  In this note we are going to prove that if we have a fibration of smooth
projective varieties $X\to S$ over a surface $S$ such that $X$ is of dimension
four and that the geometric generic fiber has finite dimensional motive and the
first \'etale cohomology of the geometric generic fiber with respect to
$\mathbb {Q}_l$ coefficients is zero and the second \'etale cohomology is
spanned by divisors, then $A^3(X)$ (codimension three algebraically trivial
cycles modulo rational equivalence) is dominated by finitely many copies of
$A_0(S)$. Meaning that there exists finitely many correspondences $\Gamma_i$ on
$S\times X$, such that $\sum_i \Gamma_i$ is surjective from $\oplus A^2(S)$ to
$A^3(X)$.
