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%% On obstructions to the Hasse principle
%% Per Salberger
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 \markboth{\qquad On obstructions to the Hasse principle \hfill}{\hfill
Per Salberger \qquad}

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\title{On obstructions to the Hasse principle}
\author{Per Salberger}
\date{\em to Sir Peter Swinnerton-Dyer}
\begin{document}
\maketitle

\addcontentsline{toc}{chapter}{Per Salberger, On obstructions to the Hasse
principle}
% \tableofcontents

\section*{Introduction} 

A basic problem in arithmetic geometry is to decide if a variety defined
over a number field $k$ has a $k$-rational point. This is only possible 
if there is a $k_v$-point on the variety for each completion $k_v$ of $k$. 
It remains to decide if there is a $k$-point on a variety with a 
$k_v$-point at each place $v$ of $k$. The first positive results were obtained
by Hasse for quadrics and varieties defined by means of certain norm forms.
A class of varieties, therefore, is
said to satisfy the Hasse principle if
each variety in the class has a $k$-point as soon as it has $k_v$-points
for all places $v$. The corresponding property for the 
smooth locus is called the smooth Hasse principle. It is also natural to
ask if weak approximation holds. This means that the set of $k$-points is
dense in the topological space of adelic points on the smooth locus.

There are counterexamples to the Hasse principle and weak approximation
already for smooth cubic curves and cubic surfaces. These counterexamples
can be explained by means of a general obstruction to the Hasse principle
due to Manin based on the Brauer group of the variety and the reciprocity
law in class field theory. Most but not all of the known counterexamples
can be explained by this obstruction (Skorobogatov [Sk]). It is likely
that Manin's obstruction is the only obstruction to the (smooth) Hasse
principle for rational varieties. But it has only been proved in very
special cases.

It is more reasonable to study the Hasse principle for 0-cycles of
degree one. For curves it is possible to relate the uniqueness of
Manin's obstruction to the finiteness of the Tate--Shafarevich group of
the jacobian, which has been proved for some elliptic curves by Kolyvagin
and Rubin. Another fairly general result is due to the author [Sa] and
concerns conic bundle surfaces over the projective line. There we proved
a difficult conjecture of \CT (Conjecture~B on p.~443 in [CT/S1]). It
says that a new kind of Shafarevich group $\Sha^1(k,M)$ defined by means
of K theory vanishes for rational surfaces. This result has several
consequences. One corollary concerns the size of the Chow group of degree
zero cycles (cf.\ [CT/S1] and [Sa]). Another corollary obtained in 1987
and announced in [Sa] is the following

\begin{thm}{0.1}{Theorem}
Let $k$ be a number field and $X$ a conic bundle surface
over $\P^1_k$. Then Manin's obstruction is the only obstruction to the Hasse 
principle for $0$-cycles of degree one.
\end{thm}

The author included in [Sa] a proof when the Brauer group $\H^2(X,\G_m)$
of $X$ contains no other elements than those coming from the Brauer group
of $k$. Then the Manin obstruction vanishes so that one obtains the
simpler statement that the Hasse principle holds for $0$-cycles of degree
one. One of the motivations for the present paper is to present a proof
of Theorem~0.1, by deducing it from our result on $\Sha^1(k,M)$. This is
an improved version of the proof found in 1987.

It is based on a generalization of the descent theory of \CT [CT/S2] for
rational points to $0$-cycles of degree one. The rest of the proof is to
show that certain diagrams commute. This is done using techniques similar
to those in Bloch [Bl] and [CT/S1].

The descent theory developed by \CT is an analog of the classical
descent theory for elliptic curves developed by Fermat, Euler, Mordell
and Weil. If $p_\al\colon \T_\al\to X$ is a class of such descent
varieties and $K$ is an overfield of $k$, then the sets $p_\al(\T_\al(K))$
form a partition of $X(K)$. The descent varieties we consider are torsors
over $X$ under commutative algebraic groups.

For varieties with finitely generated torsion-free Picard groups, \CT
[CT/S2] introduced a special kind of descent varieties called universal
torsors. These are torsors under the N\'eron--Severi torus of the
variety having a certain universal property among other torsors. One
of the most important results in their paper is the following

 \begin{thm}{0.2}{Theorem}
Let $X$ be a smooth proper rational variety with a $k_v$-point
$P_v$ in each completion of $k$. Suppose that the set of these
$k_v$-points satisfies Manin's Brauer group condition. Then there exists
a universal $X$-torsor $p\colon \T\to X$ under the N\'eron--Severi torus
$T$ of $X$ (see (1.2)) such that the $k_v$-torsors under
$T\times_k k_v$ at $P_v$ obtained by base extension are trivial
for each place $v$ of $k_v$.
 \end{thm}

This means that there are $k_v$-points $Q_v$ on $\T$ such that
$p(Q_v)=P_v$ for each place $v$ of $k$. Therefore, if the
universal torsors over $X$ satisfy the Hasse principle,
then Manin's obstruction is the only obstruction to 
the Hasse principle for $X$. There are many applications of this result.
For some classes of rational varieties $X$ it is possible to
establish the Hasse principle for the universal torsors
either directly or by means of some intermediate torsors.

The proof of Theorem~0.2 in [CT/S2] uses explicit computations of
cocycles. The aim of Section 1 is to offer a proof based on simple
functoriality properties of \'etale cohomology. It is not necessary to
assume that $X$ is rational. It suffices to assume (just as in the proof
in {\it op.\ cit.})\ that the Picard group of $X\times_k\kbar$ is
finitely generated and torsion-free for an algebraic closure $\kbar$ of
$k$. Only Brauer classes in the ``algebraic part'' $\tH^2(X,\G_m)$ of the
Brauer group of
$X$ occur. This is the kernel of the functorial map from $\H^2(X,\G_m)$
to $\H^2(X\times_k\kbar,\G_m)$. If $X$ is smooth and rational, then
$\tH^2(X,\G_m)$ is the full Brauer group of $X$.

The basic idea of the proof is to ``kill'' the nonconstant
algebraic part of the Brauer group of $X$ by considering
a fibre product $\Pi$ of a finite number of Severi--Brauer
schemes over $X$ which are trivial at the specializations
at the given $k_v$-points. The vanishing of Manin's
obstruction for the algebraic part of the Brauer group
implies that $\tH^2(\Pi,\G_m)$ contains no other
elements than those coming from the Brauer group of $k$.
The given $k_v$-points can be lifted to $k_v$-points on $\Pi$.
It is now easy to show that there exists a universal $\Pi$-torsor
which is trivial at these $k_v$-points on $\Pi$ and from this,
construct the desired universal $X$-torsor. 
(Use (1.4) and its functoriality under $\Pi\to X$.)
This gives
a natural proof of Theorem~0.2.

There is no direct extension of this proof to $0$-cycles
of degree one since such cycles cannot be lifted to 
the Severi--Brauer schemes over $X$. We therefore
replace the Severi--Brauer $X$-schemes by $X$-torsors
under tori. This makes the proof less transparent. But the r\^ole
of the auxiliary torsors is the same. They are used
to simplify the cohomological obstructions. The $X$-torsors
denoted by $\sS$ are in fact chosen in such a way that they
give rise to universal torsors over $\Pi$
after pull-back of their base with respect to
the morphism $\Pi\to X$.

The advantage of this approach is that we can generalize
Theorem~0.2 to a statement where the $k_v$-points $P_v$ are replaced by 
$0$-cycles of degree one (see Theorem~1.27). Any $0$-cycle $z_v$ on
$X\times_k k_v$ defines a natural specialization map $\rho(z_v)$ from
$\H^1(X,T)$ to $\H^1(k_v,T_v)$. Our generalization of
Theorem~0.2 says that there exists a universal $X$-torsor
$p\colon \T\to X$ such that the class $[\T]\in \H^1(X,T)$ of
$\T$ belongs to the kernel of $\rho(z_v)$ for each place $v$
of $k$. This generalization is more difficult to prove and apply than 
Theorem~0.2, since the triviality of $\rho(z_v)([\T])$
in $\H^1(k_v,T_v)$ does not guarantee that $z_v$ can be
lifted to a 0-cycle of degree one on $\T$ as in the case of $k_v$-points.

The results in Section 1 are the following. We first give precise 
criteria for when there exists a universal torsor for a large class of
varieties over a number field $k$. One necessary condition is that there
are universal torsors over the $k_v$-varieties that are obtained by base
extension from $k$ to $k_v$. A second necessary condition is given by
considering the elements in the Brauer group of the variety
that become constant after all the base extensions to local fields. We
first formulate one criterion (Proposition~1.12) without assuming that
there are $0$-cycles of degree one over the local fields $k_v$ and then,
as an application, a second criterion (Proposition~1.26) under the
assumptions that such $0$-cycles exist over each completion $k_v$. Such
criteria were first established in [CT/S2] in the case when the
$0$-cycles are $k_v$-points on $X$.

In Theorem~1.27 we then prove our generalization of Theorem~0.2 discussed
above. It is worth noting that the result also applies to varieties with
$H^1(X,\O_X)=0$ and torsion-free N\'eron--Severi group, such as K3
surfaces. But the rationality assumption in [CT/S2, Section 3] remains
essential for the conjecture that the universal torsors satisfy the
Hasse principle. The converse (ii) $\Rightarrow$ (i) of Theorem~1.27
tells us that the universal torsors contain all the information about the
obstruction coming from the algebraic part of the Brauer group.

To prove Theorem~0.1 we need a strange corollary of Theorem~1.27
(Corollary~1.45) for torsors defined over an open subset of $X$. To prove
this result, we use arguments related to the ``description locale des
torseurs'' in [CT/S2]. This corollary plays an important r\^ole in the
proof of Theorem~0.1 in Section~2.

In Section 2 we first recall the $K$-theoretic construction
of Bloch [Bl] for rational surfaces as well as some refinements
in [CT/S1] and [Sa]. A fundamental tool in [Bl] is a characteristic 
homomorphism $\phi'$ for rational surfaces from the group
$Z_0(X)^0$ of $0$-cycles of degree zero to $\H^1(k,T)$
where $T$ is the N\'eron--Severi torus of $X$. In order to prove 
Theorem~0.1 we need that this map behaves well under specializations.
This is not immediate for Bloch's map, but easy to show for
another map $\phi$ of \CT defined by means of universal torsors.
We shall therefore make use of the fact that $\phi=\phi'$ for rational surfaces.
We then prove that the vanishing of $\Sha^1(k,M)$ implies that the
Manin obstruction is the only obstruction to the Hasse principle
for $0$-cycles of degree one. This is proved for rational surfaces
and, more generally, for the class of varieties 
satisfying certain axioms (2.3)
and (2.4). In particular, we deduce Theorem~0.1 from the deep
arithmetical result on $\Sha^1(k,M)$ for rational conic bundle surfaces
in [Sa].

This paper is a slightly revised version of a manuscript from 1993 in
which I prove Theorem~0.1 for a more general class of rational varieties
with a pencil of Severi--Brauer varieties. There is also a proof of this
more general result in the paper of Colliot-Th\'el\`ene and
Swinnerton-Dyer [CT/SwD]. Their approach is different and not based on
descent theory.

I would like to express my gratitude to the referee for his careful
reading of the paper.

\clearpage

\section{Universal torsors, Brauer groups and \\ obstructions to the Hasse
principle}
 
Let $k$ be a field, $\kbar$ a separable closure of $k$ and
$\sG:=\Gal(\kbar/k)$ the absolute Galois group of $k$. There is a
contravariant equivalence (cf.\ Borel [Bo]) between the categories of
$k$-tori and the category of finitely generated torsion-free discrete
$\sG$-modules. If $S$ is a $k$-torus, then there is a natural
$\sG$-action on the character group $\Shat:=\Hom(\Sbar, \G_{m,\kbar})$ 
of the $\kbar$-torus $\Sbar=\kbar\times_k S$ such that $\Shat$ becomes a
finitely generated torsion-free discrete $\sG$-module. Conversely, if
$M$ is a finitely generated torsion-free discrete $\sG$-module, then
$D(M):=\Hom_\Z (M,\kbar^*)$ is a $\kbar$-torus with a natural
$k$-structure induced by the $\sG$-action on $M$, thereby defining a 
$k$-torus. In the sequel we identify $M$ with its bidual $\widehat{D(M)}$ 
and write $\id\colon\widehat{D(M)}\isoto M$ for the canonical
$\sG$-isomorphism.

 We recall some basic notions and results from the descent theory 
of Colliot-Th\'el\`ene and Sansuc [CT/S2]. We will consider $k$-varieties 
over a perfect field $k$ satisfying the following assumptions.
\begin{equation}
\begin{array}{l}
 \hbox{$X$ is a smooth proper $k$-variety such that $\Xbar:=\kbar\times
X$ is} \\
 \hbox{connected and $\Pic \Xbar:=\H^1(\Xbar,
\G_m)$ is finitely generated} \\
 \hbox{and torsion-free.}
\end{array} \tag{1.1} 
\end{equation}

 Let $\pi \colon \sS\to X$ be a $k$-morphism from a $k$-variety $\sS$ 
which is faithfully flat and locally of finite type over $X$. Let $S$ be
a $k$-torus. Then $\pi \colon \sS\to X$ is said to be a (left) $X$-torsor
under $S$ if there is a (left) action $\si \colon S\times\sS\to\sS$ such
that the $k$-morphism
 \[
 (\si,\pr_2)\colon S\times_X\sS \lra \sS\times_X \sS
 \]
induced by $\si$ and the second projection $\pr_2\colon
S\times_X\sS\to\sS$ is an isomorphism. We usually write $\sS$ rather than
$\pi\colon\sS\to X$ for the $X$-torsor. An $X$-torsor under a $k$-torus is
locally trivial in the \'etale topology by a theorem of Grothendieck. The
isomorphism classes of $X$-torsors under $S$ correspond to elements of
$\H^1(X,S)$.

Now let $\chi\colon \H^1(X,S)\to \Hom_\sG (\Shat, \Pic \Xbar)$ 
be the homomorphism induced
by the additive pairing $\H^1(\Xbar,\Sbar)\times\Hom(\Sbar,\G_{m,\kbar})
\to\H^1(\Xbar,\G_{m,\kbar})$.

\begin{rem}{1.2}{Definition}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item Let $\sS$ be an $X$-torsor under $S$ and $[\sS]$ its class in
$\H^1(X,S)$. Then $\chi([\sS])\in\Hom_\sG (\Shat, \Pic\Xbar )$ 
is called the {\em type} of $\sS$.
 \item The {\em N\'eron--Severi torus} $T$ of $X$ is the 
$k$-torus $D(\Pic\Xbar)$ associated to the discrete $\sG$-module 
$\Pic \Xbar$.
 \item A {\em universal torsor} over $X$ is an $X$-torsor under the
N\'eron--Severi torus $T$ whose type is $\id\colon \That\to\Pic\Xbar$.
 \end{enumerate}
\end{rem}

By considering the spectral sequence
 \begin{equation}
 \Ext_{k\et}^p(\Shat, R^q p_*\G_{m,X})\Rightarrow 
 \Ext_{X\et}^{p+q} (p^*\Shat, \G_{m,X} ) \tag{1.3}
 \end{equation}
for a $k$-torus $S$ and the structure morphism $p\colon X\to\Spec k$ (see
[CT/S2, 1.5.1]), Colliot-Th\'el\`ene and Sansuc obtained the exact
sequence:
 \begin{multline} 0 \to \H^1(k, S)\to\H^1(X, S)
\xrightarrow{\,\chi\,}\Hom_\sG(\Shat, \Pic\Xbar ) \\
\xrightarrow{\,\de\,}\H^2(k,S)\to\H^2(X,S). \tag{1.4}
 \end{multline}

The homomorphisms $\H^i(k, S )\to\H^i(X, S)$ are the functorial 
contravariant maps in \'etale cohomology. We shall not give any 
explicit description of $\de$. All we need in the proofs is that 
the sequence (1.4) is functorial under field extensions of $k$ and 
homomorphisms of $k$-tori.

Let $\tH^2(X,S):=\Ker\bigl(\H^2(X,S)\to\H^2(\Xbar,\Sbar)\bigr)$. By
analysing (1.3) further, one extends the end of (1.4)  to an exact
sequence:
\begin{multline}
\Hom_\sG (\Shat, \Pic\Xbar) \\
\xrightarrow{\,\de\,}
 \H^2(k,S)\to\tH^2(X,S)\to\Ext_\sG(\Shat, \Pic\Xbar)
\to\H^3(k,S). \tag{1.5}
 \end{multline}
In particular for $S= \G_{m,k} $, one obtains the well-known sequence:
\begin{equation}
 \H^2(k,\G_{m,k})\to\tH^2(X,\G_{m,k})\to\Ext_\sG(\Z, \Pic\Xbar)
\to\H^3(k, \G_{m,k}). \tag{1.6}
 \end{equation}
The next result is also in [CT/S2]. We include a proof, since {\em op.\
cit.} does not prove the implication (iii) $\Rightarrow$ (ii) {\it
directly}.

\begin{thm}{1.7}{Proposition} Let $k$, $X$ be as in (1.1) and
let $T$ be the N\'eron--Severi torus of $X$. Then the following
conditions are equivalent.
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item $\H^2(k,T)\to\H^2(X,T)$ is injective 
for the N\'eron--Severi torus $T$.

 \item $\H^2(k,S)\to\H^2(X,S)$ is injective for any $k$-torus $S$.

 \item There exists a universal torsor over $X$.
 \end{enumerate}
\end{thm}

\begin{pf} (ii) $\Rightarrow$ (i) is trivial and (i) $\Rightarrow$
(iii) is immediate from (1.4).  To prove (iii) $\Rightarrow$ (ii), let $S$
be a $k$-torus and $\chi\in\Hom_\sG(\Shat,\Pic\Xbar)$.  Then there is a
dual homomorphism $D(\chi)$ of $k$-tori $T\to S$ inducing a commutative
diagram
 \[
 \begin{matrix}
\H^1(X,T)&\to&\Hom_\sG(\That,\Pic\Xbar)&\to&\H^2(k,T)&\to&\H^2(X,T)\\[4pt]
\big\downarrow&&\big\downarrow&&\big\downarrow&&\big\downarrow\\[4pt]
\H^1(X,S)&\to&\Hom_\sG(\Shat,\Pic\Xbar)&\to&\H^2(k,S)&\to&\H^2(X,S)
\end{matrix}
 \]
such that $\id\in\Hom_\sG(\That,\Pic\Xbar)$ goes to $\chi$ in
$\Hom_\sG(\Shat,\Pic\Xbar)$. Hence $\de(\chi)=0$, thereby proving
(iii) $\Rightarrow$ (ii).
\end{pf}

Recall that a {\em $0$-cycle} on $X$ is a finite formal sum $z =\sum n_i
P_i$ where the $P_i$ are closed points on $X$ and the $n_i$ integers. The
integer $n:=\sum n_i[k(P_i):k]$ is called the {\em degree} of $z$. Denote
by $Z_0(X)$ the free abelian group of $0$-cycles on $X$. For each
$k$-torus $S$ and each positive integer $i$, there is a natural additive
pairing
\begin{equation}
\rho\colon Z_0(X)\times\H^i(X,S)\to\H^i(k,S) \tag{1.8}
 \end{equation}
sending a pair consisting of a closed point $P\in Z_0(X)$ and an element 
$\e\in\H^i(X, S)$ to the corestriction in $\H^i(k,S)$ 
of the pullback $\e(P)$ of $\e$ in $\H^i(k(P),S)$. 
It can be proved that this pairing factorizes through 
rational equivalence, but we do not need this.

If $z=\sum n_iP_i$ is a $0$-cycle, write $\rho(z)\colon \H^i(X,T)\to\H^i(k,T)$ 
for the homomorphism sending $\e\in\H^i(X,T)$ to
$\rho(z,\e)\in\H^i(k,T)$. This gives a retraction of the functorial map
from $\H^i(k,T)$ to $\H^i(X,T)$ when $z$ is of degree one. Then by
Proposition~1.7, there exists a universal torsor over $X$.

Let $T$ be the N\'eron--Severi torus of $X$ and $\T$ a universal 
$X$-torsor. Let $\phi_\T\colon Z_0(X)\to\H^1(k,T)$ 
be the homomorphism which sends 
$z\in Z_0(X)$ to
$\rho(z, [\T] )$ (see (1.8)), and $Z_0(X)^0$ the subgroup of $Z_0(X)$
consisting of
$0$-cycles of degree zero.

\begin{thm}{1.9}{Proposition} The restriction of $\phi_\T$ to
$Z_0(X)^0$ is independent of the choice of universal torsor $\T$.
\end{thm}

\begin{pf*} Use (1.4) and the fact that 
 \[
 Z_0(X)^0\times \Ima (\H^1(k,T)\to\H^1(X,T))\subseteq\Ker(\rho). \qed
 \]
\end{pf*}

We therefore drop the index and write $\phi$ for this map
$Z_0(X)^0\to\H^1(k,T)$. For other constructions of $\phi$ that do not
depend on the assumption that a universal torsor exists, see [CT/S1,
Section~1] and the next section.

The following almost trivial lemma from homological algebra will be useful.

\begin{thm}{1.10}{Lemma} Let $L$ be a finitely generated torsion-free
discrete $\sG$-module. Then
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item $H^1(\sG, \Z) = 0$.
 \item $H^1(\sG,L)=\Ext_\sG(\Z,L)$ is finite.
 \item Let $\e_1,\e_2,\dots,\e_r\in\Ext_\sG(\Z^{(r)},L)$ and $\e\in
\Ext_\sG(\Z^{(r)}, L)$ correspond to
$\{\e_j\}_{j=1}^r\in\bigoplus\Ext_\sG(\Z,L)$.
 \end{enumerate}
Then there is an extension of discrete $\sG$-modules
\begin{equation}
0\to L\to M\to \Z^{(r)}\to 0\tag{$*$}
 \end{equation} 
such that
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item $\Ker(\Ext_\sG(\Z,L)\to \Ext_\sG(\Z, M))$ is the subgroup
generated by $\e_1,\e_2,\ldots,
\linebreak[2]\e_r$,
\item the connecting homomorphism $\Hom_\sG (L, L)\to\Ext_\sG
(\Z^{(r)}, L)$ induced by $(*)$ sends $\id\in \Hom_\sG(L, L)$ to $\e$.
 \end{enumerate}
\end{thm}

Now let $k$ be a number field. Denote by $\Omega_k$ 
the set of places 
of $k$, and by $k_v$ the $v$-adic completion of $k$ for 
a place $v$. Choose an 
algebraic closure $\kbar_v$ of $k_v$ and an embedding $\kbar\subset
\kbar_v$ for each $v\in \Omega_k$. 
We may then regard the Galois group 
$\sG_v:=\Gal(\kbar_v / k_v)$ as a subgroup 
of $\sG = \Gal(\kbar/k)$ for each $v\in \Omega_k$.

 If $M$ is a discrete $\sG$-module and $i$ a positive integer, 
write
 \[
 \Sha^i(k,M):=
 \Ker\Bigl(H^i(\sG,M)\to \prod_{\mathrm{all\ } v} H^i(\sG_v, M)\Bigr).
 \] 
In particular, if $M$ is the group $S(\kbar)$ of $\kbar$-points on a
$k$-torus $S$, we write $\Sha^i(k,S):=\Sha^i(k,S(\kbar))$. Finally, set
 \[
 \Che^1(k,S):=\Coker\Bigl(H^1(\sG,S(\kbar))\to
\bigoplus_{\mathrm{all\ } v} H^1(\sG_v,S(\kbar_v))\Bigr).
 \]

The following result from class field theory is due to Nakayama and Tate
[Ta1]. It plays an important r\^ole in [CT/S2].

\begin{thm}{1.11}{Theorem} Let $k$ be a number field and $S$ a
$k$-torus. Then there is a perfect pairing
 \[
 \Sha^2(k,S)\times\Sha^1(k,\Shat)\lra\Q / \Z
 \]
which is functorial
under homomorphisms of $k$-tori. The kernel of the 
induced epimorphism from $\Hom(H^1(\sG,\Shat),\Q/\Z)$ to $\Sha^2(k,S)$ 
is isomorphic to $\Che^1(k, S)$. Moreover, $\H^3(k,S)=0$
for any split $k$-torus $S$.
\end{thm}

 The next proposition generalizes a result in Section 3.3 in 
[CT/S2]. If we use the word ``locally'' for a property which holds for 
$X_v:= k_v\times X$ for each place $v\in\Omega_k$, 
then we can express Proposition~1.12 in the 
following way. There exists a universal torsor over $X$ if and only if 
there exists one locally, and moreover every locally constant Azumaya
algebra over $X$ is Brauer equivalent to a product of a locally trivial
Azumaya algebra and a constant Azumaya algebra. We can replace 
$\tH^2$ by $\H^2$ in (i), 
since any ``locally'' constant Brauer class belongs to 
$\tH^2(X,\G_m)$. We prefer the formulation here since the universal torsors 
are related to $\tH^2(X,\G_m)$ rather than $\H^2(X,\G_m)$.

\begin{thm}{1.12}{Proposition} Let $k$ be a number field and $X$ a smooth
proper geo\-metrically connected variety over $k$ for which
$\Pic\Xbar$ is finitely generated and torsion-free. Then the following
statements are equivalent.
\begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item The map from $\Ker (\tH^2(X,\G_m)\to
\prod_{\mathrm{all\ } v} \tH^2(X_v,\G_m))$ to
 \[
 \Ker \Bigl(\tH^2(X,\G_m)/{\Ima\H^2(k,\G_m)}\to
\prod_{\mathrm{all\ } v} \tH^2(X_v,\G_m)/{\Ima\H^2(k_v,\G_m)}\Bigr)
 \]
is surjective, and for each place $v\in\Omega_k$ there exists a
universal torsor over $X_v$.
\item There exists a universal torsor over X.
 \end{enumerate}
\end{thm}

\paragraph{Proof} We apply Lemma~1.10 for the $\sG$-module $L
=\Pic\Xbar$  and choose 
a set of generators $\e_1,\e_2,\ldots,\e_r$ of 
$\Ker(\Ext_\sG(\Z, \Pic\Xbar)\to \prod_{\mathrm{all\ } v}
\Ext_{\sG_v}(\Z, \Pic\Xbar))$. 
Let $\e\in\Ext_\sG(\Z^{(r)},L)$ correspond to $\bigoplus_{j=1}^r\e_j\in
\Ext_\sG(\Z, L)$. 
We then obtain an exact sequence of discrete $\sG$-modules
\begin{equation}
0\to\Pic\Xbar\to M\to\Z^{(r)}\to 0\tag{1.13}
 \end{equation}
such that
 \begin{align} 
 & \enspace \Sha^1(k,\Pic\Xbar) = \Ker (H^1(\sG, \Pic\Xbar)\to
H^1(\sG,M)); \quad\hbox{and}
 \tag{1.14} \\[8pt]
 & \begin{array}{l}
 \hbox{$\id\in\Hom_\sG(\Pic\Xbar, \Pic\Xbar)$ maps to $\e$ under the
 connecting} \\
 \hbox{homomorphism
$\Hom_\sG(\Pic\Xbar,\Pic\Xbar)\to\Ext_\sG(\Z^{(r)},\Pic\Xbar)$} \\
 \hbox{induced by (1.13).}
 \end{array}
 \tag{1.15} \\[8pt]
 & \begin{array}{l}
 \hbox{the extension (1.13) is split as a sequence of $\sG_v$-modules} \\
 \hbox{for each $v\in\Omega_k$.}
 \tag{1.16}
 \end{array}
 \end{align}

 Now apply $D(\ldots)$ to (1.13) and consider the dual sequence of
$k$-tori:
\begin{equation}
 1\to R\to S\to T\to 1,\tag{1.17}
\end{equation}
where $T$ is the N\'eron--Severi torus of $X$ and 
$R =\prod_{j=1}^r\G_{m,k}$. {From} 
(1.14) and the arithmetical duality result in Theorem~1.11 we obtain 
that:
 \begin{equation}
 \Sha^2(k,S)\subseteq\Ker(\H^2(k,S)\to \H^2(k,T)) \tag{1.18}
 \end{equation}
and from (1.16) that the sequences of $k_v$-tori
\begin{equation}
1\to R_v\to S_v\to T_v\to 1 \tag{1.19}
 \end{equation} 
induced from (1.17) split for all places $v$ of $k$.

\paragraph{Proof of (i) $\Rightarrow$ (ii)} Consider the following
commutative diagram with exact rows and columns
\begin{equation}
\renewcommand{\arraystretch}{1.3}
\begin{matrix}
\H^2(k, R )&\to&\tH^2(X, R)&\to& \Ext_\sG(\Rhat,\Pic\Xbar)\\
\big\downarrow&&\big\downarrow&&\big\downarrow\\
\H^2(k, S)&\to&\tH^2(X, S)&\to& \Ext_\sG(\Shat,\Pic\Xbar)\\
\big\downarrow&&\big\downarrow&&\big\downarrow\\
\H^2(k, T)&\to&\tH^2(X, T)&\to& \Ext_\sG(\That,\Pic\Xbar)\\
\big\downarrow&&&&\\
0
\end{matrix}\tag{1.20}
 \end{equation}
where the horizontal sequences are those in (1.5) and the vertical 
sequences are induced by (1.17). The complex in the second column 
is exact since (1.17) splits over $\kbar$. The map 
$\H^2(k, S)\to\H^2(k, T)$ is 
surjective since $\H^3(k, R) = 0$ for a number field $k$ (see (1.8)).

In order to prove that there is a universal torsor, it suffices 
by Proposition~1.7 to show that $\H^2(k,T )\to \tH^2(X,T )$ 
is injective. So let $\ka\in\Ker(\H^2(k,T)\to\tH^2(X,T))$ 
and lift $\ka$ to an element $\beta\in\H^2(k,S)$. 
Then, by exactness of 
(1.20), there exists $\ga$ in 
$\Ker(\tH^2(X, R)\to\Ext_\sG (\Shat,\Pic\Xbar))$ with the 
same image as $\beta$ in $\tH^2(X,S)$. Let $\ga_v$ 
be the image of $\ga$ in $\tH^2(X_v,R)$ and consider the following commutative diagram with exact rows and columns.
\begin{equation}
\renewcommand{\arraystretch}{1.3}
\begin{array}{ccccccc}
&&0&&0&&0\\
&&\big\downarrow&&\big\downarrow&&\big\downarrow\\
0&\to&\H^2(k_v, R_v )&\to&\tH^2(X_v, R_v)&\to& 
\Ext_{\sG_v}(\widehat {R_v},\Pic\Xbar_v)\\
&&\big\downarrow&&\big\downarrow&&\big\downarrow\\
0&\to&\H^2(k_v, S_v)&\to&\tH^2(X_v, S_v)&\to& 
\Ext_{\sG_v}(\widehat {S_v},\Pic\Xbar_v)
\end{array}\tag{1.21}
 \end{equation}
The zeros in the columns come from the splitting property in (1.16) and
(1.19), and the zeros in the rows from the existence of universal
torsors over $X_v$ (see Proposition~1.7). Since $\ga$ goes to zero in 
$\Ext_\sG(\Shat, \Pic\Xbar)$, we conclude from (1.21) that
$\ga_v\in\Ima (\H^2(k_v, R_v)\to\tH^2(X_v, R_v))$ 
for each $v\in\Omega_k$. On considering the 
images of $\ga$ in $\tH^2(X, \G_m)$ under the maps from
$\tH^2(X,R)$ induced by the $r$ projections 
from $R =\prod_{j=1}^r\G_{m,k}$ to $\G_m$, 
we deduce from the first 
assumption in (i) that there exists $\alpha\in \H^2(k,R)$ 
that maps to $\ga_v$ in $\H^2(X_v,R_v)$ for each
$v\in\Omega_k$.  Let $\tilde\alpha$ be the image of $\alpha$ in 
$\H^2(k, S)$. By the choice of $\ga$, we conclude that 
$\beta -\tilde\alpha$ goes to $0$ 
in $\prod_{\mathrm{all\ } v} \tH^2(X_v, S_v)$ and by the injectivity of
the functorial maps $\H^2(k_v, S_v)\to\tH^2 (X_v, S_v)$ that 
$\beta -\tilde\alpha\in\Sha^2(k,S)$. But then the 
image $\ka$ of $\beta-\tilde\alpha$ in $\H^2(k,T)$ 
is equal to zero (see (1.18)). This completes the proof of (i)
$\Rightarrow$ (ii).

\begin{pfof}{(ii) $\Rightarrow$ (i)} Let $\T$ be a universal torsor
over $X$. Then $\T_v:=k_v\times\T$ is a universal torsor over $X_v$ for
each $v\in \Omega_k$. To prove the first part of (i), consider
the following commutative diagram with exact rows
\begin{equation}
\renewcommand{\arraycolsep}{0.35em}
\begin{matrix}
\H^1(k, T)&\to&\H^1(X,T)&\to& \Hom_{\sG}(\That,\Pic\Xbar)&\to &0\\[6pt]
\big\downarrow&&\big\downarrow&&\big\downarrow&&\\[6pt]
\H^2(k,R)&\to&\tH^2(X, R)&\to& \Ext_\sG(\Rhat,\Pic\Xbar)&
\to&0
\end{matrix}\tag{1.22}
 \end{equation}
Let $\ga\in\tH^2(X,R)$ be the image of 
$[\T]\in\H^1(X,T)$ and $\ga_1,\ga_2,\ldots,\ga_r$ the 
images of $\ga$ in $\tH^2(X,\G_m)$ under the maps from 
$\tH^2(X, R )$ induced by the 
$r$ projections from $R =\prod_{j=1}^r\G_{m,k}$ to $\G_m$.
Then $\ga_1,\ga_2,\ldots,\ga_r$ have images 
$\e_1,\e_2,\ldots,\e_r$ in $\Ext_\sG(\Rhat, \Pic\Xbar)$. 
Thus by the choice of $\e_j$ (see (1.6)) we get that the kernel of the
map
 \[
 \tH^2(X,\G_m)/{\Ima \H^2(k,\G_m)} \to \prod_{\mathrm{all\ } v}
\tH^2(X_v,\G_m)/{\Ima \H^2(k_v,\G_m)}
 \]
is generated by the images of $\ga_1,\ga_2,\ldots,\ga_r$ in
$\tH^2(X,\G_m)/{\Ima \H^2(k,\G_m)}$. To verify the first condition in
(i), it thus suffices to show that the elements 
$\ga_1,\ga_2,\ldots,\ga_r$ belong to $\Ker(\tH^2(X,\G_m)\to 
\prod_{\mathrm{all\ } v} \tH^2(X_v,\G_m))$. That is, we must prove that
$[\T]$ belongs to the kernel of the composite map:
 \[
 \H^1(X,T)\lra \tH^2(X,R)\lra \prod_{\mathrm{all\ } v} \tH^2(X_v,R_v).
 \]
But $[\T_v]\in\H^1(X_v,T_v)$ maps to zero in $\tH^2(X_v,R_v)$ since the
sequence $1\to R_v\to S_v\to T_v\to 1$ splits. This completes the proof
of Proposition~1.12.
\end{pfof}

Now suppose that we are given a $0$-cycle $z_v$ on $X_v$ for each 
place $v\in\Omega_k$.
If $S$ is a $k$-torus, let $S_v$ be the $k_v$-torus obtained 
by base extension and let
\begin{equation}
 \rho_v\colon Z_0(X_v)\times\H^i (X_v, S_v)\lra\H^i(k_v, S_v)
\tag{1.23}
 \end{equation} 
be the pairing described in (1.8). We denote this map by $\rho_v$ for 
all $k$-tori $S$ and
all positive integers $i$. 
Let $\rho_v(z_v)\colon \H^i(X_v, S_v)\to\H^i(k_v, S_v)$ be the 
homomorphism sending $\e_v\in\H^i(X_v,S_v)$ to 
$\rho_v(z_v,\e_v)\in\H^i(k_v, S_v)$.

Now recall the fundamental exact sequence of Hasse (see, for example,
Tate [Ta2])
\begin{equation}
0\to\H^2(k, \G_m)\to \bigoplus_{\mathrm{all\ } v}\H^2(k_v,\G_m)\to
\Q/\Z\to 0.\tag{1.24}
 \end{equation} 
The map from $\H^2(k,\G_m)$ is the direct sum over $v\in\Omega_k$ 
of the functorial maps
$\H^2(k, \G_m)\to\H^2(k_v, \G_m)$. 
The map to $\Q / \Z$ is the direct sum of the local 
maps $\inv_v \colon \H^2(k_v, \G_m)\to \Q / \Z$ 
which are isomorphisms for 
non-archimedean places. The fact that the sum of all local 
invariants is $0$ for an element of the Brauer group
$\H^2(k,\G_m)$ of $k$ is called the reciprocity law.

Manin [Ma] noticed that the reciprocity law gives rise to the 
following necessary condition for the existence of a $0$-cycle of 
degree $r$ on $X$.
\begin{equation}
 \begin{array}{l}
 \hbox{There exists a set of $0$-cycles $z_v$ of degree $r$ on $X_v$
indexed by} \\
 \hbox{$v\in\Omega_k$ s.t.\ $\sum_{\mathrm{all\ } v}
\inv_v(\rho_v(z_v))(\sA_v) = 0$ for all $\sA\in\H^2(X,\G_m)$.}
\end{array}
\tag{1.25}
\end{equation}

We now relate the Brauer group obstruction to the Hasse 
principle for $0$-cycles
of degree one to another obstruction based on universal torsors. 
The following result is an immediate corollary of Proposition~1.12.

\begin{thm}{1.26}{Proposition} Let $k$ be a number field and
$X$ a smooth proper geometrically connected $k$-variety for which
$\Pic\Xbar$ is finitely generated and torsion-free.
Suppose given a $0$-cycle 
of degree one $z_v$ on $X_v$ for each place $v\in\Omega_k$. 
Then the following statements are equivalent.
\begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item Manin's reciprocity condition 
$\sum_{\mathrm{all\ } v} \inv_v(\rho_v(z_v))(\sA_v) = 0$ 
holds for all
$\sA\in\Ker\bigl(\tH^2(X,\G_m)\to\prod_{\mathrm{all\ } v}
\tH^2(X_v,\G_m)/{\Ima\H^2(k_v,\G_m)}\bigr)$. 
\item There exists a universal torsor over $X$.
 \end{enumerate}
\end{thm}

\begin{pf} Given $0$-cycles $z_v$ of degree one on $X_v$ for each 
place $v\in\Omega_k$, we have to prove that the conditions (1.12i) and
(1.26i) are equivalent. It was already noticed after 
(1.8) that the existence of a 
$0$-cycle of degree one on $X_v$ implies the existence of a universal 
torsor over $X_v$. It thus suffices to show that the subgroup of 
 \[
 \Ker\Bigl(\tH^2(X,\G_m)\to
\prod_{\mathrm{all\ } v}\tH^2(X_v,\G_m)/{\Ima\H^2(k_v,\G_m)}\Bigr)
 \]
generated by 
$\Ker\bigl( \tH^2(X,\G_m) \to
\prod_{\mathrm{all\ } v} \tH^2(X_v,\G_m)\bigr)$
and $\Ima(\H^2(k,\G_m))$ equals the subgroup of 
elements $\sA$ satisfying $\sum_{\mathrm{all\ } v}
\inv_v(\rho_v(z_v))(\sA_v) = 0$. This is a formal consequence of the Hasse
exact sequence of Brauer groups (1.24) and the fact that for all places
$v$ of $k$, the map $\rho_v(z_v)$ defines a retraction of
$\H^2(k_v,\G_m)\to \tH^2(X_v, \G_m)$.
\end{pf}

We now consider Manin's obstruction to the Hasse principle for $0$-cycles
of degree one given by arbitrary elements in $\tH^2(X,\G_m)$ and relate
it to the existence of universal torsors with certain  properties. The
following result was proved in [CT/S2, 3.5.1] in the case of rational
points.

\begin{thm}{1.27}{Theorem} Let $k$ be a number field and $X$ a smooth
proper geometrically connected $k$-variety for which $\Pic\Xbar$ is
finitely generated and torsion-free. Suppose given a $0$-cycle $z_v$ of
degree one on $X_v$ for each place $v\in\Omega_k$. Then the following
statements are equivalent.
\begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item Manin's reciprocity condition 
$\sum_{\mathrm{all\ } v} \inv_v(\rho_v(z_v))(\sA_v) = 0$ 
holds for all $\sA\in \tH^2(X,\G_m)$. 
\item There exists a universal torsor $\T$ over $X$ 
such that $\rho_v(z_v)([\T_v]) = 0$ in
$\H^1(k_v,T)$ for each $v\in\Omega_k$.
 \end{enumerate}
\end{thm}

\paragraph{Proof} We again apply Lemma~1.10 for the $\sG$-module
$L=\Pic\Xbar$. Let $\e_1,\e_2,\ldots,\e_r$ be generators of $\Ext_\sG(\Z,
\Pic\Xbar)$, and let $\e\in\Ext_\sG(\Z^{(r)}, L)$ correspond to
$\{\e_j\}_{j=1}^r\in\Ext_\sG(\Z, \Pic\Xbar)$. We then obtain an
exact sequence of discrete $\sG$-modules:
\begin{equation}
0\to\Pic\Xbar\to M\to\Z^{(r)}\to 0\tag{1.28}
 \end{equation}
such that
 \begin{align}
 &\enspace H^1(\sG, M)=0; \quad\hbox{and} \tag{1.29} \\[8pt]
 &\begin{array}{l}
 \hbox{$\id\in\Hom_\sG(\Pic\Xbar,\Pic\Xbar)$ maps to $\e\in\Ext_\sG
(\Z^{(r)},\Pic\Xbar)$} \\
 \hbox{under the connecting homomorphism induced by (1.28).}
 \end{array}
 \tag{1.30}
 \end{align}

Now apply $D(\ldots)$ to (1.28) and consider the dual sequence of $k$-tori.
\begin{equation}
1\to R\to S\to T\to 1,\tag{1.31}
 \end{equation}
where $T$ is the N\'eron--Severi torus of $X$ and 
$R =\prod_{j=1}^r\G_{m,k}$. {From} (1.29) 
and the arithmetical duality result in Theorem~1.11 we obtain
\begin{equation}
\Che^1(k,S )=0\quad \hbox{and}\quad\Sha^2(k,S)=0.
\tag{1.32}
 \end{equation}

\paragraph{Proof of (i) $\Rightarrow$ (ii)} Consider the following
commutative diagram with exact rows and columns:
\begin{equation}
\renewcommand{\arraystretch}{1.3}
\begin{array}{ccccc}
\H^1(k, S)&\to&\H^1(X, S)&\to& \Hom_\sG(\Shat,\Pic\Xbar)\\
\big\downarrow&&\big\downarrow&&\big\downarrow\\
\H^1(k, T)&\to&\H^1(X, T)&\to& \Hom_\sG(\That,\Pic\Xbar)\\
\big\downarrow&&\big\downarrow&&\big\downarrow\\
\H^2(k, R )&\to& \tH^2(X, R) &\to& \Ext_\sG(\Rhat,\Pic\Xbar)\\
\big\downarrow&&\big\downarrow&&\big\downarrow\\
\H^2(k, S)&\to&\tH^2(X, S)&\to& \Ext_\sG(\Shat,\Pic\Xbar)\\
\end{array}\tag{1.33}
 \end{equation}
deduced from (1.31) and the spectral sequence in (1.3). We know 
from Proposition~1.26 that there exists a universal torsor over $X$. 
Let $[\T]\in\H^1(X,T)$ 
be the class of one such torsor $\T$ and consider the images $\ga$ 
in $\tH^2(X, R)$ and $\ga_v\in\tH^2(X_v, R_v)$,
$v\in\Omega_k$, of $[\T]$. 
Then, since $R =\prod_{j=1}^r\G_{m,k}$, we deduce from Manin's 
reciprocity condition (i) and the Hasse exact sequence (1.24) that 
there exists $\beta\in\H^2(k, R )$ that
maps to $\rho_v(z_v)(\ga_v)$ in $\H^2(k_v, R_v)$
for each $v\in\Omega_k$. But $\rho_v(z_v)(\ga_v)\in\Ker(
\H^2(k_v, R_v )\to\H^2(k_v, S_v))$ since
it is the image of $\rho_v(z_v)([\T_v])\in\H^1(k_v, T_v)$ 
in $\H^2(k_v, R_v)$. Therefore,
$\beta\in\Ker (\H^2(k, R )\to\H^2(k, S ))$ 
since $\Sha^2(k, S)=0$ (cf.\ (1.32)). Let
$\alpha\in\H^1(k, T)$ be a lifting of $\beta$ and $\alpha_v$ the image
of $\alpha$ in $\H^1(k_v,T_v)$. Then
$\rho_v(z_v)([\T_v]) - \alpha_v$ vanishes for all but finitely many 
$v\in\Omega_k$ and maps to $0$ in $\H^2(k_v,R_v)$ for all $v\in\Omega_k$. 
This combined with the fact that $\Che^1(k, S)=0$
implies that there exists $\si\in\H^1(k, S)$ whose image $\si$ in 
$\H^1(k_v,T_v)$ is $\rho_v(z_v)([\T_v]) - \alpha_v$ for each 
$v\in\Omega_k$. Let $\tilde\si$ be the image 
of $\si$ in $\H^1(X, T)$ and $\tilde\alpha$ the image of $\alpha$ in
$\H^1(X, T)$. Then, since $\tilde\alpha +\tilde\si$ belongs to the image
of $\H^1(k, T)\to\H^1(X, T)$ it follows that 
$[\tilde\T ]:= [\T ]+\tilde\alpha +\tilde\si$ is the class of 
a torsor $\tilde\T$ of the same type as $\T$. 
Further, $\rho_v(z_v)([\tilde\T_v]) = 0 $ for all $v\in\Omega_k$. 
This completes the proof of \hbox{(i) $\Rightarrow$ (ii).}

\paragraph{Proof of (ii) $\Rightarrow$ (i)} Let $\T$ be a universal
torsor over $X$ with the property that $\rho_v(z_v)([\T_v])= 0$ in
$\H^1(k_v,T)$ for all $v\in\Omega_k$. We now proceed as in the proof of
Proposition~1.12, (ii) $\Rightarrow$ (i) and consider the image $\ga$ of 
$[\T]\in\H^1(X,T)$ in $\tH^2(X, R)$ under the vertical map in (1.33), 
and the images $\ga_1,\ga_2,\ldots, \ga_r$ of $\ga$ in $\tH^2(X,\G_m)$
under the maps from $\tH^2(X, R)$ induced by the $r$ projections from $R
=\prod_{j=1}^r\G_{m,k}$ to $\G_m$. Then $\rho_v(z_v)(\ga_j) = 0$ for all
$j =1,\dots, r$ and all places $v$ of $k$. This together with the
reciprocity law (1.24) implies that $(z_v)_{v\in\Omega_k}$ satisfies
Manin's condition for any $\sA\in\tH^2(X,\G_m)$ in the subgroup $\Ga$
generated by $\ga_1,\ga_2,\ldots, \ga_r$ and the image of
$\H^2(k,\G_m)$. But the images $\e_1,\e_2,\ldots, \e_r$ in $\Ext_\sG
(\Z, \Pic\Xbar) $ of $\ga_1,\ga_2,\ldots, \ga_r$ were chosen to generate
$\Ext_\sG (\Z, \Pic\Xbar) $. Thus, $\Ga= \tH^2(X, \G_m)$, as was to be
proved.

\begin{thm}{1.34}{Corollary} Let $k$ be a number field and $X$ a smooth
proper geometrically connected $k$-variety for which $\Pic\Xbar$ is
finitely generated and torsion-free. Suppose given a
$0$-cycle $z_v$ of degree one on $X_v$ for each place $v$ such that
Manin's reciprocity condition $\sum_{{\mathrm all \ } v}
\inv_v(\rho_v(z_v))(\sA_v) = 0$ holds for all $\sA\in \tH^2(X,\G_m)$. 
Then for each $k$-torus $S$ and each element $\tau$ in $\Hom_\sG (\Shat,
\Pic\Xbar)$ there exists an $X$-torsor $\sS$ under $S$ of type $\tau$
such that $\rho_v(z_v)([\sS_v]) = 0$ in $\H^1(k_v, S )$ for all 
$v\in\Omega_k$.
\end{thm}

\paragraph{Proof} We know from (1.24) that there exists a universal torsor $\T$ 
over $X$ such that $\rho_v(z_v)([\T_v]) = 0$ in $\H^1(k_v, T_v )$ 
for each $v\in\Omega_k$. Let $\sS:=\T\times^T S$ be
the torsor under $S$ induced from $\T$ by the $k$-homomorpism 
$D(\tau)\colon T\to S$ 
dual to $\tau$. Then $\sS$ satisfies the above conditions.

\begin{thm}{1.35}{Theorem} Let $k$ be a number field and $X$ a smooth
proper geometrically connected $k$-variety for which $\Pic\Xbar$ is
finitely generated and torsion-free. Let $T$ be the N\'eron--Severi
torus of $X$ and $r$ an integer. Let $z_v$ be a $0$-cycle of degree $r$
on $X_v$ for each place $v\in\Omega_k$ such that Manin's reciprocity
condition $\sum_{\mathrm{all\ } v} \inv_v(\rho_v(z_v))(\sA_v) = 0$ holds
for all $\sA\in \tH^2(X,\G_m)$. Then for each $X$-torsor under $T$ there
exists another $X$-torsor $\T$ of the same type such that
$\rho_v(z_v)([\T_v]) = 0$ for all $v\in\Omega_k$.
\end{thm}

\begin{pf}
An examination of the proof of (i) $\Rightarrow$ (ii) 
in (1.24) reveals 
that we only used the hypothesis that $r =1$ to prove that there 
exists a universal torsor $\T$. The rest
of the arguments is valid for any $r$ and any $T$-torsor $\T$.
\end{pf}

We now make use of the ideas of [CT/S2, 2.3]. Let $k$ be a perfect 
field and let $X$ be as in (1.1). Let $U$ be an open $k$-subvariety of 
$X$ with $\Pic\Ubar= 0$. If $\sS$ is an $X$-torsor, let $\sS_U$ 
be the $U$-torsor obtained by restriction.

Consider the exact sequence of $\sG$-modules for the absolute Galois group
$\sG:=\Gal(\kbar/k)$.
\begin{equation}
0\to\kbar[U]^*/\kbar^*\to \Div_{\Zbar}\Xbar \to\Pic\Xbar\to 0,
\tag{1.36}
 \end{equation} 
where $Z$ is the complement of $U$ in $X$, and $\Div_{\Zbar}\Xbar$ 
the group of Weil divisors
on $\Xbar$ with support in $\Zbar$. 
On applying $D(\ldots)$ we obtain a dual exact 
sequence of $k$-tori
\begin{equation}
 1\to T\to N\to V\to 1. \tag{1.37}
 \end{equation}
The spectral sequence (1.3) and 
the exact sequence (1.37) give rise to the commutative diagram
\begin{equation}
\begin{matrix}
\Hom_\sG(\Vhat,\kbar[U]^*) &\xrightarrow{\,\de\,} 
&\Ext_\sG(\That,\kbar[U]^*) \\[4pt]
\kern.3em\big\downarrow{\scriptstyle\simeq} &&
\kern.3em\big\downarrow{\scriptstyle\simeq} \\[4pt]
\H^0(U,V)&\xrightarrow{\,\de\,} &\H^1(U,T)
\end{matrix}\tag{1.38}
 \end{equation}
The second vertical map is onto since $\Pic\Ubar= 0$ (see [CT/S2, 1.5.1]).

\begin{thm}{1.39}{Proposition} Let $\e\in\H^1(U,T)$. Then the
following two conditions are equivalent.
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item There exists a universal $X$-torsor $\T$ such that $[\T_U] = \e$.
 \item There is a section $\si\in\Hom_\sG(\Vhat,\kbar[U ]^*)$ of the
obvious map $\psi\colon \kbar[U]^*\to\kbar[U]^*/\kbar^*$ that maps to
$\e$ in $\H^1(U,T)$.
 \end{enumerate}
\end{thm}

\paragraph{Proof} See the ``description locale des torseurs'' in
Section~2.3 of [CT/S2].

Now assume that (1.39ii) holds. Then for any $k$-torus $S$
there is a commutative diagram
\begin{equation}
\begin{matrix}
\Ext_\sG(\Shat,\kbar^*)&\to&\Ext_\sG(\Shat,\kbar[U]^*) &\to 
& \Ext_\sG(\Shat,\kbar^*) \\[4pt]
\kern.3em\big\downarrow{\scriptstyle\simeq} &&
\kern.3em\big\downarrow{\scriptstyle\simeq} &&
\kern.3em\big\downarrow{\scriptstyle\simeq} \\[4pt]
\H^1(k,S)&\to &\H^1(U,S)&\to&\H^1(k,S)
\end{matrix}\tag{1.40}
 \end{equation}
defined in the following way. The vertical isomorphisms come from 
the spectral sequence (1.3) (see [CT/S2, 1.5.1]). The horizontal 
maps in the first square are the functorial maps and the horizontal 
map in the second square is induced by the
$\sG$-retraction $\si\psi/{\id} \colon \kbar[U]^*\to\kbar^*$ of 
the inclusion $\kbar^*\subset\kbar[U]^*$.


By completing the second square we obtain a homomorphism:
\begin{equation}
r_U\colon \H^1(U,S)\to\H^1(k,S)\tag{1.41}
 \end{equation}
which is a retraction of the functorial map from 
$\H^1(k,S)$ to $\H^1(U,S)$.

Let $r\colon \H^1(X,S)\to\H^1(k,S)$ be the composite of 
the restriction map from $\H^1(X,S)$ to 
$\H^1(U,S)$ and $r_U$.

\begin{thm}{1.42}{Proposition} Let $\T$ 
be a universal $X$-torsor and $\si\in\Hom_\sG(\Vhat,\kbar[U]^*)$
a section of $\psi \colon \kbar [U]^*\to\Vhat$ such that $\si$ 
maps to the class $[\T_U]$ of $\T_U$ in $\H^1(U,T)$ 
under the map in (1.38). Then the following hold:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item $r$ is a retraction of the functorial map from 
$\H^1(k,S)$ to $\H^1(X,S)$;
 \item $r$ is functorial under homomorphisms of $k$-tori;
 \item $r([\T]) = 0$;
 \item $r$ depends only on $[\T]$ and not on the choice of $\si$.
 \end{enumerate}
\end{thm}

\begin{pf} (c) To do this, we use the following commutative
diagram:
\begin{equation}
\renewcommand{\arraystretch}{1.3}
\begin{matrix}
\Hom_\sG(\Vhat,\kbar[U]^*)&\xrightarrow{\,\de\,}&\Ext_\sG(\That,\kbar[U]^*)\\
\big\downarrow&&\big\downarrow\\
\Hom_\sG(\Vhat,\kbar[U]^*/\kbar^*)&\xrightarrow{\,\de\,}&
\Ext_\sG(\That,\kbar[U]^*/\kbar^*)\\
\big\downarrow&&\big\downarrow\\
\Hom_\sG(\Vhat,\kbar[U]^*)&\xrightarrow{\,\de\,}&\Ext_\sG(\That,\kbar[U]^*)
\end{matrix}\tag{1.43}
 \end{equation}
where the horizontal maps are induced by (1.37) and the vertical maps by
$\psi$ and $\si$. Then $\de(\si)$ corresponds to $[\T_U]$ under the
isomorphism between $\Ext_\sG(\That,\kbar[U]^*)$ and $\H^1(U,T)$.
Therefore, $r_U ([\T_U ]) = 0$ if and only if $\de(\si)$ maps to
itself under the endomorphism of $\Ext_\sG(\That,\kbar[U ]^*)$ induced by
$\si\psi$. But this is clear from the commutative diagram (1.43).

(d) Let $\sS$ be an $X$-torsor under $S$ of type $\chi([\sS] )\in
\Hom_\sG (\Shat, \Pic\Xbar )$. Then $\sS$ is of the same type as the
$X$-torsor $\T\times^TS$ obtained from the $k$-homomorpism
$D(\tau)\colon T\to S$ dual to $\tau = \chi( [\sS] )$. Therefore, $[\sS] -
[\T\times^T S]\in\H^1(X,S))$ is the image of a unique element $\al$ in
$\H^1(k,S)$ by (1.4). Also, $r([\T\times^T S ] ) = 0$ by (b) and (c).
Hence, $r([\sS]) = \al$ by (a), thereby completing the proof.
\end{pf}

Now suppose there is a 0-cycle $z$ of degree one on $X$. Then (cf.\
(1.8)) there is a natural retraction $\rho(z)\colon
\H^1(X,S)\to\H^1(k,S)$ associated to $z$ for each $k$-torus $S$ which is
functorial under homomorphisms of $k$-tori.

\begin{thm}{1.44}{Proposition} Let $k$, $X$ be as above and suppose that
there exists a universal $X$-torsor $\T$ such that $\rho(z)([\T]) = 0$.
Let $S$ be a $k$-torus and $r$ the retraction from $\H^1(X,S)$ to
$\H^1(k,S)$ defined by $[\T]$ (see (1.42d)). Then
the two maps $\rho(z)$ and $r$ coincide.
\end{thm}

\begin{pf} The map $\rho(z)$ satisfies the same axioms
(1.42a--c) as $r$. It therefore follows from the proof of (1.42d)
that the two maps coincide.
\end{pf}

One can give another proof of Proposition~1.44 based on the
$\sG$-retraction from
$\kbar[U]^*$ to $\kbar^*$ associated to $z$.

The following result will be used in the next section in the case $S=T$.

\begin{thm}{1.45}{Corollary} Let $k$ be a number field and $X$ a smooth
proper geometrically connected $k$-variety for which $\Pic\Xbar$ is
finitely generated and torsion-free. Let $S$ be a $k$-torus. Suppose
that for each $v$ we are given a $0$-cycle $z_v$ of degree one on
$X_v:=X\times_k k_v$ and an $X_v$-torsor $\sS_v$ under $S_v$ such that
the following hold.
\begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item Manin's reciprocity condition 
$\sum_{\mathrm{all\ } v} \inv_v(\rho_v(z_v))(\sA_v) = 0$ 
holds for all $\sA\in\tH^2(X, \G_m)$.
\item There exists an element $\eta$ of 
$\H^1(k(X), S\times_k k(X))$ having the same image
as $[\sS_v]$ in $\H^1(k_v(X),S_v\times_{k_v} k_v(X))$ for each 
$v\in\Omega_k$.
 \end{enumerate}
Then there exists an element $\al\in\H^1(k, S )$ with image equal to
$\rho_v(z_v)([\sS_v])$ in $\H^1(k_v,S_v )$ for every $v\in\Omega_k$.
\end{thm}

\paragraph{Proof} Let $U$ be an open nonempty subset of $X$ and
$v\in\Omega_k$ any place of $k$. We first show that there exists a
$0$-cycle $u_v$ of degree one on $U_v:=U\times_k k_v$ with 
$\rho_v(u_v)(\sA_v) =\rho_v(z_v)(\sA_v)$ for all
$\sA_v\in\H^2(X_v,\G_m)$ and such that
$\rho_v(u_v)([\sS_v])=\rho_v(z_v)([\sS_v])$. By the additivity and
functoriality of $\rho_v$ under corestrictions it suffices to do this in
the case where $z_v$ is a $k_v$-point $P_v$. 

Let $O_v$ be an affine open neighbourhood of $P_v$. We may then represent
each element in $\H^2(X_v,\G_m)$ by an Azumaya algebra over $O_v$ (see [Mi,
p.~149]) and consider the corresponding Severi--Brauer scheme over $O_v$
(cf.\ {\em op.\ cit.}). We shall only consider elements in the finite
kernel of the specialization map from $\tH^2(X_v,\G_m)$ to
$\H^2(k_v(P_v),\G_m)$. Let $\Pi_v$ be the fibre product over $O_v$ of the
Severi--Brauer schemes corresponding to restrictions of these elements in
$\tH^2(X_v, \G_m)$. Then $\Pi_v$ is a smooth proper $O_v$-scheme and its
fibre over $P_v$ is a multiprojective space over $k_v$.

Let $W_v$ be the restriction over $O_v$ of an $X_v$-torsor of the same
type as $\sS_v$ which is trivial over $P_v$. It then follows from the
$v$-adic implicit function theorem applied to the fibre product of
$\Pi_v$ and $W_v$ over $O_v$ that there exists a $k_v$-point on $U_v\cap
O_v$ that can be lifted to $k_v$-points on $\Pi_v$ and $W_v$. This
$k_v$-point has all the desired properties. We may therefore replace
$z_v$ by a $0$-cycle on $U_v$ for each $v$ without changing the
hypothesis in Corollary~1.45.

Now choose an open subset $U$ of $X$ such that $\Pic\Ubar=0$ and such
that $\eta$ is the restriction of an element $\e\in\H^1(U, S)$. Assume,
as we may, that $z_v$ is a $0$-cycle on $U_v$ for each 
$v\in\Omega_k$.

 Now apply Theorem~1.27. Then there exists a universal torsor $\T$
over $X$ such that $\rho_v(z_v)([\T_v]) = 0$ in $\H^1(k_v,T)$ for all 
$v\in\Omega_k$. Also, let $\si$ 
be a $\sG$-module homomorphism from $\kbar[U ]^*/\kbar^*$
to $\kbar[U ]^*$ as in 
Proposition~1.39. Finally, let $r_U$ be the retraction from $\H^1(U,S)$ 
to $\H^1(k,S)$ 
in (1.41) defined by means of $\si$.

Then $\alpha=r_U(\e)\in\H^1(k,S)$ 
is the desired element with image $\rho_v(z_v)([\sS_v])$ in 
$\H^1(k_v,S_v)$ for all $v\in\Omega_k$. 
To show this, we fix one place $v$ and 
change the notation so that $k = k_v$. 
We also omit the index $v$ for 
all varieties, morphisms, cohomology groups defined over $k= k_v$. 
Thus $U$, resp.\ $\rho(z)([\sS])$, will mean $U_v$, resp.\
$\rho_v(z_v)([\sS_v])$, and $\e$, $\T$ will now mean the images after
base extension to $k_v$.  We 
shall also make use of the functoriality of $r$ and $\rho(z)$ under 
extensions of the base field without further comments.

Then we get an element $\e\in\H^1(U,S)$, a $0$-cycle $z$ 
of degree one on $U$, a universal $X$-torsor $\T$ 
with $\rho(z)([\T]) = 0$ and an $X$-torsor $\sS$ under $S$ 
satisfying the following condition:
\begin{equation}
 \begin{array}{l}
 \hbox{The image of $\e\in\H^1(U, S)$ in $\H^1(k(X), S)$ equals that of
the} \\
 \hbox{class $[\sS]\in\H^1(X,S)$ in $\H^1(k(X), S)$.}
 \end{array} \tag{$*$}
 \end{equation}

 But it follows from the commutative diagram (cf.\ (1.40))
 \[
 \begin{matrix}
\Ext_\sG(\Shat,\kbar[U]^*) &\to 
& \Ext_\sG(\Shat,\kbar(X)^*) \\[4pt]
\big\downarrow{\scriptstyle\simeq} && \big\downarrow{\scriptstyle\simeq}
 \\[4pt]
\H^1(U,S)&\to&\H^1(k(X),S)
\end{matrix}
 \]
that the restriction map from $\H^1(U, S)$ to $\H^1(k(U), S)$ 
is injective. Therefore, $\e=[\sS_U]$, and hence $r_U(\e)=r([\sS])$.
Moreover, $r([\sS])=\rho(z)([\sS])$ by Proposition~1.44.  Hence $r_U(\e)
=\rho(z)([\sS])$, as was to be proved.

In Corollary~1.45 and some other results in this section we have assumed
that the  functorial maps from $\Pic\Xbar$ to $\Pic (\kbar_v\times X)$ 
are isomorphisms for all  $v\in\Omega_k$. This was used to guarantee that
the base extensions of universal $X$-torsors to torsors  over $X_v$ remain
universal. We therefore include the following result for  which we could
find no reference.

\begin{thm}{1.46}{Proposition} Let $k$ be an algebraically closed field,
and let $X$ be a smooth and proper $k$-variety for which $\Pic X$ is
finitely generated. Then the functorial map from $\Pic X$ to $\Pic
(X\times E)$ is an isomorphism for any extension field $E$ of $k$.
\end{thm}

\paragraph{Proof} The assumption implies that $H^1(X,\O_X)=0$.
Thus $\Pic(X\times V)=\Pic X \times\Pic V$ for any (integral)
$k$-variety $V$ by the exercise on p. 292 in [Ha]. (The assumption that
$X$ is projective is not necessary since Grothendieck's theorem on
pp.~290--291 in {\it op.\ cit.}\ also holds for proper morphisms.) Now
make use of the fact that $E$ is the union of its finitely generated
$k$-subalgebras $A$. Therefore, there are canonical isomorphisms
 \begin{gather*}
 \varinjlim\Pic(\Spec A)=\Pic(E)=0 \quad\hbox{and} \\[4pt]
 \Pic(X\times E)= \varinjlim\Pic(X\times\Spec A)=
\Pic X\oplus \varinjlim\Pic( \Spec A)=\Pic X,
 \end{gather*}
as was to be proved.

\section{K theory and obstructions to the Hasse \\ principle}

Let $k$ be a perfect field, $\kbar$ an algebraic closure of $k$ and
$\sG:=\Gal(\kbar/k)$ the absolute Galois group of $k$. Let $X$ be a
smooth proper $k$-variety such that $\Xbar:=\kbar\times X$ is connected.

 Then there is a complex of discrete $\sG$-modules (cf.\ [Bl])
\begin{equation}
 \bigoplus_{\si\in\Xbar_2} K_2(\kbar(\si))
 \xrightarrow{\,\tame\,}
\bigoplus_{\ga\in\Xbar_1} \kbar(\ga)^* 
 \xrightarrow{\,\div\,}\bigoplus_{\Xbar_0} \Z, \tag{2.1}
 \end{equation} 
where $\Xbar_i$ denotes the set of points of dimension $i$. 
The first map is 
given by tame symbols and the second is the usual divisor map. Let 
$M$ be the cokernel of the first map and $\imdiv$ the
image of the second. (This notation will become natural later
after (2.4).) Then (2.1) induces a short exact sequence of 
discrete $\sG$-modules
\begin{equation}
0\to\Ker(\div)/{\Ima(\tame)}\to M\to\imdiv\to0.\tag{2.2}
 \end{equation} 
Let $Z_i(\Xbar)$ be the free abelian group of cycles of dimension $i$ on 
$\Xbar$; write $R_i(\Xbar)$ for the subgroup of $i$-cycles rationally
equivalent to zero and $\Ch_i(\Xbar ):= Z_i(\Xbar) / R_i(\Xbar)$ for the
Chow group of cycles of dimension $i$ on $\Xbar$. The degree of a
$0$-cycle on $X$ depends only on its rational equivalence class since
$\Xbar$ is proper. Let $A_0(\Xbar)$ be the subgroup of 
$\Ch_0(\Xbar )$ of $0$-cycles of degree $0$. Finally, define the map
 \[
 \pi \colon \Ch_1(\Xbar )\otimes_\Z\kbar^*\lra \Ker(\div)/{\Ima(\tame)}
 \]
by the inclusions:
 \[
 Z_1(\Xbar)\otimes_\Z\kbar^* =\bigoplus_{\Xbar_1}\kbar^*\subset\Ker(\div)
 \quad\hbox{and}\quad
 R_1(\Xbar)\otimes_\Z\kbar^*\subset\Ima(\tame).
 \]

Now let $k$, $\kbar$, $\sG$, $X$, $\Xbar$ be as above and assume in
addition that the following holds.

\begin{rem}{2.3}{Assumptions}
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\roman{enumi})}
 \item $\Ch_1(\Xbar)$ and $\Pic(\Xbar)=\Ch_{n-1}(\Xbar)$ are finitely
generated and torsion-free.
 \item The intersection pairing
$\cup\colon\Ch_1(\Xbar)\times\Ch_{n-1}(\Xbar)\to \Z$ is perfect.
 \item
$\pi\colon\Ch_1(\Xbar)\otimes_\Z\kbar^*\to\Ker(\div)/{g\Ima(\tame)}$ is
an isomorphism.
 \end{enumerate}
\end{rem}

Then $\cup$ and $\pi$ define an isomorphism between the N\'eron--Severi
torus $T = D (\Pic\Xbar)$ and $\Ker(\div) / \Ima(\tame)$. 
Suppose further that
\begin{equation}
A_0(\Xbar)=0.\tag{2.4}
 \end{equation}
 Then the Galois cohomology of (2.2) gives rise to an exact sequence
\begin{multline}
Z_0(X)^0\to H^1(\sG,T(\kbar))\to H^1(\sG,M) \\
\to\Z/{\deg(Z_0(X))}\to H^2(\sG,T(\kbar)),\tag{2.5}
 \end{multline}
where $Z_0(X)$ is the group of $0$-cycles of degree $0$. 
Denote by $\phi'$ 
the map from $Z_0(X)^0$ to $\H^1(k, T)$ 
obtained from (2.5) by identifying $H^1(\sG, T(\kbar ))$ 
with $\H^1(k,T )$.

\begin{rem}{2.6}{Example} Let $k$ be a perfect field and $X$ a
smooth proper rational geometrically connected $k$-surface. Then Bloch
[Bl]  showed that (2.3) and (2.4) hold and from that deduced the map
$\phi'$ described above. He also noticed that the values of $\phi'$ only 
depend on the rational equivalence class in $Z_0(X)$.
\end{rem}

\begin{thm}{2.7}{Proposition} Let $k$, $\kbar$, $\sG$, $X$,
$\Xbar$ be as above and assume in addition that $(2.3)$ and $(2.4)$ hold. 
Suppose that there exists a universal torsor over $X$. Then the maps
$\phi$ (see Proposition~1.9) and $\phi'$ coincide.
\end{thm}

\paragraph{Proof} This is stated and proved in [CT/S1, Section 1] for
rational surfaces, but the proof uses no other properties of rational
surfaces than (2.3) and (2.4).

Now consider a discrete valuation ring $A$ containing $k$; let $K$ be
its field of fractions and $F$ its residue field, and suppose that
these fields are perfect. For a closed point $P$ on $X_K$, write $A(P)$
for the integral closure of $A$ in $K(P)$. The valuative criterion of 
properness for $X_A\to \Spec A$ implies that there is a unique
$A$-morphism $g\colon \Spec A(P)\to X_A$ extending $P\to X_K$. Let
 \[
 \spc \colon Z_0(X_K)\to Z_0(X_F)
 \]
 be the specialization homomorphism
that sends a closed point $P$ to the cycle associated to the
$0$-dimensional closed subscheme $\Spec A(P)\times_{\Spec A} F$ of $X_F$.
Then extend $\spc$ to arbitrary $0$-cycles by additivity.

It is easy to see that $\spc$ sends $0$-cycles of degree zero to
$0$-cycles of degree zero. Denote by $\spc^0$ the associated map from 
$Z_0(X_K)^0$ to $Z_0(X_F)^0$. Then the obvious diagram
\begin{equation}
\begin{matrix}
 Z_0(X)^0& \xrightarrow{\,\id\,} &Z_0(X)^0\\[6pt]
\big\downarrow&&\big\downarrow\\[2pt]
Z_0(X_K)^0& \xrightarrow{\spc^0}
&Z_0(X_F)^0 
\end{matrix}\tag{2.8}
 \end{equation}
commutes and $\spc$ and $\spc^0$ have the expected functoriality
properties under field extensions of $k$. It can be shown that $\spc$
induces a specialization map of Chow groups of $0$-cycles, but we shall
not need this.

\begin{thm}{2.9}{Proposition} Suppose that there 
exists a universal torsor $\T$ over $X$, and let $\phi_\T$ 
be the map described in Proposition~1.9. Then the 
following holds.
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item The functorial map from $\H^1(\Spec A,T_A)$ to $\H^1(K,T_K )$ 
is injective.
 \item $\phi_\T(Z_0(X_K))\subseteq\Ima(\H^1(\Spec A,T_A)\to\H^1(K,T_K))$.
 \item The following diagram commutes
 \[
 \begin{matrix}
Z_0(X_K)& \xrightarrow{\quad\spc\quad} &Z_0(X_F)\\[6pt]
 \kern.6em\Big\downarrow\phi_{\T} && \kern.6em\Big\downarrow\phi_{\T}
\\[6pt]
\Ima(\H^1(\Spec A,T_A) &\to\H^1(K, T_K))
 \xrightarrow{\Theta} &\H^1(F,T_F) 
 \end{matrix}
 \]
for the functorial map $\Theta$ from $\H^1(\Spec A, T_A)$ (cf.\ (a)).
 \end{enumerate}
\end{thm}

\paragraph{Proof} (a) See [CT/S3, Section 4].

(b) The argument is well known (see, for example, [CT/S1, p.~428]). The 
$X_K$-torsor $\T_K$ extends to an $X_A$-torsor $\T_A$ under $T_A$, 
and any closed point $P$ on $X_K$ can be extended to a morphism 
$\Spec A(P) \to X_A$ (see the construction of $\spc$). 
Combined with the existence 
of corestriction maps from $\H^1(\Spec A(P), T_{A(P)})$ 
to $\H^1(\Spec A, T_A)$, this implies that 
$\phi(Z_0(X_K))\subseteq\H^1(\Spec A, T_A)$.

(c) The horizontal maps factorize over the completion of $K$. 
We may thus assume that $A$ is complete and hence that $A(P)$ 
is discrete for each closed point $P$. 
By using obvious functoriality properties under 
corestriction of the maps involved, one reduces
to prove that $\Theta(\phi_\T(P)) =\phi_\T (\si(P))$ 
for a rational point $P$. To see 
this, note that both composites give the pullback of $\T_A$ 
at the closed point on $X_F$ determined by $\Spec A(P)\to X_A$.

\begin{thm}{2.10}{Lemma} Let $k$ be a field of characteristic $0$, and
$X$, $Y$ two smooth, proper, geometrically connected $k$-varieties.
Suppose that $(2.3)$ and $(2.4)$ hold for $\Xbar:= X\times_k\kbar$ for any
algebraically closed field $\kbar$ containing $k$, and that there
exists a universal torsor over $X$. Then for any $0$-cycle $y$ on $Y$,
the following holds:
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item The map $\rho(y)\colon \H^1(Y,T)\to\H^1(k,T)$ factorizes through
a map $\rho'(y)$ from $\Ima\bigl(\H^1(Y,T)\to\H^1(k(Y),T\times_k
k(Y)\bigr)$ to $\H^1(k,T)$.
 \item $\phi'(Z_0(X\times_k k(Y))^0)\subseteq
\Ima\bigl(\H^1(Y,T)\to\H^1(k(Y),T\times_k k(Y))\bigr)$
 \item $\phi'(Z_0(X\times_k k(Y))^0)$ maps to
$\phi'(Z_0(X)^0)$ under $\rho'(y)$.
 \end{enumerate}
\end{thm}

\begin{pf} (a) See [CT/S2, 2.7.5].

(b) By [CT/S1, p.~428], it is known that
 \begin{multline*}
 \Ima\Bigl(\H^1(Y,T)\to\H^1(k(Y),T\times_k k(Y))\Bigr) \\
  =\bigcap_Q\Ima\Bigl(\H^1(\O_{Y,Q},T\times_k\O_{Y,Q})\to
 \H^1(k(Y), T\times_k k(Y))\Bigr),
 \end{multline*}
 where $Q$ runs over all points of codimension one on $Y$. The desired
inclusion is therefore a consequence of (2.9b) and the fact that
$\phi=\phi'$ (see Proposition~2.7).

(c) Let $y =\sum n_i y_i$, where the $y_i$ are closed points on $Y$.
Since $\rho$ is additive with respect to $Z_0(Y)$, it suffices to prove
the statement for each $\rho'(y_i)$. By factorizing $\rho(y_i)$ through
$\H^1(Y\times_k k(y_i),T\times_k k(y_i))$ and using the functoriality of
$\phi$ under extensions of the base field, we reduce further to the case
when $y$ is a rational point. We now use induction on $\dim Y$ and note
that the case $\dim Y= 0$ is trivial. If $\dim Y\ge1$, let $f \colon
\tilde Y\to Y$ be the blowup at the $k$-rational point $y$, $Z =
f^{-1}(y)$ and $A$ the stalk of $\O_{\tilde Y}$ at the generic point of
$Z$. Then $A$ is a discrete valuation ring with field of fractions $K:=
k(\tilde Y) = k(Y)$ and residue field $F:=k(Z )$.
Then by (2.9c) and Proposition~2.7 there is a commutative diagram
\begin{equation}
\begin{matrix}
Z_0(X_K)^0& \xrightarrow{\quad\spc\quad} &Z_0(X_F)^0\\[6pt]
 \kern.4em\Big\downarrow\phi' && \kern.4em\Big\downarrow\phi'
\\[6pt]
\Ima (\H^1(\Spec A,T_A) &\to\H^1(K, T_K))
 \xrightarrow{\Theta} &\H^1(F,T_F) 
\end{matrix}\tag{2.11}
 \end{equation}

Now choose a rational $K$-point $z$ on the above $Z$. Then, since $\dim
Z=\dim Y-1$, we obtain from the induction assumption that (c) holds if we
consider the pair $(Z,z)$ instead of $(Y,y)$. Further, by using the
commutativity of (2.11), we deduce from this that (c) also holds for the
pair $(\tilde Y,\tilde y)$, which in turn implies that (c) holds for
$(Y,y)$ since $K(Y)=K(\tilde Y)$ and $f(z)=y$. This finishes the proof.
\end{pf}

We shall in the sequel use the following functoriality properties of
(2.1). Let $k\subset k_1$ be an extension of perfect fields with
algebraic closures $\kbar\subset\kbar_1$. Put $\sG = \Gal(\kbar / k)$,
$\sG_1=\Gal(\kbar_1/k_1)$, 
$X_1 = X\times_k k_1$ and $\Xbar_1 =X\times_k\kbar_1$.
We may then consider (2.1) as a sequence of $\sG_1$-modules through 
the homomorphism $\sG_1\to\sG$ obtained by 
restricting the $\sG_1$-action to $\kbar$. This sequence
is the upper row in a commutative diagram of discrete 
$\sG_1$-modules where the
bottom row is given by (2.1) applied to $\Xbar_1$. 
Now suppose that $\Xbar$ and $\Xbar_1$
satisfy (2.3) and (2.4). Then we obtain the following commutative 
diagram with exact rows from the functoriality of (2.5) under 
extension of the base field:
\begin{equation}
 \renewcommand{\arraycolsep}{0.12em}
\begin{matrix}
Z_0(X)^0&\to& \H^1(k, T)&\to& H^1(\sG, M)&
\to& \Z/{\deg Z_0(X)}&\to &\H^2(k, T)\\[6pt]
\big\downarrow&&\big\downarrow&&
\big\downarrow&&\big\downarrow&&\big\downarrow\\[6pt]
Z_0(X_1)^0&\to& \H^1(k_1, T_1)&\to& H^1(\sG_1, M_1)&
\to& \Z/{\deg Z_0(X_1)}&\to &\H^2(k_1, T_1)
\end{matrix}\tag{2.12}
 \end{equation}
where $T_1=T\times_k k_1$ and $M_1$ 
is the cokernel of the tame symbol map in 
(2.1) for $\Xbar_1$. Note that $T_1$ 
can be identified with the N\'eron--Severi 
torus of $\Xbar_1$ since the functorial map gives an isomorphism from 
$\Pic(\Xbar)$ to $\Pic(\Xbar_1)$ by Proposition~1.46.
\medskip

{From} now on, let $k$ be a number field and choose algebraic 
closures $\kbar_v$ of $k_v$, and embeddings 
$\kbar\subset \kbar_v$ for each place $v$ of $k$. 
Let $\sG_v=\Gal(\kbar_v/k_v)$,
$X_v =X\times_k k_v$, $\Xbar_v=X\times_k \kbar_v$, 
and let $M_v$ be the cokernel of the tame symbol map
in (2.1) for $\Xbar_v$. 
Write $\Sha^1(k,M)$ for the kernel of the diagonal map from
$H^1(\sG, M)$ to $\prod_{\mathrm{all\ } v} H^1(\sG_v, M_v)$.

\begin{thm}{2.13}{Theorem} Let $k$ 
be a number field and $X$ a smooth proper 
geometrically connected $k$-variety such that 
$(2.3)$ and $(2.4)$ hold 
for $\Xbar:=X\times_k\kbar$ for any algebraically 
closed field $\kbar$ containing $k$. 
Suppose that for each $v\in\Omega_k$ we are given a 
$0$-cycle $z_v$ of degree 
one on $X_v$ and that Manin's reciprocity condition 
$\sum_{\mathrm{all\ } v} \inv_v(\rho_v(z_v))(\sA_v) = 0$ 
holds for all $\sA\in\tH^2(X, \G_m)$.

Then $\Sha^1(k,M)$ maps onto $\Z/{\deg(Z_0(X))}$ under the map from
$H^1(\sG, M)$ in $(2.5)$. In particular, if $\Sha^1(k,M) = 0$, then
there is a $0$-cycle of degree one on $X$.
\end{thm}

\paragraph{Proof} Let $\kbar$ be an algebraic closure of $k$, 
and $\Kbar$ an algebraic closure of the function field 
$\kbar(\Xbar)$ of $\Xbar:=X\times_k \kbar$. Then $\Kbar$ is 
also an algebraic closure of
$K:= k(X)$ and we have a natural homomorphism from 
$\HH:= \Gal(\Kbar/ K)$ to
$\sG:= \Gal(\kbar/k)$. Now consider (2.12) for $k_1=K$. 
Then $\Z/{\deg(Z_0(X_1))} = 0$ 
since the generic point of $X$ defines a $K$-rational 
point on $X_1= X_K$.

By Proposition~1.26, since Manin's reciprocity condition is satisfied,
there exists a universal torsor over $X$. In turn, this implies that (cf.\
(2.2.5) and (2.2.8) in [CT/S2]) the map from $\H^2(k,T)$ to
$\H^2(k_1,T_1)$ is injective. We thus obtain the following commutative
diagram with exact rows from (2.12):
\begin{equation}
 \renewcommand{\arraycolsep}{0.3em}
 \begin{array}{ccccccccc}
Z_0(X)^0&\to& \H^1(k, T)&\to& H^1(\sG, M)&
\to& \Z/{\deg Z_0(X)}&\to &0\\[6pt]
\big\downarrow&&\big\downarrow&&
\big\downarrow&&\big\downarrow&&\\[6pt]
Z_0(X_K)^0&\to& \H^1(K, T_K)&\to& H^1(\HH, M_K)&
\to& 0&&
\end{array} \tag{2.14}
 \end{equation}
where $M_K$ is the cokernel of the tame symbol map (see (2.1)) for 
$X\times_k\Kbar$. The assertion that $\Sha^1(k,M)$  maps onto
$\Z/{\deg(Z_0(X))}$ therefore reduces
to the assertion that $H^1(\sG, M)$ is generated by 
$\Sha^1(k,M)$ and the image of $\H^1(k,T)$.

For each place $v$ of $k$ there is a commutative diagram with exact rows:
\begin{equation}
 \renewcommand{\arraycolsep}{0.3em}
\begin{array}{ccccccccc}
Z_0(X)^0&\to& \H^1(k, T)&\to& H^1(\sG, M)&
\to& \Z/{\deg Z_0(X)}&\to &0\\[6pt]
\big\downarrow&&\big\downarrow&&
\big\downarrow&&\big\downarrow&&\\[6pt]
Z_0(X_v)^0&\to& \H^1(k_v, T_v)&\to& H^1(\sG_v, M_v)&
\to& 0&&
\end{array}\tag{2.15}
 \end{equation}
where the zero in the second row comes from the existence of a 
$0$-cycle of degree one on $X_v$. Let $K_v:= k_v(X_v)$ 
be the function field 
of $X_v$, and $\Kbar_v$ an algebraic closure 
of the function field $\kbar_v(\Xbar_v)$ of $\Xbar_v$ 
containing $\Kbar$. Then $\Kbar_v$ is also
an algebraic closure of $K_v$, and there are natural homomorphisms from
$\HH_v = \Gal(\Kbar_v /K_v)$ to $\sG_v$ and $\HH$. 
Let $M_{K_v}$ be the cokenel of 
the tame symbol map (see (2.1)) for $X\times_k \Kbar_v$. Then there are 
commutative diagrams with exact rows
\begin{equation}
 \renewcommand{\arraycolsep}{0.35em}
\begin{array}{ccccccc}
Z_0(X_K)^0&\to& \H^1(K, T_K)&\to& H^1(\HH, M_K)&\to &0\\[6pt]
\big\downarrow&&\big\downarrow&&
\big\downarrow&&\\[6pt]
Z_0(X_{K_v})^0&\to& \H^1(K_v, T_{K_v})&\to& H^1(\HH_v, M_{K_v})&
\to& 0
\end{array}\tag{2.16}
 \end{equation}
and
\begin{equation}
 \renewcommand{\arraycolsep}{0.35em}
\begin{array}{ccccccc}
Z_0(X_v)^0&\to& \H^1(k_v, T_v)&\to& H^1(\sG_v, M_v)&\to &0\\[6pt]
\big\downarrow&&\big\downarrow&&
\big\downarrow&&\\[6pt]
Z_0(X_{K_v})^0&\to& \H^1(K_v, T_{K_v})&\to& H^1(\HH_v, M_{K_v})&
\to& 0
\end{array}\tag{2.17}
 \end{equation}
and (2.14--17) are parts of a three-dimensional commutative 
diagam that also contains the commutative diagrams
 \[
 \begin{matrix}
 \renewcommand{\arraycolsep}{0.35em}
\H^1(k,T)&\to&\H^1(k_v,T_v) \\[4pt]
\big\downarrow&&\big\downarrow\\[4pt]
\H^1(K,T_K)&\to&\H^1(K_v,T_{K_v})
\end{matrix} \quad\hbox{and}\quad
\begin{matrix} 
 \renewcommand{\arraycolsep}{0.35em}
H^1(\sG, M )&\to& H^1( \sG_v, M_v)\\[4pt]
\big\downarrow&&\big\downarrow\\[4pt]
H^1(\HH, M_K)&\to& H^1(\HH_v, M_{K_v}).
\end{matrix}
 \]
Now let $\mu$ be an element of $H^1(\sG, M)$, $\mu_K$ its image in
$H^1(\HH,M_K)$, $\mu_v$ its image in $H^1(\sG_v,M_v)$ and $\mu_{K_v}$ its
image in $H^1(\HH_v,M_{K_v})$. Lift $\mu_K$ to an element $\eta$ of
$\H^1(K,T_K)$ (cf.\ (2.14)) and $\mu_v$  to an element
$\beta_v\in\H^1(k_v, T_v)$ (cf.\ (2.15)), and consider  the images
$\eta_v$ of $\eta$ and $\beta_{K_v}$ of $\beta_v$ in $\H^1(K_v,T_{K_v})$.
Then $\eta_v-\beta_{K_v}\in\Ker\bigl(\H^1(K_v,T_{K_v})\to
\H^1(\HH_v,M_{K_v})\bigr)$ which by exactness of the second  row in
(2.17) implies that $\eta_v - \beta_{K_v}\in\phi'(Z_0(X_{K_v})^0)$. Thus
by (2.10b), $\eta_v- \beta_{K_v}\in\Ima\bigl(\H^1(X_v,T_{v})\to\H^1(K_v,
T_{K_v})\bigr)$, and hence so does $\eta_v$. Choose for each place $v$
an $X_v$-torsor $\T_v$ under $T_v$  such that $[\T_v]\in\H^1(X_v, T_v)$
maps to $\eta_v$ in $\H^1(K_v,T_{K_v})$. Then since 
$\eta_v- \beta_{K_v}\in\phi'(Z_0(X_{K_v})^0)$ we conclude from (a)
and (c) of Lemma~2.10 that $\rho_v(z_v)([\T_v] - \beta_{K_v})\in
\phi'(Z_0(X_v)^0)$. This means that
$\rho_v(z_v)([\T_v])$ and $\rho_v(z_v)(\beta_{K_v}) =\beta_v$ 
have the same image $\mu_v$ in $H^1(\sG_v, M_v)$.

{From} the assumption that the $0$-cycles $(z_v)_{v\in\Omega_k}$ satisfy
Manin's reciprocity condition for all $\sA\in\tH^2(X,\G_m)$, we deduce
from Corollary~1.45 that there exists $\alpha\in\H^1(k,T)$ with image 
$\rho_v(z_v)([\T_v])$ in $\H^1(k_v,T_v)$ for each place $v\in\Omega_k$. 
Therefore, $\alpha\in\H^1(k, T )$ maps to an element in $H^1(\sG, M)$
with the same image as $\mu$ in $H^1(\sG_v, M_v)$ for each
$v\in\Omega_k$. This completes the proof.

\begin{thm}{2.18}{Theorem} Let $k$ be a number field and $X$ a
smooth proper geo\-metrically connected $k$-surface. Suppose that there 
exists a rational function $t\in k(X)$ on $X$ such that $k(X)$ 
is the function field of a Severi--Brauer curve over $k(t)$. 
Then
 \begin{enumerate}
 \renewcommand{\labelenumi}{(\alph{enumi})}
 \item $\Sha^1(k,M) = 0$,
 \item Suppose that for each $v\in\Omega_k$ 
we are given a $0$-cycle $z_v$ of 
degree one on $X_v$ such that Manin's reciprocity 
condition $\sum_{\mathrm{all\ } v} \inv_v(\rho_v(z_v))(\sA_v) = 0$ 
holds for all $\sA\in \H^2(X,\G_m)$. Then there exists 
a $0$-cycle of degree one on $X$.
 \end{enumerate}
\end{thm}

\paragraph{Proof} (a) $H^1(\sG, M)$ and $\Sha^1(k,M)$ are 
$k$-birational invariants [Sa]. 
The assumptions on $X$ implies that it is $k$-birational to a 
relatively minimal conic bundle surface over $\P^1$. 
It is therefore 
sufficient to prove that $\Sha^1(k,M)=0$ for relatively minimal conic 
bundle surface over $\P^1$. But this is the main result of [Sa].

(b) This is a consequence of (a) and the previous theorem.

\begin{thebibliography}{CT/S3}

\bibitem[Bl]{Bl} S. Bloch, On the Chow groups of certain rational
surfaces, {\it Ann. Sci. \'Ecole Norm. Sup.} {\bf 14} (1981) 41--59

\bibitem[Bo]{Bo} A. Borel, {\it Linear algebraic groups}, 2nd ed.
Graduate texts in Math. 126, Springer-Verlag

\bibitem[CT/S1]{CT/S1} J.-L. Colliot-Th\'el\`ene and J.-J. Sansuc, On the
Chow groups of certain rational surfaces: a sequel to a paper of S.
Bloch, {\it Duke Math. J.} {\bf 48} (1981) 421--447

\bibitem[CT/S2]{CT/S2} J.-L. Colliot-Th\'el\`ene et J.-J. Sansuc, La
descente sur les vari\'et\'es rationnelles. II, {\it Duke Math. J.}
{\bf54} (1987) 375--492. (cf.\ {\it C. R. Acad. Sci. Paris} {\bf 284}
S\'erie A (1977) 1215--1218)

\bibitem[CT/S3]{CT/S3} J.-L. Colliot-Th\'el\`ene and J.-J. Sansuc,
Principal homogeneous spaces under flasque tori: applications, {\it J.
Algebra} {\bf 106} (1987) 145--202

\bibitem[CT/SwD]{CT/SwD} J.-L. Colliot-Th\'el\`ene and Sir Peter
Swinnerton-Dyer, Hasse principle and
weak approximation for pencils of Severi--Brauer varieties and
similar varieties, {\it J. reine angew. Math.} {\bf 453} (1994) 49--112

\bibitem[Ha]{Ha} R. Hartshorne, {\it Algebraic Geometry}, Springer-Verlag,
1977

\bibitem[Ma]{Ma} Yu.I. Manin, Le groupe de Brauer--Grothendieck en
g\'eom\'etrie diophantienne, In: {\it Actes Congr\`es Intern. Math.}
(Nice 1970), Gauthiers-Villars, Paris 1971, Tome 1, 409--411

\bibitem[Mi]{Mi} J.S. Milne, {\it {\'E}tale cohomology}, Princeton
University Press, Princeton 1980

\bibitem[Sa]{Sa} P. Salberger, Zero-cycles on rational surfaces over 
number fields, {\it Invent. Math.} {\bf 91} (1988) 505--524

\bibitem[Sk]{Sk} A.N. Skorobogatov, Beyond the Manin obstruction, {\it
Invent. Math.} {\bf 113} (1999) 399--424

\bibitem[Ta1]{Ta1} J. Tate, The cohomology groups of tori in finite
Galois extensions of number fields, {\it Nagoya Math. J.} {\bf 27} 
(1966) 709--719

\bibitem[Ta2]{Ta2} J. Tate, Global class field theory, In: {\it Algebraic
number theory}, J.W.S. Cassels, A. Fr\"ohlich, eds., Academic Press 1967

\end{thebibliography}

\bigskip
\noindent
Per Salberger, \\
Department of Mathematics, \\
Chalmers University of Technology, \\
S--412 96 G\"oteborg, Sweden \\
e-mail: salberg@math.chalmers.se

\end{document}


p. 257, line -10:
An $X$-torsor under a $k$-torus is locally trivial in the \'etale
topology by a theorem of Grothendieck.

Either this is a theorem of Grothendieck and you should give a reference,
or it is trivial. (By definition of torsor, some pullback is a product;
therefore the morphism \sS -> X is smooth, so locally has an etale
section.)