On vanishing of unramified cohomology of geometrically rational varieties over finite fields

TL;DR

This paper proves that the third unramified cohomology group of certain geometrically rational threefolds over finite fields vanishes under specific conditions, advancing understanding of cohomological properties in algebraic geometry.

Contribution

It establishes a vanishing result for the third unramified cohomology of geometrically rational threefolds over finite fields under the $ ext{Z}_ ext{ell}$-exactness Hard Lefschetz condition, a novel theoretical insight.

Findings

Third unramified cohomology vanishes under specified conditions

Supports conjectures about cohomological behavior of rational varieties

Provides new tools for studying algebraic cycles over finite fields

Abstract

The purpose of this paper is to show that the third unramified cohomology with divisible coefficients of a smooth projective geometrically rational threefold over a finite field must vanish under $\Z_{\ell}$-exactness Hard Lefschetz condition.