{*}(({\operatorname{Br}}(X|E))\subset{\operatorname{Br}}(X^{\prime}|E^{\prime})$,
as well as the composition law, is clear from the definition of the pull-back
map. This concludes the proof. ∎

######  Proposition 8.11.

Let $f\colon(X^{\prime},D^{\prime})\to(X,D)$ be a finite morphism of modulus
pairs over $k$. Assume that $f$ is strict. Then the norm map
$f_{*}({\mathcal{O}}^{\times}_{X^{\prime}})\to{\mathcal{O}}^{\times}_{X}$
induces a homomorphism

$f_{*}\colon{\operatorname{Br}}(X^{\prime}|D^{\prime})\to{\operatorname{Br}}(X|D)$

such that
$f_{*}(({\operatorname{Br}}(X^{\prime}|E^{\prime}))\subset{\operatorname{Br}}(X|E)$
if $E\leq D$ and $E^{\prime}=f^{*}(E)$. If
$g\colon(X^{\prime\prime},D^{\prime\prime})\to(X^{\prime},D^{\prime})$ is
another finite and strict morphism of modulus pairs, then $(f\circ
g)_{*}=f_{*}\circ g_{*}$.

###### Proof.

Since $f$ is strict, we have $X^{\prime o}=f^{-1}(X^{o})$ and the norm
$f_{*}({\mathcal{O}}^{\times}_{X^{\prime}})\to{\mathcal{O}}^{\times}_{X}$
induces $f_{*}\colon{\operatorname{Br}}(X^{\prime
o})\to{\operatorname{Br}}(X^{o})$. To show that $f_{*}$ preserves the
ramification filtration, we can assume that $\text{\rm dim}(X)=\text{\rm
dim}(X^{\prime})=1$. We let $K^{\prime}$ be the function field of
$X^{\prime}$.

We let $w\in{\operatorname{Br}}(X^{\prime}|D^{\prime})$ and $x\in
D^{\dagger}$. By [4, Prop. 3.8.1, Lem. 3.8.6], there is a commutative diagram

(8.14) $\textstyle{{\operatorname{Br}}(X^{\prime
o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$$\textstyle{{\underset{y\in
f^{-1}(x)}{\prod}}\
{\operatorname{Br}}(A^{\prime}_{y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}={\rm
Cores}}$$\textstyle{{\operatorname{Br}}(X^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{Br}}(K_{x}),}$

where ${\underset{y\in f^{-1}(x)}{\prod}}A^{\prime}_{y}=X^{\prime
o}\times_{X^{o}}{\rm Spec\,}(K_{x})$ and the horizontal arrows are the pull-
back maps. We fix $y\in f^{-1}(x)$ and let $w_{y}$ be the restriction of $w$
to
${\operatorname{Br}}(A^{\prime}_{y})\cong{\operatorname{Br}}(K^{\prime}_{y})$.
It suffices to show that ${\rm Cores}_{{K^{\prime}_{y}}/{K_{x}}}(w_{y})\in{\rm
fil}_{n_{x}}{\operatorname{Br}}(K_{x})$. But this follows from Lemma 8.5
because $n_{y}=n_{x}e_{y/x}$, where $e_{y/x}$ is the ramification index of
${K^{\prime}_{y}}/{K_{x}}$ by the definition of $f^{*}(D)$. ∎

It is clear that ${\operatorname{Br}}^{\rm
div}(X|D)\cong{\operatorname{Br}}(X|D)$ when $X$ is a curve. For future study,
it will be of interest to know an answer to the following.

######  Question 8.12.

Let $(X,D)$ be a modulus pair over $k$. Is it true that
${\operatorname{Br}}^{\rm div}(X|D)\cong{\operatorname{Br}}(X|D)$?

##  9\. Brauer-Manin pairing for modulus pairs

In this section, we shall define an extension of the classical Brauer-Manin
pairing between the Picard and Brauer groups of smooth projective curves over
local fields to setting of relative Picard and Brauer groups of modulus pairs.
Throughout this section, we fix a local field $k$ of exponential
characteristic $p\geq 1$ and a connected regular projective scheme $X$ of
dimension $d\geq 1$ over $k$. We also fix a divisor $D\subset X$ and let
$D^{\dagger}=D_{\rm red}$. We let $j\colon X^{o}\hookrightarrow X$ and
$\iota\colon D\hookrightarrow X$ be the inclusions, where $X^{o}=X\setminus
D$. We let $K$ denote the function field of $X$. All other notations will be
those of § 8.3.

###  9.1. Idele class group

For any $C\in{\mathcal{C}}(X)$, we let $k(C)$ be the function field of $C$.
Given $C\in{\mathcal{C}}(X|D)$, we let
$k(C)_{\infty}={\underset{x\in\nu^{-1}(D^{\dagger})}{\bigoplus}}k(C)_{x}$,
where $\nu\colon C_{n}\to X$ is the canonical map from the normalization of
$C$. We let
$k(C)^{\times}_{\infty}={\underset{x\in\nu^{-1}(D^{\dagger})}{\bigoplus}}k(C)^{\times}_{x},\
I(C_{n}|D)={\underset{x\in\nu^{-1}(D^{\dagger})}{\bigoplus}}(1+{\mathcal{I}}_{D}{\mathcal{O}}^{h}_{C_{n},x})$
and
$\widehat{I}(C_{n}|D)={\underset{x\in\nu^{-1}(D^{\dagger})}{\bigoplus}}(1+{\mathcal{I}}_{D}\widehat{{\mathcal{O}}_{C_{n},x}})$,
where ${\mathcal{I}}_{D}\subset{\mathcal{O}}_{X}$ is the ideal sheaf defining
$D$. The inclusions $k(C)\hookrightarrow k(C)_{x}$ induce a canonical map
$\iota_{C}\colon k(C)^{\times}\to{k(C)^{\times}_{\infty}}/{I(C_{n}|D)}$. We
also have the map $\partial_{C_{n}}\colon k(C)^{\times}\xrightarrow{{\rm
div}}{\mathcal{Z}}_{0}(C_{n}^{o})\xrightarrow{\nu_{*}}{\mathcal{Z}}_{0}(X^{o})$,
where $C^{o}_{n}=\nu^{-1}(X^{o})$. We let
${\mathcal{O}}_{C_{n},\infty}={\mathcal{O}}_{C_{n},\nu^{-1}(D^{\dagger})}$.

It is easy to check that the left square in the diagram

(9.1)
$\textstyle{{\underset{C\in{\mathcal{C}}(X|D)}{\bigoplus}}{K}^{M}_{1}({\mathcal{O}}_{C_{n},\infty},{\mathcal{I}}_{D}{\mathcal{O}}_{C_{n},\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
div}}$$\textstyle{{\mathcal{Z}}_{0}(X^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{{\underset{C\in{\mathcal{C}}(X|D)}{\bigoplus}}k(C)^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\rm
div},\iota_{C})}$$\textstyle{{\mathcal{Z}}_{0}(X^{o})\bigoplus\left({\underset{C\in{\mathcal{C}}(X|D)}{\bigoplus}}\frac{k(C)^{\times}_{\infty}}{I(C_{n}|D)}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}(X|D)^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is commutative, where the left vertical arrow is the canonical inclusion, the
middle vertical arrow is the identity map of ${\mathcal{Z}}_{0}(X^{o})$ and
${\rm CH}_{0}(X|D)^{\prime}$ is the cokernel of the left horizontal arrow in
the bottom row. A straightforward application of the weak approximation
theorem (e.g., see [28, Lem. 6.3]) allows one to conclude that the left square
induces a natural isomorphism ${\rm CH}_{0}(X|D)\xrightarrow{\cong}{\rm
CH}_{0}(X|D)^{\prime}$. In other words, there is a canonical exact sequence

(9.2)
${\underset{C\in{\mathcal{C}}(X|D)}{\bigoplus}}k(C)^{\times}\xrightarrow{({\rm
div},\iota_{C})}{\mathcal{Z}}_{0}(X^{o})\bigoplus\left({\underset{C\in{\mathcal{C}}(X|D)}{\bigoplus}}\frac{k(C)^{\times}_{\infty}}{I(C_{n}|D)}\right)\to{\rm
CH}_{0}(X|D)\to 0.$

######  Remark 9.1.

We remark that the canonical map
$\frac{k(C)^{\times}_{\infty}}{I(C_{n}|D)}\to\frac{\widehat{k(C)}^{\times}_{\infty}}{\widehat{I}(C_{n}|D)}$
is an isomorphism for every $C\in{\mathcal{C}}(X|D)$. In particular, the exact
sequence (9.2) remains unchanged if we replace $k(C)^{\times}_{\infty}$ by
$\widehat{k(C)}^{\times}_{\infty}$ and $I(C_{n}|D)$ by $\widehat{I}(C_{n}|D)$
for $C\in{\mathcal{C}}(X|D)$. This can be easily verified.

######  Definition 9.2.

The idele class group $C(X^{o})$ is the cokernel of the map

${\underset{C\in{\mathcal{C}}(X|D)}{\bigoplus}}k(C)^{\times}\xrightarrow{({\rm
div},\iota_{C})}{\mathcal{Z}}_{0}(X^{o})\bigoplus\left({\underset{C\in{\mathcal{C}}(X|D)}{\bigoplus}}{\widehat{k(C)}^{\times}_{\infty}}\right).$

One can show using [15, Prop. 5.3] that $C(X^{o})$ depends only on $X^{o}$ and
not on $X$. Using Remark 9.1, it is clear that there is a canonical surjective
morphism of pro-abelian groups $\\{C(X^{o})\\}\twoheadrightarrow\\{{\rm
CH}_{0}(X|D)\\}_{D\in{\rm Div}_{D^{\dagger}}(X)}$. Taking the limit, we get a
homomorphism of abelian groups

(9.3) $\theta_{X^{o}}\colon C(X^{o})\to{\underset{D\in{\rm
Div}_{D^{\dagger}}(X)}{\varprojlim}}\ {\rm CH}_{0}(X|D).$

It is well known (e.g., see the proof of [15, Prop. 5.3]) that the sequence
(9.2) and the one in the definition of $C(X^{o})$ are covariantly functorial
for a finite morphism of modulus pairs
$f\colon(X^{\prime},D^{\prime})\to(X,D)$. Hence, they induce push-forward maps
$f_{*}\colon{\rm CH}_{0}(X^{\prime}|D^{\prime})\to{\rm CH}_{0}(X|D)$ and
$f_{*}\colon C(X^{\prime o})\to C(X^{o})$. Furthermore, (9.3) is compatible
with the push-forward maps.

###  9.2. The pairing between Brauer and Chow groups with modulus

Let the notations be as above. We shall now define a bilinear pairing between
the Brauer group and the Chow group of the modulus pair $(X,D)$. In the
following sections, we shall prove several properties of this pairing which
will be key to the proofs of our main results.

Given a closed point $x\in X^{o}$ with the inclusion $\iota_{x}\colon{\rm
Spec\,}(k(x))\hookrightarrow X^{o}$, we have the pull-back map
$\iota^{*}_{x}\colon{\operatorname{Br}}(X^{o})\to{\operatorname{Br}}(k(x))$.
By composing with ${\rm inv}_{k(x)}$, we get a canonical map
${\iota}^{*}_{x}\colon{\operatorname{Br}}(X^{o})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$.
We thus get a bilinear pairing
${\operatorname{Br}}(X^{o})\times{\mathcal{Z}}_{0}(X^{o})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$
given by
$\langle\omega,\sum_{i}n_{i}[x_{i}]\rangle=\sum_{i}n_{i}{\iota}^{*}_{x_{i}}(\omega)$.

Suppose now that $d=1$ and $x\in D$, where $D=\sum_{x}n_{x}[x]$. Since $K_{x}$
is a Henselian discrete valuation field whose residue field is a local field,
the cup product of (8.5) and Lemma 8.4(1) (we note here that Lemma 8.4(1)
holds in characteristic zero as well, see [21, p. 311]) together induce a
bilinear pairing ${\operatorname{Br}}(K_{x})\times
K^{\times}_{x}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$. By composing with the
canonical map ${\operatorname{Br}}(X^{o})\to{\operatorname{Br}}(K_{x})$, this
yields a pairing ${\operatorname{Br}}(X^{o})\times
K^{\times}_{x}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$. Moreover, it follows from
Definition 8.3 that the induced pairing ${\operatorname{Br}}(X|D)\times
K^{\times}_{x}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$ uniquely factors through
${\operatorname{Br}}(X|D)\times{K^{\times}_{x}}/{{\rm
fil}_{n_{x}}K^{\times}_{x}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$. Summing over the
points of $D$, we get a pairing
${\operatorname{Br}}(X|D)\times{K^{\times}_{\infty}}/{I(X|D)}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$.

It follows from the above discussion that for any $d\geq 1$, there exist
bilinear pairings

(9.4)
${\operatorname{Br}}(X^{o})\times\left[{\mathcal{Z}}_{0}(X^{o})\bigoplus\left({\underset{C\in{\mathcal{C}}(X|D)}{\bigoplus}}{k(C)^{\times}_{\infty}}\right)\right]\to{{\mathbb{Q}}}/{{\mathbb{Z}}};\
\ \mbox{and}$ (9.5)
${\operatorname{Br}}(X|D)\times\left[{\mathcal{Z}}_{0}(X^{o})\bigoplus\left({\underset{C\in{\mathcal{C}}(X|D)}{\bigoplus}}\frac{k(C)^{\times}_{\infty}}{I(C_{n}|D)}\right)\right]\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

One checks (cf. Remark 9.1) that the pairing (9.5) remains unchanged if we
replace $k(C)^{\times}_{\infty}$ by $\widehat{k(C)}^{\times}_{\infty}$ and
$I(C_{n}|D)$ by $\widehat{I}(C_{n}|D)$ for $C\in{\mathcal{C}}(X|D)$.

We recall the following reciprocity law due to Kato-Saito [26] when $d=1$. Let
$m\neq 0$ be an integer. Using the localization sequence, together with the
purity theorem (due to Gabber when $p\nmid m$) and Corollary 4.6 when
$m=p^{n}$), we get maps

(9.6) $H^{3}(K,{{\mathbb{Z}}}/m(2))\xrightarrow{\partial_{X}}{\underset{x\in
X_{(0)}}{\bigoplus}}H^{2}(k(x),{{\mathbb{Z}}}/m(1))\xrightarrow{\sum_{x}{\rm
inv}_{k(x)}}{{\mathbb{Z}}}/m.$

######  Lemma 9.3.

The composite map $(\sum_{x}{\rm inv}_{k(x)})\circ\partial_{X}$ in (9.6) is
zero.

###### Proof.

This is proven in [26, § 5, p. 120]. ∎

######  Proposition 9.4.

The pairings (9.4) and (9.5) descend to bilinear maps of abelian groups

${\operatorname{Br}}(X^{o})\times C(X^{o})\to{{\mathbb{Q}}}/{{\mathbb{Z}}};\
\mbox{and}$ ${\operatorname{Br}}(X|D)\times{\rm
CH}_{0}(X|D)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

###### Proof.

We let $C\in{\mathcal{C}}(X|D)$ and let $\nu\colon C_{n}\to X$ be the
canonical map from the normalization of $C$. We write $E=\nu^{*}(D)$. It
follows from Proposition 8.10 and the remark below (9.2) that there is a
diagram

(9.7)
$\textstyle{{\operatorname{Br}}(X|D)\times{\mathcal{Z}}_{0}(X^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{*}}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{id}}}$$\textstyle{{\operatorname{Br}}(C_{n}|E)\times{\mathcal{Z}}_{0}(C^{o}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{*}}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}.}$

It follows easily form the construction of (9.5) that this diagram is
commutative. Using this, and a similar commutative diagram associated to the
first pairing, it suffices to prove the proposition when $d=1$. We shall thus
assume that $X$ is a curve.

We let $w\in{\operatorname{Br}}(X^{o})$ and let
$\chi_{w}\colon{\mathcal{Z}}_{0}(X^{o})\bigoplus{K^{\times}_{\infty}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$
be the associated character. We shall show that the composite map

(9.8) $K^{\times}\to{\mathcal{Z}}_{0}(X^{o})\bigoplus
K^{\times}_{\infty}\xrightarrow{\chi_{w}}{{\mathbb{Q}}}/{{\mathbb{Z}}}$

is zero. This will automatically imply that the composite map

$K^{\times}\to{\mathcal{Z}}_{0}(X^{o})\bigoplus({K^{\times}_{\infty}}/{I(X|D)})\xrightarrow{\chi_{w}}{{\mathbb{Q}}}/{{\mathbb{Z}}}$

is zero if $w\in{\operatorname{Br}}(X|D)$.

We let $I(X):={\underset{x\in X_{(0)}}{\prod^{\prime}}}K^{\times}_{x}$ denote
the restricted product with respect to the subgroups
$({\mathcal{O}}^{h}_{X,x})^{\times}$. We consider the diagram

(9.9)
$\textstyle{K^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{({\rm
div},\iota_{X})}$$\textstyle{I(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\beta}$$\textstyle{{\rm
Hom}_{{\mathbf{Ab}}}(H^{2}(K),{{\mathbb{Q}}}/{{\mathbb{Z}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}_{0}(X^{o})\bigoplus{K^{\times}_{\infty}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{w}}$$\textstyle{{\rm
Hom}_{{\mathbf{Ab}}}({\operatorname{Br}}(X^{o}),{{\mathbb{Q}}}/{{\mathbb{Z}}}),}$

where $\alpha$ is the canonical inclusion, the bottom horizontal arrow is
induced by the pairing (9.8) and the right vertical arrow is induced by the
canonical inclusion
${\operatorname{Br}}(X^{o})\hookrightarrow{\operatorname{Br}}(K)$. To describe
$\beta$, we write $I(X)=I(X^{o})\bigoplus K^{\times}_{\infty}$. We let $\beta$
to be identity on $K^{\times}_{\infty}$ and $\beta(a_{x})=v_{x}(a_{x})$ if
$a_{x}\in K^{\times}_{x}$ with $x\in X^{o}$. This is well-defined because
$v_{x}(({\mathcal{O}}^{h}_{X,x})^{\times})=0$. It is clear that the left
triangle commutes.

To define $\gamma$, we let $\chi\in H^{2}(K)$ and $(a_{x})\in I(X)$. It
follows from Lemma 8.8(3) that $\chi\in{\operatorname{Br}}(U)$ for some dense
open subscheme $U\subset X^{o}$. We conclude by [46, Thm. 2.7(4)] that
$\langle\chi,a_{x}\rangle=0$ for all $x\in U$. In particular, the sum
$\sum_{x\in X_{(0)}}\langle\chi,a_{x}\rangle$ is finite. This shows that the
cup product of Lemma 8.4(4) induces a pairing

(9.10) $H^{2}(K)\times I(X)\to{{\mathbb{Q}}}/{{\mathbb{Z}}},$

which is clearly compatible with (9.4). Letting $\gamma$ denote the induced
map $I(X)\to{\rm
Hom}_{{\mathbf{Ab}}}(H^{2}(K),{{\mathbb{Q}}}/{{\mathbb{Z}}})$, it follows that
the right square in (9.9) is commutative. Hence, it suffices to show that
$\gamma\circ\alpha=0$.

We let $a\in K^{\times},\ \chi\in H^{2}(K)$ and write
$b=\langle\chi,a\rangle\in H^{3}(K)$. We let $a_{x}$ (resp. $\chi_{x}$) be the
image of $a$ (resp. $\chi$) in $K^{\times}_{x}$ (resp. $H^{3}(K_{x})$). We
need to show that $\sum_{x\in X_{(0)}}\langle\chi_{x},a_{x}\rangle=0$.
Equivalently, we need to show that $(\sum_{x}{\rm
inv}_{k(x)})\circ\partial_{X}(b)=0$ in (9.6). But this follows from Lemma 9.3.
∎

The following is straightforward from the construction of the pairings (9.4)
and (9.5).

######  Corollary 9.5.

There is a commutative diagram of bilinear pairings

(9.11)
$\textstyle{``{{\underset{n}{\varinjlim}}}"{\operatorname{Br}}(X|nD)\times``{{\underset{n}{\varprojlim}}}"{\rm
CH}_{0}(X|nD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{id}}}$$\textstyle{{\operatorname{Br}}(X^{o})\hskip
11.38092pt\times\hskip
11.38092ptC(X^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}}$

between ind and pro abelian groups.

The following corollary provides a quick proof of the Brauer-Manin duality
isomorphism $\beta_{X}\colon{\operatorname{Br}}(X)\xrightarrow{\cong}({\rm
CH}_{0}(X)^{\star})_{\rm tor}$ of Lichtenbaum [32] and Saito [46], assuming
that $\beta_{X}$ is injective. Recall that the main difficulty in proving that
$\beta_{X}$ is an isomorphism lies in showing its surjectivity.

######  Corollary 9.6.

Let $X$ be a geometrically connected smooth projective curve over $k$. Then
the pairing of Proposition 9.4 (with $D=\emptyset$) induces an isomorphism
$\beta_{X}\colon{\operatorname{Br}}(X)\xrightarrow{\cong}({\rm
CH}_{0}(X)^{\star})_{\rm tor}$.

###### Proof.

We fix an integer $n\geq 1$ and look at the diagram

(9.12)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\rm
CH}_{0}(X)}/{p^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{p^{n}}}$$\textstyle{H^{1}(X,W_{n}\Omega^{1}_{X,{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{p^{n}}}$$\textstyle{{}_{p^{n}}{\operatorname{Br}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{p^{n}}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(_{p^{n}}{\operatorname{Br}}(X))^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(X,W_{n}\Omega^{1}_{X,{\operatorname{log}}})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({{\rm
CH}_{0}(X)}/{p^{n}})^{\vee},}$

where $\alpha_{p^{n}}$ and $\gamma_{p^{n}}$ are induced by the pairing (9.11).
The map $\beta_{p^{n}}$ is induced by the pairing of Theorem 4.7. From the
definition of the torsion-by-profinite topology of
$H^{1}(X,W_{n}\Omega^{1}_{X,{\operatorname{log}}})$, one knows that the image
of the map ${{\rm CH}_{0}(X)}/{p^{n}}\to
H^{1}(X,W_{n}\Omega^{1}_{X,{\operatorname{log}}})$ in the top row (9.12) is
open (see (3.9) and § 3.2). In particular, the quotient topology of
${}_{p^{n}}{\operatorname{Br}}(X)$ is discrete. This shows that the bottom
arrow of (9.12) is defined and exact. It is an easy exercise that this diagram
is commutative.

The middle arrow is bijective by Theorem 4.7. Since $\gamma_{p^{n}}$ is known
to be injective (cf. Lemma 12.1), it follows that $\alpha_{p^{n}}$ is an
isomorphism. Since ${{\rm CH}_{0}(X)}/{p^{n}}$ is profinite111This is the only
place where the smoothness of $X$ is used., it follows from Lemma 2.5 and
Pontryagin duality between profinite and discrete torsion groups that
$\gamma_{p^{n}}$ maps ${}_{p^{n}}{\operatorname{Br}}(X)$ isomorphically onto
$({{\rm CH}_{0}(X)}/{p^{n}})^{\star}$. Taking the limit, we conclude that
$\beta_{X}\colon{\operatorname{Br}}(X)\\{p\\}\xrightarrow{\cong}({\rm
CH}_{0}(X)^{\star})\\{p\\}$ is an isomorphism. An identical argument (with
Theorem 4.7 replaced by Saito-Tate duality, see [15, Thm. 9.9]) shows that
$\beta_{X}\colon{\operatorname{Br}}(X)\\{p^{\prime}\\}\xrightarrow{\cong}({\rm
CH}_{0}(X)^{\star})\\{p^{\prime}\\}$. is also bijective. This concludes the
proof. ∎

We end this section with the following functorial property of the Brauer-Manin
pairings of modulus pairs.

######  Lemma 9.7.

Let $f\colon(X^{\prime},D)\to(X,D)$ be a finite and strict morphism between
modulus pairs, where $X$ and $X^{\prime}$ are connected regular projective
curves over $k$. Then we have a commutative diagram

(9.13) $\textstyle{{\operatorname{Br}}(X^{\prime}|D^{\prime})\times{\rm
CH}_{0}(X^{\prime}|D^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{id}}}$$\textstyle{{\operatorname{Br}}(X|D)\times{\rm
CH}_{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}.}$

###### Proof.

We first note that $f^{*}$ is the well-known flat pull-back map between the
Chow groups with modulus and $f_{*}$ is defined by Proposition 8.11. To show
the commutativity of (9.13), we fix a Brauer class
$w^{\prime}\in{\operatorname{Br}}(X^{\prime}|D^{\prime})$ and let
$w=f_{*}(w^{\prime})$. We need to show that the right side triangle in the
diagram

(9.14)
$\textstyle{{\mathcal{Z}}_{0}(X^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{{\rm
CH}_{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\scriptstyle{\chi_{w}}$$\textstyle{{\mathcal{Z}}_{0}(X^{\prime
o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}(X^{\prime}|D^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{w^{\prime}}}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}}$

is commutative. Since the left square is commutative and its top horizontal
arrow is surjective, it suffices to show that the outer trapezium in (9.14) is
commutative when evaluated on every free generator of
${\mathcal{Z}}_{0}(X^{o})$.

Let $x\in X^{o}$ be a closed point and let $f^{*}([x])=\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}n_{i}[y_{i}]$. Let $\iota_{x}\colon{\rm
Spec\,}(k(x))\hookrightarrow X^{o}$ and $\iota_{y_{i}}\colon{\rm
Spec\,}(k(y_{i}))\hookrightarrow X^{\prime o}$ be the inclusion maps. By the
definition of the Brauer-Manin pairing in § 9.2, we need to show that ${\rm
inv}_{k(x)}(\iota^{*}_{x}(w))=\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}n_{i}{\rm
inv}_{k_{y_{i}}}(\iota^{*}_{y_{i}}(w^{\prime}))$. But this follows by a
combination of [4, Prop. 3.8.1, Lem. 3.8.6] and [21, § 3.2, Prop. 1(2)]. ∎

##  10\. Local vs. global duality for ${\mathbb{G}}_{m}$ on compact curve

In this section, we shall establish a duality for some cohomology groups of
${\mathbb{G}}_{m}$ on a smooth projective curve over a local field and show
that it is compatible with the duality à la Kato [21] for the local cohomology
of ${\mathbb{G}}_{m}$ at closed points of the curve. This compatibility will
be another key step in the proof of Theorem 1.1.

###  10.1. Local duality for ${\mathbb{G}}_{m}$

Let $R$ be a equicharacteristic Henselian discrete valuation ring whose
residue field ${\mathfrak{f}}$ is a local field of exponential characteristic
$p\geq 1$. Let $L$ denote the quotient field of $R$. Let $x={\rm
Spec\,}({\mathfrak{f}})$ denote the closed point of $W={\rm Spec\,}(R)$. We
let $\widehat{W}={\rm Spec\,}(\widehat{R})$. We choose a common uniformizer
$\pi$ of $R$ and $\widehat{R}$, and let $W_{n}={\rm Spec\,}(R/{(\pi^{n})})$
for $n\geq 1$. The following result is due to Kato [21, § 3.5].

######  Theorem 10.1.

There is a perfect pairing of topological abelian groups

${\operatorname{Br}}(L)\times(L^{\times})^{\rm
pf}\to{{\mathbb{Q}}}/{{\mathbb{Z}}},$

where ${\operatorname{Br}}(L)$ and $L^{\times}$ are endowed with the discrete
and the Kato topologies, respectively.

We remark that Kato proved this result for $\widehat{L}$. However, it is
easily checked that the canonical maps
${\operatorname{Br}}(L)\to{\operatorname{Br}}(\widehat{L})$ and
$({L}^{\times})^{\rm pf}\to(\widehat{L}^{\times})^{\rm pf}$ are topological
isomorphisms.

Let $\gamma_{L}\colon{\operatorname{Br}}(L)\to(L^{\times})^{\star}$ denote the
map induced by the above pairing. It follows from Theorem 10.1 (see [46, Rem.
2.14]) that there is a commutative diagram of exact sequences

(10.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{Br}}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{R}}$$\textstyle{{\operatorname{Br}}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{L}}$$\textstyle{H^{3}_{{\text{\'{e}t}},x}(W,{\mathcal{O}}^{\times}_{W})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{x}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathbb{Z}}}^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(v_{L})^{\star}}$$\textstyle{(L^{\times})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(R^{\times})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the duals $(L^{\times})^{\star}$ and $(R^{\times})^{\star}$ are
considered with respect to the Kato topology.

In the above diagram, $\gamma_{R}$ is an isomorphism because it is the
composition
${\operatorname{Br}}(R)\xrightarrow{\cong}{\operatorname{Br}}({\mathfrak{f}})\xrightarrow{{\rm
inv}_{\mathfrak{f}}}{{\mathbb{Q}}}/{{\mathbb{Z}}}={{\mathbb{Z}}}^{\star}$.
$\gamma_{L}$ is an isomorphism if $p>1$ by [21, § 3.5, Rem. 4]. It maps
${\operatorname{Br}}(L)$ isomorphically onto the subgroup
$((L^{\times})^{\star})_{\rm tor}$ if $p=1$. It follows that $\gamma_{x}$ is
an isomorphism if $p>1$, and maps $H^{3}_{x}(W,{\mathcal{O}}^{\times}_{W})$
isomorphically onto $((R^{\times})^{\star})_{\rm tor}$ if $p=1$. Recall that
each ${\mathcal{O}}^{\times}(W_{n})$ is endowed with the adic topology.

######  Lemma 10.2.

For $R$ as above, we have the following.

  1. (1)

There exists a unique isomorphism $\gamma_{x}\colon
H^{3}_{x}(W,{\mathcal{O}}^{\times}_{W})\xrightarrow{\cong}((R^{\times})^{\star})_{\rm
tor}$ such that (10.1) is a commutative diagram.

  2. (2)

If $p>1$, each of the groups $(R^{\times})^{\star},\
(\widehat{R}^{\times})^{\star}$ and
$({\mathcal{O}}^{\times}({W_{n}}))^{\star}$ ($n\geq 1$) is torsion.

  3. (3)

The canonical maps $H^{3}_{x}(W,{\mathcal{O}}^{\times}_{W})\to
H^{3}_{x}(\widehat{W},{\mathcal{O}}^{\times}_{\widehat{W}})$ and
$(\widehat{R}^{\times})^{\star}\to(R^{\times})^{\star}$ are isomorphisms.

  4. (4)

If $p>1$, one has a canonical isomorphism

(10.2) $\gamma_{x}\colon
H^{3}_{x}(W,{\mathcal{O}}^{\times}_{W})\xrightarrow{\cong}{\varinjlim}_{n}\
({\mathcal{O}}^{\times}(W_{n}))^{\star},$

where the duals on the right hand side are taken with respect to the adic
topology.

  5. (5)

If $p=1$, the canonical map
$({\mathfrak{f}}^{\times})^{\star}\to(\widehat{R}^{\times})^{\star}$ induces
an isomorphism
$({\mathfrak{f}}^{\times})^{\star}\cong((\widehat{R}^{\times})^{\star})_{\rm
tor}$.

###### Proof.

To prove (2), note that the map
$({\mathcal{O}}^{\times}({W_{n}}))^{\star}\hookrightarrow(\widehat{R}^{\times})^{\star}$
is injective for all $n\geq 1$. Using the bottom row of (10.1), it suffices
therefore to show that $(L^{\times})^{\star}$ and
$(\widehat{L}^{\times})^{\star}$ are torsion groups. Using the definition of
Kato topology, it is an easy exercise that the canonical map
$(\widehat{L}^{\times})^{\star}\to(L^{\times})^{\star}$ is an isomorphism. But
[21, § 3.5, Rem. 4] says that $(\widehat{L}^{\times})^{\star}$ is a torsion
group if $p>1$. The statement (1) is already shown above. The statement (3)
follows easily by comparing (10.1) for $R$ and $\widehat{R}$ and using the
isomorphism
$(\widehat{L}^{\times})^{\star}\xrightarrow{\cong}(L^{\times})^{\star}$. To
show (4), we note that the map ${R}/{(\pi^{n})}\to{\widehat{R}}/{(\pi^{n})}$
is bijective for $n\geq 1$. Furthermore, the map
$\widehat{R}^{\times}\to{\varprojlim}_{n}\ {\mathcal{O}}^{\times}({W_{n}})$ is
an isomorphism of topological abelian groups by Lemma 7.1. It is an elementary
checking (cf. Lemma 2.6) that the induced map ${\varinjlim}_{n}\
({\mathcal{O}}^{\times}({W_{n}}))^{\star}\to(\widehat{R}^{\times})^{\star}$ is
bijective.

To prove (5), we can replace $(\widehat{R}^{\times})^{\star}$ by
$(\widehat{R}^{\times})^{\vee}$ since $\widehat{R}^{\times}$ is a discrete
group. By Lemma 7.4, we can also replace $({\mathfrak{f}}^{\times})^{\star}$
by $(({\mathfrak{f}}^{\times})^{\vee})_{\rm tor}$. We are now done because it
is clear that
$({\mathfrak{f}}^{\times})^{\vee}\to(\widehat{R}^{\times})^{\vee}$ is
injective, and it is surjective at the level of torsion subgroups by [15, Lem.
5.9]. This concludes the proof. ∎

###  10.2. Global duality for ${\mathbb{G}}_{m}$

Let $k$ be a local field of exponential characteristic $p\geq 1$ and let $X$
be a smooth and geometrically connected projective curve over $k$. Let $K$
denote the function field of $X$. The subject of this subsection is the
construction of a perfect duality between
$H^{0}(X,{\mathcal{O}}^{\times}_{X})$ and
$H^{3}(X,{\mathcal{O}}^{\times}_{X})$ which is compatible with the local
duality of § 10.1.

One can deduce from the Hochschild-Serre spectral sequence that there is an
isomorphism $H^{3}(X,{\mathcal{O}}^{\times}_{X})\cong H^{2}(k,{\rm
Pic}(X_{s}))$, where $X_{s}$ is the base change of $X$ by $k_{s}$. Using the
decomposition ${\rm Pic}(X_{s})\cong{\rm Pic}^{0}(X_{s})\oplus{\mathbb{Z}}$ as
$G_{k}$-module and the vanishing $H^{2}(k,{\rm Pic}^{0}(X_{s}))=0$ (see [37,
Thm. 7.8]), one deduces an isomorphism $\nu_{X}\colon
H^{3}(X,{\mathcal{O}}^{\times}_{X})\cong(G_{k})^{\star}\cong(k^{\times})^{\star}$.
However, this isomorphism does not serve our purpose. The reason is that it is
not known if $\nu_{X}$ is compatible with the local duality isomorphism
$\gamma_{x}$ in Lemma 10.2 for a closed point $x\in X$, in positive
characteristic. As mentioned above, this compatibility is very crucial for
proving our main results. The compatibility of $\nu_{X}$ with the local
duality seems like a challenging independent problem. The main obstacle in
solving this is the failure of purity for
$H^{2}_{x}(X,W_{n}\Omega^{1}_{X.{\operatorname{log}}})$.

To serve our purpose, we shall construct a different pairing between
$H^{0}(X,{\mathcal{O}}^{\times}_{X})$ and
$H^{3}(X,{\mathcal{O}}^{\times}_{X})$. We shall show that this pairing is
perfect and is compatible with the local duality isomorphism $\gamma_{x}$ for
$x\in X_{(0)}$. We do not have any guess whether the two duality maps
coincide. We let $H^{i}(X,\widehat{{\mathbb{Z}}}(j)):={\varprojlim}_{n}\
H^{i}(X,{{\mathbb{Z}}}/n(j))$, where the limit is taken over ${\mathbb{N}}$.
We begin with the following.

######  Lemma 10.3.

There exists a canonical bilinear pairing

(10.3) $H^{3}(X,{\mathcal{O}}^{\times}_{X})\times
H^{0}(X,{\mathcal{O}}^{\times}_{X})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

###### Proof.

By Lemma 8.1, we can replace $H^{3}(X,{\mathcal{O}}^{\times}_{X})$ by
$H^{3}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}}(1))$. Using (4.17) and the analogous
diagram for the sheaves ${{\mathbb{Z}}}/{n}(r)$ (with $n$ prime to $p$), we
get a bilinear pairing between ind-abelian and pro-abelian groups

$``{{\underset{n}{\varinjlim}}}"\
H^{i}(X,{{\mathbb{Z}}}/n(j))\times``{{\underset{n}{\varprojlim}}}"\
H^{i^{\prime}}(X,{{\mathbb{Z}}}/n(j^{\prime}))\to``{{\underset{n}{\varinjlim}}}"\
H^{i+i^{\prime}}(X,{{\mathbb{Z}}}/n(j+j^{\prime})).$

Taking the limits, we conclude by the Saito-Tate duality [47] (see also [15,
Thm. 9.9]) in the prime-to-$p$ case and Corollary 4.8 in the $p$-primary case
that there is a bilinear pairing between abelian groups

(10.4) $H^{3}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}}(1))\times
H^{1}(X,\widehat{{\mathbb{Z}}}(1))\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

Composing this with the canonical maps (see (8.1))

(10.5) $H^{0}(X,{\mathcal{O}}^{\times}_{X})\to{\varprojlim}_{n}\
{H^{0}(X,{\mathcal{O}}^{\times}_{X})}/n\hookrightarrow
H^{1}(X,\widehat{{\mathbb{Z}}}(1)),$

we get the desired pairing. ∎

In the remainder of this subsection, our goal is to prove the perfectness of
(10.3) with respect to the discrete topology on
$H^{3}(X,{\mathcal{O}}^{\times}_{X})$ and the adic topology on
$H^{0}(X,{\mathcal{O}}^{\times}_{X})\cong k^{\times}$. We begin by noting that
the diagrams in (8.2) give rise to an exact sequence of pro-abelian groups

(10.6)
$0\to\\{{H^{0}(X,{\mathcal{O}}^{\times}_{X})}/{n}\\}\to\\{H^{1}(X,{{\mathbb{Z}}}/n(1))\\}\to\\{_{n}{\rm
Pic}^{0}(X)\\}\to 0,$

whose transition maps are induced by the canonical surjections
${{\mathbb{Z}}}/{mn}\twoheadrightarrow{{\mathbb{Z}}}/{n}$. Taking the limits,
we get an exact sequence

(10.7) $0\to{\varprojlim}_{n}\ {H^{0}(X,{\mathcal{O}}^{\times}_{X})}/n\to
H^{1}(X,\widehat{{\mathbb{Z}}}(1))\to{\varprojlim}_{n}\ {}_{n}{\rm
Pic}^{0}(X)\to 0.$

Applying the dual functor in the category of abelian groups, we get an exact
sequence

(10.8) $0\to{\varinjlim}_{n}\ (_{n}{\rm
Pic}^{0}(X))^{\vee}\to{\varinjlim}_{n}\
(H^{1}(X,{{\mathbb{Z}}}/n(1)))^{\vee}\to{\varinjlim}_{n}\
({H^{0}(X,{\mathcal{O}}^{\times}_{X})}/n)^{\vee}\to 0.$

######  Lemma 10.4.

We have ${\underset{n}{\varprojlim}}\ _{n}{\rm
Pic}^{0}(X)={\underset{n}{\varinjlim}}\ (_{n}{\rm Pic}^{0}(X))^{\vee}=0$.

###### Proof.

One knows that there is a canonical inclusion ${\varprojlim}_{n}\ {}_{n}{\rm
Pic}^{0}(X)\hookrightarrow{\varprojlim}_{n}\ {}_{n}J_{X}(k)$, where $J_{X}$ is
the Jacobian variety of $X$ (e.g., see [36, Rem. 1.5]). On the other hand,
$J_{X}(k)$ is a profinite abelian group (see the proof of Corollary 6.11) and
${\varprojlim}_{n}\ {}_{n}J_{X}(k)$ coincides with the Tate module of
$J_{X}(k)$. It follows that ${\varprojlim}_{n}\ {}_{n}J_{X}(k)=0$ as the Tate
module of a profinite abelian group is zero.

To show that ${\varinjlim}_{n}\ (_{n}{\rm Pic}^{0}(X))^{\vee}$ is zero, it
suffices to show the stronger assertion that the pro-abelian group
$\\{_{n}{\rm Pic}^{0}(X)\\}$ is zero as it would imply that the ind-abelian
group $\\{(_{n}{\rm Pic}^{0}(X))^{\vee}\\}$ is zero. But this follows from our
first assertion and the well known fact that a pro-abelian group satisfying
the Mittag-Leffler condition is zero if and only if its limit is zero (e.g.,
see [54, Tag 07KV, Lem. 15.86.13]). ∎

######  Lemma 10.5.

The map $\delta_{X}\colon H^{3}(X,{\mathcal{O}}^{\times}_{X})\to
H^{0}(X,{\mathcal{O}}^{\times}_{X})^{\vee}$, induced by (10.3), is injective.

###### Proof.

By Lemma 8.1 and (10.5), $\delta_{X}$ is the same as the composite map

$H^{3}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}}(1))\to{\varinjlim}_{n}\
H^{1}(X,{{\mathbb{Z}}}/n(1))^{\vee}\to{\varinjlim}_{n}\
({H^{0}(X,{\mathcal{O}}^{\times}_{X})}/n)^{\vee}\to
H^{0}(X,{\mathcal{O}}^{\times}_{X})^{\vee}.$

The first arrow in this sequence is injective in the $p$-primary case by
Theorem 4.7 and isomorphism in the prime-to-$p$ case by [15, Thm. 9.9]. The
second arrow is injective by (10.8) and Lemma 10.4. The third arrow is easily
seen to be injective. This concludes the proof. ∎

The following shows the compatibility between the local and global duality
maps.

######  Lemma 10.6.

Let $x\in X_{(0)},\ R={\mathcal{O}}^{h}_{X,x}$ and let the notations be as in
Lemma 10.2. Then the diagram

(10.9)
$\textstyle{H^{3}_{x}(W,{\mathcal{O}}^{\times}_{W})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{x}}$$\textstyle{H^{3}(X,{\mathcal{O}}^{\times}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{X}}$$\textstyle{(R^{\times})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(H^{0}(X,{\mathcal{O}}^{\times}_{X}))^{\vee}}$

is commutative if we let the top horizontal arrow be the forget support map
and the bottom horizontal arrow be the dual of the canonical pull-back map
$H^{0}(X,{\mathcal{O}}^{\times}_{X})\to R^{\times}$.

###### Proof.

Using (10.1), the lemma is equivalent to showing that the diagram

(10.10)
$\textstyle{H^{2}(K_{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{x}}$$\scriptstyle{\gamma_{K_{x}}}$$\textstyle{H^{3}(X,{\mathcal{O}}^{\times}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{X}}$$\textstyle{(K^{\times}_{x})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{x}}$$\textstyle{(H^{0}(X,{\mathcal{O}}^{\times}_{X}))^{\vee}}$

is commutative, where $\phi_{x}$ is induced by the canonical pull-back map. To
show this, we can replace $H^{3}(X,{\mathcal{O}}^{\times}_{X})$ by
$H^{3}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}}(1))$ using Lemma 8.1. It also suffices
to prove the commutativity after replacing ${{\mathbb{Q}}}/{{\mathbb{Z}}}$ by
${{\mathbb{Z}}}/n$ for all $n\geq 1$.

We now fix an integer $n\geq 1$. We let $\alpha\in
H^{2}(K_{x},{{\mathbb{Z}}}/n(1))$ and $\beta\in
H^{0}(X,{\mathcal{O}}^{\times}_{X})$ be arbitrary elements. We look at the
diagram

(10.11) $\textstyle{H^{2}(K_{x},{{\mathbb{Z}}}/n(1))\times
H^{1}(K_{x},{{\mathbb{Z}}}/n(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{x}}$$\scriptstyle{\cup}$$\textstyle{H^{3}(K_{x},{{\mathbb{Z}}}/n(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{x}}$$\scriptstyle{\cong}$$\scriptstyle{{\rm
Tr}_{x}}$$\textstyle{{{\mathbb{Z}}}/n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{id}}}$$\textstyle{H^{3}_{x}(W,{{\mathbb{Z}}}/n(1))\times
H^{1}(W,{{\mathbb{Z}}}/n(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\cup}$$\textstyle{H^{4}_{x}(W,{{\mathbb{Z}}}/n(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H^{3}_{x}(X,{{\mathbb{Z}}}/n(1))\times
H^{1}(X,{{\mathbb{Z}}}/n(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\textstyle{H^{4}_{x}(X,{{\mathbb{Z}}}/n(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon_{x}}$$\textstyle{{{\mathbb{Z}}}/n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{id}}}$$\textstyle{H^{3}(X,{{\mathbb{Z}}}/n(1))\times
H^{1}(X,{{\mathbb{Z}}}/n(1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{id}}}$$\scriptstyle{{\operatorname{id}}}$$\scriptstyle{\cup}$$\textstyle{H^{4}(X,{{\mathbb{Z}}}/n(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Tr}_{X}}$$\textstyle{{{\mathbb{Z}}}/n}$$\textstyle{H^{3}(X,{{\mathbb{Z}}}/n(1))\times{H^{0}(X,{\mathcal{O}}^{\times}_{X})}/n\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{X}}$$\textstyle{H^{4}(X,{{\mathbb{Z}}}/n(2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{{\operatorname{id}}}$$\scriptstyle{{\rm
Tr}_{X}}$

where $\psi_{X}$ is the canonical map (see (10.6)).

All squares on the left are induced by the cup products, pull-back and the
boundary maps in étale cohomology. In particular, these squares are known to
be commutative (e.g., see (4.15) for the commutativity of the top square on
the left). The upper square on the right is commutative by the definition of
${\rm Tr}_{x}$ (see Corollary 4.6 and note that this also holds when $p\nmid
n$). The lower square on the right commutes by (4.8) when $n=p^{n}$ and by
[15, Lem. 9.7] when $p\nmid n$. The bottom right triangle clearly commutes.

We let $\theta\colon H^{2}(K_{x},{{\mathbb{Z}}}/n(1))\to
H^{3}(X,{{\mathbb{Z}}}/n(1))$ be the composition of all vertical arrows going
down on the extreme left of (10.11) and let
$\theta^{\prime}\colon{H^{0}(X,{\mathcal{O}}^{\times}_{X})}/n\to
H^{1}(K_{x},{{\mathbb{Z}}}/n(1))$ be the composition of all vertical arrows
going up in the middle of (10.11). It follows then by a diagram chase that
${\rm Tr}_{x}(\alpha\cup\theta^{\prime}(\beta))={\rm
Tr}_{X}(\theta(\alpha)\cup\psi_{X}(\beta))$. On the other hand, it is
straightforward to check that

$\langle\phi_{x}\circ\gamma_{K_{x}}(\alpha),\beta\rangle={\rm
Tr}_{x}(\alpha\cup\theta^{\prime}(\beta))\ \ \mbox{and}\ \
\langle\delta_{X}\circ\partial_{x}(\alpha),\beta\rangle={\rm
Tr}_{X}(\theta(\alpha)\cup\psi_{X}(\beta)).$

This proves the commutativity of (10.10) and concludes the proof of the lemma.
∎

We shall also need the following independent result for proving the
perfectness of (10.3) as well as in the proof of Lemma 11.3.

######  Lemma 10.7.

Let ${k^{\prime}}/{k}$ be a finite field extension of $k$. Then the inclusion
$k^{\times}\hookrightarrow k^{\prime\times}$ induces a surjective map
$(k^{\prime\times})^{\star}\twoheadrightarrow(k^{\times})^{\star}$ when the
duals are taken with respect to the adic topologies.

###### Proof.

Let $\chi\in(k^{\times})^{\star}$. Since the inclusion
$k^{\times}\hookrightarrow k^{\prime\times}$ is continuous and its image is
closed (e.g., see [20, § 7.2] or [39, Prop. II.5.7]) with respect to the adic
topologies, it follows from [9, Cor. 4.42] that there exists a continuous
homomorphism $\chi^{\prime}\colon k^{\prime\times}\to{\mathbb{T}}$ whose
restriction to $k^{\times}$ is $\chi$. It remains to show that $\chi^{\prime}$
has finite order. Since ${\mathcal{O}}^{\times}_{k^{\prime}}$ is profinite, it
is clear that $\chi^{\prime}$ has finite order when we restrict it to
${\mathcal{O}}^{\times}_{k^{\prime}}$. We let
$G=(k^{\times})\cdot({\mathcal{O}}^{\times}_{k^{\prime}})\subset
k^{\prime\times}$. Then $G$ is a closed subgroup of $k^{\prime\times}$ and
$\chi^{\prime}$ has finite order (say, $m$) on $G$.

We now look at the commutative diagram of exact sequences

(10.12)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k^{\prime\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi^{\prime}}$$\textstyle{{{\mathbb{Z}}}/{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathbb{Z}}}/m\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{{\mathbb{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $e$ is the ramification index of ${k^{\prime}}/{k}$. It follows that
$\chi^{\prime}(k^{\prime\times})$ is finite. This concludes the proof. ∎

We can now prove the main result of § 10.

######  Theorem 10.8.

The bilinear pairing of (10.3) induces a perfect pairing of topological
abelian groups

(10.13) $\delta_{X}\colon H^{3}(X,{\mathcal{O}}^{\times}_{X})\times
H^{0}(X,{\mathcal{O}}^{\times}_{X})^{\rm pf}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

###### Proof.

We can identify $H^{0}(X,{\mathcal{O}}^{\times}_{X})$ with $k^{\times}$ and
$H^{3}(X,{\mathcal{O}}^{\times}_{X})$ with
$H^{3}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}}(1))$ (cf. Lemma 8.1). In the first
step, we show that $\delta_{X}\colon
H^{3}(X,{\mathcal{O}}^{\times}_{X})\to(k^{\times})^{\vee}$ factors through
$(k^{\times})^{\star}$. Since $(k^{\times})^{\star}\cong{\rm
Hom}_{\mathbf{Tab}}(k^{\times},{{\mathbb{Q}}}/{{\mathbb{Z}}}\\{p^{\prime}\\})\bigoplus{\rm
Hom}_{\mathbf{Tab}}(k^{\times},{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})$ and
$H^{3}(X,{\mathcal{O}}^{\times}_{X})$ is a torsion group, it suffices to
consider the prime-to-$p$ and $p$-primary cases separately.

On the prime-to-$p$ torsion subgroup, $\delta_{X}$ is identified with the
composite map

(10.14)
$H^{3}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}}(1))\\{p^{\prime}\\}\to{\underset{n\in
k^{\times}}{\varinjlim}}\
(H^{1}(X,{{\mathbb{Z}}}/n(1))^{\vee}\to{\underset{n\in
k^{\times}}{\varinjlim}}\ ({k^{\times}}/n)^{\vee}\to(k^{\times})^{\vee}.$

Since the groups $H^{i}(X,{{\mathbb{Z}}}/n(j))$ are finite for $n\in
k^{\times}$ (e.g., see [15, Thm. 9.9]), we have
$H^{1}(X,{{\mathbb{Z}}}/n(1))^{\star}\cong
H^{1}(X,{{\mathbb{Z}}}/n(1))^{\vee}$. Since ${k^{\times}}/n$ is also discrete
(and finite), it follows that
$({k^{\times}}/n)^{\star}\cong({k^{\times}}/n)^{\vee}$ for every $n\geq 1$.
Hence, the above sequence of maps has a factorization

(10.15)
$H^{3}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}}(1))\\{p^{\prime}\\}\to{\underset{n\in
k^{\times}}{\varinjlim}}\
H^{1}(X,{{\mathbb{Z}}}/n(1))^{\star}\to{\underset{n\in
k^{\times}}{\varinjlim}}\ ({k^{\times}}/n)^{\star}\to(k^{\times})^{\star}.$

To prove the $p$-primary case, we can assume $p>1$. We choose a closed point
$x\in X$. We follow the notations of Lemma 10.2 and look at the commutative
diagram

(10.16)
$\textstyle{H^{3}_{x}(W,{\mathcal{O}}^{\times}_{W})\\{p\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{x}}$$\textstyle{H^{3}(X,{\mathcal{O}}^{\times}_{X})\\{p\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{X}}$$\textstyle{{\varinjlim}_{n}\
H^{0}(W_{n},{\mathcal{O}}^{\times}_{W_{n}})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(k^{\times})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(k^{\times})^{\vee}.}$

Note that $\delta_{X}$ is injective by Lemma 10.5. It is clear that the map
$H^{0}(W_{n},{\mathcal{O}}^{\times}_{W_{n}})^{\star}\to(k^{\times})^{\vee}$
factors through $(k^{\times})^{\star}$ for every $n\geq 1$. In particular, the
bottom left arrow is defined. Moreover, the outer trapezium commutes by Lemma
10.6. To show that the image of $\delta_{X}$ lies in $(k^{\times})^{\star}$,
it suffices therefore to show that the top horizontal arrow is surjective.

To show the last claim, we set $U=X\setminus\\{x\\}$. We now recall that
$H^{i}(Z,{\mathcal{O}}^{\times}_{Z})$ is a torsion group for all $i\geq 2$ if
$Z\in\\{X,U\\}$. Using the support cohomology exact sequence, this implies
that $H^{i}_{x}(W,{\mathcal{O}}^{\times}_{W})\cong
H^{i}_{x}(X,{\mathcal{O}}^{\times}_{X})$ is a torsion group for $i\geq 3$.
(the case of interest $i=3$ also follows from Lemma 10.2). In particular, the
sequence

$H^{3}_{x}(W,{\mathcal{O}}^{\times}_{W})\\{p\\}\to
H^{3}(X,{\mathcal{O}}^{\times}_{X})\\{p\\}\to
H^{3}(U,{\mathcal{O}}^{\times}_{U})\\{p\\}$

is exact.

On the other hand, there is a surjection
$H^{2}(U,W_{n}\Omega^{1}_{U,{\operatorname{log}}})\twoheadrightarrow~{}_{p^{n}}H^{3}(U,{\mathcal{O}}^{\times}_{U})$
for any $n\geq 1$, as follows from (8.1). It suffices therefore to show that
$H^{2}(U,W_{n}\Omega^{1}_{U,{\operatorname{log}}})=0$ for all $n\geq 1$. For
$n=1$, note that in the exact sequence

$0\to\Omega^{1}_{U,{\operatorname{log}}}\to\Omega^{1}_{U}\xrightarrow{\pi-\overline{F}}\frac{\Omega^{1}_{U}}{d{\mathcal{O}}_{U}}\to
0$

(see (3.1)), the middle and the term on the right are coherent
${\mathcal{O}}_{U}$-modules. In particular, their cohomological dimension is
zero as $U$ is affine. It follows that
$H^{2}(U,\Omega^{1}_{U,{\operatorname{log}}})=0$. The general case follows by
induction on $n$ using the cohomology sequence associated to (2.6).

We shall now show that $\delta_{X}\colon
H^{3}(X,{\mathcal{O}}^{\times}_{X})\to(k^{\times})^{\star}$ is bijective. It
is injective by Lemma 10.5. To prove its surjectivity between the prime-to-$p$
torsion subgroups, note that all arrows in (10.15) are isomorphisms (the first
arrow is an isomorphism by the Saito-Tate duality, see [15, Thm. 9.9]). To
prove the surjectivity of $\delta_{X}$ between the $p$-primary torsion
subgroups, note that this map is the same as the dotted arrow in (10.16). The
map $\gamma_{x}$ is bijective by Lemma 10.2(4). It suffices therefore to show
that the map ${\varinjlim}_{n}\
H^{0}(W_{n},{\mathcal{O}}^{\times}_{W_{n}})^{\star}\to(k^{\times})^{\star}$ is
surjective. Equivalently, the map
$(\widehat{R}^{\times})^{\star}\to(k^{\times})^{\star}$ is surjective, where
recall that $R={\mathcal{O}}^{h}_{X,x}$. But this latter map is induced by the
canonical inclusion $k^{\times}\hookrightarrow\widehat{R}^{\times}$. We are
now done because the composite map
$(k(x)^{\times})^{\star}\to(\widehat{R}^{\times})^{\star}\to(k^{\times})^{\star}$
is surjective by Lemma 10.7.

To prove the perfectness of (10.13), we let $F=(k^{\times})^{\rm pf}$. We now
note that the canonical map $F^{\star}\to(k^{\times})^{\star}$ is an
isomorphism, as one easily checks using [39, Prop. II.5.7] and the
profiniteness of ${\mathcal{O}}^{\times}_{k}$. It follows that (10.3) gives
rise to (10.13) and the resulting map $\delta_{X}\colon
H^{3}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}}(1))\to F^{\star}$ is an isomorphism.
Since this map is clearly continuous and $F$ is profinite, it follows that
(10.13) is a perfect pairing of topological abelian groups. This concludes the
proof. ∎

######  Remark 10.9.

We showed in the proof of Theorem 10.8 that
$H^{3}(U,{\mathcal{O}}^{\times}_{U})\\{p\\}=0$ if $U$ is the complement of a
closed point in $X$. But the proof actually shows that
$H^{i}(U,{\mathcal{O}}^{\times}_{U})\\{p\\}=0$ if $U$ is any affine open in
$X$ and $i\geq 3$ is any integer. This observation will be used in the proof
of Theorem 1.1.

######  Corollary 10.10.

For any nonempty effective Cartier divisor $D\subset X$, the forget support
map $H^{3}_{D}(X,{\mathcal{O}}^{\times}_{X})\to
H^{3}(X,{\mathcal{O}}^{\times}_{X})$ is surjective.

###### Proof.

Using Lemmas 10.2, 10.6 and Theorem 10.8, it suffices to show that the map
$(k(x)^{\times})^{\star}\to(k^{\times})^{\star}$ is surjective for any $x\in
D$. But this follows from Lemma 10.7. ∎

##  11\. Continuity of Brauer-Manin pairing

Let $k$ be a local field of exponential characteristic $p\geq 1$. The goal of
this section is to show that the Brauer-Manin pairing for a 1-dimensional
modulus pair $(X,D)$ over $k$ is continuous with respect to the discrete
topology of ${\operatorname{Br}}(X|D)$ and the adic topology of ${\rm
Pic}(X|D)$. This is an important step in the proofs of the main results. We
shall also prove few more properties of this pairing which will be used in the
proof of Theorem 1.6. We begin with the following general statement about
regular (but not necessarily smooth) curves.

######  Lemma 11.1.

Let $X$ be a connected and regular projective curve over $k$ and let $D\subset
X$ be a divisor with the complement $X^{o}$. Then the map
$\theta_{X^{o}}\colon C(X^{o})\to{\underset{D^{\prime}\in{\rm Div}_{D_{\rm
red}}(X)}{\varprojlim}}\ {\rm CH}_{0}(X|D^{\prime})$ is bijective. In
particular, the map $C(X^{o})\to{\rm CH}_{0}(X|D^{\prime})$ is surjective for
every $D^{\prime}\in{\rm Div}_{D_{\rm red}}(X)$.

###### Proof.

We can replace the right hand side by ${\varprojlim}_{n}{\rm CH}_{0}(X|nD)$.
Using Remark 9.1, we can replace ${k(X)}^{\times}_{\infty}$ (resp. $I(X|nD)$)
by $\widehat{k(X)}^{\times}_{\infty}$ (resp. $\widehat{I}(X|nD)$) in (9.2). It
suffices now to show the stronger statement that the canonical map
${\mathcal{Z}}_{0}(X^{o})\bigoplus{\widehat{k(X)}^{\times}_{\infty}}\to{\varprojlim}_{n}\left({\mathcal{Z}}_{0}(X^{o})\bigoplus\frac{\widehat{k(X)}^{\times}_{\infty}}{\widehat{I}(X|nD)}\right)$
is bijective. It is clear that the right hand side of the latter map is
${\mathcal{Z}}_{0}(X^{o})\bigoplus\left({\varprojlim}_{n}\frac{\widehat{k(X)}^{\times}_{\infty}}{\widehat{I}(X|nD)}\right)$.
It remains therefore to show that for every $x\in D$, the map
$\widehat{k(X)}_{x}^{\times}\to{\varprojlim}_{n}\frac{\widehat{k(X)}^{\times}_{x}}{{\rm
fil}_{n}\widehat{k(X)}^{\times}_{x}}$ is bijective. But this can be easily
checked using the completeness of $\widehat{{\mathcal{O}}_{X,x}}$ and the
strict exact sequence of pro-abelian groups

$0\to\left\\{\frac{\widehat{{\mathcal{O}}_{X,x}}^{\times}}{{\rm
fil}_{n}\widehat{k(X)}^{\times}_{x}}\right\\}_{n}\to\left\\{\frac{\widehat{k(X)}^{\times}_{x}}{{\rm
fil}_{n}\widehat{k(X)}^{\times}_{x}}\right\\}_{n}\xrightarrow{v_{x}}{\mathbb{Z}}\to
0.$

To see that $C(X^{o})\to{\rm CH}_{0}(X|D^{\prime})$ is surjective, we only
have to note that $\\{{\rm CH}_{0}(X|nD^{\prime}\\}_{n}$ is an inverse system
indexed by ${\mathbb{N}}$ with surjective transition maps whose limit is
$C(X^{o})$. ∎

We now let $X$ be a smooth projective geometrically integral curve over $k$
and $D\subset X$ an effective divisor with support $D^{\dagger}$. We write
$D={\underset{x\in D^{\dagger}}{\sum}}n_{x}[x]\in{\rm Div}(X)$. Let $j\colon
X^{o}=X\setminus D\hookrightarrow X$ be the inclusion. We shall consider
$C(X^{o})=\varprojlim_{n}{\rm CH}_{0}(X|nD)$ as a topological abelian group,
endowed with the inverse limit of the adic topologies of ${\rm CH}_{0}(X|nD)$
using Lemma 11.1.

######  Proposition 11.2.

The pairings

${\operatorname{Br}}(X|D)\times{\rm
CH}_{0}(X|D)\to{{\mathbb{Q}}}/{{\mathbb{Z}}},\ \ \mbox{and}\ \
{\operatorname{Br}}(X^{o})\times C(X^{o})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$

are continuous.

###### Proof.

Let $\beta_{X|D}\colon{\operatorname{Br}}(X|D)\to{\rm CH}_{0}(X|D)^{\vee}$ and
$\beta_{X^{o}}\colon{\operatorname{Br}}(X^{o})\to C(X^{o})^{\vee}$ be the
induced homomorphisms. As mentioned in § 4.2, the proposition is equivalent to
the statement that the image of $\beta_{X|D}$ (resp. $\beta_{X^{o}}$) lies in
${\rm CH}_{0}(X|D)^{\star}$ (resp. $C(X^{o})^{\star}$). Note that this uses
the discreteness of ${\operatorname{Br}}(X^{o})$. Since $\varinjlim_{n}{\rm
CH}_{0}(X|nD)^{\star}\xrightarrow{\cong}C(X^{o})^{\star}$, it suffices to show
that $\beta_{X|D}({\operatorname{Br}}(X|D))\subset{\rm CH}_{0}(X|D)^{\star}$.
To prove the latter statement, we fix $w\in{\operatorname{Br}}(X|D)$ and let
$\chi_{w}\colon{\rm CH}_{0}(X|D)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$ be the
induced character.

Assume first that $p=1$. As $nw=0$ for some $n\geq 1$, the map
$\chi_{w}\colon{\rm CH}_{0}(X|D)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$ must factor
through ${{\rm CH}_{0}(X|D)}/n$. On the other hand, ${{\rm CH}_{0}(X|D)}/n$ is
a finite discrete group, as one easily checks using (5.1), [39, Prop. II.5.7]
and the Kummer sequence. In particular, the map $\chi_{w}\colon{{\rm
CH}_{0}(X|D)}/n\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$ is continuous. It follows
that $\beta_{X|D}(w)\in{\rm CH}_{0}(X|D)^{\star}$. We shall now assume in the
rest of the proof that $p>1$.

We begin by proving the special case of the proposition when $D$ and
$X^{o}(k)$ are not empty. We fix a closed point $P\in X^{o}(k)$. As $\chi_{w}$
is a group homomorphism, it will be enough to show that it is continuous at
some point of ${\rm Pic}(X|D)$. We now note that the composition
$\chi^{1}_{w}\colon
X^{o}(k)\hookrightarrow{\mathcal{Z}}_{0}(X^{o})\twoheadrightarrow{\rm
Pic}(X|D)\xrightarrow{\chi_{w}}{{\mathbb{Q}}}/{{\mathbb{Z}}}$ (see (9.14)) is
given by $\chi^{1}_{w}(x)={\rm inv}_{k(x)}(\iota^{*}_{x}(w))$, where
$\iota_{x}\colon{\rm Spec\,}(k(x))\hookrightarrow X^{o}$ is the inclusion. It
follows from [4, Prop. 10.5.2] (see also [41, Prop. 8.2.9]) that
$\chi^{1}_{w}$ is continuous. In particular, the map
$\chi^{r}_{w}\colon(X^{o}(k))^{r}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$, given by
$\chi^{r}_{w}(x_{1},\ldots,x_{r})=\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}\chi^{1}_{w}(x_{i})$, is continuous for every
$r\geq 1$.

We let ${\rm alb}_{X|D}\colon X^{o}(k)\to{\mathbf{Pic}}^{0}(X|D)$ be the
albanese map, given by ${\rm alb}_{X|D}(x)=[x]-[P]$ (cf. (6.2)). It follows
from [4, Thm. 10.5.1] and Theorem 6.6 that for all $r\gg 0$, there exists a
Zariski open subscheme $U\subset(X^{o})^{r}$ such that the composite map
$\phi^{r}_{X|D}\colon
U(k)\hookrightarrow(X^{o}(k))^{r}\to{\mathbf{Pic}}^{0}(X|D)(k)={\rm
Pic}^{0}(X|D)$ between the adic spaces is open and has a dense image. We now
look at the commutative diagram

(11.1)
$\textstyle{U(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{r}_{X|D}}$$\scriptstyle{\chi^{r}_{w}}$$\textstyle{{\rm
Pic}^{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{w}}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-r\chi_{w}(P)}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}.}$

We showed above that the left vertical arrow is continuous. Since the bottom
horizontal arrow is clearly continuous, it follows that
$\chi_{w}\circ\phi^{r}_{X|D}$ is continuous. Since the image of
$\phi^{r}_{X|D}$ is open dense and the map $U(k)\to\phi^{r}_{X|D}(U(k))$ is
open, it follows that $\chi_{w}$ is continuous on the nonempty open subset
$\phi^{r}_{X|D}(U(k))$. This proves the proposition when $X^{o}(k)$ and $D$
are not empty.

We assume next that $D\neq\emptyset$ but $X(k)=\emptyset$. We let
$G^{\star}(X^{o})={\varinjlim}_{n}\ {\rm CH}_{0}(X|nD)^{\star}$ and
$G^{\vee}(X^{o})={\varinjlim}_{n}\ {\rm CH}_{0}(X|nD)^{\vee}$. Recall from
Lemma 8.8 that
${\varinjlim}_{n}{\operatorname{Br}}(X|nD)\xrightarrow{\cong}{\operatorname{Br}}(X^{o})$.
Suppose we know that the map $\beta_{X^{o}}\colon{\operatorname{Br}}(X^{o})\to
G^{\vee}(X^{o})$ factors through $G^{\star}(X^{o})$. Then it follows from
(9.11) that $\beta_{X|D}(\chi)\in G^{\star}(X^{o})\cap{\rm
CH}_{0}(X|D)^{\vee}$. In particular, the composite map ${\rm
CH}_{0}(X|nD)\twoheadrightarrow{\rm
CH}_{0}(X|D)\xrightarrow{\chi_{w}}{{\mathbb{Q}}}/{{\mathbb{Z}}}$ is continuous
for $n\gg 0$. This implies that $\chi_{w}$ is continuous because the first
arrow is a quotient map of adic spaces by Corollary 6.10 and [4, Thm. 10.5.1].
It remains therefore to show that
$\beta_{X^{o}}({\operatorname{Br}}(X^{o}))\in G^{\star}(X^{o})$.

We fix an element $w\in{\operatorname{Br}}(X^{o})$ and let
$\chi_{w}=\beta_{X^{o}}(w)$. By [59, Thm. 4.3.1] (see also [11, Thm. 2.2]), we
can find a finite field extension ${k^{\prime}}/k$ and a geometrically
connected smooth projective curve $Y$ over $k^{\prime}$ together with a finite
morphism $g\colon Y\to X_{k^{\prime}}$ such that the degrees of
${k^{\prime}}/k$ and $g$ are powers of $p$ and $Y$ admits a strict semi-stable
reduction over $k^{\prime}$. That is, there exists a projective and flat
morphism $\phi\colon{\mathcal{Y}}\to{\rm Spec\,}({\mathcal{O}}_{k^{\prime}})$
such that ${\mathcal{Y}}$ is regular whose generic fiber is $Y$ and whose
(scheme theoretic) closed fiber $Y_{s}$ is a reduced curve with only double
point singularities.

We let $Y^{o}$ (resp. $E^{\dagger}$) be the inverse image of $X^{o}$ (resp.
$D^{\dagger}$) under the composite map $Y\to X_{k^{\prime}}\to X$ and let
${\mathcal{Y}}^{o}$ be the complement of the closure of $E^{\dagger}$ in
${\mathcal{Y}}$. By an application of Weil conjectures over the residue field
${\mathbb{F}}_{q}$ of $k^{\prime}$, we can find a closed point
$P_{s}\in{\mathcal{Y}}^{o}\cap(Y_{s})_{\rm reg}$ whose degree over
${\mathbb{F}}_{q}$ is a power of $p$. It is easy to see that there exists a
closed point $P\in Y^{o}$ whose degree over $k^{\prime}$ is same as that of
$P_{s}$ over ${\mathbb{F}}_{q}$. We let $F=k^{\prime}(P)$ and let
$X^{\prime}=Y_{F}$. Let $f\colon X^{\prime}\to X$ be the composite finite
surjective map. We let $X^{\prime o}=f^{-1}(X^{o})$ and $E=f^{*}(D)$. It
follows then that $X^{\prime}$ is a smooth and geometrically connected curve
over the local field $F$ such that $P\in X^{\prime o}(F)$ and $\deg(f)=p^{n}$
for some $n\geq 1$.

We have the canonical maps
${\operatorname{Br}}(X^{o})\xrightarrow{f^{*}}{\operatorname{Br}}(X^{\prime
o})\xrightarrow{f_{*}}{\operatorname{Br}}(X^{o})$ whose composition is
multiplication by $p^{n}$ (e.g., see [4, § 3.8]). In particular, ${\rm
Coker}(f_{*})$ is torsion of exponent $p^{n}$. On the other hand, ${\rm
Coker}(f_{*})$ is a $p$-divisible group by [4, Thm. 3.2.3]. It follows that
$f_{*}$ is surjective. In particular, $w=f_{*}(w^{\prime})$ for some
$w^{\prime}\in{\operatorname{Br}}(X^{\prime o})$. We can assume that
$w^{\prime}\in{\operatorname{Br}}(X^{\prime}|nE)$ for some $n\gg 0$. Since
$X^{\prime o}(F)\neq\emptyset$, the previous case of the lemma shows that
$\beta_{X^{\prime}|nE}(w^{\prime})\in{\rm
CH}_{0}(X^{\prime}|nE^{\prime})^{\star}$. We let
$\chi_{w^{\prime}}=\beta_{X^{\prime}|nE}(w^{\prime})$.

We now look at the diagram

(11.2) $\textstyle{{\rm
CH}_{0}(X|nD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\scriptstyle{\chi_{w}}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}}$$\textstyle{{\rm
CH}_{0}(X^{\prime}|nE)\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\chi_{w^{\prime}}}$

It follows from Lemma 9.7 that this diagram is commutative. Since
$f^{*}\colon{\mathbf{Pic}}(X|nD)\to{\mathbf{Pic}}(X^{\prime}|nE)$ is a
morphism between locally of finite type $k$-schemes by Theorem 5.6, it follows
from [5, Prop. 5.4] that the left vertical arrow is continuous. We deduce that
$\chi_{w}$ is continuous. This concludes the proof when $D\neq\emptyset$.

To prove the proposition for $(X,\emptyset)$, we choose a nonempty effective
divisor $D\subset X$ and let
$w\in{\operatorname{Br}}(X)\subset{\operatorname{Br}}(X|D)$. We have shown
above that the composite map ${\rm CH}_{0}(X|D)\twoheadrightarrow{\rm
CH}_{0}(X)\xrightarrow{\chi_{w}}{{\mathbb{Q}}}/{{\mathbb{Z}}}$ is continuous.
On the other hand, it follows from Corollary 6.10 and [4, Thm. 10.5.1] that
the first arrow is a topological quotient map. We deduce that $\chi_{w}$ is
continuous. ∎

######  Lemma 11.3.

Let ${k^{\prime}}/k$ be a finite field extension. Let
$X^{\prime}=X_{k^{\prime}},\ D^{\prime}=D_{k^{\prime}}$ and $X^{\prime
o}=X^{\prime}\setminus D^{\prime}$. Let $f\colon X^{\prime}\to X$ be the
projection map. Then the following hold.

  1. (1)

The map $f^{*}\colon{\rm Pic}(X|D)\to{\rm Pic}(X^{\prime}|D^{\prime})$ is a
closed embedding of adic spaces.

  2. (2)

$f^{*}$ induces a surjective homomorphism $(f^{*})^{\star}\colon{\rm
Pic}(X^{\prime}|D^{\prime})^{\star}\twoheadrightarrow{\rm Pic}(X|D)^{\star}$.

###### Proof.

By Theorem 5.6, the pull-back map $f^{*}\colon{\rm Pic}(X|D)\to{\rm
Pic}(X^{\prime}|D^{\prime})$ is the canonical inclusion
${\mathbf{Pic}}(X|D)(k)\subset{\mathbf{Pic}}(X|D)(k^{\prime})$. To prove (1),
we therefore need to show that the inclusion
${\mathbf{Pic}}(X|D)(k)\subset{\mathbf{Pic}}(X|D)(k^{\prime})$ of adic spaces
is closed. But this follows from [5, Prop. 5.11(3)] since the inclusion
$k\subset k^{\prime}$ is easily seen to be closed (e.g., see [39, Prop.
II.5.7]).

Since ${{\mathbb{Q}}}/{{\mathbb{Z}}}$ is a divisible group, one easily reduces
the proof of (2) to showing that the map $(f^{*})^{\star}\colon{\rm
Pic}^{0}(X^{\prime}|D^{\prime})^{\star}\to{\rm Pic}^{0}(X|D)^{\star}$ is
surjective. Let $\chi\colon{\rm Pic}^{0}(X|D)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$
be a continuous character. By part (1) of the lemma and [9, Cor. 4.42], $\chi$
extends to a continuous character $\chi^{\prime}\colon{\rm
Pic}^{0}(X^{\prime}|D^{\prime})\to{\mathbb{T}}$. It remains to show that the
image of $\chi^{\prime}$ lies in ${{\mathbb{Q}}}/{{\mathbb{Z}}}$. We write
$F=F_{1}\times F_{2}$, where $F_{1}$ is the cokernel of the diagonal inclusion
$k^{\times}\hookrightarrow\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\prod}}}k(x_{i})^{\times}$ and
$F_{2}=\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\prod}}}{\mathbb{W}}_{n_{i}-1}(k(x_{i}))$. We then note
using (5.1) and the proof of Lemma 7.1 (see also Proposition 6.9) that there
is an exact sequence of topological groups

(11.3) $0\to F\to{\rm Pic}^{0}(X|D)\to{\rm Pic}^{0}(X)\to 0.$

We claim that $\chi(F)$ is finite.

Since $F_{2}$ is a torsion group of bounded exponent, $\chi(F_{2})$ must be
finite. For every $1\leq i\leq r$, we have an exact sequence of topological
groups

(11.4) $0\to{\mathcal{O}}^{\times}_{k(x_{i})}\to
k(x_{i})^{\times}\to{\mathbb{Z}}\to 0.$

Since ${\mathcal{O}}^{\times}_{k(x_{i})}$ is a profinite group, its image
under $\chi$ must be finite. In particular, it is a cyclic subgroup of
${{\mathbb{Q}}}/{{\mathbb{Z}}}$ of the type ${{\mathbb{Z}}}/n$ for some
integer $n\geq 1$. We now look at the commutative diagram of exact sequences

(11.5)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}^{\times}_{k(x_{i})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k(x_{i})^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\chi({\mathcal{O}}^{\times}_{k(x_{i})})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Since the image of the right vertical arrow must be finite, it follows that
$\chi(k(x_{i})^{\times})$ is finite. Summing over $1\leq i\leq r$, we get that
$\chi(F_{1})$ is finite. This proves the claim.

We let $D^{\prime}_{\rm red}=\\{y_{1},\ldots,y_{s}\\}$ and
$D^{\prime}=\sum_{i}m_{i}[y_{i}]$. We let $F^{\prime}=F^{\prime}_{1}\times
F^{\prime}_{2}$, where $F^{\prime}_{1}$ is the cokernel of the diagonal
inclusion $k^{\prime\times}\hookrightarrow\stackrel{{\scriptstyle
s}}{{\underset{i=1}{\prod}}}k(y_{i})^{\times}$ and
$F^{\prime}_{2}=\stackrel{{\scriptstyle
s}}{{\underset{i=1}{\prod}}}{\mathbb{W}}_{m_{i}-1}(k(y_{i}))$. We then have an
exact sequence similar to (11.3). Since $F^{\prime}_{2}$ is a torsion group of
bounded exponent, $\chi^{\prime}(F_{2})$ must be finite. On the other hand,
each $\chi^{\prime}(k(y_{i})^{\times})$ is finite by Lemma 10.7. Summing over
$1\leq i\leq s$, we get that $\chi^{\prime}(F^{\prime}_{1})$ is finite. We
deduce that $\chi^{\prime}(F^{\prime})$ is finite.

Finally, we look at the commutative diagram of exact sequences

(11.6)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi^{\prime}}$$\textstyle{{\rm
Pic}^{0}(X^{\prime}|D^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\chi^{\prime}}$$\textstyle{{\rm
Pic}^{0}(X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\chi}^{\prime}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathbb{Z}}}/{m^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{\prime}}$$\textstyle{{\mathbb{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

It follows from Corollary 6.10 that $\alpha$ is a quotient map. Since
$m^{\prime}\circ\chi^{\prime}$ is continuous, it follows that
$\widetilde{\chi}^{\prime}$ is a continuous homomorphism. On the other hand,
${\rm Pic}^{0}(X^{\prime})$ is profinite. It follows that the image of
$\widetilde{\chi}^{\prime}$ is finite. We conclude that the image of ${\rm
Pic}^{0}(X^{\prime}|D^{\prime})$ under $\chi^{\prime}$ is finite. We have thus
shown that $\chi^{\prime}\in{\rm Pic}^{0}(X^{\prime}|D^{\prime})^{\star}$.
This proves (2) and concludes the proof of the lemma. ∎

##  12\. Perfectness of Brauer-Manin pairing for modulus pairs

The goal of this section is to prove the perfectness of the Brauer-Manin
pairing for a one-dimensional modulus pair. We shall also prove Theorem 1.1
using the perfectness of the Brauer-Manin pairing. We let $k$ be a local field
of exponential characteristic $p\geq 1$. Let $X$ be a geometrically integral
smooth projective curve over $k$ and let $D\subset X$ be an effective divisor.
We let $j\colon X^{o}\hookrightarrow X$ be the inclusion of the complement of
$D$. We write $D=\sum_{i}n_{i}[x_{i}]$, where
$D^{\dagger}=\\{x_{1},\ldots,x_{r}\\}$ is the support of $D$ with reduced
closed subscheme structure. We let $K$ denote the function field of $X$. For
$x\in X_{(0)}$, we let $X_{x}={\rm Spec\,}({\mathcal{O}}_{X,x})$ and
$X^{h}_{x}={\rm Spec\,}({\mathcal{O}}^{h}_{X,x})$.

###  12.1. Proof of Theorem 1.6

By Proposition 11.2, the Brauer-Manin pairing with modulus (cf. Proposition
9.4) induces continuous homomorphisms
$\beta_{X|D}\colon{\operatorname{Br}}(X|D)\to{\rm CH}_{0}(X|D)^{\star}$ and
$\beta_{X^{o}}\colon{\operatorname{Br}}(X^{o})\to C(X^{o})^{\star}$. We begin
by showing the injectivity of these homomorphisms.

######  Lemma 12.1.

$\beta_{X|D}$ is a monomorphism.

###### Proof.

Let $w\in{\operatorname{Br}}(X|D)$ be such that $\chi_{w}:=\beta_{X|D}(w)=0$.
For $x\in X_{(0)}$, let $\iota_{x}\colon{\rm Spec\,}(k(x))\hookrightarrow X$
be the inclusion. We let $w_{x}$ be the image of $w$ under the canonical
composite map
${\operatorname{Br}}(X^{o})\hookrightarrow{\operatorname{Br}}(K)\to{\operatorname{Br}}(K_{x})$.
Let
$\chi^{1}_{w}\colon{\mathcal{Z}}_{0}(X^{o})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$
and $\chi^{2}_{w}\colon{K^{\times}_{\infty}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$
be the two components of $\chi_{w}$ (see § 9.2). Our assumption implies that
$\chi^{i}_{w}=0$ for $i=1,2$.

Suppose $x\in D$. Since $\chi^{2}_{w}=0$, it follows from [20, § 6, Thm. 1]
that $w_{x}=0$. The commutative diagram of exact sequences

(12.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{Br}}(X_{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{Br}}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}_{x}(X_{x},{\mathcal{O}}^{\times}_{X_{x}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{Br}}(X^{h}_{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{Br}}(K_{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}_{x}(X^{h}_{x},{\mathcal{O}}^{\times}_{X^{h}_{x}})}$

implies that as an element of ${\operatorname{Br}}(K)$, the Brauer class $w$
lies in ${\operatorname{Br}}(X_{x})$ and it dies in
${\operatorname{Br}}(X^{h}_{x})$. In particular, $\iota^{*}_{x}(w)=0$. Since
$\chi^{1}_{w}=0$, we also have $\iota^{*}_{x}(w)=0$ for every $x\in
X^{o}_{(0)}$. It follows that $w$ is a class in ${\operatorname{Br}}(X)$ such
that $\iota^{*}_{x}(w)=0$ for $x\in X_{(0)}$. We conclude from [46, Thm. 9.2]
that $w=0$. This finishes the proof. ∎

######  Lemma 12.2.

We have the following.

  1. (1)

If $p>1$, the map $\beta_{X|D}\colon{\operatorname{Br}}(X|D)\to{\rm
CH}_{0}(X|D)^{\star}$ is an isomorphism.

  2. (2)

If $p=1$, the map $\beta_{X|D}\colon{\operatorname{Br}}(X|D)\to({\rm
CH}_{0}(X|D)^{\star})_{\rm tor}$ is an isomorphism.

###### Proof.

In view of Lemma 12.1, we only need to show that $\beta_{X|D}$ is surjective.
We first prove this surjectivity under the assumption that $D^{\dagger}\subset
X(k)$. It follows from Lemma 8.8 that
${\operatorname{Br}}(X|D^{\dagger})={\operatorname{Br}}(X|D)={\operatorname{Br}}(X^{o})$
when $p=1$. Using Lemma 7.4, we can assume in this case that $D$ is reduced.
Hence, $D$ will assumed to be reduced (and $n=1$) in the following argument
when $p=1$.

We let $G^{\star}(X^{o})={\varinjlim}_{n}\ {\rm CH}_{0}(X|nD)^{\star}$. We
shall first show that the map
$\beta_{X^{o}}\colon{\operatorname{Br}}(X^{o})\to G^{\star}(X^{o})$ is
surjective. We consider the diagram

(12.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{Br}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\scriptstyle{\beta_{X}}$$\textstyle{{\operatorname{Br}}(X^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{X}}$$\scriptstyle{\beta_{X^{o}}}$$\textstyle{{\underset{x\in
D}{\oplus}}H^{3}_{x}(X,{\mathcal{O}}^{\times}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{X|D}}$$\scriptstyle{\eta_{X|D}}$$\textstyle{H^{3}(X,{\mathcal{O}}^{\times}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{X}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}(X)^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{G^{\star}(X^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\underset{x\in
D}{\oplus}}H^{0}(X^{h}_{x},{\mathcal{O}}^{\times}_{X^{h}_{x}})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{(\iota^{*}_{D})}^{\star}}$$\textstyle{(k^{\times})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $\gamma_{X|D}$ is the direct sum of maps $\gamma_{x}$ given by Lemma
10.2 and $\delta_{X}$ is given by Lemma 10.5. The top row is the localization
sequence for the étale sheaf ${\mathcal{O}}^{\times}_{X}$ and hence is exact
except that we need to explain the surjectivity of $\eta_{X|D}$. But this
follows from Corollary 10.10. The bottom row is a complex which is exact at
${\rm CH}_{0}(X)^{\star}$ and $G^{\star}(X^{o})$ by Lemmas 7.3, 7.5 and 10.2.

The left square in (12.2) is commutative by Corollary 9.5 and the right square
is commutative by Lemma 10.6. To show the commutativity of the middle square,
we look at the diagram

(12.3)
$\textstyle{{\operatorname{Br}}(X|nD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{X|nD}}$$\textstyle{{\underset{x\in
D}{\oplus}}{\operatorname{Br}}(K_{x})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{x}\gamma_{K_{x}}}$$\textstyle{{\rm
CH}_{0}(X|nD)^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\underset{x\in
D}{\oplus}}(K^{\times}_{x})^{\star}}$

for $n\geq 1$, where the dual of $K^{\times}_{x}$ is taken with respect to its
Kato topology. This diagram is commutative by the definition of $\beta_{X|D}$,
where $\gamma_{K_{x}}$ is as in (10.1). When we restrict $\beta_{X^{o}}$ to
${\operatorname{Br}}(X|nD)$, the middle square in (12.2) commutes because it
is the composition of (12.3) with the sum (over $x\in D$) of right squares in
(10.1). Taking the limit over $n\geq 1$, we see that the middle square in
(12.2) commutes.

To show that $\beta_{X^{o}}$ is an isomorphism, we note that $\beta_{X}$ is an
isomorphism by [32, Thm. 4] and [46, Thm. 9.2]. The arrow $\gamma_{X|D}$ is an
isomorphism by Lemma 10.2, and $\delta_{X}$ is an isomorphism by Theorem 10.8.
An easy diagram chase shows that the bottom row of (12.2) is exact and
$\beta_{X^{o}}$ is an isomorphism. This also proves that $\beta_{X|D}$ is
surjective when $p=1$. To pass from $\beta_{X^{o}}$ to $\beta_{X|D}$ when
$p>1$, we fix a character $\chi\in{\rm CH}_{0}(X|D)^{\star}$. It follows from
the surjectivity of $\beta_{X^{o}}$ that there exists a class
$w\in{\operatorname{Br}}(X^{o})={\underset{n}{\bigcup}}\
{\operatorname{Br}}(X|nD)$ such that $\chi=\chi_{w}:=\beta_{X^{o}}(w)$. It
follows from (9.2) and the definition of $\beta_{X|D}$ that the map
$\chi_{w}\colon K^{\times}_{\infty}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$
annihilates $I(X|D)$. Using Definition 8.9, this forces $w$ to lie in
${\operatorname{Br}}(X|D)$. In this case, we must have $\chi=\beta_{X|D}(w)$.
This finishes the proof of surjectivity of $\beta_{X|D}$ when
$D^{\dagger}\subset X(k)$.

To prove the general case, we choose a finite field extension ${k^{\prime}}/k$
such that ${\rm Supp}(D_{k^{\prime}})\subset X(k^{\prime})$. We let
$X^{\prime}=X_{k^{\prime}}$ and $D^{\prime}=D_{k^{\prime}}$. We let
$f\colon(X^{\prime},D^{\prime})\to(X,D)$ be the projection and consider the
diagram

(12.4)
$\textstyle{{\operatorname{Br}}(X^{\prime}|D^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{X^{\prime}|D^{\prime}}}$$\scriptstyle{f_{*}}$$\textstyle{{\rm
CH}_{0}(X^{\prime}|D^{\prime})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(f^{*})^{\star}}$$\textstyle{{\operatorname{Br}}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{X|D}}$$\textstyle{{\rm
CH}_{0}(X|D)^{\star}.}$

This diagram is commutative by Lemma 9.7. We have shown above that
$\beta_{X^{\prime}|D^{\prime}}$ is surjective, and Lemma 11.3 says that the
right vertical arrow is surjective. It follows that $\beta_{X|D}$ is
surjective. This concludes the proof. ∎

End of the proof of Theorem 1.6: Combine Lemmas 2.5, 8.8, 12.1, 12.2 and
Proposition 11.2. $\hfill\qed$

As an application of Theorem 1.6, we get the following result about the norm
map between the Brauer groups of regular curves over local fields.

######  Theorem 12.3.

Let $X$ be a geometrically connected regular quasi-projective curve over a
local field $k$ and let $k^{\prime}$ be a finite field extension of $k$. Let
$f\colon X_{k^{\prime}}\to X$ be the projection. Assume that $X$ is either
smooth or affine. Then the map
$f_{*}\colon{\operatorname{Br}}(X_{k^{\prime}})\to{\operatorname{Br}}(X)$ is
surjective.

###### Proof.

We choose an open embedding $j\colon X\hookrightarrow\overline{X}$, where
$\overline{X}$ is a connected regular projective curve. Then $\overline{X}$ is
necessarily geometrically connected. We let $D=\overline{X}\setminus X$ with
the reduced closed subscheme structure. We let
$\overline{X}^{\prime}=\overline{X}_{k^{\prime}},\ X^{\prime}=X_{k^{\prime}}$
and $D^{\prime}=D_{k^{\prime}}$. If $\overline{X}$ is smooth (e.g., when
$p=1$), we conclude the proof by combining Proposition 8.11, Lemma 11.3 and
Theorem 1.6. We shall now assume that $p>1$ and $X$ is affine.

If ${k^{\prime}}/k$ is purely inseparable, then ${\rm Coker}(f_{*})$ is
annihilated by some power of $p$ (e.g., see [4, § 3.8]). On the other hand,
${\rm Coker}(f_{*})$ is $p$-divisible by [4, Thm. 3.2.3]. It follows that
${\rm Coker}(f_{*})=0$. In general, we can get a factorization