Néron--Severi groups of proper schemes over finite fields

Let $X$ be a proper reduced scheme over a finite field $k$, let $\mathrm{\ell }$ be a prime different from $\mathrm{char}k$, and write $\text{ol}X=X{\times }_k\text{ol}k$ for its base change to an algebraic closure $\text{ol}k$ of $k$. Call a class in ${\text{rH}}_{\text{et}}^2(\text{ol}X,{\text{bbZ}}_{\mathrm{\ell }}(1))$ \emph{Zariski-locally trivial} if it vanishes on a Zariski-open cover of $\text{ol}X$. We prove that the first Chern class map identifies $\text{NS}(\text{ol}X)\otimes {\text{bbZ}}_{\mathrm{\ell }}$ with the group of Zariski-locally trivial classes whose image in ${\text{rH}}_{\text{et}}^2(\text{ol}X,\text{Ql}(1))$ 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.