Abstract

We show that the continuous étale cohomology groups $H_{\mathrm{cont}<!--FUNCTION\; APPLICATION-->}^n(X,{\mathbb{Z}}_l(n))$ of smooth varieties $X$ over a finite field $k$ are spanned as ${\mathbb{Z}}_l$-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 Kahn (1998) which put together the Tate and the Beilinson conjectures relative to algebraic cycles on smooth projective $k$-varieties.

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