The cycle-class form of the Tate conjecture

Let $\mathbf{F}=\mathbf{F}_q$ be a finite field, let $X$ be a smooth projective variety over $\mathbf{F}$, let $\bar X=X\times_{\mathbf{F}}\overline{\mathbf{F}}$, and let $G=\operatorname{Gal}(\overline{\mathbf{F}}/\mathbf{F})$. For $n\geq 0$, write $CH^n(X)$ for codimension-$n$ cycles modulo rational equivalence. The cycle-class form of the Tate conjecture. The cycle class map

$$CH^n(X)\otimes\mathbf{Q}_l\longrightarrow H^{2n}(\bar X,\mathbf{Q}_l(n))^G$$

is bijective for every $n\geq 0$. The text derives this by combining the cohomological Tate conjecture with Beilinson's conjecture; it gives no resolution status for the combined statement.