Le groupe de Brauer non ramifi\'e sur un corps global de caract\'eristique positive (The unramified Brauer group over a global field of positive characteristic)

TL;DR

This paper characterizes the prime-to-p torsion part of the unramified Brauer group of a smooth, geometrically integral variety over a global field of characteristic p, utilizing Gabber's alterations theorem and local point evaluations.

Contribution

It provides a new description of the prime-to-p torsion in the unramified Brauer group using alterations and local evaluations, extending understanding in positive characteristic.

Findings

Describes the prime-to-p torsion of the unramified Brauer group

Uses Gabber's alterations theorem in the proof

Connects Brauer group elements to local point evaluations

Abstract

En utilisant un th\'eor\`eme de Gabber sur les alt\'erations, on d\'emontre un r\'esultat d\'ecrivant la partie de torsion premi\`ere \`a $p$ du groupe de Brauer non ramifi\'e d'une vari\'et\'e $V$ lisse et g\'eom\'etriquement int\`egre sur un corps global de caract\'eristique $p$ au moyen de l'\'evaluation des \'el\'ements de $\Br V$ sur ses points locaux. Using a theorem of Gabber on alterations, we prove a result describing the prime-to-$p$ torsion part of the unramified Brauer group of a smooth and geometrically integral variety $V$ over a global field of characteristic $p$ by evaluating the elements of $\Br V$ at its local points.