
\documentclass[12pt,twoside,a4paper,reqno]{amsart}

\title[Cohomology of tori]{Cohomology of tori over \textit{p}-adic
curves}

\subjclass{
Primary
14G20;
secondary
14F22,
14L15,
11S25}

\keywords
 {Algebraic tori, \'etale cohomology, algebraic curves, $p$-adic
 fields, duality theorems, Lichtenbaum--Tate duality, Poincar\'e
 duality, Brauer group, Picard group}

\author
 {Claus Scheiderer}
\address
 {Institut f\"ur Mathematik \\
 Fakult\"at 4 \\
 Universit\"at Duisburg \\
 47048 Duisburg \\
 Germany}
\email
 {claus@uni-duisburg.de}

\author
 {Joost van Hamel}
\address
 {School of Mathematics and Statistics, F07 \\
 University of Sydney \\
 NSW 2006 \\
 Australia}
\email
 {vanhamel@member.ams.org}


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\begin{document}

\maketitle

\begin{abstract}
  We establish a duality in the cohomology of arbitrary tori over
  smooth but not necessarily projective curves over a $p$-adic field.
  This generalises Lichtenbaum--Tate duality between the Picard group
  and the Brauer group of a smooth projective curve.
\end{abstract}

\section*{Introduction}

Let $C$ be a connected smooth projective curve over a $p$-adic field
$k$, and let $T_0$ be a torus over the function field $K=k(C)$ of
$C$. In order to relate the Galois cohomology of $T_0$ over $K$ to
the local cohomology groups of $T_0$ in the closed points of $C$
(see \cite{Sch...}), one needs a global duality theory for `integral
models' of $T_0$, namely tori over open subcurves of $C$. The aim of
this paper is to set up such a theory.

The pattern of the result we seek is given by a theorem of
Lichtenbaum \cite{Li}, which establishes a natural non-degenerate
pairing
$$\Pic(C)\times\Br(C)\to\QmodZ$$
(`Lichtenbaum--Tate duality'). This can be seen as a pairing
$$H^1(C,\Gg_m)\times H^2(C,\Gg_m)\to\QmodZ$$
of \'etale cohomology groups.

In general, a $K$-torus $T_0$ does not necessarily have a model over
$C$, but it always admits a model $T$ over some open subset $V$ of
$C$. As is usual in cohomological duality over non-compact spaces,
one has to work with a form of cohomology with compact supports. We
define such a cohomology theory which is specially tailored for tori,
and which we denote by $H^*_{cc}(V,T)$ (see
Section~\ref{sec:cohcptsupp}). It is slightly different from the
general notion of cohomology with compact supports in \'etale
cohomology. We also need the (standard) notion of the \emph{dual
torus} $T'$ of $T$ (see Section~\ref{sec:tori}). Our main result
establishes for every integer $q$ a natural pairing
\begin{equation*}
H^q_{cc}(V,T)\times H^{3-q}(V,T')\to\QmodZ
\end{equation*}
which is non-degenerate on the left if $q\ne3$, and non-degenerate on
 the right if $q\ne2$ (see Theorem~\ref{th:main}).

We first prove our result in the case where $T$ is the split torus
$\Gg_{m,V}$. Then the general case is deduced from the split case by
a descent argument. An important role is played by the perfect (cup
product) pairings of the cohomology groups of the torsion groups,
$$H^q_c(V,\,{}_nT)\times H^{4-q}(V,\,{}_nT')\to\QmodZ$$
(see Section~\ref{sec:p-duality}) and by the Kummer exact sequences.
In fact, not only the proof that our pairings are non-degenerate, but
even their construction proceeds via these torsion pairings. For the
construction, a more elegant approach in the spirit of
\cite{vanHamel:L-T} would have been possible, but at the cost of
employing much heavier technical machinery than the approach we take
here. Moreover, and this was something of a surprise, the proof that
our pairing is non-degenerate becomes essentially a formality, after
embedding the pairing into the Kummer sequence. The crucial descent
argument is in the proof of Proposition~\ref{prop:1ststep}. Without
the Kummer sequence, a descent argument seems much more cumbersome.
\medbreak

Here is an outline of the structure of the paper. In
Section~\ref{sec:prelim} we have gathered the known constructions and
results we will need. This ranges from elementary observations
regarding abelian groups to Poincar\'e duality in the \'etale
cohomology of varieties over $p$-adic fields and Lichtenbaum--Tate
duality. In Section~\ref{sec:cohcptsupp} we define $H^*_{cc}
(V,\hvar)$, our modified cohomology of tori with compact supports,
and study how it relates to the standard definition of \'etale
cohomology with compact supports, $H^*_c(V,\hvar)$. We also establish
links between $H^*_{cc}(V,\Gg_m)$, motivic cohomology and generalised
Jacobians. In Section~\ref{sec:l-t-open-duality} we construct the
pairing $H^q_{cc}(V,\Gg_m)\times H^{3-q}(V,\Gg_m)\to\QmodZ$, using
the Kummer sequences and the cup product pairing with coefficients in
the $n$th roots of unity.  To derive our duality in this case from
Lichtenbaum--Tate duality, we use the localisation exact sequences
(see Lemma~\ref{lem:localisation}). In Example~\ref{ex:degeneracy}
we also see where the degeneracies of the pairings pop up. In
Section~\ref{sec:constr} we construct our pairings for arbitrary
tori, and prove the duality result Theorem~\ref{th:main}.

\subsection*{Acknowledgements}
This paper results from a four-month visit of the second author to
the Gerhard Mercator Universit\"at in Duisburg, financed by the
European Union in the framework of the network \emph{K-Theory and
Linear Algebraic Groups} (Contract ERB FMRX CT-97-0107). He would
like to thank the institute for its hospitality and the European
Union for the financial support.

\section{preliminaries}
\xlabel{sec:prelim}

\subsection{Abelian groups}
\xlabel{sec:abeliangps}

Let $M$ be an abelian group. Given $n\in\Nn$, we let ${}_nM$ resp.\
$M/n$ be the kernel resp.\ the cokernel of $M\labelto nM$, the
multiplication by $n$ map on $M$. We write $M^*:=\Hom(M,\QmodZ)$ for
the group of characters of $M$. If $M$ is a topological group, we
write $M^\du:=\Hom_{\textnf{cont}}(M,\QmodZ)$ for the group of
continuous characters, where $\QmodZ$ has the discrete topology.
$M\mapsto M^\du$ is an exact functor on the category of compact
abelian groups.

For a prime $\ell$ we define $\Tate_\ell(M):=\varprojlim({}_
{\ell^\nu}M)$; this is the \emph{$\ell$-adic Tate module} of $M$. It
is a torsion-free $\Zz_\ell$-module. If ${}_\ell M$ is finite, then
$\Tate_\ell(M)$ is free of finite rank. We let $\Tate(M):=\varprojlim
({}_nM)$ be the \emph{total Tate module} of $M$, and we have
$$\Tate(M)=\prod_\ell\Tate_\ell(M).$$
If ${}_\ell M$ is finite for every prime $\ell$, then $\Tate(M)$ is a
torsion-free profinite abelian group. The functor $\Tate(\hvar)$ is
left exact. If $D$ is the largest divisible torsion subgroup of $M$,
the inclusion $D\subset M$ induces an isomorphism $\Tate(D)\isoto
\Tate(M)$.

The profinite completion of $M$ is denoted by $\wh M$ (or $M\compl$).
The group $M$ is called \emph{residually finite} if the natural map
$M\to\wh M$ is injective. There is a natural map $\varprojlim M/n\to
\wh M$. It is an isomorphism if $M/\ell$ is finite for every prime
$\ell$.

Let $D(M):=\ker(M\to\varprojlim M/n)=\bigcap_nnM$, the subgroup of
divisible elements of $M$. We need the following elementary
observation:

\begin{lemma}\xlabel{divisibleelements}
If $0\to A\to B\to C$ is an exact sequence of abelian groups, and if
for every prime $\ell$ the $\ell$-primary torsion subgroup of $C$ has
finite exponent, then the sequence $0\to D(A)\to D(B)\to D(C)$ is
exact.
\end{lemma}

\begin{proof}
Let $x\in A\cap D(B)$, and fix a prime $\ell$. For every $n\in\Nn$
there is $y_n\in B$ with $\ell^ny_n=x$. By the hypothesis on $C$
there exists $r\in\Nn$ with $\ell^ry_n\in A$ for every $n$. Thus
$x\in\ell^mA$ for every $m\in\Nn$.
\end{proof}

Moreover, we need the following facts about profinite completions of
locally compact groups:

\begin{lemma}\xlabel{lem:compl of loccpt}
Let $A$ be a locally compact abelian group, and suppose that $A$
contains a profinite open subgroup $A_0$ which is (topologically)
finitely generated.
\begin{itemize}
\item[\textrm(a)]
Every subgroup of $A$ of finite index is open in $A$.
\item[\textrm(b)]
Assume further that the abelian group $A/A_0$ is finitely generated.
Then the map $A\to\wh A$ induces isomorphisms $_nA\isoto{}_n\wh A$
and $A/n\isoto\wh A/n$ for every $n\in\Nn$.
\end{itemize}
\end{lemma}

\begin{proof}
The Sylow subgroups of $A_0$ are topologically finitely generated,
and $A_0$ is their direct product. So (a) follows easily from the
corresponding fact for finitely generated pro-$p$ groups (see
\cite{CG}, I.4.2 ex.~6). As for (b), $A=A_1\times F$ where $F$ is
free abelian (of finite rank) and $A_1$ is a profinite abelian group
every finite index subgroup of which is open. Hence $\wh A=A_1\times
\wh F$, from which (b) is clear.
\end{proof}

\subsection{Pairings}
\xlabel{sec:pairings}

Let $A$ be a discrete torsion abelian group and $B$ a profinite
abelian group. Let $A\times B\to\QmodZ$ be a continuous pairing
(i.e., the annihilator of each $a\in A$ is open in $B$). The
following lemma is a standard consequence of Pontryagin duality
(\cite{Pontryagin}, see also \cite{Ribes-Zalesskii}, Sec.~2.9, or
\cite{Wilson}, Sec.~6.4).

\begin{lemma}\xlabel{left-right}
For a pairing $A\times B\to\QmodZ$ as above, we have that the induced
map $A\to B^\du$ is injective (resp., surjective, resp., bijective)
if and only if the induced map $B\to A^*$ is surjective (resp.,
injective, resp., bijective).
\end{lemma}

The pairing $A\times B\to\QmodZ$ is called \emph{perfect} if the
induced homomorphisms $A\to B^\du$ and $B\to A^*$ are bijective. A
pairing $A\times B\to\QmodZ$ between arbitrary abelian groups is said
to be \emph{non-degenerate on the left} (resp., \emph{on the right})
if the induced homomorphism $A\to B^*$ (resp., $B\to A^*$) is
injective. The pairing is called \emph{non-degenerate} if it is so on
the left and on the right.

\begin{corollary}\xlabel{perf-iff-nondeg}
A continuous pairing $A\times B\to\QmodZ$ between a discrete torsion
abelian group $A$ and a profinite abelian group $B$ is perfect if and
only if it is non-degenerate.
\end{corollary}

\begin{definition}
Let $I$ be a directed set. Let $\{A_i\}_{i\in I}$ be an inductive
system of abelian groups and $\{B_i\}_{i\in I}$ a projective system
of abelian groups, both indexed by $I$. A collection of pairings
$\langle\hvar,\hvar\rangle_i\fcolon A_i\times B_i\to\QmodZ$ indexed
by $i\in I$ will be called a \emph{system of pairings between
$\{A_i\}$ and $\{B_i\}$} if the pairings $\langle\hvar,\hvar
\rangle_i$ are compatible with the transition maps.
\end{definition}

\begin{lemma}\xlabel{lem:limit-pairing}
Let $I$ be a directed set. Let $\langle\hvar,\hvar\rangle_i$ ($i\in
I$) be a system of pairings between an inductive system of abelian
groups $\{A_i\}_{i\in I}$ and a projective system of abelian groups
$\{B_i\}_{i\in I}$.
\begin{itemize}
\item[\textrm(a)]
There is a unique continuous pairing $\langle\hvar,\hvar\rangle
\fcolon(\varinjlim A_i)\times(\varprojlim B_i)\to\QmodZ$ which is
compatible with the pairings $\langle\hvar,\hvar\rangle_i$.
\item[\textrm(b)]
If the groups $A_i$, $B_i$ are finite and the pairings $\langle\hvar,
\hvar\rangle_i$ are perfect ($i\in I$), then the limit pairing (a) is
a perfect pairing between the discrete torsion group $\varinjlim A_i$
and the profinite group $\varprojlim B_i$.
\end{itemize}
\end{lemma}

\begin{example}\xlabel{ex:induced-pairings}
Any pairing $A\times B\to\QmodZ$ of discrete abelian groups gives
rise to a continuous pairing
$$A_\tors\times\wh B\to\QmodZ,$$
and also to a continuous pairing
$$(A\otimes\QmodZ)\times\Tate(B)\to\QmodZ$$
(which is the pairing $\varinjlim(A/n)\times\varprojlim({}_nB)\to
\QmodZ$).
\end{example}

\begin{lemma}\xlabel{lem:torsion-perfect}
Let $A\times B\to\QmodZ$ be a perfect continuous pairing between a
torsion abelian group $A$ and a profinite abelian group $B$. Then for
any $n\in\Nn$, the induced pairings
$$({}_nA)\times B/n\to\QmodZ\quad\text{and}\quad A/n\times({}_nB)\to
\QmodZ$$
are perfect.
\end{lemma}

\begin{proof}
Consider the multiplication by $n$ exact sequences for $A$ and $B$,
and use exactness of the functors $(\hvar)^*$ and $(\hvar)^\du$.
\end{proof}

\begin{lemma}\xlabel{lem:gen-pairing-perfect}
Suppose we have two exact sequences $0\to A_1\to A_2\to A_3\to 0$ and
$0\ot B_1\ot B_2\ot B_3\ot 0$ of abelian groups, and pairings
$\beta_i\fcolon A_i\times B_i\to\QmodZ$ ($i=1,2,3$) that are
compatible with these sequences. Suppose further that the middle
pairing $\beta_2$ is non-degenerate.
\begin{itemize}
\item[\textrm(a)]
$\beta_1$ is non-degenerate on the left, and $\beta_3$ is
non-degenerate on the right.
\item[\textrm(b)]
$\beta_1$ is non-degenerate if and only if $\beta_3$ is
non-degenerate.
\end{itemize}
\end{lemma}

\begin{proof}
(a) is obvious, and (b) is immediate using the snake lemma.
\end{proof}

The proof of the following five lemma for pairings is immediate.

\begin{lemma}\xlabel{5lemma}
Suppose we have two exact sequences of abelian groups
$$A_1\labelto{f_1}A_2\labelto{f_2}A_3\labelto{f_3}A_4\labelto{f_4}
A_5$$
and
$$B_1\labelot{g_1}B_2\labelot{g_2}B_3\labelot{g_3}B_4\labelot{g_4}
B_5,$$
together with pairings $\beta_i\fcolon A_i\times B_i\to\QmodZ$ ($i=
2,3,4$) that are compatible with the two sequences.
\begin{itemize}
\item[\textrm(a)]
If $\beta_4$ is non-degenerate on the left and $\ker(g_1)^\bot=\im
(f_1)$, then $\beta_3$ is non-degenerate on the left.
\item[\textrm(b)]
If $\beta_2$ is non-degenerate on the right and $\ker(f_4)^\bot=\im
(g_4)$, then $\beta_3$ is non-degenerate on the right.
\end{itemize}
\end{lemma}

\subsection{Tori and their duals}
\xlabel{sec:tori}

A general reference for tori over schemes is \cite{SGA3}, tome~II.
Given a scheme $S$, a torus over $S$ (or: $S$-torus) is an $S$-group
scheme $T$ which locally with respect to the \emph{fpqc} topology is
isomorphic to $(\Gg_{m,S})^r$ for some integer $r\ge0$. The $S$-torus
$T$ is said to be split if $T\cong(\Gg_{m,S})^r$ for some $r\ge0$. If
$S$ is locally noetherian and normal, any $S$-torus is
\emph{isotrivial}, i.e., it splits over a finite \'etale covering
$S'$ of $S$ (see \cite{SGA3}, Th.~X.5.16). We will call such a
covering $S'\to S$ a \emph{trivialising covering} for $T$. If
moreover $S$ is connected, then $S'$ can be taken to be finite and
Galois over $S$.

Given an $S$-torus $T$, the \emph{character group scheme} of $T$ is
$\schX^*(T)=\sHom_{S-gr}(T,\Gg_m)$. This is a \emph{lattice group
scheme} over $S$, by which we mean an $S$-group scheme which is
locally constant free abelian of finite rank. We have $T=\sHom_{S-gr}
(\schX^*(T),\Gg_m)$, and the functor $T\mapsto\schX^*(T)$ defines a
duality between the category of $S$-tori and the category of lattice
group schemes over $S$.

Assume that $S$ is locally noetherian, normal and connected. Then the
category of $S$-tori is anti-equivalent to the category of isotrivial
group schemes over $S$ which are locally constant free abelian, by
means of the functor $\schX^*$. We can also fix a geometric point
$\bar s$ of $S$ and identify the latter category with the category of
$\pi_1(S,\bar s)$-lattices (i.e., of discrete
$\pi_1(S,\bar s)$-modules which are free abelian of finite rank as
abelian groups).

The \emph{cocharacter group scheme} of $T$ is $\schX_*(T)=
\sHom_{S-gr}(\Gg_m,T)$. We have $T=\schX_*(T)\otimes\Gg_m$. The
natural pairing
$$\schX^*(T)\times\schX_*(T)\to\sHom_{S-gr}(\Gg_m,\Gg_m)=\Zz_S$$
(of group schemes over $S$) identifies each of $\schX^*(T)$ and
$\schX_*(T)$ with the dual lattice group scheme of the other.

The \emph{dual torus} $T'$ of $T$ is defined to be
\begin{equation*}
T':=\sHom_{S-gr}(\schX_*(T),\Gg_m)=\schX^*(T)\otimes\Gg_m.
\end{equation*}
We have $\schX^*(T')=\schX_*(T)$ and $(T')'=T$ (canonical
isomorphism).

Base change gives us for any morphism $f \fcolon X \to S$ a torus
$T_X:=T\times_SX$ on $X$, the pullback of $T$ via $f$. Observe that
in general $T_X$ does not represent the \'etale sheaf $f^*T$ (e.g.,
when $f$ is the inclusion of a closed point on a curve). It does,
however, if the map $f$ is \'etale.

\subsection{Galois cohomology and duality}
\xlabel{sec:galois}

Let $F$ be a field and fix a separable closure $F_s$ of $F$. By $G_F
=\Gal(F_s/F)$ we denote the absolute Galois group of $F$. If $M$ is a
discrete $G_F$-module, Galois cohomology $H^q(G_F,M)$ is denoted by
$H^q(F,M)$. For a group scheme $G$ over $F$, we will mostly write
$H^q(F,G)$ rather than $H^q(G_F,G(F_s))$. If $M$ is a torsion module
whose torsion is prime to $\ch(F)$, then $M(i)$ is the $i$-th Tate
twist of $M$, for $i\in\Zz$.

\begin{theorem}[Tate--Nakayama]\xlabel{th:t-n-duality}
Let $T$ be a torus over a $p$-adic field~$k$. The cup product pairing
\begin{equation}\label{eq:tate-nakayama}
H^q(k,T)\times H^{2-q}(k,\galX^*(T))\to H^2(k,\Gg_m)=\QmodZ
\end{equation}
is non-degenerate for all $q\in\Zz$. Moreover, it is a perfect
pairing of finite groups for $q=1$, and induces perfect pairings
\begin{align*}
H^0(k,T)\compl\times H^2(k,\galX^*(T)) & \to\QmodZ \\
\intertext{for $q=0$ and}
H^0(k,\galX^*(T))\compl\times H^2(k,T) & \to\QmodZ
\end{align*}
for $q=2$, each between a profinite group and a discrete torsion
group.
\end{theorem}

\begin{proof}
See \cite{CG}, II.5.8, Thm.~6. (There the completion of $H^0(k,T)$ in
the $q=0$ case is taken with respect to the system of open subgroups
of finite index. By Lemma~\ref{lem:compl of loccpt}, the result is
the same as the profinite completion of the abstract (discrete) group
$H^0(k,T)$.)
\end{proof}

\subsection{Poincar\'e duality for curves over $p$-adic fields}
\xlabel{sec:p-duality}

By an algebraic variety over a field $k$, we mean a separable reduced
$k$-scheme of finite type. A curve over $k$ is a purely
one-dimensional algebraic variety over $k$. Unless mentioned
otherwise, all cohomology groups $H^*(\hvar,\hvar)$ will be \'etale
cohomology. In view of the equivalence between the Galois cohomology
and the \'etale cohomology of $\Spec k$, this is consistent with the
notation in the previous section. Let $k$ be a $p$-adic field, let
$C$ be a nonsingular projective curve over $k$ (which in general we
do not assume to be connected), let $V$ be a dense open subscheme of
$C$, and let $j\colon V\into C$ denote the inclusion. Given any
\'etale sheaf $F$ on $V$, the cohomology of $F$ with compact supports
is defined as
$$H^q_c(V,F):=H^q(C,j_!F),$$
where $j_!$ is the extension by zero. We have a trace map
\begin{equation*}
\tr\fcolon H^4_c(V,\QmodZ(2))\to H^2(k,\QmodZ(1))=\QmodZ
\end{equation*}
which is an isomorphism if (and, in fact, only if) $V$ is connected.
Indeed, $H^4_c(V,\QmodZ(2))=H^2\bigl(k,\,H^2_c(\ol V,\QmodZ(2))
\bigr)$ by the Hochschild-Serre spectral sequence, and $\tr$ is the
map induced by the trace $H^2_c(\ol V,\QmodZ(2))\to\QmodZ(1)$ (see
for example \cite{Milne:etal},\S V.2).

\begin{theorem}[Poincar\'e duality]\xlabel{th:PD}
Let $V$ be a nonsingular, not necessarily projective curve over a
$p$-adic field~$k$. Let $M$ be a locally constant \'etale sheaf on
$V$ with finite stalks. Then the cup product pairing
$$H^q_c(V,M)\times H^{4-q}(V,M^\du(2))\to H^4_c(V,\QmodZ(2))\labelto
\tr\QmodZ$$
is a perfect pairing of finite abelian groups, for every $q\in\Zz$.
\end{theorem}

\begin{proof}
This is a formal consequence of geometric Poincar\'e duality in
\'etale cohomology, and Tate duality in the Galois cohomology of $k$
with finite coefficients. See for example
\cite{Saito:global-duality}, Lemma~2.9 (or compare \cite{Milne:dual},
Th.~II.7.6).
\end{proof}

\begin{corollary}\xlabel{cor:torus-ntors-PD}
Let $V$ be as above, and let $T$ be a torus over $V$, with dual torus
$T'$. Then the cup product pairing gives for any $q\in\Zz$ and any
$n\in\Zz$ a perfect pairing
$$H^q_c(V,{}_nT)\times H^{4-q}(V,{}_nT')\to H^4_c(V,\QmodZ(2))
\labelto\tr\QmodZ$$
of finite abelian groups.
\end{corollary}

\begin{proof}
For the $n$-torsion subgroups we have ${}_nT=\sHom(\schX^*(T),\mu_n)=
M^\du(1)$ with $M:=\schX^*(T)/n$, and ${}_nT'=\schX^*(T)\otimes\mu_n=
M(1)$. Therefore, the perfectness for any $n\in\Nn$ is a particular
case of Poincar\'e duality (Theorem~\ref{th:PD}).
\end{proof}

Note that for any torus $T$ over $V$, the groups $H^q(V,T)/n$ are
finite (for all $q\in\Zz$, $n\in\Nn$), by the exact sequences $0\to
{}_nT\to T\labelto{n}T\to0$ and the finiteness of the groups $H^q
(V,{}_nT)$. In particular, we have
$$H^q(V,T)\compl=\varprojlim_n\,H^q(V,T)/n$$
for the profinite completions (cf.\ Section~\ref{sec:abeliangps}). We
will use the notation
$$H^q(V,\Tate(T)):=\varprojlim_n\,H^q(V,{}_nT)$$
and
$$H^q_c(V,\Tate(T)):=\varprojlim_n\,H^q_c(V,{}_nT)$$
for all $q\in\Zz$. These are profinite abelian groups.

\begin{corollary}\xlabel{cor:torus-tors-PD}
Let $T$ be a torus over $V$. The cup product pairings of
Corollary~\ref{cor:torus-ntors-PD} induce perfect pairings
\begin{equation}\eqlabel{eq:tors-tate-pairing}
H^q_c(V,T_\tors)\times H^{4-q}(V,\Tate(T'))\to\QmodZ
\end{equation}
and
\begin{equation}\eqlabel{eq:tate-tors-pairing}
H^q_c(V,\Tate(T'))\times H^{4-q}(V,T_\tors)\to\QmodZ
\end{equation}
(in the sense of Section~\ref{sec:pairings}) for any $q\in\Zz$.
\end{corollary}

\begin{proof}
Consider the pairings of sheaves
\begin{equation*}
{}_nT\times{}_nT'\to\Zz/n(2)\into\QmodZ(2)
\end{equation*}
constructed above for all $n\in\Nn$. It follows from the construction
that for any positive integers $m$ and $n$ the natural maps ${}_nT
\into{}_{mn}T$ and ${}_{mn}T'\onto{}_nT'$ are adjoint with respect to
these pairings. Hence the induced maps in cohomology are adjoint with
respect to the cup product pairings, and the statement is an
immediate consequence of Lemma~\ref{lem:limit-pairing} and
Corollary~\ref{cor:torus-ntors-PD}.
\end{proof}

Let $\pi\fcolon W\to V$ be a finite flat morphism of nonsingular
curves over~$k$. For any smooth group scheme $G$ over $V$ we have
trace maps
\begin{gather*}
\pi_*\fcolon H^q(W,G_W)\to H^q(V,G)\\
\intertext{and}
\pi_* \fcolon H^q_c(W,G_W)\to H^q_c(V,G)
\end{gather*}
(see \cite{SGA4}, XVII.6.3). For $M$ as in Theorem~\ref{th:PD}, these
maps are adjoint to the pullback maps with respect to the cup product
pairing. Namely, we have
$$\langle\pi_*\omega,\eta\rangle=\langle\omega,\pi^*\eta\rangle$$
for $\omega\in H^q_c(W,\pi^*M)$ and $\eta\in H^{4-q}(V,M^\du(2))$,
resp.\ for $\omega\in H^q(W,\pi^*M)$ and $\eta\in H^{4-q}_c
(V,M^\du(2))$ (see for example \cite{SGA4.5}, Dualit\'e).

\subsection{Lichtenbaum--Tate duality for curves over $p$-adic
  fields}
\xlabel{sec:l-t-duality}

\begin{theorem}[Lichtenbaum--Tate duality \cite{Li}]
 \xlabel{th:l-t-duality}
Let $C$ be a smooth projective curve over a $p$-adic field~$k$. For
every $q\in\Zz$ there is a natural non-degenerate pairing
\begin{equation}\eqlabel{eq:licht-pairing}
H^q(C,\Gg_m)\times H^{3-q}(C,\Gg_m)\to\QmodZ.
\end{equation}
The induced pairings
\begin{equation}\eqlabel{l-t-pairing0}
H^0(C,\Gg_m)\compl\times H^3(C,\Gg_m)\to\QmodZ
\end{equation}
and
\begin{equation}\eqlabel{l-t-pairing1}
H^1(C,\Gg_m)\compl\times H^2(C,\Gg_m)\to\QmodZ
\end{equation}
are perfect.
\end{theorem}

\begin{proof}
We may assume $C$ to be geometrically connected. Consider the
Hochschild--Serre spectral sequence
\begin{equation}\eqlabel{eq:hsss}
\hbox{$E_2^{ij}=H^i(k,\,H^j(\ol C,\Gg_m))$ $\Rightarrow$ $H^{i+j}
(C,\Gg_m)$.}
\end{equation}
Since $\textrm{scd}(k)=2$ and $H^j(\ol C,\Gg_m)=0$ for $j\ge2$, it
follows that $H^q(C,\Gg_m)=0$ for $q\ge4$. So only the cases $q=0$
and $q=1$ need to be considered. For $q=1$, our pairing is the
pairing $\Pic(C)\times\Br(C)\to\QmodZ$ constructed by Lichtenbaum in
\cite{Li}. He proves that $\Br(C)\to\Pic(C)^\du$ is an isomorphism,
where $\Pic(C)$ carries its natural locally compact topology (\emph
{loc.~cit.}, Thm.~4, p.~131). Since $\Pic(C)\to\wh{\Pic(C)}$ induces
an isomorphism $\bigl(\wh{\Pic(C)}\bigr)^\du\isoto\bigl(\Pic(C)\bigr)
^\du$, this implies perfectness of the pairing $\wh{\Pic(C)}\times
\Br(C)\to\QmodZ$, by Lemma~\ref{left-right}.

For $q=0$, the pairing \eqref{eq:licht-pairing} coincides with the
obvious pairing $k^*\times H^2(k,\Zz)\to\Br(k)=\QmodZ$ from local
class field theory, and the assertions are classical (it is the
special case $T=\Gg_m$, $q=0$ of Tate--Nakayama duality as treated in
Theorem~\ref{th:t-n-duality}). To see that $H^3(C,\Gg_m)=H^2(k,\Zz)$,
use the Hochschild--Serre spectral sequence~\eqref{eq:hsss} and the
fact that $H^2(k,\Pic_0(\capbar{C}))=0$ (see \cite{CG}, II.5.3).
\end{proof}

\begin{remarks}\xlabel{rem:licht-constr}
\item
For $q\ge2$, $H^q(C,\Gg_m)$ is a (discrete) abelian torsion group,
whereas for $q=0$ and $q=1$, it is an extension of a free abelian
group of finite rank by a compact $p$-adic Lie group.

\item
Lichtenbaum's construction of the pairing $\Pic(C)\times\Br(C)\to
\QmodZ$ is as follows. Suppose given a Weil divisor $D=\sum_Pn_p
\cdot P$ on $C$, representing a class $[D]\in\Pic(C)$, and a central
simple $k(C)$-algebra $A$ which is everywhere unramified on $C$,
therefore representing a class $[A]\in\Br(C)$. Let $A(P)$ denote the
residue central simple algebra over $k(P)$, for $P$ any closed point
on $C$. Then
$$\langle[D],[A]\rangle=\sum_Pn_P\cdot\inv_{k(P)}\!A(P),$$
where $\inv_K\fcolon\Br(K)\to\QmodZ$ is the invariant from local
class field theory, for $K$ any local field.

\item
For any abelian variety $A$ over a $p$-adic field $k$, with dual
abelian variety $A'$, Tate established a natural pairing $H^0(k,A)
\times H^1(k,A')\to\QmodZ$ and proved that it is perfect \cite{Tate}.
Lichtenbaum's pairing is closely related to Tate's pairing for $A:=
J$, the Jacobian of the curve $C$. In fact, Lichtenbaum's proof
proceeds by establishing this relation and then using perfectness of
Tate's pairing. His conclusion also uses a theorem of Roquette
\cite{Roq} (to which Lichtenbaum gives a new proof), to the end that
the order of $\ker\bigl(\Br(k)\to\Br(C)\bigr)$ is precisely the index
of $C$.
\end{remarks}

\begin{remark}
When $\pi\fcolon\tilde C\to C$ is a finite covering of smooth
projective curves, it follows from the construction that the trace
map $\pi_*\fcolon H^*(\tilde C,\Gg_m)\to H^*(C,\Gg_m)$ is adjoint to
the pullback map $\pi^*\fcolon H^*(C,\Gg_m)\to H^*(\tilde C,\Gg_m)$
with respect to Lichtenbaum's pairing.
\end{remark}

\subsection{Relating Poincar\'e duality to Lichtenbaum--Tate duality}
  \xlabel{sec:compare-p-l-t}

We will now check that Lichtenbaum's pairing \eqref{eq:licht-pairing}
is compatible with the cup product pairing for coefficients in the
roots of unity. Let $n\in\Nn$ and $q\in\Zz$, and consider the Kummer
exact sequence
\begin{equation}\eqlabel{eq:kummer}
  0\to H^q(C,\Gg_m)/n\labelto{\delta_n}H^{q+1}(C,\mu_n)\labelto
  {\iota_n}{}_nH^{q+1}(C,\Gg_m)\to0.
\end{equation}
We need the following observation.

\begin{lemma}\xlabel{lem:gm-bockstein-adj}
Let $C$ be a smooth projective curve over a $p$-adic field. Let $m$,
$n\in\Nn$.
\begin{theoremlist}
\item
The composite map
\begin{equation*}
H^q(C,\mu_m)\labelto{\iota_m}H^q(C,\Gg_m)\labelto{\delta_n}H^{q+1}
(C,\mu_n)
\end{equation*}
coincides with the boundary map
\begin{equation*}
\beta_{m,n}\fcolon H^q(C,\mu_m)\to H^{q+1}(C,\mu_n)
\end{equation*}
associated with the short exact sequence
\begin{equation*}
0\to\mu_n\to\mu_{mn}\labelto n\mu_m\to0.
\end{equation*}

\item
The map $\beta_{m,n}$ is adjoint to the map $\beta_{n,m}$ with
respect to the cup product pairings
\begin{gather*}
H^q(C,\mu_m)\times H^{4-q}(C,\mu_m)\to\QmodZ
\intertext{and}
H^{q+1}(C,\mu_n)\times H^{3-q}(C,\mu_n)\to\QmodZ.
\end{gather*}
\end{theoremlist}
\end{lemma}

\noindent
We leave the proof as an exercise to the reader.
\qed

\begin{proposition}\xlabel{prop:l-t-n}
Let $C$ be a smooth projective curve over a $p$-adic field. Let
$n\in\Nn$.
\begin{theoremlist}
\item
Lichtenbaum's pairing~\eqref{eq:licht-pairing} induces for every
$q\in\Zz$ a perfect pairing of finite groups
\begin{equation}\eqlabel{eq:licht-n-pairing}
H^q(C,\Gg_m)/n\times{}_nH^{3-q}(C,\Gg_m)\to\QmodZ.
\end{equation}
\item
The mappings $\delta_n$ and $\iota_n$ in the Kummer sequence
\eqref{eq:kummer} are adjoint to each other with respect to the cup
product pairings
\begin{equation}\eqlabel{eq:cup}
H^q(C,\mu_n)\times H^{4-q}(C,\mu_n)\to\QmodZ
\end{equation}
and the above pairings \eqref{eq:licht-n-pairing}.
\end{theoremlist}
\end{proposition}

\begin{proof}
\begin{prooflist}
\item
Apply Lemma~\ref{lem:torsion-perfect} to the pairings
\eqref{l-t-pairing0} and \eqref{l-t-pairing1} in
Theorem~\ref{th:l-t-duality}. Note that for $q=0$ and $q=1$, the
group $A:= H^q(C,\Gg_m)$ is of the type considered in
Lemma~\ref{lem:compl of loccpt}(b). Therefore, the torsion and
cotorsion of $\wh A$ and of $A$ are the same.

\item
We claim that
\begin{equation*}
\langle\delta_n\alpha,\omega\rangle=\langle\alpha,\iota_n\omega
\rangle
\end{equation*}
for $\alpha\in H^q(C,\Gg_m)/n$ and $\omega\in H^{3-q}(C,\mu_n)$. For
$q=0,\,3$ this is essentially a formal consequence of the
construction of Lichtenbaum's pairing, since the local class field
pairing
\begin{equation*}
H^0(k,\Gg_m)\times H^2(k,\Zz)\to H^2(k,\Gg_m)=\QmodZ
\end{equation*}
is the cup product.

For $q=1,\,2$ the situation is slightly more subtle. For $q=1$ we may
assume without loss of generality that $\alpha$ is the class $[P]$ of
a closed point $P \in C$. It is a standard property of the cycle map
that, given a diagram
\begin{equation*}
\xymatrix@C1ex{
P \ar[rr]^-{i} \ar[dr]_-{\pi} & & C \ar[dl]^-{\phi} \\
& \Spec k
}
\end{equation*}
we have
\begin{equation*}
\langle\delta_n[P],\omega\rangle=\pi_*i^*\omega
\end{equation*}
(cf.~\cite{SGA4.5}, Cycle~2.3.1).
We see from the construction of the pairing \eqref{eq:licht-pairing}
(cf.\ Remark~\ref{rem:licht-constr}.2) that the right hand side of
the equation coincides with $\langle[P],\iota_n\omega\rangle$, so we
have proved the case $q=1$.

The case $q=2$ can be reduced to the case $q=1$ by using that $H^2
(C,\Gg_m)$ is a torsion group: Indeed, $\alpha=\iota_m\eta$
(modulo~$n$) for some $m\in\Nn$ and $\eta\in H^2(C,\mu_m)$, so
Lemma~\ref{lem:gm-bockstein-adj} implies that
\begin{equation*}
\langle\delta_n\alpha,\,\omega\rangle=\langle\delta_n\comp\iota_m\,
\eta,\,\omega\rangle=\langle\eta,\,\delta_m\comp\iota_n\,\omega
\rangle,
\end{equation*}
and the case $q=1$ gives us that
\begin{equation*}
\langle\eta,\delta_m(\iota_n\omega)\rangle=\langle\iota_m\eta,\iota_n
\omega\rangle=\langle\alpha,\iota_n\omega\rangle.
\end{equation*}
\end{prooflist}
\end{proof}

Taking the inverse and direct limit of the Kummer exact sequence, we
get short exact sequences
\begin{equation*}
0 \to H^q(C,\Gg_m)\compl \labelto{\delta_\Tate} H^{q+1}(C,\wh\Zz(1))
\labelto{\iota_\Tate} \Tate\bigl(H^{q+1}(C,\Gg_m)\bigr) \to 0
\end{equation*}
of profinite groups and
\begin{equation*}
0 \leftarrow H^{3-q}(C,\Gg_m)_\tors \labelot{\iota_\infty} H^{3-q}
(C,\QmodZ(1)) \labelot {\delta_\infty} H^{2-q}(C,\Gg_m)\otimes\QmodZ
\leftarrow 0
\end{equation*}
of discrete torsion groups.

\begin{corollary}\xlabel{cor:l-t-duality-kummer}
Let $C$ be a smooth projective curve over a $p$-adic field. For every
$q\in\Zz$ we have the following diagrams with exact rows of
compatible perfect pairings into $\QmodZ$, which are induced by
Lichtenbaum's pairing~\eqref{eq:licht-pairing} and the cup product
pairing:
\begin{fitbigdiagram}
\begin{equation*}
\xymatrix@R0ex@C3ex{
0 \ar[r] & H^{q}(C,\Gg_m)\compl \ar[r]^-{\delta_{\Tate}} &
 H^{q+1}(C,\caphat{\Zz}(1)) \ar[r]^-{\iota_\Tate} &
 \Tate\left(H^{q+1}(C, \Gg_m)\right) \ar[r] & 0 \\
& \times & \times & \times & \\
0 & H^{3-q}(C, \Gg_m)_\tors \ar[l] \ar[dddr] &
 H^{3-q}(C, \QmodZ(1)) \ar[l]_-{\iota_\infty} \ar[ddd]&
 H^{2-q}(C, \Gg_m) \otimes \QmodZ \ar[l]_-{\delta_\infty} \ar[dddl] &
 0 \ar[l] \\ \\ \\
&  & \QmodZ & &}
\end{equation*}
\end{fitbigdiagram}
\end{corollary}

\begin{proof}
The compatibility of the pairings follows from
Proposition~\ref{prop:l-t-n}(b). By Theorem~\ref{th:l-t-duality}
(resp., Proposition~\ref{prop:l-t-n}(a)) and
Lemma~\ref{lem:limit-pairing}, the middle (resp., the left) column
pairing is perfect.  From Lemma~\ref{lem:gen-pairing-perfect}(b) and
Corollary~\ref{perf-iff-nondeg} it follows that the right column
pairing is perfect as well.
\end{proof}

\section{Cohomology of tori with compact supports}
   \xlabel{sec:cohcptsupp}

\subsection{Introduction}\xlabel{intro}

Let $C$ be a (not necessarily smooth) projective curve over an
arbitrary field~$F$, let $j\fcolon V\to C$ be the inclusion of an
open non-empty subset and let $i\fcolon Z\into C$ be the inclusion of
its reduced closed complement.

Let $T$ be a torus over $V$. Assume $T$ extends to $C$, i.e., we have
a torus $S$ over $C$ such that $T=S_V=j^*S$. We have two natural
possibilities of defining cohomology of $T$ with compact supports.
The first is the standard sheaf-theoretic definition
\begin{equation*}
H^q_c(V,T):= H^q(C,j_!T),
\end{equation*}
where $j_!T$ is the \'etale sheaf on $C$ characterised by the short
exact sequence
\begin{equation*}
0\to j_!T\to S\to i_*i^*S\to0.
\end{equation*}

The second, more geometric definition is
\begin{equation*}
H^q_{cc}(V,T):=H^q(C,\tilde T),
\end{equation*}
where $\tilde T$ is the \'etale sheaf on $C$ defined by the short
exact sequence
\begin{equation*}
0\to\tilde T\to S\to i_*S_Z\to0
\end{equation*}
and $S_Z$ is the pullback of $S$ to $Z$ (\ref{sec:tori}). However, if
$T$ is an arbitrary torus on $V$ which does not extend to a torus $S$
on $C$, we cannot define $\tilde T$ and $H^*_{cc}(V,T)$ in this way.

\subsection{Definitions and basic properties}

Let $F$ be a field and $V$ a (reduced) curve over $F$. An open
immersion $j\fcolon V\into C$ into a projective curve $C$ will be
called a \emph{good compactification} of $V$ if $V$ is dense in $C$
and $Z=C-V$ consists of nonsingular points of $C$. Every curve $V$
has a good compactification that is unique up to isomorphism.

\begin{definition}
\xlabel{def:hcc}
Let $j\fcolon V\to C$ be a good compactification of $V$, let $Z=C-V$
be the reduced complement and $i\fcolon Z\into C$ its inclusion in
$C$. Let $T$ be a torus on $V$.
\begin{itemize}
\item[(a)]
The \'etale sheaf $j_{!!}\Gg_m$ on $C$ is defined by the exact
sequence
\begin{equation}\eqlabel{eq:def j!!gm}
0\to j_{!!}\Gg_m\to\Gg_{m,C}\to i_*\Gg_{m,Z}\to0.
\end{equation}
\item[(b)]
The \'etale sheaf $j_{!!}T$ on $C$ is defined by $j_{!!}T:=
j_*\schX_*(T)\otimes j_{!!}\Gg_m$.
\item[(c)]
For $q\in\Zz$ we define $H^q_{cc}(V,T):=H^q(C,j_{!!}T)$.
\end{itemize}
\end{definition}

\begin{remarks}\xlabel{rem:comp-supp}
\item
The stalk of the sheaf $j_*\schX_*(T)$ at a geometric point $\ol P$
of $Z$ is the subgroup of $\schX_*(T)$ fixed by the monodromy action
at $\ol P$. In particular, if $T$ extends to a torus $S$ on all of
$C$, then $j_*\schX_*(T)=\schX_*(S)$, and thus $j_{!!}T$ coincides
with the sheaf $\tilde T$ considered in~\ref{intro} before.
(Tensoring the sequence \eqref{eq:def j!!gm} with $j_*\schX_*(T)=
\schX_*(S)$ gives the exact sequence $0\to j_{!!}T\to S\to i_*S_Z\to
1$.) At the other extreme, if no subtorus $\ne\{1\}$ of $T$ can be
extended to any larger open subcurve of $C$, then $j_{!!}T=j_!T$.

\item
The sheaf $j_{!!}\Gg_m$ appears under the name $\Gg_m(\mathfrak m)$
in \cite{SGA4}, XVIII.1.5.8, where $\mathfrak m$ is the ideal
defining the reduced closed subscheme $Z$ of $C$. In \cite{SGA4.5},
Arcata VI, the notation ${}_Z\Gg_m$ is used.
\end{remarks}

\begin{lemma}
Let $j\fcolon V\to C$ and $T$ be as above. We have a natural
injective map
\begin{equation*}
j_!T\to j_{!!}T
\end{equation*}
of sheaves on $C$ which is an isomorphism on torsion prime to $\ch
(F)$. The cokernel is a sheaf concentrated on $Z$, uniquely
divisible if $\ch(F)=0$.
\end{lemma}

\begin{proof}
The case $T=\Gg_m$ is clear, and it implies the general case since
$j_!T=j_*\schX_*(T)\otimes j_!\Gg_m$.
\end{proof}

\begin{corollary}\xlabel{cor:compare-compact}
Let $j\fcolon V\to C$ and $T$ be as above, and assume $\ch(F)=0$.
Then for $q\ne1$, the natural map
\begin{equation*}
H^q_c(V,T)\to H^q_{cc}(V,T)
\end{equation*}
is an isomorphism. For $q=1$ it is surjective, with a uniquely
divisible kernel.
\end{corollary}

\begin{proof}
Follows from the previous lemma,
together with the fact that $H^0_c(V,T)=H^0_{cc}(V,T)=0$ if $V$ is
connected and $V\ne C$.
\end{proof}

\begin{remark}
Since $(j_{!!}T)|_V=T$, there are natural maps $H^q_{cc}(V,T)\to H^q
(V,T)$,
and the composition
$$H^q_c(V,T)\to H^q_{cc}(V,T)\to H^q(V,T)$$
is the canonical map for the usual definition of cohomology with
compact supports.
\end{remark}

\begin{lemma}[Kummer sequence]
With notations as above we have, for any $n\in\Nn$ prime to $\ch(F)$
and any $q\in\Zz$, a short exact sequence
\begin{equation}\eqlabel{eq:kummer-c}
0\to H^q_{cc}(V,T)/n\to H^{q+1}_c(V,{}_nT)\to{}_nH^{q+1}_{cc}(V,T)
\to0.
\end{equation}
\end{lemma}

\begin{proof}
Clear from the exact sequence of sheaves $0\to j_!({}_nT)\to j_{!!}T
\labelto nj_{!!}T\to0$.
\end{proof}

The following lemma, together with its corollary, is obvious:

\begin{lemma}
Let $V$ be a curve over a field~$F$ and $T$ a torus over $V$. Let
$U\subset V$ be a dense open subset such that $Y=V-U$ consists of
regular points, and let $j'\fcolon U\to C$, $j\fcolon V\to C$ and
$i\fcolon Y\into C$ be the inclusions into a good compactification
$C$ of $V$. Then we have a short exact sequence of sheaves on $C$
\begin{equation*}
0\to j'_{!!}T_U\to j_{!!}T\to i_*T_Y\to0.
\end{equation*}
\end{lemma}

\begin{corollary}[Localisation]
\xlabel{lem:localisation}
With notations as above, we have a long exact sequence
\begin{equation}\eqlabel{eq:localisation}
\cdots\>H^q_{cc}(U,T_U)\labelto{j_*}H^q_{cc}(V,T)\to H^q(Y,T_Y)\>
\cdots
\end{equation}
where the second arrow is the composition $H^q_{cc}(V,T)\to H^q(V,T)
\labelto{i^*}H^q(Y,T_Y)$.
\end{corollary}

\begin{example}
When $V\ne\emptyset$ is open in a geometrically irreducible smooth
projective curve $C$ over $F$, and $Z=C-V$, we have exact sequences
\begin{equation}\eqlabel{pic-cc}
0\to F^*\to\bigoplus_{P\in Z}F(P)^*\to H^1_{cc}(V,\Gg_m)\to\Pic(C)
\to0
\end{equation}
and
\begin{equation}
0\to H^2_{cc}(V,\Gg_m)\to\Br(C)\to\bigoplus_{P\in Z}\Br F(P)\to
H^3_{cc}(V,\Gg_m)\to\cdots
\end{equation}
\end{example}

\begin{remarks}\xlabel{rem:cctrace-concrete}
\item
When $\pi\colon V'\to V$ is a finite covering of curves over $F$ and
$T$ is a torus on $V$, one defines pullback maps
$$\pi^*\fcolon H^q_{cc}(V,T)\to H^q_{cc}(V',T_{V'})$$
and trace maps
$$\pi_*\fcolon H^q_{cc}(V',T_{V'})\to H^q_{cc}(V,T)$$
in the obvious way.

\item
It is not hard to derive from the above a more explicit description
of the groups $H^1_{cc}(V, \Gg_m)$ and $H^2_{cc}(V, \Gg_m)$ when $V$
is smooth. To simplify notation we assume that $V$ is connected. As
before, let $V\into C$ be a smooth compactification and $Z=C-V$.

Denoting by $\Div(C)$ the group of divisors on $C$ and by $\divisors
\fcolon F(C)^*\to\Div(C)$ the divisor map, the group $H^1_{cc}
(V,\Gg_m)$ is the cokernel of the map
\begin{equation*}
F(C,Z)^*\labelto{\divisors}\Div(V),
\end{equation*}
where $F(C,Z)^*\subset F(C)^*$ is the subgroup consisting of all
rational functions on $C$ that take the value $1$ on all points of
$Z$ (in particular, they do not have zeroes or poles in $Z$). In
\cite{SGA4.5}, Arcata VI, the group $H^1_{cc}(V,\Gg_m)$ is denoted by
$\Pic_Z(C)$. In terms of invertible sheaves it can be described as
the group of isomorphism classes of pairs $(L,\tau)$, where $L$ is an
invertible sheaf on $C$ and $\tau\fcolon L_Z\isoto\sO_Z$ is a
trivialisation of $L$ at $Z$.

The group $H^2_{cc}(V,\Gg_m)$ is the subgroup of $\Br(C)$ of
unramified central simple $F(C)$-algebras $A$ for which the class of
the residue central simple algebra $A(P)$ in $\Br(F(P))$ is trivial
for every point $P\in Z$.
\end{remarks}

\subsection{Relations to generalised Jacobians and motivic
cohomology}
\xlabel{gendjacobs}

Now assume $C$ to be a geometrically connected smooth projective
curve over $F$ and $j\fcolon V\into C$ a dense open subset with
(reduced) complement $Z=C-V$. Let $\phi\fcolon C\to\Spec F$ be the
structure morphism. As remarked before, the sheaf $j_{!!}\Gg_m$
already appears in \cite{SGA4}, Exp.~XVIII, under the name $\Gg_m
(\mathfrak m)$, where $\mathfrak m$ is the ideal defining the reduced
closed subscheme $Z\subset C$. In \cite{SGA4}, XVIII.1.6.3, it was
observed that the higher direct image sheaf $R^1\phi_*j_{!!}\Gg_m$ is
representable by a group scheme locally of finite type over $F$
(actually, representable not just on the small \'etale site, where we
are working in the present paper, but already on the \emph{fppf}
site). This group scheme is denoted by $\Pic_{\mathfrak m,C/F}$ in
\emph{loc.~cit.}; it represents the \emph{fppf} sheaf associated to
the presheaf that sends an $F$-scheme $S$ to the set of isomorphism
classes of line bundles on $C\times S$ with a given trivialisation at
$Z\times S$. In fact, when $V\ne C$, we see from the
Hochschild--Serre spectral sequence that
\begin{equation*}
H^1_{cc}(V,\Gg_m)=\Pic_{\mathfrak m,C/F}(F).
\end{equation*}
The connected component of $\Pic_{\mathfrak m,C/F}$ containing zero
coincides with Rosenlicht's generalised Jacobian of $C$ with modulus
$\mathfrak m$ (see \cite{GACC}); it is an extension of the Jacobian
of the curve $C$ by a torus (compare the exact
sequence~\eqref{pic-cc}).

The cohomology groups $H^*_{cc}(V,\Gg_m)$ are (in degrees $\le1$)
closely related to motivic cohomology. The localisation exact
sequence for motivic cohomology $H^*_c(\hvar,\Zz(1))$ with compact
supports and coefficients in $\Zz(1)$ (see \cite{FV}, Sec.~9) and the
fact that $H^*_c(X,\Zz(1))=H^*(X,\Zz(1))=H^*_{\textnf{Zar}}(X,\Gg_m)$
when $X$ is smooth and proper (see \cite{Voe}, Cor.~3.4.3) give us
(for $V\ne C$) an exact sequence
\begin{equation*}
0\to F^*\to\Gamma(Z,\Gg_m)\to H^2_c(V,\Zz(1))\to\Pic(C)\to0
\end{equation*}
similar to \eqref{pic-cc}. So our group $H^1_{cc}(V,\Gg_m)$ should be
motivic, i.e., naturally isomorphic to $H^2_{c}(V,\Zz(1))$. We have
not actually tried to construct the map that should give this
isomorphism.

\subsection{Some properties over $p$-adic fields}
\xlabel{sec:bas-prop}

Let now $V$ be a smooth connected curve over a $p$-adic field~$k$ and
$T$ a torus over $V$. From the Kummer sequence and the finiteness of
the groups $H^q_c(V,{}_nT)$ we see that the groups $H^q_{cc}(V,T)/n$
and ${}_nH^q_{cc}(V,T)$ are finite. Therefore
\begin{equation*}
H^q_{cc}(V,T)\compl=\varprojlim_nH^q_{cc}(V,T)/n
\end{equation*}
for all $q\in\Zz$ (cf.\ Section~\ref{sec:abeliangps}).

\begin{lemma}\xlabel{semiabelian}
Let $k$ be a $p$-adic field and $G$ a commutative algebraic group
over $k$. Then the group $G(k)$ is residually finite.
\end{lemma}

\begin{proof}
Since $G(k)/n$ is finite for each $n\in\Nn$, we have to show that the
subgroup $D(G(k))$ of $G(k)$ is trivial, cf.\
Section~\ref{sec:abeliangps}. This is the case if $G$ is either
finite, or a connected linear group, or an abelian variety. From this
the general case follows, using Lemma~\ref{divisibleelements}.
\end{proof}

\begin{proposition}\xlabel{H1cc}
Let $T$ be any torus on $V$. Then the groups $H^0(V,T)$, $H^0_{cc}
(V,T)$ and $H^1_{cc}(V,T)$ are residually finite.
\end{proposition}

\begin{proof}
The first two cases are clear. There is a finite Galois covering
$W\to V$ such that the torus $T_W$ on $W$ splits. Writing $\Gamma$
for the Galois group of this covering, we have an exact sequence
$$0\to H^1(\Gamma,H^0_{cc}(W,T_W))\to H^1_{cc}(V,T)\to H^1_{cc}
(W,T_W).$$
The group $H^1_{cc}(W,T_W)$ is an extension of a free abelian group
of finite type by the group $G(k)$ of $k$-points of a semi-abelian
variety $G$ over $k$, see Section~\ref{gendjacobs}. Therefore, $H^1_
{cc}(W,T_W)$ is residually finite by Lemma~\ref{semiabelian}. Using
Lemma~\ref{divisibleelements}, it follows that $H^1_{cc}(V,T)$ is
residually finite as well.
\end{proof}

In contrast, the group $H^1(V,T)$ need not be residually finite if
$V$ is affine. For an example see~\ref{ex:degeneracy} below.

\section{Lichtenbaum--Tate duality for open curves}
\xlabel{sec:l-t-open-duality}

In the following, let $C$ be a smooth projective curve over a
$p$-adic field~$k$. Let $j\fcolon V\into C$ be a dense open subset,
and let $i\fcolon Z\into C$ be the reduced complement.

Consider the localisation exact sequences
\begin{equation}
\eqlabel{loc-c-gm}
\cdots\to H^q_{cc}(V,\Gg_m)\labelto{j_*}H^q(C,\Gg_m)\labelto{i^*}
H^q(Z,\Gg_m)\to\cdots
\end{equation}
and
\begin{equation}\eqlabel{loc-gm}
\cdots\ot H^r(V,\Gg_m)\labelot{j^*}H^r(C,\Gg_m)\labelot{i_*}H^r_Z
(C,\Gg_m)\ot\cdots
\end{equation}
Recall that $H^r_Z(C,\Gg_m)=H^{r-1}(Z,\Zz)$. We will define pairings
\begin{equation}\eqlabel{licht-c-pairing}
H^q_{cc}(V,\Gg_m)\times H^{3-q}(V,\Gg_m)\to\QmodZ
\end{equation}
such that sequences \eqref{loc-c-gm} and \eqref{loc-gm} (for $r=3-q$)
are adjoint to each other with respect to these pairings, the
Lichtenbaum--Tate pairings \eqref{eq:licht-pairing} and the
Tate--Nakayama pairings~\ref{th:t-n-duality} (for $T$ the direct
image of $\Gg_{m,Z}$ to the base $\Spec k$).

These generalised Lichtenbaum pairings will be constructed from the
cup product pairings \eqref{eq:tate-tors-pairing} and
\eqref{eq:tors-tate-pairing}, using the following two diagrams with
exact rows (for $T=T'=\Gg_m$):

\begin{fitbigdiagram}
\begin{equation}\eqlabel{kummer-limit-1}
\xymatrix@R0ex@C3ex{
0 \ar[r] & H^q_{cc}(V,T)\compl \ar[r]^-{\delta_\Tate} & H^{q+1}_c
 (V,\Tate(T)) \ar[r]^-{\iota_\Tate} & \Tate\left(H^{q+1}_{cc}(V,T)
 \right) \ar[r] & 0 \\
& & \times & & \\
0 & H^{3-q}(V,T')_\tors \ar[l] & H^{3-q}(V,T'_\tors) \ar[l]_-
 {\iota_\infty} \ar[ddd] & H^{2-q}(V,T')\otimes\QmodZ \ar[l]_-
 {\delta_\infty} & 0 \ar[l] \\
\\ \\
& & \QmodZ & &}
\end{equation}
\end{fitbigdiagram}
and
\begin{fitbigdiagram}
\begin{equation}\eqlabel{kummer-limit-2}
\xymatrix@R0ex@C3ex{
0 \ar[r] & H^{q-1}_{cc}(V,T)\otimes\QmodZ \ar[r]^-{\delta_\infty} &
 H^q_c(V,T_\tors) \ar[r]^-{\iota_\infty} & H^q_{cc}(V,T)_\tors \ar[r]
 & 0 \\
& & \times & & \\
0 & \Tate\left(H^{4-q}(V,T')\right) \ar[l] & H^{4-q}(V,\Tate(T'))
 \ar[l]_-{\iota_\Tate} \ar[ddd] & H^{3-q}(V,T')\compl \ar[l]_-
 {\delta_{\Tate}} & 0 \ar[l] \\
\\ \\
& & \QmodZ & &}
\end{equation}
\end{fitbigdiagram}
The rows of these diagrams are obtained from the Kummer sequences
\eqref{eq:kummer} and \eqref{eq:kummer-c} by taking inverse resp.\
direct limits. Note that the pairings in the middle columns are
perfect (Corollary~\ref{cor:torus-tors-PD}). We will denote the
composite maps
$$H^q_{(cc)}(V,T)\to H^q_{(cc)}(V,T)\compl\labelto{\delta_\Tate}
H^{q+1}_{(c)}(V,\Tate(T))$$
by $\delta_\Tate$ as well.

\begin{lemma}
\xlabel{lem:tors-coh}
Let $V$ be as above, and let $T$ be a torus on $V$.
\begin{itemize}
\item[\textrm(a)]
For $q\ge2$, the groups $H^q_{cc}(V,T)$ and $H^q(V,T)$ are torsion.
In particular, $H^q_{cc}(V,T)\otimes\QmodZ=H^q(V,T)\otimes\QmodZ=0$.
\item[\textrm(b)]
For $q\ge3$, the maps
\begin{align*}
H^q_c(V,T_\tors)&\labelto{\iota_\infty}H^q_{cc}(V,T) \\
\intertext{and}
H^q(V,T_\tors)&\labelto{\iota_\infty}H^q(V,T)
\end{align*}
are isomorphisms.
\item[\textrm(c)]
$H^q(V,T)=H^q_{cc}(V,T)=0$ for $q\ge4$.
\end{itemize}
\end{lemma}

\begin{proof}
For any \'etale sheaf $F$ on a curve $X$ over a field, the groups
$H^q(X,F)$ are torsion for $q\ge2$ (see~\cite{Sch} pp.\ 85-86). This
proves (a), and (a) implies (b) by the Kummer exact sequences.

It follows from Poincar\'e duality (see
Corollary~\ref{cor:torus-tors-PD}) and from (b) that for (c) it is
sufficient to show $H^0(V,\Tate(T))=H^0_c(V,\Tate(T))=0$. Since $T$
is split by a finite Galois covering $W\to V$, and since $H^0_{(c)}
(V,\Tate(T))$ embeds into $H^0_{(c)}(W,\Tate(T))$, we can assume $T=
\Gg_m$. In this case, both assertions are clear, since $\Tate(K^*)=
\varprojlim_n\mu_n(K)=0$ for any $p$-adic field~$K$.
\end{proof}

We now return to the case of split tori ($T=T'=\Gg_m$):

\begin{lemma}
\xlabel{lem:gm-orthogonal}
Let $V$ be as above. With respect to the cup product pairing, we
have for all $q\in\Zz$:
\begin{itemize}
\item[\textrm{(a)}]
The image of
\begin{equation*}
\delta_\Tate\fcolon H^{q-1}(V,\Gg_m)\compl\to H^q(V,\wh{\Zz}(1))
\end{equation*}
is orthogonal to the image of
\begin{equation*}
\delta_\infty\fcolon H^{3-q}_{cc}(V,\Gg_m)\otimes\QmodZ\to H^{4-q}_c
(V,\QmodZ(1)).
\end{equation*}
\item[\textrm{(b)}]
The image of
\begin{equation*}
\delta_\Tate\fcolon H^{q-1}_{cc}(V,\Gg_m)\compl\to H^q_c
(V,\wh{\Zz}(1))
\end{equation*}
is orthogonal to the image of
\begin{equation*}
\delta_\infty\fcolon H^{3-q}(V,\Gg_m)\otimes\QmodZ\to H^{4-q}
(V,\QmodZ(1)).
\end{equation*}
\end{itemize}
\end{lemma}

\begin{proof}
It follows from Lemma~\ref{lem:tors-coh} that for $q\le1$ or $q\ge4$,
the source of $\delta_\infty$ is zero, and that for $q=3$, the source
of $\delta_\Tate$ contains a dense torsion subgroup,
hence (a) and (b) hold when $q\ne2$. (In fact, it can be checked that
for $q=3$ the source of $\delta_\Tate$ is finite).
Remains $q=2$. Both assertions
follow once we show for any $n\in\Nn$ that the images of the Kummer
maps
$$\delta_n\fcolon H^1(V,\Gg_m)\to H^2(V,\mu_n)$$
and
$$\delta_n\fcolon H^1_{cc}(V,\Gg_m)\to H^2_c(V,\mu_n)$$
are orthogonal to each other. For $V=C$, this is true by
Proposition~\ref{prop:l-t-n}(b). The case of general $V$ follows from
this, since the restriction map $H^1(C,\Gg_m)\to H^1(V,\Gg_m)$ is
surjective and the canonical map $H^2_c(V,\mu_n)\to H^2(C,\mu_n)$ is
adjoint to the restriction map $H^2(C,\mu_n)\to H^2(V,\mu_n)$ with
respect to the cup product pairing.
\end{proof}

\begin{lemma}
\xlabel{lem:gm-inf-bockstein-adj}
Let $V$ be as above. The composite mapping
\begin{equation*}
\delta_\Tate\comp\iota_\infty\fcolon H^q_c(V,\QmodZ(1))\to H^{q+1}_c
(V,\wh{\Zz}(1))
\end{equation*}
is adjoint to the composite mapping
\begin{equation*}
\delta_\Tate\comp\iota_\infty\fcolon H^{3-q}(V,\QmodZ(1))\to H^{4-q}
(V,\wh{\Zz}(1))
\end{equation*}
with respect to the cup product pairings \eqref{eq:tors-tate-pairing}
and \eqref{eq:tate-tors-pairing}.
\end{lemma}

\begin{proof}
This follows from Lemma~\ref{lem:gm-bockstein-adj}
by taking $m$, $n\to\infty$.
\end{proof}

\begin{proposition}
Let $V$ be as above. There is a unique family of pairings
\begin{equation}\eqlabel{open-l-t-pairing}
\langle\hvar,\hvar\rangle\scolon H^q_{cc}(V,\Gg_m)\times H^{3-q}
(V,\Gg_m)\to\QmodZ
\end{equation}
($q\in\Zz$) which is compatible
\begin{itemize}
\item[\firstcondition]
with the Tate--Nakayama pairing~\ref{th:t-n-duality} (for split tori
over $p$-adic fields) and the Lichtenbaum--Tate
pairing~\ref{th:l-t-duality} for nonsingular projective curves, via
the localisation exact sequences \eqref{loc-c-gm} and \eqref{loc-gm},
and
\item[\secondcondition]
with the cup product pairing~\ref{th:PD} of torsion sheaves on $V$,
via the Kummer exact sequences.
\end{itemize}
\end{proposition}

\noindent
Recall (Example~\ref{ex:induced-pairings}) that the pairings
\eqref{open-l-t-pairing} induce pairings between the first and third
groups in the exact sequences \eqref{kummer-limit-1},
\eqref{kummer-limit-2}. By \secondcondition\ we mean that these exact
sequences are compatible with these pairings and with the already
constructed pairings between the middle groups.

\begin{proof}
We define the pairing by
\begin{equation*}
\langle x,\iota_\infty y\rangle:=\langle\delta_\Tate x,y\rangle
\end{equation*}
if $q=0,1$, and by
\begin{equation*}
\langle\iota_\infty x,y\rangle:=\langle x,\delta_\Tate y\rangle
\end{equation*}
if $q=2,3$. This gives well-defined pairings by
Lemmas~\ref{lem:tors-coh} and~\ref{lem:gm-orthogonal} and by the
Kummer exact sequences \eqref{kummer-limit-1},
\eqref{kummer-limit-2}. Now \secondcondition\ follows from the
construction and Lemma~\ref{lem:gm-inf-bockstein-adj}.
Compatibility \firstcondition\ follows from
Proposition~\ref{prop:l-t-n}(b) and the fact that the localisation
exact sequences with coefficients in $\Gg_m$ and coefficients in
$\mu_n$ are compatible with the Kummer exact sequences for all
$n\in\Nn$.
\end{proof}

\begin{theorem}[Lichtenbaum--Tate duality for open curves]
\xlabel{th:l-t-open-duality}
Let $V$ be a nonsingular curve over a $p$-adic field. The pairing
\begin{equation*}
H^q_{cc}(V,\Gg_m)\times H^{3-q}(V,\Gg_m)\to\QmodZ
\end{equation*}
is non-degenerate on the left for $q\ne3$, and non-degenerate on the
right for $q\ne2$. The induced pairings
$$H^0_{cc}(V,\Gg_m)\compl\times\,H^3(V,\Gg_m)\to\QmodZ,$$
$$H^1_{cc}(V,\Gg_m)\compl\times\,H^2(V,\Gg_m)\to\QmodZ$$
and
$$H^2_{cc}(V,\Gg_m)\times H^1(V,\Gg_m)\compl\to\QmodZ$$
are perfect.
\end{theorem}

\begin{proof}
Using Corollary~\ref{perf-iff-nondeg}, the second assertion about the
induced pairings follows from the first one and from the exact
sequences \eqref{kummer-limit-1}, \eqref{kummer-limit-2}. The first
assertion follows from the localisation exact sequences
\eqref{loc-c-gm}, \eqref{loc-gm} by Lemma~\ref{5lemma}, using the
following observation.
\end{proof}

\begin{lemma}
Let $r\in\Zz$. The maps $H^r(C,\Gg_m)\labelto{i^*}H^r(Z,\Gg_m)$ and
$H^{3-r}(C,\Gg_m)\labelot{i_*}H^{3-r}_Z(C,\Gg_m)$ are adjoint to each
other with respect to the usual pairings. For $r\ne2$, we have
\begin{theoremlist}
\item
$\ker(i_*)^\bot=\im(i^*)$
\item
$\ker(i^*)^\bot =\im(i_*)$
\end{theoremlist}
with respect to these pairings. For $r=2$, the group $\ker(i^*)^\bot/
\im(i_*)$ is divisible.
\end{lemma}

\begin{proof}
For $r=1$ resp.\ $r=3$, we have $H^r(Z,\Gg_m)=0=H^{3-r}_Z(C,\Gg_m)$,
so (a) is trivial and (b) is equivalent to the nondegeneracy on the
right of the Tate--Lichtenbaum pairing $H^r(C,\Gg_m)\times H^{3-r}
(C,\Gg_m)\to\QmodZ$. For $r=0$, the two maps are
$$i^*=\incl\fcolon k^*\to\bigoplus_{z\in Z}k(z)^*$$
and
$$i_*=\cor\fcolon\bigoplus_{z\in Z}H^2(k(z),\Zz)\to H^2(k,\Zz),$$
so both assertions follow from Nakayama-Tate duality for tori over
$k$. For $r=2$ we use that $H^{1}_Z(C,\Gg_m)=\bigoplus_{z\in Z}\Zz$,
$H^{2}_Z(C,\caphat{\Zz}(1))=\bigoplus_{z\in Z}\caphat{\Zz}$ and $H^1
(C,\Gg_m)$ injects into $H^{2}(C,\caphat{\Zz}(1))$. The statement for
$r=2$ now follows easily from Poincar\'e duality with torsion
coefficients and the fact that $\caphat{\Zz}/\Zz$ is divisible.
\end{proof}

\begin{remarks}\xlabel{rems:open-l-t-pairing}
\item
If $V$ is connected and affine (i.e., $V\ne C$), then for $q=0$ we
have the zero pairing: $H^0_{cc}(V,\Gg_m)=0=H^3(V,\Gg_m)$.

\item
The (completed) pairing
\begin{equation*}
H^3_{cc}(V,\Gg_m)\times H^0(V,\Gg_m)\compl\to\QmodZ
\end{equation*}
is non-degenerate on the right.

\item
{}From the explicit descriptions of $H^q_{cc}(V,\Gg_m)$ and $H^q
(V,\Gg_m)$ for $q=1,\,2$ (in terms of divisors and central simple
algebras, cf.\ Remark~\ref{rem:cctrace-concrete}.2), one can give
more concrete descriptions of our pairings in the spirit of
Lichtenbaum's construction (cf.\ Remark~\ref{rem:licht-constr}).
\end{remarks}

The following example shows that the conditions $q\ne3$ resp.\
$q\ne2$ in the theorem cannot be avoided:

\begin{example}\xlabel{ex:degeneracy}
Assume $V=C-\{P,Q\}$ with $P$, $Q\in C(k)$, let $x=[P]-[Q]\in\Pic_0
(C)$, and assume that $x$ generates an infinite subgroup $\langle x
\rangle$ of $\Pic_0(C)$. Then $k[V]^*=k^*$. Let $G:=\ol{\langle x
\rangle}$, and consider the two exact sequences
$$\Br(C)\labelto{i^*}\Br(k)\oplus\Br(k)\labelto\partial H^3_{cc}
(V,\Gg_m)\to H^3(C,\Gg_m)$$
and
$$\Pic(C)\ot\Zz\oplus\Zz\labelot0k[V]^*\ot k^*$$
(which are adjoint to each other). For all but finitely many primes
$q$, the group $G$ is $q$-divisible.
Choose one such prime $q$. Let $\beta\in\Br(k)$ be a class of
exponent $q$, and put $\alpha=\partial(\beta,0)$. Then $\alpha$ lies
in the left kernel of
$$H^3_{cc}(V,\Gg_m)\times H^0(V,\Gg_m)\to\QmodZ.$$
But $\alpha\ne0$. Indeed, otherwise there would be $\gamma\in\Br(C)$
with $(\beta,0)=i^*(\gamma)$. But then $\langle\gamma,\hvar\rangle$
would be a (continuous) character on $G$ which sends $x$ to $\beta$,
hence which is annihilated by $q$. This would contradict the fact
that $G$ is $q$-divisible.

This shows that in general the pairing $H^3_{cc}(V,\Gg_m)\times H^0
(V,\Gg_m)\to\QmodZ$ (and therefore also its completion $H^3_{cc}
(V,\Gg_m)\times H^0(V,\Gg_m)\compl\to\QmodZ$) has a non-trivial left
kernel.

The subgroup $G/\langle x\rangle$ of $\Pic(V)$ contains a subgroup
isomorphic to $\Zz_p/\Zz$, and hence its Tate module is $\ne0$.
Therefore $\Pic(V)$ has a non-zero Tate module. Since in the pairing
$$\Bigl(H^2_{cc}(V,\Gg_m)\otimes\QmodZ\Bigr)\times\Tate
\bigl(H^1(V,\Gg_m)\bigr)\to\QmodZ$$
the left hand group is zero ($H^2_{cc}(V,\Gg_m)$ is a torsion group),
this shows that this pairing has a non-zero right kernel.

Also, $G/\langle x\rangle$ contains non-zero divisible elements,
and hence $\Pic(V)$ is not residually finite. Therefore, the
(non-completed) pairing
\begin{equation*}
H^2_{cc}(V,\Gg_m)\times H^1(V,\Gg_m)\to\QmodZ
\end{equation*}
has a non-zero right kernel.
\end{example}

\begin{corollary}
\xlabel{cor:l-t-duality-complement}
Let $V$ be as above, let $q\in\Zz$.
\begin{itemize}
\item[\textrm(a)]
The pairing
\begin{equation*}
\Tate\bigl(H^q_{cc}(V,\Gg_m)\bigr)\times\Bigl(H^{3-q}(V,\Gg_m)\otimes
\QmodZ\Bigr)\to\QmodZ
\end{equation*}
is perfect if $q\ne3$ (and always non-degenerate on the right).
\item[\textrm(b)]
The pairing
\begin{equation*}
\Bigl(H^q_{cc}(V,\Gg_m)\otimes\QmodZ\Bigr)\times\Tate\bigl(H^{3-q}
(V,\Gg_m)\bigr)\to\QmodZ
\end{equation*}
is perfect if $q\ne2$ (and always non-degenerate on the left).
\end{itemize}
\end{corollary}

\begin{proof}
  Recall that for the pairings under consideration
  perfect is equivalent to non-degenerate
  (Corollary~\ref{perf-iff-nondeg}).  By
  Lemma~\ref{lem:gen-pairing-perfect}, this follows from sequences
  \eqref{kummer-limit-1} for~(a) (resp.\ \eqref{kummer-limit-2}
  for~(b)), using Theorem~\ref{th:l-t-open-duality}.
\end{proof}

\section{Duality for general tori}\xlabel{sec:constr}

Let $V$ be a non-singular curve over the $p$-adic field~$k$, and let
$T$ be an arbitrary torus on $V$. Recall that $T'$ denote the dual
torus of $T$. We are going to construct pairings
\begin{equation*}
H^q_{cc}(V,T)\times H^{3-q}(V,T')\to\QmodZ
\end{equation*}
($q\in\Zz$) and to investigate their duality properties. For this, we
keep referring to the Kummer exact sequences \eqref{kummer-limit-1}
and \eqref{kummer-limit-2}, together with the perfect pairings of
their middle columns.

\begin{lemma}\xlabel{lem:orthogonal}
Let $V$ and $T$ be as above.
\begin{itemize}
\item[\textrm(a)]
The image of
\begin{equation*}
\delta_\Tate\fcolon H^{q-1}_{cc}(V,T)\compl\to H^{q}_{c}(V,\Tate(T))
\end{equation*}
is orthogonal to the image of
\begin{equation*}
\delta_\infty \fcolon
H^{3-q}(V,T')\otimes\QmodZ \to H^{4-q}(V, T'_\tors).
\end{equation*}
\item[\textrm(b)]
The image of
\begin{equation*}
\delta_\Tate \fcolon H^{3-q}(V,T')\compl \to H^{4-q}(V,\Tate(T'))
\end{equation*}
is orthogonal to the image of
\begin{equation*}
\delta_\infty \fcolon
H^{q-1}_{cc}(V,T)\otimes\QmodZ \to H^q_{c}(V, T_\tors).
\end{equation*}
\end{itemize}
\end{lemma}

\begin{proof}
(a)
Let $\pi\fcolon W\to V$ be a finite \'etale map such that $T$ splits
over $W$. Consider the diagram
\begin{equation*}
\xymatrix@C0ex{
H^{q-1}_{cc}(V,T)\compl \ar[d]_-{\pi^*} & \times & \Bigl(H^{3-q}(V,T')
 \otimes\QmodZ\Bigr) & \to & \QmodZ \\
H^{q-1}_{cc}(W,T_W)\compl & \times & \Bigl(H^{3-q}(W,T'_W)\otimes
 \QmodZ\Bigr) \ar[u]^-{\pi_*} & \to & \QmodZ,}
\end{equation*}
where the pairings in both rows are defined by
$$(\alpha,\beta)\mapsto\delta_\Tate(\alpha)\cup\delta_\infty
(\beta).$$
The maps $\pi^*$ and $\pi_*$ are compatible with these
pairings.  Since $T_W$ is split, the lower pairing is zero by
Lemma~\ref{lem:gm-orthogonal}. Since
\begin{equation*}
\pi_*\comp\pi^*\fcolon H^{3-q}(V,T')\otimes\QmodZ\to H^{3-q}(V,T')
\otimes\QmodZ
\end{equation*}
is multiplication by the degree of $\pi$, and since the group
$H^{3-q}(V,T')\otimes\QmodZ$ is divisible, we see that the map
$\pi_*$ in the diagram is surjective. So the upper pairing is zero
as well.

The proof of (b) goes as the proof of (a).
\end{proof}

\begin{lemma}\xlabel{lem:inf-bockstein-adj}
Let $V$ and $T$ be as above. The composite mapping
\begin{equation*}
\delta_\Tate\comp\iota_\infty\fcolon H^q_c(V,T_\tors)\to H^{q+1}_c
(V,\Tate(T))
\end{equation*}
is adjoint to the composite mapping
\begin{equation*}
\delta_\Tate\comp\iota_\infty\fcolon H^{3-q}(V,T'_\tors)\to H^{4-q}
(V,\Tate(T'))
\end{equation*}
with respect to the cup product pairings \eqref{eq:tors-tate-pairing}
and \eqref{eq:tate-tors-pairing}.
\end{lemma}

\begin{proof}
Compare Lemma~\ref{lem:gm-inf-bockstein-adj}.
\end{proof}

\begin{proposition}\xlabel{prop:uniquepairing}
Let $V$ and $T$ be as above. For each $q\in\Zz$ there is a unique
pairing
\begin{equation}\eqlabel{eq:uniquepairing}
H^q_{cc}(V,T)\times H^{3-q}(V,T')\to\QmodZ
\end{equation}
which is compatible with the cup product pairings \eqref
{eq:tors-tate-pairing} and \eqref{eq:tate-tors-pairing} through the
Kummer sequences.
\end{proposition}

\begin{proof}
We define the pairing by
\begin{equation*}
\langle x,\iota_\infty y\rangle:=\langle\delta_\Tate x,y\rangle
\end{equation*}
if $q=0,1$ and by
\begin{equation*}
\langle\iota_\infty x,y\rangle:=\langle x,\delta_\Tate y\rangle
\end{equation*}
if $q=2,3$. This gives a well-defined pairing by
Lemma~\ref{lem:tors-coh} and Lemma~\ref{lem:orthogonal}. The
compatibility with the cup product pairings through the Kummer
sequences follows from the construction and
Lemma~\ref{lem:inf-bockstein-adj}.
\end{proof}

\begin{remark}
When $\pi\colon W\to V$ is a finite covering of smooth curves over
$k$, the pairings just constructed are compatible with the pullback
and trace maps induced by $\pi$:
$$\langle\alpha,\,\pi_*\gamma\rangle_V=\langle\pi^*\alpha,\,\gamma
\rangle_W$$
holds for $\alpha\in H^q(V,T)$ and $\gamma\in H^{3-q}_{cc}(W,T'_W)$,
resp.\ for $\alpha\in H^q_{cc}(V,T)$ and $\gamma\in H^{3-q}(W,T'_W)$.
\end{remark}

Proposition~\ref{prop:uniquepairing} allows us to fill in our
diagrams of pairings of Kummer sequences to get for every $q\in\Zz$
the following diagrams of compatible pairings, with perfect pairings
in the middle columns. Recall that $H^q(V,T)=H^q_{cc}(V,T)=0$ for
$q<0$ or $q>3$.

\begin{fitbigdiagram}
\begin{equation}\eqlabel{kummer-limit-3}
\xymatrix@R0ex@C3ex{
0 \ar[r] & H^q_{cc}(V,T)\compl \ar[r]^-{\delta_\Tate} & H^{q+1}_c
 (V,\Tate(T)) \ar[r]^-{\iota_\Tate} & \Tate\left(H^{q+1}_{cc}(V,T)
 \right) \ar[r] & 0 \\
& \times & \times & \times & \\
0 & H^{3-q}(V,T')_\tors \ar[l] \ar[dddr] & H^{3-q}(V,T'_\tors)
 \ar[l]_-{\iota_\infty} \ar[ddd] & H^{2-q}(V,T')\otimes\QmodZ
  \ar[l]_-{\delta_\infty} \ar[dddl] & 0 \ar[l] \\
\\ \\
& & \QmodZ}
\end{equation}
\end{fitbigdiagram}

\begin{fitbigdiagram}
\begin{equation}\eqlabel{kummer-limit-4}
\xymatrix@R0ex@C3ex{
0 \ar[r] & H^{q-1}_{cc}(V,T)\otimes\QmodZ \ar[r]^-{\delta_\infty} &
 H^q_c(V,T_\tors) \ar[r]^-{\iota_\infty} & H^q_{cc}(V,T)_\tors \ar[r]
 & 0 \\
& \times & \times & \times & \\
0 & \Tate\left(H^{4-q}(V,T')\right) \ar[l] \ar[dddr] & H^{4-q}
 (V,\Tate(T')) \ar[l]_-{\iota_\Tate} \ar[ddd] & H^{3-q}(V,T')\compl
 \ar[l]_-{\delta_\Tate} \ar[dddl] & 0 \ar[l] \\
\\ \\
& & \QmodZ}
\end{equation}
\end{fitbigdiagram}

\begin{proposition}\xlabel{prop:1ststep}
Let $V$ and $T$ be as above, let $q\in\Zz$.
\begin{itemize}
\item[\textrm(a)]
The pairing
\begin{equation*}
\Tate\bigl(H^q_{cc}(V,T)\bigr)\times\Bigl(H^{3-q}(V,T')\otimes\QmodZ
\Bigr)\to\QmodZ
\end{equation*}
is perfect if $q\ne3$ (and always non-degenerate on the right).
\item[\textrm(b)]
The pairing
\begin{equation*}
\Bigl(H^q_{cc}(V,T)\otimes\QmodZ\Bigr)\times\Tate\bigl(H^{3-q}(V,T')
\bigr)\to\QmodZ
\end{equation*}
is perfect if $q\ne2$ (and always non-degenerate on the left).
\end{itemize}
\end{proposition}

\begin{proof}
We use the diagrams \eqref{kummer-limit-3}, \eqref{kummer-limit-4}.
The kernels on the $\hvar\otimes\QmodZ$ side are trivial by
Lemma~\ref{lem:gen-pairing-perfect}. In order to prove the statement
on the $\Tate$ side, we lift to a trivialising cover $\pi\fcolon
W\to V$ for the torus $T$. In case (a) we get the diagram
\begin{equation*}
\xymatrix@C0ex{
\Tate\bigl(H^q_{cc}(V,T)\bigr) \ar[d]_-{\pi^*} & \times &
 \Bigl(H^{3-q}(V,T')\otimes\QmodZ\Bigr) & \to & \QmodZ \\
\Tate\bigl(H^q_{cc}(W,T_W)\bigr) & \times & \Bigl(H^{3-q}(W,T'_W)
 \otimes\QmodZ\Bigr) \ar[u]^-{\pi_*} & \to & \QmodZ.}
\end{equation*}
Since $T_W$ splits, the lower pairing is non-degenerate for $q\ne3$,
by Corollary~\ref{cor:l-t-duality-complement}. Since
$$\pi_*\comp\pi^*\fcolon\Tate\bigl(H^q_{cc}(V,T)\bigr)\to\Tate
\bigl(H^q_{cc}(V,T)\bigr)$$
is multiplication by $\deg(\pi)$, and $\Tate\bigl(H^q_{cc}(V,T)
\bigr)$ is torsion-free, it follows that the map $\pi^*$ in the above
diagram is injective. So for $q\ne3$ the upper pairing is
non-degenerate as well. The proof of the remaining part of (b) is
exactly the same.
\end{proof}

\begin{remark}
Observe that the above, together with Lemma~\ref{lem:tors-coh},
implies that the Tate modules of $H^0(V,T)$, $H^0_{cc}(V,T)$ and
$H^1_{cc}(V,T)$ are zero. This can also be derived directly from the
basic properties of the groups involved
(cf.~Section~\ref{sec:bas-prop}). Note the contrast with the
situation for $H^1(V,T)$ (cf. Example~\ref{ex:degeneracy}).
\end{remark}

\begin{corollary}\xlabel{cor:2ndstep}
Let $V$ and $T$ be as above, let $q\in\Zz$.
\begin{itemize}
\item[\textrm(a)]
The pairing
\begin{equation*}
H^q_{cc}(V,T)\compl\times H^{3-q}(V,T')_\tors\to\QmodZ
\end{equation*}
is perfect if $q\ne2$ (and always non-degenerate on the left).
\item[\textrm(b)]
The pairing
\begin{equation*}
H^q_{cc}(V,T)_\tors\times H^{3-q}(V,T')\compl\to\QmodZ
\end{equation*}
is perfect if $q\ne3$ (and always non-degenerate on the right).
\end{itemize}
\end{corollary}

\begin{proof}
Immediate from Proposition~\ref{prop:1ststep} and the Kummer exact
sequences \eqref{kummer-limit-3} (for (a)) and \eqref{kummer-limit-4}
(for (b)), by Lemma~\ref{lem:gen-pairing-perfect}.
\end{proof}

Now we can state and prove our main theorem:

\begin{theorem}\xlabel{th:main}
Let $V$ be a non-singular curve over a $p$-adic field, and let $T$ be
a torus over $V$. The natural pairing
\begin{equation*}
H^q_{cc}(V,T)\times H^{3-q}(V,T')\to\QmodZ
\end{equation*}
is non-degenerate on the left if $q\ne3$, and non-degenerate on the
right if $q\ne2$. The induced pairings
$$H^0_{cc}(V,T)\compl\times H^3(V,T')\to\QmodZ,$$
$$H^1_{cc}(V,T)\compl\times H^2(V,T')\to\QmodZ$$
and
$$H^2_{cc}(V,T)\times H^1(V,T')\compl\to\QmodZ$$
are perfect,
while the pairing
$$H^3_{cc}(V,T)\times H^0(V,T')\compl\to\QmodZ$$
is non-degenerate on the right.
\end{theorem}

\begin{proof}
Corollary~\ref{cor:2ndstep} already contains the statements about the
induced pairings. The first part of the theorem follows from these
facts, using that each of the groups $H^0_{cc}(V,T)$, $H^0(V,T)$ and
$H^1_{cc}(V,T)$ is residually finite (see Section~\ref{sec:bas-prop}).
\end{proof}

\begin{remark}
The comparison between $H^*_{cc}(V,T)$ and $H^*_c(V,T)$
(Lemma~\ref{cor:compare-compact}) gives a similar result for the
groups $H^*_c(V,T)$, which is slightly weaker in degree $q=1$.
\end{remark}

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\end{document}

