Sur la cohomologie non ramifi\'ee en degr\'e trois d'un produit

TL;DR

This paper proves the vanishing of the third unramified cohomology group for certain products of smooth projective surfaces and curves over finite fields, with applications to geometrically rational surfaces.

Contribution

It establishes new cases of vanishing for unramified cohomology in degree three for products over finite fields, especially for geometrically rational surfaces.

Findings

Third unramified cohomology vanishes for specific surface-curve products.

Results apply notably to geometrically rational surfaces.

Provides new insights into cohomological properties over finite fields.

Abstract

Let F be a finite field and let C be a smooth projective curve over F. For some smooth projective surfaces X over F we establish that the third unramified cohomology of the product of X and C vanishes. This applies in particular to geometrically rational surfaces. Soit F un corps fini et soit C une courbe projective et lisse sur F. Pour certaines surfaces projectives et lisses X sur F on \'etablit la nullit\'e du troisi\`eme groupe de cohomologie non ramifi\'ee du produit de X et C. Cela s'applique en particulier aux surfaces g\'eom\'etriquement rationnelles.