An $l$-adic norm residue epimorphism theorem

TL;DR

This paper proves that certain étale cohomology groups of smooth varieties over finite fields are generated by Milnor K-sheaves, advancing the understanding of deep conjectures linking algebraic cycles, Tate, and Beilinson conjectures.

Contribution

It establishes the first unconditional proof that these cohomology groups are locally generated by Milnor K-sheaves, supporting major conjectures in algebraic geometry.

Findings

Cohomology groups are spanned by Milnor K-sheaves for all n≥0.

Supports the conjectures linking algebraic cycles with étale cohomology.

Provides a foundational step towards the Tate and Beilinson conjectures.

Abstract

We show that the continuous \'etale 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.