Representability of Chow groups of codimension three cycles

Abstract: 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 $A^3(X)$.