An $l$-adic norm residue epimorphism theorem

We show that the continuous étale cohomology groups $H_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}^n(X,{\mathbf{Z}}_l(n))$ of smooth varieties $X$ over a finite field $k$ are spanned as ${\mathbf{Z}}_l$-modules by the $n$-th Milnor $K$-sheaf locally for the Zariski topology, for all $n\ge 0$. Here $l$ is a prime invertible in $k$. This is the first general unconditional result towards the conjectures of arXiv:math/9801017 (math.AG) which put together the Tate and the Beilinson conjectures relative to algebraic cycles on smooth projective $k$-varieties.