# Duality for cohomology of split tori on curves over local fields

Amalendu Krishna, Jitendra Rathore, Samiron Sadhukhan  Department of
Mathematics, Indian Institute of Science, Bangalore, 560012, India.
<EMAIL_ADDRESS>School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai-400005, India<EMAIL_ADDRESS>School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha
Road, Mumbai-400005, India<EMAIL_ADDRESS>
###### Key words and phrases:

Local fields, 0-cycles, Brauer group, Milnor $K$-theory

###### 2020 Mathematics Subject Classification:

Primary 14C25, 14F22; Secondary 14F30, 19D45

> _Abstract._ We prove duality theorems for the étale cohomology of
> logarithmic Hodge-Witt sheaves and split tori on smooth curves over a local
> field of positive characteristic. As an application, we obtain a description
> of the Brauer group of the function fields of curves over local fields in
> terms of the characters of the idele groups. We also show that the classical
> Brauer-Manin pairing between the Brauer and Picard groups of smooth
> projective curves over local fields has analogues for arbitrary smooth
> curves, smooth projective curves with modulus and singular projective curves
> over such fields.

###### Contents

  1. 1 Introduction
  2. 2 Recollection of Zhao’s duality
  3. 3 The topology of logarithmic Hodge-Witt cohomology
  4. 4 Duality for logarithmic Hodge-Witt cohomology
  5. 5 The relative Picard scheme
  6. 6 Relative Picard group over local fields
  7. 7 Pontryagin dual of the Chow group with modulus
  8. 8 Brauer group with modulus
  9. 9 Brauer-Manin pairing for modulus pairs
  10. 10 Local vs. global duality for ${\mathbb{G}}_{m}$ on compact curve
  11. 11 Continuity of Brauer-Manin pairing
  12. 12 Perfectness of Brauer-Manin pairing for modulus pairs
  13. 13 Brauer-Manin pairing for singular curves

##  1\. Introduction

The duality theorems of Tate (see [57], [58]) and Lichtenbaum [32] for the
étale cohomology of split tori over $p$-adic fields and smooth projective
curves over such fields are fundamental results in arithmetic geometry. In
recent years, these duality theorems have been extended by Scheiderer and van
Hamel [49] (see also [16] and [60]) to more general settings of smooth affine
curves over $p$-adic fields. These generalizations have found many
applications, especially in the study of local-global principles for
cohomology of commutative group schemes over the function fields of curves
over $p$-adic fields.

The analogue of Tate’s duality over local fields of positive characteristics
was proven by Milne [37] while the analogue of Lichtenbaum duality over such
fields was proven by Saito [46] (see also [37]). However, the analogues of the
results of Scheiderer and van Hamel for smooth affine curves and their
function fields over local fields of positive characteristics are currently
unknown. This paper is an attempt to fill this gap.

To obtain these generalizations, we prove a new duality theorem for the
logarithmic Hodge-Witt cohomology on smooth projective curves over a local
field of positive characteristic. We introduce Brauer group with modulus, and
extend the classical Brauer-Manin pairing to the setting of 0-cycles and
Brauer groups with modulus over an arbitrary local field. The duality theorem
for the logarithmic Hodge-Witt cohomology on smooth projective varieties over
finite fields was established long ago by Milne [38]. Over a local field, a
partial result was obtained by Kato-Saito [25]. Below, we describe the main
results of this paper.

###  1.1. Duality for cohomology of ${\mathbb{G}}_{m}$

We fix a local field (i.e., a complete discrete valuation field with finite
residue field) $k$ of characteristic $p>0$. Let $X$ be a geometrically
connected smooth projective curve over $k$ and let $j\colon
X^{o}\hookrightarrow X$ be a dense open immersion. Let $\iota\colon
D\hookrightarrow X$ be the inclusion of an effective Cartier divisor whose
support is the complement of $X^{o}$. We let ${\mathbb{G}}_{m}$ denote the
rank one split torus over $k$. We let
$H^{q}_{cc}(X^{o},{\mathbb{G}}_{m})={\varprojlim}_{n}\
H^{q}_{\text{\'{e}t}}(X,{\mathbb{G}}_{m,(X,nD)})$, where
${\mathbb{G}}_{m,(X,nD)}={\rm
Ker}({\mathbb{G}}_{m,X}\twoheadrightarrow\iota_{*}({\mathbb{G}}_{m,nD}))$.
This is an enriched version of the usual cohomology with compact support
$H^{q}_{{\text{\'{e}t}},c}(X^{o},{\mathbb{G}}_{m})$ of $X^{o}$ in positive
characteristic. One easily checks that the canonical map
$H^{q}_{{\text{\'{e}t}},c}(X^{o},{\mathbb{G}}_{m})\to
H^{q}_{{\text{\'{e}t}}}(X,{\mathbb{G}}_{m})$ factors through
$H^{q}_{cc}(X^{o},{\mathbb{G}}_{m})$.

In characteristic zero, the duality theorem for ${\mathbb{G}}_{m}$ on $X^{o}$
(e.g., see [49]) is essentially insensitive to the topology of the underlying
étale cohomology groups because it holds true if we simply endow these groups
with discrete topology and pass to their profinite completions. This makes the
proof of duality relatively simpler. In contrast, endowing each cohomology
group with correct topology is a challenging part of the proof of the duality
theorem in positive characteristic. We briefly explain the topologies that we
use and refer to the body of the paper for details.

Recall that ${\rm Pic}(X)$ is the group of $k$-points of the Picard scheme
${\mathbf{Pic}}(X)$. Since the latter is a locally of finite type $k$-scheme,
${\rm Pic}(X)$ is endowed with the adic topology induced by the valuation
topology of $k$. Since ${\rm Pic}(X^{o})$ is a quotient of ${\rm Pic}(X)$, it
is equipped with the quotient topology. As a key step in our proofs, we show
in this paper that the relative Picard group ${\rm Pic}(X|D)$ is also equipped
with the adic topology such that the canonical map ${\rm
Pic}(X|D)\twoheadrightarrow{\rm Pic}(X)$ is a topological quotient. We endow
$H^{1}_{cc}(X^{o},{\mathbb{G}}_{m})$ with the inverse limit topology using a
canonical isomorphism ${\rm Pic}(X|D)\cong
H^{1}_{\text{\'{e}t}}(X,{\mathbb{G}}_{m,(X,D)})$. All other cohomology groups
will be endowed with the discrete topology.

For a Hausdorff topological abelian group $G$ with the topology $\tau$, we let
$G^{{\rm pf}}$ denote the profinite $\tau$-completion of $G$. That is,
${G}^{\rm pf}$ is the inverse limit ${\varprojlim}_{U}\ {G}/{U}$ with the
inverse limit topology, where $U$ runs through $\tau$-open subgroups of finite
index in $G$. It is clear that ${G}^{{\rm pf}}$ is a profinite abelian group.
We refer to § 4.2 for the definitions of non-degenerate and perfect pairings
of topological abelian groups. In this paper, we prove the following duality
theorem for étale cohomology.

######  Theorem 1.1.

For every integer $q\neq 0$, there is a bilinear pairing

$H^{q}(X^{o},{\mathbb{G}}_{m})\times
H^{3-q}_{cc}(X^{o},{\mathbb{G}}_{m})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$

which induces perfect pairings of topological abelian groups

$H^{1}(X^{o},{\mathbb{G}}_{m})^{{\rm pf}}\times
H^{2}_{cc}(X^{o},{\mathbb{G}}_{m})\to{{\mathbb{Q}}}/{{\mathbb{Z}}},$
$H^{2}(X^{o},{\mathbb{G}}_{m})\times H^{1}_{cc}(X^{o},{\mathbb{G}}_{m})^{{\rm
pf}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$

and

$H^{3}(X^{o},{\mathbb{G}}_{m})\times H^{0}_{cc}(X^{o},{\mathbb{G}}_{m})^{{\rm
pf}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

For smooth curves over $p$-adic fields, this theorem is due to Lichtenbaum
[32] in the proper case, and to Scheiderer-van Hamel [49] (see also [16]) in
the open case. In positive characteristic, the $q=1$ case of the theorem is
due to Saito [46] (see also [37]) in the proper case.

######  Remark 1.2.

One can construct the pairing of Theorem 1.1 for $q=0$ as well but it will not
be perfect (not even non-degenerate).

###  1.2. Duality for cohomology of ${\mathbb{G}}_{m,K}$

Let $X$ be as above and let $K$ denote the function field of $X$. We let
$\widehat{I}(X)={\underset{x\in
X_{(0)}}{\prod^{\prime}}}\widehat{K}^{\times}_{x}$ denote the restricted
product with respect to the subgroups
$(\widehat{{\mathcal{O}}_{X,x}})^{\times}$, where $\widehat{K}_{x}$ is the
quotient field of $\widehat{{\mathcal{O}}_{X,x}}$. The idele class group
$C(K)$ is the cokernel of the canonical inclusion
$K^{\times}\hookrightarrow\widehat{I}(X)$. This group is endowed with the
inverse limit of the adic topologies of the relative Picard groups of $X$ (cf.
§ 12.3). We let ${\operatorname{Br}}(K)$ have the discrete topology. As an
application of Theorem 1.1, we prove the following.

######  Theorem 1.3.

There is a perfect pairing of topological abelian groups

${\operatorname{Br}}(K)\times C(K)^{{\rm
pf}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

This result provides an explicit description of ${\operatorname{Br}}(K)$ in
terms of the characters of the idele class group. In fact, we prove a duality
theorem for all étale cohomology groups of ${\mathbb{G}}_{m,K}$ in this paper
(cf. Theorem 12.8). When $K$ is the quotient field of an excellent normal
2-dimensional complete local domain with finite residue field, an analogue of
Theorem 1.3 was shown by Saito [46].

###  1.3. Duality for logarithmic Hodge-Witt cohomology

The proof of Theorem 1.1 is based on the duality theorem for the $p$-adic
étale motivic cohomology groups of smooth projective curves over $k$, which we
now describe. We fix an integer $n\geq 1$. Let $X$ be as above and let
$W_{n}\Omega^{\bullet}_{X,{\operatorname{log}}}$ denote the logarithmic Hodge-
Witt complex on $X_{\text{\'{e}t}}$ à la Bloch-Deligne-Illusie.

For smooth projective schemes (of arbitrary dimensions) over a finite field,
Milne showed that the logarithmic Hodge-Witt cohomology (i.e., the étale
cohomology of $W_{n}\Omega^{\bullet}_{X,{\operatorname{log}}}$) groups satisfy
Poincaré duality. However, the analogous result over local fields (of positive
characteristics) is presently unknown. The following result settles this
problem for curves.

To state the duality theorem, we need to recall that over finite fields,
Milne’s duality yields a perfect pairing of finite groups. However, the
logarithmic Hodge-Witt cohomology groups over local fields are generally not
finite. To offset this problem, we need to equip these groups with suitable
non-discrete topologies. In fact, dealing with the infinitude of cohomology
groups and choosing correct topology on them are perhaps the major part of
proving a duality theorem over local fields. For the topologies that we endow
these groups with, the reader is referred to § 3.2. The following theorem was
proven by Kato-Saito [25, Prop. 4] when $j=0$ by a different method.

######  Theorem 1.4.

Let $X$ be as in Theorem 1.1 and let $q,j$ be any integers. Then the cup
product on the étale cohomology of the logarithmic Hodge-Witt sheaves induces
a perfect pairing of topological abelian groups

$H^{q}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})\times
H^{2-q}(X,W_{n}\Omega^{2-j}_{X,{\operatorname{log}}})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

######  Remark 1.5.

We actually prove this theorem under weaker assumptions on $X$. Namely, we
need to assume that $X$ is regular but do not require it to be smooth over
$k$. We work under the weaker hypothesis that $X$ is generically smooth over
$k$. The validity of the theorem under generic smoothness is useful for some
concrete applications. For instance, we hope that it will allow one to prove
the duality theorem for the $p$-adic étale motivic cohomology groups with
compact support for smooth but non-proper curves. Such a result will have
applications to the class field theory of smooth open curves over local
fields. This is the topic of [31].

###  1.4. Brauer-Manin pairing for modulus pairs

Let $\iota\colon D\hookrightarrow X$ be the inclusion of an effective Cartier
divisor as above. We shall refer to the pair $(X,D)$ as a (1-dimensional)
modulus pair. Let ${\rm CH}_{0}(X|D)$ be the Chow group of 0-cycles for the
modulus pair $(X,D)$. We shall show in this paper that ${\rm CH}_{0}(X|D)$ is
naturally endowed with an adic topology such that the canonical map ${\rm
CH}_{0}(X|D)\twoheadrightarrow{\rm CH}_{0}(X)$ is a topological quotient. In
order to extend the classical Brauer-Manin pairing to the modulus setting, we
introduce the Brauer group of the modulus pair $(X,D)$ which we denote by
${\operatorname{Br}}(X|D)$ (see Definition 8.9). This is a certain subgroup of
${\operatorname{Br}}(X^{o})$ which is defined in terms of Kato’s filtration on
the étale motivic cohomology group
$H^{2}(K,{{\mathbb{Q}}}/{{\mathbb{Z}}}(1))$, where $K$ is the function field
of $X$. We endow ${\operatorname{Br}}(X|D)$ with the discrete topology. As a
key step for proving Theorem 1.1, we establish the following new Brauer-Manin
pairing.

######  Theorem 1.6.

There is a continuous pairing of topological abelian groups

${\operatorname{Br}}(X|D)\times{\rm
CH}_{0}(X|D)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$

such that the induced pairing

${\operatorname{Br}}(X|D)\times{\rm CH}_{0}(X|D)^{{\rm
pf}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$

is perfect.

The case $D=\emptyset$ of this theorem is due to Lichtenbaum [32] in
characteristic zero and Saito [46] (see also [37, Thm. III.7.8]) in positive
characteristic. In this paper, we provide an independent and short proof of
the harder part of Saito’s result as an immediate corollary of Theorem 1.4
(see Corollary 9.6).

###  1.5. Brauer-Manin pairing for singular curves

Recall that the Brauer group of a singular variety $Y$ does not satisfy the
Brauer injectivity property, i.e., the canonical map
${\operatorname{Br}}(Y)\to{\operatorname{Br}}(K)$ fails to be injective in
general, for $K$ the function field of $Y$. This defect is analogous to the
one the classical Chow group (one defined in [10]) ${\rm CH}_{0}(Y)$ suffers
from. In this paper, we introduce a refined Brauer group of $Y$ which we
denote by ${\operatorname{Br}}^{{\rm lw}}(Y)$. We refer to this as the Levine-
Weibel Brauer group. The refined Brauer group satisfies some nice properties,
including the Brauer injectivity property.

Let ${\rm CH}^{{\rm lw}}_{0}(Y)$ denote the Levine-Weibel Chow group of $Y$.
One may recall that this coincides with ${\rm Pic}(Y)$ if $\text{\rm
dim}(Y)=1$. We show in this paper that ${\rm CH}^{{\rm lw}}_{0}(Y)$ is
naturally endowed with an adic topology such that the pull-back map ${\rm
CH}^{{\rm lw}}_{0}(Y)\to{\rm CH}_{0}(Y_{n})$ is continuous if $Y$ is a
geometrically integral projective curve over $k$ with smooth normalization
$Y_{n}$. We endow ${\operatorname{Br}}^{{\rm lw}}(Y)$ with the discrete
topology and prove the following extension of the Brauer-Manin pairing to
singular curves.

######  Theorem 1.7.

Let $Y$ be a geometrically integral projective curve over a local field $k$
whose normalization is smooth over $k$. Then there is a continuous pairing of
topological abelian groups

${\operatorname{Br}}^{{\rm lw}}(Y)\times{\rm CH}^{{\rm
lw}}_{0}(Y)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$

such that the induced pairing

${\operatorname{Br}}^{{\rm lw}}(Y)\times{\rm CH}^{{\rm lw}}_{0}(Y)^{{\rm
pf}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$

is perfect.

###  1.6. A brief outline of proofs

The main ingredients for proving Theorem 1.1 are Theorems 1.4 and 1.6. A key
step in the proof of Theorem 1.4 is a duality theorem of Zhao [61]. In § 2, we
recall this duality theorem. In § 3, we equip the logarithmic Hodge-Witt
cohomology of a smooth projective curve $X$ over a local field $k$ of positive
characteristic with a suitable topology using some deductions. We construct a
trace homomorphism for the top logarithmic Hodge-Witt cohomology of $X$ in § 4
and complete the proof of Theorem 1.4. A trace map in this case was also
constructed by Kato-Saito [25, § 3, Prop. 4]. But our trace homomorphism is, a
priori, different. It has an advantage that one can easily show its
compatibility with the trace homomorphisms for closed points under the Gysin
homomorphisms. This compatibility is crucial in our proofs. In the next three
sections, we equip the relative Picard group of modulus pairs $(X,D)$ with the
adic topology. We also prove some key properties of the Albanese map for $X$
and the Pontryagin dual of the relative Picard group.

We introduce the Brauer group of modulus pairs in § 8 and establish the
Brauer-Manin pairing between the Brauer and Chow groups of a modulus pair in §
9. One of the key steps in the proof of Theorem 1.6 is a duality theorem for
the cohomology of ${\mathbb{G}}_{m}$ on $X$ and its compatibility with Kato’s
duality for 2-dimensional local fields obtained by completing the function
field of $X$ at closed points. This is done in § 10. In § 11, we complete
another step for proving Theorem 1.6, namely, we show that the Brauer-Manin
paring for a modulus pair $(X,D)$ is continuous. We complete the proof of
Theorem 1.6 in § 12, and then apply it to complete the proofs of Theorems 1.1
and Theorem 1.3. In § 13, we introduce a refined version of the Brauer group
of singular varieties and prove Theorem 1.7 as an application of Theorem 1.6.

###  1.7. Notations

We shall work over a field $k$ of characteristic $p\geq 0$ throughout this
paper. We shall let $k_{s}$ (resp. $\overline{k}$) denote a fixed separable
(resp. algebraic) closure of $k$. We let ${\operatorname{\mathbf{Sch}}}_{k}$
denote the category of separated Noetherian $k$-schemes and
${\mathbf{Sm}}_{k}$ the category of smooth (in particular, finite type)
$k$-schemes. A $k$-scheme will mean an object of
${\operatorname{\mathbf{Sch}}}_{k}$ and a finite type $k$-scheme will mean an
object of ${\operatorname{\mathbf{Sch}}}^{{\rm ft}}_{k}$. The product
$X\times_{{\rm Spec\,}(k)}Y$ in ${\operatorname{\mathbf{Sch}}}_{k}$ will be
written as $X\times Y$. We let $X^{(q)}$ (resp. $X_{(q)}$) denote the set of
points on $X$ having codimension (resp. dimension) $q$. We shall let $X_{\rm
sing}$ (resp. $X_{\rm reg}$) denote the singular locus (resp. regular locus)
of $X$ with the reduced closed subscheme structure. We let
${\mathcal{Z}}_{0}(X)$ denote the free abelian group of 0-cycles on $X$.

We let ${\operatorname{\mathbf{Sch}}}_{k/{\rm zar}}$ (resp.
${\operatorname{\mathbf{Sch}}}_{k/{\rm nis}}$, resp.
${\operatorname{\mathbf{Sch}}}_{k/{\acute{e}t}})$ denote the Zariski (resp.
Nisnevich, resp. étale) site of ${\operatorname{\mathbf{Sch}}}_{k}$. Unless we
mention the topology specifically, all cohomology groups in this paper will be
considered with respect to the étale topology. We shall let $cd_{p}(X)$ denote
the étale $p$-cohomological dimension of
$X\in{\operatorname{\mathbf{Sch}}}_{k}$ if ${\rm char}(k)=p>0$. We shall let
$cd(X)$ denote the étale cohomological dimension of torsion sheaves on $X$. We
shall let $G_{k}$ denote the abelian absolute Galois group of $k$.

For an abelian group $A$, we shall write
${\operatorname{Tor}}^{1}_{{\mathbb{Z}}}(A,{{\mathbb{Z}}}/n)$ as ${}_{n}A$ and
$A/{nA}$ as $A/n$. We shall let $A\\{p^{\prime}\\}$ denote the subgroup of
elements of $A$ which are torsion of order prime to $p$. We let $A\\{p\\}$
denote the subgroup of elements of $A$ which are torsion of order some power
of $p$. The tensor product $A\otimes_{{\mathbb{Z}}}B$ will be written as
$A\otimes B$. We let ${\mathbf{Ab}}$ denote the category of abelian groups and
${\mathbf{Tab}}$ denote the category of topological abelian groups with
continuous homomorphisms. For $A,B\in{\mathbf{Tab}}$, we shall let ${\rm
Hom}_{\rm cf}(A,B)=({\rm Hom}_{{\mathbf{Tab}}}(A,B))_{\rm tor}$. Unless a
specific topology is mentioned, we shall assume all finite abelian groups to
be endowed with the discrete topology. In this paper, we shall use the general
notation $\\{A_{n}\\}$ for pro-abelian or ind-abelian group indexed by
${\mathbb{N}}=\\{1,2,\ldots\\}$. However, we shall also use specific notations
$``{{\underset{n}{\varinjlim}}}"A_{n}$ for an ind-abelian group and
$``{{\underset{n}{\varprojlim}}}"A_{n}$ for a pro-abelian group if we need to
make a distinction between them.

##  2\. Recollection of Zhao’s duality

In this section, we recall Brauer group, Milnor $K$-theory, Hodge-Witt sheaves
and the duality theorem of Zhao [61] for the logarithmic Hodge-Witt
cohomology. This will be the key ingredient in the proof of Theorem 1.4. We
also recall the Pontryagin duality for a special class of locally compact
Hausdorff topological abelian groups.

###  2.1. Pontryagin duality for torsion-by-profinite groups

Let $G$ be a locally compact Hausdorff topological abelian group. We shall say
that $G$ is ‘torsion-by-profinite’ if there is an exact sequence

(2.1) $0\to G_{\rm pf}\to G\to G_{\rm dt}\to 0,$

where $G_{\rm pf}$ is an open and profinite subgroup of $G$ and $G_{\rm dt}$
is a torsion group with the quotient topology (necessarily discrete). We shall
call (2.1) a torsion-by-profinite presentation of $G$. Note that such a
presentation of $G$ is not unique.

Recall that $G$ admits its Pontryagin dual $G^{\star}:={\rm
Hom}_{{\mathbf{Tab}}}(G,{\mathbb{T}})$, where ${\mathbb{T}}$ is the circle
group such that the evaluation map $G\to(G^{*})^{*}$ is an isomorphism of
locally compact topological abelian groups (e.g., see [43, § 2.9] or [9, Thm.
4.32]) if $G^{\star}$ is endowed with the compact-open topology. One knows
that $G^{\star}\cong{\rm
Hom}_{{\mathbf{Tab}}}(G,{{\mathbb{Q}}}/{{\mathbb{Z}}})$ if $G$ is torsion-by-
profinite group (e.g., see [43, Lem. 2.9.2]), where
${{\mathbb{Q}}}/{{\mathbb{Z}}}={\mathbb{T}}_{{\rm tor}}$ is endowed with the
discrete topology. Furthermore, one has an isomorphism of topological abelian
groups ${\rm
Hom}_{{\mathbf{Tab}}}(G^{\star},{{\mathbb{Q}}}/{{\mathbb{Z}}})\xrightarrow{\cong}{\rm
Hom}_{{\mathbf{Tab}}}(G^{\star},{\mathbb{T}})$. In other words, letting
${\operatorname{\mathbf{Pfd}}}$ denote the category of torsion-by-profinite
topological abelian groups with continuous homomorphisms, we have the
following.

######  Lemma 2.1.

The category ${\operatorname{\mathbf{Pfd}}}$ is closed under taking Pontryagin
dual such that $G\in{\operatorname{\mathbf{Pfd}}}$ is profinite if and only if
$G^{\star}$ is a discrete torsion group. Moreover, the canonical map ${\rm
Hom}_{{\mathbf{Tab}}}(G,{{\mathbb{Q}}}/{{\mathbb{Z}}})\to G^{\star}$ is an
isomorphism of topological abelian groups for every
$G\in{\operatorname{\mathbf{Pfd}}}$.

In this paper, we shall always consider ${{\mathbb{Z}}}$ and
${{\mathbb{Q}}}/{{\mathbb{Z}}}$ as topological abelian groups with discrete
topologies. For any topological abelian group $G$, we shall denote ${\rm
Hom}_{{\mathbf{Tab}}}(G,{{\mathbb{Q}}}/{{\mathbb{Z}}})$ by $G^{\star}$. For
$G\in{\mathbf{Ab}}$, we shall let $G^{\vee}={\rm
Hom}_{\mathbf{Ab}}(G,{{\mathbb{Q}}}/{{\mathbb{Z}}})$.

######  Lemma 2.2.

If $G\in{\operatorname{\mathbf{Pfd}}}$ and $G^{\prime}\subseteq G$ is a closed
subgroup, then $G^{\prime}\in{\operatorname{\mathbf{Pfd}}}$.

###### Proof.

This is elementary and we skip the proof. ∎

######  Lemma 2.3.

Let $\beta\colon G\twoheadrightarrow G^{\prime\prime}$ be a continuous
surjective homomorphism in ${\operatorname{\mathbf{Pfd}}}$ such that
$G^{\prime\prime}_{{\rm pf}}\subseteq\beta(G_{\rm pf})$ with respect to some
torsion-by-profinite presentations of $G$ and $G^{\prime\prime}$ as in (2.1).
Then the induced map $\beta^{\prime}\colon{G}/{{\rm Ker}(\beta)}\to
G^{\prime\prime}$ is an isomorphism of topological abelian groups if
${G}/{{\rm Ker}(\beta)}$ is endowed with the quotient topology.

###### Proof.

If $G^{\prime\prime}$ is profinite, the claim is an easy application of the
fact that a continuous bijective homomorphism from a compact Hausdorff
topological abelian group to a Hausdorff topological abelian group is a
topological isomorphism. For the general case, we let
$H=\beta^{-1}(G^{\prime\prime}_{\rm pf})$. Then $H\subset G$ is an open
subgroup and $G/H$ is discrete. Furthermore, there is a surjective continuous
homomorphism $\beta\colon H\twoheadrightarrow G^{\prime\prime}_{\rm pf}$ such
that $(G_{\rm pf}\cap H)\twoheadrightarrow G^{\prime\prime}_{\rm pf}$ by our
assumption. Lemma 2.2 implies that $\beta\colon H\to G^{\prime\prime}_{\rm
pf}$ is a continuous surjective homomorphism from a torsion-by-profinite group
to a profinite group. In particular, this is a topological quotient. Since
$G^{\prime\prime}_{\rm pf}$ is open in $G^{\prime\prime}$, one deduces that
$\beta$ is an open map on $G$. ∎

######  Lemma 2.4.

Let

$0\to G^{\prime}\xrightarrow{\alpha}G\xrightarrow{\beta}G^{\prime\prime}\to 0$

be a short exact sequence in ${\operatorname{\mathbf{Pfd}}}$. Assume that with
respect to some torsion-by-profinite presentations of $G$ and
$G^{\prime\prime}$ as in (2.1), one has ${G^{\prime\prime}}_{{\rm
pf}}\subseteq\beta(G_{\rm pf})$. Then the sequence

$0\to(G^{\prime\prime})^{\star}\xrightarrow{\beta^{\star}}G^{\star}\xrightarrow{\alpha^{\star}}(G^{\prime})^{\star}\to
0$

is exact.

###### Proof.

The sequence

$0\to(G^{\prime\prime})^{\star}\xrightarrow{\beta^{\star}}G^{\star}\xrightarrow{\alpha^{\star}}(G^{\prime})^{\star}$

is exact by Lemma 2.3 and $\alpha^{\star}$ is surjective by Lemma 2.1 and [9,
Cor. 4.42]. ∎

Recall from § 1 that for a topological abelian group $G$, the profinite
completion of $G$ (with respect to its topology) is the profinite topological
abelian group $G^{\rm pf}={\varprojlim}_{U}{G}/U$ with the inverse limit
topology, where the limit is taken over all open subgroups of finite index in
$G$. The following result is elementary whose proof is left to the reader.

######  Lemma 2.5.

Let $G$ be a topological abelian group. Then the canonical map $(G^{\rm
pf})^{\star}\to G^{\star}$ induces an isomorphism $(G^{\rm
pf})^{\star}\xrightarrow{\cong}(G^{\star})_{\rm tor}$.

Suppose that $\\{G_{i}\\}_{i\in I}$ is a pro-object in ${\mathbf{Tab}}$ such
that $I$ is cofiltered. We let $G={\varprojlim}_{i}G_{i}$ and endow it with
the inverse limit topology. Let $\pi_{i}\colon G\to G_{i}$ be the projection
map. If the topology of each $G_{i}$ is generated by open subgroups, then the
same holds for $G$ as well. More precisely, $G$ admits a fundamental system of
neighborhoods of the identity of the form $\pi^{-1}_{i}(U_{i})$, where
$U_{i}\subset G_{i}$ is an open subgroup. This happens, for instance, when
each $G_{i}$ is a Hausdorff, totally disconnected and locally compact
topological abelian group. If $k$ is a local field and $X$ is a commutative
group scheme over $k$, then $X(k)$ is endowed with a topology of this kind
(cf. § 6.3). We shall use the following result in this paper.

######  Lemma 2.6.

Let $\\{G_{i}\\}_{i\in I}$ be above such that the topology of each $G_{i}$ is
generated by open subgroups. Then the canonical map
${\varinjlim}_{i}G^{\star}_{i}\to G^{\star}$ is surjective. This map is
bijective if $\pi_{i}\colon G\to G_{i}$ is surjective for each $i\in I$.

###### Proof.

Let $\chi\colon G\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$ be a continuous character
and let $H={\rm Ker}(\chi)$. Then our hypothesis implies that there exists
$i\in I$ and an open subgroup $U_{i}\subset G_{i}$ such that
$V_{i}:=\pi^{-1}_{i}(U_{i})\subset H$. It follows that $\chi$ factors through
$\chi^{\prime}\colon G/{V_{i}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$. We thus get a
diagram

(2.2)
$\textstyle{G_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{i}}$$\textstyle{{G_{i}}/{U_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{i}}$$\textstyle{G/{V_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi^{\prime}}$$\textstyle{{{\mathbb{Q}}}/{{\mathbb{Z}}},}$

whose left square is commutative. Since $G/{V_{i}}$ and ${G_{i}}/{U_{i}}$ are
discrete, we get a continuous character
$\phi^{\prime}\colon{G_{i}}/{U_{i}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$ such that
the left triangle commutes. Letting $\phi=\phi^{\prime}\circ\alpha_{i}$, we
see that $\phi$ is continuous and $\phi\circ\pi_{i}=\chi$. The second part of
the lemma is clear. ∎

###  2.2. Milnor $K$-theory and logarithmic Hodge-Witt sheaves

For a commutative ring $A$, we let $K_{*}(A)$ denote the Quillen $K$-theory of
finitely generated projective $A$-modules. We let $\widehat{K}^{M}_{*}(A)$
denote the graded commutative ring defined as the quotient of the tensor
algebra $T_{*}(A^{\times})$ by the two-sided ideal generated by the
homogeneous elements $a_{1}\otimes a_{2}$ such that $a_{1}+a_{2}=1$. We let
${K}^{M}_{*}(A)$ denote the improved Milnor $K$-theory of $A$ à la Gabber-Kerz
[27]. There are natural multiplicative homomorphisms
$\widehat{K}^{M}_{r}(A)\xrightarrow{\alpha_{A}}{K}^{M}_{r}(A)\xrightarrow{\beta_{A}}K_{r}(A)$
such that $\alpha_{A}$ is surjective (resp. an isomorphism) if $A$ is a local
ring (resp. a field). Furthermore, $\alpha_{A}$ is an isomorphism if $A$ is
local with infinite residue field and $\beta_{A}$ is an isomorphism for $r\leq
2$ if $A$ is local (see [27, Prop. 10]).

For a scheme $X$, we let ${\mathcal{K}}^{M}_{r,X}$ (resp.
$\widehat{{\mathcal{K}}}^{M}_{r,X}$, resp. ${\mathcal{K}}_{r,X}$) denote the
sheaf (in Zariski, Nisnevich or étale topology) on $X$ associated to the
presheaf $U\mapsto K^{M}_{r}({\mathcal{O}}(U))$ (resp.
$\widehat{K}^{M}_{r}({\mathcal{O}}(U))$, resp. $K_{r}({\mathcal{O}}(U))$). For
an ideal $I\subset A$ in a local ring, we let $\widehat{K}^{M}_{r}(A,I)={\rm
Ker}(\widehat{K}^{M}_{r}(A)\twoheadrightarrow\widehat{K}^{M}_{r}(A/I))$. Given
a closed immersion $\iota\colon Y\hookrightarrow X$, we let
$\widehat{{\mathcal{K}}}^{M}_{r,(X,Y)}={\rm
Ker}(\widehat{{\mathcal{K}}}^{M}_{r,X}\twoheadrightarrow\iota_{*}(\widehat{{\mathcal{K}}}^{M}_{r,Y}))$.
The group ${K}^{M}_{r}(A,I)$ and the sheaf ${{\mathcal{K}}}^{M}_{r,(X,Y)}$ are
defined in a similar manner.

Let $k$ be a field of exponential characteristic $p>1$. Recall from [19] that
for a $k$-scheme $X$, $\\{W_{m}\Omega^{\bullet}_{X}\\}_{m\geq 1}$ denotes the
pro-complex of de Rham-Witt (Nisnevich) sheaves on $X$. This is a pro-complex
of sheaves of differential graded algebras with the structure map $R$ and the
differential $d$. Let $[-]_{m}\colon{\mathcal{O}}_{X}\to
W_{m}{\mathcal{O}}_{X}$ be the multiplicative Teichmüller homomorphism. Recall
that the pro-complex $\\{W_{m}\Omega^{\bullet}_{X}\\}_{m\geq 1}$ is equipped
with the Frobenius homomorphism of graded algebras $F\colon
W_{m}\Omega^{r}_{X}\to W_{m-1}\Omega^{r}_{X}$ and the additive Verschiebung
homomorphism $V\colon W_{m}\Omega^{r}_{X}\to W_{m+1}\Omega^{r}_{X}$. We let
$ZW_{m}\Omega^{r}_{X}={\rm Ker}(d\colon W_{m}\Omega^{r}_{X}\to
W_{m}\Omega^{r+1}_{X})$ and $BW_{m}\Omega^{r}_{X}={\rm Image}(d\colon
W_{m}\Omega^{r-1}_{X}\to W_{m}\Omega^{r}_{X})$. We write
${\mathcal{H}}^{r}(W_{m}\Omega^{\bullet}_{X})={ZW_{m}\Omega^{r}_{X}}/{BW_{m}\Omega^{r}_{X}}$.

Recall from [19] that $W_{m}\Omega^{r}_{X,{\operatorname{log}}}$ is the
(Zariski, Nisnevich or étale) subsheaf of $W_{m}\Omega^{r}_{X}$ which is the
image of the map ${\rm dlog}\colon{{\mathcal{K}}^{M}_{r,X}}/{p^{m}}\to
W_{m}\Omega^{r}_{X}$, locally given by ${\rm
dlog}(\\{a_{1},\ldots,a_{r}\\})={\rm dlog}[a_{1}]_{m}\wedge\cdots\wedge{\rm
dlog}[a_{r}]_{m}$. It is easily seen that this map exists (e.g., see [33, Rem.
1.6]). Moreover, it is an isomorphism in any of the above topologies if $X$ is
regular (e.g., see [14, Lem. 2.3]). Equivalently, there is a short exact
sequence of (Zariski, Nisnevich or étale) sheaves

(2.3)
$0\to{\mathcal{K}}^{M}_{r,X}\xrightarrow{p^{n}}{\mathcal{K}}^{M}_{r,X}\to
W_{n}\Omega^{r}_{X,{\operatorname{log}}}\to 0.$

The multiplicative structure of ${{\mathcal{K}}^{M}_{*,X}}/{p^{m}}$ and
$W_{m}\Omega^{\bullet}_{X}$ together with the multplicativity of ${\rm dlog}$
induces a graded commutative ring structure on
$W_{m}\Omega^{\bullet}_{X,{\operatorname{log}}}$.

We end this subsection by recalling the Brauer group. For any Noetherian
scheme $X$, recall that ${\operatorname{Br}}(X)$ denotes the cohomological
Brauer group $H^{2}(X,{\mathcal{O}}^{\times}_{X})$. A result of Gabber (e.g.,
see [4, Thm. 4.2.1]) says that ${\operatorname{Br}}(X)$ coincides with the
Azumaya Brauer group ${\operatorname{Br}}^{\prime}(X)$ if $X$ is quasi-
projective over an affine scheme. Since all schemes in this paper will satisfy
this condition, we shall make no distinction between the Azumaya and
cohomological Brauer groups. Brauer group of curves over local fields will be
one the main objects of study in this paper.

###  2.3. The set-up

The set-up for our duality theorem will be the following. We fix a
equicharacteristic complete discrete valuation ring (cdvr) $R$ with finite
residue field ${\mathfrak{f}}$. We let ${\mathfrak{m}}$ denote the maximal
ideal of $R$ and $k$ denote the quotient field of $R$. It is well known that
$R$ is canonically isomorphic to the formal power series ring
${\mathbb{F}}_{q}[[\pi]]$, and $k={\mathbb{F}}_{q}((\pi))$, where $q=p^{n}$
for some prime number $p\geq 2$ and some positive integer $n\geq 1$. We shall
fix this isomorphism throughout our discussion. We let $S={\rm Spec\,}(R)$ and
let $\eta$ (resp. $s$) denote the generic (resp. closed) point of $S$.

We let ${\mathcal{X}}$ be a connected Noetherian regular scheme with a flat
and projective morphism ${\mathbb{F}}\colon{\mathcal{X}}\to S$ of relative
dimension $d\leq 1$. We let $X$ denote the generic fiber and
${\mathcal{X}}_{s}$ the (scheme-theoretic) closed fiber of ${\mathbb{F}}$. We
shall assume that $Y:=({\mathcal{X}}_{s})_{\rm red}$ is a simple normal
crossing divisor on ${\mathcal{X}}$. A morphism ${\mathbb{F}}$ satisfying
these properties will be called a semi-stable model for the $k$-scheme $X$. We
let $u\colon X\hookrightarrow{\mathcal{X}}$ and $\iota\colon
Y\hookrightarrow{\mathcal{X}}$ be the inclusions. We let $f\colon X\to{\rm
Spec\,}(k)$ and $g\colon{\mathcal{X}}_{s}\to{\rm Spec\,}({\mathfrak{f}})$
denote the structure maps. We shall assume that $X$ is geometrically connected
and generically smooth over $k$. Recall that the generic smoothness of a
morphism of schemes $f\colon Z\to W$ means that $f$ is smooth in a
neighborhood of each generic point of $Z$. All results in the remainder of § 2
(and also in the next two sections) will be proven under this set-up. We let
$K$ denote the function field of ${\mathcal{X}}$.

The following elementary observation will be useful in this paper. We skip its
proof.

######  Lemma 2.7.

${\mathcal{X}}_{s}$ (equivalently, $Y$) is geometrically connected over
${\mathfrak{f}}$.

###  2.4. Zhao’s duality theorem

We now recall the $d=1$ case of the duality theorem of Zhao (see [61, Cor.
1.4.10, Thm. 3.1.1]) which will be a key ingredient in the proof of Theorem
1.4. We fix an integer $n\geq 1$.

######  Theorem 2.8.

  1. (1)

There is a canonical isomorphism of ${{\mathbb{Z}}}/{p^{n}}$-modules

${{\rm Tr}}_{{\mathcal{X}}}\colon
H^{3}_{Y}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{\cong}{{\mathbb{Z}}}/{p^{n}}.$

  2. (2)

The graded commutative ring structure of
$H^{*}({\mathcal{X}},W_{n}\Omega^{\bullet}_{{\mathcal{X}},{\operatorname{log}}})$
gives rise to a pairing

$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\times
H^{3-i}_{Y}({\mathcal{X}},W_{n}\Omega^{2-j}_{{\mathcal{X}},{\operatorname{log}}})\to{{\mathbb{Z}}}/{p^{n}}$

such that the induced map

${\Phi}^{ij}_{\mathcal{X}}\colon
H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\to{\rm
Hom}_{{\mathbf{Ab}}}(H^{3-i}_{Y}({\mathcal{X}},W_{n}\Omega^{2-j}_{{\mathcal{X}},{\operatorname{log}}}),{{\mathbb{Z}}}/{p^{n}})$

is an isomorphism.

  3. (3)

The map

${\Psi}^{ij}_{{\mathcal{X}}}\colon
H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\to{\rm
Hom}_{{\mathbf{Tab}}}(H^{3-i}({\mathcal{X}},W_{n}\Omega^{2-j}_{{\mathcal{X}},{\operatorname{log}}}),{{\mathbb{Z}}}/{p^{n}})$

is a topological isomorphism if
$H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$
is endowed with the discrete topology and
$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$ is
endowed with the profinite topology by virtue of $\Phi^{ij}_{{\mathcal{X}}}$
and the Pontryagin duality.

###### Proof.

This result was proven by Zhao in [61] except that he additionally assumed
that ${\mathcal{X}}_{s}$ is reduced and $X\in{\mathbf{Sm}}_{k}$. But these
assumptions are unnecessary as we shall explain by outlining Zhao’s proof.

Recall that $\iota\colon Y\hookrightarrow{\mathcal{X}}$ is the inclusion of
the reduced special fiber, which is assumed to be a simple normal crossing
divisor in ${\mathcal{X}}$. The proof of (1) goes as follows. Using a duality
theorem of Sato for normal crossing projective schemes over ${\mathfrak{f}}$,
the proof of (1) is reduced to showing a purity isomorphism (i.e., Corollary
1.4.9 of op. cit.). The proof of this corollary has the following ingredients.
The first is Proposition 1.2.4 of op. cit. for $Y$, which is a result of Sato
that holds for any normal crossing scheme. The second is Corollary 1.3.14 of
op. cit., which is a result of Shiho that holds for any closed immersion of
regular ${\mathbb{F}}_{p}$-schemes. The third is Proposition 1.4.6 of op.
cit., which is a result of Moser that holds for any finite type
${\mathfrak{f}}$-scheme. The final step is Theorem 1.4.4 of op. cit., whose
proof also requires only the above three ingredients.

To prove (2), we let
$\iota^{\prime}\colon{\mathcal{X}}_{s}\hookrightarrow{\mathcal{X}}$ and
$\iota^{\prime\prime}\colon Y\hookrightarrow{\mathcal{X}}_{s}$ be the
inclusions. We then have a cup product pairing (see § 3.1 of op. cit.)

$H^{i}(Y,\iota^{*}(W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}))\times
H^{3-i}(Y,R\iota^{!}(W_{n}\Omega^{2-j}_{{\mathcal{X}},{\operatorname{log}}}))\to
H^{3}(Y,R\iota^{!}(W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})).$

On the other hand, there are canonical maps

(2.4)
$\begin{array}[]{lll}H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})&\xrightarrow{\iota^{\prime*}}&H^{i}({\mathcal{X}}_{s},\iota^{\prime*}(W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}))\\\
&\xrightarrow{\iota^{\prime\prime*}}&H^{i}({\mathcal{X}}_{s},\iota^{\prime\prime}_{*}\circ\iota^{\prime\prime*}\circ\iota^{\prime*}(W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}))\\\
&\cong&H^{i}({\mathcal{X}},\iota_{*}\circ\iota^{*}(W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}))\\\
&\cong&H^{i}(Y,\iota^{*}(W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})).\end{array}$

The first arrow is an isomorphism by the proper base change theorem for the
étale cohomology of torsion sheaves (e.g., see [35, Cor. VI.2.7]). The second
arrow is the canonical counit of adjunction map which is an isomorphism since
the closed immersion $\iota^{\prime\prime}$ has empty complement and
$\iota^{\prime\prime}_{*}$ is exact. Since
$H^{i}(Y,R\iota^{!}(W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}))\cong
H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$,
the above pairing can be written as

(2.5)
$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\times
H^{3-i}_{Y}({\mathcal{X}},W_{n}\Omega^{2-j}_{{\mathcal{X}},{\operatorname{log}}})\to
H^{3}_{Y}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{{{\rm
Tr}}_{{\mathcal{X}}}}{{\mathbb{Z}}}/{p^{n}}.$

The proof of the perfectness of (2.5) is easily reduced to the case when $n=1$
using the exact sequence (see [53, Prop. 2.8, 2.12])

(2.6) $0\to
W_{n-m}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}\xrightarrow{\underline{p}^{m}}W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}\xrightarrow{R^{n-m}}W_{m}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}\to
0$

and compatibility of (2.5) with respect to maps
$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{\underline{p}}H^{i}({\mathcal{X}},W_{n+1}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$
and
$H^{3-i}_{Y}({\mathcal{X}},W_{n+1}\Omega^{2-j}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{R}H^{3-i}_{Y}({\mathcal{X}},W_{n}\Omega^{2-j}_{{\mathcal{X}},{\operatorname{log}}})$.
Here, $\underline{p}^{m}$ is induced by the multiplication map
${{\mathcal{K}}^{M}_{j,{\mathcal{X}}}}/{p^{n-m}}\xrightarrow{p^{m}}{{\mathcal{K}}^{M}_{j,{\mathcal{X}}}}/{p^{n}}$
via the isomorphism ${\rm dlog}$.

The proof of the case $n=1$ is a direct consequence of Cor. 2.5.2 (which is
the coherent duality) and Prop. 3.1.2 (which proves the compatibility between
the trace maps of Sato and that of the coherent duality) of op. cit.. The
latter result is a statement only about $S$. The proof of the former result
does not depend on the special fiber at all, but it requires us to know that
the locally free sheaf $\Omega^{1}_{\mathcal{X}}$ has rank two. However, this
can be checked by restricting to the open subscheme of ${\mathcal{X}}$ where
$X$ is smooth over $k$. The generic smoothness of $X$ over $k$ suffices for
this purpose. ∎

##  3\. The topology of logarithmic Hodge-Witt cohomology

As we mentioned in § 1, we need to endow the logarithmic Hodge-Witt cohomology
of complete curves over local fields with suitable topology in order to extend
Milne’s duality for logarithmic Hodge-Witt cohomology over finite fields to
local fields. This is the goal of the present section. We shall continue to
work under the set-up of § 2.3.

###  3.1. Some finiteness results

In this subsection, we prove some technical lemmas regarding certain
logarithmic Hodge-Witt cohomology groups as preparation for proving Theorem
1.1. We let $n\geq 1$ be any integer. We begin with the following vanishing
statement.

######  Lemma 3.1.

We have $W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}=0$ for $n\geq 1$
and $j\geq 3$. Furthermore,
$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})=0$
for $i\geq 3$ and $j\geq 0$.

###### Proof.

To prove the first claim, we can assume, using (2.6), that $n=1$. By the exact
sequence (where $\overline{{\operatorname{id}}}$ is the canonical quotient map
and $\overline{F}$ is induced by $F$, see [14, § 5.2])

(3.1) $0\to W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}\to
W_{n}\Omega^{j}_{\mathcal{X}}\xrightarrow{\overline{{\operatorname{id}}}-\overline{F}}\frac{W_{n}\Omega^{j}_{\mathcal{X}}}{dV^{n-1}\Omega^{j-1}_{\mathcal{X}}}\to
0,$

it suffices to show that $\Omega^{j}_{{\mathcal{X}}}=0$ for $j\geq 3$. Since
${\mathcal{X}}$ is connected and $\Omega^{j}_{\mathcal{X}}$ is locally free
(e.g., see [61, Prop. 2.1.2]), it suffices to show that
$\Omega^{j}_{\mathcal{X}}$ is generically zero. We can thus replace
${\mathcal{X}}$ by the largest affine open subscheme
$X_{\operatorname{sm}}\subset X$ such that $X_{\operatorname{sm}}$ is smooth
over $k$. Since $X_{\operatorname{sm}}\neq\emptyset$ by our assumption, the
claim is therefore reduced to showing that $\Omega^{j}_{U}=0$ for $j\geq 3$ if
$U$ is any smooth curve over $k$.

Now, we look at the exact sequence (e.g., see [34, Thm. 25.1])

(3.2)
$0\to\Omega^{1}_{k}\otimes_{k}{\mathcal{O}}_{U}\xrightarrow{d}\Omega^{1}_{U}\to\Omega^{1}_{U/k}\to
0.$

Taking the higher exterior powers, we get
$\Omega^{j}_{U}\cong\Omega^{j-1}_{k}\otimes_{k}\Omega^{1}_{U/k}$ for $j\geq
2$. Hence, it suffices to show that $\Omega^{j}_{k}=0$ for $j\geq 2$. But this
is well known because $R={\mathbb{F}}_{q}[[t]]$ and a direct computation shows
that $R\xrightarrow{\cong}\Omega^{1}_{R}$, which implies that
$\Omega^{j}_{R}=0$ for $j\geq 2$. Note that this also shows that
$\Omega^{2}_{X}\cong\Omega^{1}_{X/k}$.

To prove the second claim, it suffices to show using (2.4) that
$H^{i}(Y,\iota^{*}(W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}))=0$
for $i\geq 3$. But this is clear since $cd_{p}(Y)\leq 2$ (e.g., see [35, Chap.
VI, Rem. 1.5(b)]). ∎

######  Lemma 3.2.

The boundary map of the localization sequence

$\partial\colon H^{2}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})\to
H^{3}_{Y}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})$

is an isomorphism.

###### Proof.

The localization sequence in question is the exact sequence

(3.3)
$H^{2}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})\to
H^{2}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})\to
H^{3}_{Y}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})\to
H^{3}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}}).$

It suffices therefore to show that
$H^{i}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})=0$
for $i\geq 2$. By (2.4), we have
$H^{i}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})\cong
H^{i}(Y,\iota^{*}(W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}}))$. But
the latter group is zero for $i\geq 3$ since $cd_{p}(Y)=2$, as we observed
before.

To prove the vanishing of
$H^{2}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})$,
it suffices to show, using Theorem 2.8, that
$H^{1}_{Y}({\mathcal{X}},{{\mathbb{Z}}}/{p^{n}})=0$. To that end, we look at
the exact sequence

(3.4) $0\to
H^{0}({\mathcal{X}},{{\mathbb{Z}}}/{p^{n}})\xrightarrow{u^{*}}H^{0}(X,{{\mathbb{Z}}}/{p^{n}})\to
H^{1}_{Y}({\mathcal{X}},{{\mathbb{Z}}}/{p^{n}})\to
H^{1}({\mathcal{X}},{{\mathbb{Z}}}/{p^{n}})\xrightarrow{u^{*}}H^{1}(X,{{\mathbb{Z}}}/{p^{n}}).$

Since ${\mathcal{X}}$ is integral and normal, the map $u^{*}$ on the left is
an isomorphism and the map $u^{*}$ on the right is injective. It follows that
the middle group is zero. ∎

######  Lemma 3.3.

The group
$H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$
is finite unless $i\in\\{2,3\\}$ and $j\in\\{0,1\\}$.

###### Proof.

In view of Lemma 3.1, we can assume $j\leq 2$. We have seen in the proof of
Lemma 3.2 that
$H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{0}_{{\mathcal{X}},{\operatorname{log}}})=0$
for $i\leq 1$. We have
$H^{0}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})=0$
for every $j\geq 0$ by the Gersten injectivity
$W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}\hookrightarrow
u_{*}(W_{n}\Omega^{j}_{X,{\operatorname{log}}})$ (see [53, Thm. 4.1]). The
finitude claim for
$H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})$
follows from Theorem 2.8 since
$H^{i}({\mathcal{X}},{{\mathbb{Z}}}/{p^{n}})\cong
H^{i}(Y,{{\mathbb{Z}}}/{p^{n}})$ (see (2.4)) and the latter group is finite by
[48, Thm. 1.2.2]. We are left with showing the finitude of
$H^{1}_{Y}({\mathcal{X}},W_{n}\Omega^{1}_{{\mathcal{X}},{\operatorname{log}}})$.

We look at the exact (localization) sequence

(3.5)
$H^{0}({\mathcal{X}},W_{n}\Omega^{1}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{u^{*}_{0}}H^{0}(X,W_{n}\Omega^{1}_{X,{\operatorname{log}}})\to
H^{1}_{Y}({\mathcal{X}},W_{n}\Omega^{1}_{{\mathcal{X}},{\operatorname{log}}})\hskip
85.35826pt$ $\hskip 199.16928pt\to
H^{1}({\mathcal{X}},W_{n}\Omega^{1}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{u^{*}_{1}}H^{1}(X,W_{n}\Omega^{1}_{X,{\operatorname{log}}}).$

It suffices to show that ${\rm Coker}(u^{*}_{0})$ and ${\rm Ker}(u^{*}_{1})$
are finite. To prove the finiteness of ${\rm Coker}(u^{*}_{0})$, note that
(2.3) yields the commutative diagram of exact sequences

(3.6)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H^{0}({\mathcal{X}},{\mathcal{O}}^{\times}_{{\mathcal{X}}})}/{p^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}}$$\textstyle{H^{0}({\mathcal{X}},W_{n}\Omega^{1}_{{\mathcal{X}},{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}_{0}}$$\textstyle{{}_{p^{n}}{\rm
Pic}({\mathcal{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{H^{0}(X,{\mathcal{O}}^{\times}_{X})}/{p^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(X,W_{n}\Omega^{1}_{X,{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{p^{n}}{\rm
Pic}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We next note that there are exact sequences

(3.7) $0\to
H^{0}({\mathcal{X}},{\mathcal{O}}^{\times}_{{\mathcal{X}}})\xrightarrow{u^{*}}H^{0}(X,{\mathcal{O}}^{\times}_{X})\xrightarrow{v_{k}}{\mathbb{Z}}\to
0$

and

(3.8) $0\to{\mathbb{Z}}\xrightarrow{\iota^{\prime}_{*}}{\underset{Z\in{\rm
Irr}(Y)}{\bigoplus}}{\mathbb{Z}}\xrightarrow{{\operatorname{\rm cyc}}}{\rm
Pic}({\mathcal{X}})\xrightarrow{u^{*}}{\rm Pic}(X)\to 0,$

where the first sequence follows from the geometric integrality condition (cf.
Lemma 2.7). The second sequence is well known (e.g., see [52]) if we let
$\iota^{\prime}_{*}(1)=[{\mathcal{X}}_{s}]$, the class of the Weil divisor
associated to the special fiber ${\mathcal{X}}_{s}$ and ${\operatorname{\rm
cyc}}$ denotes the cycle class map. It easily follows from these exact
sequences that the kernels and cokernels of the left and right vertical arrows
in (3.6) are finite. Hence, so are the kernel and cokernel of the middle
vertical arrow (the assertion about the kernel will be used only in the next
lemma).

To prove the finitude of ${\rm Ker}(u^{*}_{1})$, we look at the commutative
diagram with exact rows (following (2.3))

(3.9)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\rm
Pic}({\mathcal{X}})}/{p^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}}$$\textstyle{H^{1}({\mathcal{X}},W_{n}\Omega^{1}_{{\mathcal{X}},{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}_{1}}$$\textstyle{{}_{p^{n}}{\operatorname{Br}}({\mathcal{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u^{*}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\rm
Pic}(X)}/{p^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(X,W_{n}\Omega^{1}_{X,{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{p^{n}}{\operatorname{Br}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

It follows from [4, Thm. 5.6.1(v), 10.3.1(ii)] that

(3.10)
${\operatorname{Br}}({\mathcal{X}})={\operatorname{Br}}({\mathcal{X}}_{s})={\operatorname{Br}}(Y)=0.$

On the other hand, it follows from (3.8) that the kernel of the left vertical
arrow in (3.9) is finite. We conclude that ${\rm Ker}(u^{*}_{1})$ is finite.
We have thus shown that
$H^{1}_{Y}({\mathcal{X}},W_{n}\Omega^{1}_{{\mathcal{X}},{\operatorname{log}}})$
is finite. This concludes the proof. ∎

######  Remark 3.4.

It can be shown that each of the groups
$H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$
is infinite in general if $i\in\\{2,3\\}$ and $j\in\\{0,1\\}$.

######  Lemma 3.5.

For all $i,j\geq 0$, the kernel of $u^{*}\colon
H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\to
H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})$ is finite.

###### Proof.

Combine Theorem 2.8 and Lemma 3.3. ∎

###  3.2. Logarithmic Hodge-Witt cohomology as topological groups

We shall now endow the logarithmic Hodge-Witt cohomology of $X$ with torsion-
by-profinite topology. For $i,j\geq 0$, we consider the localization sequence

(3.11) $\cdots\to
H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{\iota_{*}}H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{u^{*}}H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})\hskip
28.45274pt$ $\hskip
256.0748pt\xrightarrow{\partial}H^{i+1}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\to\cdots.$

By Lemma 3.5, it breaks into exact sequences

(3.12)
$H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{\iota_{*}}H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{u^{*}}F_{ij}\to
0;$ (3.13) $0\to F_{ij}\to
H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})\xrightarrow{\partial}H^{i+1}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$

such that ${\rm Ker}(u^{*})$ is finite.

Recall from Theorem 2.8 that each
$H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$
is a discrete torsion group of exponent $p^{n}$ and
$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$ is
a profinite topological abelian group. It follows that ${\rm
Image}(\iota_{*})$ is a finite closed subgroup of
$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$.
In particular, its subspace topology induced from
$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$ as
well as its quotient topology induced from
$H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$
is discrete.

For $i,j\geq 0$, we endow $F_{ij}$ with the quotient topology via (3.12). Then
$F_{ij}$ becomes a quotient of the profinite group
$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})$ by
a closed subgroup. It follows that $F_{ij}$ is a profinite abelian group and
(3.12) is an exact sequence of topological abelian groups.

We endow $H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})$ with the unique
topology for which $F_{ij}$ is an open subgroup of
$H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})$ and
${H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})}/{F_{ij}}={\rm
Ker}(\iota_{*})$ is a discrete quotient of
$H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})$. This endows
$H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})$ with the structure of a
topological abelian group such that (3.13) is an exact sequence of topological
abelian groups. If we combine this with everything we have shown in § 2.4 and
§ 3.1, we get the following.

######  Proposition 3.6.

For each $i,j\geq 0$, the group
$H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})$ is equipped with the
structure of a torsion-by-profinite topological abelian group satisfying the
following.

  1. (1)

(3.11) is an exact sequence in ${\operatorname{\mathbf{Pfd}}}$ for every
$j\geq 0$.

  2. (2)

$H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})=0$ unless $0\leq i,j\leq
2$.

  3. (3)

$H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})$ is profinite unless
$i\in\\{1,2\\}$ and $j\in\\{0,1\\}$.

  4. (4)

$H^{2}(X,W_{n}\Omega^{0}_{X,{\operatorname{log}}})$ is a discrete torsion
group.

  5. (5)

$H^{1}(X,W_{n}\Omega^{0}_{X,{\operatorname{log}}})$ is an infinite discrete
torsion group.

  6. (6)

$H^{1}(X,W_{n}\Omega^{1}_{X,{\operatorname{log}}})$ is neither a profinite nor
a discrete torsion group.

###### Proof.

We only need to explain (4) and (6). But the former follows from [48, Thm.
1.2.2] and the isomorphism $H^{2}({\mathcal{X}},{{\mathbb{Z}}}/{p^{n}})\cong
H^{2}(Y,{{\mathbb{Z}}}/{p^{n}})$ while the latter follows from the perfectness
of the Brauer-Manin pairing for $X$ (e.g., see [46, § 9]). ∎

Using Lemma 2.1, Proposition 3.6 and considering the Pontryagin duals
associated to (3.11), we get a chain complex in
${\operatorname{\mathbf{Pfd}}}$:

(3.14) $\cdots\to
H^{i+1}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})^{\star}\xrightarrow{\partial^{\star}}H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})^{\star}\xrightarrow{(u^{*})^{\star}}H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})^{\star}\hskip
28.45274pt$ $\hskip
256.0748pt\xrightarrow{(\iota_{*})^{\star}}H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})^{\star}\to\cdots.$

######  Lemma 3.7.

The sequence (3.14) is exact.

###### Proof.

We look at the exact sequences

(3.15) $0\to{\rm Ker}(\iota_{*})\to
H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\to{\rm
Ker}(u^{*})\to 0;$ (3.16) $0\to{\rm Ker}(u^{*})\to
H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\to
F_{ij}\to 0;$ (3.17) $0\to F_{ij}\to
H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})\to{\rm Ker}(\iota_{*})\to
0.$

It follows from Proposition 3.6 and the definition of the topologies of the
various groups in (3.11) that the above three are exact sequences in
${\operatorname{\mathbf{Pfd}}}$ and they satisfy the hypothesis of Lemma 2.4.
We can therefore apply this lemma to get short exact sequences

(3.18) $0\to{\rm Ker}(u^{*})^{\star}\to
H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})^{\star}\to{\rm
Ker}(\iota_{*})^{\star}\to 0;$ (3.19) $0\to{\rm Ker}(\iota_{*})^{\star}\to
H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})^{\star}\to(F_{ij})^{\star}\to
0;$ (3.20) $0\to(F_{ij})^{\star}\to
H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})^{\star}\to{\rm
Ker}(u^{*})^{\star}\to 0.$

The desired result follows by piecing these exact sequences together. ∎

##  4\. Duality for logarithmic Hodge-Witt cohomology

We continue to work under the set-up of § 2.3. In this section, we shall prove
the duality theorem for the logarithmic Hodge-Witt cohomology of the proper
curve $X$ over $k$. We begin by constructing the trace map. For a scheme $Z$,
we let ${\mathcal{D}}_{\text{\'{e}t}}(Z)$ denote the derived category of étale
sheaves of abelian groups on $Z$. We let
${\mathcal{D}}_{\text{\'{e}t}}(Z,{{\mathbb{Z}}}/m)$ denote the derived
category of étale sheaves of ${{\mathbb{Z}}}/m$-modules on $Z$ with
$m\in{\mathcal{O}}^{\times}(Z)$. For a local ring $A$, we shall let $A^{h}$
(resp. $A^{sh}$, resp. $\widehat{A}$) denote the Henselization (resp. strict
Henselization, resp. completion) of $A$ with respect to its maximal ideal. For
a closed point $x\in X$, we shall let $K_{x}$ (resp. $K^{sh}_{x}$, resp.
$\widehat{K}_{x}$) denote the quotient field of the Henselization
${\mathcal{O}}^{h}_{X,x}$ (resp. strict Henselization
${\mathcal{O}}^{sh}_{X,x}$, resp. completion $\widehat{{\mathcal{O}}_{X,x}}$)
of ${\mathcal{O}}_{X,x}$. We fix an integer $n\geq 1$.

###  4.1. The trace maps

We let $j\colon{\rm Spec\,}(K)\hookrightarrow X$ be the inclusion of the
generic point of $X$ and let $\iota_{x}\colon{\rm Spec\,}(k(x))\hookrightarrow
X$ be the inclusion of a closed point. By the Gersten resolution of the
logarithmic Hodge-Witt sheaves, one has an exact sequence of étale sheaves

(4.1) $0\to W_{n}\Omega^{2}_{X,{\operatorname{log}}}\to
j_{*}(W_{n}\Omega^{2}_{K,{\operatorname{log}}})\xrightarrow{{\operatorname{res}}}{\underset{x\in
X_{(0)}}{\bigoplus}}(\iota_{x})_{*}(W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})\to
0,$

where ${\operatorname{res}}$ is the sum of residue maps in Kato’s complex (see
[53, Thm. 5.2]). It follows that there is an exact triangle

(4.2) $0\to\left({\underset{x\in
X_{(0)}}{\bigoplus}}(\iota_{x})_{*}(W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})\right)[-1]\to
W_{n}\Omega^{2}_{X,{\operatorname{log}}}\to
j_{*}(W_{n}\Omega^{2}_{K,{\operatorname{log}}})\to 0$

in ${\mathcal{D}}_{\text{\'{e}t}}(X)$. We let $\nu^{2}_{n,X}$ denote the
complex
$\left(j_{*}(W_{n}\Omega^{2}_{K,{\operatorname{log}}})\xrightarrow{{\operatorname{res}}}{\underset{x\in
X_{(0)}}{\bigoplus}}(\iota_{x})_{*}(W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})\right)$.

For every $x\in X_{(0)}$, we have the canonical trace map
$\epsilon_{x}\colon(f_{x})_{*}(W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})\to
W_{n}\Omega^{1}_{k,{\operatorname{log}}}$, induced by the norm
$N_{x}\colon(f_{x})_{*}({\mathcal{O}}^{\times}_{k(x)})\to{\mathcal{O}}^{\times}_{k}$
via the dlog map, where $f_{x}\colon{\rm Spec\,}(k(x))\to{\rm Spec\,}(k)$ is
the structure map. We let $\epsilon_{X}:=\sum_{x}\epsilon_{x}$. By the
reciprocity law for Milnor $K$-theory (e.g., see [22, Lem. 4]), it is clear
that the composite map

$g_{*}(W_{n}\Omega^{2}_{K,{\operatorname{log}}})\xrightarrow{{\operatorname{res}}}{\underset{x\in
X_{(0)}}{\bigoplus}}(f_{x})_{*}(W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})\xrightarrow{\epsilon_{X}}W_{n}\Omega^{1}_{k,{\operatorname{log}}}$

is zero if we let $g=f\circ j$, where recall that $f\colon X\to{\rm
Spec\,}(k)$ is the structure map. In other words, we have a morphism of
complexes

(4.3) $f_{*}\colon f_{*}(\nu^{2}_{n,X})\to
W_{n}\Omega^{1}_{k,{\operatorname{log}}}[-1]$

whose composition with
$(f_{x})_{*}(W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})[-1]\to
f_{*}(\nu^{2}_{n,X})$ is $\epsilon_{x}[-1]$ for every $x\in X_{(0)}$. We now
prove some key lemmas.

######  Lemma 4.1.

One has $R^{q}j_{*}(W_{n}\Omega^{2}_{K,{\operatorname{log}}})=0$ for $q\geq
1$.

###### Proof.

The stalk of $R^{q}j_{*}(W_{n}\Omega^{2}_{K,{\operatorname{log}}})$ at a
closed point $x\in X$ is
$H^{q}(K^{sh}_{x},W_{n}\Omega^{2}_{K^{sh}_{x},{\operatorname{log}}})$. To show
that this cohomology is zero, we can assume $q=1$ because
$cd_{p}(K^{sh}_{x})\leq 1$. By [24, Lem. 21], we can pass to the completion
$\widehat{K^{sh}_{x}}$ of $K^{sh}_{x}$. We let $A_{x}$ denote the completion
of ${\mathcal{O}}^{sh}_{X,x},\ F_{x}=\widehat{K^{sh}_{x}}$ and consider the
exact sequence

(4.4) $H^{1}(A_{x},W_{n}\Omega^{2}_{A_{x},{\operatorname{log}}})\to
H^{1}(F_{x},W_{n}\Omega^{2}_{F_{x},{\operatorname{log}}})\to
H^{2}_{\overline{x}}(A_{x},W_{n}\Omega^{2}_{A_{x},{\operatorname{log}}}),$

where $\overline{x}$ is the closed point of ${\rm Spec\,}(A_{x})$. By the
proper base change theorem, we have
$H^{1}(A_{x},W_{n}\Omega^{2}_{A_{x},{\operatorname{log}}})\cong
H^{1}(k(\overline{x}),\iota^{*}(W_{n}\Omega^{2}_{A_{x},{\operatorname{log}}}))$,
where $\iota\colon{\rm Spec\,}(k(\overline{x}))\hookrightarrow{\rm
Spec\,}(A_{x})$ is the inclusion of the closed point. But the latter group
vanishes since $k(\overline{x})$ is separably closed (e.g., see [35, Thm.
VI.1.1]). It remains to show that
$H^{2}_{\overline{x}}(A_{x},W_{n}\Omega^{2}_{A_{x},{\operatorname{log}}})=0$.

Since $k(\overline{x})$ is separably closed, we have the well known equality
$[k(\overline{x}):k(\overline{x})^{p}]=p$ (e.g., see [25, § 3]). Since
$F_{x}\cong k(\overline{x})((T))$, it is an elementary exercise that
$[F_{x}:(F_{x})^{p}]=p^{2}$ (e.g., see the proof of [61, Lem. 1.4.5]). We
therefore conclude from [53, Thm. 3.2] that there is a purity isomorphism
$\iota_{*}\colon
H^{1}(k(\overline{x}),W_{n}\Omega^{1}_{k(\overline{x}),{\operatorname{log}}})\xrightarrow{\cong}H^{2}_{\overline{x}}(A_{x},W_{n}\Omega^{2}_{A_{x},{\operatorname{log}}})$.
This proves our claim because
$H^{1}(k(\overline{x}),W_{n}\Omega^{1}_{k(\overline{x}),{\operatorname{log}}})$
is clearly zero. To finish the proof of the lemma, we note that the stalk of
$R^{q}j_{*}(W_{n}\Omega^{2}_{K,{\operatorname{log}}})$ at the generic point of
$X$ is $H^{q}(K_{s},W_{n}\Omega^{2}_{K_{s},{\operatorname{log}}})$, which is
zero if $q\geq 1$. ∎

######  Lemma 4.2.

The canonical map $f_{*}(\nu^{2}_{n,X})\to Rf_{*}(\nu^{2}_{n,X})$ is an
isomorphism in ${\mathcal{D}}_{\text{\'{e}t}}(k)$.

###### Proof.

In view of Lemma 4.1, it suffices to show that
$R^{q}f_{*}(j_{*}(W_{n}\Omega^{2}_{K,{\operatorname{log}}}))=0$ and
$R^{q}f_{*}((\iota_{x})_{*}(W_{n}\Omega^{1}_{k(x),{\operatorname{log}}}))=0$
for all $q\geq 1$ and $x\in X_{(0)}$. Using the exactness of $(\iota_{x})_{*}$
for $x\in X_{(0)}$ and Lemma 4.1 for the generic point of $X$, it suffices to
show that $R^{q}g_{*}(W_{n}\Omega^{2}_{K,{\operatorname{log}}})=0$ and
$R^{q}(f_{x})_{*}(W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})=0$ for all
$q\geq 1$ and $x\in X_{(0)}$. This latter claim for $x\in X_{(0)}$ is obvious
since $f_{x}$ is a finite morphism.

On the other hand, we have
$R^{q}g_{*}(W_{n}\Omega^{2}_{K,{\operatorname{log}}})\cong
H^{q}(K^{\prime},W_{n}\Omega^{2}_{K^{\prime},{\operatorname{log}}})$, where
$K^{\prime}$ is the product of the function fields of the connected components
of $X_{s}:=X\times_{{\rm Spec\,}(k)}{\rm Spec\,}(k_{s})$. We are now done
because $H^{q}(K^{\prime},W_{n}\Omega^{2}_{K^{\prime},{\operatorname{log}}})$
is clearly zero if $q\geq 2$, and its vanishing for $q=1$ was shown by Kato-
Saito [25] (see the proof of their Lemma 1(2) on p. 252). We remark that even
though the cited lemma of Kato-Saito assumes that $X$ is smooth over $k$, the
proof of the vanishing of
$H^{1}(K^{\prime},W_{n}\Omega^{2}_{K^{\prime},{\operatorname{log}}})$ only
requires $X$ to be generically smooth over $k$. This concludes the proof. ∎

######  Lemma 4.3.

There exists a canonical morphism ${{\operatorname{tr}}}_{X}\colon
Rf_{*}(W_{n}\Omega^{2}_{X,{\operatorname{log}}})\to
W_{n}\Omega^{1}_{k,{\operatorname{log}}}[-1]$ in
${\mathcal{D}}_{\text{\'{e}t}}(k)$ whose composition with the canonical
morphisms

$(f_{x})_{*}(W_{n}\Omega^{1}_{k(x),{\operatorname{log}}}[-1])\to
f_{*}(W_{n}\Omega^{2}_{X,{\operatorname{log}}})\to
Rf_{*}(W_{n}\Omega^{2}_{X,{\operatorname{log}}})$

is $\epsilon_{x}[-1]$ for every $x\in X_{(0)}$.

###### Proof.

Combine the isomorphism
$W_{n}\Omega^{2}_{X,{\operatorname{log}}}\xrightarrow{\cong}\nu^{2}_{n,X}$ in
${\mathcal{D}}_{\text{\'{e}t}}(X)$ with Lemma 4.2 and the morphism $f_{*}$ in
(4.3). ∎

Recall that there is a canonical isomorphism ${\rm
inv}_{k}\colon{\operatorname{Br}}(k)\xrightarrow{\cong}{{\mathbb{Q}}}/{{\mathbb{Z}}}$.
If ${k^{\prime}}/k$ is a finite field extension of degree $m$ and $v\colon{\rm
Spec\,}(k^{\prime})\to{\rm Spec\,}(k)$ is the projection, then the norm map
$N_{{k^{\prime}}/k}\colon
v_{*}({\mathcal{O}}^{\times}_{k^{\prime}})\to{\mathcal{O}}^{\times}_{k}$
between the étale sheaves induces the pull-back and push-forward maps
$v^{*}\colon{\operatorname{Br}}(k)\to{\operatorname{Br}}(k^{\prime})$ and
$v_{*}\colon{\operatorname{Br}}(k^{\prime})\to{\operatorname{Br}}(k)$ such
that ${\rm inv}_{k^{\prime}}\circ v^{*}=m({\rm inv}_{k})$ on
${{\mathbb{Q}}}/{{\mathbb{Z}}}$ and ${\rm inv}_{k}\circ v_{*}={\rm
inv}_{k^{\prime}}$.

Identifying ${}_{p^{n}}{\operatorname{Br}}(k^{\prime})$ with
$H^{1}(k^{\prime},W_{n}\Omega^{1}_{k^{\prime},{\operatorname{log}}})$, we get
a commutative diagram

(4.5)
$\textstyle{H^{1}(k^{\prime},W_{n}\Omega^{1}_{k^{\prime},{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{*}}$$\scriptstyle{{\rm
inv}_{k^{\prime}}}$$\textstyle{H^{1}(k,W_{n}\Omega^{1}_{k,{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
inv}_{k}}$$\textstyle{{{\mathbb{Z}}}/{p^{n}}}$

in which the horizontal and the diagonal arrows are isomorphisms. It follows
that $v_{*}$ is an isomorphism such that ${\rm inv}_{k}\circ v_{*}={\rm
inv}_{k^{\prime}}$. We shall therefore identify $v_{*}$ with the identity map
of ${{\mathbb{Z}}}/{p^{n}}$. Similarly, we shall identify
$v_{*}\colon{\operatorname{Br}}(k^{\prime})\to{\operatorname{Br}}(k)$ with the
identity map of ${{\mathbb{Q}}}/{{\mathbb{Z}}}$ throughout this paper. In the
sequel, we shall write $v_{*}=\epsilon_{x}$ if $k^{\prime}=k(x)$ for some
closed point $x$ on a curve over $k$.

######  Proposition 4.4.

There exists a unique homomorphism ${\rm Tr}_{X}\colon
H^{2}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})\to{{\mathbb{Z}}}/{p^{n}}$
such that ${\rm Tr}_{X}\circ(\iota_{x})_{*}$ is the identity map for every
$x\in X_{(0)}$. Furthermore, ${\rm Tr}_{X}$ is bijective.

###### Proof.

By (4.1) and Lemma 4.1, we have an exact sequence

(4.6)
$H^{1}(K,W_{n}\Omega^{2}_{K,{\operatorname{log}}})\xrightarrow{{\operatorname{res}}}{\underset{x\in
X_{(0)}}{\bigoplus}}H^{1}(k(x),W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})\to
H^{2}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})\to 0.$

This implies that $H^{2}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})$ is
generated by the images of $(\iota_{x})_{*}(1)$ as $x$ varies in $X_{(0)}$.
The uniqueness assertion follows immediately from this. The existence follows
directly from Lemma 4.3 and (4.5). The claim that ${\rm Tr}_{X}$ is bijective
follows from the fact that its source and target are finite of equal
cardinality by Theorem 2.8 and Lemma 3.2 while ${\rm
Tr}_{X}\circ(\iota_{x})_{*}$ is bijective. ∎

######  Remark 4.5.

The assertion that
$H^{2}_{\text{\'{e}t}}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})$ is
isomorphic to ${{\mathbb{Z}}}/{p^{n}}$ is not new (at least when $X$ is smooth
over $k$) as it was already shown by Kato-Saito in [25, Prop. 4]. It also
follows from Theorem 2.8 and Lemma 3.2. However, the above proposition
provides an explicit isomorphism which has the advantage that it commutes with
the trace maps for all closed points of $X$. This is unclear in the
construction of [25]. We shall need this explicit nature of ${\rm Tr}_{X}$
later in the paper in a critical way. Another advantage is that it does not
require $X$ to be smooth everywhere. Although this flexibility is not
important in this paper, it will be vital in the study of duality and class
field theory for open smooth curves over $k$ which do not admit smooth
compactifications. We were unable to prove if ${\rm Tr}_{X}$ coincides with
the isomorphism of Kato-Saito, whose construction, we believe, is quite
intricate.

Given $x\in X_{(0)}$, we now look at the canonical maps of étale sheaves

$(\iota_{x})_{*}(W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})\to(\iota_{x})_{*}R(\iota_{x})^{!}(W_{n}\Omega^{2}_{X,{\operatorname{log}}})\to
W_{n}\Omega^{2}_{X,{\operatorname{log}}},$

on $X$, where $[t_{x}]$ is the Teichmüller image of the chosen uniformizer
$t_{x}\in{\mathcal{O}}_{X,x}$ and the first arrow is obtained by taking cup
product with ${\rm dlog}([t_{x}])$. This arrow is an isomorphism in
${\mathcal{D}}_{\text{\'{e}t}}(X)$ by [61, Cor. 1.3.14]. The same is also true
if we replace $X$ by ${\rm Spec\,}({\mathcal{O}}^{h}_{X,x})$.

Furthermore, the diagram of the induced maps on cohomology

(4.7)
$\textstyle{H^{1}(k(x),W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{x})_{*}}$$\scriptstyle{\cup[t_{x}]}$$\scriptstyle{\cong}$$\scriptstyle{\cup[t_{x}]}$$\textstyle{H^{2}_{x}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H^{2}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(K_{x},W_{n}\Omega^{2}_{K_{x},{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{x}}$$\textstyle{H^{2}_{x}({\mathcal{O}}^{h}_{X,x},W_{n}\Omega^{2}_{{\mathcal{O}}^{h}_{X,x},{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}({\mathcal{O}}^{h}_{X,x},W_{n}\Omega^{2}_{{\mathcal{O}}^{h}_{X,x},{\operatorname{log}}})}$

is commutative, where the left vertical arrow $\cup[t_{x}]$ is the isomorphism
of [21, § 3.2, Lem. 3]. It follows from the proper base change theorem that
the group on the bottom right corner is zero. Since the other two groups in
the bottom row are isomorphic to ${{\mathbb{Z}}}/{p^{n}}$ (e.g., see [21,
Prop. 3.1]), it follows that $\partial_{x}$ is bijective.

Using Proposition 4.4 and (4.7), we get a commutative diagram

(4.8)
$\textstyle{H^{1}(k(x),W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{x})_{*}}$$\scriptstyle{h_{x}}$$\scriptstyle{\cong}$$\scriptstyle{\epsilon_{x}}$$\scriptstyle{\cong}$$\textstyle{H^{2}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{tr}}_{X}}$$\scriptstyle{{\rm
Tr}_{X}}$$\textstyle{H^{1}(K_{x},W_{n}\Omega^{2}_{K_{x},{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(k,W_{n}\Omega^{1}_{k,{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
inv}_{k}}$$\textstyle{{{\mathbb{Z}}}/{p^{n}},}$

where $h_{x}:=\cup[t_{x}]$ is an isomorphism.

Letting ${\rm Tr}_{x}:={\rm inv}_{k}\circ\epsilon_{x}\circ(h_{x})^{-1}$, we
get the following.

######  Corollary 4.6.

For every $x\in X_{(0)}$, there is a commutative diagram

(4.9)
$\textstyle{H^{1}(k(x),W_{n}\Omega^{1}_{k(x),{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{x})_{*}}$$\scriptstyle{h_{x}}$$\textstyle{H^{2}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Tr}_{X}}$$\textstyle{H^{1}(K_{x},W_{n}\Omega^{2}_{K_{x},{\operatorname{log}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Tr}_{x}}$$\textstyle{{{\mathbb{Z}}}/{p^{n}}}$

in which all arrows are isomorphisms.

###  4.2. Duality theorem for
$H^{*}(X,W_{n}\Omega^{\bullet}_{X,{\operatorname{log}}})$

Recall that a pairing between locally compact Hausdorff topological abelian
groups $\lambda\colon A\times B\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$ is called
continuous if $\lambda$ is continuous with respect to the product topology of
$A\times B$. Equivalently, either (and hence both) of the maps $A\to B^{\vee}$
and $B\to A^{\vee}$ is continuous and factors through the continuous dual
(e.g., see [18, Prop. A.14]). $\lambda$ is called non degenerate on the left
(resp. right) if the induced map $A\to B^{\vee}$ (resp. $B\to A^{\vee}$) is
injective. One says that $\lambda$ is perfect (in particular, continuous) if
it induces isomorphisms $A\xrightarrow{\cong}B^{\star}$ and
$B\xrightarrow{\cong}A^{\star}$ of topological abelian groups.

We now return to the set-up of § 2.3. Recall from Proposition 4.4 that there
is a canonical isomorphism of ${{\mathbb{Z}}}/{p^{n}}$-modules

(4.10) ${\rm Tr}_{X}\colon
H^{2}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})\xrightarrow{\cong}{{\mathbb{Z}}}/{p^{n}}.$

Using Lemma 3.2, we get a unique isomorphism ${\rm Tr}_{{\mathcal{X}}}\colon
H^{3}_{Y}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{\cong}{{\mathbb{Z}}}/{p^{n}}$
such that ${\rm Tr}_{{\mathcal{X}}}\circ\partial={\rm Tr}_{X}$ (cf. Theorem
2.8). The cup product pairing

(4.11)
$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\times
H^{3-i}_{Y}({\mathcal{X}},W_{n}\Omega^{2-j}_{{\mathcal{X}},{\operatorname{log}}})\to
H^{3}_{Y}({\mathcal{X}},W_{n}\Omega^{2}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{{\rm
Tr}_{\mathcal{X}}}{{\mathbb{Z}}}/{p^{n}}$

gives rise to maps

$H^{i}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{\Phi^{ij}_{\mathcal{X}}}H^{3-i}_{Y}({\mathcal{X}},W_{n}\Omega^{2-j}_{{\mathcal{X}},{\operatorname{log}}})^{\star};\
H^{i}_{Y}({\mathcal{X}},W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}})\xrightarrow{\Psi^{ij}_{\mathcal{X}}}H^{3-i}({\mathcal{X}},W_{n}\Omega^{2-j}_{{\mathcal{X}},{\operatorname{log}}})^{\star}$

in ${\operatorname{\mathbf{Pfd}}}$. Furthermore, it follows from Theorem 2.8
that these maps are topological isomorphisms. In other words, (4.11) is a
perfect pairing in ${\operatorname{\mathbf{Pfd}}}$.

We shall now prove our duality theorem for the logarithmic Hodge-Witt
cohomology of $X$. This will prove Theorem 1.4. This duality recovers as well
as generalizes [25, Prop. 4]. Let the notations and hypotheses be as in § 2.3.
In particular, $X$ is a geometrically connected, regular, and generically
smooth curve over the local field $k$ of exponential characteristic $p>1$.
${\mathcal{X}}$ is a projective and flat regular semi-stable model of $X$ over
${\rm Spec\,}({\mathcal{O}}_{k})$.

######  Theorem 4.7.

For every pair of integers $i,j\geq 0$, the cup product on logarithmic Hodge-
Witt cohomology induces a perfect pairing of topological abelian groups

(4.12) $H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})\times
H^{2-i}(X,W_{n}\Omega^{2-j}_{X,{\operatorname{log}}})\xrightarrow{\cup}H^{2}(X,W_{n}\Omega^{2}_{X,{\operatorname{log}}})\xrightarrow{{\rm
Tr}_{X}}{{\mathbb{Z}}}/{p^{n}}.$

###### Proof.

The existence of the bilinear pairing is clear. We shall prove its perfectness
in several steps. In order to save space, we shall write all cohomology groups
by suppressing the underlying schemes. We shall also use the short hand
${\mathcal{F}}^{j}_{{\mathcal{X}}}$ for
$W_{n}\Omega^{j}_{{\mathcal{X}},{\operatorname{log}}}$ and
${\mathcal{F}}^{j}_{X}$ for $W_{n}\Omega^{j}_{X,{\operatorname{log}}}$.

We now fix $i,j\geq 0$ and we look at the diagram

(4.13)
$\textstyle{H^{i}_{Y}({\mathcal{F}}^{j}_{{\mathcal{X}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi^{ij}_{{\mathcal{X}}}}$$\textstyle{H^{i}({\mathcal{F}}^{j}_{{\mathcal{X}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi^{ij}_{{\mathcal{X}}}}$$\scriptstyle{{2}}$$\textstyle{H^{i}({\mathcal{F}}^{j}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i+1}_{Y}({\mathcal{F}}^{j}_{{\mathcal{X}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi^{i+1j}_{{\mathcal{X}}}}$$\textstyle{H^{i+1}({\mathcal{F}}^{j}_{{\mathcal{X}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi^{i+1j}_{{\mathcal{X}}}}$$\textstyle{H^{3-i}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3-i}_{Y}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2-i}({\mathcal{F}}^{2-j}_{X})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{1}}$$\textstyle{H^{2-i}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2-i}_{Y}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3-i}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3-i}_{Y}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2-i}({\mathcal{F}}^{2-j}_{X})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2-i}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2-i}_{Y}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})^{\vee},}$

where the vertical arrows on the lower floor are the canonical inclusions. The
top and the bottom rows are clearly exact while the exactness of the middle
row is shown in Lemma 3.7. All squares on the lower floor clearly commute. We
let the composite vertical arrow in the middle be the map
$(-1)^{i}\widetilde{\Phi}^{ij}_{X}$, where $\widetilde{\Phi}^{ij}_{X}\colon
H^{i}({\mathcal{F}}^{j}_{X})\to H^{2-i}({\mathcal{F}}^{2-j}_{X})^{\vee}$ is
induced by (4.12). We now show that the composition of each of the top and the
following bottom squares in (4.13) commutes. We now show that the composition
of each of the squares on the upper floor with the one below it on the lower
floor in (4.13) commutes.

The composite squares on the left and right corners commute because they are
induced by the canonical commutative diagram

(4.14) $\textstyle{\iota_{*}R\iota^{!}{\mathcal{F}}^{j}_{{\mathcal{X}}}\hskip
14.22636pt\times\hskip
8.5359pt{\mathcal{F}}^{j^{\prime}}_{{\mathcal{X}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\iota_{*}}$$\textstyle{\iota_{*}R\iota^{!}{\mathcal{F}}^{j+j^{\prime}}_{{\mathcal{X}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hskip
17.07182pt{\mathcal{F}}^{j}_{{\mathcal{X}}}\hskip 14.22636pt\times\hskip
14.22636pt\iota_{*}R\iota^{!}{\mathcal{F}}^{j^{\prime}}_{{\mathcal{X}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{\iota_{*}}$$\textstyle{\iota_{*}R\iota^{!}{\mathcal{F}}^{j+j^{\prime}}_{{\mathcal{X}}}.}$

For the two middle composite squares, note that if we apply the classical
formula relating the cup product and the boundary maps in sheaf cohomology
(e.g., see [3, Thm. 7.1] or [56, Lem. 3.2]) to the cohomology sequences
associated to the (derived) tensor product of
${\mathcal{F}}^{j}_{{\mathcal{X}}}$ with the exact triangle

$\iota_{*}R\iota^{!}{\mathcal{F}}^{2-j}_{{\mathcal{X}}}\to{\mathcal{F}}^{2-j}_{{\mathcal{X}}}\to
Ru_{*}{\mathcal{F}}^{2-j}_{X}$

for $0\leq j\leq 2$, we get that the diagram

(4.15) $\textstyle{H^{i}({\mathcal{F}}^{j}_{{\mathcal{X}}})\times
H^{3-i}_{Y}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{u^{*}}$$\textstyle{H^{3}_{Y}({\mathcal{F}}^{2}_{{\mathcal{X}}})}$$\textstyle{H^{i}({\mathcal{F}}^{j}_{X})\times
H^{2-i}({\mathcal{F}}^{2-j}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\cup}$$\textstyle{H^{2}({\mathcal{F}}^{2}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$

is commutative up to multiplication by $(-1)^{i}$. Since ${\rm
Tr}_{{\mathcal{X}}}\circ\partial={\rm Tr}_{X}$, it follows that the two middle
composite squares commute.

Since all vertical arrows on the lower floor of (4.13) are injective and all
lower squares are commutative, it follows that the squares on the left and the
right corners on the upper floor are commutative. Next, an easy diagram chase
shows that the square labeled (1) is Cartesian. This implies that the image of
the composite middle vertical arrow lies in the subgroup
$H^{2-i}({\mathcal{F}}^{2-j}_{X})^{\star}$. Furthermore, if we let
$\Phi^{ij}_{X}\colon H^{i}({\mathcal{F}}^{j}_{X})\to
H^{2-i}({\mathcal{F}}^{2-j}_{X})^{\star}$ denote the induced map, then all
squares on the upper floor commute. Since all vertical arrows on this floor,
except possibly the middle one, are isomorphisms, it follows that
$\Phi^{ij}_{X}$ is also an isomorphism.

To finish the proof of the theorem, it remains to show that $\Phi^{ij}_{X}$ is
a continuous and open homomorphism. To show this, recall that $F_{ij}$ is a
quotient of $H^{i}({\mathcal{F}}^{j}_{{\mathcal{X}}})$ and is an open subgroup
of $H^{i}({\mathcal{F}}^{j}_{X})$ such that
${H^{i}({\mathcal{F}}^{j}_{X})}/{F_{ij}}$ is discrete. Hence, it suffices to
show in the square labeled (2) in (4.13) that the composite map
$H^{i}({\mathcal{F}}^{j}_{{\mathcal{X}}})\to
H^{3-i}_{Y}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})^{\star}\to
H^{2-i}({\mathcal{F}}^{2-j}_{X})^{\star}$ is continuous and open. Let us call
this composite map $\theta$. Now, we have already seen above that
$\Phi^{ij}_{{\mathcal{X}}}$ is a topological isomorphism. On the other hand,
$H^{3-i}_{Y}({\mathcal{F}}^{2-j}_{{\mathcal{X}}})^{\star}\to
H^{2-i}({\mathcal{F}}^{2-j}_{X})^{\star}$ is clearly continuous because the
Pontryagin dual is an auto-functor in the category of locally compact
Hausdorff topological abelian groups. This proves that $\theta$ is continuous.

To show that $\theta$ is open, it suffices to show that the map
$H^{i+1}_{Y}({\mathcal{F}}^{j}_{{\mathcal{X}}})^{\star}\to
H^{i}({\mathcal{F}}^{j}_{X})^{\star}$ is open for any $i,j\geq 0$. From (3.18)
and (3.19), we see that this map has a factorization

(4.16)
$H^{i+1}_{Y}({\mathcal{F}}^{j}_{{\mathcal{X}}})^{\star}\twoheadrightarrow{\rm
Im}(\partial^{\star})={\rm Ker}(\iota_{*})^{\star}\hookrightarrow
H^{i}({\mathcal{F}}^{j}_{X})^{\star}.$

The first arrow in this factorization is clearly open because it is a quotient
map in ${\operatorname{\mathbf{Pfd}}}$ by Lemma 2.3. On the other hand,
$(F_{ij})^{\star}$ is discrete and
$H^{i}({\mathcal{F}}^{j}_{X})^{\star}\twoheadrightarrow(F_{ij})^{\star}$ is a
continuous surjective homomorphism. It follows that its kernel ${\rm
Im}(\partial^{\star})={\rm Ker}(\iota_{*})^{\star}$ is open in
$H^{i}({\mathcal{F}}^{j}_{X})^{\star}$. We have thus shown that the composite
map in (4.16) is open. This concludes the proof. ∎

For integers $m,n\geq 1$, we have a diagram of bilinear parings

(4.17)
$\textstyle{{{\mathcal{K}}^{M}_{i,X}}/{p^{m}}\times{{\mathcal{K}}^{M}_{j,X}}/{p^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup}$$\scriptstyle{p^{n}}$$\textstyle{{{\mathcal{K}}^{M}_{i+j,X}}/{p^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{n}}$$\textstyle{{{\mathcal{K}}^{M}_{i,X}}/{p^{m+n}}\times{{\mathcal{K}}^{M}_{j,X}}/{p^{m+n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
can}}}$$\scriptstyle{\cup}$$\textstyle{{{\mathcal{K}}^{M}_{i+j,X}}/{p^{m+n}},}$

where ‘${\operatorname{\rm can}}$’ indicates the canonical surjection. It
easily follows from the bilinearity of the product in Milnor $K$-theory that
this diagram is commutative. Passing to the cohomology and using (2.3), we get
a bilinear pairing

(4.18)
$``{{\underset{n}{\varinjlim}}}"H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})\times``{{\underset{n}{\varprojlim}}}"H^{2-i}(X,W_{n}\Omega^{2-j}_{X,{\operatorname{log}}})\to``{{\underset{n}{\varinjlim}}}"{{\mathbb{Z}}}/{p^{n}}$

between ind-abelian and pro-abelian groups.

We endow ${\varinjlim}_{n}\ H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})$
with the direct limit (i.e., the weak) topology and ${\varprojlim}_{n}\
H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})$ the inverse limit topology.
Taking the limits in (4.18) and using Lemma 2.6, we get the following.

######  Corollary 4.8.

There is a pairing of topological abelian groups

${\varinjlim}_{n}\
H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})\times{\varprojlim}_{n}\
H^{2-i}(X,W_{n}\Omega^{2-j}_{X,{\operatorname{log}}})\to{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}}$

which induces an isomorphism of abelian groups

${\varprojlim}_{n}\
H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})\to\left({\varinjlim}_{n}\
H^{2-i}(X,W_{n}\Omega^{2-j}_{X,{\operatorname{log}}})\right)^{\star},$

a continuous epimorphism of topological abelian groups

${\varinjlim}_{n}\
H^{i}(X,W_{n}\Omega^{j}_{X,{\operatorname{log}}})\to\left({\varprojlim}_{n}\
H^{2-i}(X,W_{n}\Omega^{2-j}_{X,{\operatorname{log}}})\right)^{\star}.$

##  5\. The relative Picard scheme

After the duality theorem for the logarithmic Hodge-Witt cohomology, the next
key ingredients in the proof of Theorem 1.1 and the Brauer-Manin pairing for
modulus pairs are the representability and certain topological properties of
the relative Picard group. These results are of independent interest and we
shall establish them in the next two sections using [29] and [50].

We fix an arbitrary field $k$ of exponential characteristic $p\geq 1$. We let
$X$ be a smooth projective geometrically connected curve over $k$ and let
$D\subset X$ be an effective Cartier divisor whose support is a non-empty
finite closed subset $S=\\{x_{1},\ldots,x_{r}\\}$ of $X$. We shall use
${\mathfrak{m}}$ as another notation for $D$ and call it the modulus divisor,
following the terminology of [50]. We let $\iota\colon D\hookrightarrow X$ be
the inclusion. We shall call $(X,D)$ a modulus pair (of dimension one). We
write $D=\stackrel{{\scriptstyle r}}{{\underset{i=1}{\sum}}}n_{i}[x_{i}]$ as a
Weil divisor and let $\deg(D)=\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}n_{i}[k(x_{i}):k]$ denote the degree of $D$. Note
that $\deg(D)=1$ if and only if $D={\rm Spec\,}(k(x))$, where $x\in X(k)$. We
let $X^{o}=X\setminus S$. We let $A={\mathcal{O}}_{X,S}$ and $I\subset A$ the
ideal defining $D$ in ${\rm Spec\,}(A)$. We write $A_{D}=A/I$. We let $K$
denote the function field of $X$. For any $k$-scheme $Z$, we let
${\mathcal{Z}}_{0}(Z)$ denote the free abelian group on $Z_{(0)}$. We shall
let ${\mathcal{O}}(Z)$ (resp. ${\mathcal{O}}^{\times}(Z)$) denote the group
$H^{0}(Z,{\mathcal{O}}_{Z})$ (resp. $H^{0}(Z,{\mathcal{O}}^{\times}_{Z})$).

###  5.1. Relative Picard and 0-cycles with modulus

Recall that for an integral quasi-projective scheme $Y$ over $k$ and an
effective Cartier divisor $E\subset Y$, the Chow group of 0-cycles with
modulus ${\rm CH}_{0}(Y|E)$ is defined to be the quotient of
${\mathcal{Z}}_{0}(Y\setminus E)$ by the subgroup ${\mathcal{R}}_{0}(Y|E)$
generated by the 0-cycles $\nu_{*}({\rm div}(f))$, where $\nu\colon C_{n}\to
Y$ is the canonical morphism from the normalization of an integral curve
$C\subset Y$ not contained in $E$ and $f\in{\rm
Ker}({\mathcal{O}}^{\times}_{C_{n},\nu^{-1}(E)}\twoheadrightarrow{\mathcal{O}}^{\times}(\nu^{*}(E)))$.
The relative Picard group ${\rm Pic}(Y|E)$ (usually written as ${\rm
Pic}(Y,E)$ in the literature) is the set of isomorphism classes of pairs
$({\mathcal{L}},u\colon{\mathcal{O}}_{E}\xrightarrow{\cong}{\mathcal{L}}|_{E})$,
where ${\mathcal{L}}$ is an invertible sheaf on $Y$ and
${\mathcal{L}}|_{E}:={\mathcal{L}}\otimes_{{\mathcal{O}}_{Y}}{\mathcal{O}}_{E}$.
One says that $({\mathcal{L}},u)\cong({\mathcal{L}}^{\prime},u^{\prime})$ if
there is an isomorphism
$w\colon{\mathcal{L}}\xrightarrow{\cong}{\mathcal{L}}^{\prime}$ such that
$u^{\prime}=w|_{E}\circ u$. ${\rm Pic}(Y|E)$ is an abelian group under the
tensor product of invertible sheaves and their trivializations along $E$ whose
identity element is $({\mathcal{O}}_{Y},{\operatorname{id}})$. There is a
canonical homomorphism ${\rm Pic}(Y|E)\to{\rm Pic}(Y)$ which is surjective if
$\text{\rm dim}(Y)=1$.

We now specialize to the one-dimensional modulus pair $(X,D)$ over $k$ that we
fixed above. Let ${\rm Pic}^{0}(X|D)$ denote the subgroup of pairs
$({\mathcal{L}},u)$ such that $\deg({\mathcal{L}})=0$. We let ${\rm
CH}_{0}(X|D)_{0}$ be the kernel of the composite map $\deg\colon{\rm
CH}_{0}(X|D)\twoheadrightarrow{\rm CH}_{0}(X)\xrightarrow{\deg}{\mathbb{Z}}$.
There is a degree preserving cycle class map ${\operatorname{\rm
cyc}}_{X|D}\colon{\rm CH}_{0}(X|D)\to{\rm Pic}(X|D)$ (e.g., see [30, Lem.
3.1]) such that ${\operatorname{\rm
cyc}}_{X|D}([Q])=({\mathcal{O}}_{X}(Q),u(1)=f^{-1})$, where $Q\in X^{o}_{(0)}$
and $(f)\in{\mathcal{O}}_{X,S\cup\\{Q\\}}$ is the ideal of $Q$.

######  Lemma 5.1.

The homomorphism ${\operatorname{\rm cyc}}_{X|D}$ is bijective.

###### Proof.

This is a combination of [55, Lemmas 2.1, 2.3, 2.4]. ∎

From the above definitions, it is clear that there is an exact sequence

(5.1)
$0\to{\mathcal{O}}^{\times}(X)\xrightarrow{\iota^{*}}{\mathcal{O}}^{\times}(D)\xrightarrow{\partial_{X|D}}{\rm
Pic}(X|D)\xrightarrow{\vartheta_{X|D}}{\rm Pic}(X)\to 0,$

where $\iota^{*}$ is the canonical inclusion and
$\partial_{X|D}(u)=({\mathcal{O}}_{X},u\colon{\mathcal{O}}_{D}\xrightarrow{\cong}{\mathcal{O}}_{X}|_{D})$.
Note that this exact sequence holds even if $X$ is only integral. In § 7, we
shall study the exactness of the sequence obtained by taking the Pontryagin
dual of (5.1). Here, we note the following consequence.

######  Lemma 5.2.

The canonical map ${\rm Pic}(X|D)\to{\rm Pic}(X)$ is an isomorphism if
$\deg(D)=1$,

###  5.2. The singular curve attached to $(X,D)$

Assume now that $\deg(D)\geq 2$. In order to prove the representability of
${\rm Pic}(X|D)$, the strategy is to replace it with the ordinary Picard group
of a singular curve (without modulus). The following result from [50, Chap.
IV, § 1, no. 3, 4] helps us in achieving this.

######  Proposition 5.3.

There exists a unique geometrically integral $k$-scheme $X_{\mathfrak{m}}$
together with a finite morphism $\psi_{\mathfrak{m}}\colon X\to
X_{\mathfrak{m}}$ such that the following hold.

  1. (1)

$\psi_{\mathfrak{m}}(S)$ is the unique singular point $P_{0}\in
X_{\mathfrak{m}}$ which is $k$-rational.

  2. (2)

$\psi_{\mathfrak{m}}\colon
X^{o}\xrightarrow{\cong}X_{\mathfrak{m}}\setminus\\{P_{0}\\}$.

  3. (3)

If we let $I_{\mathfrak{m}}\subset{\mathcal{O}}_{X_{\mathfrak{m}},P_{0}}$
denote the maximal ideal, then the pull-back map
$\psi^{*}_{\mathfrak{m}}\colon{\mathcal{O}}_{X_{\mathfrak{m}},P_{0}}\to{\mathcal{O}}_{X,S}$
induces a bijection $I_{\mathfrak{m}}\xrightarrow{\cong}I$.

###### Proof.

Since we do not assume any condition on $k$ while the construction of
$X_{\mathfrak{m}}$ given in [50] assumes $k$ to be algebraically closed, we
sketch the proof of this proposition to explain that this additional
hypothesis is not necessary.

Since $S$ has an affine neighborhood in $X$, one easily observes that it
suffices to show the existence of $X_{\mathfrak{m}}$ when $X={\rm Spec\,}(R)$
is affine. We let $B=k+I=\\{a+b|a\in k,b\in I\\}\subset A$. Then $B$ is a
$k$-subalgebra of $A$ and there is a commutative diagram of short sequences of
$B$-modules:

(5.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{A/B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{A_{D}}/k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0.}$

Since $A_{D}$ is finite over $k$, it follows from the above diagram that there
is a finite-dimensional $k$-vector space $V\subset A$ such that $A=V+I$. It
follows that there is a surjective map of $B$-modules $(k\oplus
V)\otimes_{k}B\twoheadrightarrow A$. In particular, $A$ is a finite
$B$-module. An application of the Eakin-Nagata theorem (e.g., see [34, Thm.
3.7]) shows that $B$ is Noetherian. Using [1, Thm. 5.10], we conclude
furthermore that $B$ is a 1-dimensional local ring with maximal ideal $I$.

We now let $L$ be the fraction field of $B$ and take the tensor product of the
middle row in (5.2) with $L$ over $B$ to get a commutative diagram of short
exact sequences

(5.3)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\otimes_{B}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({A_{D}}/k)\otimes_{B}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Since ${A_{D}}/k$ is a torsion $B$-module, we get $({A_{D}}/k)\otimes_{B}L=0$.
Since $A\otimes_{B}L$ is a localization of $A$, the map $A\to A\otimes_{B}L$
is injective. It follows that $A\subset L$. This implies that the inclusion
$L\hookrightarrow K$ is a bijection. Putting everything together, we conclude
that the conditions (1) to (3) of the proposition hold if we replace $X$ by
${\rm Spec\,}(A)$. We now let $B^{\prime}$ be the intersection (in $K$) of the
localizations of $R$ at all maximal ideals except those supported on $S$ and
let $R^{\prime}=B\cap B^{\prime}$. It is then easy to check that
$X_{\mathfrak{m}}:={\rm Spec\,}(R^{\prime})$ is the desired singular curve
with the conductor ideal $I_{\mathfrak{m}}:=I$. Since
$R^{\prime}_{\overline{k}}\hookrightarrow R_{\overline{k}}$ and the latter
ring is an integral domain, it follows that $X_{\mathfrak{m}}$ is
geometrically integral. Note also that $P$ has to be a singular point of
$X_{\mathfrak{m}}$ since $\deg(k(P))=1$ while $\deg(D)\geq 2$, by our
assumption. ∎

Let $X_{\mathfrak{m}}$ be the singular curve with the unique singular point
$P_{0}$ associated to the modulus pair $(X,D)$ as obtained in Proposition 5.3.
We shall call this ‘the ${\mathfrak{m}}$-contraction of $X$’ in the sequel. We
shall identify $X_{\mathfrak{m}}\setminus\\{P_{0}\\}$ with $X^{o}$ via
$\psi_{\mathfrak{m}}$ throughout our discussion.

######  Lemma 5.4.

Let ${k^{\prime}}/k$ be any field extension and let
${\mathfrak{m}}^{\prime}:=D_{k^{\prime}}\subset X_{k^{\prime}}$. Then the
${\mathfrak{m}}^{\prime}$-contraction of $X_{k^{\prime}}$ is canonically
isomorphic to $(X_{\mathfrak{m}})_{k^{\prime}}$.

###### Proof.

This is a direct consequence of the construction of $X_{\mathfrak{m}}$ using
its property that $P_{0}\in X_{\mathfrak{m}}(k)$. ∎

Recall that the Levine-Weibel Chow group of 0-cycles ${\rm CH}^{{\rm
lw}}_{0}(Y)$ of an integral curve $Y$ over $k$ is the quotient of
${\mathcal{Z}}_{0}(Y_{\rm reg})$ by the subgroup generated by the divisors of
those rational functions on $Y$ which lie in ${\mathcal{O}}^{\times}_{Y,Y_{\rm
sing}}$. It is well known that assigning each closed point of $Y_{\rm reg}$
its class in ${\rm Pic}(Y)$ induces a canonical cycle class isomorphism
${\operatorname{\rm cyc}}_{Y}\colon{\rm CH}^{{\rm
lw}}_{0}(Y)\xrightarrow{\cong}{\rm Pic}(Y)$ (e.g., see [2, Lem. 3.12]). The
following is a key lemma.

######  Lemma 5.5.

Let ${k^{\prime}}/k$ be any field extension. Then the identity map of
${\mathcal{Z}}_{0}(X^{o})$ induces via $\psi^{*}_{\mathfrak{m}}$, a degree
preserving isomorphism

$\psi^{*}_{\mathfrak{m}}\colon{\rm
Pic}((X_{\mathfrak{m}})_{k^{\prime}})\xrightarrow{\cong}{\rm
Pic}(X_{k^{\prime}}|D_{k^{\prime}})$

whose composition with ${\rm
Pic}(X_{k^{\prime}}|D_{k^{\prime}})\twoheadrightarrow{\rm
Pic}(X_{k^{\prime}})$ is the usual pull-back between the Picard groups of
$(X_{\mathfrak{m}})_{k^{\prime}}$ and $X_{k^{\prime}}$.

###### Proof.

Since the base field $k$ is arbitrary, we can assume $k^{\prime}=k$ by virtue
of Lemma 5.4. To construct $\psi^{*}_{\mathfrak{m}}$, we can replace ${\rm
Pic}(X|D)$ and ${\rm Pic}(X_{\mathfrak{m}})$ with ${\rm CH}_{0}(X|D)$ and
${\rm CH}^{{\rm lw}}_{0}(X_{\mathfrak{m}})$, respectively, via the cycle class
isomorphisms. We first show that $\psi^{*}_{\mathfrak{m}}({\rm div}(f))=0$ if
$f\in{\mathcal{O}}^{\times}_{X_{\mathfrak{m}},P_{0}}$. To that end, we let $g$
denote the residue class of $f$ in $k(P_{0})^{\times}=k^{\times}$ and let
$\tilde{f}=fg^{-1}$. Then we see that
$\tilde{f}\in{\mathcal{O}}^{\times}_{X_{\mathfrak{m}},P_{0}}$ is such that
${\rm div}(f)={\rm div}(\tilde{f})$ in ${\mathcal{Z}}_{0}(X^{o})$. Since
$\tilde{f}\in K_{1}(A,I)$, we also see that $\psi^{*}_{\mathfrak{m}}({\rm
div}(\tilde{f}))=0$ in ${\rm CH}_{0}(X|D)$. This proves the desired
factorization of the pull-back map between the Picard groups. It is clear that
$\psi^{*}_{\mathfrak{m}}$ is degree preserving (e.g., see [54, Tag 0AYU, Lem.
33.44.4]).

To complete the proof of the lemma, we look at the commutative diagram of
short exact sequences

(5.4)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{A_{D}^{\times}}/{k^{\times}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Pic}(X_{\mathfrak{m}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{*}_{\mathfrak{m}}}$$\scriptstyle{\psi^{*}_{\mathfrak{m}}}$$\textstyle{{\rm
Pic}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{A^{\times}_{D}}/{k^{\times}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Pic}(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Pic}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

This shows that the middle vertical arrow is an isomorphism. ∎

It follows from Lemma 5.5 that there is commutative diagram

(5.5) $\textstyle{{\rm CH}^{{\rm
lw}}_{0}(X_{\mathfrak{m}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
cyc}}_{X_{\mathfrak{m}}}}$$\scriptstyle{\psi^{*}_{\mathfrak{m}}}$$\textstyle{{\rm
Pic}(X_{\mathfrak{m}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{*}_{\mathfrak{m}}}$$\textstyle{{\rm
CH}_{0}(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
cyc}}_{X|D}}$$\textstyle{{\rm Pic}(X|D),}$

in which all arrows are isomorphisms and degree preserving. We can now prove
the existence of the relative Picard variety of the modulus pair $(X,D)$.

######  Theorem 5.6.

There exists a group scheme ${\mathbf{Pic}}(X|D)$ over $k$ such that
${\mathbf{Pic}}(X|D)(k^{\prime})\cong{\rm Pic}(X_{k^{\prime}}|D_{k^{\prime}})$
for every field extension ${k^{\prime}}/{k}$. The identity component
${\mathbf{Pic}}^{0}(X|D)$ of ${\mathbf{Pic}}(X|D)$ is a smooth and irreducible
quasi-projective group scheme over $k$ of dimension equal to $\text{\rm
dim}_{k}H^{1}(X_{\mathfrak{m}},{\mathcal{O}}_{X_{\mathfrak{m}}})$ such that
${\mathbf{Pic}}^{0}(X|D)(k^{\prime})\cong{\rm
Pic}^{0}(X_{k^{\prime}}|D_{k^{\prime}})$. If $X^{o}(k)\neq\emptyset$, then the
connected components of ${\mathbf{Pic}}(X|D)$ are parameterized by
${\mathbb{Z}}$ and each component is isomorphic to ${\mathbf{Pic}}^{0}(X|D)$.
We have
${\mathbf{Pic}}(X|D)_{k^{\prime}}\cong{\mathbf{Pic}}(X_{k^{\prime}}|D_{k^{\prime}})$
for every field extension ${k^{\prime}}/{k}$. If
$f\colon(X^{\prime},D^{\prime})\to(X,D)$ is a strict morphism of modulus pairs
(i.e., $f^{*}(D)=D^{\prime}$), then there is a canonical homomorphism of group
schemes
$f^{*}\colon{\mathbf{Pic}}(X|D)\to{\mathbf{Pic}}(X^{\prime}|D^{\prime})$.

###### Proof.

If $\deg(D)=1$, then $D={\rm Spec\,}(k(x))$ for some $x\in X(k)$. In this
case, we have ${\rm Pic}(X_{k^{\prime}}|D_{k^{\prime}})\xrightarrow{\cong}{\rm
Pic}(X_{k^{\prime}})$ for every field extension ${k^{\prime}}/{k}$ by Lemma
5.2. The theorem therefore follows directly from its known case $D=\emptyset$.
We now assume $\deg(D)\geq 2$ and let $\psi_{\mathfrak{m}}\colon X\to
X_{\mathfrak{m}}$ be the ${\mathfrak{m}}$-contraction of $X$ (cf. Proposition
5.3). Since $X_{\mathfrak{m}}$ is geometrically integral, it follows from the
flat base change property of the higher direct images of coherent sheaves that
for every $k$-scheme $T$, the map
${\mathcal{O}}_{T}\to(f_{T})_{*}({\mathcal{O}}_{X_{\mathfrak{m}}\times T})$ is
an isomorphism of sheaves on $T$, where $f\colon X_{\mathfrak{m}}\to{\rm
Spec\,}(k)$ is the structure map and $f_{T}\colon X_{\mathfrak{m}}\times T\to
T$ is the base change of $f$ to $T$. Since $P_{0}\in X_{\mathfrak{m}}(k)$, the
map $f$ has a section.

Using the last two properties of $X_{\mathfrak{m}}$, we conclude from [29,
Thm. 9.2.5, 9.4.8] that there exists a group scheme
${\mathbf{Pic}}(X_{\mathfrak{m}})$ over $k$ such that
${\mathbf{Pic}}(X_{\mathfrak{m}})(T)\cong{{\rm Pic}(X_{\mathfrak{m}}\times
T)}/{{\rm Pic}(T)}$ for every $k$-scheme $T$. In particular, we have
${\mathbf{Pic}}(X_{\mathfrak{m}})(k^{\prime})\cong{\rm
Pic}((X_{\mathfrak{m}})_{k^{\prime}})$ for every field extension
${k^{\prime}}/{k}$. Furthermore, the identity component
${\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$ of ${\mathbf{Pic}}(X_{\mathfrak{m}})$
is a geometrically irreducible quasi-projective group scheme over $k$ and
${\mathbf{Pic}}^{0}(X_{\mathfrak{m}})(k^{\prime})\cong{\rm
Pic}^{0}((X_{\mathfrak{m}})_{k^{\prime}})$ by [29, Lem. 9.5.1]. It follows
from [40, Cor. 4.2] (see also [29, Prop. 9.5.19]) and the fppf descent
property of smoothness that ${\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$ is a smooth
group scheme over $k$, i.e., it is an algebraic group over $k$. In particular,
it is geometrically integral. The assertion about $\text{\rm
dim}({\mathbf{Pic}}^{0}(X|D))$ follows from its smoothness and [29, Cor.
9.5.13].

If $P\in X^{o}(k)$, then the connected components of
${\mathbf{Pic}}(X_{\mathfrak{m}})$ are parameterized by ${\mathbb{Z}}$, and
each connected component is isomorphic to
${\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$ via the transformation
${\mathcal{L}}\mapsto{\mathcal{L}}\otimes{\mathcal{O}}_{X_{\mathfrak{m}}}(-mP)$
on the scheme points, where $\deg({\mathcal{L}})=m\in{\mathbb{Z}}$. The
theorem now follows from Lemmas 5.4 and 5.5 if we let
${\mathbf{Pic}}(X|D):={\mathbf{Pic}}(X_{\mathfrak{m}})$. The last claim is
clear because the Picard scheme is a contravariant functor. ∎

###  5.3. Picard scheme of singular curves

Suppose that $Y$ is a geometrically integral singular curve with the
normalization $f\colon X\to Y$. Let $Z\subset Y$ be a conductor subscheme for
$f$ with support $Y_{\rm sing}$ and let $D=Z\times_{Y}X$. Assume that $X$ is
smooth over $k$. Then $(X,D)$ is a 1-dimensional modulus pair, and it is easy
to see from the construction in Proposition 5.3 that the
${\mathfrak{m}}$-contraction of $X$ has a unique factorization

(5.6)
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\psi_{\mathfrak{m}}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\psi}_{\mathfrak{m}}}$$\textstyle{X_{\mathfrak{m}}.}$

We conclude from Theorem 5.6 that there are morphisms of group schemes

(5.7)
${\mathbf{Pic}}(X|D)\xrightarrow{\widetilde{\psi}^{*}_{\mathfrak{m}}}{\mathbf{Pic}}(Y)\xrightarrow{f^{*}}{\mathbf{Pic}}(X)$

whose composition is the canonical morphism
${\mathbf{Pic}}(X|D)\twoheadrightarrow{\mathbf{Pic}}(X)$.

We now recall the exact sequence

(5.8) $0\to k^{\times}\to{\mathcal{O}}^{\times}(Z)\to H^{1}_{\rm
nis}(Y,{\mathcal{K}}^{M}_{1,(Y,Z)})\to H^{1}_{\rm
nis}(Y,{\mathcal{K}}^{M}_{1,Y})\to 0.$

Using the isomorphisms $H^{1}_{\rm
nis}(Y,{\mathcal{K}}^{M}_{1,(Y,Z)})\xrightarrow{\cong}H^{1}_{\rm
nis}(X,{\mathcal{K}}^{M}_{1,(X,D)})\cong{\rm Pic}(X|D)$ (see [30, Lem. 3.1]
for the second isomorphism), we get an exact sequence

(5.9) $0\to{{\mathcal{O}}^{\times}(Z)}/{k^{\times}}\to{\rm Pic}(X|D)\to{\rm
Pic}(Y)\to 0.$

Since this remains true over every algebraic extension of $k$, it follows that
the map $\widetilde{\psi}^{*}_{\mathfrak{m}}$ in (5.7) is surjective.

##  6\. Relative Picard group over local fields

The goal of this section is to prove some topological properties of the
relative Picard group and the albanese map with respect to the adic topology
that the schemes acquire when the base field is a local field. Before we do
this, we need to study the smoothness properties of the albanese morphism to
the relative Picard scheme and of the morphism from the relative to the
ordinary Picard scheme. We continue with the notations and assumptions of § 5
(in particular, $k$ is still arbitrary).

###  6.1. Generic smoothness of albanese map

Until we reach Theorem 6.6, we shall assume in this subsection that
$\deg(D)\geq 2$ and $X^{o}(k)\neq\emptyset$. We fix a point $P\in X^{o}(k)$.
For any integer $n\geq 1$, let $\mathbf{Div}^{n}(X_{\mathfrak{m}})$ be the
quasi-projective $k$-scheme which parameterizes effective Cartier divisors
$E\subset X_{\mathfrak{m}}$ of degree $n$. By [29, Defn. 9.4.6], there is a
canonical morphism of $k$-schemes ${{\rm
AJ}}^{n}_{X_{\mathfrak{m}}}\colon\mathbf{Div}^{n}(X_{\mathfrak{m}})\to{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$
which sends $E$ to ${\mathcal{O}}_{X_{\mathfrak{m}}}(E-nP)$ if the support of
$E$ lies in $X(k)$. Since $\mathbf{Div}^{1}(X_{\mathfrak{m}})\cong X^{o}$ by
[29, Exc. 9.3.8], we get a $k$-morphism ${\rm alb}_{X|D}\colon
X^{o}\to{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$ (called the albanese morphism)
which has the property that ${\rm alb}_{X|D}(x)=[x]-[P]$ for any $x\in
X^{o}(k)$. Furthermore, it induces

(6.1) ${{\rm AJ}}_{X|D}\colon{\rm CH}^{{\rm
lw}}_{0}(X_{\mathfrak{m}})_{0}\cong{\rm
Pic}^{0}(X_{\mathfrak{m}})\xrightarrow{\cong}{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})(k).$

For $n\geq 1$, the $n$-th power of the albanese map descends to a morphism
${\rm alb}^{n}_{X|D}\colon{\operatorname{\rm
Sym}}^{n}(X^{o})\to{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$, where
${\operatorname{\rm Sym}}^{n}(X^{o})$ is the $n$-th symmetric power of
$X^{o}$. We let

(6.2) $\phi^{n}_{X|D}\colon(X^{o})^{n}\xrightarrow{\pi_{n}}{\operatorname{\rm
Sym}}^{n}(X^{o})\xrightarrow{{\rm
alb}^{n}_{X|D}}{\mathbf{Pic}}^{0}(X_{\mathfrak{m}}).$

denote the composite morphism. Let $p_{a}(X_{\mathfrak{m}})=\text{\rm
dim}_{k}(H^{1}(X_{\mathfrak{m}},{\mathcal{O}}_{X_{\mathfrak{m}}}))$ denote the
arithmetic genus of $X_{\mathfrak{m}}$.

Let us now assume that $E\subset X_{\mathfrak{m}}$ is an effective Cartier
divisor disjoint from the unique singular point $P_{0}$ of $X_{\mathfrak{m}}$
and let ${\mathcal{L}}={\mathcal{O}}_{X_{\mathfrak{m}}}(E)$. Equivalently, $E$
is an effective Cartier divisor on $X$ disjoint from $S$ and
${\mathcal{L}}^{\prime}:={\mathcal{O}}_{X}(E)\cong\psi^{*}_{\mathfrak{m}}({\mathcal{L}})$.
Since $E$ is effective, there is a canonical inclusion
${\mathcal{O}}_{X}\hookrightarrow{\mathcal{L}}^{\prime}$. This yields a
canonical inclusion $k\cong H^{0}(X,{\mathcal{O}}_{X})\hookrightarrow
H^{0}(X,{\mathcal{L}}^{\prime})$ such that the composite $k\hookrightarrow
H^{0}(X,{\mathcal{L}}^{\prime})\to A/I$ is the canonical inclusion of rings
$k\hookrightarrow A/I$. Using this and the tensor product of the exact
sequence

(6.3)
$0\to{\mathcal{O}}_{X_{\mathfrak{m}}}\to(\psi_{\mathfrak{m}})_{*}({\mathcal{O}}_{X})\to{{\mathcal{O}}_{D}}/{{\mathcal{O}}_{P_{0}}}\to
0$

with ${\mathcal{L}}$, we get a commutative diagram with exact rows

(6.4)
$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(X,{\mathcal{L}}^{\prime}(-D))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(X,{\mathcal{L}}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(X_{\mathfrak{m}},{\mathcal{L}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(X,{\mathcal{L}}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{A_{D}}/k.}$

Hence, we get an exact sequence

(6.5) $0\to H^{0}(X,{\mathcal{L}}^{\prime}(-D))\to
H^{0}(X_{\mathfrak{m}},{\mathcal{L}})\to k\to 0.$

For a line bundle ${\mathcal{L}}$ on $X_{\mathfrak{m}}$, let $|{\mathcal{L}}|$
denote the complete linear system
${\operatorname{Proj}}_{k}(H^{0}(X_{\mathfrak{m}},{\mathcal{L}}))$. If
${\mathcal{L}}={\mathcal{O}}_{X_{\mathfrak{m}}}(E)$ for an effective divisor
$E$ on $X^{o}$, we let
$U_{E}=|{\mathcal{L}}|\setminus|{\mathcal{L}}^{\prime}(-D)|\subset|{\mathcal{L}}|$.

######  Lemma 6.1.

For any Cartier divisor $E$ on $X_{\mathfrak{m}}$, the linear system
$|{\mathcal{O}}_{X_{\mathfrak{m}}}(E)|$ represents effective Cartier divisors
$F$ on $X_{\mathfrak{m}}$ such that
${\mathcal{O}}_{X_{\mathfrak{m}}}(F)\cong{\mathcal{O}}_{X_{\mathfrak{m}}}(E)$
as line bundles on $X_{\mathfrak{m}}$.

###### Proof.

One checks that the proof of a similar statement for smooth schemes given in
[17, Prop. II.7.7] (see also [12, § 13.13, p. 395]) are based on two claims
both of which are met in the present case. ∎

######  Lemma 6.2.

Assume that $k$ is algebraically closed and let $E\subset X^{o}$ be an
effective Cartier divisor of degree $n\geq 1$. Then $E$ is canonically an
element of ${\operatorname{\rm Sym}}^{n}(X^{o})(k)$ and there is a canonical
isomorphism of $k$-schemes $({\rm alb}^{n}_{X|D})^{-1}({\rm
alb}^{n}_{X|D}(E))\cong U_{E}$.

###### Proof.

It is clear that ${\rm alb}^{n}_{X|D}$ has factorization ${\operatorname{\rm
Sym}}^{n}(X^{o})\hookrightarrow\mathbf{Div}^{n}(X_{\mathfrak{m}})\xrightarrow{{{\rm
AJ}}^{n}_{X|D}}{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$. As $k$ is algebraically
closed, ${{\rm AJ}}^{n}_{X|D}$ is given by
$E\mapsto[E]-[nP]\in{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$. Using Lemma 6.1, it
is clear from this that $({\rm alb}^{n}_{X|D})^{-1}({\rm
alb}^{n}_{X|D}(E))=|{\mathcal{L}}|\cap{\operatorname{\rm Sym}}^{n}(X^{o})$. It
follows from (6.4) that, up to multiplication by elements of $k^{\times}$, the
latter set consists of those sections of ${\mathcal{L}}$ which do not die in
$k$ under the map $H^{0}(X_{\mathfrak{m}},{\mathcal{L}})\to k$ in the exact
sequence (6.5). In other words, $|{\mathcal{L}}|\cap{\operatorname{\rm
Sym}}^{n}(X^{o})=U_{E}$. This finishes the proof. ∎

######  Lemma 6.3.

Assume that $k$ is algebraically closed. Then for all integers $n\gg 0$, all
non-empty fibers of the morphism ${\rm alb}^{n}_{X|D}\colon{\operatorname{\rm
Sym}}^{n}(X^{o})\to{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$ are affine spaces
over the corresponding residue fields of
${\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$, and have equal and positive
dimensions.

###### Proof.

For any effective Cartier divisor $E\subset X^{o}$, we let
${\mathcal{L}}={\mathcal{O}}_{X_{\mathfrak{m}}}(E)$ and
${\mathcal{L}}^{\prime}=\psi^{*}_{\mathfrak{m}}({\mathcal{L}})$. Since
$|{\mathcal{L}}^{\prime}(-D)|$ is a hyperplane of the projective space
$|{\mathcal{L}}|$, it follows from Lemma 6.2 that all fibers of the morphism
${\rm alb}^{n}_{X|D}\colon{\operatorname{\rm
Sym}}^{n}(X^{o})\to{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$ are affine spaces
over the corresponding residue fields of
${\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$. It remains to prove the independence
of these fiber dimensions if $n\gg 0$.

Suppose that $E$ is such that
$\deg(E)=n\geq\deg(D)+p_{a}(X_{\mathfrak{m}})+2g-1$, where $g=p_{a}(X)$
denotes the genus of $X$. Then
$\deg({\mathcal{L}}^{\prime}(-D))=\deg({\mathcal{L}}^{\prime})-\deg(D)=\deg(E)-\deg(D)\geq
2g-1$. It follows from the Serre duality and the Riemann-Roch theorem for $X$
that

(6.6) $H^{1}(X,{\mathcal{L}}^{\prime}(-D))=0=H^{1}(X,{\mathcal{L}}^{\prime}).$

Using the cohomology exact sequence associated to the sheaf exact sequence

(6.7)
$0\to{\mathcal{L}}^{\prime}(-D)\to{\mathcal{L}}^{\prime}\to{\mathcal{O}}_{D}\to
0,$

we get an exact sequence

(6.8) $0\to H^{0}(X,{\mathcal{L}}^{\prime}(-D))\to
H^{0}(X,{\mathcal{L}}^{\prime})\to H^{0}(D,{\mathcal{O}}_{D})\to 0.$

In particular, the map
$H^{0}(X,{\mathcal{L}}^{\prime})\to{H^{0}(D,{\mathcal{O}}_{D})}/k$ is
surjective. On the other hand, by tensoring (6.3) with ${\mathcal{L}}$ and
passing to cohomology, we get an exact sequence

(6.9) $0\to H^{0}(X_{\mathfrak{m}},{\mathcal{L}})\to
H^{0}(X,{\mathcal{L}}^{\prime})\to{H^{0}(D,{\mathcal{O}}_{D})}/k\to
H^{1}(X_{\mathfrak{m}},{\mathcal{L}})\to H^{1}(X,{\mathcal{L}}^{\prime})\to
0.$

We conclude that $H^{1}(X_{\mathfrak{m}},{\mathcal{L}})=0$.

Now, we use the Riemann-Roch theorem for $X_{\mathfrak{m}}$ (e.g., see [17,
Exc. IV.1.9]) to get the identity $\text{\rm
dim}_{k}(H^{0}(X_{\mathfrak{m}},{\mathcal{L}}))=n+1-p_{a}(X_{\mathfrak{m}})$.
In other words, $\text{\rm dim}_{k}(U_{E})=\text{\rm
dim}_{k}(|{\mathcal{L}}|)=n-p_{a}(X_{\mathfrak{m}})\geq\deg(D)+2g-1\geq 1$.
Since $\deg(D)+2g-1$ depends only on $(X,D)$, we conclude that all fibers of
${\rm alb}^{n}_{X|D}$ are of a fixed positive dimension. ∎

We now let $k$ be any field.

######  Proposition 6.4.

For all integers $n\gg 0$, the morphism ${\rm
alb}^{n}_{X|D}\colon{\operatorname{\rm
Sym}}^{n}(X^{o})\to{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$ is smooth and
surjective.

###### Proof.

Via the standard descent properties of the smooth and surjective morphisms
with respect to fpqc morphisms to the base scheme (e.g., see [12, Prop. 14.48,
§ 16.C]), and the commutativity of ${{\mathfrak{m}}}$-contraction and
${\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$ with the field extensions, we easily
reduce the proof to the case where we can assume $k$ to be algebraically
closed.

Since $k=\overline{k}$ and ${\rm CH}^{{\rm
lw}}_{0}(X_{\mathfrak{m}})_{0}\xrightarrow{\cong}{\rm
Pic}^{0}(X_{\mathfrak{m}})\xrightarrow{\cong}{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})(k)$,
it already follows from [7, Defn. 7.1, Thm. 7.2] that ${\rm alb}^{n}_{X|D}$ is
surjective for all $n\gg 0$. Using the fpqc descent, the smoothness claim now
follows from Lemma 6.3 and [17, Exc. III.10.9, Thm. III.10.2]. ∎

For $n\gg 0$ and $1\leq i<j\leq n$, we let
$\Delta^{n}_{ij}(X^{o})\subset(X^{o})^{n}$ be the closed subscheme (via a
permutation) $\Delta_{ij}\times(X^{o})^{n-2}\subset(X^{o})^{n}$, where
$\Delta_{ij}\subset X^{o}\times X^{o}$ is the diagonal of the product of the
$i$-th and $j$-th factors of $(X^{o})^{n}$. We let
$U_{n}=(X^{o})^{n}\setminus({\underset{1\leq i<j\leq
n}{\bigcup}}\Delta^{n}_{ij}(X^{o}))$. Then $U_{n}$ is invariant under the
action of the symmetric group $S_{n}$ and there is a Cartesian square

(6.10)
$\textstyle{U_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{\pi_{n}}$$\textstyle{(X^{o})^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n}}$$\textstyle{{U_{n}}/{S_{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{{\operatorname{\rm
Sym}}^{n}(X^{o}).}$

Furthermore, $\pi_{n}\colon U_{n}\to{U_{n}}/{S_{n}}$ is an étale
$S_{n}$-torsor (e.g., see [8, Prop. 0.9]). In particular, it is finite and
étale. We let $U_{n}(X^{o})={U_{n}}/{S_{n}}$.

######  Corollary 6.5.

For all $n\gg 0$, the morphism $\phi^{n}_{X|D}\colon
U_{n}(X^{o})\to{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$ is smooth and has dense
image.

###### Proof.

Since $\phi^{n}_{X|D}={\rm alb}^{n}_{X|D}\circ\pi_{n}$ (see (6.2)) and
$U_{n}(X^{o})\xrightarrow{\pi_{n}}{\operatorname{\rm Sym}}^{n}(X^{o})$ is
étale with dense image, the corollary follows from Proposition 6.4. ∎

We can now prove the following generic smoothness of the albanese map for a
modulus pair. We let $k$ be any field and $(X,D)$ a 1-dimensional modulus pair
over $k$ such that $X$ is connected and smooth over $k$ and
$X^{o}(k)\neq\emptyset$. We fix a point $P\in X^{o}(k)$ so that ${\rm
alb}_{X|D}$ is defined with respect to the base point $P$.

######  Theorem 6.6.

For all $n\gg 0$, the albanese map $\phi^{n}_{X|D}\colon
U_{n}(X^{o})\to{\mathbf{Pic}}^{0}(X|D)$ is smooth and has dense image.

###### Proof.

Our assumption implies that $X$ is geometrically integral over $k$. If
$\deg(D)=1$, then the canonical map
${\mathbf{Pic}}^{0}(X|D)\to{\mathbf{Pic}}^{0}(X)$ is an isomorphism of
algebraic groups over $k$ (see Lemma 5.2) and one knows classically that the
albanese map $U_{n}(X^{o})\to{\mathbf{Pic}}^{0}(X)$ is smooth and has dense
image. If $\deg(D)\geq 2$, the theorem follows from Theorem 5.6 and Corollary
6.5. ∎

###  6.2. The map from relative to ordinary Picard scheme

In order to compare the relative and ordinary Picard schemes, we need to
identify ${\mathbf{Pic}}^{0}(X|D)$ with the generalized Jacobian variety of
$(X,D)$ à la Rosenlicht-Serre [50] under some additional assumptions. We fix a
field $k$ and a 1-dimensional modulus pair $(X,D)$ over $k$ such that $X$ is
connected and smooth over $k$. We write $D=\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}n_{i}[x_{i}]$. If $\deg(D)\geq 2$, we let
$\psi_{\mathfrak{m}}\colon X\to X_{\mathfrak{m}}$ be the
${\mathfrak{m}}$-contraction of $X$. The following is the main result of
Rosenlicht-Serre for which we refer to [50, Chap. V, § 4.22, 4.23].

######  Theorem 6.7.

Assume that ${\rm Supp}(D)\subset X(k)$ and $P\in X^{o}(k)$. Then there exists
a quasi-projective algebraic group $J_{X|D}$ over $k$ satisfying the following
properties.

  1. (1)

There is a canonical isomorphism of group schemes
$J_{X_{k^{\prime}}|D_{k^{\prime}}}\cong(J_{X|D})_{k^{\prime}}$ over
$k^{\prime}$ for every field extension ${k^{\prime}}/k$.

  2. (2)

The assignment $Q\mapsto[Q]-\deg(Q)[P]$ on $X^{o}_{(0)}$ defines a morphism of
$k$-schemes (called the albanese morphism) ${\lambda}_{X|D}\colon X^{o}\to
J_{X|D}$.

  3. (3)

The albanese morphism induces a homomorphism of abelian groups

${{\rm AJ}}_{X|D}\colon{\rm CH}_{0}(X|D)_{0}\to J_{X|D}({k}).$

  4. (4)

Given any commutative group scheme $G$ over $k$ and a $k$-morphism $f\colon
X^{o}\to G$ which induces a homomorphism $f_{*}\colon{\rm CH}_{0}(X|D)_{0}\to
G(k)$, there is a unique homomorphism of group schemes $\theta\colon
J_{X|D}\to G$ over $k$ such that $f(x)=f(P)+(\theta\circ\lambda_{X|D})(x)$ for
every $x\in X^{o}$.

Assume that $\deg(D)\geq 2$. Using the albanese map ${\rm alb}_{X|D}\colon
X^{o}\to{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$ and (6.1), it follows from the
property (4) of $J_{X|D}$ that there exists a unique homomorphism (recall here
that ${\rm alb}_{X|D}(P)=0$) of group schemes $\theta_{X|D}\colon
J_{X|D}\to{\mathbf{Pic}}^{0}(X_{\mathfrak{m}})\cong{\mathbf{Pic}}^{0}(X|D)$
over $k$ such that ${\rm alb}_{X|D}=\theta_{X|D}\circ\lambda_{X|D}$.

######  Corollary 6.8.

Under the assumptions of Theorem 6.7, the albanese map $\lambda_{X|D}\colon
X^{o}\to J_{X|D}$ induces an isomorphism of algebraic groups
$\theta_{X|D}\colon J_{X|D}\xrightarrow{\cong}{\mathbf{Pic}}^{0}(X|D)$.

###### Proof.

If $\deg(D)=1$, it easily follows from [50, Chap. V, § 3.13] that the
canonical map $J_{X|D}\to J_{X}$ is an isomorphism. We have seen above that
this holds also for ${\mathbf{Pic}}^{0}(X|D)$. If $\deg(D)\geq 2$, we can
replace ${\mathbf{Pic}}^{0}(X|D)$ with ${\mathbf{Pic}}^{0}(X_{\mathfrak{m}})$.
In the latter case, we can argue as in the proof of Proposition 6.4 to reduce
the proof to the case where we can assume that $k$ is algebraically closed.
But this case is classically known. ∎

We can now compare the relative and ordinary Picard schemes of a modulus pair.
For any $n\geq 1$, let ${\mathbb{W}}_{n}$ be the étale sheaf on
${\operatorname{\mathbf{Sch}}}_{k}$ given by
${\mathbb{W}}_{n}(Z)={\mathbb{W}}_{n}({\mathcal{O}}(Z))$, where
${\mathbb{W}}_{n}(R)$ is the ring of big Witt-vectors over the ring $R$ of
length $n$ (e.g., see [44, App. A]). One knows that ${\mathbb{W}}_{n}$ is
representable by a unipotent linear algebraic group over $k$ (e.g., see [50,
Chap. V, § 3.13, Lem. 20]).

######  Proposition 6.9.

Under the assumptions of Theorem 6.7, there is an exact sequence of algebraic
groups

(6.11) $0\to{\mathbf{Pic}}^{0}(X|D)_{\rm
aff}\to{\mathbf{Pic}}^{0}(X|D)\to{\mathbf{Pic}}^{0}(X)\to 0$

over $k$, where ${\mathbf{Pic}}^{0}(X|D)_{\rm aff}$ is a linear algebraic
group canonically isomorphic to
$({\mathbb{G}}_{m})^{r-1}\times(\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\prod}}}{\mathbb{W}}_{n_{i}-1})$.

###### Proof.

Using Corollary 6.8, we can work with $J_{X|D}$, in which case the proposition
is well known (e.g., see [50, Chap. V, § 3, no. 13]). ∎

######  Corollary 6.10.

Let $X$ be a smooth projective geometrically integral curve over $k$ and let
$D\leq D^{\prime}$ be two effective Cartier divisors on $X$. Then the
canonical map ${\mathbf{Pic}}^{0}(X|D^{\prime})\to{\mathbf{Pic}}^{0}(X|D)$ is
smooth, affine and surjective. If $Y$ is a singular curve over $k$ with
normalization $X$ and $Z\subset Y$ is a conductor subscheme for the
normalization such that $D=Z\times_{Y}X$, then the map
${\mathbf{Pic}}^{0}(X|D)\to{\mathbf{Pic}}^{0}(Y)$ in (5.7) is smooth, affine
and surjective.

###### Proof.

To prove either of the statements, we can pass to the algebraic closure of
$k$. In this case, the first statement follows directly by comparing (6.11)
for ${\mathbf{Pic}}^{0}(X|D^{\prime})$ and ${\mathbf{Pic}}^{0}(X|D)$. The
second statement follows directly from (5.9). ∎

###  6.3. The case of local fields

We shall assume in this subsection that $k$ is a local field. Recall that $k$
is a topological field with respect to its valuation (also called adic)
topology such that the ring of integers ${\mathcal{O}}_{k}$ is an open
subring. One also knows that for any locally of finite type $k$-scheme $Y$,
the set $Y(k)$ has a unique structure of a totally disconnected locally
compact Hausdorff topological space (see [5, Prop. 5.4]). This topology is
characterized by the property that if $Y\subset{\mathbb{A}}^{n}_{k}$ as a
locally closed subscheme, then it coincides with the subspace topology on
$Y(k)$ induced by the product of the adic topology of $k$ on
${\mathbb{A}}^{n}_{k}(k)\cong k^{n}$. This assignment of adic topology defines
a functor ${\operatorname{\mathbf{Sch}}}_{k}\to{\mathbf{Top}}$, where the
latter is the category of topological spaces with continuous maps. We shall
call this the adic topology of $Y(k)$ and say that $Y(k)$ is an adic space.
For any property of the adic topology of $Y(k)$ that we shall use in this
paper, the reader is referred to [5] and [4, Thm. 10.5.1].

Every finite-dimensional $k$-vector space $M$ is equipped with the product
topology of the adic topology of $k$, called the adic topology of $M$. If $A$
is a finite-dimensional $k$-algebra, the adic topology of $A^{\times}$ is the
subspace topology induced from $A$. These topologies make $A$ and $A^{\times}$
into topological abelian groups. If ${k^{\prime}}/k$ is a finite field
extension and $A$ is a finite-dimensional $k^{\prime}$-vector space, then the
adic topology of $A$ (and $A^{\times}$ if $A$ is a $k^{\prime}$-algebra)
coincides with its adic topology when considered as a $k$-vector space (e.g.,
see [20, § 7.2]). In this paper, all finite-dimensional algebras over a local
field and their unit groups will be assumed to be endowed with the adic
topology unless mentioned otherwise.

We now let $(X,D)$ be as in § 6.2. We assume that $X^{o}(k)\neq\emptyset$ and
fix a point $P\in X^{o}(k)$ so that ${\rm alb}_{X|D}$ is defined with respect
to the base point $P$. Let $Y$ be a singular curve over $k$ with normalization
$X$ and a conductor subscheme $D\subset X$. Since ${\rm
Pic}(X|D)={\mathbf{Pic}}(X|D)(k)$ and ${\rm Pic}(Y)={\mathbf{Pic}}(Y)(k)$, we
see that ${\rm Pic}(X|D)$ and ${\rm Pic}(Y)$ are locally compact Hausdorff
topological abelian groups. Furthermore, it follows from [4, Thm. 10.5.1] that
${\rm Pic}^{0}(X|D)$ and ${\rm Pic}^{0}(Y)$ are open subgroups of ${\rm
Pic}(X|D)$ and ${\rm Pic}(Y)$, respectively. In the following result, we shall
use the adic topology on all sets.

######  Corollary 6.11.

We have the following.

  1. (1)