An $l$-adic norm residue epimorphism theorem
Bruno Kahn
TL;DR
This work proves, for smooth varieties over a finite field and prime $l$ invertible in the field, that the continuous $l$-adic cohomology $H^n_{ ext{cont}}(X,oldsymbol{Z}_l(n))$ is generated by Milnor $K$-theory via the $n$-th Milnor $K$-sheaf, up to the $l$-adic norm residue epimorphism. The main achievement is that the map from $oldsymbol{K}_n^M o H^n(oldsymbol{Z}_l(n)^c)$ is an epimorphism of Zariski (and, for $n eq 0$, of presheaves when $n ightarrow 2$) after suitable tensoring, providing the first general unconditional progress toward the Tate–Beilinson reformulations. The strategy blends spectral-sequence computations with purity, and modern motivic methods, notably reduced homotopic modules in Voevodsky–Déglise frameworks, together with de Jong alterations to reduce to SNC divisors. These ideas connect Milnor $K$-theory to $l$-adic cohomology in a way that advances understanding of how algebraic cycles generate cohomology and clarifies the structure of the Beilinson–Tate conjectures in the non-projective setting. The results also outline when global sections of Milnor $K$-theory can be controlled or computed, under abelian-type and Tate conjecture hypotheses, highlighting conditional bridges to motivic and birational geometry.
Abstract
We show that the continuous étale cohomology groups $H^n_{\mathrm{cont}}(X,\mathbf{Z}_l(n))$ 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.