Let $X$ be a smooth projective variety over a finite field. Since $Pic^0(X)$ is finite and $Pic^0(X)$ can be identified with numerically equivalent to zero divisors this implies that for divisors on $X$ numerical and the rational equivalences both agree (with coefficients over $\mathbb{Q}$). $NS(X)$ is also known to be finitely generated so $CH^1(X)\otimes \mathbb{Q}$ is a finite rank space. Now consider the sub-spaces of $CH^i(X)\otimes \mathbb{Q}$ generated by $i$ times intersections of divisors and let's call it $V^i$. $V^i$ is a vector space of finite rank (Since $CH^1(X)\otimes \mathbb{Q}$ was finite rank). Is it known whether the numerical and the rational equivalence match up on the cycles belonging to $V^i$? (i.e. anything numerically equivalent to zero in $V^i$ has to be zero.)
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