An l-adic norm residue epimorphism theorem

We show that the continuous étale cohomology groups $H_{cont}^{n} (X, ℤ_{l} (n))$ of smooth varieties $X$ over a finite field $k$ are spanned as $ℤ_{l}$ -modules by the $n$ -th Milnor $K$ -sheaf locally for the Zariski topology for all $n \geq 0$ . 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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Keywords

Milnor K-theory, Tate–Beilinson conjecture, motivic cohomology

Mathematical Subject Classification

Primary: 11G25, 14C35, 19E15

Milestones

Received: 3 March 2025

Revised: 6 September 2025

Accepted: 8 October 2025

Published: 28 November 2025

Authors

Bruno Kahn
CNRS, IMJ-PRG Sorbonne Université and Université Paris Cité Paris France

Vol. 10, No. 4, 2025

Bruno Kahn

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors