\section{Introduction}

Let $C$ be the spectrum of the ring of integers in a number field or a smooth projective geometrically connected curve over a finite field with function field $K$. Let $\mathcal{X}$ be a $2$-dimensional regular scheme and $\pi:\mathcal{X}\longrightarrow C$ be a proper flat morphism such that the generic fiber $X$ of $\pi$ is smooth and geometrically connected over $K$.  Artin and Grothendieck (cf.\cite[$\S$ 4]{Gro3} or \cite[Prop. 5.3]{Ulm}) proved that there is an isomorphism 
$$
{\hbox{\cyr Sh}(\mathrm{Pic}^{0}_{X/K})\cong {\mathrm{Br}}(\mathcal{X})
$$
up to finite groups. Assuming the finiteness of ${\hbox{\cyr Sh}(\mathrm{Pic}^{0}_{X/K})$ or ${\mathrm{Br}}(\mathcal{X})$, Milne \cite{Mil4}, Gonzales-Aviles \cite{Goa} and Geisser \cite{Gei1} gave a precise relation between their orders.

In this paper, we generalize the original result of Artin and Grothendieck to fibrations of high relative dimensions over arbitrary finitely generated fields.  As a consequence, we reduce the Tate conjecture for divisors on smooth projective varieties over a finitely generated field $K$ to the Tate conjecture for divisors on smooth projective varieties over the prime subfield of $K$.

For any noetherian scheme $X$,  the \emph{cohomological Brauer group}
$$
{\mathrm{Br}}(X):=H^2(X,\mathbb{G}_m)_{{\mathrm{tor}}}
$$
is defined to be the torsion part of the etale cohomology group $H^2(X,\GG_m)$.
\subsection{Tate conjecture and Brauer group}
Let us first recall the Tate conjecture for divisors over finitely generated fields.
\begin{conj}\label{Tateconj} ( Conjecture $T^1(X,\ell)$)
Let $X$ be a projective and smooth variety over a finitely generated field $k$ of characteristic $p\geq0$, and $\ell\neq p$ be a prime number. Then the cycle class map\\
$$
\mathrm{Pic}(X)\otimes_\mathbb{Z}\mathbb{Q}_\ell\longrightarrow H^2(X_{k^s},\mathbb{Q}_\ell(1))^{G_k}
$$
is surjective.
\end{conj}
The conjecture is proved for abelian varieties by Tate \cite{Tat3} (over finite fields), Zarhin \cite{Zar1, Zar2} (over positive characteristics) and Faltings \cite{Fal1, Fal2} (over characteristic zero), for K3 surfaces over characteristic zero by Andr\'e \cite{And} and Tankeev \cite{Tan4, Tan5}, for $K3$ surfaces over positive characteristics by Nygaard \cite{Nyg}, Nygaard-Ogus \cite{NO}, Artin-Swinnerton-Dyer\cite{ASD}, Maulik \cite{Mau}, Charles \cite{Cha} and Madapusi-Pera \cite{MP}.\\
\\
By a well-known result (see Proposition $\ref{prop2.1}$ ), $T^1(X,\ell)$ is equivalent to the finiteness of ${\mathrm{Br}}(X_{k^s})^{G_k}(\ell)$.
\begin{thm}\label{reduction}
Let $k$ be a prime field and $K$ be a finitely generated field over $k$. Let $\ell$ be a prime different from char$(k)$. Assuming that $T^1(\mathcal{X},\ell)$ is true for all smooth projective varieties $\mathcal{X}$ over $k$, then $T^1(X,\ell)$ holds for all smooth projective varieties $X$ over $K$.
\end{thm}
\begin{rem}
$T^1(X,\ell)$ for smooth projective varieties over finite fields has been reduced to $T^1(X,\ell)$ for smooth projective surfaces $X$ over finite fields with $H^1(X,{\mathcal {O}}_X)=0$ by the work of de Jong \cite{deJ2}, Morrow\cite[Thm. 4.3]{Mor} and Yuan \cite[Thm. 1.6]{Yua1}. 
\end{rem}
The above reduction theorem was proved by Ambrosi\cite{Amb}. The idea of our proof is to work on the obstructions of the Tate conjecture, i.e. Brauer groups directly. We obtain the following relation between Brauer groups for fibrations. 

\subsection{Brauer groups for fibrations}
For a torsion abelian group $M$ and a prime number $p$, let $M({\mathrm{non}\text{-}} p)$ denote the subgroup of elements of order prime to $p$. We also define $M({\mathrm{non}\text{-}} p):=M$ for $p=0$. For two abelian groups $M$ and $N$, we say that they are \emph{almost isomorphic} if there exists a subquotient $M_1/M_0$ of $M$(resp. $N_1/N_0$ of $N$) such that $M_0$ (resp. $N_0$) and $M/M_1$ (resp. $N/N_1$) are finite and $M_1/M_0\cong N_1/N_0$. Let $f:M\rightarrow N$ be a homomorphism with a finite cokernel and a kernel almost isomorphic to an abelian group $H$, in this situation, we say that the sequence $0\rightarrow H\rightarrow M\rightarrow N\rightarrow 0$ is exact up to finite groups.
\begin{thm}
\label{thm1.2}
Let $k$ be a finitely generated field of characteristic $p\geq 0$. Let $\mathcal{X}$ be a smooth geometrically connected variety over $k$ and $C$ be a smooth projective geometrically connected curve over $k$ with function field $K$. Let $\pi:\mathcal{X}\longrightarrow C$ be a dominant morphism such that the generic fiber $X$ is smooth projective geometrically connected over $K$. Let $K^{\prime}$ denote $Kk^s$ and write $P$ for $\mathrm{Pic}^{0}_{X/K,{\mathrm{red}}}$. Define
$${\hbox{\cyr Sh}_{K^\prime}(P):={\mathrm{Ker}}(H^1(K^\prime, P)
\longrightarrow \prod_{v\in C_{k^s}}H^1(K_{v}^{sh}, P)).$$
Then there is an exact sequence up to finite groups
$$
 0 \rightarrow {\hbox{\cyr Sh}_{K^\prime}(P)^{G_k}({\mathrm{non}\text{-}} p)\rightarrow {\mathrm{Br}}(\mathcal{X}_{k^s})^{G_k}({\mathrm{non}\text{-}} p)\rightarrow {\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)\rightarrow 0.
$$

\end{thm}
In the case that $k$ is a finite field and ${\mathcal {X}}$ is projective over $k$, the following result generalizes Artin-Grothendieck's theorem to high relative dimensions.
\begin{cor}\label{mainthm}
Let $\pi:\mathcal{X}\longrightarrow C$ be a proper flat morphism, where $C$ is a smooth projective geometrically connected curve over a finite field $k$ of characteristic $p$ with function field $K$. Assuming that $\mathcal{X}$ is smooth projective over $k$ and the generic fiber $X$ of $\pi$ is smooth projective geometrically connected over $K$, then there is an exact sequence up to finite groups
$$
0\rightarrow {\hbox{\cyr Sh}(\mathrm{Pic}_{X/K,{\mathrm{red}}}^0)({\mathrm{non}\text{-}} p)\rightarrow {\mathrm{Br}}(\mathcal{X})({\mathrm{non}\text{-}} p)\rightarrow {\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)\rightarrow 0.
$$
\end{cor}
In Theorem \ref{thm1.2}, when $k$ is finite, the canonical maps
 ${\hbox{\cyr Sh}(P)({\mathrm{non}\text{-}} p) \rightarrow {\hbox{\cyr Sh}_{K^\prime}(P)^{G_k}({\mathrm{non}\text{-}} p)$ and ${\mathrm{Br}}({\mathcal {X}})\rightarrow {\mathrm{Br}}(\mathcal{X}_{k^s})^{G_k}$ have finite kernels and cokernels (cf. Proposition \ref{agsha} and \cite[Cor. 1.4]{Yua2}). Thus, Corollary \ref{mainthm} follows from Theorem \ref{thm1.2}. It is well-known that the BSD conjecture for an abelian variety over a global function field is equivalent to the finiteness of the Tate-Shafarevich group (cf. \cite{Mil5} and \cite {Sch}). Thus, Corollary \ref{mainthm} implies the following result:
 \begin{cor}\label{Gei}
 Notations as in Corollary \ref{mainthm}, then 
$T^{1}(\mathcal{X},\ell)$ is equivalent to the BSD conjecture for $\mathrm{Pic}^{0}_{X/K,{\mathrm{red}}}$ and $T^1(X,\ell)$.
\end{cor}
\begin{rem}
Geisser\cite{Gei2} first studied fibrations of high relative dimensions over finite fields by using etale motivic cohomology theory. He proved Corollary \ref{Gei} and gave a precise relation between the orders of the Tate-Shafarevich group of the Albanese variety of the generic fiber and the Brauer group of ${\mathcal {X}}$ under the assumption of the finiteness of ${\mathrm{Br}}({\mathcal {X}})$. For fibrations over the spectrum of the ring of integers in a number field of relative dimension $>1$, the above question was first studied by Tankeev (cf.\cite{Tan1,Tan2,Tan3}). In our forthcoming work, we will prove a version of Corollary \ref{mainthm} over the spectrum of the ring of integers in a number field.
\end{rem}


\subsection{The finiteness of the Galois invariant part of geometric Brauer group}
 In \cite{SZ1}, Skorobogatov and Zarkhin conjectured that ${\mathrm{Br}}(X_{k^s})^{G_k}$ is finite for any smooth projective variety $X$ over a finitely generated field $k$. By the following result, assuming Tate conjecture for divisors, we have that ${\mathrm{Br}}(X_{k^s})^{G_k}({\mathrm{non}\text{-}} p)$ is finite if $k$ is a finitely generated field of characteristic $p>0$.
\begin{thm}\label{finitebrauer}
Let $k$ be a prime field and $K$ be a field finitely generated over $k$. Then the following statements are true.
\begin{itemize}
\item[(1)] If $k=\mathbb{Q}$ and ${\mathrm{Br}}(X_{k^s})^{G_k}$ is finite for any smooth projective variety $X$ over $k$, then ${\mathrm{Br}}(X_{K^s})^{G_K}$ is finite for any smooth projective variety $X$ over $K$.
\item[(2)] If $k=\FF_p$ and the Tate conjecture for divisors holds for any smooth projective variety over $k$, then ${\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)$ is finite for any smooth projective variety $X$ over $K$.
\end{itemize}
\end{thm}
\begin{rem}
In \cite{SZ1,SZ2}, Skorobogatov and Zarkhin proved the finiteness of ${\mathrm{Br}}(X_{k^s})^{G_k}({\mathrm{non}\text{-}} p)$ for abelian varieties and $K3$ surfaces(${\mathrm{char}}(k)\neq 2$).
\end{rem}
\subsection{Sketch of the proof of Theorem \ref{thm1.2}}
For the proof of Theorem \ref{thm1.2}, the left exactness will follow from Grothendieck's argument in \cite[$\S$ 4]{Gro3}. The tricky part is to show that the natural map
$${\mathrm{Br}}({\mathcal {X}}_{k^s})^{G_k}({\mathrm{non}\text{-}} p)\longrightarrow {\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)$$
has a finite cokernel. The key idea is to use a pull-back trick developed by Colliot-Thél\`ene and Skorobogatov (cf.\cite{CTS} or \cite{Yua2} ) to reduce it to cases of relative dimension $1$. Choose an open dense subset $U\subseteq C$ such that $\pi^{-1}(U)\longrightarrow U$ is smooth and proper. Write ${\mathcal {V}}$ for $\pi^{-1}(U)$. By purity for Brauer groups, the natural map
$${\mathrm{Br}}({\mathcal {X}}_{k^s})^{G_k}({\mathrm{non}\text{-}} p)\longrightarrow {\mathrm{Br}}({\mathcal {V}}_{k^s})^{G_k}({\mathrm{non}\text{-}} p)$$
is injective and has a finite cokernel. It suffices to show that
$${\mathrm{Br}}({\mathcal {V}})({\mathrm{non}\text{-}} p)\longrightarrow {\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)$$
has a finite cokernel. For simplicity, we assume that $X(K)$ is not empty. By the spectral sequence
$$ E^{p,q}_2=H^p(U,R^q\pi_*\GG_m)\Rightarrow H^{p+q}({\mathcal {V}},\GG_m),$$
we get an exact sequence
$$ H^2({\mathcal {V}},\GG_m)\longrightarrow H^0(U,R^2\pi_*\GG_m)\stackrel{d_2^{0,2}}{\longrightarrow} H^2(U,R^1\pi_*\GG_m).$$
And we will show there is a canonical isomorphism
 $$H^0(U,R^2\pi_*\GG_m)({\mathrm{non}\text{-}} p)\cong{\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p).$$
So the question is reduced to show the vanishing of $d^{0,2}_{2}$. In the case of relative dimension $1$, this is obvious since $R^2\pi_*\GG_m=0
 $ by Artin's theorem \cite[Cor. 3.2]{Gro3}. In general, taking a smooth projective curve $Y\subseteq X$, then spreading out, we get a relative curve $\pi^\prime:{\mathcal {Y}}\longrightarrow C$  and a $C$-morphism ${\mathcal {Y}}\longrightarrow {\mathcal {X}}$. This gives a commutative diagram
\begin{displaymath}
\xymatrix{ 
H^0(U,R^2\pi_*\GG_m)\ar[r] \ar[d]& H^2(U,R^1\pi_*\GG_m) \ar[d] 
\\
H^0(U,R^2\pi^\prime_*\GG_m) \ar[r] & H^2(U,R^1\pi^\prime_*\GG_m)
}
\end{displaymath}
We hope that the second column is injective. In fact, we need finitely many $\pi^{i}:{\mathcal {Y}}_i\longrightarrow C$ such that the induced morphism of abelian sheaves
$$R^1\pi_*\GG_m\longrightarrow \bigoplus_i R^1\pi_*^{i}\GG_m$$
is split up to abelian sheaves of finite exponent. The splitting
implies that the natural map
$$H^2(U,R^1\pi_*\GG_m)\longrightarrow \bigoplus_i H^2(U,R^1\pi^i_*\GG_m)$$
has a kernel of finite exponent. It follows that $d^{0,2}_2$ has an image of finite exponent.

To get this splitness, we will choose $Y_i$ as the complete intersection of very ample divisors on $X$ such that there exists an abelian variety $A/K$ and a $G_K$-equivariant map
$$\mathrm{Pic}(X_{K^s})\times A(K^s)\longrightarrow \bigoplus_i\mathrm{Pic}(Y_{i,K^s})$$
with finite kernel and cokernel (cf. \cite[\S 3.3, p17]{Yua2}). 

\subsection{Notation and Terminology}
\subsubsection*{Fields}
By a \emph{finitely generated field}, we mean a field which is finitely generated over a prime field.\\
For any field $k$, denote by $k^s$ the separable closure. Denote by $G_k={\mathrm{Gal}}(k^s/k)$ the absolute Galois group of $k$.
\subsubsection*{Henselization}
Let $R$ be a noetherian local ring, denote by $R^h$ (resp. $R^{sh}$) the henselization (resp. strict henselization) of $R$ at the maximal ideal. If $R$
is a domain, denote by $K^h$ (resp. $K^{sh}$) the fraction field of $R^{h}$ (resp. $R^{sh}$).
\subsubsection*{Varieties}
By a \emph{variety} over a field $k$, we mean a scheme which is separated and of finite type over $k$.

For a smooth proper geometrically connected variety $X$ over a field $k$, we use $\mathrm{Pic}^0_{X/k}$ to denote the identity component of the Picard scheme $\mathrm{Pic}_{X/k}$. Denote by $\mathrm{Pic}^0_{X/k,{\mathrm{red}}}$ the underlying reduced closed subscheme of $\mathrm{Pic}^0_{X/k}$(which is known to be an abelian variety cf. FGA 236-17, Cor. 3.2.)
\subsubsection*{Cohomology}
The default sheaves and cohomology over schemes are with respect to the
small \'etale site. So $H^i$ is the abbreviation of $H_{\et}^i$.
\subsubsection*{Brauer groups}
For any noetherian scheme $X$, denote the \emph{cohomological Brauer group}
$$
{\mathrm{Br}}(X):=H^2(X,\mathbb{G}_m)_{{\mathrm{tor}}}.
$$
\subsubsection*{Tate-Shafarevich group}
For an abelian variety $A$ defined over a global field $K$, denote the  \emph{Tate–Shafarevich group}

$$
{\hbox{\cyr Sh}(A):={\mathrm{Ker}}(H^1(G_K, A)\longrightarrow \prod _v H^1(G_{K_v}, A)),
$$
where $K_v$ denotes the completion of $K$ at a place $v$. 
\subsubsection*{Abelian group}
For any abelian group $M$, integer $m$ and prime $\ell$, we set\\
$$M[m]=\{x\in M| mx=0\},\quad M_{{\mathrm{tor}}}=\bigcup\limits_{m\geq 1}M[m],\quad  M(\ell)=\bigcup\limits_{n\geq 1}M[\ell^n], $$
$$T_\ell M={\mathrm{Hom}}_\mathbb{Z}(\mathbb{Q}_\ell/\mathbb{Z}_\ell, M)=\lim \limits_{\substack{\leftarrow \\ n}}M[\ell^n],\quad V_\ell M= T_\ell(M)\otimes_{\mathbb{Z}_\ell}\mathbb{Q}_\ell.$$
For any torsion abelian group $M$ and prime number $p$, we set
$$M({\mathrm{non}\text{-}} p)=\bigcup\limits_{p\nmid m}M[m].$$ 
For two abelian groups $M$ and $N$, we say that they are \emph{almost isomorphic} if there exist filtrations $M_0\subseteq M_1\subseteq M$ and $N_0\subseteq N_1\subseteq N$ such that $M_0$ (resp. $N_0$) and $M/M_1$ (resp. $N/N_1$) are finite and $M_1/M_0$ is isomorphic to $N_1/N_0$.



\bigskip
\bigskip
\noindent {\bf Acknowledgements}:

\noindent I would like to thank my advisor Xinyi Yuan for suggesting this question to me and for telling me the pull-back trick. I also want to thank Jean-Louis Colliot-Thélène and Alexei N. Skorobogatov for helpful suggestions on references.













\newpage













\section{Geometric Brauer groups and Tate-Shafarevich groups}
\subsection{Preliminary on Tate's conjecture and Brauer groups}
\begin{prop}\label{prop2.1}
Let $X$ be a projective and smooth variety over a finitely generated field $k$ of characteristic $p\geq0$, and $\ell\neq p$ be a prime number. Then the cycle class map\\
$$
\mathrm{Pic}(X)\otimes_\mathbb{Z}\mathbb{Q}_\ell\longrightarrow H^2(X_{k^s},\mathbb{Q}_\ell(1))^{G_k}
$$
is surjective if and only if ${\mathrm{Br}}(X_{k^s})^{G_k}(\ell)$ is finite.
\end{prop}
\begin{proof}
See \cite[Thm. 1.2]{Yua2}.
\end{proof}
\begin{lem}
Let $X$ be a projective and smooth variety over a field $k$ of characteristic $p\geq0$, and $\ell\neq p$ be a prime number. Then there is a canonical $G_k$-splitting exact sequence
$$
0\longrightarrow {\mathrm{NS}}(X_{k^s})\otimes_\ZZ\mathbb{Q}_\ell\longrightarrow H^2(X_{k^s},\mathbb{Q}_\ell(1)) \longrightarrow V_\ell {\mathrm{Br}}(X_{k^s})\longrightarrow 0
$$
\end{lem}
\begin{proof}
See \cite[\S 2.2]{Yua2}.
\end{proof}












\subsection{The Galois invariant part of geometric Brauer group}
In this section, we will show that the Galois invariant part of the geometric Brauer group of a smooth variety is a birational invariant up to finite groups. This will allow us to shrink the base of a fibration without changing the question.

The following results in the case of characteristic $0$ were proved in \cite{CTS} (cf. \cite[Prop. 6.1 and Thm. 6.2 (iii)]{CTS}).
\begin{prop}\label{purity}
Let $X$ be a smooth geometrically connected variety over a finitely generated field $k$ of characteristic $p\geq 0$. Let $U\subseteq X$ be an open dense subset. Then the natural map
$$ {\mathrm{Br}}(X_{k^s})^{G_k}({\mathrm{non}\text{-}} p)\longrightarrow {\mathrm{Br}}(U_{k^s})^{G_k}({\mathrm{non}\text{-}} p)$$
is injectie and has a finite cokernel.
\end{prop}
\begin{proof}
The injectivity follows from that $X_{k^s}$ is regular and irreducible. To show that the cokernel is finite, we need the lemma below. Let $Y$ denote $X-U$ with reduced scheme structure. By purity for Brauer group (cf. \cite[Thm. 1.1]{Ces}), removing a close subset of codimension $\geq 2$ will not change the Brauer group. Thus, by shrinking $X$, we might assume that $Y$ is regular and of codimension $1$ in $X$.  Let $\ell \neq p$ be a prime. By purity for Brauer group (cf. \cite[\S 6]{Gro3} and \cite{Fuj}), there is a canonical exact sequence
$$0\longrightarrow {\mathrm{Br}}(X_{k^s})(\ell)\longrightarrow {\mathrm{Br}}(U_{k^s})(\ell)\longrightarrow H^1(Y_{k^s},\mathbb{Q}_{\ell}/\ZZ_{\ell}).
$$
Thus, it suffices to show that the group
$$H^1(Y_{k^s},\mathbb{Q}_{\ell}/\ZZ_\ell)^{G_k}$$
is finite and vanishes for all but finitely many $\ell$. This follows from the lemma below.
\end{proof}

\begin{lem}
Let $U$ be a regular variety over a finitely generated field $k$. Let $\ell\neq {\mathrm{char}}(k)$ be a prime. Then 
$$H^1(U_{k^s},\mathbb{Q}_{\ell}/\ZZ_\ell)^{G_k}$$
is finite and vanishes for all but finitely many $\ell$.
\end{lem}
\begin{proof}
Firstly, we will show that it suffices to prove the claim for smooth varieties over $k$. Notice that we can replace $k$ by a finite separable extension. Thus, we might assume that $U_{k^s}$ is irreducible. Let $V$ be an open dense subset of $U_{k^s}$. By semi-purity (cf. \cite[\S 8]{Fuj}),
$$H^1(U_{k^s},\ZZ/\ell^n)\longrightarrow H^1(V,\ZZ/\ell^n)$$
is injective. We might assume that $V$ is defined over $k$. Therefore, it suffices to prove the claim for an open dense subset. Since $U_{k^s}$ and $U_{\bar{k}}$ have the same underlying toplological space, by shrinking $U_{k^s}$, we might assume that $(U_{\bar{k}})_{{\mathrm{red}}}$ is irreducible and smooth over $\bar{k}$. So there exists a finite extension $l/k$ such that $(U_{l})_{\mathrm{red}}$ is irreducible and smooth over $l$. We might assume that $l/k$ is purely inseparable. Then $l^s=l\otimes_{k}k^s$ and $G_l=G_k$. Let $V$ denote $(U_{l})_{\mathrm{red}}$. Thus $V_{l^s}=V\times_{\Spec k} \Spec k^s$. The $k$-morphism $V\longrightarrow U$ induces a $G_k$-equivariant isomorphism
$$ H^1(U_{k^s}, \ZZ/\ell^n)\cong H^1(V_{l^s},\ZZ/\ell^n),$$
so it suffices to prove the claim for $V$. Thus we might assume that $U$ is smooth and geometrically connected over $k$.\\
By above arguments, we can always replace $k$ by a finite extension and shrink $U$. By de Jong's alteration theorem, we might assume that there is a finite flat morphism $f:V\longrightarrow U$ such that $V$ admits a smooth projective geometrically connected compactification over $k$. Since the kernel of 
$$H^1(U_{k^s},\ZZ/\ell^n)\longrightarrow H^1(V_{k^s},\ZZ/\ell^n)$$
is killed by the degree of $f$ (cf. Lemma \ref{finiteker} below), it suffices to prove the claim for $V$. Thus we might assume that $U$ is an open subvariety of a smooth projective geometrically connected variety $X$ over $k$. Let $Y_i$ be irreducible components of $X-U$ of codimension $1$. Let $D_i$ be the regular locus of $Y_i$. By extending $k$ to a finite separable extension, we might assume that $D_i$ is geometrically irreducible. By purity, there is a canonical exact sequence

$$ 0\longrightarrow H^1(X_{k^s},\ZZ/\ell^n)\longrightarrow H^1(U_{k^s},\ZZ/\ell^n) \longrightarrow\bigoplus_{i}H^0(D_{i,k^s},\ZZ/\ell^n(-1))
$$
Taking $G_k$-invariants, we get an exact sequence
$$ 0\longrightarrow H^1(X_{k^s},\ZZ/\ell^n)^{G_k}\longrightarrow H^1(U_{k^s},\ZZ/\ell^n)^{G_k} \longrightarrow\bigoplus_{i}H^0(D_{i,k^s},\ZZ/\ell^n(-1))^{G_k}.
$$
It suffices to show that the size of the first and third group are bounded independent of $n$ and is equal to $1$ for all but finitely many $\ell$. 
Since $D_{i,k^s}$ is connected, we have
$$H^0(D_{i,k^s},\ZZ/\ell^n(-1))={\mathrm{Hom}}(\mu_{\ell^n},\ZZ/\ell^n).$$
By the lemma below, the size of the third group is bounded independent of $n$ and is equal to $1$ for all but finitely many $\ell$.
Since
$$ H^1(X_{k^s},\ZZ/\ell^n)\cong \mathrm{Pic}_{X/K}[\ell^n](-1)$$
and
$$0\longrightarrow\mathrm{Pic}^0_{X/K,{\mathrm{red}}}[\ell^n]\longrightarrow \mathrm{Pic}_{X/K}[\ell^n]\longrightarrow {\mathrm{NS}}(X_{k^s})[\ell^n]$$
is exact, it suffices to show that the size of $(\mathrm{Pic}^0_{X/K,{\mathrm{red}}}[\ell^n](-1))^{G_k}$ is bounded independent of $n$ and is equal to $1$ for all but finitely many $\ell$.
Let $A$ be the dual of $\mathrm{Pic}^0_{X/K,{\mathrm{red}}}$. By the Weil pairing, 
$$\mathrm{Pic}^0_{X/K,{\mathrm{red}}}[\ell^n](-1)\cong {\mathrm{Hom}}(A[\ell^n],\ZZ/\ell^n).$$
Taking $G_k$-invariants, we get
$$(\mathrm{Pic}^0_{X/K,{\mathrm{red}}}[\ell^n](-1))^{G_k}\cong {\mathrm{Hom}}(A[\ell^n],\ZZ/\ell^n)^{G_k}.$$
By the lemma below, the claim follows.
\end{proof}
\begin{lem}
Let $A$ be an abelian variety over a finitely generated field $K$ of characteristic $p\geq 0$. Let $\ell\neq p$ be a prime.  Then the sizes of 

$${\mathrm{Hom}}(A[\ell^n],\ZZ/\ell^n)^{G_K} \quad \mathrm{and} \quad  {\mathrm{Hom}}(\mu_{\ell^n},\ZZ/\ell^n)^{G_K} $$
are bounded independent of $n$. Moreover, for all but finitely many $\ell$, these two groups vanish for any $n$.
\end{lem}
\begin{proof}
We will only prove the claim for the first group, since the second one follows from the same arguments. Firstly, we assume that $K$ is finite, then we will use a specialization technique to reduce the general case to the finite field case.

If $K$ is finite, then we have
$${\mathrm{Hom}}(A[\ell^n],\ZZ/\ell^n)^{G_K}={\mathrm{Hom}}(A[\ell^n]_{G_K},\ZZ/\ell^n),$$
which has the same size as $A[\ell^n]^{G_{K}}$. Then the claim follows from the finiteness of $A(K)$. 

In general, choose an integral regular scheme $S$ of finite type over $\Spec \ZZ$ with function field $K$. By Shrinking $S$, we might assume $A/K$ extends to an abelian scheme ${\mathscr{A}}/S$. Fix a closed point $s\in S$. For any $\ell\neq {\mathrm{char}}(k(s))$, we can shrink $S$ such that $\ell$ is invertible on $S$ and $s\in S$. Then the etale sheaf ${\mathscr{A}}[\ell^n]$ is a locally constant sheaf of $\ZZ/\ell^n$-module since ${\mathscr{A}}[\ell^n]$ is finite etale over $S$. Thus
$${\mathscr{H}} om({\mathscr{A}}[\ell^n],\ZZ/\ell^n)$$
is also a locally constant sheaf of $\ZZ/\ell^n$-module and its stalk at the generic point can be identified with
$$ {\mathrm{Hom}}(A[\ell^n],\ZZ/\ell^n).$$
Set $\SF={\mathscr{H}} om({\mathscr{A}}[\ell^n],\ZZ/\ell^n)$, we have
$${\mathrm{Hom}}(A[\ell^n],\ZZ/\ell^n)^{G_K}=H^0(S,\SF)\hookrightarrow H^0(s,\SF).
$$
Since $H^0(s,\SF)={\mathrm{Hom}}({\mathscr{A}}_{s}[\ell^n],\ZZ/\ell^n)^{G_{k(s)}}$ and $k(s)$ is finite, thus the claim holds for $\ell\neq {\mathrm{char}}(k(s))$. In the case ${\mathrm{char}}(K)=0$, we can choose another closed point $s^\prime$ such that ${\mathrm{char}}(k(s^\prime))\neq {\mathrm{char}}(k(s))$, then by the same argument, the claim also holds for $\ell={\mathrm{char}}(k(s))$. This completes the proof.
\end{proof}

\begin{lem}\label{finiteker}
Let $f:X\longrightarrow Y$ be a dominant morphism between regular integral schemes. Assuming that $K(X)/K(Y)$ is a finite extension of degree $d$, then the induced map
$${\mathrm{Br}}(Y)\longrightarrow{\mathrm{Br}}(X)$$
has a kernel killed by $d$. Moreover, if $\ell$ is a prime invertible on $Y$, then the induced map
$$H^1(Y,\mu_{\ell^n})\longrightarrow H^1(X,\mu_{\ell^n})$$
also has a kernel killed by $d$.
\end{lem}
\begin{proof}
We will only prove the first claim since the second one follows from the same arguments. Since $X$ and $Y$ are regular, we have
$${\mathrm{Br}}(X)\longrightarrow{\mathrm{Br}}(K(X)) \quad \mathrm{and} \quad {\mathrm{Br}}(Y)\longrightarrow{\mathrm{Br}}(K(Y))
$$
are injective. It suffices to show
$${\mathrm{Br}}(K(Y))\longrightarrow{\mathrm{Br}}(K(X))$$
has a kernel killed by $d$. Let $L$ (resp. $K$) denote $K(X)$ (resp. $K(Y)$).
Since $f:\Spec L\longrightarrow \Spec K$
is finite, we have 
$$H^2(\Spec L, \GG_m)=H^2(\Spec K,f_*\GG_m)=H^2(K,(K^s\otimes_K L)^{\times}).$$
There is a norm map $(K^s\otimes_K L)^{\times} \longrightarrow (K^s)^{\times}$ such that the composition of
$(K^s)^{\times} \longrightarrow (K^s\otimes_K L)^{\times} \longrightarrow (K^s)^{\times}$
is the $d$-th power map. Taking cohomology, the composition of

$$H^2(K,(K^s)^{\times}) \longrightarrow H^2(K, (K^s\otimes_K L)^{\times}) \longrightarrow H^2(K,(K^s)^{\times})$$
is equal to multiplication by $d$. It follows that
$${\mathrm{Ker}} ({\mathrm{Br}}(K)\longrightarrow{\mathrm{Br}}(L))$$
is killed by $d$.
\end{proof}
\subsection{Tate-Shafarevich group}
In this section, first, we will study a geometric version of the Tate-Shafarevich group for an abelian variety over a function field with a base field $k$. Then, we will prove that the Galois fixed part of the geometric Tate-Shafarevich group is canonically isomorphic to Tate-Shafarevich group up to finite groups in the case that $k$ is a finite field. The idea is reducing the question to the relation between arithmetic Brauer groups and geometric Brauer groups which was studied in \cite{CTS}(cf. \cite{Yua2}).
\begin{prop}\label{geosha}
Let $C$ be a smooth projective geometrically connected curve defined over a finitely generated field $k$ of characteristic $p\geq 0$. Let $K$ be the function field of $C$ and $A$ be an abelian variety over $K$. Denote $Kk^s$ by $K^\prime$.  Let $U\subseteq C$ be an open dense subscheme. Define 
$${\hbox{\cyr Sh}_{U_{k^s}}(A):={\mathrm{Ker}}(H^1(K^\prime,A)\longrightarrow \prod_{v\in U_{k^s}} H^1(K_v^{sh}, A))
$$
Then the natural map 
$$ 
{\hbox{\cyr Sh}_{C_{k^s}}(A)^{G_k}({\mathrm{non}\text{-}} p)\longrightarrow {\hbox{\cyr Sh}_{U_{k^s}}(A)^{G_{k}}({\mathrm{non}\text{-}} p)
$$
is injective and has a cokernel of finite exponent.
\end{prop}
\begin{proof}
By definitions, the injectivity is obvious. It suffices to show that the cokernel is of finite exponent. Since there exists an abelian variety $B/K$ such that $A\times B$ is isogenous to $\mathrm{Pic}^0_{X/K}$ for some smooth projective geometrically connected curve $X$ over $K$, it suffices to prove the claim for $A=\mathrm{Pic}^0_{X/K}$. Without loss of generality, we might replace $k$ by a finite extension $l/k$. This is obvious if $l/k$ is separable. For $l/k$ a purely inseparable extension of degree $p^n$, the quotient $A(K^sl^s)/A(K^s)$ is killed by $p^n$. Then $H^1(K^\prime,A) \longrightarrow H^1(Kl^s,A)$ has a kernel and a cokernel killed by $p^n$. It follows that ${\hbox{\cyr Sh}_{U_{k^s}}(A)\longrightarrow {\hbox{\cyr Sh}_{U_{l^s}}(A)$ has a kernel and a cokernel killed by some power of $p$. By resolution of singularity of surfaces, $X\longrightarrow \Spec K$ admits a proper regular model $\pi:{\mathcal {X}}\longrightarrow C$. By extending $k$, we might assume that ${\mathcal {X}}$ is smooth proper geometrically connected over $k$. Write $V$ for $\pi^{-1}(U)$. By the Leray spectral sequence
$$E^{p,q}_2=H^p(U_{k^s},R^q\pi_*\GG_m)\Rightarrow H^{p+q}(V_{k^s},\GG_m),
$$
we get a long exact sequence\\
$$H^2(U_{k^s},\mathbb{G}_m)\longrightarrow {\mathrm{Ker}}(H^2(V_{k^s},\mathbb{G}_m)
\longrightarrow H^0(U_{k^s},R^2\pi_*\mathbb{G}_m))$$
$$\longrightarrow H^1(U_{k^s},R^1\pi_*\mathbb{G}_m)\longrightarrow H^3(U_{k^s},\mathbb{G}_m).
$$
By \cite[Cor. 3.2 and Lem. 3.2.1]{Gro3}, $R^2\pi_*\GG_m=0$ and $H^i(U_{k^s},\GG_m)({\mathrm{non}\text{-}} p)=0$ for $i\geq 2$. Thus, we have a canonical isomoprhism
$$ H^2(V_{k^s},\GG_m)({\mathrm{non}\text{-}} p)\cong H^1(U_{k^s},R^1\pi_*\GG_m)({\mathrm{non}\text{-}} p).$$
Let $j:\Spec K^\prime \longrightarrow C_{k^s}$ be the generic point.
By the spectral sequence
$$H^p(U_{k^s},R^qj_*(j^*R^1\pi_*\GG_m))\Rightarrow H^{p+q}(K^\prime, \mathrm{Pic}(X_{K^s})),$$
we have
$$H^1(U_{k^s},j_*j^*R^1\pi_*\GG_m)={\mathrm{Ker}}(H^1(K^\prime,\mathrm{Pic}(X_{K^s}))\longrightarrow \prod_{v\in U_{k^s}} H^1(K_v^{sh}, \mathrm{Pic}(X_{K^s}))).$$
Since $\mathrm{Pic}(X_{K^s})=\mathrm{Pic}^0_{X/K}(K^s)\oplus \ZZ$ as $G_K$-modules, we get a natural isomorphism
$$H^1(U_{k^s},j_*j^*R^1\pi_*\GG_m)\cong{\hbox{\cyr Sh}_{U_{k^s}}(\mathrm{Pic}^0_{X/K}).$$
Without loss of generality, we might shrink $U$ such that $\pi$ is smooth on $\pi^{-1}(U)$. Then, by Lemma \ref{picard}, the natural map
$$R^1\pi_*\GG_m\longrightarrow j_*j^*R^1\pi_*\GG_m$$
is an isomorphism on $U$. It follows that
$$H^2(V_{k^s},\GG_m)({\mathrm{non}\text{-}} p)\cong H^1(U_{k^s},R^1\pi_*\GG_m)({\mathrm{non}\text{-}} p)$$
$$\cong H^1(U_{k^s},j_*j^*R^1\pi_*\GG_m)({\mathrm{non}\text{-}} p)\cong {\hbox{\cyr Sh}_{U_{k^s}}(\mathrm{Pic}^0_{X/K})({\mathrm{non}\text{-}} p) .
$$
Consider the following commutative diagram\\
\begin{displaymath}
\xymatrix{ {\mathrm{Br}}({\mathcal {X}}_{k^s})({\mathrm{non}\text{-}} p)\ar[r] \ar[d] &H^1(C_{k^s},j_*j^*R^1\pi_*\GG_m)({\mathrm{non}\text{-}} p)\ar[d]\ar[r] & {\hbox{\cyr Sh}_{C_{k^s}}(\mathrm{Pic}^0_{X/K})({\mathrm{non}\text{-}} p) \ar[d]\\
	{\mathrm{Br}}(V_{k^s})({\mathrm{non}\text{-}} p)\ar[r]  &H^1(U_{k^s},j_*j^*R^1\pi_*\GG_m)({\mathrm{non}\text{-}} p) \ar[r] &{\hbox{\cyr Sh}_{U_{k^s}}(\mathrm{Pic}^0_{X/K})({\mathrm{non}\text{-}} p)}
\end{displaymath}
Since maps on the bottom are isomorphisms, it suffices to show that
the natural map
$$ {\mathrm{Br}}({\mathcal {X}}_{k^s})^{G_k}({\mathrm{non}\text{-}} p)\longrightarrow {\mathrm{Br}}(V_{k^s})^{G_k}({\mathrm{non}\text{-}} p) $$
has a finite cokernel. This follows from Proposition \ref{purity}.
\end{proof}

\begin{prop}\label{agsha}
Notations as in the above proposition. Assuming that $k$ is a finite field, define 
$${\hbox{\cyr Sh}_{K^\prime}(A):={\mathrm{Ker}}(H^1(K^\prime,A)\longrightarrow \prod_{v\in C_{k^s}} H^1(K_v^{sh}, A)).
$$
Then the natural map 
$${\hbox{\cyr Sh}(A)({\mathrm{non}\text{-}} p)\longrightarrow {\hbox{\cyr Sh}_{K^\prime}(A)^{G_k}({\mathrm{non}\text{-}} p)
$$
has a kernel and a cokernel of finite exponent.
\end{prop}
\begin{proof}
We use the same arguments as in the proof of the previous lemma. It suffices to prove the claim for $A=\mathrm{Pic}^0_{Y/K}$ where $Y$ is a smooth projective geometrically connected curve over $k$. $Y$ admits a projective regular model $ {\mathcal {Y}}\longrightarrow C$. Then we have
$$ {\mathrm{Br}}(\mathcal{Y}) \cong {\hbox{\cyr Sh}(\mathrm{Pic}_{Y/K}^0) $$ 
and 
$$ {\mathrm{Br}}(\mathcal{Y}_{k^s}) \cong  {\hbox{\cyr Sh}_{K^\prime}(\mathrm{Pic}_{Y/K}^0) $$
up to finite groups. Thus the question is reduced to show that
$$
{\mathrm{Br}}(\mathcal{Y})\longrightarrow {\mathrm{Br}}(\mathcal{Y}_{k^s})^{G_k}
$$ 
has a finite kernel and a finite cokernel. This follows from \cite[Cor. 1.4]{Yua2}.
\end{proof}
\subsection{Cofiniteness of Brauer groups and Tate-Shafarevich groups}
Let $\ell$ be a prime number. Recall that a $\ell$-torsion abelian group $M$ is of \emph{cofinite type} if $M[\ell]$ is finite. This is also equivalent to that $M$ can be written as $(\mathbb{Q}_\ell/\ZZ_\ell)^{r}\oplus M_0$ for some integer $r\geq 0$ and some finite group $M_0$. We also say that a torsion abelian group $M$ is of cofinite type if $M(\ell)$ is of cofinite type for all primes $\ell$. It is easy to see that a morphism $M\rightarrow N$ between abelian groups of cofinite type has a kernel and a cokernel of finite exponent if and only if the kernel and cokernel are finite.

In the following, we will show the cofinitness of geometric Tate-Shafarevich groups and geometric Brauer groups defined in previous sections.

\begin{lem}\label{cofsha}
The group ${\hbox{\cyr Sh}_{U_{k^s}}(A)(\ell)$ defined in the Proposition \ref{geosha} is of cofinite type for any $\ell\neq p$.
\end{lem}
\begin{proof}
We might shrink $U$ such that $A$ extends to an abelian scheme ${\mathscr{A}}$ over $U$. By the exact sequence
$$0\longrightarrow{\mathscr{A}}[\ell]\longrightarrow{\mathscr{A}}\stackrel{\ell}{\longrightarrow}{\mathscr{A}}\longrightarrow 0$$
we get a surjection
$$H^1(U_{k^s},{\mathscr{A}}[\ell])\longrightarrow H^1(U_{k^s}, {\mathscr{A}})[\ell].$$
Since $H^1(U_{k^s},{\mathscr{A}}[\ell])$ is finite, $H^1(U_{k^s}, {\mathscr{A}})[\ell]$ is also finite. Thus, it suffices to show
$$H^1(U_{k^s}, {\mathscr{A}})\cong{\hbox{\cyr Sh}_{U_{k^s}}(A). $$
This follows from  ${\mathscr{A}}\cong j_*A$, since ${\mathscr{A}}$ is a N\'eron model of $A$ over $U$, where $j:\Spec K\longrightarrow U$ is the generic point.

\end{proof}
\begin{lem}\label{cofbr}
Let $X$ be a smooth variety over a separable closed field $k$ of characteristic $p\geq 0$. Let $\ell\neq p$ be a prime. Then ${\mathrm{Br}}(X)(\ell)$
is of cofinite type.
\end{lem}
\begin{proof}
The Kummer exact sequence 
$$0\longrightarrow \mu_\ell \longrightarrow \GG_m\stackrel{\ell}{\longrightarrow} \GG_m \longrightarrow 0$$
induces a surjection
$$H^2(X,\mu_\ell)\longrightarrow {\mathrm{Br}}(X)[\ell].$$
Then the claim follows from the finiteness of $H^2(X,\mu_\ell)$.

\end{proof}
\section{Proof of the main theorem }
\subsection{The left exactness in Theorem \ref{thm1.2}}
In this section, we will prove the left exactness of the sequence in Theorem \ref{thm1.2} by following Grothendieck's arguments in \cite[\S 4]{Gro3}.
\begin{lem}\label{picard}
Let $U$ be an irreducible regular scheme of dimension $1$ with function field $K$. Let $\pi:{\mathcal {X}}\longrightarrow U$ be a smooth proper morphism with a generic fiber geometrically connected over $K$. Let $j:\Spec K\longrightarrow U$ be the generic point of $U$. Then we have
\begin{itemize}
\item[(a)]
the natural map
$$R^1\pi_*\GG_m\longrightarrow j_*j^*R^1\pi_*\GG_m$$
is an isomorphism,
\item[(b)]
the natural map
$$R^2\pi_*\GG_m(\ell)\longrightarrow j_*j^*R^2\pi_*\GG_m(\ell)$$
is an isomorphism for any prime $\ell$ invertible on $U$.
\end{itemize}
\end{lem}
\begin{proof}
It suffices to show that the induced maps on stalks are isomorphism. Thus, we might assume that $U=\Spec R$ where $R$ is a strictly henselian DVR.

Let $X$ denote the generic fiber. Let $s \in U$ be the closed point. Then we have
 $$(R^1\pi_*\GG_m)_{\bar{s}}=\mathrm{Pic}({\mathcal {X}}) \quad \mathrm{and} \quad (j_*j^*R^1\pi_*\GG_m)_{\bar{s}}=\mathrm{Pic}_{X/K}(K).$$ 
Since ${\mathcal {X}}_s$ admits a section $s\longrightarrow {\mathcal {X}}_s$ and $\pi$ is smooth, the section can be extended to a section $U\longrightarrow {\mathcal {X}}$. Thus $X(K)$ is not empty. So
$$\mathrm{Pic}_{X/K}(K)=\mathrm{Pic}(X).$$
Since ${\mathcal {X}}$ is regular, the natural map
$$\mathrm{Pic}({\mathcal {X}})\longrightarrow\mathrm{Pic}(X)$$
is surjective and has a kernel generated by vertical divisors. It suffices to show that ${\mathcal {X}}_s$ is connected. This actually follows from $\pi_*{\mathscr{O}}_{{\mathcal {X}}}={\mathscr{O}}_U$ (cf. \cite[Chap. III, Cor. 11.3]{Har}). This proves $(a)$.

Let $I$ denote $G_K$. For $(b)$, the induced map on the stalk at $s$ is
$${\mathrm{Br}}({\mathcal {X}})(\ell)\longrightarrow {\mathrm{Br}}(X_{K^s})^I(\ell).
$$
Since $\pi$ is smooth and proper, we have 
$$ H^2({\mathcal {X}},\mu_{\ell^\infty})\cong H^2(X_{K^s}, \mu_{\ell^\infty})= H^2(X_{K^s}, \mu_{\ell^\infty})^I.
$$
Consider\\
\begin{displaymath}
\xymatrix{
0\ar[r] & \mathrm{Pic}({\mathcal {X}})\otimes \mathbb{Q}_\ell /\ZZ_\ell\ar[r] \ar[d] & H^2({\mathcal {X}} , \mu_{\ell^\infty}) \ar[r] \ar[d] & {\mathrm{Br}}({\mathcal {X}})(\ell) \ar[r]\ar[d] & 0  \\
0\ar[r] & {\mathrm{NS}}(X_{K^s})\otimes \mathbb{Q}_\ell /\ZZ_\ell \ar[r]  & H^2(X_{K^s}, \mu_{\ell^\infty}) \ar[r]  & {\mathrm{Br}}(X_{K^s})(\ell) \ar[r] & 0 }
\end{displaymath}
Since ${\mathrm{NS}}(X_{K^s})\otimes \mathbb{Q}_\ell /\ZZ_\ell$ is $I$-invariant and $\mathrm{Pic}({\mathcal {X}})=\mathrm{Pic}(X)$. It suffices to show that
$$\mathrm{Pic}(X)\otimes \mathbb{Q}_\ell /\ZZ_\ell \longrightarrow ({\mathrm{NS}}(X_{K^s})\otimes \mathbb{Q}_\ell /\ZZ_\ell)^I
$$ 
is surjective. Write ${\mathrm{NS}}(X_{K^s})_{free}$ for ${\mathrm{NS}}(X_{K^s})/{\mathrm{NS}}(X_{K^s})_{{\mathrm{tor}}}$. The action of $I$ on ${\mathrm{NS}}(X_{K^s})$ factors through a finite quotient $I^\prime$. Consider the exact sequence
$$ 0\longrightarrow ({\mathrm{NS}}(X_{K^s})_{free})^{I^\prime} \otimes \ZZ_{\ell}\longrightarrow  {\mathrm{NS}}(X_{K^s})^{I^\prime}\otimes \mathbb{Q}_{\ell}\longrightarrow  ({\mathrm{NS}}(X_{\bar{K}}) \otimes \mathbb{Q}_{\ell}/\ZZ_\ell)^I$$
$$
\longrightarrow H^1(I^\prime,{\mathrm{NS}}(X_{K^s})_{free} \otimes \ZZ_{\ell}).
$$
$H^1(I^\prime,{\mathrm{NS}}(X_{K^s})_{free} \otimes \ZZ_{\ell})$ is killed by the order of $I^\prime$. Since $({\mathrm{NS}}(X_{K^s}) \otimes \mathbb{Q}_{\ell}/\ZZ_\ell)^I={\mathrm{NS}}(X_{K^s}) \otimes \mathbb{Q}_{\ell}/\ZZ_\ell$ is divisible, so the image of the last map is zero. Since 
$$\mathrm{Pic}(X)\otimes\mathbb{Q}_\ell \longrightarrow {\mathrm{NS}}(X_{K^s})^I \otimes \mathbb{Q}_{\ell}
$$
is surjective (cf. \cite[\S2.2]{Yua2}), the claim follows. By Snake Lemma, the natural map
$${\mathrm{Br}}({\mathcal {X}})(\ell) \longrightarrow {\mathrm{Br}}(X_{K^s})^{I}(\ell)$$
is an isomorphism.

\end{proof}

\begin{lem}\label{exact}
Let $U$ be a smooth geometrically connected curve over a field $k$ of characteristic $p\geq 0$ with function field $K$. Let $\pi:{\mathcal {X}}\longrightarrow U$ be a smooth proper morphism with a generic fiber $X$ geometrically connected over $K$. Let $K^\prime$ denote $Kk^s$.
Define
 $${\hbox{\cyr Sh}_{U_{k^s}}(\mathrm{Pic}_{X/K}):={\mathrm{Ker}}(H^1(K^\prime,\mathrm{Pic}(X_{K^s}))\longrightarrow \prod_{v\in U_{k^s}} H^1(K_v^{sh}, \mathrm{Pic}(X_{K^s}))).
$$
Then we have a canonical exact sequence
$$0\rightarrow({\hbox{\cyr Sh}_{U_{k^s}}(\mathrm{Pic}_{X/K}))^{G_k}({\mathrm{non}\text{-}} p)\rightarrow {\mathrm{Br}}({\mathcal {X}}_{k^s})^{G_k}({\mathrm{non}\text{-}} p)\rightarrow {\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p).$$
Moreover, the natural map
$$({\hbox{\cyr Sh}_{U_{k^s}}(\mathrm{Pic}^0_{X/K,{\mathrm{red}}}))^{G_k}\longrightarrow ({\hbox{\cyr Sh}_{U_{k^s}}(\mathrm{Pic}_{X/K}))^{G_k}
$$
has a kernel and cokernel of finite exponent.

\end{lem}
\begin{proof}
We might assume that $k=k^s$. 
By the Leray spectral sequence
$$E^{p,q}_2=H^p(U,R^q\pi_*\GG_m)\Rightarrow H^{p+q}({\mathcal {X}},\GG_m),
$$
we get a long exact sequence\\
$$H^2(U,\mathbb{G}_m)\longrightarrow {\mathrm{Ker}}(H^2({\mathcal {X}},\mathbb{G}_m)
\longrightarrow H^0(U, R^2\pi_*\mathbb{G}_m))$$
$$\longrightarrow H^1(U,R^1\pi_*\mathbb{G}_m)\longrightarrow H^3(U,\mathbb{G}_m)
$$
By \cite[Lem. 3.2.1]{Gro3}, $H^i(U,\GG_m)({\mathrm{non}\text{-}} p)=0$ for $i\geq 2$. Thus, we get a canonical exact sequence
$$ 0\rightarrow H^1(U,R^1\pi_*\GG_m)({\mathrm{non}\text{-}} p) \rightarrow H^2({\mathcal {X}},\GG_m) ({\mathrm{non}\text{-}} p)\rightarrow H^0(U,R^2\pi_*\GG_m)({\mathrm{non}\text{-}} p).$$
Let $j:\Spec K \longrightarrow U$ be the generic point. By Lemma \ref{picard}, we have  canonical isomorphisms
$$H^1(U,R^1\pi_*\GG_m)({\mathrm{non}\text{-}} p)\cong H^1(U,j_*j^*R^1\pi_*\GG_m)({\mathrm{non}\text{-}} p)$$ 
and
$$H^0(U,R^2\pi_*\GG_m)({\mathrm{non}\text{-}} p)\cong H^0(U,j_*j^*R^2\pi_*\GG_m)({\mathrm{non}\text{-}} p).
$$
Since $j^*R^i\pi_*\GG_m$ corresponds to the $G_K$-module $H^i(X_{K^s},\GG_m)$, we have
$$H^1(U,j_*j^*R^1\pi_*\GG_m)={\hbox{\cyr Sh}_{U}(\mathrm{Pic}_{X/K})\quad \mathrm{and} \quad 
H^0(U,j_*j^*R^2\pi_*\GG_m)= {\mathrm{Br}}(X_{K^s})^{G_K}.
$$
This proves the first claim.

For the second claim, consider the exact sequence
$$0\longrightarrow \mathrm{Pic}^0(X_{K^s})\longrightarrow \mathrm{Pic}(X_{K^s})\longrightarrow {\mathrm{NS}}(X_{K^s})\longrightarrow 0.$$
Since ${\mathrm{NS}}(X_{K^s})$ is finitely generated, there exists a finite Galois extension $L/K$ such that
$\mathrm{Pic}(X_{L})\longrightarrow{\mathrm{NS}}(X_{K^s})$
is surjective. Taking Galois cohomoloy, we get a long exact sequence
$$0\longrightarrow H^0(K,\mathrm{Pic}^0(X_{K^s}))\longrightarrow H^0(K,\mathrm{Pic}(X_{K^s}))\longrightarrow H^0(K,{\mathrm{NS}}(X_{K^s}))$$
$$\stackrel{a_K}{\longrightarrow} H^1(K,\mathrm{Pic}^0(X_{K^s}))\longrightarrow H^1(K,\mathrm{Pic}(X_{K^s}))\longrightarrow H^1(K,{\mathrm{NS}}(X_{K^s})).$$
We have a similar long exact sequence for $H^i(L,-)$. Since $\mathrm{Pic}(X_L)\rightarrow {\mathrm{NS}}(X_{K^s})$ is surjective, we have $a_L=0$ and 
$$H^1(L,{\mathrm{NS}}(X_{K^s}))={\mathrm{Hom}}(G_L,{\mathrm{NS}}(X_{K^s}))={\mathrm{Hom}}(G_L,{\mathrm{NS}}(X_{K^s})_{{\mathrm{tor}}}).$$
$a_L=0$ implies that the image of $a_K$ is contained in
$$ {\mathrm{Ker}}(H^1(K,\mathrm{Pic}^0(X_{K^s}))\longrightarrow H^1(L,\mathrm{Pic}^0(X_{K^s})))$$
which is killed by $[L:K]$. Similarly, one can show that $H^1(K,{\mathrm{NS}}(X_{K^s}))$ is killed by $[L:K]|{\mathrm{NS}}(X_{K^s})_{{\mathrm{tor}}}|$. Therefore, the kernel and cokernel of 
$$H^1(K,\mathrm{Pic}^0(X_{K^s}))\longrightarrow H^1(K,\mathrm{Pic}(X_{K^s}))$$
are killed by $[L:K]|{\mathrm{NS}}(X_{K^s})_{{\mathrm{tor}}}|$. The claim also holds for 

$$H^1(K_v^{sh},\mathrm{Pic}^0(X_{K^s}))\longrightarrow H^1(K^{sh}_v,\mathrm{Pic}(X_{K^s})).$$
By diagram chasings, the kernel and cokernel of
$${\hbox{\cyr Sh}_{U}(\mathrm{Pic}^0_{X/K,{\mathrm{red}}})\longrightarrow {\hbox{\cyr Sh}_{U}(\mathrm{Pic}_{X/K})
$$
is of finite exponent. This completes the proof.
\end{proof}

\subsection{The pull-back trick}
In this section, we will introduce a pull-back trick initially developed by Colliot-Thél\`ene and Skorobogatov in \cite{CTS} which allows us to reduce the question to cases of relative dimension $1$.

Let $U$ be a regular integral excellent scheme of dimension $1$ with function field $K$. Let $\pi:{\mathcal {X}}\longrightarrow U$ be a smooth projective morphism with the generic fiber $X$ geometrically connected over $K$. The Leray spectral sequence 
$$E_2^{p,q}=H^p(U,R^q\pi_*\GG_m)\Rightarrow H^{p+q}({\mathcal {X}},\GG_m)$$
induces canonical maps
$$d^{1,1}_2:E_2^{1,1}\longrightarrow E_2^{3,0},$$
$$d_3^{0,2}:E_3^{0,2}\longrightarrow E_3^{3,0},$$
$$d_2^{0,2}:E_2^{0,2}\longrightarrow E_2^{2,1}.$$
\begin{lem}\label{keylem}
Assuming that $X(K)$ is not empty, then the canonical maps $d^{1,1}_2$ and $d_3^{0,2}$ vanish and $E_3^{3,0}=E_2^{3,0}$. Moreover, there exists an open dense subscheme $V\subseteq U$ such that the canonical map $d_2^{0,2}$ has an image of finite exponent when replacing $U$ by $V$. As a result, the natural map
$$H^2(\pi^{-1}(V),\GG_m)\longrightarrow H^0(V,R^2\pi_*\GG_m)$$
has a cokernel of finite exponent.
\end{lem}
\begin{proof}
Let $s\in X(K)$.  Since $\pi$ is proper, it extends to a section $s:U\longrightarrow {\mathcal {X}}$. Let $\tilde{E}_2^{p,q}$ denote the Leray spectral sequence for the identy map $U\longrightarrow U$. Then $s$ induces a commutative diagram
\begin{displaymath}
\xymatrix{E_2^{1,1}\ar[r]^{d_2^{1,1}}\ar[d]& E_2^{3,0}\ar[d] \\
\tilde{E}_2^{1,1}\ar[r]& \tilde{E}_2^{3,0}}
\end{displaymath}
The second column is an isomorphism since $E_2^{3,0}=H^3(U,\GG_m)=\tilde{E}_2^{3,0}$. Since $\tilde{E}_2^{1,1}=0$, thus $d_2^{1,1}=0$. By definition, $d_2^{3,0}=0$, it follows that $E_3^{3,0}=E_2^{3,0}$. By the same arguments, we have $d_3^{0,2}=0$.

For the proof of the second claim, we will use the pullback method for finitely many $U$-morphisms ${\mathcal {Y}}_i\longrightarrow{\mathcal {X}}$ where ${\mathcal {Y}}_i$ is of relative dimension $1$ over $U$. Let $^iE^{p,q}_2$ denote the Leray spectral sequence for $\pi_i:{\mathcal {Y}}_i\longrightarrow U$. There is a commutative diagram
\begin{displaymath}
\xymatrix{E_2^{0,2} \ar[r]^{d_2^{0,2}}\ar[d]& E_2^{2,1}\ar[d] \\
\oplus_i \,^iE_2^{0,2}\ar[r]& \oplus_i \,^iE_2^{2,1}}
\end{displaymath}
By shrinking $U$, we can assume that $\pi_i$ is smooth and projective for all $i$. By Artin's theorem \cite[Cor. 3.2]{Gro3}, $R^2\pi_{i,*}\GG_m=0$. Thus, to show that $d_2^{0,2}$ has an image of finite exponent, it suffices to show that the second column has a kernel of finite exponent. The second column is the map
$$ H^2(U,R^1\pi_*\GG_m)\longrightarrow \bigoplus_iH^2(U,R^1\pi_{i,*}\GG_m).$$
By Lemma \ref{picard}$(a)$, we have a canonical isomorphism
$$R^1\pi_{i,*}\GG_m\cong j_*j^*R^1\pi_{i,*}\GG_m,$$
where $j:\Spec K\longrightarrow U$ is the generic point. It suffices to show that there exists an etale sheaf $\SF$ on $\Spec K$ and a morphism $\SF\longrightarrow \oplus_ij^*R^1\pi_{i,*}\GG_m$ such that the induced map
\begin{equation}\label{splitness}
j^*R^1\pi_{*}\GG_m\oplus\SF \longrightarrow \bigoplus_ij^*R^1\pi_{i,*}\GG_m
\end{equation}
has a kernel and a cokernel killed by some positive integer. This will imply that the induced map
$$H^2(U,j_*j^*R^1\pi_{*}\GG_m)\oplus H^2(U,j_*\SF) \longrightarrow \bigoplus_i H^2(U,j_*j^*R^1\pi_{i,*}\GG_m)
$$
has a kernel and a cokernel of finite exponent. Since $j^*R^1\pi_*\GG_m$ corresponds to the $G_K$-module $\mathrm{Pic}(X_{K^s})$, (\ref{splitness}) can be interpreted as that there is a $G_K$-module $M$ and a $G_K$-equivariant map $M\longrightarrow \oplus_{i}\mathrm{Pic}(Y_{i,K^s})$ such that
$$\mathrm{Pic}(X_{K^s})\oplus M\longrightarrow \bigoplus_{i}\mathrm{Pic}(Y_{i,K^s})$$
has a kernel and a cokernel of finite exponent. In \cite[\S 3.3, p17]{Yua2},
Yuan proved that there exist smooth projective curves $Y_i$ in $X$ which are obtained by taking intersection of very ample divisors and an abelian subvariety $A$ of $\prod_i\mathrm{Pic}^0_{Y_i/K}$ such that the induced morphism
$$\mathrm{Pic}(X_{K^s})\times A(K^s)\longrightarrow \prod_i\mathrm{Pic}(Y_{i,K^s})$$
has a finite kernel and a finite cokernel. Taking ${\mathcal {Y}}_i$ to be the Zariski closure of $Y_i$ in ${\mathcal {X}}$, then shrinking $U$, we get smooth and proper morphisms $\pi:{\mathcal {Y}}_i\longrightarrow U$. This proves the second claim.

For the last claim, consider the following canonical exact sequences induced by the Leray spectral sequence,
$$E^2\longrightarrow E_{4}^{0,2}\longrightarrow 0,$$
$$0\longrightarrow E_{4}^{0,2}\longrightarrow E_{3}^{0,2}\stackrel{d_3^{0,2}}{\longrightarrow} E_{3}^{3,0},$$
$$0\longrightarrow E_{3}^{0,2}\longrightarrow E_{2}^{0,2}\stackrel{d_2^{0,2}}{\longrightarrow} E_{2}^{2,1}.$$
Since $d_3^{0,2}$ vanishes and $d_2^{0,2}$ has an image of finite exponent, it follows that
$$E^2\longrightarrow E_2^{0,2}$$
has a cokernel of finite exponent. This completes the proof.


\end{proof}
\subsection{Proof of Theorem \ref{thm1.2}}
Now we prove Theorem \ref{thm1.2}. Let $U$ be an open dense subscheme of $C$ such that $\pi$ is smooth and projective over $U$. By Lemma \ref{exact}, the natural map
$$({\hbox{\cyr Sh}_{U_{k^s}}(\mathrm{Pic}^0_{X/K,{\mathrm{red}}}))^{G_k}({\mathrm{non}\text{-}} p)\rightarrow {\mathrm{Ker}}({\mathrm{Br}}(\pi^{-1}(U_{k^s}))^{G_k}({\mathrm{non}\text{-}} p)\rightarrow{\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p))$$
has a kernel of finite exponent. By Lemma \ref{cofsha} and \ref{cofbr}, all groups here are of cofinite type, thus the kernel is actually finite. By Proposition \ref{purity} and \ref{geosha}, it suffices to show that the natural map
$${\mathrm{Br}}(\pi^{-1}(U_{k^s}))^{G_k}({\mathrm{non}\text{-}} p)\longrightarrow {\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)$$
has a cokernel of finite exponent.

Firstly, we will prove this under the assumption that $X(K)$ is not empty.
We still use ${\mathcal {X}}$ to denote $\pi^{-1}(U)$. By Proposition \ref{purity}, without loss of generality, we can always replace $U$ by an open dense subset. By Lemma \ref{picard}$(b)$, we have
$$H^0(U, R^2\pi_*\GG_m)({\mathrm{non}\text{-}} p)\cong H^0(U, j_*j^*R^2\pi_*\GG_m)({\mathrm{non}\text{-}} p)={\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p),$$
so it suffices to show that the natural map induced by the Leray spectral sequence for $\pi$
$$ H^2({\mathcal {X}},\GG_m)\longrightarrow H^0(U, R^2\pi_*\GG_m)$$
has a cokernel of finite exponent. By shrinking $U$, the claim follows from Lemma \ref{keylem}.

Secondly, we will show that the question can be reduced to the case that $X(K)$ is not empty. There exists a finite Galois extenion $L/K$ such that $X(L)$ is not empty. Let $W$ be a smooth curve over $k$ with function field $L$. By shrinking $U$ and $W$, the map $\Spec L \longrightarrow \Spec K$ extends to a finite etale Galois covering map $W\longrightarrow U$. Let ${\mathcal {X}}_W \longrightarrow W$ be the base change of ${\mathcal {X}}\longrightarrow U$ to $W$. By the above arguments, the natural map
$$ {\mathrm{Br}}({\mathcal {X}}_W)({\mathrm{non}\text{-}} p)\longrightarrow{\mathrm{Br}}(X_{K^s})^{G_L}({\mathrm{non}\text{-}} p)$$
has a cokernel of finite exponent. Let $G$ denote ${\mathrm{Gal}}(L/K)$. The above map is compatible with the $G$-action. Taking $G$-invariant, we have that
$${\mathrm{Br}}({\mathcal {X}}_W)^{G}({\mathrm{non}\text{-}} p)\longrightarrow{\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)$$
has a cokernel of finite exponent. Then the question is reduced to show that the natural map
$${\mathrm{Br}}({\mathcal {X}})\longrightarrow{\mathrm{Br}}({\mathcal {X}}_W)^{G}$$
has cokernel of finite exponent. Consider the spectral sequence
$$H^p(G,H^q({\mathcal {X}}_W,\GG_m))\Rightarrow H^{p+q}({\mathcal {X}},\GG_m).$$
Since $H^p(G,-)$ is killed by the order of $G$, by similar arguments as in Lemma \ref{keylem}, we can conclude that the cokernel of
$$H^2({\mathcal {X}},\GG_m)\longrightarrow H^2({\mathcal {X}}_{W},\GG_m)^{G}$$
is of finite exponent. This completes the proof of the theorem.
\section{Applications}
\subsection{Proof of Theorem \ref{finitebrauer}}
Firstly, note that in the case that $k$ is finite, by Milne's theorems in \cite{Mil5,Mil6}, the finiteness of ${\mathrm{Br}}({\mathcal {X}}_{k^s})^{G_k}({\mathrm{non}\text{-}} p)$ is equivalent to Tate conjecture for divisors on ${\mathcal {X}}$.

We will prove that the finiteness of ${\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)$ by induction on the transcendence degree of $K/k$. If $K/k$ is of transcendence degree $0$, a smooth projective variety over $K$ can be view as a smooth projective variety over $k$. So the claim is true by assumptions.

Assume  that the claim is true for all extensions $l/k$ of transcendence degree $n$. Let $K/k$ be a finitely generated extension of transcendence degree $n+1$. Let $X/K$ be a smooth projective connected variety over $K$, by extending $K$, we might assume that $X/K$ is geometrically connected. $K$ can be regarded as the function field of a smooth projective geometrically connected curve $C$ over a field $l$ of transcendence degree $n$ over $k$. $X\longrightarrow \Spec K$ can be extended to a smooth morphism $\pi:{\mathcal {X}}\longrightarrow C$. By Theorem \ref{thm1.2}, 
$${\mathrm{Br}}({\mathcal {X}}_{l^s})^{G_k}({\mathrm{non}\text{-}} p)\longrightarrow {\mathrm{Br}}({\mathcal {X}}_{K^s})^{G_K}({\mathrm{non}\text{-}} p)
$$
has a finite cokernel. It suffices to show that ${\mathrm{Br}}({\mathcal {X}}_{l^s})^{G_l}({\mathrm{non}\text{-}} p)
$ is finite. By Proposition \ref{purity}, we might shrink ${\mathcal {X}}$ such that there is an alteration $f:{\mathcal {X}}^{\prime}\longrightarrow {\mathcal {X}}$ such that ${\mathcal {X}}^{\prime}$ admits a smooth projective compactification over a finite extension of $l$. Replacing $l$ by a finite extension will not change the question, so we might assume that ${\mathcal {X}}^{\prime}$ admits a smooth projective compactification over $l$. By shrinking ${\mathcal {X}}$, we might assume that $f$ is finite flat. Then the kernel of 
$${\mathrm{Br}}({\mathcal {X}}_{l^s})({\mathrm{non}\text{-}} p)\longrightarrow{\mathrm{Br}}({\mathcal {X}}^\prime_{l^s})({\mathrm{non}\text{-}} p)$$
is killed by the degree of $f$ (cf. Lemma \ref{finiteker}) and therefore is finite. Since ${\mathrm{Br}}({\mathcal {X}}^\prime_{l^s})^{G_l}({\mathrm{non}\text{-}} p)$ is finite by induction, so ${\mathrm{Br}}({\mathcal {X}}_{l^s})^{G_l}({\mathrm{non}\text{-}} p)$ is finite. This completes the proof.

\subsection{Proof of Theorem \ref{reduction}}
Let $\ell\neq p$ be a prime, the the claim in Theorem \ref{finitebrauer} also holds when replacing the prime to $p$ part by the $\ell$ primary part in the statement. So by the same argument as in the proof of Theorem \ref{finitebrauer}, Theorem \ref{reduction} is true.

\subsection{Brauer groups of integral models of abelian varieties and $K3$ surfaces }
\begin{prop}
Let $C$ be a smooth projective geometrically connected curve over a finite field $k$ of characteristic $p>2$. Let $\pi:{\mathcal {X}} \longrightarrow C$ be a projective flat morphism. Let $X$ denote the generic fiber of $\pi$. Assuming that
${\mathcal {X}}$ is regular and $X_{K^s}$ is an abelian variety over $K^s$, then there is an isomorphism up to finite groups
$$ {\hbox{\cyr Sh}(\mathrm{Pic}_{X/K,{\mathrm{red}}}^{0})({\mathrm{non}\text{-}} p)\cong {\mathrm{Br}}({\mathcal {X}})({\mathrm{non}\text{-}} p).
$$

\end{prop}
\begin{proof}
By \cite[Thm. 1.1]{SZ1}, ${\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)$ is finite, the claim follows directly from Corollary \ref{mainthm}.
\end{proof}

\begin{prop}
Let $C$ be a smooth projective geometrically connected curve over a finite field $k$ of characteristic $p$. Let $\pi:{\mathcal {X}} \longrightarrow C$ be a projective flat morphism. Assuming that ${\mathcal {X}}$ is regular and the generic fiber $X$ of $\pi$ is a smooth projective geometrically connected surface over $K$ with $H^1(X_{K^s},\mathbb{Q}_\ell)=0$ for some prime $\ell\neq p$, then the natural map
$${\mathrm{Br}}({\mathcal {X}})({\mathrm{non}\text{-}} p)\longrightarrow {\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)
$$
has a finite kernel and cokernel. As a result, ${\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)$ is finite if and only if ${\mathrm{Br}}(X_{K^s})^{G_K}(\ell)$ is finite for some prime $\ell\neq p$.
\end{prop}
\begin{proof}
$\mathrm{Pic}^0_{X/K,{\mathrm{red}}}=0$ since $H^1(X_{K^s},\mathbb{Q}_\ell)=0$. It follows that
$${\hbox{\cyr Sh}(\mathrm{Pic}_{X/K,{\mathrm{red}}}^{0})=0.$$
Then the claim follows from Corollary \ref{mainthm}.
\end{proof}
\begin{cor}
Notations as above, if $X/K$ is a $K3$ surface and $p>2$, then 
${\mathrm{Br}}({\mathcal {X}})$ is finite.
\end{cor}
\begin{proof}
Since Tate conjecture for $X$ is known by work of lots of people (cf. Conjecture \ref{Tateconj} ), so ${\mathrm{Br}}(X_{K^s})^{G_K}(\ell)$ is finite for some prime $\ell\neq p$. By the above proposition, ${\mathrm{Br}}({\mathcal {X}})(\ell)$ is also finite. Then by Milne's theorems \cite{Mil5,Mil6}, ${\mathrm{Br}}({\mathcal {X}})$ is finite.
\end{proof}
\begin{prop}
Let $X$ be a $K3$ surface over a global field $K$ of characteristic $p>2$.   Then  ${\mathrm{Br}}(X_{K^s})^{G_K}({\mathrm{non}\text{-}} p)$ is finite.
\end{prop}
\begin{proof} 
By previous results, it suffices to show that $X$ admits a projective regular model. By resolution of singularity of threefolds(cf. \cite{CP}), $X$ does admit a projective regular model.
\end{proof}






















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