Selmer group associated to the Chow group of certain codimension two cycles

Let $X$ be a surface with geometric genus and irregularity zero which is defined over a number field $K$. Let $\mathcal{X}$ denote a smooth spread of $X$ over the spectrum of a Zariski open subset in the spectrum of the ring of integers and $A^2$ stands for the group of algebraically trivial cycles on schemes modulo rational equivalence. If $j^{\ast }:A^2(\mathcal{X})\to A^2(X)$ be the flat pull-back corresponding to the embedding $j:X\hookrightarrow \mathscr{X}$thenweprovethat$\im(j^*)(K)/A^2(\mathscr{X})(K)$ is a torsion group. Here $\text{im}(j^{\ast })(K)$, $A^2(\mathcal{X})(K)$ stand for the cycles fixed under the action of the absolute Galois group.