Néron--Severi groups of proper schemes over finite fields

Let $X$ be a proper reduced scheme over a finite field $k$, let $\ell$ be a prime different from $\operatorname{char} k$, and write $\ol X=X\times_{k}\ol k$ for its base change to an algebraic closure $\ol k$ of $k$. Call a class in $\rH^{2}_{\et}(\ol X,\bbZ_{\ell}(1))$ \emph{Zariski-locally trivial} if it vanishes on a Zariski-open cover of $\ol X$. We prove that the first Chern class map identifies $\NS(\ol X)\otimes\bbZ_{\ell}$ with the group of Zariski-locally trivial classes whose image in $\rH^{2}_{\et}(\ol X,\Ql(1))$ has weight zero. This is the finite-field analogue of a theorem of Barbieri-Viale--Rosenschon--Srinivas for proper seminormal complex varieties. In the finite-field setting neither seminormality nor irreducibility is needed.